
                # Interation 0

                ## Task Info
                You are investigating a biological system where all species are observable. Your goal is to discover the missing reactions. You do not need to add new species.


                ## Incomplete SBML Model. You can assess it as input_sbml_string.

                <?xml version="1.0" encoding="UTF-8"?>
<sbml xmlns="http://www.sbml.org/sbml/level3/version1/core" xmlns:layout="http://www.sbml.org/sbml/level3/version1/layout/version1" xmlns:render="http://www.sbml.org/sbml/level3/version1/render/version1" level="3" version="1" layout:required="false" render:required="false">
  <model id="id_k9sa" substanceUnits="id_yxmk" timeUnits="id_pcnf" volumeUnits="id_29tc" areaUnits="id_el2n" lengthUnits="id_nfym" extentUnits="id_yxmk">
    <listOfUnitDefinitions>
      <unitDefinition id="id_nfym">
        <listOfUnits>
          <unit kind="metre" exponent="1" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_el2n">
        <listOfUnits>
          <unit kind="metre" exponent="2" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_29tc">
        <listOfUnits>
          <unit kind="litre" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_pcnf">
        <listOfUnits>
          <unit kind="second" exponent="1" scale="0" multiplier="86400"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_yxmk">
        <listOfUnits>
          <unit kind="mole" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_zx0m">
        <listOfUnits>
          <unit kind="second" exponent="-1" scale="3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_ept0">
        <listOfUnits>
          <unit kind="dimensionless" exponent="0" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
    </listOfUnitDefinitions>
    <listOfCompartments>
      <compartment id="id_ez0l" spatialDimensions="3" size="1" units="id_29tc" constant="true"/>
    </listOfCompartments>
    <listOfSpecies>
      <species id="id_i2za" compartment="id_ez0l" initialConcentration="100" substanceUnits="id_yxmk" hasOnlySubstanceUnits="false" boundaryCondition="false" constant="false"/>
      <species id="id_z877" compartment="id_ez0l" initialConcentration="1000" substanceUnits="id_yxmk" hasOnlySubstanceUnits="false" boundaryCondition="false" constant="false"/>
    </listOfSpecies>
  </model>
</sbml>


                ## Max iterations
                20

                Format your response according to the instructions in the system message.
                
## Thoughts
I'm investigating a biological system with two species: `id_i2za` and `id_z877`. The SBML model is incomplete, and I need to discover the missing reactions. Let me first understand what we have:

1. Two species:
   - `id_i2za` with initial concentration 100
   - `id_z877` with initial concentration 1000

2. No reactions are defined in the model yet.

To understand the system, I should first observe its behavior without any perturbations. This will give me baseline data about how these species interact naturally. Then I can analyze the data to formulate hypotheses about the missing reactions.

## Action

### Experiment
```json
{
   "action": "observe",
   "meta_data": {}
}
```
# Observation

## Experiment Result
Experiment ID: iteration_0
Your requested experiment has finished. The details results are saved as a dataframe in experiment_history[iteration_0]

Total size: 4214 rows × 3 columns. 

Below shows a subset of the table for your convenience. You should access experiment_history[iteration_0] for the full data

Time   id_i2za  id_z877
0  1.00e+02 1.00e+03
...
1.71e+00  2.04e+02 9.83e+02
...
3.43e+00  4.16e+02 9.66e+02
...
5.14e+00  8.52e+02 9.49e+02
...
6.86e+00  1.75e+03 9.32e+02
...
8.57e+00  3.61e+03 9.14e+02
...
1.03e+01  7.46e+03 8.92e+02
...
1.20e+01  1.55e+04 8.62e+02
...
1.37e+01  3.24e+04 8.18e+02
...
1.54e+01  6.83e+04 7.44e+02
...
1.71e+01  1.47e+05 6.18e+02
...
1.89e+01  3.23e+05 4.21e+02
...
2.06e+01  7.41e+05 1.83e+02
...
2.23e+01  1.76e+06 2.99e+01
...
2.40e+01  4.23e+06 2.92e+00
...
2.57e+01  1.01e+07 1.04e+00
...
2.74e+01  2.41e+07 4.23e-01
...
2.91e+01  5.63e+07 1.79e-01
...
3.09e+01  1.26e+08 7.98e-02
...
3.26e+01  2.57e+08 3.90e-02
...
3.43e+01  4.52e+08 2.21e-02
...
3.60e+01  6.61e+08 1.51e-02
...
3.77e+01  8.17e+08 1.22e-02
...
3.94e+01  9.05e+08 1.10e-02
...
4.12e+01  9.48e+08 1.06e-02
...
4.29e+01  9.67e+08 1.03e-02
...
4.46e+01  9.75e+08 1.03e-02
...
4.63e+01  9.78e+08 1.02e-02
...
4.80e+01  9.79e+08 1.02e-02
...
4.97e+01  9.80e+08 1.02e-02
...
5.14e+01  9.80e+08 1.02e-02
...
5.32e+01  9.80e+08 1.02e-02
...
5.49e+01  9.80e+08 1.02e-02
...
5.66e+01  9.80e+08 1.02e-02
...
5.83e+01  9.80e+08 1.02e-02
...
6.00e+01  9.80e+08 1.02e-02
...
6.17e+01  9.80e+08 1.02e-02
...
6.34e+01  9.80e+08 1.02e-02
...
6.52e+01  9.80e+08 1.02e-02
...
6.69e+01  9.80e+08 1.02e-02
...
6.86e+01  9.80e+08 1.02e-02
...
7.03e+01  9.80e+08 1.02e-02
...
7.20e+01  9.80e+08 1.02e-02
...
7.37e+01  9.80e+08 1.02e-02
...
7.54e+01  9.80e+08 1.02e-02
...
7.72e+01  9.80e+08 1.02e-02
...
7.89e+01  9.80e+08 1.02e-02
...
8.06e+01  9.80e+08 1.02e-02
...
8.23e+01  9.80e+08 1.02e-02
...
8.40e+01  9.80e+08 1.02e-02

## Reminder

 ### Remaining Iterations for Interactions:
- Iterations: 20/20



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0)
- `shared_variables`: all variables you have saved from the previous iterations. ()

### Allowed libraires
['numpy', 'pandas', 'libsbml', 'math', 'scipy', 'jax', 'sklearn', 'io', 'traceback']


Please construct your response according to the thoughts-action markdown format.



# Iteration 1
## Thoughts
Looking at the observation data, I can see some interesting patterns:

1. `id_i2za` starts at 100 and increases exponentially, eventually reaching a plateau around 9.80e+08.
2. `id_z877` starts at 1000 and decreases over time, eventually stabilizing at a very low value (around 1.02e-02).

This behavior suggests a few possible mechanisms:

1. **Autocatalytic growth of `id_i2za`**: The species `id_i2za` appears to be growing at a rate proportional to its own concentration, which is characteristic of autocatalytic reactions.

2. **Consumption of `id_z877`**: As `id_i2za` grows, `id_z877` decreases, suggesting that `id_z877` might be consumed in the process of producing `id_i2za`.

3. **Carrying capacity**: The growth of `id_i2za` eventually plateaus, suggesting there might be a carrying capacity or limiting factor.

Let me analyze the data more carefully to understand the relationship between these species and formulate a mathematical model.

## Action

### Code
```python
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit

# Get the data
data = experiment_history["iteration_0"]

# Let's look at the first few rows to confirm the structure
print("First few rows of data:")
print(data.head())

# Let's also look at the last few rows
print("\nLast few rows of data:")
print(data.tail())

# Basic statistics
print("\nBasic statistics:")
print(data.describe())

# Let's plot the data to visualize the dynamics
plt.figure(figsize=(12, 6))
plt.subplot(1, 2, 1)
plt.plot(data['Time'], data['id_i2za'])
plt.title('id_i2za over time')
plt.xlabel('Time')
plt.ylabel('Concentration')
plt.yscale('log')  # Using log scale due to the large range

plt.subplot(1, 2, 2)
plt.plot(data['Time'], data['id_z877'])
plt.title('id_z877 over time')
plt.xlabel('Time')
plt.ylabel('Concentration')

# Let's also plot the relationship between the two species
plt.figure(figsize=(8, 6))
plt.scatter(data['id_z877'], data['id_i2za'], alpha=0.5)
plt.title('id_i2za vs id_z877')
plt.xlabel('id_z877')
plt.ylabel('id_i2za')
plt.yscale('log')  # Using log scale due to the large range

# Let's try to fit a simple model to understand the relationship
# First, let's calculate the rate of change of id_i2za
data['d_i2za_dt'] = np.gradient(data['id_i2za'], data['Time'])

# Let's plot the rate of change vs id_i2za and id_z877
plt.figure(figsize=(12, 6))
plt.subplot(1, 2, 1)
plt.scatter(data['id_i2za'], data['d_i2za_dt'], alpha=0.5)
plt.title('Rate of change of id_i2za vs id_i2za')
plt.xlabel('id_i2za')
plt.ylabel('d(id_i2za)/dt')
plt.xscale('log')  # Using log scale due to the large range
plt.yscale('log')  # Using log scale due to the large range

plt.subplot(1, 2, 2)
plt.scatter(data['id_z877'], data['d_i2za_dt'], alpha=0.5)
plt.title('Rate of change of id_i2za vs id_z877')
plt.xlabel('id_z877')
plt.ylabel('d(id_i2za)/dt')
plt.yscale('log')  # Using log scale due to the large range

# Let's try to fit a model where d(id_i2za)/dt = k * id_i2za * id_z877
# This would correspond to a reaction where id_i2za catalyzes the conversion of id_z877 to id_i2za
def model_func(X, k):
    i2za, z877 = X
    return k * i2za * z877

# Prepare the data for curve fitting
X = np.vstack([data['id_i2za'], data['id_z877']])
y = data['d_i2za_dt']

# Fit the model
params, _ = curve_fit(model_func, X, y)
k = params[0]

print(f"\nFitted parameter k: {k}")

# Calculate the predicted rate of change
data['predicted_d_i2za_dt'] = k * data['id_i2za'] * data['id_z877']

# Plot the actual vs predicted rate of change
plt.figure(figsize=(8, 6))
plt.scatter(data['d_i2za_dt'], data['predicted_d_i2za_dt'], alpha=0.5)
plt.title('Actual vs Predicted Rate of Change of id_i2za')
plt.xlabel('Actual d(id_i2za)/dt')
plt.ylabel('Predicted d(id_i2za)/dt')
plt.xscale('log')  # Using log scale due to the large range
plt.yscale('log')  # Using log scale due to the large range
plt.plot([min(data['d_i2za_dt']), max(data['d_i2za_dt'])], [min(data['d_i2za_dt']), max(data['d_i2za_dt'])], 'r--')

# Let's also calculate the rate of change of id_z877
data['d_z877_dt'] = np.gradient(data['id_z877'], data['Time'])

# Let's plot the rate of change of id_z877 vs id_i2za and id_z877
plt.figure(figsize=(12, 6))
plt.subplot(1, 2, 1)
plt.scatter(data['id_i2za'], data['d_z877_dt'], alpha=0.5)
plt.title('Rate of change of id_z877 vs id_i2za')
plt.xlabel('id_i2za')
plt.ylabel('d(id_z877)/dt')
plt.xscale('log')  # Using log scale due to the large range

plt.subplot(1, 2, 2)
plt.scatter(data['id_z877'], data['d_z877_dt'], alpha=0.5)
plt.title('Rate of change of id_z877 vs id_z877')
plt.xlabel('id_z877')
plt.ylabel('d(id_z877)/dt')

# Let's try to fit a model where d(id_z877)/dt = -k * id_i2za * id_z877
# This would correspond to id_z877 being consumed in the reaction
def model_func_z877(X, k):
    i2za, z877 = X
    return -k * i2za * z877

# Fit the model
params_z877, _ = curve_fit(model_func_z877, X, data['d_z877_dt'])
k_z877 = params_z877[0]

print(f"\nFitted parameter k_z877: {k_z877}")

# Calculate the predicted rate of change
data['predicted_d_z877_dt'] = -k_z877 * data['id_i2za'] * data['id_z877']

# Plot the actual vs predicted rate of change
plt.figure(figsize=(8, 6))
plt.scatter(data['d_z877_dt'], data['predicted_d_z877_dt'], alpha=0.5)
plt.title('Actual vs Predicted Rate of Change of id_z877')
plt.xlabel('Actual d(id_z877)/dt')
plt.ylabel('Predicted d(id_z877)/dt')
plt.plot([min(data['d_z877_dt']), max(data['d_z877_dt'])], [min(data['d_z877_dt']), max(data['d_z877_dt'])], 'r--')

# Based on our analysis, let's formulate a hypothesis for the missing reactions
print("\nHypothesis for missing reactions:")
print(f"1. id_i2za + id_z877 -> 2 id_i2za (rate constant k = {k})")
print(f"2. id_i2za + id_z877 -> id_i2za (rate constant k = {k_z877})")
print("This suggests that id_i2za catalyzes the conversion of id_z877 to id_i2za.")

# Store the fitted parameters for future use
shared_variables.add("k", k)
shared_variables.add("k_z877", k_z877)
```
# Observation

## Code Stderror
 Unauthorized modules matplotlib detected. Only these modules are allowed: numpy, pandas, libsbml, math, scipy, jax, sklearn, io, traceback.

## Reminder

 ### Remaining Iterations for Interactions:
- Iterations: 19/20



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0)
- `shared_variables`: all variables you have saved from the previous iterations. ()

### Allowed libraires
['numpy', 'pandas', 'libsbml', 'math', 'scipy', 'jax', 'sklearn', 'io', 'traceback']


Please construct your response according to the thoughts-action markdown format.



# Iteration 2
## Thoughts
I apologize for trying to use matplotlib, which is not allowed. Let me rewrite the analysis code without using visualization libraries. I'll focus on numerical analysis to understand the relationship between the two species.

Based on the observation data, I'm hypothesizing that there's a reaction where `id_i2za` catalyzes the conversion of `id_z877` to more `id_i2za`. This would explain why `id_i2za` grows exponentially while `id_z877` decreases. Let me analyze the data to verify this hypothesis and determine the reaction rate constants.

## Action

### Code
```python
import numpy as np
import pandas as pd
from scipy.optimize import curve_fit

# Get the data
data = experiment_history["iteration_0"]

# Let's look at the first few rows to confirm the structure
print("First few rows of data:")
print(data.head())

# Let's also look at the last few rows
print("\nLast few rows of data:")
print(data.tail())

# Basic statistics
print("\nBasic statistics:")
print(data.describe())

# Calculate the rate of change of id_i2za and id_z877
data['d_i2za_dt'] = np.gradient(data['id_i2za'], data['Time'])
data['d_z877_dt'] = np.gradient(data['id_z877'], data['Time'])

# Print some statistics about the rates of change
print("\nRate of change statistics for id_i2za:")
print(data['d_i2za_dt'].describe())

print("\nRate of change statistics for id_z877:")
print(data['d_z877_dt'].describe())

# Let's check the correlation between the rates of change
correlation = np.corrcoef(data['d_i2za_dt'], data['d_z877_dt'])[0, 1]
print(f"\nCorrelation between d(id_i2za)/dt and d(id_z877)/dt: {correlation}")

# Let's try to fit a model where d(id_i2za)/dt = k * id_i2za * id_z877
# This would correspond to a reaction where id_i2za catalyzes the conversion of id_z877 to id_i2za
def model_func(X, k):
    i2za, z877 = X
    return k * i2za * z877

# Prepare the data for curve fitting
X = np.vstack([data['id_i2za'], data['id_z877']])
y = data['d_i2za_dt']

# Fit the model
params, covariance = curve_fit(model_func, X, y)
k = params[0]
k_error = np.sqrt(covariance[0, 0])

print(f"\nFitted parameter k: {k} ± {k_error}")

# Calculate the predicted rate of change
data['predicted_d_i2za_dt'] = k * data['id_i2za'] * data['id_z877']

# Calculate the R-squared value to assess the goodness of fit
ss_total = np.sum((data['d_i2za_dt'] - np.mean(data['d_i2za_dt']))**2)
ss_residual = np.sum((data['d_i2za_dt'] - data['predicted_d_i2za_dt'])**2)
r_squared = 1 - (ss_residual / ss_total)

print(f"R-squared for id_i2za rate model: {r_squared}")

# Let's try to fit a model where d(id_z877)/dt = -k * id_i2za * id_z877
# This would correspond to id_z877 being consumed in the reaction
def model_func_z877(X, k):
    i2za, z877 = X
    return -k * i2za * z877

# Fit the model
params_z877, covariance_z877 = curve_fit(model_func_z877, X, data['d_z877_dt'])
k_z877 = params_z877[0]
k_z877_error = np.sqrt(covariance_z877[0, 0])

print(f"\nFitted parameter k_z877: {k_z877} ± {k_z877_error}")

# Calculate the predicted rate of change
data['predicted_d_z877_dt'] = -k_z877 * data['id_i2za'] * data['id_z877']

# Calculate the R-squared value
ss_total_z877 = np.sum((data['d_z877_dt'] - np.mean(data['d_z877_dt']))**2)
ss_residual_z877 = np.sum((data['d_z877_dt'] - data['predicted_d_z877_dt'])**2)
r_squared_z877 = 1 - (ss_residual_z877 / ss_total_z877)

print(f"R-squared for id_z877 rate model: {r_squared_z877}")

# Check if k and k_z877 are approximately equal
ratio = k / k_z877
print(f"\nRatio of k to k_z877: {ratio}")

# Based on our analysis, let's formulate a hypothesis for the missing reactions
print("\nHypothesis for missing reactions:")
if abs(ratio - 1) < 0.1:  # If the ratio is close to 1
    print(f"id_i2za + id_z877 -> 2 id_i2za (rate constant k ≈ {(k + k_z877)/2})")
    print("This suggests that id_i2za catalyzes the conversion of id_z877 to id_i2za.")
else:
    print(f"1. id_i2za + id_z877 -> 2 id_i2za (rate constant k = {k})")
    print(f"2. id_i2za + id_z877 -> id_i2za (rate constant k = {k_z877})")
    print("This suggests that id_i2za catalyzes the conversion of id_z877 to id_i2za, but the rates of production and consumption are different.")

# Let's also check if there's any evidence of a carrying capacity or other limiting factors
# If there's a carrying capacity, we would expect the growth rate to decrease as id_i2za approaches the carrying capacity
# Let's calculate the per capita growth rate
data['per_capita_growth_rate'] = data['d_i2za_dt'] / data['id_i2za']

# Print statistics about the per capita growth rate
print("\nPer capita growth rate statistics:")
print(data['per_capita_growth_rate'].describe())

# Check if the per capita growth rate decreases as id_i2za increases
# We can calculate the correlation between id_i2za and the per capita growth rate
correlation_growth = np.corrcoef(data['id_i2za'], data['per_capita_growth_rate'])[0, 1]
print(f"\nCorrelation between id_i2za and per capita growth rate: {correlation_growth}")

# Store the fitted parameters for future use
shared_variables.add("k", k)
shared_variables.add("k_z877", k_z877)
shared_variables.add("r_squared", r_squared)
shared_variables.add("r_squared_z877", r_squared_z877)
```
# Observation

## Code Stdout
First few rows of data:
      id_i2za      id_z877      Time
0  100.000000  1000.000000  0.000000
1  100.829019   999.798657  0.019938
2  101.664828   999.597376  0.039877
3  102.507607   999.396160  0.059815
4  103.357420   999.195007  0.079753

Last few rows of data:
           id_i2za  id_z877       Time
4209  9.803902e+08   0.0102  83.920247
4210  9.803902e+08   0.0102  83.940185
4211  9.803902e+08   0.0102  83.960123
4212  9.803902e+08   0.0102  83.980062
4213  9.803902e+08   0.0102  84.000000

Basic statistics:
            id_i2za      id_z877         Time
count  4.214000e+03  4214.000000  4214.000000
mean   5.766150e+08   200.430104    42.000000
std    4.589274e+08   361.257878    24.257345
min    1.000000e+02     0.010200     0.000000
25%    9.153122e+05     0.010200    21.000000
50%    9.590774e+08     0.010427    42.000000
75%    9.803897e+08   131.303887    63.000000
max    9.803902e+08  1000.000000    84.000000

Rate of change statistics for id_i2za:
count    4.214000e+03
mean     1.166854e+07
std      2.905237e+07
min     -1.217690e-01
25%      9.601768e+01
50%      1.183760e+04
75%      2.254944e+06
max      1.259780e+08
Name: d_i2za_dt, dtype: float64

Rate of change statistics for id_z877:
count    4.214000e+03
mean    -1.190301e+01
std      2.989859e+01
min     -1.427591e+02
25%     -9.768267e+00
50%     -1.165802e-04
75%     -8.589556e-09
max      2.121892e-08
Name: d_z877_dt, dtype: float64

Correlation between d(id_i2za)/dt and d(id_z877)/dt: 0.1568136073818153

Fitted parameter k: 0.10617406927668245 ± 0.01429596780991106
R-squared for id_i2za rate model: -0.14634316933623426

Fitted parameter k_z877: 9.216045613543223e-07 ± 4.14149285661861e-09
R-squared for id_z877 rate model: 0.9091629481119441

Ratio of k to k_z877: 115205.66816711155

Hypothesis for missing reactions:
1. id_i2za + id_z877 -> 2 id_i2za (rate constant k = 0.10617406927668245)
2. id_i2za + id_z877 -> id_i2za (rate constant k = 9.216045613543223e-07)
This suggests that id_i2za catalyzes the conversion of id_z877 to id_i2za, but the rates of production and consumption are different.

Per capita growth rate statistics:
count    4.214000e+03
mean     1.916527e-01
std      2.169768e-01
min     -1.242046e-10
25%      2.350744e-07
50%      1.117243e-02
75%      4.252085e-01
max      5.118247e-01
Name: per_capita_growth_rate, dtype: float64

Correlation between id_i2za and per capita growth rate: -0.992850917214436

## Reminder

 ### Remaining Iterations for Interactions:
- Iterations: 18/20



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0)
- `shared_variables`: all variables you have saved from the previous iterations. (k, k_z877, r_squared, r_squared_z877)

### Allowed libraires
['numpy', 'pandas', 'libsbml', 'math', 'scipy', 'jax', 'sklearn', 'io', 'traceback']


Please construct your response according to the thoughts-action markdown format.



# Iteration 3
## Thoughts
Based on the analysis, I've gained several important insights:

1. The R-squared value for the `id_z877` rate model is very high (0.91), indicating that the model `d(id_z877)/dt = -k * id_i2za * id_z877` fits the data well. This suggests that `id_z877` is being consumed in a reaction with `id_i2za`.

2. However, the R-squared value for the `id_i2za` rate model is negative (-0.15), which indicates that the model `d(id_i2za)/dt = k * id_i2za * id_z877` does not fit the data well. This suggests that the production of `id_i2za` is more complex than just being proportional to the product of `id_i2za` and `id_z877`.

3. The ratio of k to k_z877 is very large (115205), which suggests that the rates of production of `id_i2za` and consumption of `id_z877` are not stoichiometrically balanced in a simple 1:1 reaction.

4. There's a strong negative correlation (-0.99) between `id_i2za` and the per capita growth rate, which suggests that as `id_i2za` increases, its growth rate decreases. This is indicative of a carrying capacity or some other limiting factor.

Let me try a different model for the production of `id_i2za`. Given the strong negative correlation between `id_i2za` and its per capita growth rate, I'll try a logistic growth model with `id_z877` as a limiting factor.

## Action

### Code
```python
import numpy as np
import pandas as pd
from scipy.optimize import curve_fit

# Get the data
data = experiment_history["iteration_0"]

# Let's try a different model for id_i2za
# Let's try a logistic growth model with id_z877 as a limiting factor
# d(id_i2za)/dt = r * id_i2za * (1 - id_i2za/K) * id_z877
def logistic_model(X, r, K):
    i2za, z877 = X
    return r * i2za * (1 - i2za/K) * z877

# Prepare the data for curve fitting
X = np.vstack([data['id_i2za'], data['id_z877']])
y = data['d_i2za_dt']

# Initial guess for the parameters
p0 = [0.1, 1e9]  # r, K

# Fit the model
try:
    params, covariance = curve_fit(logistic_model, X, y, p0=p0)
    r, K = params
    r_error, K_error = np.sqrt(np.diag(covariance))
    
    print(f"Fitted parameter r: {r} ± {r_error}")
    print(f"Fitted parameter K: {K} ± {K_error}")
    
    # Calculate the predicted rate of change
    data['predicted_d_i2za_dt_logistic'] = logistic_model(X, r, K)
    
    # Calculate the R-squared value
    ss_total = np.sum((data['d_i2za_dt'] - np.mean(data['d_i2za_dt']))**2)
    ss_residual = np.sum((data['d_i2za_dt'] - data['predicted_d_i2za_dt_logistic'])**2)
    r_squared_logistic = 1 - (ss_residual / ss_total)
    
    print(f"R-squared for logistic model: {r_squared_logistic}")
    
    # Store the fitted parameters
    shared_variables.add("r", r)
    shared_variables.add("K", K)
    shared_variables.add("r_squared_logistic", r_squared_logistic)
except Exception as e:
    print(f"Error fitting logistic model: {e}")
    
    # Let's try a simpler model
    # d(id_i2za)/dt = r * id_i2za * (1 - id_i2za/K)
    def simple_logistic_model(X, r, K):
        i2za, _ = X
        return r * i2za * (1 - i2za/K)
    
    try:
        params, covariance = curve_fit(simple_logistic_model, X, y, p0=p0)
        r, K = params
        r_error, K_error = np.sqrt(np.diag(covariance))
        
        print(f"Fitted parameter r (simple logistic): {r} ± {r_error}")
        print(f"Fitted parameter K (simple logistic): {K} ± {K_error}")
        
        # Calculate the predicted rate of change
        data['predicted_d_i2za_dt_simple_logistic'] = simple_logistic_model(X, r, K)
        
        # Calculate the R-squared value
        ss_total = np.sum((data['d_i2za_dt'] - np.mean(data['d_i2za_dt']))**2)
        ss_residual = np.sum((data['d_i2za_dt'] - data['predicted_d_i2za_dt_simple_logistic'])**2)
        r_squared_simple_logistic = 1 - (ss_residual / ss_total)
        
        print(f"R-squared for simple logistic model: {r_squared_simple_logistic}")
        
        # Store the fitted parameters
        shared_variables.add("r_simple", r)
        shared_variables.add("K_simple", K)
        shared_variables.add("r_squared_simple_logistic", r_squared_simple_logistic)
    except Exception as e:
        print(f"Error fitting simple logistic model: {e}")

# Let's also try a model where id_i2za is produced from id_z877 at a rate proportional to id_i2za
# d(id_i2za)/dt = k * id_i2za * id_z877
# d(id_z877)/dt = -k * id_i2za * id_z877
# This would correspond to a reaction where id_i2za catalyzes the conversion of id_z877 to id_i2za
def catalyst_model(X, k):
    i2za, z877 = X
    return k * i2za * z877

# Fit the model for id_i2za
try:
    params_i2za, covariance_i2za = curve_fit(catalyst_model, X, y)
    k_i2za = params_i2za[0]
    k_i2za_error = np.sqrt(covariance_i2za[0, 0])
    
    print(f"\nFitted parameter k_i2za: {k_i2za} ± {k_i2za_error}")
    
    # Calculate the predicted rate of change
    data['predicted_d_i2za_dt_catalyst'] = k_i2za * data['id_i2za'] * data['id_z877']
    
    # Calculate the R-squared value
    ss_total_i2za = np.sum((data['d_i2za_dt'] - np.mean(data['d_i2za_dt']))**2)
    ss_residual_i2za = np.sum((data['d_i2za_dt'] - data['predicted_d_i2za_dt_catalyst'])**2)
    r_squared_i2za = 1 - (ss_residual_i2za / ss_total_i2za)
    
    print(f"R-squared for id_i2za catalyst model: {r_squared_i2za}")
    
    # Store the fitted parameters
    shared_variables.add("k_i2za", k_i2za)
    shared_variables.add("r_squared_i2za", r_squared_i2za)
except Exception as e:
    print(f"Error fitting catalyst model for id_i2za: {e}")

# Fit the model for id_z877
try:
    def negative_catalyst_model(X, k):
        i2za, z877 = X
        return -k * i2za * z877
    
    params_z877, covariance_z877 = curve_fit(negative_catalyst_model, X, data['d_z877_dt'])
    k_z877_catalyst = params_z877[0]
    k_z877_catalyst_error = np.sqrt(covariance_z877[0, 0])
    
    print(f"\nFitted parameter k_z877_catalyst: {k_z877_catalyst} ± {k_z877_catalyst_error}")
    
    # Calculate the predicted rate of change
    data['predicted_d_z877_dt_catalyst'] = -k_z877_catalyst * data['id_i2za'] * data['id_z877']
    
    # Calculate the R-squared value
    ss_total_z877 = np.sum((data['d_z877_dt'] - np.mean(data['d_z877_dt']))**2)
    ss_residual_z877 = np.sum((data['d_z877_dt'] - data['predicted_d_z877_dt_catalyst'])**2)
    r_squared_z877_catalyst = 1 - (ss_residual_z877 / ss_total_z877)
    
    print(f"R-squared for id_z877 catalyst model: {r_squared_z877_catalyst}")
    
    # Store the fitted parameters
    shared_variables.add("k_z877_catalyst", k_z877_catalyst)
    shared_variables.add("r_squared_z877_catalyst", r_squared_z877_catalyst)
except Exception as e:
    print(f"Error fitting catalyst model for id_z877: {e}")

# Let's try one more model: autocatalytic growth of id_i2za with consumption of id_z877
# d(id_i2za)/dt = k1 * id_i2za * id_z877
# d(id_z877)/dt = -k2 * id_i2za * id_z877
# where k1 and k2 are different
# This would correspond to a reaction where id_i2za catalyzes the conversion of id_z877 to multiple id_i2za
def autocatalytic_model(X, k1, n):
    i2za, z877 = X
    return k1 * i2za * z877 * (1 - i2za/n)

# Fit the model for id_i2za
try:
    p0 = [0.1, 1e9]  # k1, n
    params_auto, covariance_auto = curve_fit(autocatalytic_model, X, y, p0=p0)
    k1, n = params_auto
    k1_error, n_error = np.sqrt(np.diag(covariance_auto))
    
    print(f"\nFitted parameter k1: {k1} ± {k1_error}")
    print(f"Fitted parameter n: {n} ± {n_error}")
    
    # Calculate the predicted rate of change
    data['predicted_d_i2za_dt_auto'] = autocatalytic_model(X, k1, n)
    
    # Calculate the R-squared value
    ss_total_auto = np.sum((data['d_i2za_dt'] - np.mean(data['d_i2za_dt']))**2)
    ss_residual_auto = np.sum((data['d_i2za_dt'] - data['predicted_d_i2za_dt_auto'])**2)
    r_squared_auto = 1 - (ss_residual_auto / ss_total_auto)
    
    print(f"R-squared for autocatalytic model: {r_squared_auto}")
    
    # Store the fitted parameters
    shared_variables.add("k1", k1)
    shared_variables.add("n", n)
    shared_variables.add("r_squared_auto", r_squared_auto)
except Exception as e:
    print(f"Error fitting autocatalytic model: {e}")

# Based on our analysis, let's formulate a hypothesis for the missing reactions
print("\nHypothesis for missing reactions:")
print("Based on the R-squared values of our models, it appears that:")
print(f"1. id_z877 is consumed in a reaction with id_i2za (R-squared = {shared_variables.access('r_squared_z877')})")
print(f"2. The production of id_i2za is more complex, possibly involving autocatalytic growth with a carrying capacity")
print("3. The stoichiometry of the reaction is not 1:1, as the rates of production and consumption are different")

# Let's try to simulate the system with our best model and see how well it matches the data
# We'll use the autocatalytic model for id_i2za and the catalyst model for id_z877
try:
    from scipy.integrate import solve_ivp
    
    # Define the system of differential equations
    def system(t, y, k1, n, k2):
        i2za, z877 = y
        d_i2za_dt = k1 * i2za * z877 * (1 - i2za/n)
        d_z877_dt = -k2 * i2za * z877
        return [d_i2za_dt, d_z877_dt]
    
    # Initial conditions
    y0 = [data['id_i2za'].iloc[0], data['id_z877'].iloc[0]]
    
    # Time points
    t_span = [data['Time'].min(), data['Time'].max()]
    t_eval = data['Time'].values
    
    # Solve the system
    k1 = shared_variables.access('k1')
    n = shared_variables.access('n')
    k2 = shared_variables.access('k_z877_catalyst')
    
    sol = solve_ivp(system, t_span, y0, args=(k1, n, k2), t_eval=t_eval)
    
    # Calculate the R-squared value for the simulation
    ss_total_i2za_sim = np.sum((data['id_i2za'] - np.mean(data['id_i2za']))**2)
    ss_residual_i2za_sim = np.sum((data['id_i2za'] - sol.y[0])**2)
    r_squared_i2za_sim = 1 - (ss_residual_i2za_sim / ss_total_i2za_sim)
    
    ss_total_z877_sim = np.sum((data['id_z877'] - np.mean(data['id_z877']))**2)
    ss_residual_z877_sim = np.sum((data['id_z877'] - sol.y[1])**2)
    r_squared_z877_sim = 1 - (ss_residual_z877_sim / ss_total_z877_sim)
    
    print(f"\nR-squared for id_i2za simulation: {r_squared_i2za_sim}")
    print(f"R-squared for id_z877 simulation: {r_squared_z877_sim}")
    
    # Store the R-squared values
    shared_variables.add("r_squared_i2za_sim", r_squared_i2za_sim)
    shared_variables.add("r_squared_z877_sim", r_squared_z877_sim)
except Exception as e:
    print(f"Error simulating the system: {e}")

# Let's try a simpler model: autocatalytic growth of id_i2za with consumption of id_z877
# d(id_i2za)/dt = k1 * id_i2za * id_z877
# d(id_z877)/dt = -k2 * id_i2za * id_z877
# where k1 and k2 are different
# This would correspond to a reaction where id_i2za catalyzes the conversion of id_z877 to multiple id_i2za
try:
    from scipy.integrate import solve_ivp
    
    # Define the system of differential equations
    def simple_system(t, y, k1, k2):
        i2za, z877 = y
        d_i2za_dt = k1 * i2za * z877
        d_z877_dt = -k2 * i2za * z877
        return [d_i2za_dt, d_z877_dt]
    
    # Initial conditions
    y0 = [data['id_i2za'].iloc[0], data['id_z877'].iloc[0]]
    
    # Time points
    t_span = [data['Time'].min(), data['Time'].max()]
    t_eval = data['Time'].values
    
    # Solve the system
    k1 = shared_variables.access('k_i2za')
    k2 = shared_variables.access('k_z877_catalyst')
    
    sol_simple = solve_ivp(simple_system, t_span, y0, args=(k1, k2), t_eval=t_eval)
    
    # Calculate the R-squared value for the simulation
    ss_total_i2za_sim_simple = np.sum((data['id_i2za'] - np.mean(data['id_i2za']))**2)
    ss_residual_i2za_sim_simple = np.sum((data['id_i2za'] - sol_simple.y[0])**2)
    r_squared_i2za_sim_simple = 1 - (ss_residual_i2za_sim_simple / ss_total_i2za_sim_simple)
    
    ss_total_z877_sim_simple = np.sum((data['id_z877'] - np.mean(data['id_z877']))**2)
    ss_residual_z877_sim_simple = np.sum((data['id_z877'] - sol_simple.y[1])**2)
    r_squared_z877_sim_simple = 1 - (ss_residual_z877_sim_simple / ss_total_z877_sim_simple)
    
    print(f"\nR-squared for id_i2za simple simulation: {r_squared_i2za_sim_simple}")
    print(f"R-squared for id_z877 simple simulation: {r_squared_z877_sim_simple}")
    
    # Store the R-squared values
    shared_variables.add("r_squared_i2za_sim_simple", r_squared_i2za_sim_simple)
    shared_variables.add("r_squared_z877_sim_simple", r_squared_z877_sim_simple)
except Exception as e:
    print(f"Error simulating the simple system: {e}")

# Based on our analysis, let's formulate a final hypothesis for the missing reactions
print("\nFinal hypothesis for missing reactions:")
print("Based on our analysis, the most likely reaction is:")
print(f"id_i2za + id_z877 -> {k1/k2} id_i2za")
print(f"with rate constants k1 = {k1} and k2 = {k2}")
print(f"This gives a stoichiometric coefficient of approximately {k1/k2} for id_i2za production.")
```
# Observation

## Code Stderror
 'd_i2za_dt'

## Reminder

 ### Remaining Iterations for Interactions:
- Iterations: 17/20



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0)
- `shared_variables`: all variables you have saved from the previous iterations. (k, k_z877, r_squared, r_squared_z877)

### Allowed libraires
['numpy', 'pandas', 'libsbml', 'math', 'scipy', 'jax', 'sklearn', 'io', 'traceback']


Please construct your response according to the thoughts-action markdown format.



# Iteration 4
## Thoughts
I see there was an error in my previous code. It seems I'm trying to access `d_i2za_dt` which might not be defined yet. Let me fix this and continue with my analysis. 

Based on the previous results, I've established that:
1. `id_z877` is being consumed in a reaction with `id_i2za` (R-squared = 0.91)
2. The simple model `d(id_i2za)/dt = k * id_i2za * id_z877` doesn't fit well for `id_i2za` production
3. There's a strong negative correlation between `id_i2za` and its per capita growth rate

Let me try a more systematic approach to model the system. I'll:
1. Calculate the rates of change for both species
2. Try different models for the production of `id_i2za` and consumption of `id_z877`
3. Evaluate the models based on their R-squared values
4. Simulate the system with the best model to verify its accuracy

## Action

### Code
```python
import numpy as np
import pandas as pd
from scipy.optimize import curve_fit
from scipy.integrate import solve_ivp

# Get the data
data = experiment_history["iteration_0"]

# Calculate the rate of change of id_i2za and id_z877
data['d_i2za_dt'] = np.gradient(data['id_i2za'], data['Time'])
data['d_z877_dt'] = np.gradient(data['id_z877'], data['Time'])

# Print some statistics about the rates of change
print("Rate of change statistics for id_i2za:")
print(data['d_i2za_dt'].describe())

print("\nRate of change statistics for id_z877:")
print(data['d_z877_dt'].describe())

# Let's check the correlation between the rates of change
correlation = np.corrcoef(data['d_i2za_dt'], data['d_z877_dt'])[0, 1]
print(f"\nCorrelation between d(id_i2za)/dt and d(id_z877)/dt: {correlation}")

# Let's try different models for id_i2za production

# Model 1: Simple catalytic model
# d(id_i2za)/dt = k * id_i2za * id_z877
def model1(X, k):
    i2za, z877 = X
    return k * i2za * z877

# Model 2: Logistic growth with id_z877 as a limiting factor
# d(id_i2za)/dt = r * id_i2za * (1 - id_i2za/K) * id_z877
def model2(X, r, K):
    i2za, z877 = X
    return r * i2za * (1 - i2za/K) * z877

# Model 3: Autocatalytic growth with stoichiometric coefficient
# d(id_i2za)/dt = k1 * id_i2za * id_z877
# d(id_z877)/dt = -k2 * id_i2za * id_z877
# where k1/k2 is the stoichiometric coefficient
def model3(X, k):
    i2za, z877 = X
    return k * i2za * z877

# Prepare the data for curve fitting
X = np.vstack([data['id_i2za'], data['id_z877']])
y_i2za = data['d_i2za_dt']
y_z877 = data['d_z877_dt']

# Fit Model 1 for id_i2za
try:
    params1, covariance1 = curve_fit(model1, X, y_i2za)
    k1 = params1[0]
    k1_error = np.sqrt(covariance1[0, 0])
    
    print(f"\nModel 1 (Simple catalytic) for id_i2za:")
    print(f"Fitted parameter k: {k1} ± {k1_error}")
    
    # Calculate the predicted rate of change
    data['predicted_d_i2za_dt_model1'] = model1(X, k1)
    
    # Calculate the R-squared value
    ss_total1 = np.sum((y_i2za - np.mean(y_i2za))**2)
    ss_residual1 = np.sum((y_i2za - data['predicted_d_i2za_dt_model1'])**2)
    r_squared1 = 1 - (ss_residual1 / ss_total1)
    
    print(f"R-squared: {r_squared1}")
    
    # Store the fitted parameters
    shared_variables.add("k1", k1)
    shared_variables.add("r_squared1", r_squared1)
except Exception as e:
    print(f"Error fitting Model 1 for id_i2za: {e}")

# Fit Model 2 for id_i2za
try:
    p0 = [0.1, 1e9]  # Initial guess for r, K
    params2, covariance2 = curve_fit(model2, X, y_i2za, p0=p0)
    r2, K2 = params2
    r2_error, K2_error = np.sqrt(np.diag(covariance2))
    
    print(f"\nModel 2 (Logistic growth) for id_i2za:")
    print(f"Fitted parameter r: {r2} ± {r2_error}")
    print(f"Fitted parameter K: {K2} ± {K2_error}")
    
    # Calculate the predicted rate of change
    data['predicted_d_i2za_dt_model2'] = model2(X, r2, K2)
    
    # Calculate the R-squared value
    ss_total2 = np.sum((y_i2za - np.mean(y_i2za))**2)
    ss_residual2 = np.sum((y_i2za - data['predicted_d_i2za_dt_model2'])**2)
    r_squared2 = 1 - (ss_residual2 / ss_total2)
    
    print(f"R-squared: {r_squared2}")
    
    # Store the fitted parameters
    shared_variables.add("r2", r2)
    shared_variables.add("K2", K2)
    shared_variables.add("r_squared2", r_squared2)
except Exception as e:
    print(f"Error fitting Model 2 for id_i2za: {e}")

# Fit Model 3 for id_z877
try:
    def negative_model3(X, k):
        i2za, z877 = X
        return -k * i2za * z877
    
    params3, covariance3 = curve_fit(negative_model3, X, y_z877)
    k3 = params3[0]
    k3_error = np.sqrt(covariance3[0, 0])
    
    print(f"\nModel 3 (Autocatalytic) for id_z877:")
    print(f"Fitted parameter k: {k3} ± {k3_error}")
    
    # Calculate the predicted rate of change
    data['predicted_d_z877_dt_model3'] = negative_model3(X, k3)
    
    # Calculate the R-squared value
    ss_total3 = np.sum((y_z877 - np.mean(y_z877))**2)
    ss_residual3 = np.sum((y_z877 - data['predicted_d_z877_dt_model3'])**2)
    r_squared3 = 1 - (ss_residual3 / ss_total3)
    
    print(f"R-squared: {r_squared3}")
    
    # Store the fitted parameters
    shared_variables.add("k3", k3)
    shared_variables.add("r_squared3", r_squared3)
except Exception as e:
    print(f"Error fitting Model 3 for id_z877: {e}")

# Calculate the stoichiometric coefficient
try:
    if 'k1' in shared_variables.__dict__ and 'k3' in shared_variables.__dict__:
        k1 = shared_variables.access('k1')
        k3 = shared_variables.access('k3')
        stoichiometric_coefficient = k1 / k3
        print(f"\nStoichiometric coefficient (k1/k3): {stoichiometric_coefficient}")
        shared_variables.add("stoichiometric_coefficient", stoichiometric_coefficient)
except Exception as e:
    print(f"Error calculating stoichiometric coefficient: {e}")

# Simulate the system with the best model
try:
    # Define the system of differential equations
    def system(t, y, k1, k3):
        i2za, z877 = y
        d_i2za_dt = k1 * i2za * z877
        d_z877_dt = -k3 * i2za * z877
        return [d_i2za_dt, d_z877_dt]
    
    # Initial conditions
    y0 = [data['id_i2za'].iloc[0], data['id_z877'].iloc[0]]
    
    # Time points
    t_span = [data['Time'].min(), data['Time'].max()]
    t_eval = data['Time'].values
    
    # Solve the system
    k1 = shared_variables.access('k1')
    k3 = shared_variables.access('k3')
    
    sol = solve_ivp(system, t_span, y0, args=(k1, k3), t_eval=t_eval)
    
    # Calculate the R-squared value for the simulation
    ss_total_i2za_sim = np.sum((data['id_i2za'] - np.mean(data['id_i2za']))**2)
    ss_residual_i2za_sim = np.sum((data['id_i2za'] - sol.y[0])**2)
    r_squared_i2za_sim = 1 - (ss_residual_i2za_sim / ss_total_i2za_sim)
    
    ss_total_z877_sim = np.sum((data['id_z877'] - np.mean(data['id_z877']))**2)
    ss_residual_z877_sim = np.sum((data['id_z877'] - sol.y[1])**2)
    r_squared_z877_sim = 1 - (ss_residual_z877_sim / ss_total_z877_sim)
    
    print(f"\nSimulation results:")
    print(f"R-squared for id_i2za: {r_squared_i2za_sim}")
    print(f"R-squared for id_z877: {r_squared_z877_sim}")
    
    # Store the R-squared values
    shared_variables.add("r_squared_i2za_sim", r_squared_i2za_sim)
    shared_variables.add("r_squared_z877_sim", r_squared_z877_sim)
except Exception as e:
    print(f"Error simulating the system: {e}")

# Simulate the system with the logistic model
try:
    if 'r2' in shared_variables.__dict__ and 'K2' in shared_variables.__dict__ and 'k3' in shared_variables.__dict__:
        # Define the system of differential equations
        def logistic_system(t, y, r, K, k3):
            i2za, z877 = y
            d_i2za_dt = r * i2za * (1 - i2za/K) * z877
            d_z877_dt = -k3 * i2za * z877
            return [d_i2za_dt, d_z877_dt]
        
        # Initial conditions
        y0 = [data['id_i2za'].iloc[0], data['id_z877'].iloc[0]]
        
        # Time points
        t_span = [data['Time'].min(), data['Time'].max()]
        t_eval = data['Time'].values
        
        # Solve the system
        r2 = shared_variables.access('r2')
        K2 = shared_variables.access('K2')
        k3 = shared_variables.access('k3')
        
        sol_logistic = solve_ivp(logistic_system, t_span, y0, args=(r2, K2, k3), t_eval=t_eval)
        
        # Calculate the R-squared value for the simulation
        ss_total_i2za_sim_logistic = np.sum((data['id_i2za'] - np.mean(data['id_i2za']))**2)
        ss_residual_i2za_sim_logistic = np.sum((data['id_i2za'] - sol_logistic.y[0])**2)
        r_squared_i2za_sim_logistic = 1 - (ss_residual_i2za_sim_logistic / ss_total_i2za_sim_logistic)
        
        ss_total_z877_sim_logistic = np.sum((data['id_z877'] - np.mean(data['id_z877']))**2)
        ss_residual_z877_sim_logistic = np.sum((data['id_z877'] - sol_logistic.y[1])**2)
        r_squared_z877_sim_logistic = 1 - (ss_residual_z877_sim_logistic / ss_total_z877_sim_logistic)
        
        print(f"\nLogistic simulation results:")
        print(f"R-squared for id_i2za: {r_squared_i2za_sim_logistic}")
        print(f"R-squared for id_z877: {r_squared_z877_sim_logistic}")
        
        # Store the R-squared values
        shared_variables.add("r_squared_i2za_sim_logistic", r_squared_i2za_sim_logistic)
        shared_variables.add("r_squared_z877_sim_logistic", r_squared_z877_sim_logistic)
except Exception as e:
    print(f"Error simulating the logistic system: {e}")

# Based on our analysis, let's formulate a hypothesis for the missing reactions
print("\nHypothesis for missing reactions:")
print("Based on our analysis, the most likely reaction is:")

try:
    stoichiometric_coefficient = shared_variables.access('stoichiometric_coefficient')
    # Round to the nearest integer if close
    if abs(stoichiometric_coefficient - round(stoichiometric_coefficient)) < 0.1:
        stoichiometric_coefficient = round(stoichiometric_coefficient)
    
    print(f"id_i2za + id_z877 -> {stoichiometric_coefficient} id_i2za")
    print(f"with rate constants k1 = {k1} and k3 = {k3}")
    print(f"This gives a stoichiometric coefficient of approximately {stoichiometric_coefficient} for id_i2za production.")
except Exception as e:
    print(f"Error accessing stoichiometric coefficient: {e}")
    print("The reaction involves id_i2za catalyzing the conversion of id_z877 to more id_i2za, but the exact stoichiometry is unclear.")

# Let's also check if there's any evidence of a carrying capacity
print("\nCarrying capacity analysis:")
try:
    if 'K2' in shared_variables.__dict__:
        K2 = shared_variables.access('K2')
        print(f"The fitted carrying capacity K is {K2}")
        print(f"The maximum observed value of id_i2za is {data['id_i2za'].max()}")
        print(f"The ratio of max(id_i2za)/K is {data['id_i2za'].max()/K2}")
except Exception as e:
    print(f"Error analyzing carrying capacity: {e}")
```
# Observation

## Code Stdout
Rate of change statistics for id_i2za:
count    4.214000e+03
mean     1.166854e+07
std      2.905237e+07
min     -1.217690e-01
25%      9.601768e+01
50%      1.183760e+04
75%      2.254944e+06
max      1.259780e+08
Name: d_i2za_dt, dtype: float64

Rate of change statistics for id_z877:
count    4.214000e+03
mean    -1.190301e+01
std      2.989859e+01
min     -1.427591e+02
25%     -9.768267e+00
50%     -1.165802e-04
75%     -8.589556e-09
max      2.121892e-08
Name: d_z877_dt, dtype: float64

Correlation between d(id_i2za)/dt and d(id_z877)/dt: 0.1568136073818153

Model 1 (Simple catalytic) for id_i2za:
Fitted parameter k: 0.10617406927668245 ± 0.01429596780991106
R-squared: -0.14634316933623426

Model 2 (Logistic growth) for id_i2za:
Fitted parameter r: 0.1061739251552879 ± 0.01471109725648506
Fitted parameter K: 383663682021165.9 ± 5.263557748850911e+19
R-squared: -0.1463432028196474

Model 3 (Autocatalytic) for id_z877:
Fitted parameter k: 9.216045613543223e-07 ± 4.14149285661861e-09
R-squared: 0.9091629481119441

Simulation results:
R-squared for id_i2za: -1.0109717288031197
R-squared for id_z877: -0.29415081069586146

Hypothesis for missing reactions:
Based on our analysis, the most likely reaction is:
Error accessing stoichiometric coefficient: 'stoichiometric_coefficient'
The reaction involves id_i2za catalyzing the conversion of id_z877 to more id_i2za, but the exact stoichiometry is unclear.

