
                # 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/level2/version4" level="2" version="4">
  <model id="id_b301">
    <listOfUnitDefinitions>
      <unitDefinition id="id_pv52">
        <listOfUnits>
          <unit kind="litre" exponent="1" scale="-3" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_9whl">
        <listOfUnits>
          <unit kind="second" exponent="1" scale="0" multiplier="86400"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_l1cp">
        <listOfUnits>
          <unit kind="item" exponent="1" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
    </listOfUnitDefinitions>
    <listOfCompartments>
      <compartment id="id_ag6d" spatialDimensions="3" size="1" constant="true"/>
    </listOfCompartments>
    <listOfSpecies>
      <species id="id_l6et" compartment="id_ag6d" initialConcentration="1000" boundaryCondition="false" constant="false"/>
      <species id="id_ep2m" compartment="id_ag6d" initialConcentration="200" boundaryCondition="false" constant="false"/>
      <species id="id_wnsa" compartment="id_ag6d" initialConcentration="1" boundaryCondition="false" constant="false"/>
      <species id="id_2ary" compartment="id_ag6d" initialConcentration="1" boundaryCondition="false" constant="false"/>
    </listOfSpecies>
  </model>
</sbml>


                ## Max iterations
                20

                Format your response according to the instructions in the system message.
                
## Thoughts
The initial SBML model contains four species but no reactions. To understand the system's dynamics, I need to observe how the concentrations of these species change over time in the real system. This will tell me if any reactions are occurring and give me clues about which species are involved.

My first action will be to request an 'observe' experiment. This will run the system with the default initial concentrations provided in the SBML model and record the time-series data.

## 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: 1001 rows × 5 columns. 

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

Time   id_l6et  id_ep2m  id_wnsa  id_2ary
0  1.00e+03 2.00e+02 1.00e+00 1.00e+00
...
1.60e+01  8.74e+02 5.18e-08 2.50e+00 6.15e-10
...
3.20e+01  7.36e+02 1.18e+01 8.01e+00 1.09e-01
...
4.80e+01  7.75e+02 6.95e-07 1.62e+00 8.36e-09
...
6.40e+01  6.44e+02 1.55e-01 6.21e+00 1.52e-03
...
8.00e+01  6.97e+02 4.67e-06 1.25e+00 5.63e-08
...
9.60e+01  5.71e+02 3.15e-01 5.38e+00 3.13e-03
...
1.12e+02  6.34e+02 2.31e-05 1.09e+00 2.77e-07
...
1.28e+02  5.15e+02 2.09e+00 4.85e+00 2.10e-02
...
1.44e+02  5.87e+02 1.57e-04 9.79e-01 1.87e-06
...
1.60e+02  4.77e+02 4.34e+00 4.23e+00 4.44e-02
...
1.76e+02  5.54e+02 1.90e-03 8.55e-01 2.26e-05
...
1.92e+02  4.55e+02 1.21e+00 3.40e+00 1.27e-02
...
2.08e+02  5.36e+02 4.89e-02 6.87e-01 5.84e-04
...
2.24e+02  4.50e+02 1.30e-01 2.43e+00 1.43e-03
...
2.40e+02  5.31e+02 4.00e+00 4.92e-01 4.81e-02
...
2.56e+02  4.58e+02 5.12e-02 1.53e+00 5.83e-04
...
2.72e+02  4.99e+02 1.33e+03 6.15e-01 1.58e+01
...
2.88e+02  4.78e+02 5.24e-01 8.41e-01 6.16e-03
...
3.04e+02  4.21e+02 3.82e+00 2.02e+00 4.23e-02
...
3.20e+02  4.97e+02 2.95e+02 4.82e-01 3.53e+00
...
3.36e+02  4.61e+02 2.65e+00 8.80e-01 3.10e-02
...
3.52e+02  4.19e+02 7.85e+00 1.70e+00 8.80e-02
...
3.68e+02  4.34e+02 1.31e+03 1.20e+00 1.51e+01
...
3.84e+02  4.68e+02 5.34e+01 6.45e-01 6.31e-01
...
4.00e+02  4.40e+02 1.03e+01 1.03e+00 1.19e-01
...
4.16e+02  4.14e+02 2.91e+01 1.56e+00 3.28e-01
...
4.32e+02  4.06e+02 5.21e+02 1.72e+00 5.83e+00
...
4.48e+02  4.52e+02 5.05e+02 7.56e-01 5.93e+00
...
4.64e+02  4.51e+02 9.96e+01 7.59e-01 1.17e+00
...
4.80e+02  4.37e+02 4.27e+01 9.69e-01 4.95e-01
...
4.96e+02  4.22e+02 4.67e+01 1.23e+00 5.34e-01
...
5.12e+02  4.11e+02 1.10e+02 1.47e+00 1.25e+00
...
5.28e+02  4.10e+02 3.45e+02 1.49e+00 3.90e+00
...
5.44e+02  4.24e+02 5.67e+02 1.15e+00 6.52e+00
...
5.60e+02  4.38e+02 3.90e+02 9.04e-01 4.54e+00
...
5.76e+02  4.40e+02 2.11e+02 8.74e-01 2.46e+00
...
5.92e+02  4.35e+02 1.31e+02 9.42e-01 1.53e+00
...
6.08e+02  4.29e+02 1.06e+02 1.05e+00 1.22e+00
...
6.24e+02  4.23e+02 1.11e+02 1.16e+00 1.28e+00
...
6.40e+02  4.18e+02 1.45e+02 1.26e+00 1.66e+00
...
6.56e+02  4.16e+02 2.12e+02 1.31e+00 2.42e+00
...
6.72e+02  4.17e+02 3.01e+02 1.28e+00 3.43e+00
...
6.88e+02  4.21e+02 3.60e+02 1.18e+00 4.13e+00
...
7.04e+02  4.26e+02 3.51e+02 1.08e+00 4.04e+00
...
7.20e+02  4.30e+02 2.96e+02 1.02e+00 3.42e+00
...
7.36e+02  4.30e+02 2.38e+02 1.01e+00 2.75e+00
...
7.52e+02  4.29e+02 1.98e+02 1.03e+00 2.29e+00
...
7.68e+02  4.27e+02 1.77e+02 1.07e+00 2.04e+00
...
8.00e+02  4.22e+02 1.84e+02 1.16e+00 2.11e+00

## 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
The observation data from `iteration_0` shows dynamic changes in all species concentrations, confirming that reactions are happening. The concentrations of `id_wnsa` and `id_2ary` generally increase, while `id_l6et` generally decreases. `id_ep2m` shows significant fluctuations, including sharp increases and decreases.

To better understand the relationships between the species and infer potential reactions, I will visualize the time-series data. Plotting the concentrations over time should help identify correlations or anti-correlations between species, which can suggest reactant-product relationships. The large spikes in `id_ep2m` are particularly noteworthy and might indicate a non-linear reaction or a reaction with a high rate constant that is only active under certain conditions.

I will use the code interpreter to load the data from `experiment_history['iteration_0']` and create a plot.

## Action

### Code
```python
import matplotlib.pyplot as plt
import pandas as pd

# Load the data from the first experiment
df = experiment_history['iteration_0']

# Plot the time series data
plt.figure(figsize=(12, 8))
plt.plot(df['Time'], df['id_l6et'], label='id_l6et')
plt.plot(df['Time'], df['id_ep2m'], label='id_ep2m')
plt.plot(df['Time'], df['id_wnsa'], label='id_wnsa')
plt.plot(df['Time'], df['id_2ary'], label='id_2ary')
plt.xlabel('Time')
plt.ylabel('Concentration')
plt.title('Species Concentrations Over Time (Observation)')
plt.legend()
plt.grid(True)
plt.show()

# Print the column names to confirm species IDs
print(df.columns)
```
# 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
My apologies, I used an unauthorized library (`matplotlib`). I need to stick to the allowed libraries (`numpy`, `pandas`, `libsbml`, `math`, `scipy`, `jax`, `sklearn`, `io`, `traceback`).

While I cannot directly plot, I can still analyze the data numerically using pandas and numpy. I can look at the changes in concentrations over time, calculate rates of change, and look for correlations between species concentrations.

Let's examine the data more closely in the code interpreter. I'll calculate the change in concentration between consecutive time points for each species and look at the relationships between these changes. This might help identify potential reactions and their stoichiometry.

## Action

### Code
```python
import pandas as pd
import numpy as np

# Load the data from the first experiment
df = experiment_history['iteration_0']

# Calculate the difference between consecutive time points
df_diff = df.diff()

# Drop the first row which contains NaNs
df_diff = df_diff.dropna()

# Calculate the ratio of changes between species
# This can give clues about stoichiometry
# Let's look at changes relative to id_l6et, as it seems to be decreasing
# Avoid division by zero or very small numbers by adding a small epsilon
epsilon = 1e-9
df_diff_ratio_l6et = df_diff.drop(columns=['Time']).apply(lambda x: x / (df_diff['id_l6et'] + epsilon))

# Print some statistics about the changes and ratios
print("Mean changes in concentrations:")
print(df_diff.mean())

print("\nCorrelation matrix of concentration changes:")
print(df_diff.corr())

# Print some rows of the difference and ratio dataframes to inspect
print("\nFirst 5 rows of concentration changes:")
print(df_diff.head())

print("\nFirst 5 rows of concentration change ratios relative to id_l6et:")
print(df_diff_ratio_l6et.head())

# Let's also look at the points where id_ep2m has large changes
large_ep2m_changes = df_diff[abs(df_diff['id_ep2m']) > 100]
print("\nRows with large changes in id_ep2m:")
print(large_ep2m_changes)
```
# Observation

## Code Stdout
Mean changes in concentrations:
id_l6et   -0.577508
id_ep2m   -0.016256
id_wnsa    0.000159
id_2ary    0.001109
Time       0.800000
dtype: float64

Correlation matrix of concentration changes:
          id_l6et   id_ep2m   id_wnsa   id_2ary      Time
id_l6et  1.000000 -0.052041 -0.830168 -0.008970  0.001592
id_ep2m -0.052041  1.000000 -0.343970  0.996051  0.013476
id_wnsa -0.830168 -0.343970  1.000000 -0.384979 -0.000407
id_2ary -0.008970  0.996051 -0.384979  1.000000  0.013273
Time     0.001592  0.013476 -0.000407  0.013273  1.000000

First 5 rows of concentration changes:
     id_l6et      id_ep2m   id_wnsa    id_2ary  Time
1 -14.103043   488.660317  0.180227   7.719520   0.8
2 -70.508243  1820.513322  2.545016  20.287286   0.8
3 -56.582137 -2251.112413  5.575030 -26.658683   0.8
4  -1.417493  -252.636112 -0.328155  -2.298316   0.8
5   1.079064    -5.260516 -0.681579  -0.048252   0.8

First 5 rows of concentration change ratios relative to id_l6et:
   id_l6et     id_ep2m   id_wnsa   id_2ary
1      1.0  -34.649281 -0.012779 -0.547366
2      1.0  -25.819865 -0.036095 -0.287729
3      1.0   39.784860 -0.098530  0.471150
4      1.0  178.227412  0.231503  1.621395
5      1.0   -4.875074 -0.631639 -0.044716

Rows with large changes in id_ep2m:
       id_l6et      id_ep2m   id_wnsa    id_2ary  Time
1   -14.103043   488.660317  0.180227   7.719520   0.8
2   -70.508243  1820.513322  2.545016  20.287286   0.8
3   -56.582137 -2251.112413  5.575030 -26.658683   0.8
4    -1.417493  -252.636112 -0.328155  -2.298316   0.8
36   -1.653804   114.890673 -0.026567   1.453513   0.8
..         ...          ...       ...        ...   ...
638  -0.358547  -102.831578  0.013199  -1.160759   0.8
655  -4.386992   105.990596  0.064801   1.217974   0.8
659  -5.177175  -102.944547  0.123332  -1.206792   0.8
660  -2.901176  -117.062162  0.075866  -1.342461   0.8
683  -2.624342  -101.502791  0.067092  -1.165515   0.8

[146 rows x 5 columns]

## 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. ()

### 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
The analysis of concentration changes provides some interesting clues.
- `id_l6et` generally decreases (mean change is negative).
- `id_wnsa` and `id_2ary` generally increase (mean changes are positive, though small).
- `id_ep2m` has a near-zero mean change, but the "Rows with large changes in id_ep2m" output shows very large positive and negative changes, indicating rapid fluctuations.
- The correlation matrix shows a strong negative correlation between `id_l6et` and `id_wnsa` (-0.83), suggesting `id_l6et` might be consumed while `id_wnsa` is produced in a reaction.
- There is a very strong positive correlation between `id_ep2m` and `id_2ary` (0.996), suggesting they are produced or consumed together, possibly in the same reaction or a pair of tightly coupled reactions.
- The correlation between `id_l6et` and `id_ep2m` is weak (-0.05), and between `id_l6et` and `id_2ary` is also weak (-0.008). This suggests that the reaction(s) involving `id_ep2m` and `id_2ary` might not directly involve `id_l6et` as a primary reactant or product, or the relationship is more complex (e.g., catalytic).
- The correlation between `id_wnsa` and `id_ep2m` (-0.34) and `id_wnsa` and `id_2ary` (-0.38) is negative but weaker than the `l6et`/`wnsa` and `ep2m`/`2ary` correlations.

Based on these observations, I can hypothesize the following:
1. A reaction where `id_l6et` is consumed and `id_wnsa` is produced. Let's call this Reaction 1.
2. A reaction (or reactions) involving `id_ep2m` and `id_2ary` where they are produced or consumed together. The rapid fluctuations in `id_ep2m` suggest a potentially reversible reaction or a reaction with complex kinetics. The strong positive correlation suggests they are on the same side of a reaction (both products or both reactants) or produced/consumed in a 1:1 ratio in separate reactions. Given the overall increase in `id_2ary` and the large fluctuations in `id_ep2m`, it's plausible they are products of a reaction, or `id_ep2m` is an intermediate or catalyst in a cycle that produces `id_2ary`.

Let's start by modeling the first hypothesis: `id_l6et` -> `id_wnsa`. I'll add a simple irreversible mass-action reaction for this. I'll need to use `libsbml` to modify the `input_sbml_string`. I'll add a parameter for the rate constant, but I won't know the value yet. I'll start with a placeholder value.

After adding this reaction, I will simulate the updated model and compare the simulation results with the experimental data, particularly focusing on `id_l6et` and `id_wnsa`.

## Action

### Code
```python
import libsbml
import pandas as pd

# Load the initial SBML model
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Add a parameter for the rate constant
k1 = model.createParameter()
k1.setId("k1")
k1.setValue(0.1) # Placeholder value
k1.setConstant(False) # Rate constants are typically constant, but let's keep it flexible for now

# Create the first reaction: id_l6et -> id_wnsa
reaction1 = model.createReaction()
reaction1.setId("reaction1")
reaction1.setReversible(False)
reaction1.setFast(False)

# Add reactant: id_l6et
reactant1 = reaction1.createReactant()
reactant1.setSpecies("id_l6et")
reactant1.setStoichiometry(1.0)
reactant1.setConstant(False)

# Add product: id_wnsa
product1 = reaction1.createProduct()
product1.setSpecies("id_wnsa")
product1.setStoichiometry(1.0)
product1.setConstant(False)

# Set the kinetic law for reaction1 (mass action)
kinetic_law1 = reaction1.createKineticLaw()
kinetic_law1.setFormula("k1 * id_l6et")

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

# Simulate the updated model
sim_df = simulate(updated_sbml)

# Load the experimental data
exp_df = experiment_history['iteration_0']

# Compare simulation and experiment (visually or numerically)
# For now, let's print the head of the simulation data
print("Simulation results (first 5 rows):")
print(sim_df.head())

# We can also calculate the difference between simulation and experiment
# Need to merge/align dataframes by time
merged_df = pd.merge(sim_df, exp_df, on='Time', suffixes=('_sim', '_exp'))
print("\nDifference between simulation and experiment (first 5 rows):")
print(merged_df.head())

# Store the updated SBML for future use
shared_variables.add("current_sbml", updated_sbml)
```
# Observation

## Code Stdout
Simulation results (first 5 rows):
       id_l6et  id_ep2m     id_wnsa  id_2ary  Time
0  1000.000000    200.0    1.000000      1.0   0.0
1   923.116381    200.0   77.883619      1.0   0.8
2   852.144279    200.0  148.855721      1.0   1.6
3   786.628585    200.0  214.371415      1.0   2.4
4   726.149471    200.0  274.850529      1.0   3.2

Difference between simulation and experiment (first 5 rows):
   id_l6et_sim  id_ep2m_sim  id_wnsa_sim  ...  id_ep2m_exp  id_wnsa_exp  id_2ary_exp
0  1000.000000        200.0     1.000000  ...   200.000000     1.000000     1.000000
1   923.116381        200.0    77.883619  ...   688.660317     1.180227     8.719520
2   852.144279        200.0   148.855721  ...  2509.173639     3.725243    29.006806
3   786.628585        200.0   214.371415  ...   258.061226     9.300273     2.348123
4   726.149471        200.0   274.850529  ...     5.425115     8.972119     0.049807

[5 rows x 9 columns]

## 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. (current_sbml)

### 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
The simulation results with the first hypothesized reaction (`id_l6et` -> `id_wnsa`) show that `id_l6et` decreases and `id_wnsa` increases, which is consistent with the general trend in the experimental data. However, the simulated concentrations of `id_ep2m` and `id_2ary` remain constant at their initial values because no reactions involving them have been added yet. This clearly shows that the model is incomplete.

The experimental data showed a strong correlation between `id_ep2m` and `id_2ary`, and rapid fluctuations in `id_ep2m`. This suggests a reaction or set of reactions involving these two species. Given the strong positive correlation, they are likely produced or consumed together. The fluctuations in `id_ep2m` could be due to a reversible reaction or a reaction with non-linear kinetics.

Let's consider the possibility of a reversible reaction involving `id_ep2m` and `id_2ary`. Since their concentrations generally increase or fluctuate together, perhaps they are products of a reaction, or involved in a cycle. The large spikes in `id_ep2m` followed by decreases are intriguing.

Let's hypothesize a second reaction where `id_l6et` is consumed to produce `id_ep2m` and `id_2ary`. This would explain the decrease in `id_l6et` and the increase in `id_ep2m` and `id_2ary`. The fluctuations in `id_ep2m` might then be explained by a subsequent reaction that consumes `id_ep2m`.

Let's try adding a second irreversible reaction: `id_l6et` -> `id_ep2m` + `id_2ary`. I'll use mass action kinetics again with a new rate constant `k2`. I'll keep the first reaction (`id_l6et` -> `id_wnsa`) as well.

I will update the SBML model stored in `shared_variables['current_sbml']`, add the new reaction and parameter, and then simulate the updated model.

## Action

### Code
```python
import libsbml
import pandas as pd

# Load the current SBML model from shared_variables
sbml_doc = libsbml.readSBMLFromString(shared_variables.access("current_sbml"))
model = sbml_doc.getModel()

# Add a parameter for the rate constant of the second reaction
k2 = model.createParameter()
k2.setId("k2")
k2.setValue(0.05) # Placeholder value
k2.setConstant(False)

# Create the second reaction: id_l6et -> id_ep2m + id_2ary
reaction2 = model.createReaction()
reaction2.setId("reaction2")
reaction2.setReversible(False)
reaction2.setFast(False)

# Add reactant: id_l6et
reactant2 = reaction2.createReactant()
reactant2.setSpecies("id_l6et")
reactant2.setStoichiometry(1.0)
reactant2.setConstant(False)

# Add product: id_ep2m
product2_ep2m = reaction2.createProduct()
product2_ep2m.setSpecies("id_ep2m")
product2_ep2m.setStoichiometry(1.0)
product2_ep2m.setConstant(False)

# Add product: id_2ary
product2_2ary = reaction2.createProduct()
product2_2ary.setSpecies("id_2ary")
product2_2ary.setStoichiometry(1.0)
product2_2ary.setConstant(False)


# Set the kinetic law for reaction2 (mass action)
kinetic_law2 = reaction2.createKineticLaw()
kinetic_law2.setFormula("k2 * id_l6et")

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

# Simulate the updated model
sim_df = simulate(updated_sbml)

# Load the experimental data
exp_df = experiment_history['iteration_0']

# Compare simulation and experiment (visually or numerically)
# For now, let's print the head of the simulation data
print("Simulation results (first 5 rows):")
print(sim_df.head())

# Calculate the difference between simulation and experiment
merged_df = pd.merge(sim_df, exp_df, on='Time', suffixes=('_sim', '_exp'))
print("\nDifference between simulation and experiment (first 5 rows):")
print(merged_df.head())

# Store the updated SBML for future use
shared_variables.add("current_sbml", updated_sbml)
```
# Observation

## Code Stdout
Simulation results (first 5 rows):
       id_l6et     id_ep2m     id_wnsa     id_2ary  Time
0  1000.000000  200.000000    1.000000    1.000000   0.0
1   886.920813  237.693062   76.386124   38.693062   0.8
2   786.628195  271.123935  143.247870   72.123935   1.6
3   697.676677  300.774441  202.548882  101.774441   2.4
4   618.783507  327.072164  255.144329  128.072164   3.2

Difference between simulation and experiment (first 5 rows):
   id_l6et_sim  id_ep2m_sim  id_wnsa_sim  ...  id_ep2m_exp  id_wnsa_exp  id_2ary_exp
0  1000.000000   200.000000     1.000000  ...   200.000000     1.000000     1.000000
1   886.920813   237.693062    76.386124  ...   688.660317     1.180227     8.719520
2   786.628195   271.123935   143.247870  ...  2509.173639     3.725243    29.006806
3   697.676677   300.774441   202.548882  ...   258.061226     9.300273     2.348123
4   618.783507   327.072164   255.144329  ...     5.425115     8.972119     0.049807

[5 rows x 9 columns]

## 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. (current_sbml)

### 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
The simulation results with the two irreversible reactions (`id_l6et` -> `id_wnsa` and `id_l6et` -> `id_ep2m` + `id_2ary`) show that `id_l6et` decreases, and `id_wnsa`, `id_ep2m`, and `id_2ary` increase. This is a better match to the general trends than the previous model, but it doesn't capture the rapid fluctuations observed in `id_ep2m` and `id_2ary` in the experimental data. The simulated concentrations increase smoothly, unlike the experimental data which shows sharp peaks and dips, especially for `id_ep2m`.

The strong positive correlation between `id_ep2m` and `id_2ary` and their fluctuating behavior, particularly the sharp increases and decreases in `id_ep2m`, strongly suggest a more complex interaction than simple irreversible production from `id_l6et`. This kind of behavior could be indicative of:
1. A reversible reaction involving `id_ep2m` and `id_2ary`.
2. A reaction where `id_ep2m` acts as a catalyst or intermediate in the production or consumption of `id_2ary`.
3. A reaction with non-linear kinetics, perhaps involving thresholds or feedback loops.

Let's reconsider the possibility of a reversible reaction. If `id_ep2m` and `id_2ary` are produced together, perhaps they can also react to form something else, or revert back to a previous state. Given the strong correlation, a reversible reaction between `id_ep2m` and `id_2ary` themselves seems less likely unless they are interconverted. A reversible reaction where they are products might look like `Something <=> id_ep2m + id_2ary`. What could "Something" be? `id_l6et` is being consumed, so perhaps the second reaction is reversible: `id_l6et <=> id_ep2m + id_2ary`.

Let's modify the second reaction to be reversible: `id_l6et <=> id_ep2m + id_2ary`. This would require a forward rate constant (`k2`) and a reverse rate constant (`k_reverse2`). The kinetic law for a reversible mass-action reaction `A <=> B + C` is typically `k_forward * [A] - k_reverse * [B] * [C]`.

I will update the SBML model again, changing `reaction2` to be reversible and updating its kinetic law. I'll add a new parameter `k_reverse2`.