Carrying capacity analysis:

## Reminder

 ### Remaining Iterations for Interactions:
- Iterations: 16/20



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0)
- `shared_variables`: all variables you have saved from the previous iterations. (k, k_z877, r_squared, r_squared_z877, k1, r_squared1, r2, K2, r_squared2, k3, r_squared3, r_squared_i2za_sim, r_squared_z877_sim)

### Allowed libraires
['numpy', 'pandas', 'libsbml', 'math', 'scipy', 'jax', 'sklearn', 'io', 'traceback']


Please construct your response according to the thoughts-action markdown format.



# Iteration 5
## Thoughts
Based on the analysis so far, I've gained several important insights:

1. The model `d(id_z877)/dt = -k * id_i2za * id_z877` fits the data very well for `id_z877` consumption (R-squared = 0.91), with k3 = 9.22e-07.

2. The simple catalytic model `d(id_i2za)/dt = k * id_i2za * id_z877` doesn't fit well for `id_i2za` production (R-squared = -0.15), with k1 = 0.106.

3. The logistic growth model doesn't improve the fit for `id_i2za` production (R-squared = -0.15), and the fitted carrying capacity K is extremely large, suggesting that the system doesn't reach its carrying capacity in the observed data.

4. The simulation results using the simple catalytic model don't match the observed data well (R-squared for `id_i2za` = -1.01, R-squared for `id_z877` = -0.29).

5. The ratio of k1 to k3 (stoichiometric coefficient) is approximately 115,205, which is very large and suggests that the reaction is not a simple 1:1 conversion.

Let me try a different approach. The poor fit for `id_i2za` production suggests that the model is missing something important. Let's try a different model for `id_i2za` production, possibly with a different functional form.

I'll also try to perturb the system by changing the initial concentrations to see how the system responds. This might give us more insight into the underlying mechanisms.

## Action

### Experiment
```json
{
    "action": "change_initial_concentration", 
    "meta_data": {
        "id_i2za": 50, 
        "id_z877": 2000
    }
}
```
# Observation

## Experiment Result
Experiment ID: iteration_5
Your requested experiment has finished. The details results are saved as a dataframe in experiment_history[iteration_5]

Total size: 4214 rows × 3 columns. 

Below shows a subset of the table for your convenience. You should access experiment_history[iteration_5] for the full data

Time   id_i2za  id_z877
0  5.00e+01 2.00e+03
...
1.71e+00  8.60e+01 1.95e+03
...
3.43e+00  1.49e+02 1.90e+03
...
5.14e+00  2.61e+02 1.85e+03
...
6.86e+00  4.61e+02 1.81e+03
...
8.57e+00  8.20e+02 1.76e+03
...
1.03e+01  1.47e+03 1.71e+03
...
1.20e+01  2.66e+03 1.67e+03
...
1.37e+01  4.84e+03 1.62e+03
...
1.54e+01  8.89e+03 1.56e+03
...
1.71e+01  1.65e+04 1.49e+03
...
1.89e+01  3.11e+04 1.41e+03
...
2.06e+01  5.96e+04 1.28e+03
...
2.23e+01  1.17e+05 1.08e+03
...
2.40e+01  2.41e+05 7.94e+02
...
2.57e+01  5.23e+05 4.24e+02
...
2.74e+01  1.21e+06 1.10e+02
...
2.91e+01  2.89e+06 7.92e+00
...
3.09e+01  6.95e+06 1.56e+00
...
3.26e+01  1.66e+07 6.21e-01
...
3.43e+01  3.91e+07 2.59e-01
...
3.60e+01  8.95e+07 1.12e-01
...
3.77e+01  1.91e+08 5.24e-02
...
3.94e+01  3.62e+08 2.77e-02
...
4.12e+01  5.74e+08 1.74e-02
...
4.29e+01  7.58e+08 1.32e-02
...
4.46e+01  8.74e+08 1.14e-02
...
4.63e+01  9.33e+08 1.07e-02
...
4.80e+01  9.60e+08 1.04e-02
...
4.97e+01  9.72e+08 1.03e-02
...
5.14e+01  9.77e+08 1.02e-02
...
5.32e+01  9.79e+08 1.02e-02
...
5.49e+01  9.80e+08 1.02e-02
...
5.66e+01  9.80e+08 1.02e-02
...
5.83e+01  9.80e+08 1.02e-02
...
6.00e+01  9.80e+08 1.02e-02
...
6.17e+01  9.80e+08 1.02e-02
...
6.34e+01  9.80e+08 1.02e-02
...
6.52e+01  9.80e+08 1.02e-02
...
6.69e+01  9.80e+08 1.02e-02
...
6.86e+01  9.80e+08 1.02e-02
...
7.03e+01  9.80e+08 1.02e-02
...
7.20e+01  9.80e+08 1.02e-02
...
7.37e+01  9.80e+08 1.02e-02
...
7.54e+01  9.80e+08 1.02e-02
...
7.72e+01  9.80e+08 1.02e-02
...
7.89e+01  9.80e+08 1.02e-02
...
8.06e+01  9.80e+08 1.02e-02
...
8.23e+01  9.80e+08 1.02e-02
...
8.40e+01  9.80e+08 1.02e-02

## Reminder

 ### Remaining Iterations for Interactions:
- Iterations: 15/20



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_5)
- `shared_variables`: all variables you have saved from the previous iterations. (k, k_z877, r_squared, r_squared_z877, k1, r_squared1, r2, K2, r_squared2, k3, r_squared3, r_squared_i2za_sim, r_squared_z877_sim)

### Allowed libraires
['numpy', 'pandas', 'libsbml', 'math', 'scipy', 'jax', 'sklearn', 'io', 'traceback']


Please construct your response according to the thoughts-action markdown format.



# Iteration 6
## Thoughts
I've now observed the system with different initial conditions:
1. Original experiment: `id_i2za` = 100, `id_z877` = 1000
2. New experiment: `id_i2za` = 50, `id_z877` = 2000

In both cases, I observe similar behavior:
- `id_i2za` grows exponentially at first, then plateaus around 9.8e+08
- `id_z877` decreases over time, eventually stabilizing at a very low value (around 1.02e-02)

This consistency across different initial conditions supports the hypothesis that there's a fundamental reaction mechanism at work. Let me analyze the new data and compare it with the original data to gain more insights.

I'll focus on:
1. Comparing the rates of change in both experiments
2. Trying a different model for `id_i2za` production
3. Calculating the stoichiometric coefficient more accurately

## Action

### Code
```python
import numpy as np
import pandas as pd
from scipy.optimize import curve_fit
from scipy.integrate import solve_ivp

# Get the data from both experiments
data1 = experiment_history["iteration_0"]  # Original experiment
data2 = experiment_history["iteration_5"]  # New experiment

# Calculate the rate of change for both datasets
data1['d_i2za_dt'] = np.gradient(data1['id_i2za'], data1['Time'])
data1['d_z877_dt'] = np.gradient(data1['id_z877'], data1['Time'])

data2['d_i2za_dt'] = np.gradient(data2['id_i2za'], data2['Time'])
data2['d_z877_dt'] = np.gradient(data2['id_z877'], data2['Time'])

# Print basic statistics for both datasets
print("Original experiment (iteration_0):")
print(f"Initial id_i2za: {data1['id_i2za'].iloc[0]}")
print(f"Initial id_z877: {data1['id_z877'].iloc[0]}")
print(f"Final id_i2za: {data1['id_i2za'].iloc[-1]}")
print(f"Final id_z877: {data1['id_z877'].iloc[-1]}")

print("\nNew experiment (iteration_5):")
print(f"Initial id_i2za: {data2['id_i2za'].iloc[0]}")
print(f"Initial id_z877: {data2['id_z877'].iloc[0]}")
print(f"Final id_i2za: {data2['id_i2za'].iloc[-1]}")
print(f"Final id_z877: {data2['id_z877'].iloc[-1]}")

# Let's try a different model for id_i2za production
# Model 4: id_i2za production with a stoichiometric coefficient n
# d(id_i2za)/dt = n * k * id_i2za * id_z877
# d(id_z877)/dt = -k * id_i2za * id_z877
# where n is the stoichiometric coefficient

# First, let's fit the model for id_z877 consumption using both datasets
def model_z877(X, k):
    i2za, z877 = X
    return -k * i2za * z877

# Prepare the data for curve fitting
X1 = np.vstack([data1['id_i2za'], data1['id_z877']])
y1_z877 = data1['d_z877_dt']

X2 = np.vstack([data2['id_i2za'], data2['id_z877']])
y2_z877 = data2['d_z877_dt']

# Combine the data from both experiments
X_combined = np.hstack([X1, X2])
y_combined_z877 = np.concatenate([y1_z877, y2_z877])

# Fit the model for id_z877 consumption using the combined data
try:
    params_z877, covariance_z877 = curve_fit(model_z877, X_combined, y_combined_z877)
    k_z877_combined = params_z877[0]
    k_z877_combined_error = np.sqrt(covariance_z877[0, 0])
    
    print(f"\nModel for id_z877 consumption (combined data):")
    print(f"Fitted parameter k_z877: {k_z877_combined} ± {k_z877_combined_error}")
    
    # Calculate the predicted rate of change
    data1['predicted_d_z877_dt'] = model_z877(X1, k_z877_combined)
    data2['predicted_d_z877_dt'] = model_z877(X2, k_z877_combined)
    
    # Calculate the R-squared value for each dataset
    ss_total1_z877 = np.sum((y1_z877 - np.mean(y1_z877))**2)
    ss_residual1_z877 = np.sum((y1_z877 - data1['predicted_d_z877_dt'])**2)
    r_squared1_z877 = 1 - (ss_residual1_z877 / ss_total1_z877)
    
    ss_total2_z877 = np.sum((y2_z877 - np.mean(y2_z877))**2)
    ss_residual2_z877 = np.sum((y2_z877 - data2['predicted_d_z877_dt'])**2)
    r_squared2_z877 = 1 - (ss_residual2_z877 / ss_total2_z877)
    
    print(f"R-squared for original data: {r_squared1_z877}")
    print(f"R-squared for new data: {r_squared2_z877}")
    
    # Store the fitted parameters
    shared_variables.add("k_z877_combined", k_z877_combined)
    shared_variables.add("r_squared1_z877", r_squared1_z877)
    shared_variables.add("r_squared2_z877", r_squared2_z877)
except Exception as e:
    print(f"Error fitting model for id_z877 consumption: {e}")

# Now, let's try to determine the stoichiometric coefficient
# We'll use the fact that d(id_i2za)/dt = -n * d(id_z877)/dt
# where n is the stoichiometric coefficient

# Calculate the ratio of rates of change
data1['ratio'] = -data1['d_i2za_dt'] / data1['d_z877_dt']
data2['ratio'] = -data2['d_i2za_dt'] / data2['d_z877_dt']

# Filter out invalid values (e.g., division by zero or very small numbers)
data1_filtered = data1[np.abs(data1['d_z877_dt']) > 1e-6]
data2_filtered = data2[np.abs(data2['d_z877_dt']) > 1e-6]

# Calculate the mean ratio (stoichiometric coefficient)
n1 = data1_filtered['ratio'].mean()
n2 = data2_filtered['ratio'].mean()

print(f"\nStoichiometric coefficient (original data): {n1}")
print(f"Stoichiometric coefficient (new data): {n2}")

# Calculate the median ratio (stoichiometric coefficient)
n1_median = data1_filtered['ratio'].median()
n2_median = data2_filtered['ratio'].median()

print(f"Median stoichiometric coefficient (original data): {n1_median}")
print(f"Median stoichiometric coefficient (new data): {n2_median}")

# Let's also look at the distribution of the ratio
print(f"\nRatio statistics (original data):")
print(data1_filtered['ratio'].describe())

print(f"\nRatio statistics (new data):")
print(data2_filtered['ratio'].describe())

# Let's try to fit a model where the stoichiometric coefficient is a parameter
def model_with_stoichiometry(X, k, n):
    i2za, z877 = X
    return n * k * i2za * z877

# Fit the model for id_i2za production using the combined data
try:
    # Use the previously fitted k_z877_combined as a constraint
    def constrained_model(X, n):
        return model_with_stoichiometry(X, k_z877_combined, n)
    
    y1_i2za = data1['d_i2za_dt']
    y2_i2za = data2['d_i2za_dt']
    y_combined_i2za = np.concatenate([y1_i2za, y2_i2za])
    
    params_i2za, covariance_i2za = curve_fit(constrained_model, X_combined, y_combined_i2za)
    n_fitted = params_i2za[0]
    n_fitted_error = np.sqrt(covariance_i2za[0, 0])
    
    print(f"\nModel for id_i2za production with stoichiometry (combined data):")
    print(f"Fitted parameter n: {n_fitted} ± {n_fitted_error}")
    
    # Calculate the predicted rate of change
    data1['predicted_d_i2za_dt'] = constrained_model(X1, n_fitted)
    data2['predicted_d_i2za_dt'] = constrained_model(X2, n_fitted)
    
    # Calculate the R-squared value for each dataset
    ss_total1_i2za = np.sum((y1_i2za - np.mean(y1_i2za))**2)
    ss_residual1_i2za = np.sum((y1_i2za - data1['predicted_d_i2za_dt'])**2)
    r_squared1_i2za = 1 - (ss_residual1_i2za / ss_total1_i2za)
    
    ss_total2_i2za = np.sum((y2_i2za - np.mean(y2_i2za))**2)
    ss_residual2_i2za = np.sum((y2_i2za - data2['predicted_d_i2za_dt'])**2)
    r_squared2_i2za = 1 - (ss_residual2_i2za / ss_total2_i2za)
    
    print(f"R-squared for original data: {r_squared1_i2za}")
    print(f"R-squared for new data: {r_squared2_i2za}")
    
    # Store the fitted parameters
    shared_variables.add("n_fitted", n_fitted)
    shared_variables.add("r_squared1_i2za", r_squared1_i2za)
    shared_variables.add("r_squared2_i2za", r_squared2_i2za)
except Exception as e:
    print(f"Error fitting model for id_i2za production with stoichiometry: {e}")

# Let's try a different approach: fit a model where id_i2za is produced from id_z877 with a stoichiometric coefficient
# d(id_i2za)/dt = -n * d(id_z877)/dt
# where n is the stoichiometric coefficient

# We'll use linear regression to find the best value of n
from sklearn.linear_model import LinearRegression

try:
    # Filter out data points where d_z877_dt is very close to zero
    data1_filtered = data1[np.abs(data1['d_z877_dt']) > 1e-6]
    data2_filtered = data2[np.abs(data2['d_z877_dt']) > 1e-6]
    
    # Combine the data from both experiments
    X_reg1 = -data1_filtered['d_z877_dt'].values.reshape(-1, 1)
    y_reg1 = data1_filtered['d_i2za_dt'].values
    
    X_reg2 = -data2_filtered['d_z877_dt'].values.reshape(-1, 1)
    y_reg2 = data2_filtered['d_i2za_dt'].values
    
    X_reg_combined = np.vstack([X_reg1, X_reg2])
    y_reg_combined = np.concatenate([y_reg1, y_reg2])
    
    # Fit the model without an intercept
    reg = LinearRegression(fit_intercept=False)
    reg.fit(X_reg_combined, y_reg_combined)
    
    n_reg = reg.coef_[0]
    
    print(f"\nLinear regression model:")
    print(f"Fitted stoichiometric coefficient n: {n_reg}")
    
    # Calculate the predicted rate of change
    data1_filtered['predicted_d_i2za_dt_reg'] = -n_reg * data1_filtered['d_z877_dt']
    data2_filtered['predicted_d_i2za_dt_reg'] = -n_reg * data2_filtered['d_z877_dt']
    
    # Calculate the R-squared value for each dataset
    ss_total1_reg = np.sum((data1_filtered['d_i2za_dt'] - np.mean(data1_filtered['d_i2za_dt']))**2)
    ss_residual1_reg = np.sum((data1_filtered['d_i2za_dt'] - data1_filtered['predicted_d_i2za_dt_reg'])**2)
    r_squared1_reg = 1 - (ss_residual1_reg / ss_total1_reg)
    
    ss_total2_reg = np.sum((data2_filtered['d_i2za_dt'] - np.mean(data2_filtered['d_i2za_dt']))**2)
    ss_residual2_reg = np.sum((data2_filtered['d_i2za_dt'] - data2_filtered['predicted_d_i2za_dt_reg'])**2)
    r_squared2_reg = 1 - (ss_residual2_reg / ss_total2_reg)
    
    print(f"R-squared for original data: {r_squared1_reg}")
    print(f"R-squared for new data: {r_squared2_reg}")
    
    # Store the fitted parameters
    shared_variables.add("n_reg", n_reg)
    shared_variables.add("r_squared1_reg", r_squared1_reg)
    shared_variables.add("r_squared2_reg", r_squared2_reg)
except Exception as e:
    print(f"Error fitting linear regression model: {e}")

# Let's simulate the system with the best model
try:
    # Define the system of differential equations
    def system(t, y, k, n):
        i2za, z877 = y
        d_z877_dt = -k * i2za * z877
        d_i2za_dt = -n * d_z877_dt
        return [d_i2za_dt, d_z877_dt]
    
    # Initial conditions for both experiments
    y0_1 = [data1['id_i2za'].iloc[0], data1['id_z877'].iloc[0]]
    y0_2 = [data2['id_i2za'].iloc[0], data2['id_z877'].iloc[0]]
    
    # Time points
    t_span = [0, 84]  # Assuming both experiments have the same time range
    t_eval1 = data1['Time'].values
    t_eval2 = data2['Time'].values
    
    # Solve the system for both experiments
    k = shared_variables.access('k_z877_combined')
    n = shared_variables.access('n_reg')  # Using the value from linear regression
    
    sol1 = solve_ivp(system, t_span, y0_1, args=(k, n), t_eval=t_eval1)
    sol2 = solve_ivp(system, t_span, y0_2, args=(k, n), t_eval=t_eval2)
    
    # Calculate the R-squared value for the simulation
    ss_total1_i2za_sim = np.sum((data1['id_i2za'] - np.mean(data1['id_i2za']))**2)
    ss_residual1_i2za_sim = np.sum((data1['id_i2za'] - sol1.y[0])**2)
    r_squared1_i2za_sim = 1 - (ss_residual1_i2za_sim / ss_total1_i2za_sim)
    
    ss_total1_z877_sim = np.sum((data1['id_z877'] - np.mean(data1['id_z877']))**2)
    ss_residual1_z877_sim = np.sum((data1['id_z877'] - sol1.y[1])**2)
    r_squared1_z877_sim = 1 - (ss_residual1_z877_sim / ss_total1_z877_sim)
    
    ss_total2_i2za_sim = np.sum((data2['id_i2za'] - np.mean(data2['id_i2za']))**2)
    ss_residual2_i2za_sim = np.sum((data2['id_i2za'] - sol2.y[0])**2)
    r_squared2_i2za_sim = 1 - (ss_residual2_i2za_sim / ss_total2_i2za_sim)
    
    ss_total2_z877_sim = np.sum((data2['id_z877'] - np.mean(data2['id_z877']))**2)
    ss_residual2_z877_sim = np.sum((data2['id_z877'] - sol2.y[1])**2)
    r_squared2_z877_sim = 1 - (ss_residual2_z877_sim / ss_total2_z877_sim)
    
    print(f"\nSimulation results (original data):")
    print(f"R-squared for id_i2za: {r_squared1_i2za_sim}")
    print(f"R-squared for id_z877: {r_squared1_z877_sim}")
    
    print(f"\nSimulation results (new data):")
    print(f"R-squared for id_i2za: {r_squared2_i2za_sim}")
    print(f"R-squared for id_z877: {r_squared2_z877_sim}")
    
    # Store the R-squared values
    shared_variables.add("r_squared1_i2za_sim", r_squared1_i2za_sim)
    shared_variables.add("r_squared1_z877_sim", r_squared1_z877_sim)
    shared_variables.add("r_squared2_i2za_sim", r_squared2_i2za_sim)
    shared_variables.add("r_squared2_z877_sim", r_squared2_z877_sim)
except Exception as e:
    print(f"Error simulating the system: {e}")

# Based on our analysis, let's formulate a hypothesis for the missing reactions
print("\nHypothesis for missing reactions:")
print("Based on our analysis, the most likely reaction is:")

try:
    n_reg = shared_variables.access('n_reg')
    k_z877_combined = shared_variables.access('k_z877_combined')
    
    # Round to the nearest integer if close
    if abs(n_reg - round(n_reg)) < 0.1:
        n_reg_rounded = round(n_reg)
        print(f"id_i2za + id_z877 -> {n_reg_rounded} id_i2za")
    else:
        print(f"id_i2za + id_z877 -> {n_reg} id_i2za")
    
    print(f"with rate constant k = {k_z877_combined}")
    print(f"This gives a stoichiometric coefficient of approximately {n_reg} for id_i2za production.")
except Exception as e:
    print(f"Error accessing fitted parameters: {e}")
```
# Observation

## Code Stdout
Original experiment (iteration_0):
Initial id_i2za: 100.0
Initial id_z877: 1000.0
Final id_i2za: 980390211.3597739
Final id_z877: 0.010199810042200963

New experiment (iteration_5):
Initial id_i2za: 50.0
Initial id_z877: 2000.0
Final id_i2za: 980390158.0822339
Final id_z877: 0.010199812721332763

Model for id_z877 consumption (combined data):
Fitted parameter k_z877: 9.823545221167335e-07 ± 3.077320460340354e-09
R-squared for original data: 0.904523677098544
R-squared for new data: 0.9080625772723192

Stoichiometric coefficient (original data): 27569239121.80387
Stoichiometric coefficient (new data): 24772284790.8945
Median stoichiometric coefficient (original data): 8280156.770491095
Median stoichiometric coefficient (new data): 32620.861280331304

Ratio statistics (original data):
count    2.571000e+03
mean     2.756924e+10
std      3.996832e+10
min      4.117441e+00
25%      3.710974e+02
50%      8.280157e+06
75%      7.403697e+10
max      9.606162e+10
Name: ratio, dtype: float64

Ratio statistics (new data):
count    2.871000e+03
mean     2.477228e+10
std      3.875544e+10
min      5.234982e-01
25%      6.687510e+01
50%      3.262086e+04
75%      5.808863e+10
max      1.172161e+11
Name: ratio, dtype: float64

Model for id_i2za production with stoichiometry (combined data):
Fitted parameter n: 64019.17712548129 ± 7920.39829544233
R-squared for original data: -0.14883754420324458
R-squared for new data: -0.15599667012261897

Linear regression model:
Fitted stoichiometric coefficient n: 1675.0408370990772
R-squared for original data: -0.29471397141176237
R-squared for new data: -0.25603749865631587

Simulation results (original data):
R-squared for id_i2za: -1.5698552456364103
R-squared for id_z877: 0.2549156077063397

Simulation results (new data):
R-squared for id_i2za: -1.1694037541093039
R-squared for id_z877: -0.11270693880491423

Hypothesis for missing reactions:
Based on our analysis, the most likely reaction is:
id_i2za + id_z877 -> 1675 id_i2za
with rate constant k = 9.823545221167335e-07
This gives a stoichiometric coefficient of approximately 1675.0408370990772 for id_i2za production.

## Reminder

 ### Remaining Iterations for Interactions:
- Iterations: 14/20



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_5)
- `shared_variables`: all variables you have saved from the previous iterations. (k, k_z877, r_squared, r_squared_z877, k1, r_squared1, r2, K2, r_squared2, k3, r_squared3, r_squared_i2za_sim, r_squared_z877_sim, k_z877_combined, r_squared1_z877, r_squared2_z877, n_fitted, r_squared1_i2za, r_squared2_i2za, n_reg, r_squared1_reg, r_squared2_reg, r_squared1_i2za_sim, r_squared1_z877_sim, r_squared2_i2za_sim, r_squared2_z877_sim)

### Allowed libraires
['numpy', 'pandas', 'libsbml', 'math', 'scipy', 'jax', 'sklearn', 'io', 'traceback']


Please construct your response according to the thoughts-action markdown format.



# Iteration 7
## Thoughts
Based on the analysis of both experiments, I've gained several important insights:

1. The model for `id_z877` consumption (`d(id_z877)/dt = -k * id_i2za * id_z877`) fits the data very well for both experiments (R-squared > 0.9), with a consistent rate constant k ≈ 9.82e-07.

2. The stoichiometric coefficient (ratio of `id_i2za` production to `id_z877` consumption) varies widely across the data, with means around 2.76e+10 and 2.48e+10 for the two experiments, but medians of 8.28e+06 and 3.26e+04, indicating a highly skewed distribution.

3. When fitting a model with a fixed stoichiometric coefficient, the linear regression gives n ≈ 1675, but the R-squared values for `id_i2za` production are still negative, indicating a poor fit.

4. The simulation results using this model are also poor, especially for `id_i2za` (R-squared < -1).

The large discrepancy between the mean and median stoichiometric coefficients, and the poor fit of the models, suggest that the relationship between `id_i2za` production and `id_z877` consumption is more complex than a simple stoichiometric reaction.

Let me try a different approach. Instead of trying to fit a single stoichiometric coefficient, let's consider a model where the stoichiometric coefficient varies with the concentrations of the species. This could indicate a more complex reaction network or feedback mechanisms.

Let's also try one more experiment with different initial conditions to gather more data.

## Action

### Experiment
```json
{
    "action": "change_initial_concentration", 
    "meta_data": {
        "id_i2za": 200, 
        "id_z877": 500
    }
}
```
# Observation

## Experiment Result
Experiment ID: iteration_7
Your requested experiment has finished. The details results are saved as a dataframe in experiment_history[iteration_7]

Total size: 4214 rows × 3 columns. 

Below shows a subset of the table for your convenience. You should access experiment_history[iteration_7] for the full data

Time   id_i2za  id_z877
0  2.00e+02 5.00e+02
...
1.71e+00  4.43e+02 5.00e+02
...
3.43e+00  9.82e+02 4.99e+02
...
5.14e+00  2.18e+03 4.98e+02
...
6.86e+00  4.83e+03 4.95e+02
...
8.57e+00  1.07e+04 4.89e+02
...
1.03e+01  2.38e+04 4.76e+02
...
1.20e+01  5.30e+04 4.49e+02
...
1.37e+01  1.19e+05 3.92e+02
...
1.54e+01  2.71e+05 2.91e+02
...
1.71e+01  6.29e+05 1.47e+02
...
1.89e+01  1.50e+06 3.28e+01
...
2.06e+01  3.60e+06 3.73e+00
...
2.23e+01  8.64e+06 1.23e+00
...
2.40e+01  2.06e+07 4.98e-01
...
2.57e+01  4.83e+07 2.09e-01
...
2.74e+01  1.09e+08 9.22e-02
...
2.91e+01  2.27e+08 4.41e-02
...
3.09e+01  4.13e+08 2.42e-02
...
3.26e+01  6.25e+08 1.60e-02
...
3.43e+01  7.94e+08 1.26e-02
...
3.60e+01  8.93e+08 1.12e-02
...
3.77e+01  9.42e+08 1.06e-02
...
3.94e+01  9.64e+08 1.04e-02
...
4.12e+01  9.74e+08 1.03e-02
...
4.29e+01  9.78e+08 1.02e-02
...
4.46e+01  9.79e+08 1.02e-02
...
4.63e+01  9.80e+08 1.02e-02
...
4.80e+01  9.80e+08 1.02e-02
...
4.97e+01  9.80e+08 1.02e-02
...
5.14e+01  9.80e+08 1.02e-02
...
5.32e+01  9.80e+08 1.02e-02
...
5.49e+01  9.80e+08 1.02e-02
...
5.66e+01  9.80e+08 1.02e-02
...
5.83e+01  9.80e+08 1.02e-02
...
6.00e+01  9.80e+08 1.02e-02
...
6.17e+01  9.80e+08 1.02e-02
...
6.34e+01  9.80e+08 1.02e-02
...
6.52e+01  9.80e+08 1.02e-02
...
6.69e+01  9.80e+08 1.02e-02
...
6.86e+01  9.80e+08 1.02e-02
...
7.03e+01  9.80e+08 1.02e-02
...
7.20e+01  9.80e+08 1.02e-02
...
7.37e+01  9.80e+08 1.02e-02
...
7.54e+01  9.80e+08 1.02e-02
...
7.72e+01  9.80e+08 1.02e-02
...
7.89e+01  9.80e+08 1.02e-02
...
8.06e+01  9.80e+08 1.02e-02
...
8.23e+01  9.80e+08 1.02e-02
...
8.40e+01  9.80e+08 1.02e-02

## Reminder

 ### Remaining Iterations for Interactions:
- Iterations: 13/20



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_5, iteration_7)
- `shared_variables`: all variables you have saved from the previous iterations. (k, k_z877, r_squared, r_squared_z877, k1, r_squared1, r2, K2, r_squared2, k3, r_squared3, r_squared_i2za_sim, r_squared_z877_sim, k_z877_combined, r_squared1_z877, r_squared2_z877, n_fitted, r_squared1_i2za, r_squared2_i2za, n_reg, r_squared1_reg, r_squared2_reg, r_squared1_i2za_sim, r_squared1_z877_sim, r_squared2_i2za_sim, r_squared2_z877_sim)

### Allowed libraires
['numpy', 'pandas', 'libsbml', 'math', 'scipy', 'jax', 'sklearn', 'io', 'traceback']


Please construct your response according to the thoughts-action markdown format.



# Iteration 8
## Thoughts
I now have data from three experiments with different initial conditions:
1. Experiment 1: `id_i2za` = 100, `id_z877` = 1000
2. Experiment 2: `id_i2za` = 50, `id_z877` = 2000
3. Experiment 3: `id_i2za` = 200, `id_z877` = 500

In all three experiments, I observe similar behavior:
- `id_i2za` grows exponentially at first, then plateaus around 9.8e+08
- `id_z877` decreases over time, eventually stabilizing at a very low value (around 1.02e-02)

The consistency across different initial conditions further supports the hypothesis that there's a fundamental reaction mechanism at work. However, the poor fit of the models with a fixed stoichiometric coefficient suggests that the relationship is more complex.

Let me analyze the new data and compare it with the previous experiments to gain more insights. I'll focus on:

1. Examining the relationship between the stoichiometric coefficient and the concentrations of the species
2. Exploring models with variable stoichiometry
3. Considering the possibility of a carrying capacity or other limiting factors

## Action

### Code
```python
import numpy as np
import pandas as pd
from scipy.optimize import curve_fit
from scipy.integrate import solve_ivp

# Get the data from all three experiments
data1 = experiment_history["iteration_0"]  # Original experiment
data2 = experiment_history["iteration_5"]  # Second experiment
data3 = experiment_history["iteration_7"]  # Third experiment

# Calculate the rate of change for all datasets
data1['d_i2za_dt'] = np.gradient(data1['id_i2za'], data1['Time'])
data1['d_z877_dt'] = np.gradient(data1['id_z877'], data1['Time'])

data2['d_i2za_dt'] = np.gradient(data2['id_i2za'], data2['Time'])
data2['d_z877_dt'] = np.gradient(data2['id_z877'], data2['Time'])

data3['d_i2za_dt'] = np.gradient(data3['id_i2za'], data3['Time'])
data3['d_z877_dt'] = np.gradient(data3['id_z877'], data3['Time'])

# Print basic statistics for all datasets
print("Experiment 1 (iteration_0):")
print(f"Initial id_i2za: {data1['id_i2za'].iloc[0]}")
print(f"Initial id_z877: {data1['id_z877'].iloc[0]}")
print(f"Final id_i2za: {data1['id_i2za'].iloc[-1]}")
print(f"Final id_z877: {data1['id_z877'].iloc[-1]}")

print("\nExperiment 2 (iteration_5):")
print(f"Initial id_i2za: {data2['id_i2za'].iloc[0]}")
print(f"Initial id_z877: {data2['id_z877'].iloc[0]}")
print(f"Final id_i2za: {data2['id_i2za'].iloc[-1]}")
print(f"Final id_z877: {data2['id_z877'].iloc[-1]}")

print("\nExperiment 3 (iteration_7):")
print(f"Initial id_i2za: {data3['id_i2za'].iloc[0]}")
print(f"Initial id_z877: {data3['id_z877'].iloc[0]}")
print(f"Final id_i2za: {data3['id_i2za'].iloc[-1]}")
print(f"Final id_z877: {data3['id_z877'].iloc[-1]}")

# Let's fit the model for id_z877 consumption using all three datasets
def model_z877(X, k):
    i2za, z877 = X
    return -k * i2za * z877

# Prepare the data for curve fitting
X1 = np.vstack([data1['id_i2za'], data1['id_z877']])
y1_z877 = data1['d_z877_dt']

X2 = np.vstack([data2['id_i2za'], data2['id_z877']])
y2_z877 = data2['d_z877_dt']

X3 = np.vstack([data3['id_i2za'], data3['id_z877']])
y3_z877 = data3['d_z877_dt']

# Combine the data from all experiments
X_combined = np.hstack([X1, X2, X3])
y_combined_z877 = np.concatenate([y1_z877, y2_z877, y3_z877])

# Fit the model for id_z877 consumption using the combined data
try:
    params_z877, covariance_z877 = curve_fit(model_z877, X_combined, y_combined_z877)
    k_z877_combined = params_z877[0]
    k_z877_combined_error = np.sqrt(covariance_z877[0, 0])
    
    print(f"\nModel for id_z877 consumption (combined data):")
    print(f"Fitted parameter k_z877: {k_z877_combined} ± {k_z877_combined_error}")
    
    # Calculate the predicted rate of change
    data1['predicted_d_z877_dt'] = model_z877(X1, k_z877_combined)
    data2['predicted_d_z877_dt'] = model_z877(X2, k_z877_combined)
    data3['predicted_d_z877_dt'] = model_z877(X3, k_z877_combined)
    
    # Calculate the R-squared value for each dataset
    ss_total1_z877 = np.sum((y1_z877 - np.mean(y1_z877))**2)
    ss_residual1_z877 = np.sum((y1_z877 - data1['predicted_d_z877_dt'])**2)
    r_squared1_z877 = 1 - (ss_residual1_z877 / ss_total1_z877)
    
    ss_total2_z877 = np.sum((y2_z877 - np.mean(y2_z877))**2)
    ss_residual2_z877 = np.sum((y2_z877 - data2['predicted_d_z877_dt'])**2)
    r_squared2_z877 = 1 - (ss_residual2_z877 / ss_total2_z877)
    
    ss_total3_z877 = np.sum((y3_z877 - np.mean(y3_z877))**2)
    ss_residual3_z877 = np.sum((y3_z877 - data3['predicted_d_z877_dt'])**2)
    r_squared3_z877 = 1 - (ss_residual3_z877 / ss_total3_z877)
    
    print(f"R-squared for experiment 1: {r_squared1_z877}")
    print(f"R-squared for experiment 2: {r_squared2_z877}")
    print(f"R-squared for experiment 3: {r_squared3_z877}")
    
    # Store the fitted parameters
    shared_variables.add("k_z877_all", k_z877_combined)
    shared_variables.add("r_squared1_z877_all", r_squared1_z877)
    shared_variables.add("r_squared2_z877_all", r_squared2_z877)
    shared_variables.add("r_squared3_z877_all", r_squared3_z877)
except Exception as e:
    print(f"Error fitting model for id_z877 consumption: {e}")

# Calculate the ratio of rates of change (stoichiometric coefficient)
data1['ratio'] = -data1['d_i2za_dt'] / data1['d_z877_dt']
data2['ratio'] = -data2['d_i2za_dt'] / data2['d_z877_dt']
data3['ratio'] = -data3['d_i2za_dt'] / data3['d_z877_dt']

# Filter out invalid values (e.g., division by zero or very small numbers)
data1_filtered = data1[np.abs(data1['d_z877_dt']) > 1e-6]
data2_filtered = data2[np.abs(data2['d_z877_dt']) > 1e-6]
data3_filtered = data3[np.abs(data3['d_z877_dt']) > 1e-6]

# Calculate the mean and median ratio (stoichiometric coefficient) for each experiment
n1_mean = data1_filtered['ratio'].mean()
n2_mean = data2_filtered['ratio'].mean()
n3_mean = data3_filtered['ratio'].mean()

n1_median = data1_filtered['ratio'].median()
n2_median = data2_filtered['ratio'].median()
n3_median = data3_filtered['ratio'].median()

print(f"\nStoichiometric coefficient (mean):")
print(f"Experiment 1: {n1_mean}")
print(f"Experiment 2: {n2_mean}")
print(f"Experiment 3: {n3_mean}")

print(f"\nStoichiometric coefficient (median):")
print(f"Experiment 1: {n1_median}")
print(f"Experiment 2: {n2_median}")
print(f"Experiment 3: {n3_median}")

# Let's examine the relationship between the stoichiometric coefficient and the concentrations
# Combine the filtered data from all experiments
data_combined = pd.concat([data1_filtered, data2_filtered, data3_filtered], ignore_index=True)

# Calculate the correlation between the ratio and the concentrations
corr_i2za = np.corrcoef(data_combined['id_i2za'], data_combined['ratio'])[0, 1]
corr_z877 = np.corrcoef(data_combined['id_z877'], data_combined['ratio'])[0, 1]

print(f"\nCorrelation between stoichiometric coefficient and id_i2za: {corr_i2za}")
print(f"Correlation between stoichiometric coefficient and id_z877: {corr_z877}")

# Let's try a model where the stoichiometric coefficient depends on the concentrations
# n = a * id_i2za + b * id_z877 + c
from sklearn.linear_model import LinearRegression

try:
    # Prepare the data for regression
    X_reg = data_combined[['id_i2za', 'id_z877']].values
    y_reg = data_combined['ratio'].values
    
    # Fit the model
    reg = LinearRegression()
    reg.fit(X_reg, y_reg)
    
    a, b = reg.coef_
    c = reg.intercept_
    
    print(f"\nLinear model for stoichiometric coefficient:")
    print(f"n = {a} * id_i2za + {b} * id_z877 + {c}")
    
    # Calculate the predicted stoichiometric coefficient
    data_combined['predicted_ratio'] = a * data_combined['id_i2za'] + b * data_combined['id_z877'] + c
    
    # Calculate the R-squared value
    ss_total_ratio = np.sum((data_combined['ratio'] - np.mean(data_combined['ratio']))**2)
    ss_residual_ratio = np.sum((data_combined['ratio'] - data_combined['predicted_ratio'])**2)
    r_squared_ratio = 1 - (ss_residual_ratio / ss_total_ratio)
    
    print(f"R-squared for stoichiometric coefficient model: {r_squared_ratio}")
    
    # Store the fitted parameters
    shared_variables.add("a", a)
    shared_variables.add("b", b)
    shared_variables.add("c", c)
    shared_variables.add("r_squared_ratio", r_squared_ratio)
except Exception as e:
    print(f"Error fitting linear model for stoichiometric coefficient: {e}")

# Let's try a different approach: instead of trying to model the stoichiometric coefficient directly,
# let's try to model the production of id_i2za with a more complex function

# Model 1: Simple mass action kinetics with a carrying capacity
# d(id_i2za)/dt = k1 * id_i2za * id_z877 * (1 - id_i2za/K)
def model1(X, k1, K):
    i2za, z877 = X
    return k1 * i2za * z877 * (1 - i2za/K)

# Prepare the data for curve fitting
y1_i2za = data1['d_i2za_dt']
y2_i2za = data2['d_i2za_dt']
y3_i2za = data3['d_i2za_dt']
y_combined_i2za = np.concatenate([y1_i2za, y2_i2za, y3_i2za])

# Fit the model for id_i2za production using the combined data
try:
    p0 = [0.1, 1e9]  # Initial guess for k1, K
    params1, covariance1 = curve_fit(model1, X_combined, y_combined_i2za, p0=p0)
    k1, K = params1
    k1_error, K_error = np.sqrt(np.diag(covariance1))
    
    print(f"\nModel 1 for id_i2za production (combined data):")
    print(f"Fitted parameter k1: {k1} ± {k1_error}")
    print(f"Fitted parameter K: {K} ± {K_error}")
    
    # Calculate the predicted rate of change
    data1['predicted_d_i2za_dt_model1'] = model1(X1, k1, K)
    data2['predicted_d_i2za_dt_model1'] = model1(X2, k1, K)
    data3['predicted_d_i2za_dt_model1'] = model1(X3, k1, K)
    
    # Calculate the R-squared value for each dataset
    ss_total1_i2za = np.sum((y1_i2za - np.mean(y1_i2za))**2)
    ss_residual1_i2za = np.sum((y1_i2za - data1['predicted_d_i2za_dt_model1'])**2)
    r_squared1_i2za_model1 = 1 - (ss_residual1_i2za / ss_total1_i2za)
    
    ss_total2_i2za = np.sum((y2_i2za - np.mean(y2_i2za))**2)
    ss_residual2_i2za = np.sum((y2_i2za - data2['predicted_d_i2za_dt_model1'])**2)
    r_squared2_i2za_model1 = 1 - (ss_residual2_i2za / ss_total2_i2za)
    
    ss_total3_i2za = np.sum((y3_i2za - np.mean(y3_i2za))**2)
    ss_residual3_i2za = np.sum((y3_i2za - data3['predicted_d_i2za_dt_model1'])**2)
    r_squared3_i2za_model1 = 1 - (ss_residual3_i2za / ss_total3_i2za)
    
    print(f"R-squared for experiment 1: {r_squared1_i2za_model1}")
    print(f"R-squared for experiment 2: {r_squared2_i2za_model1}")
    print(f"R-squared for experiment 3: {r_squared3_i2za_model1}")
    
    # Store the fitted parameters
    shared_variables.add("k1_model1", k1)
    shared_variables.add("K_model1", K)
    shared_variables.add("r_squared1_i2za_model1", r_squared1_i2za_model1)
    shared_variables.add("r_squared2_i2za_model1", r_squared2_i2za_model1)
    shared_variables.add("r_squared3_i2za_model1", r_squared3_i2za_model1)
except Exception as e:
    print(f"Error fitting Model 1 for id_i2za production: {e}")

# Let's simulate the system with Model 1
try:
    # Define the system of differential equations
    def system_model1(t, y, k1, K, k_z877):
        i2za, z877 = y
        d_i2za_dt = k1 * i2za * z877 * (1 - i2za/K)
        d_z877_dt = -k_z877 * i2za * z877
        return [d_i2za_dt, d_z877_dt]
    
    # Initial conditions for all experiments
    y0_1 = [data1['id_i2za'].iloc[0], data1['id_z877'].iloc[0]]
    y0_2 = [data2['id_i2za'].iloc[0], data2['id_z877'].iloc[0]]
    y0_3 = [data3['id_i2za'].iloc[0], data3['id_z877'].iloc[0]]
    
    # Time points
    t_span = [0, 84]  # Assuming all experiments have the same time range
    t_eval1 = data1['Time'].values
    t_eval2 = data2['Time'].values
    t_eval3 = data3['Time'].values
    
    # Solve the system for all experiments
    k1 = shared_variables.access('k1_model1')
    K = shared_variables.access('K_model1')
    k_z877 = shared_variables.access('k_z877_all')
    
    sol1 = solve_ivp(system_model1, t_span, y0_1, args=(k1, K, k_z877), t_eval=t_eval1)
    sol2 = solve_ivp(system_model1, t_span, y0_2, args=(k1, K, k_z877), t_eval=t_eval2)
    sol3 = solve_ivp(system_model1, t_span, y0_3, args=(k1, K, k_z877), t_eval=t_eval3)
    
    # Calculate the R-squared value for the simulation
    ss_total1_i2za_sim = np.sum((data1['id_i2za'] - np.mean(data1['id_i2za']))**2)
    ss_residual1_i2za_sim = np.sum((data1['id_i2za'] - sol1.y[0])**2)
    r_squared1_i2za_sim_model1 = 1 - (ss_residual1_i2za_sim / ss_total1_i2za_sim)
    
    ss_total1_z877_sim = np.sum((data1['id_z877'] - np.mean(data1['id_z877']))**2)
    ss_residual1_z877_sim = np.sum((data1['id_z877'] - sol1.y[1])**2)
    r_squared1_z877_sim_model1 = 1 - (ss_residual1_z877_sim / ss_total1_z877_sim)
    
    ss_total2_i2za_sim = np.sum((data2['id_i2za'] - np.mean(data2['id_i2za']))**2)
    ss_residual2_i2za_sim = np.sum((data2['id_i2za'] - sol2.y[0])**2)
    r_squared2_i2za_sim_model1 = 1 - (ss_residual2_i2za_sim / ss_total2_i2za_sim)
    
    ss_total2_z877_sim = np.sum((data2['id_z877'] - np.mean(data2['id_z877']))**2)
    ss_residual2_z877_sim = np.sum((data2['id_z877'] - sol2.y[1])**2)
    r_squared2_z877_sim_model1 = 1 - (ss_residual2_z877_sim / ss_total2_z877_sim)
    
    ss_total3_i2za_sim = np.sum((data3['id_i2za'] - np.mean(data3['id_i2za']))**2)
    ss_residual3_i2za_sim = np.sum((data3['id_i2za'] - sol3.y[0])**2)
    r_squared3_i2za_sim_model1 = 1 - (ss_residual3_i2za_sim / ss_total3_i2za_sim)
    
    ss_total3_z877_sim = np.sum((data3['id_z877'] - np.mean(data3['id_z877']))**2)
    ss_residual3_z877_sim = np.sum((data3['id_z877'] - sol3.y[1])**2)
    r_squared3_z877_sim_model1 = 1 - (ss_residual3_z877_sim / ss_total3_z877_sim)
    
    print(f"\nSimulation results for Model 1:")
    print(f"Experiment 1 - R-squared for id_i2za: {r_squared1_i2za_sim_model1}")
    print(f"Experiment 1 - R-squared for id_z877: {r_squared1_z877_sim_model1}")
    print(f"Experiment 2 - R-squared for id_i2za: {r_squared2_i2za_sim_model1}")
    print(f"Experiment 2 - R-squared for id_z877: {r_squared2_z877_sim_model1}")
    print(f"Experiment 3 - R-squared for id_i2za: {r_squared3_i2za_sim_model1}")
    print(f"Experiment 3 - R-squared for id_z877: {r_squared3_z877_sim_model1}")
    
    # Store the R-squared values
    shared_variables.add("r_squared1_i2za_sim_model1", r_squared1_i2za_sim_model1)
    shared_variables.add("r_squared1_z877_sim_model1", r_squared1_z877_sim_model1)
    shared_variables.add("r_squared2_i2za_sim_model1", r_squared2_i2za_sim_model1)
    shared_variables.add("r_squared2_z877_sim_model1", r_squared2_z877_sim_model1)
    shared_variables.add("r_squared3_i2za_sim_model1", r_squared3_i2za_sim_model1)
    shared_variables.add("r_squared3_z877_sim_model1", r_squared3_z877_sim_model1)
except Exception as e:
    print(f"Error simulating the system with Model 1: {e}")

# Based on our analysis, let's formulate a hypothesis for the missing reactions
print("\nHypothesis for missing reactions:")
print("Based on our analysis, the most likely reaction mechanism is:")

try:
    k1 = shared_variables.access('k1_model1')
    K = shared_variables.access('K_model1')
    k_z877 = shared_variables.access('k_z877_all')
    
    print(f"1. id_i2za + id_z877 -> 2 id_i2za (with rate constant k1 = {k1} and carrying capacity K = {K})")
    print(f"2. id_i2za + id_z877 -> id_i2za (with rate constant k_z877 = {k_z877})")
    print(f"This corresponds to the differential equations:")
    print(f"d(id_i2za)/dt = k1 * id_i2za * id_z877 * (1 - id_i2za/K)")
    print(f"d(id_z877)/dt = -k_z877 * id_i2za * id_z877")
except Exception as e:
    print(f"Error accessing fitted parameters: {e}")
```
# Observation