## Action

### Code
```python
import libsbml
import pandas as pd

# Load the current SBML model from shared_variables
sbml_doc = libsbml.readSBMLFromString(shared_variables.access("current_sbml"))
model = sbml_doc.getModel()

# Find reaction2 and modify it
reaction2 = model.getReaction("reaction2")
if reaction2 is not None:
    reaction2.setReversible(True)

    # Add a parameter for the reverse rate constant
    k_reverse2 = model.createParameter()
    k_reverse2.setId("k_reverse2")
    k_reverse2.setValue(0.001) # Placeholder value
    k_reverse2.setConstant(False)

    # Update the kinetic law for reaction2
    kinetic_law2 = reaction2.getKineticLaw()
    # Assuming mass action for both forward and reverse
    # Forward: k2 * id_l6et
    # Reverse: k_reverse2 * id_ep2m * id_2ary
    kinetic_law2.setFormula("k2 * id_l6et - k_reverse2 * id_ep2m * id_2ary")

    # Ensure species are correctly listed as reactants/products with appropriate stoichiometry
    # Remove existing reactants/products first to avoid duplicates or incorrect settings
    while reaction2.getNumReactants() > 0:
        reaction2.removeReactant(0)
    while reaction2.getNumProducts() > 0:
        reaction2.removeProduct(0)

    # Add reactant: id_l6et
    reactant2 = reaction2.createReactant()
    reactant2.setSpecies("id_l6et")
    reactant2.setStoichiometry(1.0)
    reactant2.setConstant(False)

    # Add product: id_ep2m
    product2_ep2m = reaction2.createProduct()
    product2_ep2m.setSpecies("id_ep2m")
    product2_ep2m.setStoichiometry(1.0)
    product2_ep2m.setConstant(False)

    # Add product: id_2ary
    product2_2ary = reaction2.createProduct()
    product2_2ary.setSpecies("id_2ary")
    product2_2ary.setStoichiometry(1.0)
    product2_2ary.setConstant(False)

else:
    print("Reaction 'reaction2' not found. Cannot make it reversible.")


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

# Simulate the updated model
sim_df = simulate(updated_sbml)

# Load the experimental data
exp_df = experiment_history['iteration_0']

# Compare simulation and experiment
print("Simulation results (first 5 rows):")
print(sim_df.head())

merged_df = pd.merge(sim_df, exp_df, on='Time', suffixes=('_sim', '_exp'))
print("\nDifference between simulation and experiment (first 5 rows):")
print(merged_df.head())

# Store the updated SBML for future use
shared_variables.add("current_sbml", updated_sbml)
```
# Observation

## Code Stdout
Simulation results (first 5 rows):
       id_l6et     id_ep2m     id_wnsa    id_2ary  Time
0  1000.000000  200.000000    1.000000   1.000000   0.0
1   890.191233  234.332194   76.476572  35.332194   0.8
2   798.598789  258.484198  143.917013  59.484198   1.6
3   722.037267  274.309649  204.653083  75.309649   2.4
4   657.389216  283.850560  259.760224  84.850560   3.2

Difference between simulation and experiment (first 5 rows):
   id_l6et_sim  id_ep2m_sim  id_wnsa_sim  ...  id_ep2m_exp  id_wnsa_exp  id_2ary_exp
0  1000.000000   200.000000     1.000000  ...   200.000000     1.000000     1.000000
1   890.191233   234.332194    76.476572  ...   688.660317     1.180227     8.719520
2   798.598789   258.484198   143.917013  ...  2509.173639     3.725243    29.006806
3   722.037267   274.309649   204.653083  ...   258.061226     9.300273     2.348123
4   657.389216   283.850560   259.760224  ...     5.425115     8.972119     0.049807

[5 rows x 9 columns]

## 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)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml)

### 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
Making the second reaction reversible (`id_l6et <=> id_ep2m + id_2ary`) still doesn't reproduce the sharp fluctuations in `id_ep2m` and `id_2ary`. The simulated concentrations continue to change relatively smoothly. This suggests that a simple reversible mass-action kinetic might not be sufficient to capture the observed dynamics, especially the rapid spikes.

The strong correlation between `id_ep2m` and `id_2ary` and the large, rapid changes in `id_ep2m` could indicate a reaction that is highly sensitive to concentrations or involves some form of autocatalysis or activation/inhibition.

Let's look back at the correlation matrix of concentration changes. `id_ep2m` and `id_2ary` changes are highly correlated (0.996). `id_l6et` changes are negatively correlated with `id_wnsa` changes (-0.83). The changes in `id_ep2m`/`id_2ary` are weakly correlated with changes in `id_l6et` and `id_wnsa`.

This pattern suggests two relatively independent processes are occurring:
1. `id_l6et` is converted to `id_wnsa`.
2. `id_ep2m` and `id_2ary` are interconverted or produced/consumed together in a process that doesn't directly involve `id_l6et` or `id_wnsa` as primary reactants/products, or at least not with simple stoichiometry.

The rapid fluctuations in `id_ep2m` and `id_2ary` could be due to a reaction between `id_ep2m` and `id_2ary` themselves, or a reaction that is catalyzed by one of them, or a reaction that has a non-mass-action kinetic law.

Let's consider a reaction where `id_ep2m` and `id_2ary` are interconverted or react with each other. Given the strong positive correlation in their *changes*, it's more likely they are produced/consumed together. What if there's a reaction that consumes `id_ep2m` and produces `id_2ary`, or vice versa? The overall trend shows `id_2ary` increasing, while `id_ep2m` fluctuates but doesn't necessarily have a strong overall increase like `id_2ary`.

Let's hypothesize a third reaction that consumes `id_ep2m` and produces `id_2ary`. This could potentially explain the decrease in `id_ep2m` after the spikes and the continued increase in `id_2ary`. What could be the reactant that reacts with `id_ep2m`? Perhaps `id_l6et` is involved in this reaction too, but in a different way.

Let's try adding a third irreversible reaction: `id_ep2m` -> `id_2ary`. This doesn't seem right given the strong positive correlation in *changes*. If `ep2m` goes down, `2ary` goes up, which is a negative correlation. The data shows a positive correlation in changes.

Let's reconsider the possibility that `id_ep2m` and `id_2ary` are produced from something, and then `id_ep2m` is consumed. The first reaction `id_l6et` -> `id_wnsa` seems plausible based on the strong negative correlation between their changes. The second process involves `id_ep2m` and `id_2ary`.

What if `id_l6et` produces `id_wnsa` (Reaction 1), and `id_l6et` also produces `id_ep2m` and `id_2ary` (Reaction 2, possibly reversible or with complex kinetics)? The reversible reaction `id_l6et <=> id_ep2m + id_2ary` didn't produce the fluctuations.

Let's think about what could cause sharp spikes and drops. This often involves positive and negative feedback or a reaction that is highly sensitive to substrate concentration (e.g., Michaelis-Menten kinetics with a low Km, or Hill kinetics).

Could `id_ep2m` be an enzyme or catalyst? If `id_ep2m` catalyzes the conversion of `id_l6et` to `id_2ary`, that might explain some things. Reaction: `id_l6et` --(catalyzed by `id_ep2m`)--> `id_2ary`. This doesn't explain the production of `id_ep2m` itself or the strong `ep2m`/`2ary` correlation.

Let's go back to the idea of two main processes.
Process 1: `id_l6et` -> `id_wnsa` (Reaction 1, seems plausible).
Process 2: Involves `id_ep2m` and `id_2ary`. They are produced/consumed together. The fluctuations suggest a dynamic equilibrium or a process with varying rates.

What if `id_l6et` is converted into an intermediate, which then breaks down into `id_ep2m` and `id_2ary`, and the breakdown is somehow regulated? We are told not to add new species, so this intermediate idea is out.

Let's look at the large change points again. When `id_ep2m` increases sharply, `id_l6et` decreases, and `id_2ary` increases sharply. When `id_ep2m` decreases sharply, `id_l6et` might increase slightly or decrease less, and `id_2ary` decreases sharply. This reinforces the strong positive correlation between `id_ep2m` and `id_2ary` changes.

Could there be a reaction `id_l6et -> id_ep2m + id_2ary` (Reaction 2), and then another reaction that consumes `id_ep2m`? What would it produce? Maybe it converts `id_ep2m` back to `id_l6et`? `id_ep2m -> id_l6et`? This would be a cycle. `id_l6et -> id_ep2m + id_2ary` and `id_ep2m -> id_l6et`. This doesn't seem right as `id_l6et` is generally decreasing.

What if `id_ep2m` is consumed in a reaction that produces `id_wnsa`? `id_ep2m -> id_wnsa`? This would mean `id_ep2m` and `id_wnsa` changes are negatively correlated, which they are (-0.34), but the correlation is not as strong as `l6et`/`wnsa` or `ep2m`/`2ary`.

Let's reconsider the reversible reaction `id_l6et <=> id_ep2m + id_2ary`. The mass action kinetic might be too simple. What if the reverse reaction rate is not simply `k_reverse2 * id_ep2m * id_2ary`?

Let's try a different approach. Instead of guessing reaction types and kinetics, let's use the data to infer the rate laws. The rate of change of each species is the sum of the rates of reactions it participates in, multiplied by its stoichiometry.
d[l6et]/dt = - Rate1 - Rate2
d[ep2m]/dt = + Rate2 - Rate3? ...
d[wnsa]/dt = + Rate1
d[2ary]/dt = + Rate2 - Rate4? ...

From the correlation analysis, Rate1 seems to be proportional to `id_l6et` (mass action for `id_l6et` -> `id_wnsa`).
d[wnsa]/dt ≈ k1 * [id_l6et]
d[l6et]/dt ≈ - k1 * [id_l6et] - Rate2

The strong correlation between changes in `id_ep2m` and `id_2ary` suggests Rate2 involves both of them with positive stoichiometry.
d[ep2m]/dt = + Rate2 + ...
d[2ary]/dt = + Rate2 + ...

Let's assume Reaction 1 is `id_l6et` -> `id_wnsa` with rate `k1 * id_l6et`.
Let's assume Reaction 2 is `id_l6et` -> `id_ep2m` + `id_2ary` with rate `Rate2`.
d[l6et]/dt = - k1 * id_l6et - Rate2
d[wnsa]/dt = k1 * id_l6et
d[ep2m]/dt = Rate2 + Rate3?
d[2ary]/dt = Rate2 + Rate4?

If Reaction 2 is `id_l6et <=> id_ep2m + id_2ary`, then
d[l6et]/dt = - k1 * id_l6et - (k2 * id_l6et - k_reverse2 * id_ep2m * id_2ary)
d[wnsa]/dt = k1 * id_l6et
d[ep2m]/dt = k2 * id_l6et - k_reverse2 * id_ep2m * id_2ary
d[2ary]/dt = k2 * id_l6et - k_reverse2 * id_ep2m * id_2ary

The simulation with this model didn't produce fluctuations. This suggests either the stoichiometry is wrong, the kinetic law is wrong, or there's another reaction.

Let's look at the large change points again. When `id_ep2m` and `id_2ary` spike, `id_l6et` drops significantly. This supports `id_l6et` being a reactant. When `id_ep2m` and `id_2ary` drop, what happens? `id_l6et` seems to recover slightly or its decrease slows down.

Could there be a reaction that consumes `id_ep2m` and `id_2ary` and produces `id_l6et`? This is the reverse of Reaction 2. So the reversible reaction `id_l6et <=> id_ep2m + id_2ary` is still a strong candidate, but the mass-action kinetic might be the issue.

What if the reverse rate is not simply proportional to `id_ep2m * id_2ary`? What if it's proportional to `id_ep2m` or `id_2ary` individually, or has a more complex form?

Let's try to fit the parameters `k1`, `k2`, and `k_reverse2` to the experimental data using `scipy.optimize`. This might give us better parameter values and show if this model structure can fit the data at all.

I will define a function that takes the parameters, builds the SBML model, simulates it, and returns the sum of squared errors between the simulation and the experimental data. Then I will use an optimization routine to minimize this error.

First, I need to extract the experimental data into numpy arrays for easier comparison.

## Action

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

# Load experimental data
exp_df = experiment_history['iteration_0']
exp_time = exp_df['Time'].values
exp_species_data = exp_df[['id_l6et', 'id_ep2m', 'id_wnsa', 'id_2ary']].values

# Get initial concentrations from the experimental data at Time 0
initial_concentrations = exp_species_data[0, :]
species_ids = ['id_l6et', 'id_ep2m', 'id_wnsa', 'id_2ary']

# Function to build SBML model with given parameters
def build_sbml_model(k1, k2, k_reverse2):
    sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
    model = sbml_doc.getModel()

    # Clear existing reactions and parameters (if any were added in previous steps)
    while model.getNumReactions() > 0:
        model.removeReaction(0)
    while model.getNumParameters() > 0:
        model.removeParameter(0)

    # Add parameters
    p_k1 = model.createParameter()
    p_k1.setId("k1")
    p_k1.setValue(k1)
    p_k1.setConstant(True)

    p_k2 = model.createParameter()
    p_k2.setId("k2")
    p_k2.setValue(k2)
    p_k2.setConstant(True)

    p_k_reverse2 = model.createParameter()
    p_k_reverse2.setId("k_reverse2")
    p_k_reverse2.setValue(k_reverse2)
    p_k_reverse2.setConstant(True)

    # Reaction 1: id_l6et -> id_wnsa (Mass action)
    reaction1 = model.createReaction()
    reaction1.setId("reaction1")
    reaction1.setReversible(False)
    reaction1.setFast(False)
    r1_reactant = reaction1.createReactant()
    r1_reactant.setSpecies("id_l6et")
    r1_reactant.setStoichiometry(1.0)
    r1_reactant.setConstant(False)
    r1_product = reaction1.createProduct()
    r1_product.setSpecies("id_wnsa")
    r1_product.setStoichiometry(1.0)
    r1_product.setConstant(False)
    kl1 = reaction1.createKineticLaw()
    kl1.setFormula("k1 * id_l6et")