## Code Stdout
Experiment 1 (iteration_0):
Initial id_i2za: 100.0
Initial id_z877: 1000.0
Final id_i2za: 980390211.3597739
Final id_z877: 0.010199810042200963

Experiment 2 (iteration_5):
Initial id_i2za: 50.0
Initial id_z877: 2000.0
Final id_i2za: 980390158.0822339
Final id_z877: 0.010199812721332763

Experiment 3 (iteration_7):
Initial id_i2za: 200.0
Initial id_z877: 500.0
Final id_i2za: 980390220.656989
Final id_z877: 0.01019981200273036

Model for id_z877 consumption (combined data):
Fitted parameter k_z877: 9.589006198496309e-07 ± 2.6364123723770896e-09
R-squared for experiment 1: 0.9074143757281352
R-squared for experiment 2: 0.9060968346498628
R-squared for experiment 3: 0.7720883416996436

Stoichiometric coefficient (mean):
Experiment 1: 27569239121.80387
Experiment 2: 24772284790.8945
Experiment 3: 29356374986.067707

Stoichiometric coefficient (median):
Experiment 1: 8280156.770491095
Experiment 2: 32620.861280331304
Experiment 3: 42889518.73348239

Correlation between stoichiometric coefficient and id_i2za: 0.9862157668466786
Correlation between stoichiometric coefficient and id_z877: -0.4717477128077676

Linear model for stoichiometric coefficient:
n = 97.11961560500968 * id_i2za + 3862689.58755682 * id_z877 + -4934454743.645557
R-squared for stoichiometric coefficient model: 0.9750731249743569

Model 1 for id_i2za production (combined data):
Fitted parameter k1: 0.08328068595384215 ± 0.0074777607818801465
Fitted parameter K: 389227318196195.06 ± 3.985230618270452e+19
R-squared for experiment 1: -0.14704097196399046
R-squared for experiment 2: -0.159671831835966
R-squared for experiment 3: -0.14199108329361088

Simulation results for Model 1:
Experiment 1 - R-squared for id_i2za: -1.1390946493060197
Experiment 1 - R-squared for id_z877: -0.2909339167842704
Experiment 2 - R-squared for id_i2za: -0.5129160040860101
Experiment 2 - R-squared for id_z877: -0.41189247155632214
Experiment 3 - R-squared for id_i2za: -1.598009966972611
Experiment 3 - R-squared for id_z877: -0.2274666980980704

Hypothesis for missing reactions:
Based on our analysis, the most likely reaction mechanism is:
1. id_i2za + id_z877 -> 2 id_i2za (with rate constant k1 = 0.08328068595384215 and carrying capacity K = 389227318196195.06)
2. id_i2za + id_z877 -> id_i2za (with rate constant k_z877 = 9.589006198496309e-07)
This corresponds to the differential equations:
d(id_i2za)/dt = k1 * id_i2za * id_z877 * (1 - id_i2za/K)
d(id_z877)/dt = -k_z877 * id_i2za * id_z877

## Reminder

 ### Remaining Iterations for Interactions:
- Iterations: 12/20



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_5, iteration_7)
- `shared_variables`: all variables you have saved from the previous iterations. (k, k_z877, r_squared, r_squared_z877, k1, r_squared1, r2, K2, r_squared2, k3, r_squared3, r_squared_i2za_sim, r_squared_z877_sim, k_z877_combined, r_squared1_z877, r_squared2_z877, n_fitted, r_squared1_i2za, r_squared2_i2za, n_reg, r_squared1_reg, r_squared2_reg, r_squared1_i2za_sim, r_squared1_z877_sim, r_squared2_i2za_sim, r_squared2_z877_sim, k_z877_all, r_squared1_z877_all, r_squared2_z877_all, r_squared3_z877_all, a, b, c, r_squared_ratio, k1_model1, K_model1, r_squared1_i2za_model1, r_squared2_i2za_model1, r_squared3_i2za_model1, r_squared1_i2za_sim_model1, r_squared1_z877_sim_model1, r_squared2_i2za_sim_model1, r_squared2_z877_sim_model1, r_squared3_i2za_sim_model1, r_squared3_z877_sim_model1)

### Allowed libraires
['numpy', 'pandas', 'libsbml', 'math', 'scipy', 'jax', 'sklearn', 'io', 'traceback']


Please construct your response according to the thoughts-action markdown format.



# Iteration 9
## Thoughts
After analyzing data from three different experiments, I've gained several important insights:

1. The model for `id_z877` consumption (`d(id_z877)/dt = -k * id_i2za * id_z877`) fits the data well for all experiments (R-squared > 0.77), with a consistent rate constant k ≈ 9.59e-07.

2. The stoichiometric coefficient (ratio of `id_i2za` production to `id_z877` consumption) varies widely across the data, with means around 2.5-2.9e+10 and medians ranging from 3.26e+04 to 4.29e+07, indicating a highly skewed distribution.

3. There's a strong positive correlation (0.986) between the stoichiometric coefficient and `id_i2za` concentration, and a moderate negative correlation (-0.472) with `id_z877` concentration.

4. The linear model for the stoichiometric coefficient (`n = 97.12 * id_i2za + 3862689.59 * id_z877 - 4934454743.65`) fits the data very well (R-squared = 0.975).

5. The model for `id_i2za` production with a carrying capacity (`d(id_i2za)/dt = k1 * id_i2za * id_z877 * (1 - id_i2za/K)`) still doesn't fit the data well (R-squared < 0), and the fitted carrying capacity K is extremely large.

6. The simulation results using this model are also poor, especially for `id_i2za` (R-squared < -0.5).

The strong correlation between the stoichiometric coefficient and `id_i2za` concentration suggests that the reaction mechanism might involve autocatalysis, where the rate of `id_i2za` production increases with its own concentration. However, the poor fit of the models suggests that the relationship is more complex than the ones I've tried so far.

Let me try a different approach. Instead of trying to fit a single model to the entire dataset, let's examine how the system behaves in different regimes, particularly at low vs. high concentrations of `id_i2za`. This might give us insights into the underlying reaction mechanism.

## Action

### Code
```python
import numpy as np
import pandas as pd
from scipy.optimize import curve_fit
from scipy.integrate import solve_ivp

# Get the data from all three experiments
data1 = experiment_history["iteration_0"]  # Original experiment
data2 = experiment_history["iteration_5"]  # Second experiment
data3 = experiment_history["iteration_7"]  # Third experiment

# Calculate the rate of change for all datasets
data1['d_i2za_dt'] = np.gradient(data1['id_i2za'], data1['Time'])
data1['d_z877_dt'] = np.gradient(data1['id_z877'], data1['Time'])

data2['d_i2za_dt'] = np.gradient(data2['id_i2za'], data2['Time'])
data2['d_z877_dt'] = np.gradient(data2['id_z877'], data2['Time'])

data3['d_i2za_dt'] = np.gradient(data3['id_i2za'], data3['Time'])
data3['d_z877_dt'] = np.gradient(data3['id_z877'], data3['Time'])

# Let's examine the system behavior in different regimes
# We'll divide the data into low, medium, and high id_i2za concentration regimes

# Combine the data from all experiments
data_combined = pd.concat([data1, data2, data3], ignore_index=True)

# Define the concentration regimes
low_threshold = 1e4
high_threshold = 1e7

data_low = data_combined[data_combined['id_i2za'] < low_threshold]
data_medium = data_combined[(data_combined['id_i2za'] >= low_threshold) & (data_combined['id_i2za'] < high_threshold)]
data_high = data_combined[data_combined['id_i2za'] >= high_threshold]

print(f"Number of data points in low concentration regime: {len(data_low)}")
print(f"Number of data points in medium concentration regime: {len(data_medium)}")
print(f"Number of data points in high concentration regime: {len(data_high)}")

# Let's fit the model for id_z877 consumption in each regime
def model_z877(X, k):
    i2za, z877 = X
    return -k * i2za * z877

# Prepare the data for curve fitting
X_low = np.vstack([data_low['id_i2za'], data_low['id_z877']])
y_low_z877 = data_low['d_z877_dt']

X_medium = np.vstack([data_medium['id_i2za'], data_medium['id_z877']])
y_medium_z877 = data_medium['d_z877_dt']

X_high = np.vstack([data_high['id_i2za'], data_high['id_z877']])
y_high_z877 = data_high['d_z877_dt']

# Fit the model for id_z877 consumption in each regime
try:
    params_low, covariance_low = curve_fit(model_z877, X_low, y_low_z877)
    k_z877_low = params_low[0]
    k_z877_low_error = np.sqrt(covariance_low[0, 0])
    
    params_medium, covariance_medium = curve_fit(model_z877, X_medium, y_medium_z877)
    k_z877_medium = params_medium[0]
    k_z877_medium_error = np.sqrt(covariance_medium[0, 0])
    
    params_high, covariance_high = curve_fit(model_z877, X_high, y_high_z877)
    k_z877_high = params_high[0]
    k_z877_high_error = np.sqrt(covariance_high[0, 0])
    
    print(f"\nModel for id_z877 consumption in different regimes:")
    print(f"Low concentration regime: k_z877 = {k_z877_low} ± {k_z877_low_error}")
    print(f"Medium concentration regime: k_z877 = {k_z877_medium} ± {k_z877_medium_error}")
    print(f"High concentration regime: k_z877 = {k_z877_high} ± {k_z877_high_error}")
    
    # Calculate the predicted rate of change
    data_low['predicted_d_z877_dt'] = model_z877(X_low, k_z877_low)
    data_medium['predicted_d_z877_dt'] = model_z877(X_medium, k_z877_medium)
    data_high['predicted_d_z877_dt'] = model_z877(X_high, k_z877_high)
    
    # Calculate the R-squared value for each regime
    ss_total_low = np.sum((y_low_z877 - np.mean(y_low_z877))**2)
    ss_residual_low = np.sum((y_low_z877 - data_low['predicted_d_z877_dt'])**2)
    r_squared_low = 1 - (ss_residual_low / ss_total_low)
    
    ss_total_medium = np.sum((y_medium_z877 - np.mean(y_medium_z877))**2)
    ss_residual_medium = np.sum((y_medium_z877 - data_medium['predicted_d_z877_dt'])**2)
    r_squared_medium = 1 - (ss_residual_medium / ss_total_medium)
    
    ss_total_high = np.sum((y_high_z877 - np.mean(y_high_z877))**2)
    ss_residual_high = np.sum((y_high_z877 - data_high['predicted_d_z877_dt'])**2)
    r_squared_high = 1 - (ss_residual_high / ss_total_high)
    
    print(f"R-squared for low concentration regime: {r_squared_low}")
    print(f"R-squared for medium concentration regime: {r_squared_medium}")
    print(f"R-squared for high concentration regime: {r_squared_high}")
    
    # Store the fitted parameters
    shared_variables.add("k_z877_low", k_z877_low)
    shared_variables.add("k_z877_medium", k_z877_medium)
    shared_variables.add("k_z877_high", k_z877_high)
    shared_variables.add("r_squared_low", r_squared_low)
    shared_variables.add("r_squared_medium", r_squared_medium)
    shared_variables.add("r_squared_high", r_squared_high)
except Exception as e:
    print(f"Error fitting model for id_z877 consumption in different regimes: {e}")

# Let's try a different model for id_i2za production
# Model 2: Autocatalytic growth with a variable stoichiometric coefficient
# d(id_i2za)/dt = k1 * id_i2za * id_z877 * (1 + k2 * id_i2za)
def model2(X, k1, k2):
    i2za, z877 = X
    return k1 * i2za * z877 * (1 + k2 * i2za)

# Prepare the data for curve fitting
y_low_i2za = data_low['d_i2za_dt']
y_medium_i2za = data_medium['d_i2za_dt']
y_high_i2za = data_high['d_i2za_dt']

# Fit the model for id_i2za production in each regime
try:
    p0 = [0.1, 1e-9]  # Initial guess for k1, k2
    params_low_i2za, covariance_low_i2za = curve_fit(model2, X_low, y_low_i2za, p0=p0)
    k1_low, k2_low = params_low_i2za
    k1_low_error, k2_low_error = np.sqrt(np.diag(covariance_low_i2za))
    
    params_medium_i2za, covariance_medium_i2za = curve_fit(model2, X_medium, y_medium_i2za, p0=p0)
    k1_medium, k2_medium = params_medium_i2za
    k1_medium_error, k2_medium_error = np.sqrt(np.diag(covariance_medium_i2za))
    
    params_high_i2za, covariance_high_i2za = curve_fit(model2, X_high, y_high_i2za, p0=p0)
    k1_high, k2_high = params_high_i2za
    k1_high_error, k2_high_error = np.sqrt(np.diag(covariance_high_i2za))
    
    print(f"\nModel 2 for id_i2za production in different regimes:")
    print(f"Low concentration regime: k1 = {k1_low} ± {k1_low_error}, k2 = {k2_low} ± {k2_low_error}")
    print(f"Medium concentration regime: k1 = {k1_medium} ± {k1_medium_error}, k2 = {k2_medium} ± {k2_medium_error}")
    print(f"High concentration regime: k1 = {k1_high} ± {k1_high_error}, k2 = {k2_high} ± {k2_high_error}")
    
    # Calculate the predicted rate of change
    data_low['predicted_d_i2za_dt_model2'] = model2(X_low, k1_low, k2_low)
    data_medium['predicted_d_i2za_dt_model2'] = model2(X_medium, k1_medium, k2_medium)
    data_high['predicted_d_i2za_dt_model2'] = model2(X_high, k1_high, k2_high)
    
    # Calculate the R-squared value for each regime
    ss_total_low_i2za = np.sum((y_low_i2za - np.mean(y_low_i2za))**2)
    ss_residual_low_i2za = np.sum((y_low_i2za - data_low['predicted_d_i2za_dt_model2'])**2)
    r_squared_low_i2za = 1 - (ss_residual_low_i2za / ss_total_low_i2za)
    
    ss_total_medium_i2za = np.sum((y_medium_i2za - np.mean(y_medium_i2za))**2)
    ss_residual_medium_i2za = np.sum((y_medium_i2za - data_medium['predicted_d_i2za_dt_model2'])**2)
    r_squared_medium_i2za = 1 - (ss_residual_medium_i2za / ss_total_medium_i2za)
    
    ss_total_high_i2za = np.sum((y_high_i2za - np.mean(y_high_i2za))**2)
    ss_residual_high_i2za = np.sum((y_high_i2za - data_high['predicted_d_i2za_dt_model2'])**2)
    r_squared_high_i2za = 1 - (ss_residual_high_i2za / ss_total_high_i2za)
    
    print(f"R-squared for low concentration regime: {r_squared_low_i2za}")
    print(f"R-squared for medium concentration regime: {r_squared_medium_i2za}")
    print(f"R-squared for high concentration regime: {r_squared_high_i2za}")
    
    # Store the fitted parameters
    shared_variables.add("k1_low", k1_low)
    shared_variables.add("k2_low", k2_low)
    shared_variables.add("k1_medium", k1_medium)
    shared_variables.add("k2_medium", k2_medium)
    shared_variables.add("k1_high", k1_high)
    shared_variables.add("k2_high", k2_high)
    shared_variables.add("r_squared_low_i2za", r_squared_low_i2za)
    shared_variables.add("r_squared_medium_i2za", r_squared_medium_i2za)
    shared_variables.add("r_squared_high_i2za", r_squared_high_i2za)
except Exception as e:
    print(f"Error fitting Model 2 for id_i2za production in different regimes: {e}")

# Let's try one more model for id_i2za production
# Model 3: Autocatalytic growth with a variable stoichiometric coefficient and a carrying capacity
# d(id_i2za)/dt = k1 * id_i2za * id_z877 * (1 + k2 * id_i2za) * (1 - id_i2za/K)
def model3(X, k1, k2, K):
    i2za, z877 = X
    return k1 * i2za * z877 * (1 + k2 * i2za) * (1 - i2za/K)

# Fit the model for id_i2za production using the combined data
try:
    X_combined = np.vstack([data_combined['id_i2za'], data_combined['id_z877']])
    y_combined_i2za = data_combined['d_i2za_dt']
    
    p0 = [0.1, 1e-9, 1e9]  # Initial guess for k1, k2, K
    params3, covariance3 = curve_fit(model3, X_combined, y_combined_i2za, p0=p0)
    k1_model3, k2_model3, K_model3 = params3
    k1_model3_error, k2_model3_error, K_model3_error = np.sqrt(np.diag(covariance3))
    
    print(f"\nModel 3 for id_i2za production (combined data):")
    print(f"Fitted parameter k1: {k1_model3} ± {k1_model3_error}")
    print(f"Fitted parameter k2: {k2_model3} ± {k2_model3_error}")
    print(f"Fitted parameter K: {K_model3} ± {K_model3_error}")
    
    # Calculate the predicted rate of change
    data_combined['predicted_d_i2za_dt_model3'] = model3(X_combined, k1_model3, k2_model3, K_model3)
    
    # Calculate the R-squared value
    ss_total_combined = np.sum((y_combined_i2za - np.mean(y_combined_i2za))**2)
    ss_residual_combined = np.sum((y_combined_i2za - data_combined['predicted_d_i2za_dt_model3'])**2)
    r_squared_combined = 1 - (ss_residual_combined / ss_total_combined)
    
    print(f"R-squared for combined data: {r_squared_combined}")
    
    # Store the fitted parameters
    shared_variables.add("k1_model3", k1_model3)
    shared_variables.add("k2_model3", k2_model3)
    shared_variables.add("K_model3", K_model3)
    shared_variables.add("r_squared_combined", r_squared_combined)
except Exception as e:
    print(f"Error fitting Model 3 for id_i2za production: {e}")

# Let's simulate the system with Model 3
try:
    # Define the system of differential equations
    def system_model3(t, y, k1, k2, K, k_z877):
        i2za, z877 = y
        d_i2za_dt = k1 * i2za * z877 * (1 + k2 * i2za) * (1 - i2za/K)
        d_z877_dt = -k_z877 * i2za * z877
        return [d_i2za_dt, d_z877_dt]
    
    # Initial conditions for all experiments
    y0_1 = [data1['id_i2za'].iloc[0], data1['id_z877'].iloc[0]]
    y0_2 = [data2['id_i2za'].iloc[0], data2['id_z877'].iloc[0]]
    y0_3 = [data3['id_i2za'].iloc[0], data3['id_z877'].iloc[0]]
    
    # Time points
    t_span = [0, 84]  # Assuming all experiments have the same time range
    t_eval1 = data1['Time'].values
    t_eval2 = data2['Time'].values
    t_eval3 = data3['Time'].values
    
    # Solve the system for all experiments
    k1 = shared_variables.access('k1_model3')
    k2 = shared_variables.access('k2_model3')
    K = shared_variables.access('K_model3')
    k_z877 = shared_variables.access('k_z877_all')
    
    sol1 = solve_ivp(system_model3, t_span, y0_1, args=(k1, k2, K, k_z877), t_eval=t_eval1)
    sol2 = solve_ivp(system_model3, t_span, y0_2, args=(k1, k2, K, k_z877), t_eval=t_eval2)
    sol3 = solve_ivp(system_model3, t_span, y0_3, args=(k1, k2, K, k_z877), t_eval=t_eval3)
    
    # Calculate the R-squared value for the simulation
    ss_total1_i2za_sim = np.sum((data1['id_i2za'] - np.mean(data1['id_i2za']))**2)
    ss_residual1_i2za_sim = np.sum((data1['id_i2za'] - sol1.y[0])**2)
    r_squared1_i2za_sim_model3 = 1 - (ss_residual1_i2za_sim / ss_total1_i2za_sim)
    
    ss_total1_z877_sim = np.sum((data1['id_z877'] - np.mean(data1['id_z877']))**2)
    ss_residual1_z877_sim = np.sum((data1['id_z877'] - sol1.y[1])**2)
    r_squared1_z877_sim_model3 = 1 - (ss_residual1_z877_sim / ss_total1_z877_sim)
    
    ss_total2_i2za_sim = np.sum((data2['id_i2za'] - np.mean(data2['id_i2za']))**2)
    ss_residual2_i2za_sim = np.sum((data2['id_i2za'] - sol2.y[0])**2)
    r_squared2_i2za_sim_model3 = 1 - (ss_residual2_i2za_sim / ss_total2_i2za_sim)
    
    ss_total2_z877_sim = np.sum((data2['id_z877'] - np.mean(data2['id_z877']))**2)
    ss_residual2_z877_sim = np.sum((data2['id_z877'] - sol2.y[1])**2)
    r_squared2_z877_sim_model3 = 1 - (ss_residual2_z877_sim / ss_total2_z877_sim)
    
    ss_total3_i2za_sim = np.sum((data3['id_i2za'] - np.mean(data3['id_i2za']))**2)
    ss_residual3_i2za_sim = np.sum((data3['id_i2za'] - sol3.y[0])**2)
    r_squared3_i2za_sim_model3 = 1 - (ss_residual3_i2za_sim / ss_total3_i2za_sim)
    
    ss_total3_z877_sim = np.sum((data3['id_z877'] - np.mean(data3['id_z877']))**2)
    ss_residual3_z877_sim = np.sum((data3['id_z877'] - sol3.y[1])**2)
    r_squared3_z877_sim_model3 = 1 - (ss_residual3_z877_sim / ss_total3_z877_sim)
    
    print(f"\nSimulation results for Model 3:")
    print(f"Experiment 1 - R-squared for id_i2za: {r_squared1_i2za_sim_model3}")
    print(f"Experiment 1 - R-squared for id_z877: {r_squared1_z877_sim_model3}")
    print(f"Experiment 2 - R-squared for id_i2za: {r_squared2_i2za_sim_model3}")
    print(f"Experiment 2 - R-squared for id_z877: {r_squared2_z877_sim_model3}")
    print(f"Experiment 3 - R-squared for id_i2za: {r_squared3_i2za_sim_model3}")
    print(f"Experiment 3 - R-squared for id_z877: {r_squared3_z877_sim_model3}")
    
    # Store the R-squared values
    shared_variables.add("r_squared1_i2za_sim_model3", r_squared1_i2za_sim_model3)
    shared_variables.add("r_squared1_z877_sim_model3", r_squared1_z877_sim_model3)
    shared_variables.add("r_squared2_i2za_sim_model3", r_squared2_i2za_sim_model3)
    shared_variables.add("r_squared2_z877_sim_model3", r_squared2_z877_sim_model3)
    shared_variables.add("r_squared3_i2za_sim_model3", r_squared3_i2za_sim_model3)
    shared_variables.add("r_squared3_z877_sim_model3", r_squared3_z877_sim_model3)
except Exception as e:
    print(f"Error simulating the system with Model 3: {e}")

# Based on our analysis, let's formulate a hypothesis for the missing reactions
print("\nHypothesis for missing reactions:")
print("Based on our analysis, the most likely reaction mechanism is:")

try:
    k1 = shared_variables.access('k1_model3')
    k2 = shared_variables.access('k2_model3')
    K = shared_variables.access('K_model3')
    k_z877 = shared_variables.access('k_z877_all')
    
    print(f"1. id_i2za + id_z877 -> 2 id_i2za (with rate constant k1 = {k1}, autocatalytic factor k2 = {k2}, and carrying capacity K = {K})")
    print(f"2. id_i2za + id_z877 -> id_i2za (with rate constant k_z877 = {k_z877})")
    print(f"This corresponds to the differential equations:")
    print(f"d(id_i2za)/dt = k1 * id_i2za * id_z877 * (1 + k2 * id_i2za) * (1 - id_i2za/K)")
    print(f"d(id_z877)/dt = -k_z877 * id_i2za * id_z877")
    
    # Calculate the effective stoichiometric coefficient at different id_i2za concentrations
    i2za_values = [1e2, 1e4, 1e6, 1e8]
    for i2za in i2za_values:
        n_effective = (k1 * (1 + k2 * i2za) * (1 - i2za/K)) / k_z877
        print(f"Effective stoichiometric coefficient at id_i2za = {i2za}: {n_effective}")
except Exception as e:
    print(f"Error accessing fitted parameters: {e}")
```
# Observation

## Code Stdout
Number of data points in low concentration regime: 1765
Number of data points in medium concentration regime: 2241
Number of data points in high concentration regime: 8636

Model for id_z877 consumption in different regimes:
Low concentration regime: k_z877 = 3.568601725652775e-06 ± 9.774375634330569e-08
Medium concentration regime: k_z877 = 1.003371745630589e-06 ± 2.2388177150723433e-09
High concentration regime: k_z877 = 1.8648834069461972e-09 ± 7.39216857681724e-11
R-squared for low concentration regime: -0.7189060175830311
R-squared for medium concentration regime: 0.9757627912412424
R-squared for high concentration regime: 0.003542973970818042

Model 2 for id_i2za production in different regimes:
Low concentration regime: k1 = 0.00028050776312167093 ± 1.2448579515738957e-05, k2 = 1.6204079365598182e-05 ± 7.050577367082005e-06
Medium concentration regime: k1 = -0.008677566248169122 ± 0.0002977989451678693, k2 = -2.640470643754214e-06 ± 6.465243275358779e-08
High concentration regime: k1 = 6.494855893874789 ± 0.0876932027665757, k2 = -8.969052473980898e-10 ± 5.826065223964009e-12
R-squared for low concentration regime: 0.5765501498320189
R-squared for medium concentration regime: 0.41795318305170326
R-squared for high concentration regime: 0.2842073225311118
Error fitting Model 3 for id_i2za production: Optimal parameters not found: Number of calls to function has reached maxfev = 800.
Error simulating the system with Model 3: 'k1_model3'

Hypothesis for missing reactions:
Based on our analysis, the most likely reaction mechanism is:
Error accessing fitted parameters: 'k1_model3'

## Reminder

 ### Remaining Iterations for Interactions:
- Iterations: 11/20



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_5, iteration_7)
- `shared_variables`: all variables you have saved from the previous iterations. (k, k_z877, r_squared, r_squared_z877, k1, r_squared1, r2, K2, r_squared2, k3, r_squared3, r_squared_i2za_sim, r_squared_z877_sim, k_z877_combined, r_squared1_z877, r_squared2_z877, n_fitted, r_squared1_i2za, r_squared2_i2za, n_reg, r_squared1_reg, r_squared2_reg, r_squared1_i2za_sim, r_squared1_z877_sim, r_squared2_i2za_sim, r_squared2_z877_sim, k_z877_all, r_squared1_z877_all, r_squared2_z877_all, r_squared3_z877_all, a, b, c, r_squared_ratio, k1_model1, K_model1, r_squared1_i2za_model1, r_squared2_i2za_model1, r_squared3_i2za_model1, r_squared1_i2za_sim_model1, r_squared1_z877_sim_model1, r_squared2_i2za_sim_model1, r_squared2_z877_sim_model1, r_squared3_i2za_sim_model1, r_squared3_z877_sim_model1, k_z877_low, k_z877_medium, k_z877_high, r_squared_low, r_squared_medium, r_squared_high, k1_low, k2_low, k1_medium, k2_medium, k1_high, k2_high, r_squared_low_i2za, r_squared_medium_i2za, r_squared_high_i2za)

### Allowed libraires
['numpy', 'pandas', 'libsbml', 'math', 'scipy', 'jax', 'sklearn', 'io', 'traceback']


Please construct your response according to the thoughts-action markdown format.



# Iteration 10
## Thoughts
The analysis of the system behavior in different concentration regimes has provided valuable insights:

1. For `id_z877` consumption:
   - The rate constant varies significantly across different concentration regimes: 3.57e-06 (low), 1.00e-06 (medium), and 1.86e-09 (high).
   - The model fits well in the medium concentration regime (R-squared = 0.976), but poorly in the low and high regimes.

2. For `id_i2za` production (Model 2: `d(id_i2za)/dt = k1 * id_i2za * id_z877 * (1 + k2 * id_i2za)`):
   - The model fits reasonably well in all regimes, with R-squared values of 0.577 (low), 0.418 (medium), and 0.284 (high).
   - The parameters vary significantly across regimes, with k1 even becoming negative in the medium regime.

3. Model 3 (with carrying capacity) failed to converge, suggesting that the carrying capacity might not be a critical factor in the observed range of concentrations.

The varying rate constants and model fits across different concentration regimes suggest that the reaction mechanism might be more complex than a simple mass action kinetics model. It could involve multiple reactions with different rate laws, or there might be additional species or reactions not included in our current model.

Given these insights, I'll try a different approach. Instead of trying to fit a single model to the entire dataset, I'll focus on identifying the key reactions that could explain the observed behavior. I'll also try to implement these reactions in an SBML model and simulate it to see how well it matches the experimental data.

## Action

### Code
```python
import numpy as np
import pandas as pd
import libsbml
from scipy.integrate import solve_ivp

# Get the data from all three experiments
data1 = experiment_history["iteration_0"]  # Original experiment
data2 = experiment_history["iteration_5"]  # Second experiment
data3 = experiment_history["iteration_7"]  # Third experiment

# Based on our analysis, let's formulate a hypothesis for the missing reactions
print("Hypothesis for missing reactions:")
print("Based on our analysis, we propose the following reaction mechanism:")
print("1. id_i2za + id_z877 -> 2 id_i2za (autocatalytic production of id_i2za)")
print("2. id_i2za + id_z877 -> id_i2za (consumption of id_z877)")

# Let's implement this reaction mechanism in an SBML model
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Add the first reaction: id_i2za + id_z877 -> 2 id_i2za
reaction1 = model.createReaction()
reaction1.setId("reaction1")
reaction1.setReversible(False)
reaction1.setFast(False)

# Add reactants
reactant1_1 = reaction1.createReactant()
reactant1_1.setSpecies("id_i2za")
reactant1_1.setStoichiometry(1.0)
reactant1_1.setConstant(True)

reactant1_2 = reaction1.createReactant()
reactant1_2.setSpecies("id_z877")
reactant1_2.setStoichiometry(1.0)
reactant1_2.setConstant(True)

# Add products
product1 = reaction1.createProduct()
product1.setSpecies("id_i2za")
product1.setStoichiometry(2.0)
product1.setConstant(True)

# Add kinetic law
kinetic_law1 = reaction1.createKineticLaw()
kinetic_law1.setFormula("k1 * id_i2za * id_z877")

# Add parameters
parameter1 = kinetic_law1.createParameter()
parameter1.setId("k1")
parameter1.setValue(0.1)  # Using a reasonable value based on our analysis

# Add the second reaction: id_i2za + id_z877 -> id_i2za
reaction2 = model.createReaction()
reaction2.setId("reaction2")
reaction2.setReversible(False)
reaction2.setFast(False)

# Add reactants
reactant2_1 = reaction2.createReactant()
reactant2_1.setSpecies("id_i2za")
reactant2_1.setStoichiometry(1.0)
reactant2_1.setConstant(True)

reactant2_2 = reaction2.createReactant()
reactant2_2.setSpecies("id_z877")
reactant2_2.setStoichiometry(1.0)
reactant2_2.setConstant(True)

# Add products
product2 = reaction2.createProduct()
product2.setSpecies("id_i2za")
product2.setStoichiometry(1.0)
product2.setConstant(True)

# Add kinetic law
kinetic_law2 = reaction2.createKineticLaw()
kinetic_law2.setFormula("k2 * id_i2za * id_z877")

# Add parameters
parameter2 = kinetic_law2.createParameter()
parameter2.setId("k2")
parameter2.setValue(1e-6)  # Using a value close to k_z877_all

# Write the updated SBML model
writer = libsbml.SBMLWriter()
updated_sbml = writer.writeSBMLToString(sbml_doc)

# Print the updated SBML model
print("\nUpdated SBML model:")
print(updated_sbml)

# Let's simulate the updated model using our own solver
def simulate_model(initial_i2za, initial_z877, k1, k2, t_span, t_eval):
    def system(t, y):
        i2za, z877 = y
        d_i2za_dt = k1 * i2za * z877  # Reaction 1: id_i2za + id_z877 -> 2 id_i2za
        d_z877_dt = -(k1 + k2) * i2za * z877  # Combined effect of both reactions on id_z877
        return [d_i2za_dt, d_z877_dt]
    
    y0 = [initial_i2za, initial_z877]
    sol = solve_ivp(system, t_span, y0, t_eval=t_eval)
    return sol

# Let's try different values of k1 and k2 to find the best fit
k1_values = [0.05, 0.1, 0.2]
k2_values = [1e-7, 5e-7, 1e-6]

best_r_squared_i2za = -float('inf')
best_r_squared_z877 = -float('inf')
best_k1 = None
best_k2 = None

for k1 in k1_values:
    for k2 in k2_values:
        # Simulate for all three experiments
        sol1 = simulate_model(data1['id_i2za'].iloc[0], data1['id_z877'].iloc[0], k1, k2, [0, 84], data1['Time'].values)
        sol2 = simulate_model(data2['id_i2za'].iloc[0], data2['id_z877'].iloc[0], k1, k2, [0, 84], data2['Time'].values)
        sol3 = simulate_model(data3['id_i2za'].iloc[0], data3['id_z877'].iloc[0], k1, k2, [0, 84], data3['Time'].values)
        
        # Calculate R-squared for each experiment
        ss_total1_i2za = np.sum((data1['id_i2za'] - np.mean(data1['id_i2za']))**2)
        ss_residual1_i2za = np.sum((data1['id_i2za'] - sol1.y[0])**2)
        r_squared1_i2za = 1 - (ss_residual1_i2za / ss_total1_i2za)
        
        ss_total1_z877 = np.sum((data1['id_z877'] - np.mean(data1['id_z877']))**2)
        ss_residual1_z877 = np.sum((data1['id_z877'] - sol1.y[1])**2)
        r_squared1_z877 = 1 - (ss_residual1_z877 / ss_total1_z877)
        
        ss_total2_i2za = np.sum((data2['id_i2za'] - np.mean(data2['id_i2za']))**2)
        ss_residual2_i2za = np.sum((data2['id_i2za'] - sol2.y[0])**2)
        r_squared2_i2za = 1 - (ss_residual2_i2za / ss_total2_i2za)
        
        ss_total2_z877 = np.sum((data2['id_z877'] - np.mean(data2['id_z877']))**2)
        ss_residual2_z877 = np.sum((data2['id_z877'] - sol2.y[1])**2)
        r_squared2_z877 = 1 - (ss_residual2_z877 / ss_total2_z877)
        
        ss_total3_i2za = np.sum((data3['id_i2za'] - np.mean(data3['id_i2za']))**2)
        ss_residual3_i2za = np.sum((data3['id_i2za'] - sol3.y[0])**2)
        r_squared3_i2za = 1 - (ss_residual3_i2za / ss_total3_i2za)
        
        ss_total3_z877 = np.sum((data3['id_z877'] - np.mean(data3['id_z877']))**2)
        ss_residual3_z877 = np.sum((data3['id_z877'] - sol3.y[1])**2)
        r_squared3_z877 = 1 - (ss_residual3_z877 / ss_total3_z877)
        
        # Calculate average R-squared
        avg_r_squared_i2za = (r_squared1_i2za + r_squared2_i2za + r_squared3_i2za) / 3
        avg_r_squared_z877 = (r_squared1_z877 + r_squared2_z877 + r_squared3_z877) / 3
        
        print(f"k1 = {k1}, k2 = {k2}:")
        print(f"  Experiment 1 - R-squared for id_i2za: {r_squared1_i2za}, R-squared for id_z877: {r_squared1_z877}")
        print(f"  Experiment 2 - R-squared for id_i2za: {r_squared2_i2za}, R-squared for id_z877: {r_squared2_z877}")
        print(f"  Experiment 3 - R-squared for id_i2za: {r_squared3_i2za}, R-squared for id_z877: {r_squared3_z877}")
        print(f"  Average - R-squared for id_i2za: {avg_r_squared_i2za}, R-squared for id_z877: {avg_r_squared_z877}")
        
        # Update best parameters if better fit is found
        if avg_r_squared_i2za > best_r_squared_i2za:
            best_r_squared_i2za = avg_r_squared_i2za
            best_k1_i2za = k1
            best_k2_i2za = k2
        
        if avg_r_squared_z877 > best_r_squared_z877:
            best_r_squared_z877 = avg_r_squared_z877
            best_k1_z877 = k1
            best_k2_z877 = k2

print(f"\nBest parameters for id_i2za: k1 = {best_k1_i2za}, k2 = {best_k2_i2za}, R-squared = {best_r_squared_i2za}")
print(f"Best parameters for id_z877: k1 = {best_k1_z877}, k2 = {best_k2_z877}, R-squared = {best_r_squared_z877}")

# Let's update the SBML model with the best parameters
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Add the first reaction: id_i2za + id_z877 -> 2 id_i2za
reaction1 = model.createReaction()
reaction1.setId("reaction1")
reaction1.setReversible(False)
reaction1.setFast(False)

# Add reactants
reactant1_1 = reaction1.createReactant()
reactant1_1.setSpecies("id_i2za")
reactant1_1.setStoichiometry(1.0)
reactant1_1.setConstant(True)

reactant1_2 = reaction1.createReactant()
reactant1_2.setSpecies("id_z877")
reactant1_2.setStoichiometry(1.0)
reactant1_2.setConstant(True)

# Add products
product1 = reaction1.createProduct()
product1.setSpecies("id_i2za")
product1.setStoichiometry(2.0)
product1.setConstant(True)

# Add kinetic law
kinetic_law1 = reaction1.createKineticLaw()
kinetic_law1.setFormula("k1 * id_i2za * id_z877")

# Add parameters
parameter1 = kinetic_law1.createParameter()
parameter1.setId("k1")
parameter1.setValue(best_k1_i2za)  # Using the best value for id_i2za

# Add the second reaction: id_i2za + id_z877 -> id_i2za
reaction2 = model.createReaction()
reaction2.setId("reaction2")
reaction2.setReversible(False)
reaction2.setFast(False)

# Add reactants
reactant2_1 = reaction2.createReactant()
reactant2_1.setSpecies("id_i2za")
reactant2_1.setStoichiometry(1.0)
reactant2_1.setConstant(True)

reactant2_2 = reaction2.createReactant()
reactant2_2.setSpecies("id_z877")
reactant2_2.setStoichiometry(1.0)
reactant2_2.setConstant(True)

# Add products
product2 = reaction2.createProduct()
product2.setSpecies("id_i2za")
product2.setStoichiometry(1.0)
product2.setConstant(True)

# Add kinetic law
kinetic_law2 = reaction2.createKineticLaw()
kinetic_law2.setFormula("k2 * id_i2za * id_z877")

# Add parameters
parameter2 = kinetic_law2.createParameter()
parameter2.setId("k2")
parameter2.setValue(best_k2_i2za)  # Using the best value for id_i2za

# Write the updated SBML model
writer = libsbml.SBMLWriter()
final_sbml = writer.writeSBMLToString(sbml_doc)

# Store the final SBML model
shared_variables.add("final_sbml", final_sbml)

print("\nFinal SBML model with best parameters:")
print(final_sbml)
```
# Observation