    # Reaction 2: id_l6et <=> id_ep2m + id_2ary (Reversible Mass action)
    reaction2 = model.createReaction()
    reaction2.setId("reaction2")
    reaction2.setReversible(True)
    reaction2.setFast(False)
    r2_reactant = reaction2.createReactant()
    r2_reactant.setSpecies("id_l6et")
    r2_reactant.setStoichiometry(1.0)
    r2_reactant.setConstant(False)
    r2_product1 = reaction2.createProduct()
    r2_product1.setSpecies("id_ep2m")
    r2_product1.setStoichiometry(1.0)
    r2_product1.setConstant(False)
    r2_product2 = reaction2.createProduct()
    r2_product2.setSpecies("id_2ary")
    r2_product2.setStoichiometry(1.0)
    r2_product2.setConstant(False)
    kl2 = reaction2.createKineticLaw()
    kl2.setFormula("k2 * id_l6et - k_reverse2 * id_ep2m * id_2ary")


    writer = libsbml.SBMLWriter()
    return writer.writeSBMLToString(sbml_doc)

# Function to simulate the model and return species concentrations
def simulate_model(sbml_string, time_points):
    # This function calls the provided simulate function
    sim_df = simulate(sbml_string)
    # Ensure the simulation output has the same time points as experiment for comparison
    # This might require interpolation if simulate doesn't hit the exact points
    # For simplicity now, let's assume simulate returns data at exp_time points or more
    # We will merge and handle missing points later if needed
    return sim_df

# Objective function to minimize (Sum of Squared Errors)
def objective_function(params):
    k1, k2, k_reverse2 = params
    if k1 < 0 or k2 < 0 or k_reverse2 < 0:
        return np.inf # Penalize negative rate constants

    try:
        sbml_string = build_sbml_model(k1, k2, k_reverse2)
        sim_df = simulate_model(sbml_string, exp_time)

        # Merge simulation and experimental data
        merged_df = pd.merge(sim_df, exp_df, on='Time', suffixes=('_sim', '_exp'))

        # Calculate SSE for the species we are trying to fit
        # Exclude Time and focus on species concentrations
        sse = 0
        for species_id in species_ids:
             sse += np.sum((merged_df[species_id + '_sim'] - merged_df[species_id + '_exp'])**2)

        return sse
    except Exception as e:
        print(f"Simulation or merge failed: {e}")
        return np.inf # Penalize simulation errors

# Initial guess for parameters (based on previous placeholder values)
initial_params = [0.1, 0.05, 0.001]

# Perform optimization
# Using 'Nelder-Mead' as it doesn't require gradients and is relatively robust
# bounds = [(0, None), (0, None), (0, None)] # Ensure parameters are non-negative
result = minimize(objective_function, initial_params, method='Nelder-Mead', tol=1e-6) #, bounds=bounds)

print("Optimization Result:")
print(result)

# Get the best parameters
best_params = result.x
print(f"\nBest parameters: k1={best_params[0]}, k2={best_params[1]}, k_reverse2={best_params[2]}")

# Build the SBML model with the best parameters
best_sbml_string = build_sbml_model(*best_params)

# Simulate with the best parameters and compare visually (by printing data)
sim_df_best = simulate(best_sbml_string)
merged_df_best = pd.merge(sim_df_best, exp_df, on='Time', suffixes=('_sim', '_exp'))

print("\nSimulation with best parameters vs Experiment (first 10 rows):")
print(merged_df_best.head(10))

# Store the best SBML model
shared_variables.add("best_sbml_reversible_massaction", best_sbml_string)
```
# Observation

## Code Stdout
Optimization Result:
       message: Optimization terminated successfully.
       success: True
        status: 0
           fun: 260242790.0014993
             x: [ 5.703e-04  6.792e-02  1.355e-03]
           nit: 221
          nfev: 494
 final_simplex: (array([[ 5.703e-04,  6.792e-02,  1.355e-03],
                       [ 5.703e-04,  6.792e-02,  1.355e-03],
                       [ 5.703e-04,  6.792e-02,  1.355e-03],
                       [ 5.703e-04,  6.792e-02,  1.355e-03]]), array([ 2.602e+08,  2.602e+08,  2.602e+08,  2.602e+08]))

Best parameters: k1=0.0005703295852496197, k2=0.06791860618156326, k_reverse2=0.0013551623348530858

Simulation with best parameters vs Experiment (first 10 rows):
   id_l6et_sim  id_ep2m_sim  ...  id_wnsa_exp   id_2ary_exp
0  1000.000000   200.000000  ...     1.000000  1.000000e+00
1   952.981541   246.573498  ...     1.180227  8.719520e+00
2   920.340494   278.787683  ...     3.725243  2.900681e+01
3   899.174396   299.539060  ...     9.300273  2.348123e+00
4   886.010837   312.295605  ...     8.972119  4.980706e-02
5   877.994902   319.909266  ...     8.290540  1.555331e-03
6   873.136149   324.368623  ...     7.653392  7.377733e-05
7   870.159993   326.947131  ...     7.064983  5.178048e-06
8   868.288089   328.422472  ...     6.521802  5.237403e-07
9   867.057968   329.256723  ...     6.020382  7.445900e-08

[10 rows x 9 columns]

## 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)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml, best_sbml_reversible_massaction)

### 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
The optimization found parameters for the model with two reactions (`id_l6et` -> `id_wnsa` and `id_l6et` <=> `id_ep2m` + `id_2ary`). The resulting sum of squared errors is quite large (around 2.6e+08), and looking at the first few rows of the simulation vs experiment data, the simulated values still don't match the experimental data well, especially for `id_ep2m` and `id_2ary`. The simulation shows smooth changes, while the experiment shows rapid fluctuations.

This confirms that the current model structure, even with optimized mass-action kinetics for the reversible reaction, is not sufficient to capture the observed dynamics. The sharp spikes and drops in `id_ep2m` and `id_2ary` are the key features that the current model fails to reproduce.

This type of behavior often arises from non-linear kinetics, such as:
- Autocatalysis: A product catalyzes its own formation.
- Activation/Inhibition: A species activates or inhibits a reaction.
- Thresholds: Reaction rates change drastically when a species concentration crosses a certain threshold.

Given the strong positive correlation between `id_ep2m` and `id_2ary` changes, they are likely produced together. The spikes in `id_ep2m` could be due to a reaction that produces `id_ep2m` and `id_2ary` and is somehow activated or accelerated when `id_ep2m` or `id_2ary` (or another species) reaches a certain level.

Let's reconsider the reactions:
Reaction 1: `id_l6et` -> `id_wnsa` (seems relatively simple, maybe mass action is okay here).
Reaction 2: Produces `id_ep2m` and `id_2ary`. What if this reaction is catalyzed by `id_ep2m` itself? An autocatalytic reaction.
A simple autocatalytic reaction producing `id_ep2m` and `id_2ary` from `id_l6et` could look like:
`id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary`
Net change: `id_l6et -> id_ep2m + id_2ary`
The rate could be proportional to `id_l6et * id_ep2m`.

Let's try replacing Reaction 2 with an autocatalytic reaction: `id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary`. The kinetic law would be `k2 * id_l6et * id_ep2m`. This reaction consumes `id_l6et` and `id_ep2m` and produces `id_ep2m` (net production of 1 `id_ep2m`) and `id_2ary`.

This might explain the rapid increase in `id_ep2m` and `id_2ary` once `id_ep2m` reaches a certain level. What about the subsequent drop? This autocatalytic reaction alone would lead to exponential growth of `id_ep2m` if `id_l6et` is abundant. There must be a consumption mechanism for `id_ep2m` and `id_2ary`.

Perhaps there is a third reaction that consumes `id_ep2m` and `id_2ary`. What could it produce? Maybe it converts them back to `id_l6et`? `id_ep2m + id_2ary -> id_l6et`. This would complete a cycle and could potentially lead to oscillatory behavior or fluctuations depending on the rates.

Let's propose the following model structure:
Reaction 1: `id_l6et` -> `id_wnsa` (Rate = `k1 * id_l6et`)
Reaction 2: `id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary` (Rate = `k2 * id_l6et * id_ep2m`) - Autocatalytic production of `id_ep2m` and `id_2ary`
Reaction 3: `id_ep2m + id_2ary -> id_l6et` (Rate = `k3 * id_ep2m * id_2ary`) - Consumption of `id_ep2m` and `id_2ary`, regeneration of `id_l6et`

Let's implement this model structure in SBML and try to simulate it with some initial placeholder parameters. I'll use the `current_sbml` from the shared variables as a base, but I'll need to remove the old reactions and add the new ones.

## Action

### Code
```python
import libsbml
import pandas as pd

# Load the initial SBML model structure (without reactions)
# This ensures we start clean before adding the new reaction set
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Clear existing reactions and parameters
while model.getNumReactions() > 0:
    model.removeReaction(0)
while model.getNumParameters() > 0:
    model.removeParameter(0)

# Add parameters for the new reactions
k1 = model.createParameter()
k1.setId("k1")
k1.setValue(0.001) # Placeholder
k1.setConstant(True)

k2 = model.createParameter()
k2.setId("k2")
k2.setValue(0.0001) # Placeholder
k2.setConstant(True)

k3 = model.createParameter()
k3.setId("k3")
k3.setValue(0.00001) # Placeholder
k3.setConstant(True)

# Reaction 1: id_l6et -> id_wnsa (Mass action)
reaction1 = model.createReaction()
reaction1.setId("reaction1")
reaction1.setReversible(False)
reaction1.setFast(False)
r1_reactant = reaction1.createReactant()
r1_reactant.setSpecies("id_l6et")
r1_reactant.setStoichiometry(1.0)
r1_reactant.setConstant(False)
r1_product = reaction1.createProduct()
r1_product.setSpecies("id_wnsa")
r1_product.setStoichiometry(1.0)
r1_product.setConstant(False)
kl1 = reaction1.createKineticLaw()
kl1.setFormula("k1 * id_l6et")

# Reaction 2: id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary (Autocatalytic)
reaction2 = model.createReaction()
reaction2.setId("reaction2")
reaction2.setReversible(False)
reaction2.setFast(False)
r2_reactant1 = reaction2.createReactant()
r2_reactant1.setSpecies("id_l6et")
r2_reactant1.setStoichiometry(1.0)
r2_reactant1.setConstant(False)
r2_reactant2 = reaction2.createReactant()
r2_reactant2.setSpecies("id_ep2m")
r2_reactant2.setStoichiometry(1.0)
r2_reactant2.setConstant(False)
r2_product1 = reaction2.createProduct()
r2_product1.setSpecies("id_ep2m")
r2_product1.setStoichiometry(2.0) # Produces 2 molecules for every 1 consumed
r2_product1.setConstant(False)
r2_product2 = reaction2.createProduct()
r2_product2.setSpecies("id_2ary")
r2_product2.setStoichiometry(1.0)
r2_product2.setConstant(False)
kl2 = reaction2.createKineticLaw()
kl2.setFormula("k2 * id_l6et * id_ep2m") # Rate proportional to reactants

# Reaction 3: id_ep2m + id_2ary -> id_l6et (Consumption/Regeneration)
reaction3 = model.createReaction()
reaction3.setId("reaction3")
reaction3.setReversible(False)
reaction3.setFast(False)
r3_reactant1 = reaction3.createReactant()
r3_reactant1.setSpecies("id_ep2m")
r3_reactant1.setStoichiometry(1.0)
r3_reactant1.setConstant(False)
r3_reactant2 = reaction3.createReactant()
r3_reactant2.setSpecies("id_2ary")
r3_reactant2.setStoichiometry(1.0)
r3_reactant2.setConstant(False)
r3_product = reaction3.createProduct()
r3_product.setSpecies("id_l6et")
r3_product.setStoichiometry(1.0)
r3_product.setConstant(False)
kl3 = reaction3.createKineticLaw()
kl3.setFormula("k3 * id_ep2m * id_2ary") # Rate proportional to reactants


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

# Simulate the updated model
sim_df = simulate(updated_sbml)

# Load the experimental data
exp_df = experiment_history['iteration_0']

# Compare simulation and experiment
print("Simulation results (first 10 rows):")
print(sim_df.head(10))

merged_df = pd.merge(sim_df, exp_df, on='Time', suffixes=('_sim', '_exp'))
print("\nDifference between simulation and experiment (first 10 rows):")
print(merged_df.head(10))

# Store the updated SBML for future use
shared_variables.add("current_sbml_autocatalytic", updated_sbml)
```
# Observation

## Code Stdout
Simulation results (first 10 rows):
       id_l6et     id_ep2m   id_wnsa     id_2ary  Time
0  1000.000000  200.000000  1.000000    1.000000   0.0
1   982.713631  216.493220  1.793149   17.493220   0.8
2   964.446277  233.981645  2.572079   34.981645   1.6
3   945.196880  252.467119  3.336001   53.467119   2.4
4   924.973770  271.942096  4.084134   72.942096   3.2
5   903.796295  292.388000  4.815705   93.388000   4.0
6   881.695148  313.774888  5.529964  114.774888   4.8
7   858.713026  336.060787  6.226187  137.060787   5.6
8   834.905164  359.191145  6.903691  160.191145   6.4
9   810.339229  383.098934  7.561837  184.098934   7.2

Difference between simulation and experiment (first 10 rows):
   id_l6et_sim  id_ep2m_sim  ...  id_wnsa_exp   id_2ary_exp
0  1000.000000   200.000000  ...     1.000000  1.000000e+00
1   982.713631   216.493220  ...     1.180227  8.719520e+00
2   964.446277   233.981645  ...     3.725243  2.900681e+01
3   945.196880   252.467119  ...     9.300273  2.348123e+00
4   924.973770   271.942096  ...     8.972119  4.980706e-02
5   903.796295   292.388000  ...     8.290540  1.555331e-03
6   881.695148   313.774888  ...     7.653392  7.377733e-05
7   858.713026   336.060787  ...     7.064983  5.178048e-06
8   834.905164   359.191145  ...     6.521802  5.237403e-07
9   810.339229   383.098934  ...     6.020382  7.445900e-08

[10 rows x 9 columns]

## 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)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml, best_sbml_reversible_massaction, current_sbml_autocatalytic)

### 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
The simulation results with the autocatalytic model structure (Reaction 1: `id_l6et` -> `id_wnsa`, Reaction 2: `id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary`, Reaction 3: `id_ep2m + id_2ary -> id_l6et`) still show smooth changes in concentrations, not the sharp fluctuations observed in the experimental data for `id_ep2m` and `id_2ary`. The placeholder parameter values are likely not optimal, but the lack of any oscillatory or spiking behavior suggests the model structure might still be missing something or the kinetics are not simple mass action.

The autocatalytic reaction `id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary` does introduce a non-linearity that can lead to faster growth of `id_ep2m` and `id_2ary` when `id_ep2m` is present. The third reaction `id_ep2m + id_2ary -> id_l6et` provides a consumption mechanism for `id_ep2m` and `id_2ary`. Together, these reactions form a cycle that consumes `id_l6et` and produces `id_wnsa`.

Let's try to optimize the parameters (`k1`, `k2`, `k3`) for this autocatalytic model structure using the experimental data. This will tell us if this model, with the best possible mass-action rate constants, can reproduce the experimental observations, including the fluctuations.

I will use the same optimization approach as before, minimizing the sum of squared errors between the simulation and the experimental data.

## Action

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

# Load experimental data
exp_df = experiment_history['iteration_0']
exp_time = exp_df['Time'].values
exp_species_data = exp_df[['id_l6et', 'id_ep2m', 'id_wnsa', 'id_2ary']].values

# Get initial concentrations from the experimental data at Time 0
initial_concentrations = exp_species_data[0, :]
species_ids = ['id_l6et', 'id_ep2m', 'id_wnsa', 'id_2ary']

# Function to build SBML model with given parameters (Autocatalytic structure)
def build_sbml_model_autocatalytic(k1, k2, k3):
    sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
    model = sbml_doc.getModel()

    # Clear existing reactions and parameters
    while model.getNumReactions() > 0:
        model.removeReaction(0)
    while model.getNumParameters() > 0:
        model.removeParameter(0)

    # Add parameters
    p_k1 = model.createParameter()
    p_k1.setId("k1")
    p_k1.setValue(k1)
    p_k1.setConstant(True)

    p_k2 = model.createParameter()
    p_k2.setId("k2")
    p_k2.setValue(k2)
    p_k2.setConstant(True)

    p_k3 = model.createParameter()
    p_k3.setId("k3")
    p_k3.setValue(k3)
    p_k3.setConstant(True)

    # Reaction 1: id_l6et -> id_wnsa (Mass action)
    reaction1 = model.createReaction()
    reaction1.setId("reaction1")
    reaction1.setReversible(False)
    reaction1.setFast(False)
    r1_reactant = reaction1.createReactant()
    r1_reactant.setSpecies("id_l6et")
    r1_reactant.setStoichiometry(1.0)
    r1_reactant.setConstant(False)
    r1_product = reaction1.createProduct()
    r1_product.setSpecies("id_wnsa")
    r1_product.setStoichiometry(1.0)
    r1_product.setConstant(False)
    kl1 = reaction1.createKineticLaw()
    kl1.setFormula("k1 * id_l6et")

    # Reaction 2: id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary (Autocatalytic)
    reaction2 = model.createReaction()
    reaction2.setId("reaction2")
    reaction2.setReversible(False)
    reaction2.setFast(False)
    r2_reactant1 = reaction2.createReactant()
    r2_reactant1.setSpecies("id_l6et")
    r2_reactant1.setStoichiometry(1.0)
    r2_reactant1.setConstant(False)
    r2_reactant2 = reaction2.createReactant()
    r2_reactant2.setSpecies("id_ep2m")
    r2_reactant2.setStoichiometry(1.0)
    r2_reactant2.setConstant(False)
    r2_product1 = reaction2.createProduct()
    r2_product1.setSpecies("id_ep2m")
    r2_product1.setStoichiometry(2.0)
    r2_product1.setConstant(False)
    r2_product2 = reaction2.createProduct()
    r2_product2.setSpecies("id_2ary")
    r2_product2.setStoichiometry(1.0)
    r2_product2.setConstant(False)
    kl2 = reaction2.createKineticLaw()
    kl2.setFormula("k2 * id_l6et * id_ep2m")

    # Reaction 3: id_ep2m + id_2ary -> id_l6et (Consumption/Regeneration)
    reaction3 = model.createReaction()
    reaction3.setId("reaction3")
    reaction3.setReversible(False)
    reaction3.setFast(False)
    r3_reactant1 = reaction3.createReactant()
    r3_reactant1.setSpecies("id_ep2m")
    r3_reactant1.setStoichiometry(1.0)
    r3_reactant1.setConstant(False)
    r3_reactant2 = reaction3.createReactant()
    r3_reactant2.setSpecies("id_2ary")
    r3_reactant2.setStoichiometry(1.0)
    r3_reactant2.setConstant(False)
    r3_product = reaction3.createProduct()
    r3_product.setSpecies("id_l6et")
    r3_product.setStoichiometry(1.0)
    r3_product.setConstant(False)
    kl3 = reaction3.createKineticLaw()
    kl3.setFormula("k3 * id_ep2m * id_2ary")

    writer = libsbml.SBMLWriter()
    return writer.writeSBMLToString(sbml_doc)

# Function to simulate the model and return species concentrations
def simulate_model(sbml_string, time_points):
    # This function calls the provided simulate function
    sim_df = simulate(sbml_string)
    return sim_df

# Objective function to minimize (Sum of Squared Errors)
def objective_function_autocatalytic(params):
    k1, k2, k3 = params
    if k1 < 0 or k2 < 0 or k3 < 0:
        return np.inf # Penalize negative rate constants

    try:
        sbml_string = build_sbml_model_autocatalytic(k1, k2, k3)
        sim_df = simulate_model(sbml_string, exp_time)

        # Merge simulation and experimental data
        merged_df = pd.merge(sim_df, exp_df, on='Time', suffixes=('_sim', '_exp'))

        # Calculate SSE for all species
        sse = 0
        for species_id in species_ids:
             sse += np.sum((merged_df[species_id + '_sim'] - merged_df[species_id + '_exp'])**2)

        return sse
    except Exception as e:
        # print(f"Simulation or merge failed: {e}") # Suppress frequent printing during optimization
        return np.inf # Penalize simulation errors

# Initial guess for parameters (based on previous placeholder values)
# Let's try a range of initial guesses to see if it affects the result
initial_params_guesses = [
    [0.001, 0.0001, 0.00001],
    [0.01, 0.001, 0.0001],
    [0.0005, 0.00005, 0.000005]
]

best_sse = np.inf
best_params = None
best_result = None

for initial_params in initial_params_guesses:
    print(f"Attempting optimization with initial guess: {initial_params}")
    result = minimize(objective_function_autocatalytic, initial_params, method='Nelder-Mead', tol=1e-6) #, bounds=[(0, None)]*3)

    print(f"Result for guess {initial_params}:")
    print(result)

    if result.success and result.fun < best_sse:
        best_sse = result.fun
        best_params = result.x
        best_result = result

print("\nOptimization finished.")
print("Best Result:")
print(best_result)
print(f"\nBest parameters: k1={best_params[0]}, k2={best_params[1]}, k3={best_params[2]}")

# Build the SBML model with the best parameters
best_sbml_string_autocatalytic = build_sbml_model_autocatalytic(*best_params)

# Simulate with the best parameters and compare visually (by printing data)
sim_df_best = simulate(best_sbml_string_autocatalytic)
merged_df_best = pd.merge(sim_df_best, exp_df, on='Time', suffixes=('_sim', '_exp'))

print("\nSimulation with best parameters (autocatalytic model) vs Experiment (first 10 rows):")
print(merged_df_best.head(10))

# Store the best SBML model
shared_variables.add("best_sbml_autocatalytic", best_sbml_string_autocatalytic)
```
# Observation

## Code Stdout
Attempting optimization with initial guess: [0.001, 0.0001, 1e-05]
Result for guess [0.001, 0.0001, 1e-05]:
       message: Maximum number of function evaluations has been exceeded.
       success: False
        status: 1
           fun: 256576318.29782325
             x: [ 5.673e-04  7.810e-06  4.227e-05]
           nit: 304
          nfev: 600
 final_simplex: (array([[ 5.673e-04,  7.810e-06,  4.227e-05],
                       [ 5.673e-04,  7.810e-06,  4.227e-05],
                       [ 5.673e-04,  7.810e-06,  4.227e-05],
                       [ 5.673e-04,  7.810e-06,  4.227e-05]]), array([ 2.566e+08,  2.566e+08,  2.566e+08,  2.566e+08]))
Attempting optimization with initial guess: [0.01, 0.001, 0.0001]
Result for guess [0.01, 0.001, 0.0001]:
       message: Optimization terminated successfully.
       success: True
        status: 0
           fun: 259979796.56665286
             x: [ 5.689e-04  4.793e-05  2.989e-04]
           nit: 177
          nfev: 402
 final_simplex: (array([[ 5.689e-04,  4.793e-05,  2.989e-04],
                       [ 5.689e-04,  4.793e-05,  2.989e-04],
                       [ 5.689e-04,  4.793e-05,  2.989e-04],
                       [ 5.689e-04,  4.793e-05,  2.989e-04]]), array([ 2.600e+08,  2.600e+08,  2.600e+08,  2.600e+08]))
Attempting optimization with initial guess: [0.0005, 5e-05, 5e-06]
Result for guess [0.0005, 5e-05, 5e-06]:
       message: Optimization terminated successfully.
       success: True
        status: 0
           fun: 256189774.52635932
             x: [ 5.523e-04  5.277e-06  2.720e-05]
           nit: 232
          nfev: 509
 final_simplex: (array([[ 5.523e-04,  5.277e-06,  2.720e-05],
                       [ 5.523e-04,  5.277e-06,  2.720e-05],
                       [ 5.523e-04,  5.277e-06,  2.720e-05],
                       [ 5.523e-04,  5.277e-06,  2.720e-05]]), array([ 2.562e+08,  2.562e+08,  2.562e+08,  2.562e+08]))

Optimization finished.
Best Result:
       message: Optimization terminated successfully.
       success: True
        status: 0
           fun: 256189774.52635932
             x: [ 5.523e-04  5.277e-06  2.720e-05]
           nit: 232
          nfev: 509
 final_simplex: (array([[ 5.523e-04,  5.277e-06,  2.720e-05],
                       [ 5.523e-04,  5.277e-06,  2.720e-05],
                       [ 5.523e-04,  5.277e-06,  2.720e-05],
                       [ 5.523e-04,  5.277e-06,  2.720e-05]]), array([ 2.562e+08,  2.562e+08,  2.562e+08,  2.562e+08]))

Best parameters: k1=0.0005523066521139892, k2=5.276601390363442e-06, k3=2.7195764626845956e-05

Simulation with best parameters (autocatalytic model) vs Experiment (first 10 rows):
   id_l6et_sim  id_ep2m_sim  ...  id_wnsa_exp   id_2ary_exp
0  1000.000000   200.000000  ...     1.000000  1.000000e+00
1   998.719148   200.839291  ...     1.180227  8.719520e+00
2   997.440107   201.677335  ...     3.725243  2.900681e+01
3   996.162914   202.514096  ...     9.300273  2.348123e+00
4   994.887608   203.349534  ...     8.972119  4.980706e-02
5   993.614225   204.183612  ...     8.290540  1.555331e-03
6   992.342801   205.016293  ...     7.653392  7.377733e-05
7   991.073372   205.847540  ...     7.064983  5.178048e-06
8   989.805975   206.677316  ...     6.521802  5.237403e-07
9   988.540643   207.505586  ...     6.020382  7.445900e-08

[10 rows x 9 columns]

## 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)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml, best_sbml_reversible_massaction, current_sbml_autocatalytic, best_sbml_autocatalytic)

### 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
Optimizing the parameters for the autocatalytic model structure did not significantly improve the fit, and the simulated data still does not show the characteristic fluctuations of `id_ep2m` and `id_2ary`. The SSE is still very high, and the initial rows of the simulation data show smooth, slow changes, unlike the rapid initial changes in the experiment.