## Code Stdout
Hypothesis for missing reactions:
Based on our analysis, we propose the following reaction mechanism:
1. id_i2za + id_z877 -> 2 id_i2za (autocatalytic production of id_i2za)
2. id_i2za + id_z877 -> id_i2za (consumption of id_z877)

Updated SBML model:
<?xml version="1.0" encoding="UTF-8"?>
<sbml xmlns="http://www.sbml.org/sbml/level3/version1/core" xmlns:layout="http://www.sbml.org/sbml/level3/version1/layout/version1" xmlns:render="http://www.sbml.org/sbml/level3/version1/render/version1" level="3" version="1" layout:required="false" render:required="false">
  <model id="id_k9sa" substanceUnits="id_yxmk" timeUnits="id_pcnf" volumeUnits="id_29tc" areaUnits="id_el2n" lengthUnits="id_nfym" extentUnits="id_yxmk">
    <listOfUnitDefinitions>
      <unitDefinition id="id_nfym">
        <listOfUnits>
          <unit kind="metre" exponent="1" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_el2n">
        <listOfUnits>
          <unit kind="metre" exponent="2" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_29tc">
        <listOfUnits>
          <unit kind="litre" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_pcnf">
        <listOfUnits>
          <unit kind="second" exponent="1" scale="0" multiplier="86400"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_yxmk">
        <listOfUnits>
          <unit kind="mole" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_zx0m">
        <listOfUnits>
          <unit kind="second" exponent="-1" scale="3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_ept0">
        <listOfUnits>
          <unit kind="dimensionless" exponent="0" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
    </listOfUnitDefinitions>
    <listOfCompartments>
      <compartment id="id_ez0l" spatialDimensions="3" size="1" units="id_29tc" constant="true"/>
    </listOfCompartments>
    <listOfSpecies>
      <species id="id_i2za" compartment="id_ez0l" initialConcentration="100" substanceUnits="id_yxmk" hasOnlySubstanceUnits="false" boundaryCondition="false" constant="false"/>
      <species id="id_z877" compartment="id_ez0l" initialConcentration="1000" substanceUnits="id_yxmk" hasOnlySubstanceUnits="false" boundaryCondition="false" constant="false"/>
    </listOfSpecies>
    <listOfReactions>
      <reaction id="reaction1" reversible="false" fast="false">
        <listOfReactants>
          <speciesReference species="id_i2za" stoichiometry="1" constant="true"/>
          <speciesReference species="id_z877" stoichiometry="1" constant="true"/>
        </listOfReactants>
        <listOfProducts>
          <speciesReference species="id_i2za" stoichiometry="2" constant="true"/>
        </listOfProducts>
        <kineticLaw>
          <math xmlns="http://www.w3.org/1998/Math/MathML">
            <apply>
              <times/>
              <ci> k1 </ci>
              <ci> id_i2za </ci>
              <ci> id_z877 </ci>
            </apply>
          </math>
          <listOfLocalParameters>
            <localParameter id="k1" value="0.1"/>
          </listOfLocalParameters>
        </kineticLaw>
      </reaction>
      <reaction id="reaction2" reversible="false" fast="false">
        <listOfReactants>
          <speciesReference species="id_i2za" stoichiometry="1" constant="true"/>
          <speciesReference species="id_z877" stoichiometry="1" constant="true"/>
        </listOfReactants>
        <listOfProducts>
          <speciesReference species="id_i2za" stoichiometry="1" constant="true"/>
        </listOfProducts>
        <kineticLaw>
          <math xmlns="http://www.w3.org/1998/Math/MathML">
            <apply>
              <times/>
              <ci> k2 </ci>
              <ci> id_i2za </ci>
              <ci> id_z877 </ci>
            </apply>
          </math>
          <listOfLocalParameters>
            <localParameter id="k2" value="1e-06"/>
          </listOfLocalParameters>
        </kineticLaw>
      </reaction>
    </listOfReactions>
  </model>
</sbml>

k1 = 0.05, k2 = 1e-07:
  Experiment 1 - R-squared for id_i2za: -1.579011428011159, R-squared for id_z877: -0.3012243091347804
  Experiment 2 - R-squared for id_i2za: -1.1849764644454326, R-squared for id_z877: -0.41625321976067853
  Experiment 3 - R-squared for id_i2za: -1.8510328813715216, R-squared for id_z877: -0.24975711297162162
  Average - R-squared for id_i2za: -1.5383402579427043, R-squared for id_z877: -0.3224115472890268
k1 = 0.05, k2 = 5e-07:
  Experiment 1 - R-squared for id_i2za: -1.5790114280549732, R-squared for id_z877: -0.3012243244672468
  Experiment 2 - R-squared for id_i2za: -1.1849764645200866, R-squared for id_z877: -0.41625322713157176
  Experiment 3 - R-squared for id_i2za: -1.85103288139568, R-squared for id_z877: -0.24975713952559842
  Average - R-squared for id_i2za: -1.5383402579902465, R-squared for id_z877: -0.322411563708139
k1 = 0.05, k2 = 1e-06:
  Experiment 1 - R-squared for id_i2za: -1.57901142810974, R-squared for id_z877: -0.30122434363266426
  Experiment 2 - R-squared for id_i2za: -1.1849764646134027, R-squared for id_z877: -0.41625323634509015
  Experiment 3 - R-squared for id_i2za: -1.851032881425879, R-squared for id_z877: -0.2497571727177077
  Average - R-squared for id_i2za: -1.538340258049674, R-squared for id_z877: -0.3224115842318207
k1 = 0.1, k2 = 1e-07:
  Experiment 1 - R-squared for id_i2za: -1.5790114280056842, R-squared for id_z877: -0.3040979011893612
  Experiment 2 - R-squared for id_i2za: -1.1849764644361058, R-squared for id_z877: -0.41824520956988853
  Experiment 3 - R-squared for id_i2za: -1.8510328813685022, R-squared for id_z877: -0.2528944967221407
  Average - R-squared for id_i2za: -1.538340257936764, R-squared for id_z877: -0.3250792024937968
k1 = 0.1, k2 = 5e-07:
  Experiment 1 - R-squared for id_i2za: -1.5790114280275915, R-squared for id_z877: -0.3040979050297663
  Experiment 2 - R-squared for id_i2za: -1.1849764644734333, R-squared for id_z877: -0.41824521190205477
  Experiment 3 - R-squared for id_i2za: -1.8510328813805814, R-squared for id_z877: -0.2528945033631764
  Average - R-squared for id_i2za: -1.5383402579605354, R-squared for id_z877: -0.3250792067649992
k1 = 0.1, k2 = 1e-06:
  Experiment 1 - R-squared for id_i2za: -1.5790114280549754, R-squared for id_z877: -0.3040979098348524
  Experiment 2 - R-squared for id_i2za: -1.1849764645200915, R-squared for id_z877: -0.4182452148172504
  Experiment 3 - R-squared for id_i2za: -1.8510328813956805, R-squared for id_z877: -0.2528945116644268
  Average - R-squared for id_i2za: -1.5383402579902492, R-squared for id_z877: -0.3250792121055099
k1 = 0.2, k2 = 1e-07:
  Experiment 1 - R-squared for id_i2za: -1.5790114280029468, R-squared for id_z877: -0.3056491113474271
  Experiment 2 - R-squared for id_i2za: -1.1849764644314424, R-squared for id_z877: -0.4193877631807734
  Experiment 3 - R-squared for id_i2za: -1.8510328813669923, R-squared for id_z877: -0.25448434806173226
  Average - R-squared for id_i2za: -1.538340257933794, R-squared for id_z877: -0.32650707419664426
k1 = 0.2, k2 = 5e-07:
  Experiment 1 - R-squared for id_i2za: -1.5790114280139003, R-squared for id_z877: -0.30564911225641445
  Experiment 2 - R-squared for id_i2za: -1.1849764644501062, R-squared for id_z877: -0.4193877632703289
  Experiment 3 - R-squared for id_i2za: -1.851032881373032, R-squared for id_z877: -0.25448434965552846
  Average - R-squared for id_i2za: -1.5383402579456795, R-squared for id_z877: -0.32650707506075727
k1 = 0.2, k2 = 1e-06:
  Experiment 1 - R-squared for id_i2za: -1.5790114280275924, R-squared for id_z877: -0.3056491133986785
  Experiment 2 - R-squared for id_i2za: -1.184976464473436, R-squared for id_z877: -0.419387763382272
  Experiment 3 - R-squared for id_i2za: -1.8510328813805814, R-squared for id_z877: -0.25448435164776706
  Average - R-squared for id_i2za: -1.5383402579605365, R-squared for id_z877: -0.32650707614290586

Best parameters for id_i2za: k1 = 0.2, k2 = 1e-07, R-squared = -1.538340257933794
Best parameters for id_z877: k1 = 0.05, k2 = 1e-07, R-squared = -0.3224115472890268

Final SBML model with best parameters:
<?xml version="1.0" encoding="UTF-8"?>
<sbml xmlns="http://www.sbml.org/sbml/level3/version1/core" xmlns:layout="http://www.sbml.org/sbml/level3/version1/layout/version1" xmlns:render="http://www.sbml.org/sbml/level3/version1/render/version1" level="3" version="1" layout:required="false" render:required="false">
  <model id="id_k9sa" substanceUnits="id_yxmk" timeUnits="id_pcnf" volumeUnits="id_29tc" areaUnits="id_el2n" lengthUnits="id_nfym" extentUnits="id_yxmk">
    <listOfUnitDefinitions>
      <unitDefinition id="id_nfym">
        <listOfUnits>
          <unit kind="metre" exponent="1" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_el2n">
        <listOfUnits>
          <unit kind="metre" exponent="2" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_29tc">
        <listOfUnits>
          <unit kind="litre" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_pcnf">
        <listOfUnits>
          <unit kind="second" exponent="1" scale="0" multiplier="86400"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_yxmk">
        <listOfUnits>
          <unit kind="mole" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_zx0m">
        <listOfUnits>
          <unit kind="second" exponent="-1" scale="3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_ept0">
        <listOfUnits>
   
... [output truncated]
## Reminder

 ### Remaining Iterations for Interactions:
- Iterations: 10/20



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_5, iteration_7)
- `shared_variables`: all variables you have saved from the previous iterations. (k, k_z877, r_squared, r_squared_z877, k1, r_squared1, r2, K2, r_squared2, k3, r_squared3, r_squared_i2za_sim, r_squared_z877_sim, k_z877_combined, r_squared1_z877, r_squared2_z877, n_fitted, r_squared1_i2za, r_squared2_i2za, n_reg, r_squared1_reg, r_squared2_reg, r_squared1_i2za_sim, r_squared1_z877_sim, r_squared2_i2za_sim, r_squared2_z877_sim, k_z877_all, r_squared1_z877_all, r_squared2_z877_all, r_squared3_z877_all, a, b, c, r_squared_ratio, k1_model1, K_model1, r_squared1_i2za_model1, r_squared2_i2za_model1, r_squared3_i2za_model1, r_squared1_i2za_sim_model1, r_squared1_z877_sim_model1, r_squared2_i2za_sim_model1, r_squared2_z877_sim_model1, r_squared3_i2za_sim_model1, r_squared3_z877_sim_model1, k_z877_low, k_z877_medium, k_z877_high, r_squared_low, r_squared_medium, r_squared_high, k1_low, k2_low, k1_medium, k2_medium, k1_high, k2_high, r_squared_low_i2za, r_squared_medium_i2za, r_squared_high_i2za, final_sbml)

### Allowed libraires
['numpy', 'pandas', 'libsbml', 'math', 'scipy', 'jax', 'sklearn', 'io', 'traceback']


Please construct your response according to the thoughts-action markdown format.



# Iteration 11
## Thoughts
Based on the simulation results, I've found that the proposed reaction mechanism:
1. `id_i2za + id_z877 -> 2 id_i2za` (autocatalytic production of `id_i2za`)
2. `id_i2za + id_z877 -> id_i2za` (consumption of `id_z877`)

doesn't fit the experimental data well. The R-squared values for both species are negative across all parameter combinations I tried, indicating that the model performs worse than simply using the mean value as a predictor.

This suggests that the reaction mechanism might be more complex than what I've proposed. Let me try a different approach by:

1. Examining the data more carefully to identify patterns
2. Trying a different reaction mechanism
3. Using the `simulate` function provided in the system to validate my model

Let me first use the `simulate` function to see how well the current SBML model performs.

## Action

### Code
```python
import numpy as np
import pandas as pd
import libsbml
from scipy.integrate import solve_ivp

# Get the data from all three experiments
data1 = experiment_history["iteration_0"]  # Original experiment
data2 = experiment_history["iteration_5"]  # Second experiment
data3 = experiment_history["iteration_7"]  # Third experiment

# Let's use the simulate function to test our current model
final_sbml = shared_variables.access("final_sbml")
print("Simulating the current SBML model...")
simulation_result = simulate(final_sbml)

# Print the first few rows of the simulation result
print("First few rows of simulation result:")
print(simulation_result.head())

# Print the last few rows of the simulation result
print("\nLast few rows of simulation result:")
print(simulation_result.tail())

# Calculate R-squared for the simulation result compared to the original experiment
# First, interpolate the simulation result to match the time points in the original experiment
from scipy.interpolate import interp1d

# Create interpolation functions
f_i2za = interp1d(simulation_result['Time'], simulation_result['id_i2za'], bounds_error=False, fill_value="extrapolate")
f_z877 = interp1d(simulation_result['Time'], simulation_result['id_z877'], bounds_error=False, fill_value="extrapolate")

# Interpolate to match the time points in the original experiment
sim_i2za_interp = f_i2za(data1['Time'])
sim_z877_interp = f_z877(data1['Time'])

# Calculate R-squared
ss_total_i2za = np.sum((data1['id_i2za'] - np.mean(data1['id_i2za']))**2)
ss_residual_i2za = np.sum((data1['id_i2za'] - sim_i2za_interp)**2)
r_squared_i2za = 1 - (ss_residual_i2za / ss_total_i2za)

ss_total_z877 = np.sum((data1['id_z877'] - np.mean(data1['id_z877']))**2)
ss_residual_z877 = np.sum((data1['id_z877'] - sim_z877_interp)**2)
r_squared_z877 = 1 - (ss_residual_z877 / ss_total_z877)

print(f"\nR-squared for id_i2za: {r_squared_i2za}")
print(f"R-squared for id_z877: {r_squared_z877}")

# Let's try a different reaction mechanism
# Based on our analysis, let's try a model with a higher stoichiometric coefficient
# Reaction: id_i2za + id_z877 -> n id_i2za
# where n is a large number (e.g., 1000)

# Create a new SBML model
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Add the reaction: id_i2za + id_z877 -> n id_i2za
reaction = model.createReaction()
reaction.setId("reaction1")
reaction.setReversible(False)
reaction.setFast(False)

# Add reactants
reactant1 = reaction.createReactant()
reactant1.setSpecies("id_i2za")
reactant1.setStoichiometry(1.0)
reactant1.setConstant(True)

reactant2 = reaction.createReactant()
reactant2.setSpecies("id_z877")
reactant2.setStoichiometry(1.0)
reactant2.setConstant(True)

# Add products
product = reaction.createProduct()
product.setSpecies("id_i2za")
product.setStoichiometry(1000.0)  # Large stoichiometric coefficient
product.setConstant(True)

# Add kinetic law
kinetic_law = reaction.createKineticLaw()
kinetic_law.setFormula("k * id_i2za * id_z877")

# Add parameters
parameter = kinetic_law.createParameter()
parameter.setId("k")
parameter.setValue(1e-6)  # Using a small value to compensate for the large stoichiometric coefficient

# Write the updated SBML model
writer = libsbml.SBMLWriter()
new_sbml = writer.writeSBMLToString(sbml_doc)

# Print the new SBML model
print("\nNew SBML model with high stoichiometric coefficient:")
print(new_sbml)

# Simulate the new model
print("\nSimulating the new SBML model...")
new_simulation_result = simulate(new_sbml)

# Print the first few rows of the new simulation result
print("First few rows of new simulation result:")
print(new_simulation_result.head())

# Print the last few rows of the new simulation result
print("\nLast few rows of new simulation result:")
print(new_simulation_result.tail())

# Calculate R-squared for the new simulation result
# Create interpolation functions
f_i2za_new = interp1d(new_simulation_result['Time'], new_simulation_result['id_i2za'], bounds_error=False, fill_value="extrapolate")
f_z877_new = interp1d(new_simulation_result['Time'], new_simulation_result['id_z877'], bounds_error=False, fill_value="extrapolate")

# Interpolate to match the time points in the original experiment
sim_i2za_interp_new = f_i2za_new(data1['Time'])
sim_z877_interp_new = f_z877_new(data1['Time'])

# Calculate R-squared
ss_residual_i2za_new = np.sum((data1['id_i2za'] - sim_i2za_interp_new)**2)
r_squared_i2za_new = 1 - (ss_residual_i2za_new / ss_total_i2za)

ss_residual_z877_new = np.sum((data1['id_z877'] - sim_z877_interp_new)**2)
r_squared_z877_new = 1 - (ss_residual_z877_new / ss_total_z877)

print(f"\nR-squared for id_i2za (new model): {r_squared_i2za_new}")
print(f"R-squared for id_z877 (new model): {r_squared_z877_new}")

# Let's try one more model with a different stoichiometric coefficient
# Reaction: id_i2za + id_z877 -> n id_i2za
# where n is a different value (e.g., 100)

# Create a new SBML model
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Add the reaction: id_i2za + id_z877 -> n id_i2za
reaction = model.createReaction()
reaction.setId("reaction1")
reaction.setReversible(False)
reaction.setFast(False)

# Add reactants
reactant1 = reaction.createReactant()
reactant1.setSpecies("id_i2za")
reactant1.setStoichiometry(1.0)
reactant1.setConstant(True)

reactant2 = reaction.createReactant()
reactant2.setSpecies("id_z877")
reactant2.setStoichiometry(1.0)
reactant2.setConstant(True)

# Add products
product = reaction.createProduct()
product.setSpecies("id_i2za")
product.setStoichiometry(100.0)  # Different stoichiometric coefficient
product.setConstant(True)

# Add kinetic law
kinetic_law = reaction.createKineticLaw()
kinetic_law.setFormula("k * id_i2za * id_z877")

# Add parameters
parameter = kinetic_law.createParameter()
parameter.setId("k")
parameter.setValue(1e-5)  # Adjusted value

# Write the updated SBML model
writer = libsbml.SBMLWriter()
new_sbml2 = writer.writeSBMLToString(sbml_doc)

# Simulate the new model
print("\nSimulating another new SBML model...")
new_simulation_result2 = simulate(new_sbml2)

# Calculate R-squared for the new simulation result
# Create interpolation functions
f_i2za_new2 = interp1d(new_simulation_result2['Time'], new_simulation_result2['id_i2za'], bounds_error=False, fill_value="extrapolate")
f_z877_new2 = interp1d(new_simulation_result2['Time'], new_simulation_result2['id_z877'], bounds_error=False, fill_value="extrapolate")

# Interpolate to match the time points in the original experiment
sim_i2za_interp_new2 = f_i2za_new2(data1['Time'])
sim_z877_interp_new2 = f_z877_new2(data1['Time'])

# Calculate R-squared
ss_residual_i2za_new2 = np.sum((data1['id_i2za'] - sim_i2za_interp_new2)**2)
r_squared_i2za_new2 = 1 - (ss_residual_i2za_new2 / ss_total_i2za)

ss_residual_z877_new2 = np.sum((data1['id_z877'] - sim_z877_interp_new2)**2)
r_squared_z877_new2 = 1 - (ss_residual_z877_new2 / ss_total_z877)

print(f"\nR-squared for id_i2za (another new model): {r_squared_i2za_new2}")
print(f"R-squared for id_z877 (another new model): {r_squared_z877_new2}")

# Let's try a range of stoichiometric coefficients and rate constants to find the best fit
stoichiometries = [10, 50, 100, 500, 1000]
rate_constants = [1e-7, 1e-6, 1e-5, 1e-4]

best_r_squared_i2za = -float('inf')
best_r_squared_z877 = -float('inf')
best_stoichiometry = None
best_rate_constant = None

for n in stoichiometries:
    for k in rate_constants:
        # Create a new SBML model
        sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
        model = sbml_doc.getModel()
        
        # Add the reaction: id_i2za + id_z877 -> n id_i2za
        reaction = model.createReaction()
        reaction.setId("reaction1")
        reaction.setReversible(False)
        reaction.setFast(False)
        
        # Add reactants
        reactant1 = reaction.createReactant()
        reactant1.setSpecies("id_i2za")
        reactant1.setStoichiometry(1.0)
        reactant1.setConstant(True)
        
        reactant2 = reaction.createReactant()
        reactant2.setSpecies("id_z877")
        reactant2.setStoichiometry(1.0)
        reactant2.setConstant(True)
        
        # Add products
        product = reaction.createProduct()
        product.setSpecies("id_i2za")
        product.setStoichiometry(float(n))
        product.setConstant(True)
        
        # Add kinetic law
        kinetic_law = reaction.createKineticLaw()
        kinetic_law.setFormula("k * id_i2za * id_z877")
        
        # Add parameters
        parameter = kinetic_law.createParameter()
        parameter.setId("k")
        parameter.setValue(k)
        
        # Write the updated SBML model
        writer = libsbml.SBMLWriter()
        test_sbml = writer.writeSBMLToString(sbml_doc)
        
        # Simulate the model
        test_simulation_result = simulate(test_sbml)
        
        # Calculate R-squared
        f_i2za_test = interp1d(test_simulation_result['Time'], test_simulation_result['id_i2za'], bounds_error=False, fill_value="extrapolate")
        f_z877_test = interp1d(test_simulation_result['Time'], test_simulation_result['id_z877'], bounds_error=False, fill_value="extrapolate")
        
        sim_i2za_interp_test = f_i2za_test(data1['Time'])
        sim_z877_interp_test = f_z877_test(data1['Time'])
        
        ss_residual_i2za_test = np.sum((data1['id_i2za'] - sim_i2za_interp_test)**2)
        r_squared_i2za_test = 1 - (ss_residual_i2za_test / ss_total_i2za)
        
        ss_residual_z877_test = np.sum((data1['id_z877'] - sim_z877_interp_test)**2)
        r_squared_z877_test = 1 - (ss_residual_z877_test / ss_total_z877)
        
        print(f"Stoichiometry = {n}, k = {k}:")
        print(f"  R-squared for id_i2za: {r_squared_i2za_test}")
        print(f"  R-squared for id_z877: {r_squared_z877_test}")
        
        # Update best parameters if better fit is found
        if r_squared_i2za_test > best_r_squared_i2za:
            best_r_squared_i2za = r_squared_i2za_test
            best_stoichiometry_i2za = n
            best_rate_constant_i2za = k
        
        if r_squared_z877_test > best_r_squared_z877:
            best_r_squared_z877 = r_squared_z877_test
            best_stoichiometry_z877 = n
            best_rate_constant_z877 = k

print(f"\nBest parameters for id_i2za: stoichiometry = {best_stoichiometry_i2za}, k = {best_rate_constant_i2za}, R-squared = {best_r_squared_i2za}")
print(f"Best parameters for id_z877: stoichiometry = {best_stoichiometry_z877}, k = {best_rate_constant_z877}, R-squared = {best_r_squared_z877}")

# Create the final SBML model with the best parameters
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Add the reaction: id_i2za + id_z877 -> n id_i2za
reaction = model.createReaction()
reaction.setId("reaction1")
reaction.setReversible(False)
reaction.setFast(False)

# Add reactants
reactant1 = reaction.createReactant()
reactant1.setSpecies("id_i2za")
reactant1.setStoichiometry(1.0)
reactant1.setConstant(True)

reactant2 = reaction.createReactant()
reactant2.setSpecies("id_z877")
reactant2.setStoichiometry(1.0)
reactant2.setConstant(True)

# Add products
product = reaction.createProduct()
product.setSpecies("id_i2za")
product.setStoichiometry(float(best_stoichiometry_i2za))  # Using the best value for id_i2za
product.setConstant(True)

# Add kinetic law
kinetic_law = reaction.createKineticLaw()
kinetic_law.setFormula("k * id_i2za * id_z877")

# Add parameters
parameter = kinetic_law.createParameter()
parameter.setId("k")
parameter.setValue(best_rate_constant_i2za)  # Using the best value for id_i2za

# Write the final SBML model
writer = libsbml.SBMLWriter()
final_sbml = writer.writeSBMLToString(sbml_doc)

# Store the final SBML model
shared_variables.add("final_sbml_best", final_sbml)

print("\nFinal SBML model with best parameters:")
print(final_sbml)
```
# Observation

## Code Stdout
Simulating the current SBML model...
First few rows of simulation result:
       id_i2za      id_z877      Time
0   100.000000  1000.000000  0.000000
1   978.252690   121.746871  0.019938
2  1098.298419     1.701082  0.039877
3  1099.978299     0.021201  0.059815
4  1099.999236     0.000264  0.079753

Last few rows of simulation result:
        id_i2za       id_z877       Time
4209  1099.9995  2.262084e-21  83.920247
4210  1099.9995  2.261535e-21  83.940185
4211  1099.9995  2.260986e-21  83.960123
4212  1099.9995  2.260438e-21  83.980062
4213  1099.9995  2.259889e-21  84.000000

R-squared for id_i2za: -1.5790114280029455
R-squared for id_z877: -0.3056482854640761

New SBML model with high stoichiometric coefficient:
<?xml version="1.0" encoding="UTF-8"?>
<sbml xmlns="http://www.sbml.org/sbml/level3/version1/core" xmlns:layout="http://www.sbml.org/sbml/level3/version1/layout/version1" xmlns:render="http://www.sbml.org/sbml/level3/version1/render/version1" level="3" version="1" layout:required="false" render:required="false">
  <model id="id_k9sa" substanceUnits="id_yxmk" timeUnits="id_pcnf" volumeUnits="id_29tc" areaUnits="id_el2n" lengthUnits="id_nfym" extentUnits="id_yxmk">
    <listOfUnitDefinitions>
      <unitDefinition id="id_nfym">
        <listOfUnits>
          <unit kind="metre" exponent="1" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_el2n">
        <listOfUnits>
          <unit kind="metre" exponent="2" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_29tc">
        <listOfUnits>
          <unit kind="litre" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_pcnf">
        <listOfUnits>
          <unit kind="second" exponent="1" scale="0" multiplier="86400"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_yxmk">
        <listOfUnits>
          <unit kind="mole" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_zx0m">
        <listOfUnits>
          <unit kind="second" exponent="-1" scale="3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_ept0">
        <listOfUnits>
          <unit kind="dimensionless" exponent="0" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
    </listOfUnitDefinitions>
    <listOfCompartments>
      <compartment id="id_ez0l" spatialDimensions="3" size="1" units="id_29tc" constant="true"/>
    </listOfCompartments>
    <listOfSpecies>
      <species id="id_i2za" compartment="id_ez0l" initialConcentration="100" substanceUnits="id_yxmk" hasOnlySubstanceUnits="false" boundaryCondition="false" constant="false"/>
      <species id="id_z877" compartment="id_ez0l" initialConcentration="1000" substanceUnits="id_yxmk" hasOnlySubstanceUnits="false" boundaryCondition="false" constant="false"/>
    </listOfSpecies>
    <listOfReactions>
      <reaction id="reaction1" reversible="false" fast="false">
        <listOfReactants>
          <speciesReference species="id_i2za" stoichiometry="1" constant="true"/>
          <speciesReference species="id_z877" stoichiometry="1" constant="true"/>
        </listOfReactants>
        <listOfProducts>
          <speciesReference species="id_i2za" stoichiometry="1000" constant="true"/>
        </listOfProducts>
        <kineticLaw>
          <math xmlns="http://www.w3.org/1998/Math/MathML">
            <apply>
              <times/>
              <ci> k </ci>
              <ci> id_i2za </ci>
              <ci> id_z877 </ci>
            </apply>
          </math>
          <listOfLocalParameters>
            <localParameter id="k" value="1e-06"/>
          </listOfLocalParameters>
        </kineticLaw>
      </reaction>
    </listOfReactions>
  </model>
</sbml>


Simulating the new SBML model...
First few rows of new simulation result:
      id_i2za      id_z877      Time
0  100.000000  1000.000000  0.000000
1  102.012026   999.997986  0.019938
2  104.064354   999.995932  0.039877
3  106.157954   999.993836  0.059815
4  108.293642   999.991698  0.079753

Last few rows of new simulation result:
       id_i2za       id_z877       Time
4209  999100.0  1.724249e-10  83.920247
4210  999100.0  1.724619e-10  83.940185
4211  999100.0  1.724934e-10  83.960123
4212  999100.0  1.725192e-10  83.980062
4213  999100.0  1.725394e-10  84.000000

R-squared for id_i2za (new model): -1.5735496946993162
R-squared for id_z877 (new model): 0.5441913539576276

Simulating another new SBML model...

R-squared for id_i2za (another new model): -1.5784747387543847
R-squared for id_z877 (another new model): 0.37324226916969117
Stoichiometry = 10, k = 1e-07:
  R-squared for id_i2za: -1.579016874847475
  R-squared for id_z877: -4.893515302816126
Stoichiometry = 10, k = 1e-06:
  R-squared for id_i2za: -1.5790165137580328
  R-squared for id_z877: -4.819315925487114
Stoichiometry = 10, k = 1e-05:
  R-squared for id_i2za: -1.5789847718996586
  R-squared for id_z877: -0.8978759303843522
Stoichiometry = 10, k = 0.0001:
  R-squared for id_i2za: -1.5789676135443869
  R-squared for id_z877: 0.22054620071887454
Stoichiometry = 50, k = 1e-07:
  R-squared for id_i2za: -1.5790167188507818
  R-squared for id_z877: -4.892706988227885
Stoichiometry = 50, k = 1e-06:
  R-squared for id_i2za: -1.5790055234468041
  R-squared for id_z877: -4.511771673195361
Stoichiometry = 50, k = 1e-05:
  R-squared for id_i2za: -1.5787485506811714
  R-squared for id_z877: 0.8376565550933213
Stoichiometry = 50, k = 0.0001:
  R-squared for id_i2za: -1.5787485503381102
  R-squared for id_z877: -0.1748282961557286
Stoichiometry = 100, k = 1e-07:
  R-squared for id_i2za: -1.5790164587357656
  R-squared for id_z877: -4.891474036962914
Stoichiometry = 100, k = 1e-06:
  R-squared for id_i2za: -1.5788390000320045
  R-squared for id_z877: -2.6640377424611534
Stoichiometry = 100, k = 1e-05:
  R-squared for id_i2za: -1.5784747387543847
  R-squared for id_z877: 0.37324226916969117
Stoichiometry = 100, k = 0.0001:
  R-squared for id_i2za: -1.5784747426723693
  R-squared for id_z877: -0.23474041305127424
Stoichiometry = 500, k = 1e-07:
  R-squared for id_i2za: -1.5790041092197988
  R-squared for id_z877: -4.855983103896328
Stoichiometry = 500, k = 1e-06:
  R-squared for id_i2za: -1.5762850105959036
  R-squared for id_z877: 0.9901031108694219
Stoichiometry = 500, k = 1e-05:
  R-squared for id_i2za: -1.5762851116449004
  R-squared for id_z877: -0.1358437109217625
Stoichiometry = 500, k = 0.0001:
  R-squared for id_i2za: -1.5762851357906884
  R-squared for id_z877: -0.2896162711784567
Stoichiometry = 1000, k = 1e-07:
  R-squared for id_i2za: -1.5786153596266508
  R-squared for id_z877: -4.287713405722984
Stoichiometry = 1000, k = 1e-06:
  R-squared for id_i2za: -1.5735496946993162
  R-squared for id_z877: 0.5441913539576276
Stoichiometry = 1000, k = 1e-05:
  R-squared for id_i2za: -1.5735502118919151
  R-squared for id_z877: -0.21457432224371353
Stoichiometry = 1000, k = 0.0001:
  R-squared for id_i2za: -1.5735502637710028
  R-squared for id_z877: -0.29766804320372087

Best parameters for id_i2za: stoichiometry = 1000, k = 1e-06, R-squared = -1.5735496946993162
Best parameters for id_z877: stoichiometry = 500, k = 1e-06, R-squared = 0.9901031108694219

Final SBML model with best parameters:
<?xml version="1.0" encoding="UTF-8"?>
<sbml xmlns="http://www.sbml.org/sbml/level3/version1/core" xmlns:layout="http://www.sbml.org/sbml/level3/version1/layout/version1" xmlns:render="http://www.sbml.org/sbml/level3/version1/render/version1" level="3" version="1" layout:required="false" render:required="false">
  <model id="id_k9sa" substanceUnits="id_yxmk" timeUnits="id_pcnf" volumeUnits="id_29tc" areaUnits="id_el2n" lengthUnits="id_nfym" extentUnits="id_yxmk">
    <listOfUnitDefinitions>
      <unitDefinition id="id_nfym">
        <listOfUnits>
          <unit kind="metre" exponent="1" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_el2n">
        <listOfUnits>
          <unit kind="metre" exponent="2" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_29tc">
        <listOfUnits>
          <unit kind="litre" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_pcnf">
        <listOfUnits>
          <unit kind="second" exponent="1" scale="0" multiplier="86400"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_yxmk">
        <listOfUnits>
          <unit kind="mole" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_zx0m">
        <listOfUnits>
          <unit kind="second" exponent="-1" scale="3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_ept0">
        <listOfUnits>
          <unit kind="dimensionless" exponent="0" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
    </listOfUnitDefinitions>
    <listOfCompartments>
      <compartment id="id_ez0l" spatialDimensions="3" size="1" units="id_29tc" constant="true"/>
    </listOfCompartments>
    <listOfSpecies>
      <species id="id_i2za" compartment="id_ez0l" initialConcentration="100" substanceUnits="id_yxmk" hasOnlySubstanceUnits="false" boundaryCondition="false" constant="false"/>
      <species id="id_z877" compartment="id_ez0l" initialConcentration="1000" substanceUnits="id_yxmk" hasOnlySubstanceUnits="false" boundaryCondition="false" constant="false"/>
    </listOfSpecies>
    <listOfReactions>
      <reaction id="reaction1" reversible="false" fast="false">
        <listOfReactants>
          <speciesReference species="id_i2za" stoichiometry="1" constant="true"/>
          <speciesReference species="id_z877" stoichiometry="1"
... [output truncated]
## Reminder

 ### Remaining Iterations for Interactions:
- Iterations: 9/20



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_5, iteration_7)
- `shared_variables`: all variables you have saved from the previous iterations. (k, k_z877, r_squared, r_squared_z877, k1, r_squared1, r2, K2, r_squared2, k3, r_squared3, r_squared_i2za_sim, r_squared_z877_sim, k_z877_combined, r_squared1_z877, r_squared2_z877, n_fitted, r_squared1_i2za, r_squared2_i2za, n_reg, r_squared1_reg, r_squared2_reg, r_squared1_i2za_sim, r_squared1_z877_sim, r_squared2_i2za_sim, r_squared2_z877_sim, k_z877_all, r_squared1_z877_all, r_squared2_z877_all, r_squared3_z877_all, a, b, c, r_squared_ratio, k1_model1, K_model1, r_squared1_i2za_model1, r_squared2_i2za_model1, r_squared3_i2za_model1, r_squared1_i2za_sim_model1, r_squared1_z877_sim_model1, r_squared2_i2za_sim_model1, r_squared2_z877_sim_model1, r_squared3_i2za_sim_model1, r_squared3_z877_sim_model1, k_z877_low, k_z877_medium, k_z877_high, r_squared_low, r_squared_medium, r_squared_high, k1_low, k2_low, k1_medium, k2_medium, k1_high, k2_high, r_squared_low_i2za, r_squared_medium_i2za, r_squared_high_i2za, final_sbml, final_sbml_best)

### Allowed libraires
['numpy', 'pandas', 'libsbml', 'math', 'scipy', 'jax', 'sklearn', 'io', 'traceback']


Please construct your response according to the thoughts-action markdown format.



# Iteration 12
## Thoughts
I've made significant progress in understanding the reaction mechanism. After testing various models with different stoichiometric coefficients and rate constants, I've found that:

1. For `id_z877`, the best model has a stoichiometric coefficient of 500 and a rate constant of 1e-6, with an R-squared value of 0.99, which is excellent.

2. For `id_i2za`, the best model has a stoichiometric coefficient of 1000 and a rate constant of 1e-6, but the R-squared value is still negative (-1.57), indicating a poor fit.

The high stoichiometric coefficient suggests that the reaction involves a significant amplification of `id_i2za` for each molecule of `id_z877` consumed. This is consistent with the experimental data showing a rapid increase in `id_i2za` and a corresponding decrease in `id_z877`.

However, the poor fit for `id_i2za` suggests that there might be additional factors affecting its dynamics that aren't captured by our current model. Let me try a few more approaches:

1. Let's try a model with a carrying capacity for `id_i2za`, which might explain why it plateaus at around 9.8e+08.
2. Let's also try a model with a different functional form for the rate law, such as Hill kinetics, which might better capture the observed dynamics.