This suggests that either the autocatalytic model structure with simple mass-action kinetics is incorrect, or the optimization is getting stuck in a local minimum. However, given the fundamental difference in dynamics (smooth vs. fluctuating), it's more likely the model structure is wrong or incomplete.

The strong positive correlation between `id_ep2m` and `id_2ary` changes, coupled with the rapid fluctuations, is a strong indicator of a process that generates both species together, and this process is likely non-linear or involves feedback.

Let's reconsider the possibility of a reversible reaction with non-mass-action kinetics, or a set of reactions that create a cycle with non-linear steps.

Another possibility for fluctuations is a predator-prey type interaction or an activator-inhibitor system.
- If `id_ep2m` activates its own production (autocatalysis) and is consumed by `id_2ary`, while `id_2ary` is also produced in the autocatalytic step and consumed elsewhere, this could lead to oscillations.

Let's look at the experimental data again, focusing on the timing of the spikes. `id_ep2m` and `id_2ary` spike together. `id_l6et` drops during the spike. `id_wnsa` increases relatively smoothly.

This reinforces the idea of two main processes:
1. `id_l6et` -> `id_wnsa` (likely simple kinetics).
2. A process involving `id_l6et`, `id_ep2m`, and `id_2ary` that generates the fluctuations.

What if `id_l6et` is converted to `id_ep2m` and `id_2ary` in a reaction, and then `id_ep2m` and `id_2ary` are consumed in another reaction that regenerates `id_l6et`, but one of these reactions has non-linear kinetics?

Consider the Lotka-Volterra like system structure, which can produce oscillations:
Prey (X) + Predator (Y) -> 2 * Predator (Y) (Predator consumes Prey and reproduces)
Predator (Y) -> Death (Predator dies)

Mapping this to our species:
If `id_l6et` is the "Prey" and `id_ep2m` is the "Predator":
`id_l6et + id_ep2m -> 2 * id_ep2m` (Rate proportional to `id_l6et * id_ep2m`)
`id_ep2m -> Something` (Consumption of `id_ep2m`)

Where does `id_2ary` fit in? `id_2ary` is strongly correlated with `id_ep2m`. Maybe `id_2ary` is also produced in the first reaction:
`id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary` (Rate `k2 * id_l6et * id_ep2m`)
And maybe `id_ep2m` and `id_2ary` are consumed together:
`id_ep2m + id_2ary -> Something`

If `id_ep2m + id_2ary -> id_l6et` (Reaction 3 from the previous attempt, Rate `k3 * id_ep2m * id_2ary`), this completes a cycle.
Reaction 1: `id_l6et` -> `id_wnsa` (Rate `k1 * id_l6et`)
Reaction 2: `id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary` (Rate `k2 * id_l6et * id_ep2m`)
Reaction 3: `id_ep2m + id_2ary -> id_l6et` (Rate `k3 * id_ep2m * id_2ary`)

This is the exact model structure I just optimized, and it didn't produce fluctuations. This suggests that either the mass-action kinetics are wrong for Reaction 2 or 3, or there's another reaction involved.

What if the consumption of `id_ep2m` and `id_2ary` (Reaction 3) is not a simple bimolecular reaction? What if it's a first-order decay of `id_ep2m` and `id_2ary`? But they are strongly correlated, so they are likely consumed together.

Let's consider the possibility that the reverse reaction of `id_l6et <=> id_ep2m + id_2ary` is not mass action. What if the rate of `id_ep2m + id_2ary -> id_l6et` is proportional to `id_ep2m` only, or `id_2ary` only, or some other function?

Let's go back to the reversible reaction idea: `id_l6et <=> id_ep2m + id_2ary`.
Forward rate: `k2 * id_l6et` (mass action)
Reverse rate: What if the reverse rate is proportional to `id_ep2m`? Rate = `k_reverse2 * id_ep2m`.
This doesn't involve `id_2ary` in the reverse rate, which contradicts the strong correlation between `id_ep2m` and `id_2ary` changes.

What if the reverse rate is proportional to `id_ep2m` and `id_2ary` but with different stoichiometry in the reverse reaction? E.g., `id_ep2m + id_2ary -> id_l6et` with rate `k_reverse2 * id_ep2m * id_2ary` (this is what I tried).

Let's consider the possibility that the autocatalytic reaction is the key to the spikes.
Reaction 1: `id_l6et` -> `id_wnsa` (Rate `k1 * id_l6et`)
Reaction 2: `id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary` (Rate `k2 * id_l6et * id_ep2m`)
What if there's a simple decay of `id_ep2m` and `id_2ary`?
Reaction 3: `id_ep2m -> Something`
Reaction 4: `id_2ary -> Something`
But they are consumed together.

Let's try a different structure based on the observation that `id_l6et` decreases, `id_wnsa` increases, and `id_ep2m`/`id_2ary` fluctuate together.
Reaction 1: `id_l6et` -> `id_wnsa` (Rate `k1 * id_l6et`)
Reaction 2: `id_l6et` -> `id_ep2m` + `id_2ary` (Rate `k2 * id_l6et`) - Simple production
Reaction 3: `id_ep2m + id_2ary` -> `id_ep2m` (Rate `k3 * id_ep2m * id_2ary`) - `id_2ary` is consumed, `id_ep2m` acts catalytically or is regenerated. Net: `id_2ary` consumed. This doesn't produce fluctuations.

Let's go back to the autocatalytic idea, but consider if the consumption reaction is different.
Reaction 1: `id_l6et` -> `id_wnsa` (Rate `k1 * id_l6et`)
Reaction 2: `id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary` (Rate `k2 * id_l6et * id_ep2m`)
Reaction 3: `id_ep2m -> Something` (e.g., decay, Rate `k3 * id_ep2m`)
Reaction 4: `id_2ary -> Something` (e.g., decay, Rate `k4 * id_2ary`)
This doesn't explain the coupled consumption of `id_ep2m` and `id_2ary`.

What if Reaction 3 consumes `id_ep2m` and `id_2ary` and produces nothing (or something outside the system, like a boundary species)?
Reaction 3: `id_ep2m + id_2ary -> ` (Rate `k3 * id_ep2m * id_2ary`)
Let's try this structure.

Model Structure Attempt 3:
Reaction 1: `id_l6et` -> `id_wnsa` (Rate = `k1 * id_l6et`)
Reaction 2: `id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary` (Rate = `k2 * id_l6et * id_ep2m`)
Reaction 3: `id_ep2m + id_2ary -> ` (Rate = `k3 * id_ep2m * id_2ary`)

Let's implement this and simulate with placeholder parameters.

## Action

### Code
```python
import libsbml
import pandas as pd

# Load the initial SBML model structure (without reactions)
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Clear existing reactions and parameters
while model.getNumReactions() > 0:
    model.removeReaction(0)
while model.getNumParameters() > 0:
    model.removeParameter(0)

# Add parameters for the new reactions
k1 = model.createParameter()
k1.setId("k1")
k1.setValue(0.001) # Placeholder
k1.setConstant(True)

k2 = model.createParameter()
k2.setId("k2")
k2.setValue(0.0001) # Placeholder
k2.setConstant(True)

k3 = model.createParameter()
k3.setId("k3")
k3.setValue(0.00001) # Placeholder
k3.setConstant(True)

# Reaction 1: id_l6et -> id_wnsa (Mass action)
reaction1 = model.createReaction()
reaction1.setId("reaction1")
reaction1.setReversible(False)
reaction1.setFast(False)
r1_reactant = reaction1.createReactant()
r1_reactant.setSpecies("id_l6et")
r1_reactant.setStoichiometry(1.0)
r1_reactant.setConstant(False)
r1_product = reaction1.createProduct()
r1_product.setSpecies("id_wnsa")
r1_product.setStoichiometry(1.0)
r1_product.setConstant(False)
kl1 = reaction1.createKineticLaw()
kl1.setFormula("k1 * id_l6et")

# Reaction 2: id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary (Autocatalytic production)
reaction2 = model.createReaction()
reaction2.setId("reaction2")
reaction2.setReversible(False)
reaction2.setFast(False)
r2_reactant1 = reaction2.createReactant()
r2_reactant1.setSpecies("id_l6et")
r2_reactant1.setStoichiometry(1.0)
r2_reactant1.setConstant(False)
r2_reactant2 = reaction2.createReactant()
r2_reactant2.setSpecies("id_ep2m")
r2_reactant2.setStoichiometry(1.0)
r2_reactant2.setConstant(False)
r2_product1 = reaction2.createProduct()
r2_product1.setSpecies("id_ep2m")
r2_product1.setStoichiometry(2.0)
r2_product1.setConstant(False)
r2_product2 = reaction2.createProduct()
r2_product2.setSpecies("id_2ary")
r2_product2.setStoichiometry(1.0)
r2_product2.setConstant(False)
kl2 = reaction2.createKineticLaw()
kl2.setFormula("k2 * id_l6et * id_ep2m")

# Reaction 3: id_ep2m + id_2ary -> (Consumption)
reaction3 = model.createReaction()
reaction3.setId("reaction3")
reaction3.setReversible(False)
reaction3.setFast(False)
r3_reactant1 = reaction3.createReactant()
r3_reactant1.setSpecies("id_ep2m")
r3_reactant1.setStoichiometry(1.0)
r3_reactant1.setConstant(False)
r3_reactant2 = reaction3.createReactant()
r3_reactant2.setSpecies("id_2ary")
r3_reactant2.setStoichiometry(1.0)
r3_reactant2.setConstant(False)
# No products for this reaction

kl3 = reaction3.createKineticLaw()
kl3.setFormula("k3 * id_ep2m * id_2ary")


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

# Simulate the updated model
sim_df = simulate(updated_sbml)

# Load the experimental data
exp_df = experiment_history['iteration_0']

# Compare simulation and experiment
print("Simulation results (first 10 rows):")
print(sim_df.head(10))

merged_df = pd.merge(sim_df, exp_df, on='Time', suffixes=('_sim', '_exp'))
print("\nDifference between simulation and experiment (first 10 rows):")
print(merged_df.head(10))

# Store the updated SBML for future use
shared_variables.add("current_sbml_autocatalytic_consumption", updated_sbml)
```
# Observation

## Code Stdout
Simulation results (first 10 rows):
       id_l6et     id_ep2m   id_wnsa     id_2ary  Time
0  1000.000000  200.000000  1.000000    1.000000   0.0
1   982.698271  216.493128  1.793144   17.493128   0.8
2   964.384303  233.980878  2.572045   34.980878   1.6
3   945.050972  252.464266  3.335888   53.464266   2.4
4   924.700100  271.934561  4.083856   72.934561   3.2
5   903.344095  292.371547  4.815140   93.371547   4.0
6   881.006399  313.743132  5.528947  114.743132   4.8
7   857.722250  336.004591  6.224505  137.004591   5.6
8   833.539767  359.097414  6.901069  160.097414   6.4
9   808.519156  382.950414  7.557945  183.950414   7.2

Difference between simulation and experiment (first 10 rows):
   id_l6et_sim  id_ep2m_sim  ...  id_wnsa_exp   id_2ary_exp
0  1000.000000   200.000000  ...     1.000000  1.000000e+00
1   982.698271   216.493128  ...     1.180227  8.719520e+00
2   964.384303   233.980878  ...     3.725243  2.900681e+01
3   945.050972   252.464266  ...     9.300273  2.348123e+00
4   924.700100   271.934561  ...     8.972119  4.980706e-02
5   903.344095   292.371547  ...     8.290540  1.555331e-03
6   881.006399   313.743132  ...     7.653392  7.377733e-05
7   857.722250   336.004591  ...     7.064983  5.178048e-06
8   833.539767   359.097414  ...     6.521802  5.237403e-07
9   808.519156   382.950414  ...     6.020382  7.445900e-08

[10 rows x 9 columns]

## 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)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml, best_sbml_reversible_massaction, current_sbml_autocatalytic, best_sbml_autocatalytic, current_sbml_autocatalytic_consumption)

### 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
Simulating the model with the autocatalytic production (Reaction 2: `id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary`) and bimolecular consumption (Reaction 3: `id_ep2m + id_2ary -> `) still results in smooth concentration changes, not the observed fluctuations. The initial placeholder parameters are likely not correct, but the fundamental dynamic behavior is missing.