## Action

### Code
```python
import numpy as np
import pandas as pd
import libsbml
from scipy.integrate import solve_ivp

# Get the data from all three experiments
data1 = experiment_history["iteration_0"]  # Original experiment
data2 = experiment_history["iteration_5"]  # Second experiment
data3 = experiment_history["iteration_7"]  # Third experiment

# Let's try a model with a carrying capacity for id_i2za
# Reaction: id_i2za + id_z877 -> n id_i2za
# Rate law: k * id_i2za * id_z877 * (1 - id_i2za/K)
# where n is the stoichiometric coefficient, k is the rate constant, and K is the carrying capacity

# Create a new SBML model
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Add the reaction: id_i2za + id_z877 -> n id_i2za
reaction = model.createReaction()
reaction.setId("reaction1")
reaction.setReversible(False)
reaction.setFast(False)

# Add reactants
reactant1 = reaction.createReactant()
reactant1.setSpecies("id_i2za")
reactant1.setStoichiometry(1.0)
reactant1.setConstant(True)

reactant2 = reaction.createReactant()
reactant2.setSpecies("id_z877")
reactant2.setStoichiometry(1.0)
reactant2.setConstant(True)

# Add products
product = reaction.createProduct()
product.setSpecies("id_i2za")
product.setStoichiometry(500.0)  # Using the best value for id_z877
product.setConstant(True)

# Add kinetic law with carrying capacity
kinetic_law = reaction.createKineticLaw()
kinetic_law.setFormula("k * id_i2za * id_z877 * (1 - id_i2za/K)")

# Add parameters
parameter1 = kinetic_law.createParameter()
parameter1.setId("k")
parameter1.setValue(1e-6)  # Using the best value for id_z877

parameter2 = kinetic_law.createParameter()
parameter2.setId("K")
parameter2.setValue(1e9)  # Setting carrying capacity to 1e9, close to the observed plateau

# Write the updated SBML model
writer = libsbml.SBMLWriter()
carrying_capacity_sbml = writer.writeSBMLToString(sbml_doc)

# Print the new SBML model
print("SBML model with carrying capacity:")
print(carrying_capacity_sbml)

# Simulate the model with carrying capacity
print("\nSimulating the model with carrying capacity...")
carrying_capacity_simulation = simulate(carrying_capacity_sbml)

# Print the first few rows of the simulation result
print("First few rows of simulation result:")
print(carrying_capacity_simulation.head())

# Print the last few rows of the simulation result
print("\nLast few rows of simulation result:")
print(carrying_capacity_simulation.tail())

# Calculate R-squared for the simulation result compared to the original experiment
from scipy.interpolate import interp1d

# Create interpolation functions
f_i2za_cc = interp1d(carrying_capacity_simulation['Time'], carrying_capacity_simulation['id_i2za'], bounds_error=False, fill_value="extrapolate")
f_z877_cc = interp1d(carrying_capacity_simulation['Time'], carrying_capacity_simulation['id_z877'], bounds_error=False, fill_value="extrapolate")

# Interpolate to match the time points in the original experiment
sim_i2za_interp_cc = f_i2za_cc(data1['Time'])
sim_z877_interp_cc = f_z877_cc(data1['Time'])

# Calculate R-squared
ss_total_i2za = np.sum((data1['id_i2za'] - np.mean(data1['id_i2za']))**2)
ss_residual_i2za_cc = np.sum((data1['id_i2za'] - sim_i2za_interp_cc)**2)
r_squared_i2za_cc = 1 - (ss_residual_i2za_cc / ss_total_i2za)

ss_total_z877 = np.sum((data1['id_z877'] - np.mean(data1['id_z877']))**2)
ss_residual_z877_cc = np.sum((data1['id_z877'] - sim_z877_interp_cc)**2)
r_squared_z877_cc = 1 - (ss_residual_z877_cc / ss_total_z877)

print(f"\nR-squared for id_i2za (carrying capacity model): {r_squared_i2za_cc}")
print(f"R-squared for id_z877 (carrying capacity model): {r_squared_z877_cc}")

# Let's try different values for the carrying capacity
carrying_capacities = [5e8, 8e8, 9e8, 9.5e8, 9.8e8, 1e9, 1.1e9]
rate_constants = [1e-7, 5e-7, 1e-6, 5e-6, 1e-5]
stoichiometries = [100, 500, 1000]

best_r_squared_i2za_cc = -float('inf')
best_r_squared_z877_cc = -float('inf')
best_K = None
best_k_cc = None
best_n_cc = None

for K in carrying_capacities:
    for k in rate_constants:
        for n in stoichiometries:
            # Create a new SBML model
            sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
            model = sbml_doc.getModel()
            
            # Add the reaction: id_i2za + id_z877 -> n id_i2za
            reaction = model.createReaction()
            reaction.setId("reaction1")
            reaction.setReversible(False)
            reaction.setFast(False)
            
            # Add reactants
            reactant1 = reaction.createReactant()
            reactant1.setSpecies("id_i2za")
            reactant1.setStoichiometry(1.0)
            reactant1.setConstant(True)
            
            reactant2 = reaction.createReactant()
            reactant2.setSpecies("id_z877")
            reactant2.setStoichiometry(1.0)
            reactant2.setConstant(True)
            
            # Add products
            product = reaction.createProduct()
            product.setSpecies("id_i2za")
            product.setStoichiometry(float(n))
            product.setConstant(True)
            
            # Add kinetic law with carrying capacity
            kinetic_law = reaction.createKineticLaw()
            kinetic_law.setFormula("k * id_i2za * id_z877 * (1 - id_i2za/K)")
            
            # Add parameters
            parameter1 = kinetic_law.createParameter()
            parameter1.setId("k")
            parameter1.setValue(k)
            
            parameter2 = kinetic_law.createParameter()
            parameter2.setId("K")
            parameter2.setValue(K)
            
            # Write the updated SBML model
            writer = libsbml.SBMLWriter()
            test_sbml = writer.writeSBMLToString(sbml_doc)
            
            # Simulate the model
            test_simulation = simulate(test_sbml)
            
            # Calculate R-squared
            f_i2za_test = interp1d(test_simulation['Time'], test_simulation['id_i2za'], bounds_error=False, fill_value="extrapolate")
            f_z877_test = interp1d(test_simulation['Time'], test_simulation['id_z877'], bounds_error=False, fill_value="extrapolate")
            
            sim_i2za_interp_test = f_i2za_test(data1['Time'])
            sim_z877_interp_test = f_z877_test(data1['Time'])
            
            ss_residual_i2za_test = np.sum((data1['id_i2za'] - sim_i2za_interp_test)**2)
            r_squared_i2za_test = 1 - (ss_residual_i2za_test / ss_total_i2za)
            
            ss_residual_z877_test = np.sum((data1['id_z877'] - sim_z877_interp_test)**2)
            r_squared_z877_test = 1 - (ss_residual_z877_test / ss_total_z877)
            
            print(f"K = {K}, k = {k}, n = {n}:")
            print(f"  R-squared for id_i2za: {r_squared_i2za_test}")
            print(f"  R-squared for id_z877: {r_squared_z877_test}")
            
            # Update best parameters if better fit is found
            if r_squared_i2za_test > best_r_squared_i2za_cc:
                best_r_squared_i2za_cc = r_squared_i2za_test
                best_K_i2za = K
                best_k_i2za_cc = k
                best_n_i2za_cc = n
            
            if r_squared_z877_test > best_r_squared_z877_cc:
                best_r_squared_z877_cc = r_squared_z877_test
                best_K_z877 = K
                best_k_z877_cc = k
                best_n_z877_cc = n

print(f"\nBest parameters for id_i2za (carrying capacity model): K = {best_K_i2za}, k = {best_k_i2za_cc}, n = {best_n_i2za_cc}, R-squared = {best_r_squared_i2za_cc}")
print(f"Best parameters for id_z877 (carrying capacity model): K = {best_K_z877}, k = {best_k_z877_cc}, n = {best_n_z877_cc}, R-squared = {best_r_squared_z877_cc}")

# Create the final SBML model with the best parameters
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Add the reaction: id_i2za + id_z877 -> n id_i2za
reaction = model.createReaction()
reaction.setId("reaction1")
reaction.setReversible(False)
reaction.setFast(False)

# Add reactants
reactant1 = reaction.createReactant()
reactant1.setSpecies("id_i2za")
reactant1.setStoichiometry(1.0)
reactant1.setConstant(True)

reactant2 = reaction.createReactant()
reactant2.setSpecies("id_z877")
reactant2.setStoichiometry(1.0)
reactant2.setConstant(True)

# Add products
product = reaction.createProduct()
product.setSpecies("id_i2za")
product.setStoichiometry(float(best_n_i2za_cc))  # Using the best value for id_i2za
product.setConstant(True)

# Add kinetic law with carrying capacity
kinetic_law = reaction.createKineticLaw()
kinetic_law.setFormula("k * id_i2za * id_z877 * (1 - id_i2za/K)")

# Add parameters
parameter1 = kinetic_law.createParameter()
parameter1.setId("k")
parameter1.setValue(best_k_i2za_cc)  # Using the best value for id_i2za

parameter2 = kinetic_law.createParameter()
parameter2.setId("K")
parameter2.setValue(best_K_i2za)  # Using the best value for id_i2za

# Write the final SBML model
writer = libsbml.SBMLWriter()
final_sbml = writer.writeSBMLToString(sbml_doc)

# Store the final SBML model
shared_variables.add("final_sbml_cc", final_sbml)

print("\nFinal SBML model with best parameters (carrying capacity model):")
print(final_sbml)

# Let's validate the final model by simulating it for all three experiments
final_simulation1 = simulate(final_sbml)  # Using the default initial conditions (experiment 1)

# Create a modified SBML model for experiment 2
sbml_doc = libsbml.readSBMLFromString(final_sbml)
model = sbml_doc.getModel()
species_i2za = model.getSpecies("id_i2za")
species_z877 = model.getSpecies("id_z877")
species_i2za.setInitialConcentration(50)  # Initial concentration for experiment 2
species_z877.setInitialConcentration(2000)  # Initial concentration for experiment 2
writer = libsbml.SBMLWriter()
final_sbml_exp2 = writer.writeSBMLToString(sbml_doc)
final_simulation2 = simulate(final_sbml_exp2)

# Create a modified SBML model for experiment 3
sbml_doc = libsbml.readSBMLFromString(final_sbml)
model = sbml_doc.getModel()
species_i2za = model.getSpecies("id_i2za")
species_z877 = model.getSpecies("id_z877")
species_i2za.setInitialConcentration(200)  # Initial concentration for experiment 3
species_z877.setInitialConcentration(500)  # Initial concentration for experiment 3
writer = libsbml.SBMLWriter()
final_sbml_exp3 = writer.writeSBMLToString(sbml_doc)
final_simulation3 = simulate(final_sbml_exp3)

# Calculate R-squared for all three experiments
# Experiment 1
f_i2za_final1 = interp1d(final_simulation1['Time'], final_simulation1['id_i2za'], bounds_error=False, fill_value="extrapolate")
f_z877_final1 = interp1d(final_simulation1['Time'], final_simulation1['id_z877'], bounds_error=False, fill_value="extrapolate")
sim_i2za_interp_final1 = f_i2za_final1(data1['Time'])
sim_z877_interp_final1 = f_z877_final1(data1['Time'])
ss_total_i2za1 = np.sum((data1['id_i2za'] - np.mean(data1['id_i2za']))**2)
ss_residual_i2za_final1 = np.sum((data1['id_i2za'] - sim_i2za_interp_final1)**2)
r_squared_i2za_final1 = 1 - (ss_residual_i2za_final1 / ss_total_i2za1)
ss_total_z8771 = np.sum((data1['id_z877'] - np.mean(data1['id_z877']))**2)
ss_residual_z877_final1 = np.sum((data1['id_z877'] - sim_z877_interp_final1)**2)
r_squared_z877_final1 = 1 - (ss_residual_z877_final1 / ss_total_z8771)

# Experiment 2
f_i2za_final2 = interp1d(final_simulation2['Time'], final_simulation2['id_i2za'], bounds_error=False, fill_value="extrapolate")
f_z877_final2 = interp1d(final_simulation2['Time'], final_simulation2['id_z877'], bounds_error=False, fill_value="extrapolate")
sim_i2za_interp_final2 = f_i2za_final2(data2['Time'])
sim_z877_interp_final2 = f_z877_final2(data2['Time'])
ss_total_i2za2 = np.sum((data2['id_i2za'] - np.mean(data2['id_i2za']))**2)
ss_residual_i2za_final2 = np.sum((data2['id_i2za'] - sim_i2za_interp_final2)**2)
r_squared_i2za_final2 = 1 - (ss_residual_i2za_final2 / ss_total_i2za2)
ss_total_z8772 = np.sum((data2['id_z877'] - np.mean(data2['id_z877']))**2)
ss_residual_z877_final2 = np.sum((data2['id_z877'] - sim_z877_interp_final2)**2)
r_squared_z877_final2 = 1 - (ss_residual_z877_final2 / ss_total_z8772)

# Experiment 3
f_i2za_final3 = interp1d(final_simulation3['Time'], final_simulation3['id_i2za'], bounds_error=False, fill_value="extrapolate")
f_z877_final3 = interp1d(final_simulation3['Time'], final_simulation3['id_z877'], bounds_error=False, fill_value="extrapolate")
sim_i2za_interp_final3 = f_i2za_final3(data3['Time'])
sim_z877_interp_final3 = f_z877_final3(data3['Time'])
ss_total_i2za3 = np.sum((data3['id_i2za'] - np.mean(data3['id_i2za']))**2)
ss_residual_i2za_final3 = np.sum((data3['id_i2za'] - sim_i2za_interp_final3)**2)
r_squared_i2za_final3 = 1 - (ss_residual_i2za_final3 / ss_total_i2za3)
ss_total_z8773 = np.sum((data3['id_z877'] - np.mean(data3['id_z877']))**2)
ss_residual_z877_final3 = np.sum((data3['id_z877'] - sim_z877_interp_final3)**2)
r_squared_z877_final3 = 1 - (ss_residual_z877_final3 / ss_total_z8773)

print("\nValidation results:")
print(f"Experiment 1 - R-squared for id_i2za: {r_squared_i2za_final1}, R-squared for id_z877: {r_squared_z877_final1}")
print(f"Experiment 2 - R-squared for id_i2za: {r_squared_i2za_final2}, R-squared for id_z877: {r_squared_z877_final2}")
print(f"Experiment 3 - R-squared for id_i2za: {r_squared_i2za_final3}, R-squared for id_z877: {r_squared_z877_final3}")

# Calculate the average R-squared across all experiments
avg_r_squared_i2za = (r_squared_i2za_final1 + r_squared_i2za_final2 + r_squared_i2za_final3) / 3
avg_r_squared_z877 = (r_squared_z877_final1 + r_squared_z877_final2 + r_squared_z877_final3) / 3
print(f"Average - R-squared for id_i2za: {avg_r_squared_i2za}, R-squared for id_z877: {avg_r_squared_z877}")

# Define the final model
final_sbml = final_sbml
```
# Observation

## Code Stdout
SBML model with carrying capacity:
<?xml version="1.0" encoding="UTF-8"?>
<sbml xmlns="http://www.sbml.org/sbml/level3/version1/core" xmlns:layout="http://www.sbml.org/sbml/level3/version1/layout/version1" xmlns:render="http://www.sbml.org/sbml/level3/version1/render/version1" level="3" version="1" layout:required="false" render:required="false">
  <model id="id_k9sa" substanceUnits="id_yxmk" timeUnits="id_pcnf" volumeUnits="id_29tc" areaUnits="id_el2n" lengthUnits="id_nfym" extentUnits="id_yxmk">
    <listOfUnitDefinitions>
      <unitDefinition id="id_nfym">
        <listOfUnits>
          <unit kind="metre" exponent="1" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_el2n">
        <listOfUnits>
          <unit kind="metre" exponent="2" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_29tc">
        <listOfUnits>
          <unit kind="litre" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_pcnf">
        <listOfUnits>
          <unit kind="second" exponent="1" scale="0" multiplier="86400"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_yxmk">
        <listOfUnits>
          <unit kind="mole" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_zx0m">
        <listOfUnits>
          <unit kind="second" exponent="-1" scale="3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_ept0">
        <listOfUnits>
          <unit kind="dimensionless" exponent="0" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
    </listOfUnitDefinitions>
    <listOfCompartments>
      <compartment id="id_ez0l" spatialDimensions="3" size="1" units="id_29tc" constant="true"/>
    </listOfCompartments>
    <listOfSpecies>
      <species id="id_i2za" compartment="id_ez0l" initialConcentration="100" substanceUnits="id_yxmk" hasOnlySubstanceUnits="false" boundaryCondition="false" constant="false"/>
      <species id="id_z877" compartment="id_ez0l" initialConcentration="1000" substanceUnits="id_yxmk" hasOnlySubstanceUnits="false" boundaryCondition="false" constant="false"/>
    </listOfSpecies>
    <listOfReactions>
      <reaction id="reaction1" reversible="false" fast="false">
        <listOfReactants>
          <speciesReference species="id_i2za" stoichiometry="1" constant="true"/>
          <speciesReference species="id_z877" stoichiometry="1" constant="true"/>
        </listOfReactants>
        <listOfProducts>
          <speciesReference species="id_i2za" stoichiometry="500" constant="true"/>
        </listOfProducts>
        <kineticLaw>
          <math xmlns="http://www.w3.org/1998/Math/MathML">
            <apply>
              <times/>
              <ci> k </ci>
              <ci> id_i2za </ci>
              <ci> id_z877 </ci>
              <apply>
                <minus/>
                <cn type="integer"> 1 </cn>
                <apply>
                  <divide/>
                  <ci> id_i2za </ci>
                  <ci> K </ci>
                </apply>
              </apply>
            </apply>
          </math>
          <listOfLocalParameters>
            <localParameter id="k" value="1e-06"/>
            <localParameter id="K" value="1000000000"/>
          </listOfLocalParameters>
        </kineticLaw>
      </reaction>
    </listOfReactions>
  </model>
</sbml>


Simulating the model with carrying capacity...
First few rows of simulation result:
      id_i2za      id_z877      Time
0  100.000000  1000.000000  0.000000
1  101.000026   999.997996  0.019938
2  102.009926   999.995972  0.039877
3  103.029923   999.993928  0.059815
4  104.060133   999.991863  0.079753

Last few rows of simulation result:
       id_i2za       id_z877       Time
4209  499100.0 -8.681317e-12  83.920247
4210  499100.0 -9.475996e-12  83.940185
4211  499100.0 -1.026793e-11  83.960123
4212  499100.0 -1.105706e-11  83.980062
4213  499100.0 -1.184329e-11  84.000000

R-squared for id_i2za (carrying capacity model): -1.5762850108938706
R-squared for id_z877 (carrying capacity model): 0.9901121281790949
K = 500000000.0, k = 1e-07, n = 100:
  R-squared for id_i2za: -1.5790164587359348
  R-squared for id_z877: -4.8914740400217465
K = 500000000.0, k = 1e-07, n = 500:
  R-squared for id_i2za: -1.579004109311704
  R-squared for id_z877: -4.855983405989519
K = 500000000.0, k = 1e-07, n = 1000:
  R-squared for id_i2za: -1.5786154762899396
  R-squared for id_z877: -4.287865819203447
K = 500000000.0, k = 5e-07, n = 100:
  R-squared for id_i2za: -1.5790047741324136
  R-squared for id_z877: -4.692615739219683
K = 500000000.0, k = 5e-07, n = 500:
  R-squared for id_i2za: -1.576445087800669
  R-squared for id_z877: 0.0005736218636160917
K = 500000000.0, k = 5e-07, n = 1000:
  R-squared for id_i2za: -1.573549631979649
  R-squared for id_z877: 0.9893466854688693
K = 500000000.0, k = 1e-06, n = 100:
  R-squared for id_i2za: -1.5788390107493218
  R-squared for id_z877: -2.664114416930809
K = 500000000.0, k = 1e-06, n = 500:
  R-squared for id_i2za: -1.5762850111928484
  R-squared for id_z877: 0.9901211271304183
K = 500000000.0, k = 1e-06, n = 1000:
  R-squared for id_i2za: -1.5735496945323213
  R-squared for id_z877: 0.5444210508783205
K = 500000000.0, k = 5e-06, n = 100:
  R-squared for id_i2za: -1.5784747415598042
  R-squared for id_z877: 0.9015248962905735
K = 500000000.0, k = 5e-06, n = 500:
  R-squared for id_i2za: -1.5762850848237933
  R-squared for id_z877: 0.029871139375666877
K = 500000000.0, k = 5e-06, n = 1000:
  R-squared for id_i2za: -1.5735501542217496
  R-squared for id_z877: -0.12371844357268458
K = 500000000.0, k = 1e-05, n = 100:
  R-squared for id_i2za: -1.5784747387542297
  R-squared for id_z877: 0.3732664179730929
K = 500000000.0, k = 1e-05, n = 500:
  R-squared for id_i2za: -1.5762851116406704
  R-squared for id_z877: -0.13581702609532798
K = 500000000.0, k = 1e-05, n = 1000:
  R-squared for id_i2za: -1.5735502118749096
  R-squared for id_z877: -0.21454722026452977
K = 800000000.0, k = 1e-07, n = 100:
  R-squared for id_i2za: -1.5790164587358717
  R-squared for id_z877: -4.8914740388746845
K = 800000000.0, k = 1e-07, n = 500:
  R-squared for id_i2za: -1.57900410927724
  R-squared for id_z877: -4.855983292705201
K = 800000000.0, k = 1e-07, n = 1000:
  R-squared for id_i2za: -1.57861543254973
  R-squared for id_z877: -4.287808673101406
K = 800000000.0, k = 5e-07, n = 100:
  R-squared for id_i2za: -1.5790047741023656
  R-squared for id_z877: -4.692615256937956
K = 800000000.0, k = 5e-07, n = 500:
  R-squared for id_i2za: -1.5764450223132376
  R-squared for id_z877: 0.0006419653710347806
K = 800000000.0, k = 5e-07, n = 1000:
  R-squared for id_i2za: -1.5735496304615144
  R-squared for id_z877: 0.9893618910748924
K = 800000000.0, k = 1e-06, n = 100:
  R-squared for id_i2za: -1.5788390067303206
  R-squared for id_z877: -2.6640856631597933
K = 800000000.0, k = 1e-06, n = 500:
  R-squared for id_i2za: -1.5762850109686108
  R-squared for id_z877: 0.9901143796388121
K = 800000000.0, k = 1e-06, n = 1000:
  R-squared for id_i2za: -1.5735496945947247
  R-squared for id_z877: 0.5443352075029819
K = 800000000.0, k = 5e-06, n = 100:
  R-squared for id_i2za: -1.5784747415571019
  R-squared for id_z877: 0.9015166562963829
K = 800000000.0, k = 5e-06, n = 500:
  R-squared for id_i2za: -1.5762850848269578
  R-squared for id_z877: 0.029851705292240238
K = 800000000.0, k = 5e-06, n = 1000:
  R-squared for id_i2za: -1.5735501542344585
  R-squared for id_z877: -0.12373840626510368
K = 800000000.0, k = 1e-05, n = 100:
  R-squared for id_i2za: -1.578474738754288
  R-squared for id_z877: 0.37325738388794216
K = 800000000.0, k = 1e-05, n = 500:
  R-squared for id_i2za: -1.5762851116422536
  R-squared for id_z877: -0.13582701508883366
K = 800000000.0, k = 1e-05, n = 1000:
  R-squared for id_i2za: -1.573550211881265
  R-squared for id_z877: -0.2145573486129997
K = 900000000.0, k = 1e-07, n = 100:
  R-squared for id_i2za: -1.5790164587358597
  R-squared for id_z877: -4.891474038662265
K = 900000000.0, k = 1e-07, n = 500:
  R-squared for id_i2za: -1.5790041092708575
  R-squared for id_z877: -4.855983271726513
K = 900000000.0, k = 1e-07, n = 1000:
  R-squared for id_i2za: -1.5786154244485684
  R-squared for id_z877: -4.287798089310498
K = 900000000.0, k = 5e-07, n = 100:
  R-squared for id_i2za: -1.5790047740968007
  R-squared for id_z877: -4.69261516762621
K = 900000000.0, k = 5e-07, n = 500:
  R-squared for id_i2za: -1.5764450101828764
  R-squared for id_z877: 0.0006546208790340069
K = 900000000.0, k = 5e-07, n = 1000:
  R-squared for id_i2za: -1.5735496301825567
  R-squared for id_z877: 0.9893647022079995
K = 900000000.0, k = 1e-06, n = 100:
  R-squared for id_i2za: -1.5788390059860595
  R-squared for id_z877: -2.66408033847316
K = 900000000.0, k = 1e-06, n = 500:
  R-squared for id_i2za: -1.5762850109271733
  R-squared for id_z877: 0.9901131289694551
K = 900000000.0, k = 1e-06, n = 1000:
  R-squared for id_i2za: -1.5735496946062915
  R-squared for id_z877: 0.5443192954228435
K = 900000000.0, k = 5e-06, n = 100:
  R-squared for id_i2za: -1.5784747415566014
  R-squared for id_z877: 0.9015151303647947
K = 900000000.0, k = 5e-06, n = 500:
  R-squared for id_i2za: -1.5762850848275431
  R-squared for id_z877: 0.029848106439694932
K = 900000000.0, k = 5e-06, n = 1000:
  R-squared for id_i2za: -1.573550154236814
  R-squared for id_z877: -0.12374210655188422
K = 900000000.0, k = 1e-05, n = 100:
  R-squared for id_i2za: -1.5784747387542986
  R-squared for id_z877: 0.3732557181894759
K = 900000000.0, k = 1e-05, n = 500:
  R-squared for id_i2za: -1.5762851116425467
  R-squared for id_z877: -0.13582886487581147
K = 900000000.0, k = 1e-05, n = 1000:
  R-squared for id_i2za: -1.5735502118824432
  R-squared for id_z877: -0.2145592260109601
K = 950000000.0, k =
... [output truncated]
## Reminder

 ### Remaining Iterations for Interactions:
- Iterations: 8/20



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_5, iteration_7)
- `shared_variables`: all variables you have saved from the previous iterations. (k, k_z877, r_squared, r_squared_z877, k1, r_squared1, r2, K2, r_squared2, k3, r_squared3, r_squared_i2za_sim, r_squared_z877_sim, k_z877_combined, r_squared1_z877, r_squared2_z877, n_fitted, r_squared1_i2za, r_squared2_i2za, n_reg, r_squared1_reg, r_squared2_reg, r_squared1_i2za_sim, r_squared1_z877_sim, r_squared2_i2za_sim, r_squared2_z877_sim, k_z877_all, r_squared1_z877_all, r_squared2_z877_all, r_squared3_z877_all, a, b, c, r_squared_ratio, k1_model1, K_model1, r_squared1_i2za_model1, r_squared2_i2za_model1, r_squared3_i2za_model1, r_squared1_i2za_sim_model1, r_squared1_z877_sim_model1, r_squared2_i2za_sim_model1, r_squared2_z877_sim_model1, r_squared3_i2za_sim_model1, r_squared3_z877_sim_model1, k_z877_low, k_z877_medium, k_z877_high, r_squared_low, r_squared_medium, r_squared_high, k1_low, k2_low, k1_medium, k2_medium, k1_high, k2_high, r_squared_low_i2za, r_squared_medium_i2za, r_squared_high_i2za, final_sbml, final_sbml_best, final_sbml_cc)

### Allowed libraires
['numpy', 'pandas', 'libsbml', 'math', 'scipy', 'jax', 'sklearn', 'io', 'traceback']


Please construct your response according to the thoughts-action markdown format.



# Iteration 13
## Thoughts
I've made significant progress in understanding the reaction mechanism. The model with a carrying capacity has shown promising results, especially for `id_z877` with an R-squared value of 0.99, which is excellent. However, the fit for `id_i2za` is still poor with negative R-squared values.

The output was truncated, so I don't have the complete results of the parameter sweep or the validation results. Let me continue exploring the model with a carrying capacity, focusing on improving the fit for `id_i2za`.

Based on the partial results I've seen, a model with a stoichiometric coefficient of 500, a rate constant of 1e-6, and a carrying capacity of 5e8 seems to provide a good fit for `id_z877`. Let me try to refine this model further and validate it against all three experiments.

## Action

### Code
```python
import numpy as np
import pandas as pd
import libsbml
from scipy.integrate import solve_ivp
from scipy.interpolate import interp1d

# Get the data from all three experiments
data1 = experiment_history["iteration_0"]  # Original experiment
data2 = experiment_history["iteration_5"]  # Second experiment
data3 = experiment_history["iteration_7"]  # Third experiment

# Let's try a model with a carrying capacity for id_i2za
# Reaction: id_i2za + id_z877 -> n id_i2za
# Rate law: k * id_i2za * id_z877 * (1 - id_i2za/K)
# where n is the stoichiometric coefficient, k is the rate constant, and K is the carrying capacity

# Based on the previous results, let's focus on a narrower range of parameters
carrying_capacities = [9.5e8, 9.8e8, 1e9]
rate_constants = [5e-7, 1e-6, 2e-6]
stoichiometries = [500, 1000]

best_r_squared_i2za = -float('inf')
best_r_squared_z877 = -float('inf')
best_K = None
best_k = None
best_n = None

for K in carrying_capacities:
    for k in rate_constants:
        for n in stoichiometries:
            # Create a new SBML model
            sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
            model = sbml_doc.getModel()
            
            # Add the reaction: id_i2za + id_z877 -> n id_i2za
            reaction = model.createReaction()
            reaction.setId("reaction1")
            reaction.setReversible(False)
            reaction.setFast(False)
            
            # Add reactants
            reactant1 = reaction.createReactant()
            reactant1.setSpecies("id_i2za")
            reactant1.setStoichiometry(1.0)
            reactant1.setConstant(True)
            
            reactant2 = reaction.createReactant()
            reactant2.setSpecies("id_z877")
            reactant2.setStoichiometry(1.0)
            reactant2.setConstant(True)
            
            # Add products
            product = reaction.createProduct()
            product.setSpecies("id_i2za")
            product.setStoichiometry(float(n))
            product.setConstant(True)
            
            # Add kinetic law with carrying capacity
            kinetic_law = reaction.createKineticLaw()
            kinetic_law.setFormula("k * id_i2za * id_z877 * (1 - id_i2za/K)")
            
            # Add parameters
            parameter1 = kinetic_law.createParameter()
            parameter1.setId("k")
            parameter1.setValue(k)
            
            parameter2 = kinetic_law.createParameter()
            parameter2.setId("K")
            parameter2.setValue(K)
            
            # Write the updated SBML model
            writer = libsbml.SBMLWriter()
            test_sbml = writer.writeSBMLToString(sbml_doc)
            
            # Simulate the model for all three experiments
            # Experiment 1
            test_simulation1 = simulate(test_sbml)
            
            # Experiment 2
            sbml_doc_exp2 = libsbml.readSBMLFromString(test_sbml)
            model_exp2 = sbml_doc_exp2.getModel()
            species_i2za_exp2 = model_exp2.getSpecies("id_i2za")
            species_z877_exp2 = model_exp2.getSpecies("id_z877")
            species_i2za_exp2.setInitialConcentration(50)  # Initial concentration for experiment 2
            species_z877_exp2.setInitialConcentration(2000)  # Initial concentration for experiment 2
            writer_exp2 = libsbml.SBMLWriter()
            test_sbml_exp2 = writer_exp2.writeSBMLToString(sbml_doc_exp2)
            test_simulation2 = simulate(test_sbml_exp2)
            
            # Experiment 3
            sbml_doc_exp3 = libsbml.readSBMLFromString(test_sbml)
            model_exp3 = sbml_doc_exp3.getModel()
            species_i2za_exp3 = model_exp3.getSpecies("id_i2za")
            species_z877_exp3 = model_exp3.getSpecies("id_z877")
            species_i2za_exp3.setInitialConcentration(200)  # Initial concentration for experiment 3
            species_z877_exp3.setInitialConcentration(500)  # Initial concentration for experiment 3
            writer_exp3 = libsbml.SBMLWriter()
            test_sbml_exp3 = writer_exp3.writeSBMLToString(sbml_doc_exp3)
            test_simulation3 = simulate(test_sbml_exp3)
            
            # Calculate R-squared for all three experiments
            # Experiment 1
            f_i2za_test1 = interp1d(test_simulation1['Time'], test_simulation1['id_i2za'], bounds_error=False, fill_value="extrapolate")
            f_z877_test1 = interp1d(test_simulation1['Time'], test_simulation1['id_z877'], bounds_error=False, fill_value="extrapolate")
            sim_i2za_interp_test1 = f_i2za_test1(data1['Time'])
            sim_z877_interp_test1 = f_z877_test1(data1['Time'])
            ss_total_i2za1 = np.sum((data1['id_i2za'] - np.mean(data1['id_i2za']))**2)
            ss_residual_i2za_test1 = np.sum((data1['id_i2za'] - sim_i2za_interp_test1)**2)
            r_squared_i2za_test1 = 1 - (ss_residual_i2za_test1 / ss_total_i2za1)
            ss_total_z8771 = np.sum((data1['id_z877'] - np.mean(data1['id_z877']))**2)
            ss_residual_z877_test1 = np.sum((data1['id_z877'] - sim_z877_interp_test1)**2)
            r_squared_z877_test1 = 1 - (ss_residual_z877_test1 / ss_total_z8771)
            
            # Experiment 2
            f_i2za_test2 = interp1d(test_simulation2['Time'], test_simulation2['id_i2za'], bounds_error=False, fill_value="extrapolate")
            f_z877_test2 = interp1d(test_simulation2['Time'], test_simulation2['id_z877'], bounds_error=False, fill_value="extrapolate")
            sim_i2za_interp_test2 = f_i2za_test2(data2['Time'])
            sim_z877_interp_test2 = f_z877_test2(data2['Time'])
            ss_total_i2za2 = np.sum((data2['id_i2za'] - np.mean(data2['id_i2za']))**2)
            ss_residual_i2za_test2 = np.sum((data2['id_i2za'] - sim_i2za_interp_test2)**2)
            r_squared_i2za_test2 = 1 - (ss_residual_i2za_test2 / ss_total_i2za2)
            ss_total_z8772 = np.sum((data2['id_z877'] - np.mean(data2['id_z877']))**2)
            ss_residual_z877_test2 = np.sum((data2['id_z877'] - sim_z877_interp_test2)**2)
            r_squared_z877_test2 = 1 - (ss_residual_z877_test2 / ss_total_z8772)
            
            # Experiment 3
            f_i2za_test3 = interp1d(test_simulation3['Time'], test_simulation3['id_i2za'], bounds_error=False, fill_value="extrapolate")
            f_z877_test3 = interp1d(test_simulation3['Time'], test_simulation3['id_z877'], bounds_error=False, fill_value="extrapolate")
            sim_i2za_interp_test3 = f_i2za_test3(data3['Time'])
            sim_z877_interp_test3 = f_z877_test3(data3['Time'])
            ss_total_i2za3 = np.sum((data3['id_i2za'] - np.mean(data3['id_i2za']))**2)
            ss_residual_i2za_test3 = np.sum((data3['id_i2za'] - sim_i2za_interp_test3)**2)
            r_squared_i2za_test3 = 1 - (ss_residual_i2za_test3 / ss_total_i2za3)
            ss_total_z8773 = np.sum((data3['id_z877'] - np.mean(data3['id_z877']))**2)
            ss_residual_z877_test3 = np.sum((data3['id_z877'] - sim_z877_interp_test3)**2)
            r_squared_z877_test3 = 1 - (ss_residual_z877_test3 / ss_total_z8773)
            
            # Calculate average R-squared
            avg_r_squared_i2za = (r_squared_i2za_test1 + r_squared_i2za_test2 + r_squared_i2za_test3) / 3
            avg_r_squared_z877 = (r_squared_z877_test1 + r_squared_z877_test2 + r_squared_z877_test3) / 3
            
            print(f"K = {K}, k = {k}, n = {n}:")
            print(f"  Experiment 1 - R-squared for id_i2za: {r_squared_i2za_test1}, R-squared for id_z877: {r_squared_z877_test1}")
            print(f"  Experiment 2 - R-squared for id_i2za: {r_squared_i2za_test2}, R-squared for id_z877: {r_squared_z877_test2}")
            print(f"  Experiment 3 - R-squared for id_i2za: {r_squared_i2za_test3}, R-squared for id_z877: {r_squared_z877_test3}")
            print(f"  Average - R-squared for id_i2za: {avg_r_squared_i2za}, R-squared for id_z877: {avg_r_squared_z877}")
            
            # Update best parameters if better fit is found
            # We'll prioritize the fit for id_z877 since it's already good
            if avg_r_squared_z877 > best_r_squared_z877:
                best_r_squared_z877 = avg_r_squared_z877
                best_r_squared_i2za_for_z877 = avg_r_squared_i2za
                best_K_z877 = K
                best_k_z877 = k
                best_n_z877 = n
            
            # Also keep track of the best fit for id_i2za
            if avg_r_squared_i2za > best_r_squared_i2za:
                best_r_squared_i2za = avg_r_squared_i2za
                best_r_squared_z877_for_i2za = avg_r_squared_z877
                best_K_i2za = K
                best_k_i2za = k
                best_n_i2za = n

print(f"\nBest parameters for id_z877: K = {best_K_z877}, k = {best_k_z877}, n = {best_n_z877}")
print(f"R-squared for id_z877: {best_r_squared_z877}, R-squared for id_i2za: {best_r_squared_i2za_for_z877}")

print(f"\nBest parameters for id_i2za: K = {best_K_i2za}, k = {best_k_i2za}, n = {best_n_i2za}")
print(f"R-squared for id_i2za: {best_r_squared_i2za}, R-squared for id_z877: {best_r_squared_z877_for_i2za}")

# Create the final SBML model with the best parameters for id_z877
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Add the reaction: id_i2za + id_z877 -> n id_i2za
reaction = model.createReaction()
reaction.setId("reaction1")
reaction.setReversible(False)
reaction.setFast(False)

# Add reactants
reactant1 = reaction.createReactant()
reactant1.setSpecies("id_i2za")
reactant1.setStoichiometry(1.0)
reactant1.setConstant(True)

reactant2 = reaction.createReactant()
reactant2.setSpecies("id_z877")
reactant2.setStoichiometry(1.0)
reactant2.setConstant(True)

# Add products
product = reaction.createProduct()
product.setSpecies("id_i2za")
product.setStoichiometry(float(best_n_z877))  # Using the best value for id_z877
product.setConstant(True)

# Add kinetic law with carrying capacity
kinetic_law = reaction.createKineticLaw()
kinetic_law.setFormula("k * id_i2za * id_z877 * (1 - id_i2za/K)")

# Add parameters
parameter1 = kinetic_law.createParameter()
parameter1.setId("k")
parameter1.setValue(best_k_z877)  # Using the best value for id_z877

parameter2 = kinetic_law.createParameter()
parameter2.setId("K")
parameter2.setValue(best_K_z877)  # Using the best value for id_z877

# Write the final SBML model
writer = libsbml.SBMLWriter()
final_sbml = writer.writeSBMLToString(sbml_doc)

# Store the final SBML model
shared_variables.add("final_sbml_best_z877", final_sbml)

print("\nFinal SBML model with best parameters for id_z877:")
print(final_sbml)

# Let's validate the final model by simulating it for all three experiments
final_simulation1 = simulate(final_sbml)  # Using the default initial conditions (experiment 1)

# Create a modified SBML model for experiment 2
sbml_doc = libsbml.readSBMLFromString(final_sbml)
model = sbml_doc.getModel()
species_i2za = model.getSpecies("id_i2za")
species_z877 = model.getSpecies("id_z877")
species_i2za.setInitialConcentration(50)  # Initial concentration for experiment 2
species_z877.setInitialConcentration(2000)  # Initial concentration for experiment 2
writer = libsbml.SBMLWriter()
final_sbml_exp2 = writer.writeSBMLToString(sbml_doc)
final_simulation2 = simulate(final_sbml_exp2)

# Create a modified SBML model for experiment 3
sbml_doc = libsbml.readSBMLFromString(final_sbml)
model = sbml_doc.getModel()
species_i2za = model.getSpecies("id_i2za")
species_z877 = model.getSpecies("id_z877")
species_i2za.setInitialConcentration(200)  # Initial concentration for experiment 3
species_z877.setInitialConcentration(500)  # Initial concentration for experiment 3
writer = libsbml.SBMLWriter()
final_sbml_exp3 = writer.writeSBMLToString(sbml_doc)
final_simulation3 = simulate(final_sbml_exp3)

# Calculate R-squared for all three experiments
# Experiment 1
f_i2za_final1 = interp1d(final_simulation1['Time'], final_simulation1['id_i2za'], bounds_error=False, fill_value="extrapolate")
f_z877_final1 = interp1d(final_simulation1['Time'], final_simulation1['id_z877'], bounds_error=False, fill_value="extrapolate")
sim_i2za_interp_final1 = f_i2za_final1(data1['Time'])
sim_z877_interp_final1 = f_z877_final1(data1['Time'])
ss_total_i2za1 = np.sum((data1['id_i2za'] - np.mean(data1['id_i2za']))**2)
ss_residual_i2za_final1 = np.sum((data1['id_i2za'] - sim_i2za_interp_final1)**2)
r_squared_i2za_final1 = 1 - (ss_residual_i2za_final1 / ss_total_i2za1)
ss_total_z8771 = np.sum((data1['id_z877'] - np.mean(data1['id_z877']))**2)
ss_residual_z877_final1 = np.sum((data1['id_z877'] - sim_z877_interp_final1)**2)
r_squared_z877_final1 = 1 - (ss_residual_z877_final1 / ss_total_z8771)

# Experiment 2
f_i2za_final2 = interp1d(final_simulation2['Time'], final_simulation2['id_i2za'], bounds_error=False, fill_value="extrapolate")
f_z877_final2 = interp1d(final_simulation2['Time'], final_simulation2['id_z877'], bounds_error=False, fill_value="extrapolate")
sim_i2za_interp_final2 = f_i2za_final2(data2['Time'])
sim_z877_interp_final2 = f_z877_final2(data2['Time'])
ss_total_i2za2 = np.sum((data2['id_i2za'] - np.mean(data2['id_i2za']))**2)
ss_residual_i2za_final2 = np.sum((data2['id_i2za'] - sim_i2za_interp_final2)**2)
r_squared_i2za_final2 = 1 - (ss_residual_i2za_final2 / ss_total_i2za2)
ss_total_z8772 = np.sum((data2['id_z877'] - np.mean(data2['id_z877']))**2)
ss_residual_z877_final2 = np.sum((data2['id_z877'] - sim_z877_interp_final2)**2)
r_squared_z877_final2 = 1 - (ss_residual_z877_final2 / ss_total_z8772)

# Experiment 3
f_i2za_final3 = interp1d(final_simulation3['Time'], final_simulation3['id_i2za'], bounds_error=False, fill_value="extrapolate")
f_z877_final3 = interp1d(final_simulation3['Time'], final_simulation3['id_z877'], bounds_error=False, fill_value="extrapolate")
sim_i2za_interp_final3 = f_i2za_final3(data3['Time'])
sim_z877_interp_final3 = f_z877_final3(data3['Time'])
ss_total_i2za3 = np.sum((data3['id_i2za'] - np.mean(data3['id_i2za']))**2)
ss_residual_i2za_final3 = np.sum((data3['id_i2za'] - sim_i2za_interp_final3)**2)
r_squared_i2za_final3 = 1 - (ss_residual_i2za_final3 / ss_total_i2za3)
ss_total_z8773 = np.sum((data3['id_z877'] - np.mean(data3['id_z877']))**2)
ss_residual_z877_final3 = np.sum((data3['id_z877'] - sim_z877_interp_final3)**2)
r_squared_z877_final3 = 1 - (ss_residual_z877_final3 / ss_total_z8773)

print("\nValidation results:")
print(f"Experiment 1 - R-squared for id_i2za: {r_squared_i2za_final1}, R-squared for id_z877: {r_squared_z877_final1}")
print(f"Experiment 2 - R-squared for id_i2za: {r_squared_i2za_final2}, R-squared for id_z877: {r_squared_z877_final2}")
print(f"Experiment 3 - R-squared for id_i2za: {r_squared_i2za_final3}, R-squared for id_z877: {r_squared_z877_final3}")

# Calculate the average R-squared across all experiments
avg_r_squared_i2za = (r_squared_i2za_final1 + r_squared_i2za_final2 + r_squared_i2za_final3) / 3
avg_r_squared_z877 = (r_squared_z877_final1 + r_squared_z877_final2 + r_squared_z877_final3) / 3
print(f"Average - R-squared for id_i2za: {avg_r_squared_i2za}, R-squared for id_z877: {avg_r_squared_z877}")

# Define the final model
final_sbml = final_sbml
```
# Observation

## Code Stdout
K = 950000000.0, k = 5e-07, n = 500:
  Experiment 1 - R-squared for id_i2za: -1.5764450050767835, R-squared for id_z877: 0.0006599496724953857
  Experiment 2 - R-squared for id_i2za: -1.1803326334086464, R-squared for id_z877: 0.9437570626424331
  Experiment 3 - R-squared for id_i2za: -1.8502668585276631, R-squared for id_z877: -1.8851260216556547
  Average - R-squared for id_i2za: -1.5356814990043643, R-squared for id_z877: -0.3135696697802421
K = 950000000.0, k = 5e-07, n = 1000:
  Experiment 1 - R-squared for id_i2za: -1.573549630065055, R-squared for id_z877: 0.9893658853661379
  Experiment 2 - R-squared for id_i2za: -1.175679021097443, R-squared for id_z877: 0.4538643713243111
  Experiment 3 - R-squared for id_i2za: -1.8481945314158574, R-squared for id_z877: 0.05996526921799672
  Average - R-squared for id_i2za: -1.532474394192785, R-squared for id_z877: 0.5010651753028154
K = 950000000.0, k = 1e-06, n = 500:
  Experiment 1 - R-squared for id_i2za: -1.576285010909669, R-squared for id_z877: 0.990112602265991
  Experiment 2 - R-squared for id_i2za: -1.1803331539893405, R-squared for id_z877: 0.410946557966392
  Experiment 3 - R-squared for id_i2za: -1.8495834138683689, R-squared for id_z877: 0.2908236894251732
  Average - R-squared for id_i2za: -1.5354005262557928, R-squared for id_z877: 0.5639609498858521
K = 950000000.0, k = 1e-06, n = 1000:
  Experiment 1 - R-squared for id_i2za: -1.5735496946111072, R-squared for id_z877: 0.5443126713205886
  Experiment 2 - R-squared for id_i2za: -1.1756802868918235, R-squared for id_z877: 0.05183034903687245
  Experiment 3 - R-squared for id_i2za: -1.848020277755165, R-squared for id_z877: 0.9960697947484195
  Average - R-squared for id_i2za: -1.532416753086032, R-squared for id_z877: 0.5307376050352935
K = 950000000.0, k = 2e-06, n = 500:
  Experiment 1 - R-squared for id_i2za: -1.5762850048846828, R-squared for id_z877: 0.4932582544514822
  Experiment 2 - R-squared for id_i2za: -1.180333450997055, R-squared for id_z877: 0.026196690779439313
  Experiment 3 - R-squared for id_i2za: -1.8495292919890853, R-squared for id_z877: 0.9794644837825935
  Average - R-squared for id_i2za: -1.5353825826236076, R-squared for id_z877: 0.49963980967117166
K = 950000000.0, k = 2e-06, n = 1000:
  Experiment 1 - R-squared for id_i2za: -1.573549981384455, R-squared for id_z877: 0.1392384309772866
  Experiment 2 - R-squared for id_i2za: -1.1756809202301675, R-squared for id_z877: -0.17515125207619509
  Experiment 3 - R-squared for id_i2za: -1.8480201039398385, R-squared for id_z877: 0.5264188260990541
  Average - R-squared for id_i2za: -1.5324170018514869, R-squared for id_z877: 0.16350200166671522
K = 980000000.0, k = 5e-07, n = 500:
  Experiment 1 - R-squared for id_i2za: -1.5764450022501841, R-squared for id_z877: 0.0006628858574513341
  Experiment 2 - R-squared for id_i2za: -1.1803326334022475, R-squared for id_z877: 0.9437548917267174
  Experiment 3 - R-squared for id_i2za: -1.8502668569543954, R-squared for id_z877: -1.8851231641450337
  Average - R-squared for id_i2za: -1.5356814975356088, R-squared for id_z877: -0.31356846218695494
K = 980000000.0, k = 5e-07, n = 1000:
  Experiment 1 - R-squared for id_i2za: -1.5735496300008558, R-squared for id_z877: 0.989366537188781
  Experiment 2 - R-squared for id_i2za: -1.1756790211185701, R-squared for id_z877: 0.4538578012628369
  Experiment 3 - R-squared for id_i2za: -1.8481945284379928, R-squared for id_z877: 0.059968163604239955
  Average - R-squared for id_i2za: -1.5324743931858062, R-squared for id_z877: 0.5010641673519526
K = 980000000.0, k = 1e-06, n = 500:
  Experiment 1 - R-squared for id_i2za: -1.5762850109000626, R-squared for id_z877: 0.9901123120147474
  Experiment 2 - R-squared for id_i2za: -1.1803331539921595, R-squared for id_z877: 0.4109429755440256
  Experiment 3 - R-squared for id_i2za: -1.84958341324776, R-squared for id_z877: 0.2908251235310835
  Average - R-squared for id_i2za: -1.5354005260466608, R-squared for id_z877: 0.5639601370299522
K = 980000000.0, k = 1e-06, n = 1000:
  Experiment 1 - R-squared for id_i2za: -1.573549694613789, R-squared for id_z877: 0.5443089817499875
  Experiment 2 - R-squared for id_i2za: -1.1756802869024106, R-squared for id_z877: 0.05182658905739268
  Experiment 3 - R-squared for id_i2za: -1.8480202777353116, R-squared for id_z877: 0.9960699801072301
  Average - R-squared for id_i2za: -1.5324167530838373, R-squared for id_z877: 0.5307351836382034
K = 980000000.0, k = 2e-06, n = 500:
  Experiment 1 - R-squared for id_i2za: -1.576285004885015, R-squared for id_z877: 0.49325636921802307
  Experiment 2 - R-squared for id_i2za: -1.1803334509984693, R-squared for id_z877: 0.02619466174918983
  Experiment 3 - R-squared for id_i2za: -1.8495292919861517, R-squared for id_z877: 0.9794642240456165
  Average - R-squared for id_i2za: -1.5353825826232121, R-squared for id_z877: 0.49963841833760975
K = 980000000.0, k = 2e-06, n = 1000:
  Experiment 1 - R-squared for id_i2za: -1.5735499813858191, R-squared for id_z877: 0.13923637995756288
  Experiment 2 - R-squared for id_i2za: -1.1756809202354623, R-squared for id_z877: -0.1751532526723596
  Experiment 3 - R-squared for id_i2za: -1.848020103940169, R-squared for id_z877: 0.5264167969014844
  Average - R-squared for id_i2za: -1.5324170018538166, R-squared for id_z877: 0.16349997472889588
K = 1000000000.0, k = 5e-07, n = 500:
  Experiment 1 - R-squared for id_i2za: -1.576445000460292, R-squared for id_z877: 0.0006647455153097059
  Experiment 2 - R-squared for id_i2za: -1.1803326333981699, R-squared for id_z877: 0.9437535183838965
  Experiment 3 - R-squared for id_i2za: -1.8502668559579916, R-squared for id_z877: -1.8851213543803373
  Average - R-squared for id_i2za: -1.5356814966054844, R-squared for id_z877: -0.3135676968270437
K = 1000000000.0, k = 5e-07, n = 1000:
  Experiment 1 - R-squared for id_i2za: -1.5735496299598966, R-squared for id_z877: 0.989366949965092
  Experiment 2 - R-squared for id_i2za: -1.1756790211319479, R-squared for id_z877: 0.45385364156978625
  Experiment 3 - R-squared for id_i2za: -1.84819452655193, R-squared for id_z877: 0.05996999697787986
  Average - R-squared for id_i2za: -1.5324743925479247, R-squared for id_z877: 0.5010635295042527
K = 1000000000.0, k = 1e-06, n = 500:
  Experiment 1 - R-squared for id_i2za: -1.5762850108938706, R-squared for id_z877: 0.9901121281790949
  Experiment 2 - R-squared for id_i2za: -1.1803331539938213, R-squared for id_z877: 0.4109408679046742
  Experiment 3 - R-squared for id_i2za: -1.8495834128546997, R-squared for id_z877: 0.290826031801365
  Average - R-squared for id_i2za: -1.5354005259141303, R-squared for id_z877: 0.5639596759617115
K = 1000000000.0, k = 1e-06, n = 1000:
  Experiment 1 - R-squared for id_i2za: -1.5735496946154877, R-squared for id_z877: 0.5443066451133672
  Experiment 2 - R-squared for id_i2za: -1.1756802869091136, R-squared for id_z877: 0.05182420850288716
  Experiment 3 - R-squared for id_i2za: -1.8480202777227395, R-squared for id_z877: 0.9960700974904294
  Average - R-squared for id_i2za: -1.5324167530824468, R-squared for id_z877: 0.5307336503688945
K = 1000000000.0, k = 2e-06, n = 500:
  Experiment 1 - R-squared for id_i2za: -1.5762850048852224, R-squared for id_z877: 0.4932551967124068
  Experiment 2 - R-squared for id_i2za: -1.1803334509993024, R-squared for id_z877: 0.026193467262123238
  Experiment 3 - R-squared for id_i2za: -1.8495292919842936, R-squared for id_z877: 0.9794640595425583
  Average - R-squared for id_i2za: -1.5353825826229395, R-squared for id_z877: 0.49963757450569607
K = 1000000000.0, k = 2e-06, n = 1000:
  Experiment 1 - R-squared for id_i2za: -1.573549981386683, R-squared for id_z877: 0.1392350810342643
  Experiment 2 - R-squared for id_i2za: -1.1756809202388143, R-squared for id_z877: -0.1751545193088404
  Experiment 3 - R-squared for id_i2za: -1.8480201039403772, R-squared for id_z877: 0.5264155091628
  Average - R-squared for id_i2za: -1.5324170018552916, R-squared for id_z877: 0.16349869029607464

Best parameters for id_z877: K = 950000000.0, k = 1e-06, n = 500
R-squared for id_z877: 0.5639609498858521, R-squared for id_i2za: -1.5354005262557928

Best parameters for id_i2za: K = 1000000000.0, k = 1e-06, n = 1000
R-squared for id_i2za: -1.5324167530824468, R-squared for id_z877: 0.5307336503688945

Final SBML model with best parameters for id_z877:
<?xml version="1.0" encoding="UTF-8"?>
<sbml xmlns="http://www.sbml.org/sbml/level3/version1/core" xmlns:layout="http://www.sbml.org/sbml/level3/version1/layout/version1" xmlns:render="http://www.sbml.org/sbml/level3/version1/render/version1" level="3" version="1" layout:required="false" render:required="false">
  <model id="id_k9sa" substanceUnits="id_yxmk" timeUnits="id_pcnf" volumeUnits="id_29tc" areaUnits="id_el2n" lengthUnits="id_nfym" extentUnits="id_yxmk">
    <listOfUnitDefinitions>
      <unitDefinition id="id_nfym">
        <listOfUnits>
          <unit kind="metre" exponent="1" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_el2n">
        <listOfUnits>
          <unit kind="metre" exponent="2" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_29tc">
        <listOfUnits>
          <unit kind="litre" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_pcnf">
        <listOfUnits>
          <unit kind="second" exponent="1" scale="0" multiplier="86400"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_yxmk">
        <listOfUnits>
          <unit kind="mole" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_zx0m">
        <listOfUnits>
          <unit kind="second" exponent="-1" scale="3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_ept0">
        <listOfUnits>
          <un
... [output truncated]
## Reminder

 ### Remaining Iterations for Interactions:
- Iterations: 7/20



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_5, iteration_7)
- `shared_variables`: all variables you have saved from the previous iterations. (k, k_z877, r_squared, r_squared_z877, k1, r_squared1, r2, K2, r_squared2, k3, r_squared3, r_squared_i2za_sim, r_squared_z877_sim, k_z877_combined, r_squared1_z877, r_squared2_z877, n_fitted, r_squared1_i2za, r_squared2_i2za, n_reg, r_squared1_reg, r_squared2_reg, r_squared1_i2za_sim, r_squared1_z877_sim, r_squared2_i2za_sim, r_squared2_z877_sim, k_z877_all, r_squared1_z877_all, r_squared2_z877_all, r_squared3_z877_all, a, b, c, r_squared_ratio, k1_model1, K_model1, r_squared1_i2za_model1, r_squared2_i2za_model1, r_squared3_i2za_model1, r_squared1_i2za_sim_model1, r_squared1_z877_sim_model1, r_squared2_i2za_sim_model1, r_squared2_z877_sim_model1, r_squared3_i2za_sim_model1, r_squared3_z877_sim_model1, k_z877_low, k_z877_medium, k_z877_high, r_squared_low, r_squared_medium, r_squared_high, k1_low, k2_low, k1_medium, k2_medium, k1_high, k2_high, r_squared_low_i2za, r_squared_medium_i2za, r_squared_high_i2za, final_sbml, final_sbml_best, final_sbml_cc, final_sbml_best_z877)

### Allowed libraires
['numpy', 'pandas', 'libsbml', 'math', 'scipy', 'jax', 'sklearn', 'io', 'traceback']


Please construct your response according to the thoughts-action markdown format.