The fact that the autocatalytic model with mass-action kinetics doesn't produce fluctuations suggests that either the stoichiometry is wrong, the kinetic laws are more complex, or there's another missing reaction or regulatory mechanism.

Let's revisit the experimental data and the correlation analysis. The strong positive correlation between `id_ep2m` and `id_2ary` changes is a key feature. The negative correlation between `id_l6et` and `id_wnsa` changes is also strong.

This reinforces the idea of two somewhat separate modules or pathways:
Module 1: `id_l6et` -> `id_wnsa`
Module 2: Involves `id_l6et`, `id_ep2m`, and `id_2ary`, responsible for the fluctuations.

What if the production of `id_ep2m` and `id_2ary` from `id_l6et` is not autocatalytic in the way I modeled it? What if it's a reaction that is activated by one of the products, but the products are formed with different stoichiometry?

Let's consider the possibility of a Michaelis-Menten type kinetic, which can introduce non-linearity and saturation. However, without knowing the specific enzyme or mechanism, this is speculative.

Let's think about minimal systems that produce oscillations. The Brusselator and Oregonator are classic examples, often involving autocatalysis and feedback. The structure `A -> X`, `2X + Y -> 3X`, `B + X -> Y + C`, `X -> D` can oscillate. Mapping this to our species is difficult without more information.

Let's look at the stoichiometry implied by the large changes in the experimental data. When `id_ep2m` increases by a large amount, `id_2ary` increases by a similarly large amount, and `id_l6et` decreases. The ratio of changes between `id_ep2m` and `id_2ary` is close to 1:1 during these spikes. The ratio of `id_l6et` decrease to `id_ep2m`/`id_2ary` increase is not constant, suggesting complex stoichiometry or multiple reactions contributing.

Let's try to simplify and focus on the core fluctuating module. Assume Reaction 1 (`id_l6et` -> `id_wnsa`) is correct and simple. The fluctuations come from the interactions between `id_l6et`, `id_ep2m`, and `id_2ary`.

What if `id_l6et` is converted to `id_ep2m` and `id_2ary` (Reaction 2), and then `id_ep2m` and `id_2ary` are converted back to `id_l6et` (Reaction 3), but the rates are non-linear?

Let's consider the possibility that the autocatalytic step involves only `id_ep2m` and `id_l6et`, producing `id_ep2m` and `id_2ary`, and the consumption step involves `id_ep2m` and `id_2ary` producing `id_l6et`. This is the structure I just tried.

Perhaps the issue is with the mass-action assumption for the autocatalytic step. What if the rate of production of `id_ep2m` and `id_2ary` is not simply `k2 * id_l6et * id_ep2m`?

Let's consider the possibility that `id_ep2m` and `id_2ary` are produced from `id_l6et` in a reaction that is *activated* by `id_ep2m` or `id_2ary`. This is similar to autocatalysis but the kinetic law might be different. For example, a Hill-like activation. However, implementing complex kinetics like Hill functions directly in the SBML kinetic law string can be tricky and requires defining parameters for Km, Hill coefficient, etc.

Let's try a different angle. What if there's a reaction that produces `id_ep2m` and `id_2ary` from `id_l6et`, and another reaction that consumes `id_ep2m` and `id_2ary`?
Reaction 1: `id_l6et` -> `id_wnsa` (Rate `k1 * id_l6et`)
Reaction 2: `id_l6et` -> `id_ep2m` + `id_2ary` (Rate `k2 * id_l6et`)
Reaction 3: `id_ep2m + id_2ary` -> `id_l6et` (Rate `k3 * id_ep2m * id_2ary`)
This is the reversible mass-action model, which didn't work.

What if Reaction 2 is `id_l6et -> id_ep2m` and `id_l6et -> id_2ary`? This doesn't explain the strong correlation between `id_ep2m` and `id_2ary` changes.

Let's go back to the autocatalytic model structure (Attempt 3) and try to optimize the parameters again, but perhaps with different initial guesses or a different optimization method if available (though Nelder-Mead is generally robust). The previous optimization attempt hit the maximum number of function evaluations in one case, suggesting it might not have fully converged or the landscape is difficult.

Let's try optimizing again, maybe with slightly different bounds or initial guesses, and allow more iterations if possible (though I'm limited by the total iteration count). I'll reuse the optimization code from Iteration 8.

## Action

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

# Load experimental data
exp_df = experiment_history['iteration_0']
exp_time = exp_df['Time'].values
exp_species_data = exp_df[['id_l6et', 'id_ep2m', 'id_wnsa', 'id_2ary']].values

# Get initial concentrations from the experimental data at Time 0
initial_concentrations = exp_species_data[0, :]
species_ids = ['id_l6et', 'id_ep2m', 'id_wnsa', 'id_2ary']

# Function to build SBML model with given parameters (Autocatalytic structure - Attempt 3)
def build_sbml_model_autocatalytic_consumption(k1, k2, k3):
    sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
    model = sbml_doc.getModel()

    # Clear existing reactions and parameters
    while model.getNumReactions() > 0:
        model.removeReaction(0)
    while model.getNumParameters() > 0:
        model.removeParameter(0)

    # Add parameters
    p_k1 = model.createParameter()
    p_k1.setId("k1")
    p_k1.setValue(k1)
    p_k1.setConstant(True)

    p_k2 = model.createParameter()
    p_k2.setId("k2")
    p_k2.setValue(k2)
    p_k2.setConstant(True)

    p_k3 = model.createParameter()
    p_k3.setId("k3")
    p_k3.setValue(k3)
    p_k3.setConstant(True)

    # Reaction 1: id_l6et -> id_wnsa (Mass action)
    reaction1 = model.createReaction()
    reaction1.setId("reaction1")
    reaction1.setReversible(False)
    reaction1.setFast(False)
    r1_reactant = reaction1.createReactant()
    r1_reactant.setSpecies("id_l6et")
    r1_reactant.setStoichiometry(1.0)
    r1_reactant.setConstant(False)
    r1_product = reaction1.createProduct()
    r1_product.setSpecies("id_wnsa")
    r1_product.setStoichiometry(1.0)
    r1_product.setConstant(False)
    kl1 = reaction1.createKineticLaw()
    kl1.setFormula("k1 * id_l6et")

    # Reaction 2: id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary (Autocatalytic production)
    reaction2 = model.createReaction()
    reaction2.setId("reaction2")
    reaction2.setReversible(False)
    reaction2.setFast(False)
    r2_reactant1 = reaction2.createReactant()
    r2_reactant1.setSpecies("id_l6et")
    r2_reactant1.setStoichiometry(1.0)
    r2_reactant1.setConstant(False)
    r2_reactant2 = reaction2.createReactant()
    r2_reactant2.setSpecies("id_ep2m")
    r2_reactant2.setStoichiometry(1.0)
    r2_reactant2.setConstant(False)
    r2_product1 = reaction2.createProduct()
    r2_product1.setSpecies("id_ep2m")
    r2_product1.setStoichiometry(2.0)
    r2_product1.setConstant(False)
    r2_product2 = reaction2.createProduct()
    r2_product2.setSpecies("id_2ary")
    r2_product2.setStoichiometry(1.0)
    r2_product2.setConstant(False)
    kl2 = reaction2.createKineticLaw()
    kl2.setFormula("k2 * id_l6et * id_ep2m")

    # Reaction 3: id_ep2m + id_2ary -> (Consumption)
    reaction3 = model.createReaction()
    reaction3.setId("reaction3")
    reaction3.setReversible(False)
    reaction3.setFast(False)
    r3_reactant1 = reaction3.createReactant()
    r3_reactant1.setSpecies("id_ep2m")
    r3_reactant1.setStoichiometry(1.0)
    r3_reactant1.setConstant(False)
    r3_reactant2 = reaction3.createReactant()
    r3_reactant2.setSpecies("id_2ary")
    r3_reactant2.setStoichiometry(1.0)
    r3_reactant2.setConstant(False)
    # No products for this reaction

    kl3 = reaction3.createKineticLaw()
    kl3.setFormula("k3 * id_ep2m * id_2ary")

    writer = libsbml.SBMLWriter()
    return writer.writeSBMLToString(sbml_doc)

# Function to simulate the model and return species concentrations
def simulate_model(sbml_string, time_points):
    sim_df = simulate(sbml_string)
    return sim_df

# Objective function to minimize (Sum of Squared Errors)
def objective_function_autocatalytic_consumption(params):
    k1, k2, k3 = params
    if k1 < 0 or k2 < 0 or k3 < 0:
        return np.inf # Penalize negative rate constants

    try:
        sbml_string = build_sbml_model_autocatalytic_consumption(k1, k2, k3)
        sim_df = simulate_model(sbml_string, exp_time)

        # Merge simulation and experimental data
        merged_df = pd.merge(sim_df, exp_df, on='Time', suffixes=('_sim', '_exp'))

        # Calculate SSE for all species
        sse = 0
        for species_id in species_ids:
             sse += np.sum((merged_df[species_id + '_sim'] - merged_df[species_id + '_exp'])**2)

        return sse
    except Exception as e:
        # print(f"Simulation or merge failed: {e}") # Suppress frequent printing during optimization
        return np.inf # Penalize simulation errors

# Initial guess for parameters
initial_params_guesses = [
    [0.0005, 5e-06, 2.7e-05], # Best from previous attempt
    [0.001, 1e-05, 5e-05],
    [0.0001, 1e-07, 1e-06]
]

best_sse = np.inf
best_params = None
best_result = None

for initial_params in initial_params_guesses:
    print(f"Attempting optimization with initial guess: {initial_params}")
    # Increase maxiter slightly if possible, but constrained by total iterations
    result = minimize(objective_function_autocatalytic_consumption, initial_params, method='Nelder-Mead', tol=1e-7) #, bounds=[(0, None)]*3)

    print(f"Result for guess {initial_params}:")
    print(result)

    if result.success and result.fun < best_sse:
        best_sse = result.fun
        best_params = result.x
        best_result = result

print("\nOptimization finished.")
print("Best Result:")
print(best_result)
print(f"\nBest parameters: k1={best_params[0]}, k2={best_params[1]}, k3={best_params[2]}")

# Build the SBML model with the best parameters
best_sbml_string_autocatalytic_consumption = build_sbml_model_autocatalytic_consumption(*best_params)

# Simulate with the best parameters and compare visually (by printing data)
sim_df_best = simulate(best_sbml_string_autocatalytic_consumption)
merged_df_best = pd.merge(sim_df_best, exp_df, on='Time', suffixes=('_sim', '_exp'))

print("\nSimulation with best parameters (autocatalytic consumption model) vs Experiment (first 10 rows):")
print(merged_df_best.head(10))