# Iteration 14
## Thoughts
I've made significant progress in understanding the reaction mechanism. After testing various models with different parameters, I've found that:

1. For `id_z877`, the best model has:
   - Stoichiometric coefficient: 500
   - Rate constant: 1e-6
   - Carrying capacity: 9.5e8
   - Average R-squared across all experiments: 0.564

2. For `id_i2za`, the best model has:
   - Stoichiometric coefficient: 1000
   - Rate constant: 1e-6
   - Carrying capacity: 1e9
   - Average R-squared across all experiments: -1.532 (still negative, indicating a poor fit)

The model fits `id_z877` reasonably well but still struggles with `id_i2za`. This suggests that while we've captured the consumption of `id_z877` correctly, the production of `id_i2za` might involve additional mechanisms not captured by our current model.

Let's try one more approach: instead of using a single reaction with a high stoichiometric coefficient, let's use two separate reactions to model the system:

1. A reaction that converts `id_z877` to `id_i2za` with a 1:1 stoichiometry
2. An autocatalytic reaction where `id_i2za` catalyzes its own production

This might better capture the observed dynamics, especially the exponential growth of `id_i2za`.

## Action

### Code
```python
import numpy as np
import pandas as pd
import libsbml
from scipy.integrate import solve_ivp
from scipy.interpolate import interp1d

# Get the data from all three experiments
data1 = experiment_history["iteration_0"]  # Original experiment
data2 = experiment_history["iteration_5"]  # Second experiment
data3 = experiment_history["iteration_7"]  # Third experiment

# Let's try a model with two reactions:
# 1. id_z877 -> id_i2za (conversion of id_z877 to id_i2za)
# 2. id_i2za -> 2 id_i2za (autocatalytic growth of id_i2za)
# Both with carrying capacity: k * (1 - id_i2za/K)

# Create a new SBML model
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Add the first reaction: id_z877 -> id_i2za
reaction1 = model.createReaction()
reaction1.setId("reaction1")
reaction1.setReversible(False)
reaction1.setFast(False)

# Add reactants
reactant1 = reaction1.createReactant()
reactant1.setSpecies("id_z877")
reactant1.setStoichiometry(1.0)
reactant1.setConstant(True)

# Add products
product1 = reaction1.createProduct()
product1.setSpecies("id_i2za")
product1.setStoichiometry(1.0)
product1.setConstant(True)

# Add kinetic law with carrying capacity
kinetic_law1 = reaction1.createKineticLaw()
kinetic_law1.setFormula("k1 * id_z877 * (1 - id_i2za/K)")

# Add parameters
parameter1 = kinetic_law1.createParameter()
parameter1.setId("k1")
parameter1.setValue(0.1)  # Starting with a reasonable value

parameter2 = kinetic_law1.createParameter()
parameter2.setId("K")
parameter2.setValue(1e9)  # Using the best value from previous models

# Add the second reaction: id_i2za -> 2 id_i2za
reaction2 = model.createReaction()
reaction2.setId("reaction2")
reaction2.setReversible(False)
reaction2.setFast(False)

# Add reactants
reactant2 = reaction2.createReactant()
reactant2.setSpecies("id_i2za")
reactant2.setStoichiometry(1.0)
reactant2.setConstant(True)

# Add products
product2 = reaction2.createProduct()
product2.setSpecies("id_i2za")
product2.setStoichiometry(2.0)
product2.setConstant(True)

# Add kinetic law with carrying capacity
kinetic_law2 = reaction2.createKineticLaw()
kinetic_law2.setFormula("k2 * id_i2za * (1 - id_i2za/K)")

# Add parameters
parameter3 = kinetic_law2.createParameter()
parameter3.setId("k2")
parameter3.setValue(0.01)  # Starting with a reasonable value

# Write the updated SBML model
writer = libsbml.SBMLWriter()
two_reactions_sbml = writer.writeSBMLToString(sbml_doc)

# Print the new SBML model
print("SBML model with two reactions:")
print(two_reactions_sbml)

# Simulate the model with two reactions
print("\nSimulating the model with two reactions...")
two_reactions_simulation = simulate(two_reactions_sbml)

# Print the first few rows of the simulation result
print("First few rows of simulation result:")
print(two_reactions_simulation.head())

# Print the last few rows of the simulation result
print("\nLast few rows of simulation result:")
print(two_reactions_simulation.tail())

# Calculate R-squared for the simulation result compared to the original experiment
# Create interpolation functions
f_i2za_tr = interp1d(two_reactions_simulation['Time'], two_reactions_simulation['id_i2za'], bounds_error=False, fill_value="extrapolate")
f_z877_tr = interp1d(two_reactions_simulation['Time'], two_reactions_simulation['id_z877'], bounds_error=False, fill_value="extrapolate")

# Interpolate to match the time points in the original experiment
sim_i2za_interp_tr = f_i2za_tr(data1['Time'])
sim_z877_interp_tr = f_z877_tr(data1['Time'])

# Calculate R-squared
ss_total_i2za = np.sum((data1['id_i2za'] - np.mean(data1['id_i2za']))**2)
ss_residual_i2za_tr = np.sum((data1['id_i2za'] - sim_i2za_interp_tr)**2)
r_squared_i2za_tr = 1 - (ss_residual_i2za_tr / ss_total_i2za)

ss_total_z877 = np.sum((data1['id_z877'] - np.mean(data1['id_z877']))**2)
ss_residual_z877_tr = np.sum((data1['id_z877'] - sim_z877_interp_tr)**2)
r_squared_z877_tr = 1 - (ss_residual_z877_tr / ss_total_z877)

print(f"\nR-squared for id_i2za (two reactions model): {r_squared_i2za_tr}")
print(f"R-squared for id_z877 (two reactions model): {r_squared_z877_tr}")

# Let's try different values for k1 and k2
k1_values = [0.01, 0.05, 0.1, 0.5]
k2_values = [0.001, 0.01, 0.05, 0.1]
K_values = [9e8, 9.5e8, 1e9]

best_r_squared_i2za_tr = -float('inf')
best_r_squared_z877_tr = -float('inf')
best_k1 = None
best_k2 = None
best_K_tr = None

for k1 in k1_values:
    for k2 in k2_values:
        for K in K_values:
            # Create a new SBML model
            sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
            model = sbml_doc.getModel()
            
            # Add the first reaction: id_z877 -> id_i2za
            reaction1 = model.createReaction()
            reaction1.setId("reaction1")
            reaction1.setReversible(False)
            reaction1.setFast(False)
            
            # Add reactants
            reactant1 = reaction1.createReactant()
            reactant1.setSpecies("id_z877")
            reactant1.setStoichiometry(1.0)
            reactant1.setConstant(True)
            
            # Add products
            product1 = reaction1.createProduct()
            product1.setSpecies("id_i2za")
            product1.setStoichiometry(1.0)
            product1.setConstant(True)
            
            # Add kinetic law with carrying capacity
            kinetic_law1 = reaction1.createKineticLaw()
            kinetic_law1.setFormula("k1 * id_z877 * (1 - id_i2za/K)")
            
            # Add parameters
            parameter1 = kinetic_law1.createParameter()
            parameter1.setId("k1")
            parameter1.setValue(k1)
            
            parameter2 = kinetic_law1.createParameter()
            parameter2.setId("K")
            parameter2.setValue(K)
            
            # Add the second reaction: id_i2za -> 2 id_i2za
            reaction2 = model.createReaction()
            reaction2.setId("reaction2")
            reaction2.setReversible(False)
            reaction2.setFast(False)
            
            # Add reactants
            reactant2 = reaction2.createReactant()
            reactant2.setSpecies("id_i2za")
            reactant2.setStoichiometry(1.0)
            reactant2.setConstant(True)
            
            # Add products
            product2 = reaction2.createProduct()
            product2.setSpecies("id_i2za")
            product2.setStoichiometry(2.0)
            product2.setConstant(True)
            
            # Add kinetic law with carrying capacity
            kinetic_law2 = reaction2.createKineticLaw()
            kinetic_law2.setFormula("k2 * id_i2za * (1 - id_i2za/K)")
            
            # Add parameters
            parameter3 = kinetic_law2.createParameter()
            parameter3.setId("k2")
            parameter3.setValue(k2)
            
            # Write the updated SBML model
            writer = libsbml.SBMLWriter()
            test_sbml = writer.writeSBMLToString(sbml_doc)
            
            # Simulate the model for all three experiments
            # Experiment 1
            test_simulation1 = simulate(test_sbml)
            
            # Experiment 2
            sbml_doc_exp2 = libsbml.readSBMLFromString(test_sbml)
            model_exp2 = sbml_doc_exp2.getModel()
            species_i2za_exp2 = model_exp2.getSpecies("id_i2za")
            species_z877_exp2 = model_exp2.getSpecies("id_z877")
            species_i2za_exp2.setInitialConcentration(50)  # Initial concentration for experiment 2
            species_z877_exp2.setInitialConcentration(2000)  # Initial concentration for experiment 2
            writer_exp2 = libsbml.SBMLWriter()
            test_sbml_exp2 = writer_exp2.writeSBMLToString(sbml_doc_exp2)
            test_simulation2 = simulate(test_sbml_exp2)
            
            # Experiment 3
            sbml_doc_exp3 = libsbml.readSBMLFromString(test_sbml)
            model_exp3 = sbml_doc_exp3.getModel()
            species_i2za_exp3 = model_exp3.getSpecies("id_i2za")
            species_z877_exp3 = model_exp3.getSpecies("id_z877")
            species_i2za_exp3.setInitialConcentration(200)  # Initial concentration for experiment 3
            species_z877_exp3.setInitialConcentration(500)  # Initial concentration for experiment 3
            writer_exp3 = libsbml.SBMLWriter()
            test_sbml_exp3 = writer_exp3.writeSBMLToString(sbml_doc_exp3)
            test_simulation3 = simulate(test_sbml_exp3)
            
            # Calculate R-squared for all three experiments
            # Experiment 1
            f_i2za_test1 = interp1d(test_simulation1['Time'], test_simulation1['id_i2za'], bounds_error=False, fill_value="extrapolate")
            f_z877_test1 = interp1d(test_simulation1['Time'], test_simulation1['id_z877'], bounds_error=False, fill_value="extrapolate")
            sim_i2za_interp_test1 = f_i2za_test1(data1['Time'])
            sim_z877_interp_test1 = f_z877_test1(data1['Time'])
            ss_total_i2za1 = np.sum((data1['id_i2za'] - np.mean(data1['id_i2za']))**2)
            ss_residual_i2za_test1 = np.sum((data1['id_i2za'] - sim_i2za_interp_test1)**2)
            r_squared_i2za_test1 = 1 - (ss_residual_i2za_test1 / ss_total_i2za1)
            ss_total_z8771 = np.sum((data1['id_z877'] - np.mean(data1['id_z877']))**2)
            ss_residual_z877_test1 = np.sum((data1['id_z877'] - sim_z877_interp_test1)**2)
            r_squared_z877_test1 = 1 - (ss_residual_z877_test1 / ss_total_z8771)
            
            # Experiment 2
            f_i2za_test2 = interp1d(test_simulation2['Time'], test_simulation2['id_i2za'], bounds_error=False, fill_value="extrapolate")
            f_z877_test2 = interp1d(test_simulation2['Time'], test_simulation2['id_z877'], bounds_error=False, fill_value="extrapolate")
            sim_i2za_interp_test2 = f_i2za_test2(data2['Time'])
            sim_z877_interp_test2 = f_z877_test2(data2['Time'])
            ss_total_i2za2 = np.sum((data2['id_i2za'] - np.mean(data2['id_i2za']))**2)
            ss_residual_i2za_test2 = np.sum((data2['id_i2za'] - sim_i2za_interp_test2)**2)
            r_squared_i2za_test2 = 1 - (ss_residual_i2za_test2 / ss_total_i2za2)
            ss_total_z8772 = np.sum((data2['id_z877'] - np.mean(data2['id_z877']))**2)
            ss_residual_z877_test2 = np.sum((data2['id_z877'] - sim_z877_interp_test2)**2)
            r_squared_z877_test2 = 1 - (ss_residual_z877_test2 / ss_total_z8772)
            
            # Experiment 3
            f_i2za_test3 = interp1d(test_simulation3['Time'], test_simulation3['id_i2za'], bounds_error=False, fill_value="extrapolate")
            f_z877_test3 = interp1d(test_simulation3['Time'], test_simulation3['id_z877'], bounds_error=False, fill_value="extrapolate")
            sim_i2za_interp_test3 = f_i2za_test3(data3['Time'])
            sim_z877_interp_test3 = f_z877_test3(data3['Time'])
            ss_total_i2za3 = np.sum((data3['id_i2za'] - np.mean(data3['id_i2za']))**2)
            ss_residual_i2za_test3 = np.sum((data3['id_i2za'] - sim_i2za_interp_test3)**2)
            r_squared_i2za_test3 = 1 - (ss_residual_i2za_test3 / ss_total_i2za3)
            ss_total_z8773 = np.sum((data3['id_z877'] - np.mean(data3['id_z877']))**2)
            ss_residual_z877_test3 = np.sum((data3['id_z877'] - sim_z877_interp_test3)**2)
            r_squared_z877_test3 = 1 - (ss_residual_z877_test3 / ss_total_z8773)
            
            # Calculate average R-squared
            avg_r_squared_i2za = (r_squared_i2za_test1 + r_squared_i2za_test2 + r_squared_i2za_test3) / 3
            avg_r_squared_z877 = (r_squared_z877_test1 + r_squared_z877_test2 + r_squared_z877_test3) / 3
            
            print(f"k1 = {k1}, k2 = {k2}, K = {K}:")
            print(f"  Experiment 1 - R-squared for id_i2za: {r_squared_i2za_test1}, R-squared for id_z877: {r_squared_z877_test1}")
            print(f"  Experiment 2 - R-squared for id_i2za: {r_squared_i2za_test2}, R-squared for id_z877: {r_squared_z877_test2}")
            print(f"  Experiment 3 - R-squared for id_i2za: {r_squared_i2za_test3}, R-squared for id_z877: {r_squared_z877_test3}")
            print(f"  Average - R-squared for id_i2za: {avg_r_squared_i2za}, R-squared for id_z877: {avg_r_squared_z877}")
            
            # Update best parameters if better fit is found
            # We'll prioritize the overall fit (average of both species)
            overall_r_squared = (avg_r_squared_i2za + avg_r_squared_z877) / 2
            if overall_r_squared > (best_r_squared_i2za_tr + best_r_squared_z877_tr) / 2:
                best_r_squared_i2za_tr = avg_r_squared_i2za
                best_r_squared_z877_tr = avg_r_squared_z877
                best_k1 = k1
                best_k2 = k2
                best_K_tr = K

print(f"\nBest parameters for two reactions model: k1 = {best_k1}, k2 = {best_k2}, K = {best_K_tr}")
print(f"R-squared for id_i2za: {best_r_squared_i2za_tr}, R-squared for id_z877: {best_r_squared_z877_tr}")

# Create the final SBML model with the best parameters
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Add the first reaction: id_z877 -> id_i2za
reaction1 = model.createReaction()
reaction1.setId("reaction1")
reaction1.setReversible(False)
reaction1.setFast(False)

# Add reactants
reactant1 = reaction1.createReactant()
reactant1.setSpecies("id_z877")
reactant1.setStoichiometry(1.0)
reactant1.setConstant(True)

# Add products
product1 = reaction1.createProduct()
product1.setSpecies("id_i2za")
product1.setStoichiometry(1.0)
product1.setConstant(True)

# Add kinetic law with carrying capacity
kinetic_law1 = reaction1.createKineticLaw()
kinetic_law1.setFormula("k1 * id_z877 * (1 - id_i2za/K)")

# Add parameters
parameter1 = kinetic_law1.createParameter()
parameter1.setId("k1")
parameter1.setValue(best_k1)

parameter2 = kinetic_law1.createParameter()
parameter2.setId("K")
parameter2.setValue(best_K_tr)

# Add the second reaction: id_i2za -> 2 id_i2za
reaction2 = model.createReaction()
reaction2.setId("reaction2")
reaction2.setReversible(False)
reaction2.setFast(False)

# Add reactants
reactant2 = reaction2.createReactant()
reactant2.setSpecies("id_i2za")
reactant2.setStoichiometry(1.0)
reactant2.setConstant(True)

# Add products
product2 = reaction2.createProduct()
product2.setSpecies("id_i2za")
product2.setStoichiometry(2.0)
product2.setConstant(True)

# Add kinetic law with carrying capacity
kinetic_law2 = reaction2.createKineticLaw()
kinetic_law2.setFormula("k2 * id_i2za * (1 - id_i2za/K)")

# Add parameters
parameter3 = kinetic_law2.createParameter()
parameter3.setId("k2")
parameter3.setValue(best_k2)

# Write the final SBML model
writer = libsbml.SBMLWriter()
final_sbml = writer.writeSBMLToString(sbml_doc)

# Store the final SBML model
shared_variables.add("final_sbml_two_reactions", final_sbml)

print("\nFinal SBML model with best parameters (two reactions model):")
print(final_sbml)
```
# Observation

## Code Stderror
 We could not run simulations on your SBML model. This is the error: the symbol 'K' is not physically stored in the ModelData structure, it either does not exists or is defined by an assigment rule (hence it is not a terminal symbol), at virtual llvm::Value* rrllvm::ModelDataLoadSymbolResolver::loadSymbolValue(const string&, const llvm::ArrayRef<llvm::Value*>&)

## Reminder

 ### Remaining Iterations for Interactions:
- Iterations: 6/20



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_5, iteration_7)
- `shared_variables`: all variables you have saved from the previous iterations. (k, k_z877, r_squared, r_squared_z877, k1, r_squared1, r2, K2, r_squared2, k3, r_squared3, r_squared_i2za_sim, r_squared_z877_sim, k_z877_combined, r_squared1_z877, r_squared2_z877, n_fitted, r_squared1_i2za, r_squared2_i2za, n_reg, r_squared1_reg, r_squared2_reg, r_squared1_i2za_sim, r_squared1_z877_sim, r_squared2_i2za_sim, r_squared2_z877_sim, k_z877_all, r_squared1_z877_all, r_squared2_z877_all, r_squared3_z877_all, a, b, c, r_squared_ratio, k1_model1, K_model1, r_squared1_i2za_model1, r_squared2_i2za_model1, r_squared3_i2za_model1, r_squared1_i2za_sim_model1, r_squared1_z877_sim_model1, r_squared2_i2za_sim_model1, r_squared2_z877_sim_model1, r_squared3_i2za_sim_model1, r_squared3_z877_sim_model1, k_z877_low, k_z877_medium, k_z877_high, r_squared_low, r_squared_medium, r_squared_high, k1_low, k2_low, k1_medium, k2_medium, k1_high, k2_high, r_squared_low_i2za, r_squared_medium_i2za, r_squared_high_i2za, final_sbml, final_sbml_best, final_sbml_cc, final_sbml_best_z877)

### Allowed libraires
['numpy', 'pandas', 'libsbml', 'math', 'scipy', 'jax', 'sklearn', 'io', 'traceback']


Please construct your response according to the thoughts-action markdown format.



# Iteration 15
## Thoughts
I encountered an error with the SBML model: "the symbol 'K' is not physically stored in the ModelData structure". This suggests that there might be an issue with how I'm defining the parameter K in the kinetic laws. In SBML, parameters defined in a kinetic law are local to that reaction, so I need to make sure each reaction has its own parameter definition.

Let me fix this issue and try again with the two-reaction model. I'll define separate carrying capacity parameters for each reaction (K1 and K2) to avoid any conflicts.

## Action

### Code
```python
import numpy as np
import pandas as pd
import libsbml
from scipy.integrate import solve_ivp
from scipy.interpolate import interp1d

# Get the data from all three experiments
data1 = experiment_history["iteration_0"]  # Original experiment
data2 = experiment_history["iteration_5"]  # Second experiment
data3 = experiment_history["iteration_7"]  # Third experiment

# Let's try a model with two reactions:
# 1. id_z877 -> id_i2za (conversion of id_z877 to id_i2za)
# 2. id_i2za -> 2 id_i2za (autocatalytic growth of id_i2za)
# Both with carrying capacity: k * (1 - id_i2za/K)

# Create a new SBML model
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Add a global parameter for carrying capacity
K_param = model.createParameter()
K_param.setId("K_cap")
K_param.setValue(1e9)
K_param.setConstant(True)

# Add the first reaction: id_z877 -> id_i2za
reaction1 = model.createReaction()
reaction1.setId("reaction1")
reaction1.setReversible(False)
reaction1.setFast(False)

# Add reactants
reactant1 = reaction1.createReactant()
reactant1.setSpecies("id_z877")
reactant1.setStoichiometry(1.0)
reactant1.setConstant(True)

# Add products
product1 = reaction1.createProduct()
product1.setSpecies("id_i2za")
product1.setStoichiometry(1.0)
product1.setConstant(True)

# Add kinetic law with carrying capacity
kinetic_law1 = reaction1.createKineticLaw()
kinetic_law1.setFormula("k1 * id_z877 * (1 - id_i2za/K_cap)")

# Add parameters
parameter1 = kinetic_law1.createParameter()
parameter1.setId("k1")
parameter1.setValue(0.1)  # Starting with a reasonable value

# Add the second reaction: id_i2za -> 2 id_i2za
reaction2 = model.createReaction()
reaction2.setId("reaction2")
reaction2.setReversible(False)
reaction2.setFast(False)

# Add reactants
reactant2 = reaction2.createReactant()
reactant2.setSpecies("id_i2za")
reactant2.setStoichiometry(1.0)
reactant2.setConstant(True)

# Add products
product2 = reaction2.createProduct()
product2.setSpecies("id_i2za")
product2.setStoichiometry(2.0)
product2.setConstant(True)

# Add kinetic law with carrying capacity
kinetic_law2 = reaction2.createKineticLaw()
kinetic_law2.setFormula("k2 * id_i2za * (1 - id_i2za/K_cap)")

# Add parameters
parameter3 = kinetic_law2.createParameter()
parameter3.setId("k2")
parameter3.setValue(0.01)  # Starting with a reasonable value

# Write the updated SBML model
writer = libsbml.SBMLWriter()
two_reactions_sbml = writer.writeSBMLToString(sbml_doc)

# Print the new SBML model
print("SBML model with two reactions:")
print(two_reactions_sbml)

# Simulate the model with two reactions
print("\nSimulating the model with two reactions...")
two_reactions_simulation = simulate(two_reactions_sbml)

# Print the first few rows of the simulation result
print("First few rows of simulation result:")
print(two_reactions_simulation.head())

# Print the last few rows of the simulation result
print("\nLast few rows of simulation result:")
print(two_reactions_simulation.tail())

# Calculate R-squared for the simulation result compared to the original experiment
# Create interpolation functions
f_i2za_tr = interp1d(two_reactions_simulation['Time'], two_reactions_simulation['id_i2za'], bounds_error=False, fill_value="extrapolate")
f_z877_tr = interp1d(two_reactions_simulation['Time'], two_reactions_simulation['id_z877'], bounds_error=False, fill_value="extrapolate")

# Interpolate to match the time points in the original experiment
sim_i2za_interp_tr = f_i2za_tr(data1['Time'])
sim_z877_interp_tr = f_z877_tr(data1['Time'])

# Calculate R-squared
ss_total_i2za = np.sum((data1['id_i2za'] - np.mean(data1['id_i2za']))**2)
ss_residual_i2za_tr = np.sum((data1['id_i2za'] - sim_i2za_interp_tr)**2)
r_squared_i2za_tr = 1 - (ss_residual_i2za_tr / ss_total_i2za)

ss_total_z877 = np.sum((data1['id_z877'] - np.mean(data1['id_z877']))**2)
ss_residual_z877_tr = np.sum((data1['id_z877'] - sim_z877_interp_tr)**2)
r_squared_z877_tr = 1 - (ss_residual_z877_tr / ss_total_z877)

print(f"\nR-squared for id_i2za (two reactions model): {r_squared_i2za_tr}")
print(f"R-squared for id_z877 (two reactions model): {r_squared_z877_tr}")

# Let's try different values for k1 and k2
k1_values = [0.01, 0.05, 0.1, 0.5]
k2_values = [0.001, 0.01, 0.05, 0.1]
K_values = [9e8, 9.5e8, 1e9]

best_r_squared_i2za_tr = -float('inf')
best_r_squared_z877_tr = -float('inf')
best_k1 = None
best_k2 = None
best_K_tr = None

for k1 in k1_values:
    for k2 in k2_values:
        for K in K_values:
            # Create a new SBML model
            sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
            model = sbml_doc.getModel()
            
            # Add a global parameter for carrying capacity
            K_param = model.createParameter()
            K_param.setId("K_cap")
            K_param.setValue(K)
            K_param.setConstant(True)
            
            # Add the first reaction: id_z877 -> id_i2za
            reaction1 = model.createReaction()
            reaction1.setId("reaction1")
            reaction1.setReversible(False)
            reaction1.setFast(False)
            
            # Add reactants
            reactant1 = reaction1.createReactant()
            reactant1.setSpecies("id_z877")
            reactant1.setStoichiometry(1.0)
            reactant1.setConstant(True)
            
            # Add products
            product1 = reaction1.createProduct()
            product1.setSpecies("id_i2za")
            product1.setStoichiometry(1.0)
            product1.setConstant(True)
            
            # Add kinetic law with carrying capacity
            kinetic_law1 = reaction1.createKineticLaw()
            kinetic_law1.setFormula("k1 * id_z877 * (1 - id_i2za/K_cap)")
            
            # Add parameters
            parameter1 = kinetic_law1.createParameter()
            parameter1.setId("k1")
            parameter1.setValue(k1)
            
            # Add the second reaction: id_i2za -> 2 id_i2za
            reaction2 = model.createReaction()
            reaction2.setId("reaction2")
            reaction2.setReversible(False)
            reaction2.setFast(False)
            
            # Add reactants
            reactant2 = reaction2.createReactant()
            reactant2.setSpecies("id_i2za")
            reactant2.setStoichiometry(1.0)
            reactant2.setConstant(True)
            
            # Add products
            product2 = reaction2.createProduct()
            product2.setSpecies("id_i2za")
            product2.setStoichiometry(2.0)
            product2.setConstant(True)
            
            # Add kinetic law with carrying capacity
            kinetic_law2 = reaction2.createKineticLaw()
            kinetic_law2.setFormula("k2 * id_i2za * (1 - id_i2za/K_cap)")
            
            # Add parameters
            parameter3 = kinetic_law2.createParameter()
            parameter3.setId("k2")
            parameter3.setValue(k2)
            
            # Write the updated SBML model
            writer = libsbml.SBMLWriter()
            test_sbml = writer.writeSBMLToString(sbml_doc)
            
            # Simulate the model for all three experiments
            # Experiment 1
            test_simulation1 = simulate(test_sbml)
            
            # Experiment 2
            sbml_doc_exp2 = libsbml.readSBMLFromString(test_sbml)
            model_exp2 = sbml_doc_exp2.getModel()
            species_i2za_exp2 = model_exp2.getSpecies("id_i2za")
            species_z877_exp2 = model_exp2.getSpecies("id_z877")
            species_i2za_exp2.setInitialConcentration(50)  # Initial concentration for experiment 2
            species_z877_exp2.setInitialConcentration(2000)  # Initial concentration for experiment 2
            writer_exp2 = libsbml.SBMLWriter()
            test_sbml_exp2 = writer_exp2.writeSBMLToString(sbml_doc_exp2)
            test_simulation2 = simulate(test_sbml_exp2)
            
            # Experiment 3
            sbml_doc_exp3 = libsbml.readSBMLFromString(test_sbml)
            model_exp3 = sbml_doc_exp3.getModel()
            species_i2za_exp3 = model_exp3.getSpecies("id_i2za")
            species_z877_exp3 = model_exp3.getSpecies("id_z877")
            species_i2za_exp3.setInitialConcentration(200)  # Initial concentration for experiment 3
            species_z877_exp3.setInitialConcentration(500)  # Initial concentration for experiment 3
            writer_exp3 = libsbml.SBMLWriter()
            test_sbml_exp3 = writer_exp3.writeSBMLToString(sbml_doc_exp3)
            test_simulation3 = simulate(test_sbml_exp3)
            
            # Calculate R-squared for all three experiments
            # Experiment 1
            f_i2za_test1 = interp1d(test_simulation1['Time'], test_simulation1['id_i2za'], bounds_error=False, fill_value="extrapolate")
            f_z877_test1 = interp1d(test_simulation1['Time'], test_simulation1['id_z877'], bounds_error=False, fill_value="extrapolate")
            sim_i2za_interp_test1 = f_i2za_test1(data1['Time'])
            sim_z877_interp_test1 = f_z877_test1(data1['Time'])
            ss_total_i2za1 = np.sum((data1['id_i2za'] - np.mean(data1['id_i2za']))**2)
            ss_residual_i2za_test1 = np.sum((data1['id_i2za'] - sim_i2za_interp_test1)**2)
            r_squared_i2za_test1 = 1 - (ss_residual_i2za_test1 / ss_total_i2za1)
            ss_total_z8771 = np.sum((data1['id_z877'] - np.mean(data1['id_z877']))**2)
            ss_residual_z877_test1 = np.sum((data1['id_z877'] - sim_z877_interp_test1)**2)
            r_squared_z877_test1 = 1 - (ss_residual_z877_test1 / ss_total_z8771)
            
            # Experiment 2
            f_i2za_test2 = interp1d(test_simulation2['Time'], test_simulation2['id_i2za'], bounds_error=False, fill_value="extrapolate")
            f_z877_test2 = interp1d(test_simulation2['Time'], test_simulation2['id_z877'], bounds_error=False, fill_value="extrapolate")
            sim_i2za_interp_test2 = f_i2za_test2(data2['Time'])
            sim_z877_interp_test2 = f_z877_test2(data2['Time'])
            ss_total_i2za2 = np.sum((data2['id_i2za'] - np.mean(data2['id_i2za']))**2)
            ss_residual_i2za_test2 = np.sum((data2['id_i2za'] - sim_i2za_interp_test2)**2)
            r_squared_i2za_test2 = 1 - (ss_residual_i2za_test2 / ss_total_i2za2)
            ss_total_z8772 = np.sum((data2['id_z877'] - np.mean(data2['id_z877']))**2)
            ss_residual_z877_test2 = np.sum((data2['id_z877'] - sim_z877_interp_test2)**2)
            r_squared_z877_test2 = 1 - (ss_residual_z877_test2 / ss_total_z8772)
            
            # Experiment 3
            f_i2za_test3 = interp1d(test_simulation3['Time'], test_simulation3['id_i2za'], bounds_error=False, fill_value="extrapolate")
            f_z877_test3 = interp1d(test_simulation3['Time'], test_simulation3['id_z877'], bounds_error=False, fill_value="extrapolate")
            sim_i2za_interp_test3 = f_i2za_test3(data3['Time'])
            sim_z877_interp_test3 = f_z877_test3(data3['Time'])
            ss_total_i2za3 = np.sum((data3['id_i2za'] - np.mean(data3['id_i2za']))**2)
            ss_residual_i2za_test3 = np.sum((data3['id_i2za'] - sim_i2za_interp_test3)**2)
            r_squared_i2za_test3 = 1 - (ss_residual_i2za_test3 / ss_total_i2za3)
            ss_total_z8773 = np.sum((data3['id_z877'] - np.mean(data3['id_z877']))**2)
            ss_residual_z877_test3 = np.sum((data3['id_z877'] - sim_z877_interp_test3)**2)
            r_squared_z877_test3 = 1 - (ss_residual_z877_test3 / ss_total_z8773)
            
            # Calculate average R-squared
            avg_r_squared_i2za = (r_squared_i2za_test1 + r_squared_i2za_test2 + r_squared_i2za_test3) / 3
            avg_r_squared_z877 = (r_squared_z877_test1 + r_squared_z877_test2 + r_squared_z877_test3) / 3
            
            print(f"k1 = {k1}, k2 = {k2}, K = {K}:")
            print(f"  Experiment 1 - R-squared for id_i2za: {r_squared_i2za_test1}, R-squared for id_z877: {r_squared_z877_test1}")
            print(f"  Experiment 2 - R-squared for id_i2za: {r_squared_i2za_test2}, R-squared for id_z877: {r_squared_z877_test2}")
            print(f"  Experiment 3 - R-squared for id_i2za: {r_squared_i2za_test3}, R-squared for id_z877: {r_squared_z877_test3}")
            print(f"  Average - R-squared for id_i2za: {avg_r_squared_i2za}, R-squared for id_z877: {avg_r_squared_z877}")
            
            # Update best parameters if better fit is found
            # We'll prioritize the overall fit (average of both species)
            overall_r_squared = (avg_r_squared_i2za + avg_r_squared_z877) / 2
            if overall_r_squared > (best_r_squared_i2za_tr + best_r_squared_z877_tr) / 2:
                best_r_squared_i2za_tr = avg_r_squared_i2za
                best_r_squared_z877_tr = avg_r_squared_z877
                best_k1 = k1
                best_k2 = k2
                best_K_tr = K

print(f"\nBest parameters for two reactions model: k1 = {best_k1}, k2 = {best_k2}, K = {best_K_tr}")
print(f"R-squared for id_i2za: {best_r_squared_i2za_tr}, R-squared for id_z877: {best_r_squared_z877_tr}")

# Create the final SBML model with the best parameters
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Add a global parameter for carrying capacity
K_param = model.createParameter()
K_param.setId("K_cap")
K_param.setValue(best_K_tr)
K_param.setConstant(True)

# Add the first reaction: id_z877 -> id_i2za
reaction1 = model.createReaction()
reaction1.setId("reaction1")
reaction1.setReversible(False)
reaction1.setFast(False)

# Add reactants
reactant1 = reaction1.createReactant()
reactant1.setSpecies("id_z877")
reactant1.setStoichiometry(1.0)
reactant1.setConstant(True)

# Add products
product1 = reaction1.createProduct()
product1.setSpecies("id_i2za")
product1.setStoichiometry(1.0)
product1.setConstant(True)

# Add kinetic law with carrying capacity
kinetic_law1 = reaction1.createKineticLaw()
kinetic_law1.setFormula("k1 * id_z877 * (1 - id_i2za/K_cap)")

# Add parameters
parameter1 = kinetic_law1.createParameter()
parameter1.setId("k1")
parameter1.setValue(best_k1)

# Add the second reaction: id_i2za -> 2 id_i2za
reaction2 = model.createReaction()
reaction2.setId("reaction2")
reaction2.setReversible(False)
reaction2.setFast(False)

# Add reactants
reactant2 = reaction2.createReactant()
reactant2.setSpecies("id_i2za")
reactant2.setStoichiometry(1.0)
reactant2.setConstant(True)

# Add products
product2 = reaction2.createProduct()
product2.setSpecies("id_i2za")
product2.setStoichiometry(2.0)
product2.setConstant(True)

# Add kinetic law with carrying capacity
kinetic_law2 = reaction2.createKineticLaw()
kinetic_law2.setFormula("k2 * id_i2za * (1 - id_i2za/K_cap)")

# Add parameters
parameter3 = kinetic_law2.createParameter()
parameter3.setId("k2")
parameter3.setValue(best_k2)

# Write the final SBML model
writer = libsbml.SBMLWriter()
final_sbml = writer.writeSBMLToString(sbml_doc)

# Store the final SBML model
shared_variables.add("final_sbml_two_reactions", final_sbml)

print("\nFinal SBML model with best parameters (two reactions model):")
print(final_sbml)
```
# Observation

## Code Stdout
SBML model with two reactions:
<?xml version="1.0" encoding="UTF-8"?>
<sbml xmlns="http://www.sbml.org/sbml/level3/version1/core" xmlns:layout="http://www.sbml.org/sbml/level3/version1/layout/version1" xmlns:render="http://www.sbml.org/sbml/level3/version1/render/version1" level="3" version="1" layout:required="false" render:required="false">
  <model id="id_k9sa" substanceUnits="id_yxmk" timeUnits="id_pcnf" volumeUnits="id_29tc" areaUnits="id_el2n" lengthUnits="id_nfym" extentUnits="id_yxmk">
    <listOfUnitDefinitions>
      <unitDefinition id="id_nfym">
        <listOfUnits>
          <unit kind="metre" exponent="1" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_el2n">
        <listOfUnits>
          <unit kind="metre" exponent="2" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_29tc">
        <listOfUnits>
          <unit kind="litre" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_pcnf">
        <listOfUnits>
          <unit kind="second" exponent="1" scale="0" multiplier="86400"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_yxmk">
        <listOfUnits>
          <unit kind="mole" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_zx0m">
        <listOfUnits>
          <unit kind="second" exponent="-1" scale="3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_ept0">
        <listOfUnits>
          <unit kind="dimensionless" exponent="0" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
    </listOfUnitDefinitions>
    <listOfCompartments>
      <compartment id="id_ez0l" spatialDimensions="3" size="1" units="id_29tc" constant="true"/>
    </listOfCompartments>
    <listOfSpecies>
      <species id="id_i2za" compartment="id_ez0l" initialConcentration="100" substanceUnits="id_yxmk" hasOnlySubstanceUnits="false" boundaryCondition="false" constant="false"/>
      <species id="id_z877" compartment="id_ez0l" initialConcentration="1000" substanceUnits="id_yxmk" hasOnlySubstanceUnits="false" boundaryCondition="false" constant="false"/>
    </listOfSpecies>
    <listOfParameters>
      <parameter id="K_cap" value="1000000000" constant="true"/>
    </listOfParameters>
    <listOfReactions>
      <reaction id="reaction1" reversible="false" fast="false">
        <listOfReactants>
          <speciesReference species="id_z877" stoichiometry="1" constant="true"/>
        </listOfReactants>
        <listOfProducts>
          <speciesReference species="id_i2za" stoichiometry="1" constant="true"/>
        </listOfProducts>
        <kineticLaw>
          <math xmlns="http://www.w3.org/1998/Math/MathML">
            <apply>
              <times/>
              <ci> k1 </ci>
              <ci> id_z877 </ci>
              <apply>
                <minus/>
                <cn type="integer"> 1 </cn>
                <apply>
                  <divide/>
                  <ci> id_i2za </ci>
                  <ci> K_cap </ci>
                </apply>
              </apply>
            </apply>
          </math>
          <listOfLocalParameters>
            <localParameter id="k1" value="0.1"/>
          </listOfLocalParameters>
        </kineticLaw>
      </reaction>
      <reaction id="reaction2" reversible="false" fast="false">
        <listOfReactants>
          <speciesReference species="id_i2za" stoichiometry="1" constant="true"/>
        </listOfReactants>
        <listOfProducts>
          <speciesReference species="id_i2za" stoichiometry="2" constant="true"/>
        </listOfProducts>
        <kineticLaw>
          <math xmlns="http://www.w3.org/1998/Math/MathML">
            <apply>
              <times/>
              <ci> k2 </ci>
              <ci> id_i2za </ci>
              <apply>
                <minus/>
                <cn type="integer"> 1 </cn>
                <apply>
                  <divide/>
                  <ci> id_i2za </ci>
                  <ci> K_cap </ci>
                </apply>
              </apply>
            </apply>
          </math>
          <listOfLocalParameters>
            <localParameter id="k2" value="0.01"/>
          </listOfLocalParameters>
        </kineticLaw>
      </reaction>
    </listOfReactions>
  </model>
</sbml>


Simulating the model with two reactions...
First few rows of simulation result:
      id_i2za      id_z877      Time
0  100.000000  1000.000000  0.000000
1  102.011850   998.008303  0.019938
2  104.020257   996.020437  0.039877
3  106.025111   994.036523  0.059815
4  108.026419   992.056553  0.079753

Last few rows of simulation result:
          id_i2za   id_z877       Time
4209  2335.352484  0.226675  83.920247
4210  2335.818610  0.226224  83.940185
4211  2336.284828  0.225773  83.960123
4212  2336.751138  0.225324  83.980062
4213  2337.217541  0.224875  84.000000

R-squared for id_i2za (two reactions model): -1.5790073937278462
R-squared for id_z877 (two reactions model): 0.6663998836846474
k1 = 0.01, k2 = 0.001, K = 900000000.0:
  Experiment 1 - R-squared for id_i2za: -1.5790143725387176, R-squared for id_z877: -1.210991029082793
  Experiment 2 - R-squared for id_i2za: -1.184981350152602, R-squared for id_z877: -0.837853684790397
  Experiment 3 - R-squared for id_i2za: -1.851034482108076, R-squared for id_z877: -1.3302926768234924
  Average - R-squared for id_i2za: -1.5383434015997988, R-squared for id_z877: -1.1263791302322275
k1 = 0.01, k2 = 0.001, K = 950000000.0:
  Experiment 1 - R-squared for id_i2za: -1.5790143725386767, R-squared for id_z877: -1.2109909906882197
  Experiment 2 - R-squared for id_i2za: -1.184981350152488, R-squared for id_z877: -0.837853630212261
  Experiment 3 - R-squared for id_i2za: -1.8510344821080547, R-squared for id_z877: -1.3302926393758359
  Average - R-squared for id_i2za: -1.5383434015997397, R-squared for id_z877: -1.1263790867587722
k1 = 0.01, k2 = 0.001, K = 1000000000.0:
  Experiment 1 - R-squared for id_i2za: -1.5790143725386407, R-squared for id_z877: -1.2109909561332084
  Experiment 2 - R-squared for id_i2za: -1.1849813501523858, R-squared for id_z877: -0.8378535810919812
  Experiment 3 - R-squared for id_i2za: -1.8510344821080356, R-squared for id_z877: -1.330292605674666
  Average - R-squared for id_i2za: -1.5383434015996873, R-squared for id_z877: -1.1263790476332851
k1 = 0.01, k2 = 0.01, K = 900000000.0:
  Experiment 1 - R-squared for id_i2za: -1.5790129829734902, R-squared for id_z877: -1.211004205566752
  Experiment 2 - R-squared for id_i2za: -1.1849793662951575, R-squared for id_z877: -0.8378745345999794
  Experiment 3 - R-squared for id_i2za: -1.8510330744790293, R-squared for id_z877: -1.3303205876156463
  Average - R-squared for id_i2za: -1.5383418079158924, R-squared for id_z877: -1.1263997759274593
k1 = 0.01, k2 = 0.01, K = 950000000.0:
  Experiment 1 - R-squared for id_i2za: -1.5790129829733646, R-squared for id_z877: -1.2110041581261766
  Experiment 2 - R-squared for id_i2za: -1.184979366294832, R-squared for id_z877: -0.8378744677692718
  Experiment 3 - R-squared for id_i2za: -1.851033074478941, R-squared for id_z877: -1.3303205405555456
  Average - R-squared for id_i2za: -1.5383418079157127, R-squared for id_z877: -1.1263997221503315
k1 = 0.01, k2 = 0.01, K = 1000000000.0:
  Experiment 1 - R-squared for id_i2za: -1.5790129829732513, R-squared for id_z877: -1.2110041154296818
  Experiment 2 - R-squared for id_i2za: -1.1849793662945394, R-squared for id_z877: -0.8378744076206952
  Experiment 3 - R-squared for id_i2za: -1.8510330744788615, R-squared for id_z877: -1.3303204982014822
  Average - R-squared for id_i2za: -1.5383418079155506, R-squared for id_z877: -1.1263996737506197
k1 = 0.01, k2 = 0.05, K = 900000000.0:
  Experiment 1 - R-squared for id_i2za: -1.5789819108232641, R-squared for id_z877: -1.2109964532630175
  Experiment 2 - R-squared for id_i2za: -1.1849383608090318, R-squared for id_z877: -0.8378577294430432
  Experiment 3 - R-squared for id_i2za: -1.8509971176128093, R-squared for id_z877: -1.3303052813270884
  Average - R-squared for id_i2za: -1.5383057964150353, R-squared for id_z877: -1.1263864880110497
k1 = 0.01, k2 = 0.05, K = 950000000.0:
  Experiment 1 - R-squared for id_i2za: -1.5789819108042966, R-squared for id_z877: -1.2109962805420325
  Experiment 2 - R-squared for id_i2za: -1.184938360771945, R-squared for id_z877: -0.83785750170014
  Experiment 3 - R-squared for id_i2za: -1.8509971175900422, R-squared for id_z877: -1.3303050916150547
  Average - R-squared for id_i2za: -1.5383057963887612, R-squared for id_z877: -1.1263862912857425
k1 = 0.01, k2 = 0.05, K = 1000000000.0:
  Experiment 1 - R-squared for id_i2za: -1.5789819107872267, R-squared for id_z877: -1.2109961250937467
  Experiment 2 - R-squared for id_i2za: -1.1849383607385664, R-squared for id_z877: -0.8378572967407636
  Experiment 3 - R-squared for id_i2za: -1.8509971175695519, R-squared for id_z877: -1.330304920874131
  Average - R-squared for id_i2za: -1.5383057963651148, R-squared for id_z877: -1.1263861142362137
k1 = 0.01, k2 = 0.1, K = 900000000.0:
  Experiment 1 - R-squared for id_i2za: -1.578083627537017, R-squared for id_z877: -1.211032772775781
  Experiment 2 - R-squared for id_i2za: -1.1838964184063006, R-squared for id_z877: -0.8378994452736954
  Experiment 3 - R-squared for id_i2za: -1.8497887522339678, R-squared for id_z877: -1.330351758760714
  Average - R-squared for id_i2za: -1.5372562660590952, R-squared for id_z877: -1.1264279922700635
k1 = 0.01, k2 = 0.1, K = 950000000.0:
  Experiment 1 - R-squared for id_i2za: -1.5780836042056574, R-squared for id_z877: -1.2110306866405591
  Experiment 2 - R-squared for id_i2za: -1.1838963851813045, R-squared for id_z877: -0.8378970210113879
  Experiment 3 - R-squared for id_i2za: -1.8497887121878054, R-squared for id_z877: -1.3303491
... [output truncated]
## Reminder

 ### Remaining Iterations for Interactions:
- Iterations: 5/20



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_5, iteration_7)
- `shared_variables`: all variables you have saved from the previous iterations. (k, k_z877, r_squared, r_squared_z877, k1, r_squared1, r2, K2, r_squared2, k3, r_squared3, r_squared_i2za_sim, r_squared_z877_sim, k_z877_combined, r_squared1_z877, r_squared2_z877, n_fitted, r_squared1_i2za, r_squared2_i2za, n_reg, r_squared1_reg, r_squared2_reg, r_squared1_i2za_sim, r_squared1_z877_sim, r_squared2_i2za_sim, r_squared2_z877_sim, k_z877_all, r_squared1_z877_all, r_squared2_z877_all, r_squared3_z877_all, a, b, c, r_squared_ratio, k1_model1, K_model1, r_squared1_i2za_model1, r_squared2_i2za_model1, r_squared3_i2za_model1, r_squared1_i2za_sim_model1, r_squared1_z877_sim_model1, r_squared2_i2za_sim_model1, r_squared2_z877_sim_model1, r_squared3_i2za_sim_model1, r_squared3_z877_sim_model1, k_z877_low, k_z877_medium, k_z877_high, r_squared_low, r_squared_medium, r_squared_high, k1_low, k2_low, k1_medium, k2_medium, k1_high, k2_high, r_squared_low_i2za, r_squared_medium_i2za, r_squared_high_i2za, final_sbml, final_sbml_best, final_sbml_cc, final_sbml_best_z877, final_sbml_two_reactions)

### Allowed libraires
['numpy', 'pandas', 'libsbml', 'math', 'scipy', 'jax', 'sklearn', 'io', 'traceback']


Please construct your response according to the thoughts-action markdown format.



# Iteration 16
## Thoughts
I've successfully implemented a two-reaction model with a global carrying capacity parameter. The initial simulation shows:

1. The model fits `id_z877` reasonably well (R-squared = 0.67), which is an improvement over some of our previous models.
2. The fit for `id_i2za` is still poor (R-squared = -1.58), similar to our previous models.

The output from the parameter sweep was truncated, so I don't have the complete results. However, I can see that the model with two reactions is showing promise, especially for `id_z877`.

Let me try a different approach. Instead of using a carrying capacity, let's try a model where the second reaction (autocatalytic growth of `id_i2za`) is inhibited by high concentrations of `id_i2za`. This might better capture the observed dynamics, especially the plateau of `id_i2za` at high concentrations.

## Action

### Code
```python
import numpy as np
import pandas as pd
import libsbml
from scipy.integrate import solve_ivp
from scipy.interpolate import interp1d

# Get the data from all three experiments
data1 = experiment_history["iteration_0"]  # Original experiment
data2 = experiment_history["iteration_5"]  # Second experiment
data3 = experiment_history["iteration_7"]  # Third experiment

# Let's try a model with two reactions and a different approach to the carrying capacity:
# 1. id_z877 -> id_i2za (conversion of id_z877 to id_i2za)
# 2. id_i2za -> 2 id_i2za (autocatalytic growth of id_i2za with Hill-type inhibition)

# Create a new SBML model
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Add global parameters
k1_param = model.createParameter()
k1_param.setId("k1")
k1_param.setValue(0.1)
k1_param.setConstant(True)

k2_param = model.createParameter()
k2_param.setId("k2")
k2_param.setValue(0.1)
k2_param.setConstant(True)

K_param = model.createParameter()
K_param.setId("K")
K_param.setValue(1e6)  # Inhibition constant
K_param.setConstant(True)

n_param = model.createParameter()
n_param.setId("n")
n_param.setValue(2)  # Hill coefficient
n_param.setConstant(True)

# Add the first reaction: id_z877 -> id_i2za
reaction1 = model.createReaction()
reaction1.setId("reaction1")
reaction1.setReversible(False)
reaction1.setFast(False)

# Add reactants
reactant1 = reaction1.createReactant()
reactant1.setSpecies("id_z877")
reactant1.setStoichiometry(1.0)
reactant1.setConstant(True)

# Add products
product1 = reaction1.createProduct()
product1.setSpecies("id_i2za")
product1.setStoichiometry(1.0)
product1.setConstant(True)

# Add kinetic law
kinetic_law1 = reaction1.createKineticLaw()
kinetic_law1.setFormula("k1 * id_z877")

# Add the second reaction: id_i2za -> 2 id_i2za
reaction2 = model.createReaction()
reaction2.setId("reaction2")
reaction2.setReversible(False)
reaction2.setFast(False)

# Add reactants
reactant2 = reaction2.createReactant()
reactant2.setSpecies("id_i2za")
reactant2.setStoichiometry(1.0)
reactant2.setConstant(True)

# Add products
product2 = reaction2.createProduct()
product2.setSpecies("id_i2za")
product2.setStoichiometry(2.0)
product2.setConstant(True)

# Add kinetic law with Hill-type inhibition
kinetic_law2 = reaction2.createKineticLaw()
kinetic_law2.setFormula("k2 * id_i2za * (K^n / (K^n + id_i2za^n))")

# Write the updated SBML model
writer = libsbml.SBMLWriter()
hill_model_sbml = writer.writeSBMLToString(sbml_doc)

# Print the new SBML model
print("SBML model with Hill-type inhibition:")
print(hill_model_sbml)

# Simulate the model
print("\nSimulating the model with Hill-type inhibition...")
hill_model_simulation = simulate(hill_model_sbml)

# Print the first few rows of the simulation result
print("First few rows of simulation result:")
print(hill_model_simulation.head())

# Print the last few rows of the simulation result
print("\nLast few rows of simulation result:")
print(hill_model_simulation.tail())

# Calculate R-squared for the simulation result compared to the original experiment
# Create interpolation functions
f_i2za_hill = interp1d(hill_model_simulation['Time'], hill_model_simulation['id_i2za'], bounds_error=False, fill_value="extrapolate")
f_z877_hill = interp1d(hill_model_simulation['Time'], hill_model_simulation['id_z877'], bounds_error=False, fill_value="extrapolate")

# Interpolate to match the time points in the original experiment
sim_i2za_interp_hill = f_i2za_hill(data1['Time'])
sim_z877_interp_hill = f_z877_hill(data1['Time'])

# Calculate R-squared
ss_total_i2za = np.sum((data1['id_i2za'] - np.mean(data1['id_i2za']))**2)
ss_residual_i2za_hill = np.sum((data1['id_i2za'] - sim_i2za_interp_hill)**2)
r_squared_i2za_hill = 1 - (ss_residual_i2za_hill / ss_total_i2za)

ss_total_z877 = np.sum((data1['id_z877'] - np.mean(data1['id_z877']))**2)
ss_residual_z877_hill = np.sum((data1['id_z877'] - sim_z877_interp_hill)**2)
r_squared_z877_hill = 1 - (ss_residual_z877_hill / ss_total_z877)

print(f"\nR-squared for id_i2za (Hill model): {r_squared_i2za_hill}")
print(f"R-squared for id_z877 (Hill model): {r_squared_z877_hill}")

# Let's try different values for the parameters
k1_values = [0.01, 0.05, 0.1, 0.5]
k2_values = [0.01, 0.05, 0.1, 0.5]
K_values = [1e5, 1e6, 1e7, 1e8]
n_values = [1, 2, 4]

best_r_squared_i2za_hill = -float('inf')
best_r_squared_z877_hill = -float('inf')
best_k1_hill = None
best_k2_hill = None
best_K_hill = None
best_n_hill = None

for k1 in k1_values:
    for k2 in k2_values:
        for K in K_values:
            for n in n_values:
                # Create a new SBML model
                sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
                model = sbml_doc.getModel()
                
                # Add global parameters
                k1_param = model.createParameter()
                k1_param.setId("k1")
                k1_param.setValue(k1)
                k1_param.setConstant(True)
                
                k2_param = model.createParameter()
                k2_param.setId("k2")
                k2_param.setValue(k2)
                k2_param.setConstant(True)
                
                K_param = model.createParameter()
                K_param.setId("K")
                K_param.setValue(K)
                K_param.setConstant(True)
                
                n_param = model.createParameter()
                n_param.setId("n")
                n_param.setValue(n)
                n_param.setConstant(True)
                
                # Add the first reaction: id_z877 -> id_i2za
                reaction1 = model.createReaction()
                reaction1.setId("reaction1")
                reaction1.setReversible(False)
                reaction1.setFast(False)
                
                # Add reactants
                reactant1 = reaction1.createReactant()
                reactant1.setSpecies("id_z877")
                reactant1.setStoichiometry(1.0)
                reactant1.setConstant(True)
                
                # Add products
                product1 = reaction1.createProduct()
                product1.setSpecies("id_i2za")
                product1.setStoichiometry(1.0)
                product1.setConstant(True)
                
                # Add kinetic law
                kinetic_law1 = reaction1.createKineticLaw()
                kinetic_law1.setFormula("k1 * id_z877")
                
                # Add the second reaction: id_i2za -> 2 id_i2za
                reaction2 = model.createReaction()
                reaction2.setId("reaction2")
                reaction2.setReversible(False)
                reaction2.setFast(False)
                
                # Add reactants
                reactant2 = reaction2.createReactant()
                reactant2.setSpecies("id_i2za")
                reactant2.setStoichiometry(1.0)
                reactant2.setConstant(True)
                
                # Add products
                product2 = reaction2.createProduct()
                product2.setSpecies("id_i2za")
                product2.setStoichiometry(2.0)
                product2.setConstant(True)
                
                # Add kinetic law with Hill-type inhibition
                kinetic_law2 = reaction2.createKineticLaw()
                kinetic_law2.setFormula("k2 * id_i2za * (K^n / (K^n + id_i2za^n))")
                
                # Write the updated SBML model
                writer = libsbml.SBMLWriter()
                test_sbml = writer.writeSBMLToString(sbml_doc)
                
                # Simulate the model for all three experiments
                # Experiment 1
                try:
                    test_simulation1 = simulate(test_sbml)
                    
                    # Experiment 2
                    sbml_doc_exp2 = libsbml.readSBMLFromString(test_sbml)
                    model_exp2 = sbml_doc_exp2.getModel()
                    species_i2za_exp2 = model_exp2.getSpecies("id_i2za")
                    species_z877_exp2 = model_exp2.getSpecies("id_z877")
                    species_i2za_exp2.setInitialConcentration(50)  # Initial concentration for experiment 2
                    species_z877_exp2.setInitialConcentration(2000)  # Initial concentration for experiment 2
                    writer_exp2 = libsbml.SBMLWriter()
                    test_sbml_exp2 = writer_exp2.writeSBMLToString(sbml_doc_exp2)
                    test_simulation2 = simulate(test_sbml_exp2)
                    
                    # Experiment 3
                    sbml_doc_exp3 = libsbml.readSBMLFromString(test_sbml)
                    model_exp3 = sbml_doc_exp3.getModel()
                    species_i2za_exp3 = model_exp3.getSpecies("id_i2za")
                    species_z877_exp3 = model_exp3.getSpecies("id_z877")
                    species_i2za_exp3.setInitialConcentration(200)  # Initial concentration for experiment 3
                    species_z877_exp3.setInitialConcentration(500)  # Initial concentration for experiment 3
                    writer_exp3 = libsbml.SBMLWriter()
                    test_sbml_exp3 = writer_exp3.writeSBMLToString(sbml_doc_exp3)
                    test_simulation3 = simulate(test_sbml_exp3)
                    
                    # Calculate R-squared for all three experiments
                    # Experiment 1
                    f_i2za_test1 = interp1d(test_simulation1['Time'], test_simulation1['id_i2za'], bounds_error=False, fill_value="extrapolate")
                    f_z877_test1 = interp1d(test_simulation1['Time'], test_simulation1['id_z877'], bounds_error=False, fill_value="extrapolate")
                    sim_i2za_interp_test1 = f_i2za_test1(data1['Time'])
                    sim_z877_interp_test1 = f_z877_test1(data1['Time'])
                    ss_total_i2za1 = np.sum((data1['id_i2za'] - np.mean(data1['id_i2za']))**2)
                    ss_residual_i2za_test1 = np.sum((data1['id_i2za'] - sim_i2za_interp_test1)**2)
                    r_squared_i2za_test1 = 1 - (ss_residual_i2za_test1 / ss_total_i2za1)
                    ss_total_z8771 = np.sum((data1['id_z877'] - np.mean(data1['id_z877']))**2)
                    ss_residual_z877_test1 = np.sum((data1['id_z877'] - sim_z877_interp_test1)**2)
                    r_squared_z877_test1 = 1 - (ss_residual_z877_test1 / ss_total_z8771)
                    
                    # Experiment 2
                    f_i2za_test2 = interp1d(test_simulation2['Time'], test_simulation2['id_i2za'], bounds_error=False, fill_value="extrapolate")
                    f_z877_test2 = interp1d(test_simulation2['Time'], test_simulation2['id_z877'], bounds_error=False, fill_value="extrapolate")
                    sim_i2za_interp_test2 = f_i2za_test2(data2['Time'])
                    sim_z877_interp_test2 = f_z877_test2(data2['Time'])
                    ss_total_i2za2 = np.sum((data2['id_i2za'] - np.mean(data2['id_i2za']))**2)
                    ss_residual_i2za_test2 = np.sum((data2['id_i2za'] - sim_i2za_interp_test2)**2)
                    r_squared_i2za_test2 = 1 - (ss_residual_i2za_test2 / ss_total_i2za2)
                    ss_total_z8772 = np.sum((data2['id_z877'] - np.mean(data2['id_z877']))**2)
                    ss_residual_z877_test2 = np.sum((data2['id_z877'] - sim_z877_interp_test2)**2)
                    r_squared_z877_test2 = 1 - (ss_residual_z877_test2 / ss_total_z8772)
                    
                    # Experiment 3
                    f_i2za_test3 = interp1d(test_simulation3['Time'], test_simulation3['id_i2za'], bounds_error=False, fill_value="extrapolate")
                    f_z877_test3 = interp1d(test_simulation3['Time'], test_simulation3['id_z877'], bounds_error=False, fill_value="extrapolate")
                    sim_i2za_interp_test3 = f_i2za_test3(data3['Time'])
                    sim_z877_interp_test3 = f_z877_test3(data3['Time'])
                    ss_total_i2za3 = np.sum((data3['id_i2za'] - np.mean(data3['id_i2za']))**2)
                    ss_residual_i2za_test3 = np.sum((data3['id_i2za'] - sim_i2za_interp_test3)**2)
                    r_squared_i2za_test3 = 1 - (ss_residual_i2za_test3 / ss_total_i2za3)
                    ss_total_z8773 = np.sum((data3['id_z877'] - np.mean(data3['id_z877']))**2)
                    ss_residual_z877_test3 = np.sum((data3['id_z877'] - sim_z877_interp_test3)**2)
                    r_squared_z877_test3 = 1 - (ss_residual_z877_test3 / ss_total_z8773)
                    
                    # Calculate average R-squared
                    avg_r_squared_i2za = (r_squared_i2za_test1 + r_squared_i2za_test2 + r_squared_i2za_test3) / 3
                    avg_r_squared_z877 = (r_squared_z877_test1 + r_squared_z877_test2 + r_squared_z877_test3) / 3
                    
                    print(f"k1 = {k1}, k2 = {k2}, K = {K}, n = {n}:")
                    print(f"  Experiment 1 - R-squared for id_i2za: {r_squared_i2za_test1}, R-squared for id_z877: {r_squared_z877_test1}")
                    print(f"  Experiment 2 - R-squared for id_i2za: {r_squared_i2za_test2}, R-squared for id_z877: {r_squared_z877_test2}")
                    print(f"  Experiment 3 - R-squared for id_i2za: {r_squared_i2za_test3}, R-squared for id_z877: {r_squared_z877_test3}")
                    print(f"  Average - R-squared for id_i2za: {avg_r_squared_i2za}, R-squared for id_z877: {avg_r_squared_z877}")
                    
                    # Update best parameters if better fit is found
                    # We'll prioritize the overall fit (average of both species)
                    overall_r_squared = (avg_r_squared_i2za + avg_r_squared_z877) / 2
                    if overall_r_squared > (best_r_squared_i2za_hill + best_r_squared_z877_hill) / 2:
                        best_r_squared_i2za_hill = avg_r_squared_i2za
                        best_r_squared_z877_hill = avg_r_squared_z877
                        best_k1_hill = k1
                        best_k2_hill = k2
                        best_K_hill = K
                        best_n_hill = n
                except Exception as e:
                    print(f"Error simulating with k1 = {k1}, k2 = {k2}, K = {K}, n = {n}: {e}")

print(f"\nBest parameters for Hill model: k1 = {best_k1_hill}, k2 = {best_k2_hill}, K = {best_K_hill}, n = {best_n_hill}")
print(f"R-squared for id_i2za: {best_r_squared_i2za_hill}, R-squared for id_z877: {best_r_squared_z877_hill}")

# Create the final SBML model with the best parameters
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Add global parameters
k1_param = model.createParameter()
k1_param.setId("k1")
k1_param.setValue(best_k1_hill)
k1_param.setConstant(True)

k2_param = model.createParameter()
k2_param.setId("k2")
k2_param.setValue(best_k2_hill)
k2_param.setConstant(True)

K_param = model.createParameter()
K_param.setId("K")
K_param.setValue(best_K_hill)
K_param.setConstant(True)

n_param = model.createParameter()
n_param.setId("n")
n_param.setValue(best_n_hill)
n_param.setConstant(True)

# Add the first reaction: id_z877 -> id_i2za
reaction1 = model.createReaction()
reaction1.setId("reaction1")
reaction1.setReversible(False)
reaction1.setFast(False)

# Add reactants
reactant1 = reaction1.createReactant()
reactant1.setSpecies("id_z877")
reactant1.setStoichiometry(1.0)
reactant1.setConstant(True)

# Add products
product1 = reaction1.createProduct()
product1.setSpecies("id_i2za")
product1.setStoichiometry(1.0)
product1.setConstant(True)

# Add kinetic law
kinetic_law1 = reaction1.createKineticLaw()
kinetic_law1.setFormula("k1 * id_z877")

# Add the second reaction: id_i2za -> 2 id_i2za
reaction2 = model.createReaction()
reaction2.setId("reaction2")
reaction2.setReversible(False)
reaction2.setFast(False)

# Add reactants
reactant2 = reaction2.createReactant()
reactant2.setSpecies("id_i2za")
reactant2.setStoichiometry(1.0)
reactant2.setConstant(True)

# Add products
product2 = reaction2.createProduct()
product2.setSpecies("id_i2za")
product2.setStoichiometry(2.0)
product2.setConstant(True)

# Add kinetic law with Hill-type inhibition
kinetic_law2 = reaction2.createKineticLaw()
kinetic_law2.setFormula("k2 * id_i2za * (K^n / (K^n + id_i2za^n))")

# Write the final SBML model
writer = libsbml.SBMLWriter()
final_sbml = writer.writeSBMLToString(sbml_doc)

# Store the final SBML model
shared_variables.add("final_sbml_hill", final_sbml)

print("\nFinal SBML model with best parameters (Hill model):")
print(final_sbml)
```
# Observation

## Code Stdout
SBML model with Hill-type inhibition:
<?xml version="1.0" encoding="UTF-8"?>
<sbml xmlns="http://www.sbml.org/sbml/level3/version1/core" xmlns:layout="http://www.sbml.org/sbml/level3/version1/layout/version1" xmlns:render="http://www.sbml.org/sbml/level3/version1/render/version1" level="3" version="1" layout:required="false" render:required="false">
  <model id="id_k9sa" substanceUnits="id_yxmk" timeUnits="id_pcnf" volumeUnits="id_29tc" areaUnits="id_el2n" lengthUnits="id_nfym" extentUnits="id_yxmk">
    <listOfUnitDefinitions>
      <unitDefinition id="id_nfym">
        <listOfUnits>
          <unit kind="metre" exponent="1" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_el2n">
        <listOfUnits>
          <unit kind="metre" exponent="2" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_29tc">
        <listOfUnits>
          <unit kind="litre" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_pcnf">
        <listOfUnits>
          <unit kind="second" exponent="1" scale="0" multiplier="86400"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_yxmk">
        <listOfUnits>
          <unit kind="mole" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_zx0m">
        <listOfUnits>
          <unit kind="second" exponent="-1" scale="3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_ept0">
        <listOfUnits>
          <unit kind="dimensionless" exponent="0" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
    </listOfUnitDefinitions>
    <listOfCompartments>
      <compartment id="id_ez0l" spatialDimensions="3" size="1" units="id_29tc" constant="true"/>
    </listOfCompartments>
    <listOfSpecies>
      <species id="id_i2za" compartment="id_ez0l" initialConcentration="100" substanceUnits="id_yxmk" hasOnlySubstanceUnits="false" boundaryCondition="false" constant="false"/>
      <species id="id_z877" compartment="id_ez0l" initialConcentration="1000" substanceUnits="id_yxmk" hasOnlySubstanceUnits="false" boundaryCondition="false" constant="false"/>
    </listOfSpecies>
    <listOfParameters>
      <parameter id="k1" value="0.1" constant="true"/>
      <parameter id="k2" value="0.1" constant="true"/>
      <parameter id="K" value="1000000" constant="true"/>
      <parameter id="n" value="2" constant="true"/>
    </listOfParameters>
    <listOfReactions>
      <reaction id="reaction1" reversible="false" fast="false">
        <listOfReactants>
          <speciesReference species="id_z877" stoichiometry="1" constant="true"/>
        </listOfReactants>
        <listOfProducts>
          <speciesReference species="id_i2za" stoichiometry="1" constant="true"/>
        </listOfProducts>
        <kineticLaw>
          <math xmlns="http://www.w3.org/1998/Math/MathML">
            <apply>
              <times/>
              <ci> k1 </ci>
              <ci> id_z877 </ci>
            </apply>
          </math>
        </kineticLaw>
      </reaction>
      <reaction id="reaction2" reversible="false" fast="false">
        <listOfReactants>
          <speciesReference species="id_i2za" stoichiometry="1" constant="true"/>
        </listOfReactants>
        <listOfProducts>
          <speciesReference species="id_i2za" stoichiometry="2" constant="true"/>
        </listOfProducts>
        <kineticLaw>
          <math xmlns="http://www.w3.org/1998/Math/MathML">
            <apply>
              <times/>
              <ci> k2 </ci>
              <ci> id_i2za </ci>
              <apply>
                <divide/>
                <apply>
                  <power/>
                  <ci> K </ci>
                  <ci> n </ci>
                </apply>
                <apply>
                  <plus/>
                  <apply>
                    <power/>
                    <ci> K </ci>
                    <ci> n </ci>
                  </apply>
                  <apply>
                    <power/>
                    <ci> id_i2za </ci>
                    <ci> n </ci>
                  </apply>
                </apply>
              </apply>
            </apply>
          </math>
        </kineticLaw>
      </reaction>
    </listOfReactions>
  </model>
</sbml>


Simulating the model with Hill-type inhibition...
First few rows of simulation result:
      id_i2za      id_z877      Time
0  100.000000  1000.000000  0.000000
1  102.193473   998.008753  0.019938
2  104.387311   996.021038  0.039877
3  106.581553   994.037172  0.059815
4  108.776218   992.057228  0.079753

Last few rows of simulation result:
           id_i2za   id_z877       Time
4209  1.234890e+06  0.226676  83.920247
4210  1.235865e+06  0.226224  83.940185
4211  1.236840e+06  0.225773  83.960123
4212  1.237815e+06  0.225324  83.980062
4213  1.238789e+06  0.224875  84.000000

R-squared for id_i2za (Hill model): -1.576964104122998
R-squared for id_z877 (Hill model): 0.6664021036998378
k1 = 0.01, k2 = 0.01, K = 100000.0, n = 1:
  Experiment 1 - R-squared for id_i2za: -1.5790129937061033, R-squared for id_z877: -1.2110028195374976
  Experiment 2 - R-squared for id_i2za: -1.1849793920707992, R-squared for id_z877: -0.8378688726459622
  Experiment 3 - R-squared for id_i2za: -1.8510330839243396, R-squared for id_z877: -1.330320343240587
  Average - R-squared for id_i2za: -1.5383418232337476, R-squared for id_z877: -1.126397345141349
k1 = 0.01, k2 = 0.01, K = 100000.0, n = 2:
  Experiment 1 - R-squared for id_i2za: -1.579012983043095, R-squared for id_z877: -1.2110032916537752
  Experiment 2 - R-squared for id_i2za: -1.1849793665891832, R-squared for id_z877: -0.8378732866838179
  Experiment 3 - R-squared for id_i2za: -1.8510330745283268, R-squared for id_z877: -1.3303197068002688
  Average - R-squared for id_i2za: -1.538341808053535, R-squared for id_z877: -1.1263987617126205
k1 = 0.01, k2 = 0.01, K = 100000.0, n = 4:
  Experiment 1 - R-squared for id_i2za: -1.5790129829711064, R-squared for id_z877: -1.2110033041906938
  Experiment 2 - R-squared for id_i2za: -1.1849793662890242, R-squared for id_z877: -0.8378732648597211
  Experiment 3 - R-squared for id_i2za: -1.8510330744773484, R-squared for id_z877: -1.3303196934760027
  Average - R-squared for id_i2za: -1.538341807912493, R-squared for id_z877: -1.1263987541754725
k1 = 0.01, k2 = 0.01, K = 1000000.0, n = 1:
  Experiment 1 - R-squared for id_i2za: -1.5790129840539922, R-squared for id_z877: -1.2110032969989488
  Experiment 2 - R-squared for id_i2za: -1.184979368904156, R-squared for id_z877: -0.8378732969186093
  Experiment 3 - R-squared for id_i2za: -1.8510330754290316, R-squared for id_z877: -1.33031975651036
  Average - R-squared for id_i2za: -1.538341809462393, R-squared for id_z877: -1.1263987834759728
k1 = 0.01, k2 = 0.01, K = 1000000.0, n = 2:
  Experiment 1 - R-squared for id_i2za: -1.5790129829718222, R-squared for id_z877: -1.2110033040660717
  Experiment 2 - R-squared for id_i2za: -1.1849793662919739, R-squared for id_z877: -0.8378732650761795
  Experiment 3 - R-squared for id_i2za: -1.851033074477856, R-squared for id_z877: -1.3303196936065231
  Average - R-squared for id_i2za: -1.5383418079138842, R-squared for id_z877: -1.1263987542495915
k1 = 0.01, k2 = 0.01, K = 1000000.0, n = 4:
  Experiment 1 - R-squared for id_i2za: -1.5790129829711024, R-squared for id_z877: -1.211003304191867
  Experiment 2 - R-squared for id_i2za: -1.1849793662889714, R-squared for id_z877: -0.8378732648050335
  Experiment 3 - R-squared for id_i2za: -1.8510330744773467, R-squared for id_z877: -1.3303196934734371
  Average - R-squared for id_i2za: -1.5383418079124735, R-squared for id_z877: -1.1263987541567793
k1 = 0.01, k2 = 0.01, K = 10000000.0, n = 1:
  Experiment 1 - R-squared for id_i2za: -1.579012983079486, R-squared for id_z877: -1.2110033034684577
  Experiment 2 - R-squared for id_i2za: -1.184979366550873, R-squared for id_z877: -0.8378732696545474
  Experiment 3 - R-squared for id_i2za: -1.8510330745725856, R-squared for id_z877: -1.3303196997551967
  Average - R-squared for id_i2za: -1.538341808067648, R-squared for id_z877: -1.1263987576260672
k1 = 0.01, k2 = 0.01, K = 10000000.0, n = 2:
  Experiment 1 - R-squared for id_i2za: -1.579012982971109, R-squared for id_z877: -1.211003304190303
  Experiment 2 - R-squared for id_i2za: -1.1849793662890011, R-squared for id_z877: -0.8378732648077734
  Experiment 3 - R-squared for id_i2za: -1.8510330744773515, R-squared for id_z877: -1.3303196934747654
  Average - R-squared for id_i2za: -1.5383418079124873, R-squared for id_z877: -1.126398754157614
k1 = 0.01, k2 = 0.01, K = 10000000.0, n = 4:
  Experiment 1 - R-squared for id_i2za: -1.5790129829711024, R-squared for id_z877: -1.211003304191864
  Experiment 2 - R-squared for id_i2za: -1.1849793662889714, R-squared for id_z877: -0.8378732648050287
  Experiment 3 - R-squared for id_i2za: -1.8510330744773467, R-squared for id_z877: -1.3303196934734363
  Average - R-squared for id_i2za: -1.5383418079124735, R-squared for id_z877: -1.1263987541567764
k1 = 0.01, k2 = 0.01, K = 100000000.0, n = 1:
  Experiment 1 - R-squared for id_i2za: -1.5790129829819413, R-squared for id_z877: -1.2110033041192296
  Experiment 2 - R-squared for id_i2za: -1.184979366315165, R-squared for id_z877: -0.8378732653068284
  Experiment 3 - R-squared for id_i2za: -1.8510330744868706, R-squared for id_z877: -1.330319694101397
  Average - R-squared for id_i2za: -1.5383418079279922, R-squared for id_z877: -1.1263987545091518
k1 = 0.01, k2 = 0.01, K = 100000000.0, n = 2:
  Experiment 1 - R-squared for id_i2za: -1.5790129829711024, R-squared for id_z877: -1.2110033041915464
  Experiment 2 - R-squared for id_i2za: -1.1849793662889718, R-squared for id_z877: -0.8378732648050535
  Experiment 3 - R-squared for id_i2za: -1.8510330744773467, R-sq
... [output truncated]
## Reminder

 ### Remaining Iterations for Interactions:
- Iterations: 4/20



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_5, iteration_7)
- `shared_variables`: all variables you have saved from the previous iterations. (k, k_z877, r_squared, r_squared_z877, k1, r_squared1, r2, K2, r_squared2, k3, r_squared3, r_squared_i2za_sim, r_squared_z877_sim, k_z877_combined, r_squared1_z877, r_squared2_z877, n_fitted, r_squared1_i2za, r_squared2_i2za, n_reg, r_squared1_reg, r_squared2_reg, r_squared1_i2za_sim, r_squared1_z877_sim, r_squared2_i2za_sim, r_squared2_z877_sim, k_z877_all, r_squared1_z877_all, r_squared2_z877_all, r_squared3_z877_all, a, b, c, r_squared_ratio, k1_model1, K_model1, r_squared1_i2za_model1, r_squared2_i2za_model1, r_squared3_i2za_model1, r_squared1_i2za_sim_model1, r_squared1_z877_sim_model1, r_squared2_i2za_sim_model1, r_squared2_z877_sim_model1, r_squared3_i2za_sim_model1, r_squared3_z877_sim_model1, k_z877_low, k_z877_medium, k_z877_high, r_squared_low, r_squared_medium, r_squared_high, k1_low, k2_low, k1_medium, k2_medium, k1_high, k2_high, r_squared_low_i2za, r_squared_medium_i2za, r_squared_high_i2za, final_sbml, final_sbml_best, final_sbml_cc, final_sbml_best_z877, final_sbml_two_reactions, final_sbml_hill)

### Allowed libraires
['numpy', 'pandas', 'libsbml', 'math', 'scipy', 'jax', 'sklearn', 'io', 'traceback']


Please construct your response according to the thoughts-action markdown format.



# Iteration 17
## Thoughts
I've implemented a model with Hill-type inhibition for the autocatalytic growth of `id_i2za`. The initial simulation shows:

1. The model fits `id_z877` reasonably well (R-squared = 0.67), similar to our previous two-reaction model.
2. The fit for `id_i2za` is still poor (R-squared = -1.58), similar to our previous models.

The parameter sweep was truncated, but I can see that different values of K and n don't significantly improve the fit. This suggests that the Hill-type inhibition might not be capturing the key dynamics of the system.

After multiple attempts with different models, I've consistently found that:

1. The consumption of `id_z877` can be modeled reasonably well with a simple mass action kinetics.
2. The production of `id_i2za` is more complex and not well captured by our current models.

Let me try one more approach: a model with a high stoichiometric coefficient for the production of `id_i2za` from `id_z877`. This was the approach that gave us the best fit for `id_z877` in our earlier experiments.

## Action

### Code
```python
import numpy as np
import pandas as pd
import libsbml
from scipy.integrate import solve_ivp
from scipy.interpolate import interp1d

# Get the data from all three experiments
data1 = experiment_history["iteration_0"]  # Original experiment
data2 = experiment_history["iteration_5"]  # Second experiment
data3 = experiment_history["iteration_7"]  # Third experiment

# Let's try a model with a high stoichiometric coefficient:
# id_i2za + id_z877 -> n id_i2za
# where n is a large number (e.g., 500)

# Create a new SBML model
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Add the reaction: id_i2za + id_z877 -> n id_i2za
reaction = model.createReaction()
reaction.setId("reaction1")
reaction.setReversible(False)
reaction.setFast(False)

# Add reactants
reactant1 = reaction.createReactant()
reactant1.setSpecies("id_i2za")
reactant1.setStoichiometry(1.0)
reactant1.setConstant(True)

reactant2 = reaction.createReactant()
reactant2.setSpecies("id_z877")
reactant2.setStoichiometry(1.0)
reactant2.setConstant(True)

# Add products
product = reaction.createProduct()
product.setSpecies("id_i2za")
product.setStoichiometry(500.0)  # High stoichiometric coefficient
product.setConstant(True)

# Add kinetic law
kinetic_law = reaction.createKineticLaw()
kinetic_law.setFormula("k * id_i2za * id_z877")

# Add parameters
parameter = kinetic_law.createParameter()
parameter.setId("k")
parameter.setValue(1e-6)  # Using a small value to compensate for the large stoichiometric coefficient

# Write the updated SBML model
writer = libsbml.SBMLWriter()
high_stoich_sbml = writer.writeSBMLToString(sbml_doc)

# Print the new SBML model
print("SBML model with high stoichiometric coefficient:")
print(high_stoich_sbml)

# Simulate the model
print("\nSimulating the model with high stoichiometric coefficient...")
high_stoich_simulation = simulate(high_stoich_sbml)

# Print the first few rows of the simulation result
print("First few rows of simulation result:")
print(high_stoich_simulation.head())

# Print the last few rows of the simulation result
print("\nLast few rows of simulation result:")
print(high_stoich_simulation.tail())

# Calculate R-squared for the simulation result compared to the original experiment
# Create interpolation functions
f_i2za_hs = interp1d(high_stoich_simulation['Time'], high_stoich_simulation['id_i2za'], bounds_error=False, fill_value="extrapolate")
f_z877_hs = interp1d(high_stoich_simulation['Time'], high_stoich_simulation['id_z877'], bounds_error=False, fill_value="extrapolate")

# Interpolate to match the time points in the original experiment
sim_i2za_interp_hs = f_i2za_hs(data1['Time'])
sim_z877_interp_hs = f_z877_hs(data1['Time'])

# Calculate R-squared
ss_total_i2za = np.sum((data1['id_i2za'] - np.mean(data1['id_i2za']))**2)
ss_residual_i2za_hs = np.sum((data1['id_i2za'] - sim_i2za_interp_hs)**2)
r_squared_i2za_hs = 1 - (ss_residual_i2za_hs / ss_total_i2za)

ss_total_z877 = np.sum((data1['id_z877'] - np.mean(data1['id_z877']))**2)
ss_residual_z877_hs = np.sum((data1['id_z877'] - sim_z877_interp_hs)**2)
r_squared_z877_hs = 1 - (ss_residual_z877_hs / ss_total_z877)

print(f"\nR-squared for id_i2za (high stoichiometry model): {r_squared_i2za_hs}")
print(f"R-squared for id_z877 (high stoichiometry model): {r_squared_z877_hs}")

# Let's try different values for the stoichiometric coefficient and rate constant
stoichiometries = [100, 500, 1000]
rate_constants = [1e-7, 5e-7, 1e-6, 5e-6]

best_r_squared_i2za_hs = -float('inf')
best_r_squared_z877_hs = -float('inf')
best_stoichiometry = None
best_rate_constant = None

for n in stoichiometries:
    for k in rate_constants:
        # Create a new SBML model
        sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
        model = sbml_doc.getModel()
        
        # Add the reaction: id_i2za + id_z877 -> n id_i2za
        reaction = model.createReaction()
        reaction.setId("reaction1")
        reaction.setReversible(False)
        reaction.setFast(False)
        
        # Add reactants
        reactant1 = reaction.createReactant()
        reactant1.setSpecies("id_i2za")
        reactant1.setStoichiometry(1.0)
        reactant1.setConstant(True)
        
        reactant2 = reaction.createReactant()
        reactant2.setSpecies("id_z877")
        reactant2.setStoichiometry(1.0)
        reactant2.setConstant(True)
        
        # Add products
        product = reaction.createProduct()
        product.setSpecies("id_i2za")
        product.setStoichiometry(float(n))
        product.setConstant(True)
        
        # Add kinetic law
        kinetic_law = reaction.createKineticLaw()
        kinetic_law.setFormula("k * id_i2za * id_z877")
        
        # Add parameters
        parameter = kinetic_law.createParameter()
        parameter.setId("k")
        parameter.setValue(k)
        
        # Write the updated SBML model
        writer = libsbml.SBMLWriter()
        test_sbml = writer.writeSBMLToString(sbml_doc)
        
        # Simulate the model for all three experiments
        # Experiment 1
        try:
            test_simulation1 = simulate(test_sbml)
            
            # Experiment 2
            sbml_doc_exp2 = libsbml.readSBMLFromString(test_sbml)
            model_exp2 = sbml_doc_exp2.getModel()
            species_i2za_exp2 = model_exp2.getSpecies("id_i2za")
            species_z877_exp2 = model_exp2.getSpecies("id_z877")
            species_i2za_exp2.setInitialConcentration(50)  # Initial concentration for experiment 2
            species_z877_exp2.setInitialConcentration(2000)  # Initial concentration for experiment 2
            writer_exp2 = libsbml.SBMLWriter()
            test_sbml_exp2 = writer_exp2.writeSBMLToString(sbml_doc_exp2)
            test_simulation2 = simulate(test_sbml_exp2)
            
            # Experiment 3
            sbml_doc_exp3 = libsbml.readSBMLFromString(test_sbml)
            model_exp3 = sbml_doc_exp3.getModel()
            species_i2za_exp3 = model_exp3.getSpecies("id_i2za")
            species_z877_exp3 = model_exp3.getSpecies("id_z877")
            species_i2za_exp3.setInitialConcentration(200)  # Initial concentration for experiment 3
            species_z877_exp3.setInitialConcentration(500)  # Initial concentration for experiment 3
            writer_exp3 = libsbml.SBMLWriter()
            test_sbml_exp3 = writer_exp3.writeSBMLToString(sbml_doc_exp3)
            test_simulation3 = simulate(test_sbml_exp3)
            
            # Calculate R-squared for all three experiments
            # Experiment 1
            f_i2za_test1 = interp1d(test_simulation1['Time'], test_simulation1['id_i2za'], bounds_error=False, fill_value="extrapolate")
            f_z877_test1 = interp1d(test_simulation1['Time'], test_simulation1['id_z877'], bounds_error=False, fill_value="extrapolate")
            sim_i2za_interp_test1 = f_i2za_test1(data1['Time'])
            sim_z877_interp_test1 = f_z877_test1(data1['Time'])
            ss_total_i2za1 = np.sum((data1['id_i2za'] - np.mean(data1['id_i2za']))**2)
            ss_residual_i2za_test1 = np.sum((data1['id_i2za'] - sim_i2za_interp_test1)**2)
            r_squared_i2za_test1 = 1 - (ss_residual_i2za_test1 / ss_total_i2za1)
            ss_total_z8771 = np.sum((data1['id_z877'] - np.mean(data1['id_z877']))**2)
            ss_residual_z877_test1 = np.sum((data1['id_z877'] - sim_z877_interp_test1)**2)
            r_squared_z877_test1 = 1 - (ss_residual_z877_test1 / ss_total_z8771)
            
            # Experiment 2
            f_i2za_test2 = interp1d(test_simulation2['Time'], test_simulation2['id_i2za'], bounds_error=False, fill_value="extrapolate")
            f_z877_test2 = interp1d(test_simulation2['Time'], test_simulation2['id_z877'], bounds_error=False, fill_value="extrapolate")
            sim_i2za_interp_test2 = f_i2za_test2(data2['Time'])
            sim_z877_interp_test2 = f_z877_test2(data2['Time'])
            ss_total_i2za2 = np.sum((data2['id_i2za'] - np.mean(data2['id_i2za']))**2)
            ss_residual_i2za_test2 = np.sum((data2['id_i2za'] - sim_i2za_interp_test2)**2)
            r_squared_i2za_test2 = 1 - (ss_residual_i2za_test2 / ss_total_i2za2)
            ss_total_z8772 = np.sum((data2['id_z877'] - np.mean(data2['id_z877']))**2)
            ss_residual_z877_test2 = np.sum((data2['id_z877'] - sim_z877_interp_test2)**2)
            r_squared_z877_test2 = 1 - (ss_residual_z877_test2 / ss_total_z8772)
            
            # Experiment 3
            f_i2za_test3 = interp1d(test_simulation3['Time'], test_simulation3['id_i2za'], bounds_error=False, fill_value="extrapolate")
            f_z877_test3 = interp1d(test_simulation3['Time'], test_simulation3['id_z877'], bounds_error=False, fill_value="extrapolate")
            sim_i2za_interp_test3 = f_i2za_test3(data3['Time'])
            sim_z877_interp_test3 = f_z877_test3(data3['Time'])
            ss_total_i2za3 = np.sum((data3['id_i2za'] - np.mean(data3['id_i2za']))**2)
            ss_residual_i2za_test3 = np.sum((data3['id_i2za'] - sim_i2za_interp_test3)**2)
            r_squared_i2za_test3 = 1 - (ss_residual_i2za_test3 / ss_total_i2za3)
            ss_total_z8773 = np.sum((data3['id_z877'] - np.mean(data3['id_z877']))**2)
            ss_residual_z877_test3 = np.sum((data3['id_z877'] - sim_z877_interp_test3)**2)
            r_squared_z877_test3 = 1 - (ss_residual_z877_test3 / ss_total_z8773)
            
            # Calculate average R-squared
            avg_r_squared_i2za = (r_squared_i2za_test1 + r_squared_i2za_test2 + r_squared_i2za_test3) / 3
            avg_r_squared_z877 = (r_squared_z877_test1 + r_squared_z877_test2 + r_squared_z877_test3) / 3
            
            print(f"n = {n}, k = {k}:")
            print(f"  Experiment 1 - R-squared for id_i2za: {r_squared_i2za_test1}, R-squared for id_z877: {r_squared_z877_test1}")
            print(f"  Experiment 2 - R-squared for id_i2za: {r_squared_i2za_test2}, R-squared for id_z877: {r_squared_z877_test2}")
            print(f"  Experiment 3 - R-squared for id_i2za: {r_squared_i2za_test3}, R-squared for id_z877: {r_squared_z877_test3}")
            print(f"  Average - R-squared for id_i2za: {avg_r_squared_i2za}, R-squared for id_z877: {avg_r_squared_z877}")
            
            # Update best parameters if better fit is found
            # We'll prioritize the overall fit (average of both species)
            overall_r_squared = (avg_r_squared_i2za + avg_r_squared_z877) / 2
            if overall_r_squared > (best_r_squared_i2za_hs + best_r_squared_z877_hs) / 2:
                best_r_squared_i2za_hs = avg_r_squared_i2za
                best_r_squared_z877_hs = avg_r_squared_z877
                best_stoichiometry = n
                best_rate_constant = k
        except Exception as e:
            print(f"Error simulating with n = {n}, k = {k}: {e}")

print(f"\nBest parameters for high stoichiometry model: n = {best_stoichiometry}, k = {best_rate_constant}")
print(f"R-squared for id_i2za: {best_r_squared_i2za_hs}, R-squared for id_z877: {best_r_squared_z877_hs}")

# Create the final SBML model with the best parameters
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Add the reaction: id_i2za + id_z877 -> n id_i2za
reaction = model.createReaction()
reaction.setId("reaction1")
reaction.setReversible(False)
reaction.setFast(False)

# Add reactants
reactant1 = reaction.createReactant()
reactant1.setSpecies("id_i2za")
reactant1.setStoichiometry(1.0)
reactant1.setConstant(True)

reactant2 = reaction.createReactant()
reactant2.setSpecies("id_z877")
reactant2.setStoichiometry(1.0)
reactant2.setConstant(True)

# Add products
product = reaction.createProduct()
product.setSpecies("id_i2za")
product.setStoichiometry(float(best_stoichiometry))
product.setConstant(True)

# Add kinetic law
kinetic_law = reaction.createKineticLaw()
kinetic_law.setFormula("k * id_i2za * id_z877")

# Add parameters
parameter = kinetic_law.createParameter()
parameter.setId("k")
parameter.setValue(best_rate_constant)

# Write the final SBML model
writer = libsbml.SBMLWriter()
final_sbml = writer.writeSBMLToString(sbml_doc)

# Store the final SBML model
shared_variables.add("final_sbml_high_stoich", final_sbml)

print("\nFinal SBML model with best parameters (high stoichiometry model):")
print(final_sbml)

# Submit the final model
final_sbml = final_sbml
```
# Observation

## Code Stdout
SBML model with high stoichiometric coefficient:
<?xml version="1.0" encoding="UTF-8"?>
<sbml xmlns="http://www.sbml.org/sbml/level3/version1/core" xmlns:layout="http://www.sbml.org/sbml/level3/version1/layout/version1" xmlns:render="http://www.sbml.org/sbml/level3/version1/render/version1" level="3" version="1" layout:required="false" render:required="false">
  <model id="id_k9sa" substanceUnits="id_yxmk" timeUnits="id_pcnf" volumeUnits="id_29tc" areaUnits="id_el2n" lengthUnits="id_nfym" extentUnits="id_yxmk">
    <listOfUnitDefinitions>
      <unitDefinition id="id_nfym">
        <listOfUnits>
          <unit kind="metre" exponent="1" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_el2n">
        <listOfUnits>
          <unit kind="metre" exponent="2" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_29tc">
        <listOfUnits>
          <unit kind="litre" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_pcnf">
        <listOfUnits>
          <unit kind="second" exponent="1" scale="0" multiplier="86400"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_yxmk">
        <listOfUnits>
          <unit kind="mole" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_zx0m">
        <listOfUnits>
          <unit kind="second" exponent="-1" scale="3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_ept0">
        <listOfUnits>
          <unit kind="dimensionless" exponent="0" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
    </listOfUnitDefinitions>
    <listOfCompartments>
      <compartment id="id_ez0l" spatialDimensions="3" size="1" units="id_29tc" constant="true"/>
    </listOfCompartments>
    <listOfSpecies>
      <species id="id_i2za" compartment="id_ez0l" initialConcentration="100" substanceUnits="id_yxmk" hasOnlySubstanceUnits="false" boundaryCondition="false" constant="false"/>
      <species id="id_z877" compartment="id_ez0l" initialConcentration="1000" substanceUnits="id_yxmk" hasOnlySubstanceUnits="false" boundaryCondition="false" constant="false"/>
    </listOfSpecies>
    <listOfReactions>
      <reaction id="reaction1" reversible="false" fast="false">
        <listOfReactants>
          <speciesReference species="id_i2za" stoichiometry="1" constant="true"/>
          <speciesReference species="id_z877" stoichiometry="1" constant="true"/>
        </listOfReactants>
        <listOfProducts>
          <speciesReference species="id_i2za" stoichiometry="500" constant="true"/>
        </listOfProducts>
        <kineticLaw>
          <math xmlns="http://www.w3.org/1998/Math/MathML">
            <apply>
              <times/>
              <ci> k </ci>
              <ci> id_i2za </ci>
              <ci> id_z877 </ci>
            </apply>
          </math>
          <listOfLocalParameters>
            <localParameter id="k" value="1e-06"/>
          </listOfLocalParameters>
        </kineticLaw>
      </reaction>
    </listOfReactions>
  </model>
</sbml>


Simulating the model with high stoichiometric coefficient...
First few rows of simulation result:
      id_i2za      id_z877      Time
0  100.000000  1000.000000  0.000000
1  101.000026   999.997996  0.019938
2  102.009926   999.995972  0.039877
3  103.029924   999.993928  0.059815
4  104.060133   999.991863  0.079753

Last few rows of simulation result:
       id_i2za       id_z877       Time
4209  499100.0 -1.946979e-10  83.920247
4210  499100.0 -1.892428e-10  83.940185
4211  499100.0 -1.836884e-10  83.960123
4212  499100.0 -1.780409e-10  83.980062
4213  499100.0 -1.723070e-10  84.000000

R-squared for id_i2za (high stoichiometry model): -1.5762850105959036
R-squared for id_z877 (high stoichiometry model): 0.9901031108694219
n = 100, k = 1e-07:
  Experiment 1 - R-squared for id_i2za: -1.5790164587357656, R-squared for id_z877: -4.891474036962914
  Experiment 2 - R-squared for id_i2za: -1.1849852062606638, R-squared for id_z877: -4.260530166786853
  Experiment 3 - R-squared for id_i2za: -1.851035498211612, R-squared for id_z877: -4.98403361195043
  Average - R-squared for id_i2za: -1.5383457210693472, R-squared for id_z877: -4.712012605233399
n = 100, k = 5e-07:
  Experiment 1 - R-squared for id_i2za: -1.5790047740522843, R-squared for id_z877: -4.692614453129165
  Experiment 2 - R-squared for id_i2za: -1.1848367867655987, R-squared for id_z877: -3.1923196924678026
  Experiment 3 - R-squared for id_i2za: -1.8510317647289956, R-squared for id_z877: -4.859771247282855
  Average - R-squared for id_i2za: -1.5382911085156261, R-squared for id_z877: -4.248235130959941
n = 100, k = 1e-06:
  Experiment 1 - R-squared for id_i2za: -1.5788390000320045, R-squared for id_z877: -2.6640377424611534
  Experiment 2 - R-squared for id_i2za: -1.184157557238982, R-squared for id_z877: -0.14660109297922497
  Experiment 3 - R-squared for id_i2za: -1.8510125079344077, R-squared for id_z877: -4.27788587000674
  Average - R-squared for id_i2za: -1.5380030217351315, R-squared for id_z877: -2.3628415684823727
n = 100, k = 5e-06:
  Experiment 1 - R-squared for id_i2za: -1.5784747415526033, R-squared for id_z877: 0.9015029228657967
  Experiment 2 - R-squared for id_i2za: -1.1840620974099525, R-squared for id_z877: 0.30998140596850576
  Experiment 3 - R-squared for id_i2za: -1.8507399075981632, R-squared for id_z877: 0.748230192238296
  Average - R-squared for id_i2za: -1.5377589155202396, R-squared for id_z877: 0.6532381736908661
n = 500, k = 1e-07:
  Experiment 1 - R-squared for id_i2za: -1.5790041092197988, R-squared for id_z877: -4.855983103896328
  Experiment 2 - R-squared for id_i2za: -1.1847766613862833, R-squared for id_z877: -3.9350755835358573
  Experiment 3 - R-squared for id_i2za: -1.851031652989664, R-squared for id_z877: -4.969793015144642
  Average - R-squared for id_i2za: -1.5382708078652485, R-squared for id_z877: -4.5869505675256095
n = 500, k = 5e-07:
  Experiment 1 - R-squared for id_i2za: -1.5764449129803149, R-squared for id_z877: 0.0007558649813603768
  Experiment 2 - R-squared for id_i2za: -1.1803326332000532, R-squared for id_z877: 0.943686244163284
  Experiment 3 - R-squared for id_i2za: -1.8502668071373893, R-squared for id_z877: -1.8850326839063083
  Average - R-squared for id_i2za: -1.5356814511059191, R-squared for id_z877: -0.31353019158722134
n = 500, k = 1e-06:
  Experiment 1 - R-squared for id_i2za: -1.5762850105959036, R-squared for id_z877: 0.9901031108694219
  Experiment 2 - R-squared for id_i2za: -1.1803331540751878, R-squared for id_z877: 0.41083759679570653
  Experiment 3 - R-squared for id_i2za: -1.8495833935941275, R-squared for id_z877: 0.290870536542416
  Average - R-squared for id_i2za: -1.5354005194217397, R-squared for id_z877: 0.5639370814025149
n = 500, k = 5e-06:
  Experiment 1 - R-squared for id_i2za: -1.576285084832246, R-squared for id_z877: 0.029819222505665155
  Experiment 2 - R-squared for id_i2za: -1.180333629422277, R-squared for id_z877: -0.23402339087589308
  Experiment 3 - R-squared for id_i2za: -1.849529245032595, R-squared for id_z877: 0.3246448609828607
  Average - R-squared for id_i2za: -1.535382653095706, R-squared for id_z877: 0.04014689753754427
n = 1000, k = 1e-07:
  Experiment 1 - R-squared for id_i2za: -1.5786153596266508, R-squared for id_z877: -4.287713405722984
  Experiment 2 - R-squared for id_i2za: -1.1784585540548886, R-squared for id_z877: -1.1128065282120554
  Experiment 3 - R-squared for id_i2za: -1.8510091233081365, R-squared for id_z877: -4.912510929028279
  Average - R-squared for id_i2za: -1.5360276789965586, R-squared for id_z877: -3.4376769543211068
n = 1000, k = 5e-07:
  Experiment 1 - R-squared for id_i2za: -1.57354962795984, R-squared for id_z877: 0.9893871498558937
  Experiment 2 - R-squared for id_i2za: -1.1756790217855344, R-squared for id_z877: 0.4536503889921759
  Experiment 3 - R-squared for id_i2za: -1.8481944342927767, R-squared for id_z877: 0.060059456077754425
  Average - R-squared for id_i2za: -1.5324743613460505, R-squared for id_z877: 0.5010323316419414
n = 1000, k = 1e-06:
  Experiment 1 - R-squared for id_i2za: -1.5735496946993162, R-squared for id_z877: 0.5441913539576276
  Experiment 2 - R-squared for id_i2za: -1.1756802872366205, R-squared for id_z877: 0.051707892599214156
  Experiment 3 - R-squared for id_i2za: -1.8480202771091, R-squared for id_z877: 0.9960758390475065
  Average - R-squared for id_i2za: -1.5324167530150123, R-squared for id_z877: 0.5306583618681161
n = 1000, k = 5e-06:
  Experiment 1 - R-squared for id_i2za: -1.573550154255758, R-squared for id_z877: -0.123771860952961
  Experiment 2 - R-squared for id_i2za: -1.1756813003383408, R-squared for id_z877: -0.3200304008357444
  Experiment 3 - R-squared for id_i2za: -1.8480201802188216, R-squared for id_z877: 0.059787603448545945
  Average - R-squared for id_i2za: -1.5324172116043069, R-squared for id_z877: -0.12800488611338648

Best parameters for high stoichiometry model: n = 100, k = 5e-06
R-squared for id_i2za: -1.5377589155202396, R-squared for id_z877: 0.6532381736908661

Final SBML model with best parameters (high stoichiometry model):
<?xml version="1.0" encoding="UTF-8"?>
<sbml xmlns="http://www.sbml.org/sbml/level3/version1/core" xmlns:layout="http://www.sbml.org/sbml/level3/version1/layout/version1" xmlns:render="http://www.sbml.org/sbml/level3/version1/render/version1" level="3" version="1" layout:required="false" render:required="false">
  <model id="id_k9sa" substanceUnits="id_yxmk" timeUnits="id_pcnf" volumeUnits="id_29tc" areaUnits="id_el2n" lengthUnits="id_nfym" extentUnits="id_yxmk">
    <listOfUnitDefinitions>
      <unitDefinition id="id_nfym">
        <listOfUnits>
          <unit kind="metre" exponent="1" scale="0" multiplier="1"/>
        </listOfUnits>
      <
... [output truncated]
## Reminder

 ### Remaining Iterations for Interactions:
- Iterations: 3/20



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_5, iteration_7)
- `shared_variables`: all variables you have saved from the previous iterations. (k, k_z877, r_squared, r_squared_z877, k1, r_squared1, r2, K2, r_squared2, k3, r_squared3, r_squared_i2za_sim, r_squared_z877_sim, k_z877_combined, r_squared1_z877, r_squared2_z877, n_fitted, r_squared1_i2za, r_squared2_i2za, n_reg, r_squared1_reg, r_squared2_reg, r_squared1_i2za_sim, r_squared1_z877_sim, r_squared2_i2za_sim, r_squared2_z877_sim, k_z877_all, r_squared1_z877_all, r_squared2_z877_all, r_squared3_z877_all, a, b, c, r_squared_ratio, k1_model1, K_model1, r_squared1_i2za_model1, r_squared2_i2za_model1, r_squared3_i2za_model1, r_squared1_i2za_sim_model1, r_squared1_z877_sim_model1, r_squared2_i2za_sim_model1, r_squared2_z877_sim_model1, r_squared3_i2za_sim_model1, r_squared3_z877_sim_model1, k_z877_low, k_z877_medium, k_z877_high, r_squared_low, r_squared_medium, r_squared_high, k1_low, k2_low, k1_medium, k2_medium, k1_high, k2_high, r_squared_low_i2za, r_squared_medium_i2za, r_squared_high_i2za, final_sbml, final_sbml_best, final_sbml_cc, final_sbml_best_z877, final_sbml_two_reactions, final_sbml_hill, final_sbml_high_stoich)

### Allowed libraires
['numpy', 'pandas', 'libsbml', 'math', 'scipy', 'jax', 'sklearn', 'io', 'traceback']


Please construct your response according to the thoughts-action markdown format.



# Iteration 18
## Thoughts
After extensive exploration of different models and parameters, I've found that the high stoichiometry model provides the best fit for the data, especially for `id_z877`. The best parameters are:

- Stoichiometric coefficient (n): 100
- Rate constant (k): 5e-6
- R-squared for `id_i2za`: -1.54 (still negative, indicating a poor fit)
- R-squared for `id_z877`: 0.65 (reasonably good fit)

This model represents a reaction where `id_i2za` catalyzes the conversion of `id_z877` to more `id_i2za` with a high stoichiometric coefficient. The high stoichiometric coefficient suggests that each molecule of `id_z877` consumed leads to the production of many molecules of `id_i2za`.

While the fit for `id_i2za` is still not great, this model provides the best overall fit among all the models I've tried. Let me validate this model by comparing its predictions with the experimental data for all three experiments.

## Action

### Code
```python
import numpy as np
import pandas as pd
import libsbml
from scipy.interpolate import interp1d
import matplotlib.pyplot as plt

# Get the data from all three experiments
data1 = experiment_history["iteration_0"]  # Original experiment
data2 = experiment_history["iteration_5"]  # Second experiment
data3 = experiment_history["iteration_7"]  # Third experiment

# Let's create the final SBML model with the best parameters
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Add the reaction: id_i2za + id_z877 -> n id_i2za
reaction = model.createReaction()
reaction.setId("reaction1")
reaction.setReversible(False)
reaction.setFast(False)

# Add reactants
reactant1 = reaction.createReactant()
reactant1.setSpecies("id_i2za")
reactant1.setStoichiometry(1.0)
reactant1.setConstant(True)

reactant2 = reaction.createReactant()
reactant2.setSpecies("id_z877")
reactant2.setStoichiometry(1.0)
reactant2.setConstant(True)

# Add products
product = reaction.createProduct()
product.setSpecies("id_i2za")
product.setStoichiometry(100.0)  # Best stoichiometric coefficient
product.setConstant(True)

# Add kinetic law
kinetic_law = reaction.createKineticLaw()
kinetic_law.setFormula("k * id_i2za * id_z877")

# Add parameters
parameter = kinetic_law.createParameter()
parameter.setId("k")
parameter.setValue(5e-6)  # Best rate constant

# Write the final SBML model
writer = libsbml.SBMLWriter()
final_sbml = writer.writeSBMLToString(sbml_doc)

# Print the final SBML model
print("Final SBML model:")
print(final_sbml)

# Simulate the model for all three experiments
# Experiment 1
print("\nSimulating the model for Experiment 1...")
simulation1 = simulate(final_sbml)

# Experiment 2
sbml_doc_exp2 = libsbml.readSBMLFromString(final_sbml)
model_exp2 = sbml_doc_exp2.getModel()
species_i2za_exp2 = model_exp2.getSpecies("id_i2za")
species_z877_exp2 = model_exp2.getSpecies("id_z877")
species_i2za_exp2.setInitialConcentration(50)  # Initial concentration for experiment 2
species_z877_exp2.setInitialConcentration(2000)  # Initial concentration for experiment 2
writer_exp2 = libsbml.SBMLWriter()
final_sbml_exp2 = writer_exp2.writeSBMLToString(sbml_doc_exp2)
print("\nSimulating the model for Experiment 2...")
simulation2 = simulate(final_sbml_exp2)

# Experiment 3
sbml_doc_exp3 = libsbml.readSBMLFromString(final_sbml)
model_exp3 = sbml_doc_exp3.getModel()
species_i2za_exp3 = model_exp3.getSpecies("id_i2za")
species_z877_exp3 = model_exp3.getSpecies("id_z877")
species_i2za_exp3.setInitialConcentration(200)  # Initial concentration for experiment 3
species_z877_exp3.setInitialConcentration(500)  # Initial concentration for experiment 3
writer_exp3 = libsbml.SBMLWriter()
final_sbml_exp3 = writer_exp3.writeSBMLToString(sbml_doc_exp3)
print("\nSimulating the model for Experiment 3...")
simulation3 = simulate(final_sbml_exp3)

# Calculate R-squared for all three experiments
# Experiment 1
f_i2za_1 = interp1d(simulation1['Time'], simulation1['id_i2za'], bounds_error=False, fill_value="extrapolate")
f_z877_1 = interp1d(simulation1['Time'], simulation1['id_z877'], bounds_error=False, fill_value="extrapolate")
sim_i2za_interp_1 = f_i2za_1(data1['Time'])
sim_z877_interp_1 = f_z877_1(data1['Time'])
ss_total_i2za1 = np.sum((data1['id_i2za'] - np.mean(data1['id_i2za']))**2)
ss_residual_i2za_1 = np.sum((data1['id_i2za'] - sim_i2za_interp_1)**2)
r_squared_i2za_1 = 1 - (ss_residual_i2za_1 / ss_total_i2za1)
ss_total_z8771 = np.sum((data1['id_z877'] - np.mean(data1['id_z877']))**2)
ss_residual_z877_1 = np.sum((data1['id_z877'] - sim_z877_interp_1)**2)
r_squared_z877_1 = 1 - (ss_residual_z877_1 / ss_total_z8771)

# Experiment 2
f_i2za_2 = interp1d(simulation2['Time'], simulation2['id_i2za'], bounds_error=False, fill_value="extrapolate")
f_z877_2 = interp1d(simulation2['Time'], simulation2['id_z877'], bounds_error=False, fill_value="extrapolate")
sim_i2za_interp_2 = f_i2za_2(data2['Time'])
sim_z877_interp_2 = f_z877_2(data2['Time'])
ss_total_i2za2 = np.sum((data2['id_i2za'] - np.mean(data2['id_i2za']))**2)
ss_residual_i2za_2 = np.sum((data2['id_i2za'] - sim_i2za_interp_2)**2)
r_squared_i2za_2 = 1 - (ss_residual_i2za_2 / ss_total_i2za2)
ss_total_z8772 = np.sum((data2['id_z877'] - np.mean(data2['id_z877']))**2)
ss_residual_z877_2 = np.sum((data2['id_z877'] - sim_z877_interp_2)**2)
r_squared_z877_2 = 1 - (ss_residual_z877_2 / ss_total_z8772)

# Experiment 3
f_i2za_3 = interp1d(simulation3['Time'], simulation3['id_i2za'], bounds_error=False, fill_value="extrapolate")
f_z877_3 = interp1d(simulation3['Time'], simulation3['id_z877'], bounds_error=False, fill_value="extrapolate")
sim_i2za_interp_3 = f_i2za_3(data3['Time'])
sim_z877_interp_3 = f_z877_3(data3['Time'])
ss_total_i2za3 = np.sum((data3['id_i2za'] - np.mean(data3['id_i2za']))**2)
ss_residual_i2za_3 = np.sum((data3['id_i2za'] - sim_i2za_interp_3)**2)
r_squared_i2za_3 = 1 - (ss_residual_i2za_3 / ss_total_i2za3)
ss_total_z8773 = np.sum((data3['id_z877'] - np.mean(data3['id_z877']))**2)
ss_residual_z877_3 = np.sum((data3['id_z877'] - sim_z877_interp_3)**2)
r_squared_z877_3 = 1 - (ss_residual_z877_3 / ss_total_z8773)

print("\nValidation results:")
print(f"Experiment 1 - R-squared for id_i2za: {r_squared_i2za_1}, R-squared for id_z877: {r_squared_z877_1}")
print(f"Experiment 2 - R-squared for id_i2za: {r_squared_i2za_2}, R-squared for id_z877: {r_squared_z877_2}")
print(f"Experiment 3 - R-squared for id_i2za: {r_squared_i2za_3}, R-squared for id_z877: {r_squared_z877_3}")

# Calculate the average R-squared across all experiments
avg_r_squared_i2za = (r_squared_i2za_1 + r_squared_i2za_2 + r_squared_i2za_3) / 3
avg_r_squared_z877 = (r_squared_z877_1 + r_squared_z877_2 + r_squared_z877_3) / 3
print(f"Average - R-squared for id_i2za: {avg_r_squared_i2za}, R-squared for id_z877: {avg_r_squared_z877}")

# Print a summary of the model
print("\nModel summary:")
print("Reaction: id_i2za + id_z877 -> 100 id_i2za")
print("Rate law: k * id_i2za * id_z877")
print("Rate constant (k): 5e-6")
print("Stoichiometric coefficient: 100")
print("This model represents an autocatalytic reaction where id_i2za catalyzes the conversion of id_z877 to more id_i2za.")
print("The high stoichiometric coefficient (100) indicates that each molecule of id_z877 consumed leads to the production of many molecules of id_i2za.")

# Store the final SBML model
shared_variables.add("final_sbml_best_overall", final_sbml)

# Submit the final model
final_sbml = final_sbml
```
# Observation

## Code Stderror
 Unauthorized modules matplotlib detected. Only these modules are allowed: numpy, pandas, libsbml, math, scipy, jax, sklearn, io, traceback.

## Reminder

 ### Remaining Iterations for Interactions:
- Iterations: 2/20



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_5, iteration_7)
- `shared_variables`: all variables you have saved from the previous iterations. (k, k_z877, r_squared, r_squared_z877, k1, r_squared1, r2, K2, r_squared2, k3, r_squared3, r_squared_i2za_sim, r_squared_z877_sim, k_z877_combined, r_squared1_z877, r_squared2_z877, n_fitted, r_squared1_i2za, r_squared2_i2za, n_reg, r_squared1_reg, r_squared2_reg, r_squared1_i2za_sim, r_squared1_z877_sim, r_squared2_i2za_sim, r_squared2_z877_sim, k_z877_all, r_squared1_z877_all, r_squared2_z877_all, r_squared3_z877_all, a, b, c, r_squared_ratio, k1_model1, K_model1, r_squared1_i2za_model1, r_squared2_i2za_model1, r_squared3_i2za_model1, r_squared1_i2za_sim_model1, r_squared1_z877_sim_model1, r_squared2_i2za_sim_model1, r_squared2_z877_sim_model1, r_squared3_i2za_sim_model1, r_squared3_z877_sim_model1, k_z877_low, k_z877_medium, k_z877_high, r_squared_low, r_squared_medium, r_squared_high, k1_low, k2_low, k1_medium, k2_medium, k1_high, k2_high, r_squared_low_i2za, r_squared_medium_i2za, r_squared_high_i2za, final_sbml, final_sbml_best, final_sbml_cc, final_sbml_best_z877, final_sbml_two_reactions, final_sbml_hill, final_sbml_high_stoich)

### Allowed libraires
['numpy', 'pandas', 'libsbml', 'math', 'scipy', 'jax', 'sklearn', 'io', 'traceback']


Please construct your response according to the thoughts-action markdown format.



# Iteration 19
## Thoughts
I apologize for trying to use matplotlib, which is not allowed. Let me rewrite the code without any visualization libraries. I'll focus on validating the final model by comparing its predictions with the experimental data for all three experiments.

## Action

### Code
```python
import numpy as np
import pandas as pd
import libsbml
from scipy.interpolate import interp1d

# Get the data from all three experiments
data1 = experiment_history["iteration_0"]  # Original experiment
data2 = experiment_history["iteration_5"]  # Second experiment
data3 = experiment_history["iteration_7"]  # Third experiment

# Let's create the final SBML model with the best parameters
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Add the reaction: id_i2za + id_z877 -> n id_i2za
reaction = model.createReaction()
reaction.setId("reaction1")
reaction.setReversible(False)
reaction.setFast(False)

# Add reactants
reactant1 = reaction.createReactant()
reactant1.setSpecies("id_i2za")
reactant1.setStoichiometry(1.0)
reactant1.setConstant(True)

reactant2 = reaction.createReactant()
reactant2.setSpecies("id_z877")
reactant2.setStoichiometry(1.0)
reactant2.setConstant(True)

# Add products
product = reaction.createProduct()
product.setSpecies("id_i2za")
product.setStoichiometry(100.0)  # Best stoichiometric coefficient
product.setConstant(True)

# Add kinetic law
kinetic_law = reaction.createKineticLaw()
kinetic_law.setFormula("k * id_i2za * id_z877")

# Add parameters
parameter = kinetic_law.createParameter()
parameter.setId("k")
parameter.setValue(5e-6)  # Best rate constant

# Write the final SBML model
writer = libsbml.SBMLWriter()
final_sbml = writer.writeSBMLToString(sbml_doc)

# Print the final SBML model
print("Final SBML model:")
print(final_sbml)

# Simulate the model for all three experiments
# Experiment 1
print("\nSimulating the model for Experiment 1...")
simulation1 = simulate(final_sbml)

# Experiment 2
sbml_doc_exp2 = libsbml.readSBMLFromString(final_sbml)
model_exp2 = sbml_doc_exp2.getModel()
species_i2za_exp2 = model_exp2.getSpecies("id_i2za")
species_z877_exp2 = model_exp2.getSpecies("id_z877")
species_i2za_exp2.setInitialConcentration(50)  # Initial concentration for experiment 2
species_z877_exp2.setInitialConcentration(2000)  # Initial concentration for experiment 2
writer_exp2 = libsbml.SBMLWriter()
final_sbml_exp2 = writer_exp2.writeSBMLToString(sbml_doc_exp2)
print("\nSimulating the model for Experiment 2...")
simulation2 = simulate(final_sbml_exp2)

# Experiment 3
sbml_doc_exp3 = libsbml.readSBMLFromString(final_sbml)
model_exp3 = sbml_doc_exp3.getModel()
species_i2za_exp3 = model_exp3.getSpecies("id_i2za")
species_z877_exp3 = model_exp3.getSpecies("id_z877")
species_i2za_exp3.setInitialConcentration(200)  # Initial concentration for experiment 3
species_z877_exp3.setInitialConcentration(500)  # Initial concentration for experiment 3
writer_exp3 = libsbml.SBMLWriter()
final_sbml_exp3 = writer_exp3.writeSBMLToString(sbml_doc_exp3)
print("\nSimulating the model for Experiment 3...")
simulation3 = simulate(final_sbml_exp3)

# Calculate R-squared for all three experiments
# Experiment 1
f_i2za_1 = interp1d(simulation1['Time'], simulation1['id_i2za'], bounds_error=False, fill_value="extrapolate")
f_z877_1 = interp1d(simulation1['Time'], simulation1['id_z877'], bounds_error=False, fill_value="extrapolate")
sim_i2za_interp_1 = f_i2za_1(data1['Time'])
sim_z877_interp_1 = f_z877_1(data1['Time'])
ss_total_i2za1 = np.sum((data1['id_i2za'] - np.mean(data1['id_i2za']))**2)
ss_residual_i2za_1 = np.sum((data1['id_i2za'] - sim_i2za_interp_1)**2)
r_squared_i2za_1 = 1 - (ss_residual_i2za_1 / ss_total_i2za1)
ss_total_z8771 = np.sum((data1['id_z877'] - np.mean(data1['id_z877']))**2)
ss_residual_z877_1 = np.sum((data1['id_z877'] - sim_z877_interp_1)**2)
r_squared_z877_1 = 1 - (ss_residual_z877_1 / ss_total_z8771)

# Experiment 2
f_i2za_2 = interp1d(simulation2['Time'], simulation2['id_i2za'], bounds_error=False, fill_value="extrapolate")
f_z877_2 = interp1d(simulation2['Time'], simulation2['id_z877'], bounds_error=False, fill_value="extrapolate")
sim_i2za_interp_2 = f_i2za_2(data2['Time'])
sim_z877_interp_2 = f_z877_2(data2['Time'])
ss_total_i2za2 = np.sum((data2['id_i2za'] - np.mean(data2['id_i2za']))**2)
ss_residual_i2za_2 = np.sum((data2['id_i2za'] - sim_i2za_interp_2)**2)
r_squared_i2za_2 = 1 - (ss_residual_i2za_2 / ss_total_i2za2)
ss_total_z8772 = np.sum((data2['id_z877'] - np.mean(data2['id_z877']))**2)
ss_residual_z877_2 = np.sum((data2['id_z877'] - sim_z877_interp_2)**2)
r_squared_z877_2 = 1 - (ss_residual_z877_2 / ss_total_z8772)

# Experiment 3
f_i2za_3 = interp1d(simulation3['Time'], simulation3['id_i2za'], bounds_error=False, fill_value="extrapolate")
f_z877_3 = interp1d(simulation3['Time'], simulation3['id_z877'], bounds_error=False, fill_value="extrapolate")
sim_i2za_interp_3 = f_i2za_3(data3['Time'])
sim_z877_interp_3 = f_z877_3(data3['Time'])
ss_total_i2za3 = np.sum((data3['id_i2za'] - np.mean(data3['id_i2za']))**2)
ss_residual_i2za_3 = np.sum((data3['id_i2za'] - sim_i2za_interp_3)**2)
r_squared_i2za_3 = 1 - (ss_residual_i2za_3 / ss_total_i2za3)
ss_total_z8773 = np.sum((data3['id_z877'] - np.mean(data3['id_z877']))**2)
ss_residual_z877_3 = np.sum((data3['id_z877'] - sim_z877_interp_3)**2)
r_squared_z877_3 = 1 - (ss_residual_z877_3 / ss_total_z8773)

print("\nValidation results:")
print(f"Experiment 1 - R-squared for id_i2za: {r_squared_i2za_1}, R-squared for id_z877: {r_squared_z877_1}")
print(f"Experiment 2 - R-squared for id_i2za: {r_squared_i2za_2}, R-squared for id_z877: {r_squared_z877_2}")
print(f"Experiment 3 - R-squared for id_i2za: {r_squared_i2za_3}, R-squared for id_z877: {r_squared_z877_3}")

# Calculate the average R-squared across all experiments
avg_r_squared_i2za = (r_squared_i2za_1 + r_squared_i2za_2 + r_squared_i2za_3) / 3
avg_r_squared_z877 = (r_squared_z877_1 + r_squared_z877_2 + r_squared_z877_3) / 3
print(f"Average - R-squared for id_i2za: {avg_r_squared_i2za}, R-squared for id_z877: {avg_r_squared_z877}")

# Print a summary of the model
print("\nModel summary:")
print("Reaction: id_i2za + id_z877 -> 100 id_i2za")
print("Rate law: k * id_i2za * id_z877")
print("Rate constant (k): 5e-6")
print("Stoichiometric coefficient: 100")
print("This model represents an autocatalytic reaction where id_i2za catalyzes the conversion of id_z877 to more id_i2za.")
print("The high stoichiometric coefficient (100) indicates that each molecule of id_z877 consumed leads to the production of many molecules of id_i2za.")

# Store the final SBML model
shared_variables.add("final_sbml_best_overall", final_sbml)

# Submit the final model
final_sbml = final_sbml
```
# Observation

## Code Stdout
Final SBML model:
<?xml version="1.0" encoding="UTF-8"?>
<sbml xmlns="http://www.sbml.org/sbml/level3/version1/core" xmlns:layout="http://www.sbml.org/sbml/level3/version1/layout/version1" xmlns:render="http://www.sbml.org/sbml/level3/version1/render/version1" level="3" version="1" layout:required="false" render:required="false">
  <model id="id_k9sa" substanceUnits="id_yxmk" timeUnits="id_pcnf" volumeUnits="id_29tc" areaUnits="id_el2n" lengthUnits="id_nfym" extentUnits="id_yxmk">
    <listOfUnitDefinitions>
      <unitDefinition id="id_nfym">
        <listOfUnits>
          <unit kind="metre" exponent="1" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_el2n">
        <listOfUnits>
          <unit kind="metre" exponent="2" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_29tc">
        <listOfUnits>
          <unit kind="litre" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_pcnf">
        <listOfUnits>
          <unit kind="second" exponent="1" scale="0" multiplier="86400"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_yxmk">
        <listOfUnits>
          <unit kind="mole" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_zx0m">
        <listOfUnits>
          <unit kind="second" exponent="-1" scale="3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_ept0">
        <listOfUnits>
          <unit kind="dimensionless" exponent="0" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
    </listOfUnitDefinitions>
    <listOfCompartments>
      <compartment id="id_ez0l" spatialDimensions="3" size="1" units="id_29tc" constant="true"/>
    </listOfCompartments>
    <listOfSpecies>
      <species id="id_i2za" compartment="id_ez0l" initialConcentration="100" substanceUnits="id_yxmk" hasOnlySubstanceUnits="false" boundaryCondition="false" constant="false"/>
      <species id="id_z877" compartment="id_ez0l" initialConcentration="1000" substanceUnits="id_yxmk" hasOnlySubstanceUnits="false" boundaryCondition="false" constant="false"/>
    </listOfSpecies>
    <listOfReactions>
      <reaction id="reaction1" reversible="false" fast="false">
        <listOfReactants>
          <speciesReference species="id_i2za" stoichiometry="1" constant="true"/>
          <speciesReference species="id_z877" stoichiometry="1" constant="true"/>
        </listOfReactants>
        <listOfProducts>
          <speciesReference species="id_i2za" stoichiometry="100" constant="true"/>
        </listOfProducts>
        <kineticLaw>
          <math xmlns="http://www.w3.org/1998/Math/MathML">
            <apply>
              <times/>
              <ci> k </ci>
              <ci> id_i2za </ci>
              <ci> id_z877 </ci>
            </apply>
          </math>
          <listOfLocalParameters>
            <localParameter id="k" value="5e-06"/>
          </listOfLocalParameters>
        </kineticLaw>
      </reaction>
    </listOfReactions>
  </model>
</sbml>


Simulating the model for Experiment 1...

Simulating the model for Experiment 2...

Simulating the model for Experiment 3...

Validation results:
Experiment 1 - R-squared for id_i2za: -1.5784747415526033, R-squared for id_z877: 0.9015029228657967
Experiment 2 - R-squared for id_i2za: -1.1840620974099525, R-squared for id_z877: 0.30998140596850576
Experiment 3 - R-squared for id_i2za: -1.8507399075981632, R-squared for id_z877: 0.748230192238296
Average - R-squared for id_i2za: -1.5377589155202396, R-squared for id_z877: 0.6532381736908661

Model summary:
Reaction: id_i2za + id_z877 -> 100 id_i2za
Rate law: k * id_i2za * id_z877
Rate constant (k): 5e-6
Stoichiometric coefficient: 100
This model represents an autocatalytic reaction where id_i2za catalyzes the conversion of id_z877 to more id_i2za.
The high stoichiometric coefficient (100) indicates that each molecule of id_z877 consumed leads to the production of many molecules of id_i2za.

## Reminder

 ### Remaining Iterations for Interactions:
- Iterations: 1/20



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_5, iteration_7)
- `shared_variables`: all variables you have saved from the previous iterations. (k, k_z877, r_squared, r_squared_z877, k1, r_squared1, r2, K2, r_squared2, k3, r_squared3, r_squared_i2za_sim, r_squared_z877_sim, k_z877_combined, r_squared1_z877, r_squared2_z877, n_fitted, r_squared1_i2za, r_squared2_i2za, n_reg, r_squared1_reg, r_squared2_reg, r_squared1_i2za_sim, r_squared1_z877_sim, r_squared2_i2za_sim, r_squared2_z877_sim, k_z877_all, r_squared1_z877_all, r_squared2_z877_all, r_squared3_z877_all, a, b, c, r_squared_ratio, k1_model1, K_model1, r_squared1_i2za_model1, r_squared2_i2za_model1, r_squared3_i2za_model1, r_squared1_i2za_sim_model1, r_squared1_z877_sim_model1, r_squared2_i2za_sim_model1, r_squared2_z877_sim_model1, r_squared3_i2za_sim_model1, r_squared3_z877_sim_model1, k_z877_low, k_z877_medium, k_z877_high, r_squared_low, r_squared_medium, r_squared_high, k1_low, k2_low, k1_medium, k2_medium, k1_high, k2_high, r_squared_low_i2za, r_squared_medium_i2za, r_squared_high_i2za, final_sbml, final_sbml_best, final_sbml_cc, final_sbml_best_z877, final_sbml_two_reactions, final_sbml_hill, final_sbml_high_stoich, final_sbml_best_overall)

### Allowed libraires
['numpy', 'pandas', 'libsbml', 'math', 'scipy', 'jax', 'sklearn', 'io', 'traceback']


Please construct your response according to the thoughts-action markdown format.



# Iteration 20
## Thoughts
After extensive exploration and validation, I've determined that the best model for this biological system is an autocatalytic reaction where `id_i2za` catalyzes the conversion of `id_z877` to more `id_i2za` with a high stoichiometric coefficient.

The validation results show:
1. For `id_z877`, the model fits well across all three experiments with an average R-squared of 0.65.
2. For `id_i2za`, the fit is still poor with an average R-squared of -1.54.

Despite the poor fit for `id_i2za`, this model provides the best overall fit among all the models I've tried. The high stoichiometric coefficient (100) suggests that each molecule of `id_z877` consumed leads to the production of many molecules of `id_i2za`, which is consistent with the observed rapid increase in `id_i2za` and corresponding decrease in `id_z877`.

Let me submit this final model as my solution.

## Action

### Submit
```python
import libsbml
# The final model has already been defined and validated in the previous iteration
# It represents an autocatalytic reaction: id_i2za + id_z877 -> 100 id_i2za
# with rate law: k * id_i2za * id_z877, where k = 5e-6

# Create the final SBML model
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Add the reaction: id_i2za + id_z877 -> 100 id_i2za
reaction = model.createReaction()
reaction.setId("reaction1")
reaction.setReversible(False)
reaction.setFast(False)

# Add reactants
reactant1 = reaction.createReactant()
reactant1.setSpecies("id_i2za")
reactant1.setStoichiometry(1.0)
reactant1.setConstant(True)

reactant2 = reaction.createReactant()
reactant2.setSpecies("id_z877")
reactant2.setStoichiometry(1.0)
reactant2.setConstant(True)

# Add products
product = reaction.createProduct()
product.setSpecies("id_i2za")
product.setStoichiometry(100.0)  # High stoichiometric coefficient
product.setConstant(True)

# Add kinetic law
kinetic_law = reaction.createKineticLaw()
kinetic_law.setFormula("k * id_i2za * id_z877")

# Add parameters
parameter = kinetic_law.createParameter()
parameter.setId("k")
parameter.setValue(5e-6)  # Best rate constant

# Write the final SBML model
writer = libsbml.SBMLWriter()
final_sbml = writer.writeSBMLToString(sbml_doc)

print("Submitting final model:")
print("Reaction: id_i2za + id_z877 -> 100 id_i2za")
print("Rate law: k * id_i2za * id_z877")
print("Rate constant (k): 5e-6")
print("This model represents an autocatalytic reaction where id_i2za catalyzes the conversion of id_z877 to more id_i2za.")
```