# Store the best SBML model
shared_variables.add("best_sbml_autocatalytic_consumption", best_sbml_string_autocatalytic_consumption)
```
# Observation

## Code Stdout
Attempting optimization with initial guess: [0.0005, 5e-06, 2.7e-05]
Result for guess [0.0005, 5e-06, 2.7e-05]:
       message: Optimization terminated successfully.
       success: True
        status: 0
           fun: 162845444.64912766
             x: [ 6.414e-14  8.697e-06  4.373e-04]
           nit: 248
          nfev: 531
 final_simplex: (array([[ 6.414e-14,  8.697e-06,  4.373e-04],
                       [ 6.414e-14,  8.697e-06,  4.373e-04],
                       [ 6.414e-14,  8.697e-06,  4.373e-04],
                       [ 6.414e-14,  8.697e-06,  4.373e-04]]), array([ 1.628e+08,  1.628e+08,  1.628e+08,  1.628e+08]))
Attempting optimization with initial guess: [0.001, 1e-05, 5e-05]
Result for guess [0.001, 1e-05, 5e-05]:
       message: Optimization terminated successfully.
       success: True
        status: 0
           fun: 162845418.58702332
             x: [ 4.466e-07  8.693e-06  4.342e-04]
           nit: 239
          nfev: 532
 final_simplex: (array([[ 4.466e-07,  8.693e-06,  4.342e-04],
                       [ 4.466e-07,  8.693e-06,  4.342e-04],
                       [ 4.466e-07,  8.693e-06,  4.342e-04],
                       [ 4.466e-07,  8.693e-06,  4.342e-04]]), array([ 1.628e+08,  1.628e+08,  1.628e+08,  1.628e+08]))
Attempting optimization with initial guess: [0.0001, 1e-07, 1e-06]
Result for guess [0.0001, 1e-07, 1e-06]:
       message: Maximum number of function evaluations has been exceeded.
       success: False
        status: 1
           fun: 162845342.98360798
             x: [ 1.536e-07  8.688e-06  4.316e-04]
           nit: 295
          nfev: 600
 final_simplex: (array([[ 1.536e-07,  8.688e-06,  4.316e-04],
                       [ 1.536e-07,  8.688e-06,  4.316e-04],
                       [ 1.536e-07,  8.688e-06,  4.316e-04],
                       [ 1.536e-07,  8.688e-06,  4.316e-04]]), array([ 1.628e+08,  1.628e+08,  1.628e+08,  1.628e+08]))

Optimization finished.
Best Result:
       message: Optimization terminated successfully.
       success: True
        status: 0
           fun: 162845418.58702332
             x: [ 4.466e-07  8.693e-06  4.342e-04]
           nit: 239
          nfev: 532
 final_simplex: (array([[ 4.466e-07,  8.693e-06,  4.342e-04],
                       [ 4.466e-07,  8.693e-06,  4.342e-04],
                       [ 4.466e-07,  8.693e-06,  4.342e-04],
                       [ 4.466e-07,  8.693e-06,  4.342e-04]]), array([ 1.628e+08,  1.628e+08,  1.628e+08,  1.628e+08]))

Best parameters: k1=4.4661901409027125e-07, k2=8.693023080443279e-06, k3=0.0004341999877793394

Simulation with best parameters (autocatalytic consumption model) vs Experiment (first 10 rows):
   id_l6et_sim  id_ep2m_sim  ...  id_wnsa_exp   id_2ary_exp
0  1000.000000   200.000000  ...     1.000000  1.000000e+00
1   998.605235   201.279590  ...     1.180227  8.719520e+00
2   997.203824   202.478249  ...     3.725243  2.900681e+01
3   995.796347   203.600012  ...     9.300273  2.348123e+00
4   994.383352   204.648835  ...     8.972119  4.980706e-02
5   992.965358   205.628588  ...     8.290540  1.555331e-03
6   991.542858   206.543055  ...     7.653392  7.377733e-05
7   990.116315   207.395892  ...     7.064983  5.178048e-06
8   988.686165   208.190627  ...     6.521802  5.237403e-07
9   987.252814   208.930648  ...     6.020382  7.445900e-08

[10 rows x 9 columns]

## 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)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml, best_sbml_reversible_massaction, current_sbml_autocatalytic, best_sbml_autocatalytic, current_sbml_autocatalytic_consumption, best_sbml_autocatalytic_consumption)

### 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
Optimizing the parameters for the autocatalytic production and bimolecular consumption model (Attempt 3) resulted in a lower SSE (around 1.6e+08) compared to previous attempts, but the simulated dynamics still do not show the characteristic fluctuations of `id_ep2m` and `id_2ary`. The simulated concentrations change very slowly and smoothly in the initial phase, which is a poor match to the rapid changes observed in the experiment.

This strongly suggests that the current model structure, even with optimized mass-action kinetics, is fundamentally incapable of reproducing the observed behavior. The missing mechanism is likely related to the generation of these rapid fluctuations.

Let's reconsider the nature of the fluctuations. They are sharp increases followed by sharp decreases. This "boom and bust" cycle is characteristic of systems with positive feedback (leading to rapid growth) coupled with a limiting factor or negative feedback (leading to a crash).

The autocatalytic reaction `id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary` provides positive feedback on `id_ep2m`. The consumption reaction `id_ep2m + id_2ary -> ` provides a consumption mechanism. However, the combination with mass-action kinetics doesn't seem sufficient.

What if the consumption reaction is not simply bimolecular mass action? What if it's somehow dependent on the concentration of `id_l6et` or another species?

Let's look at the experimental data again. When `id_ep2m` and `id_2ary` concentrations are high, they drop rapidly. This drop coincides with `id_l6et` concentration being relatively low (compared to the start). When `id_ep2m` and `id_2ary` are low, they start increasing again, which happens when `id_l6et` has recovered somewhat. This hints at a dependency on `id_l6et`.

Could the consumption of `id_ep2m` and `id_2ary` be inhibited by `id_l6et`? Or perhaps the production is activated by `id_l6et` (which is already in the autocatalytic term `k2 * id_l6et * id_ep2m`), and the consumption is simply proportional to `id_ep2m` and `id_2ary`?

Let's consider the possibility that the stoichiometry of the autocatalytic reaction is different. What if it's `id_l6et + id_ep2m -> 3 * id_ep2m + id_2ary`? This would increase the positive feedback. Or `id_l6et + id_ep2m -> 2 * id_ep2m + 2 * id_2ary`? This would maintain the 1:1 production ratio.

Let's try adjusting the stoichiometry of the products in the autocatalytic reaction (Reaction 2) to see if it can induce fluctuations. Let's try producing 2 `id_2ary` for every 1 `id_ep2m` net production.
Reaction 2: `id_l6et + id_ep2m -> 2 * id_ep2m + 2 * id_2ary` (Rate = `k2 * id_l6et * id_ep2m`)
Reaction 3: `id_ep2m + id_2ary -> ` (Rate = `k3 * id_ep2m * id_2ary`)

Let's implement this modified structure and simulate with placeholder parameters.

## Action

### Code
```python
import libsbml
import pandas as pd

# Load the initial SBML model structure (without reactions)
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Clear existing reactions and parameters
while model.getNumReactions() > 0:
    model.removeReaction(0)
while model.getNumParameters() > 0:
    model.removeParameter(0)

# Add parameters for the new reactions
k1 = model.createParameter()
k1.setId("k1")
k1.setValue(0.001) # Placeholder
k1.setConstant(True)

k2 = model.createParameter()
k2.setId("k2")
k2.setValue(0.0001) # Placeholder
k2.setConstant(True)

k3 = model.createParameter()
k3.setId("k3")
k3.setValue(0.00001) # Placeholder
k3.setConstant(True)

# Reaction 1: id_l6et -> id_wnsa (Mass action)
reaction1 = model.createReaction()
reaction1.setId("reaction1")
reaction1.setReversible(False)
reaction1.setFast(False)
r1_reactant = reaction1.createReactant()
r1_reactant.setSpecies("id_l6et")
r1_reactant.setStoichiometry(1.0)
r1_reactant.setConstant(False)
r1_product = reaction1.createProduct()
r1_product.setSpecies("id_wnsa")
r1_product.setStoichiometry(1.0)
r1_product.setConstant(False)
kl1 = reaction1.createKineticLaw()
kl1.setFormula("k1 * id_l6et")

# Reaction 2: id_l6et + id_ep2m -> 2 * id_ep2m + 2 * id_2ary (Autocatalytic production, modified stoichiometry)
reaction2 = model.createReaction()
reaction2.setId("reaction2")
reaction2.setReversible(False)
reaction2.setFast(False)
r2_reactant1 = reaction2.createReactant()
r2_reactant1.setSpecies("id_l6et")
r2_reactant1.setStoichiometry(1.0)
r2_reactant1.setConstant(False)
r2_reactant2 = reaction2.createReactant()
r2_reactant2.setSpecies("id_ep2m")
r2_reactant2.setStoichiometry(1.0)
r2_reactant2.setConstant(False)
r2_product1 = reaction2.createProduct()
r2_product1.setSpecies("id_ep2m")
r2_product1.setStoichiometry(2.0) # Produces 2 molecules for every 1 consumed
r2_product1.setConstant(False)
r2_product2 = reaction2.createProduct()
r2_product2.setSpecies("id_2ary")
r2_product2.setStoichiometry(2.0) # Modified stoichiometry
r2_product2.setConstant(False)
kl2 = reaction2.createKineticLaw()
kl2.setFormula("k2 * id_l6et * id_ep2m")

# Reaction 3: id_ep2m + id_2ary -> (Consumption)
reaction3 = model.createReaction()
reaction3.setId("reaction3")
reaction3.setReversible(False)
reaction3.setFast(False)
r3_reactant1 = reaction3.createReactant()
r3_reactant1.setSpecies("id_ep2m")
r3_reactant1.setStoichiometry(1.0)
r3_reactant1.setConstant(False)
r3_reactant2 = reaction3.createReactant()
r3_reactant2.setSpecies("id_2ary")
r3_reactant2.setStoichiometry(1.0)
r3_reactant2.setConstant(False)
# No products for this reaction

kl3 = reaction3.createKineticLaw()
kl3.setFormula("k3 * id_ep2m * id_2ary")


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

# Simulate the updated model
sim_df = simulate(updated_sbml)

# Load the experimental data
exp_df = experiment_history['iteration_0']

# Compare simulation and experiment
print("Simulation results (first 10 rows):")
print(sim_df.head(10))

merged_df = pd.merge(sim_df, exp_df, on='Time', suffixes=('_sim', '_exp'))
print("\nDifference between simulation and experiment (first 10 rows):")
print(merged_df.head(10))

# Store the updated SBML for future use
shared_variables.add("current_sbml_autocatalytic_consumption_stoich", updated_sbml)
```
# Observation

## Code Stdout
Simulation results (first 10 rows):
       id_l6et     id_ep2m   id_wnsa     id_2ary  Time
0  1000.000000  200.000000  1.000000    1.000000   0.0
1   982.698664  216.478961  1.793146   33.987151   0.8
2   964.387402  233.918501  2.572048   68.959051   1.6
3   945.061922  252.309967  3.335895  105.912149   2.4
4   924.727358  271.633705  4.083878  144.822468   3.2
5   903.399769  291.857452  4.815194  185.642488   4.0
6   881.106942  312.935600  5.529061  228.299596   4.8
7   857.888728  334.808988  6.224721  272.695538   5.6
8   833.797437  357.404630  6.901454  318.705738   6.4
9   808.897976  380.635801  7.558586  366.179239   7.2

Difference between simulation and experiment (first 10 rows):
   id_l6et_sim  id_ep2m_sim  ...  id_wnsa_exp   id_2ary_exp
0  1000.000000   200.000000  ...     1.000000  1.000000e+00
1   982.698664   216.478961  ...     1.180227  8.719520e+00
2   964.387402   233.918501  ...     3.725243  2.900681e+01
3   945.061922   252.309967  ...     9.300273  2.348123e+00
4   924.727358   271.633705  ...     8.972119  4.980706e-02
5   903.399769   291.857452  ...     8.290540  1.555331e-03
6   881.106942   312.935600  ...     7.653392  7.377733e-05
7   857.888728   334.808988  ...     7.064983  5.178048e-06
8   833.797437   357.404630  ...     6.521802  5.237403e-07
9   808.897976   380.635801  ...     6.020382  7.445900e-08

[10 rows x 9 columns]

## 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)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml, best_sbml_reversible_massaction, current_sbml_autocatalytic, best_sbml_autocatalytic, current_sbml_autocatalytic_consumption, best_sbml_autocatalytic_consumption, current_sbml_autocatalytic_consumption_stoich)

### 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
Modifying the stoichiometry in the autocatalytic production reaction (Reaction 2: `id_l6et + id_ep2m -> 2 * id_ep2m + 2 * id_2ary`) still did not produce the desired fluctuations in the simulation. The concentrations continue to change smoothly. This reinforces the conclusion that simple mass-action kinetics, even with autocatalysis and consumption, is likely not sufficient to capture the observed dynamics.

The persistent failure to reproduce the fluctuations suggests that the missing mechanism involves a stronger form of non-linearity or feedback than simple mass action can provide. This could involve:
- More complex kinetic laws (e.g., Michaelis-Menten, Hill kinetics, or other forms of regulation).
- A different set of reactions that create a system capable of oscillations (like a relaxation oscillator).

Let's reconsider the experimental data and the correlations. The strong positive correlation between `id_ep2m` and `id_2ary` changes is the most striking feature of the fluctuating behavior. This suggests they are produced and consumed together in roughly equal proportions during these events.

The autocatalytic production reaction `id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary` (or `+ 2*id_2ary`) captures the idea of `id_ep2m` promoting its own formation along with `id_2ary` from `id_l6et`. The consumption reaction `id_ep2m + id_2ary -> ` captures the coupled removal.

What if the rate of the autocatalytic reaction is not simply proportional to `id_l6et * id_ep2m`? What if it has a threshold or switch-like behavior? Or what if the consumption reaction is regulated?

Let's consider the possibility that the consumption reaction `id_ep2m + id_2ary -> ` is somehow dependent on the concentration of `id_l6et`. For instance, if high `id_l6et` inhibits this consumption, or low `id_l6et` promotes it. Looking at the data, the drops in `id_ep2m` and `id_2ary` seem to occur when `id_l6et` is relatively low. This hints that low `id_l6et` might *promote* the consumption of `id_ep2m` and `id_2ary`.

Let's try modifying the kinetic law of Reaction 3 (`id_ep2m + id_2ary -> `). Instead of `k3 * id_ep2m * id_2ary`, let's make the rate inversely proportional to `id_l6et` (with some offset to avoid division by zero). Rate = `k3 * id_ep2m * id_2ary / (id_l6et + Km_l6et)`. This would mean consumption is faster when `id_l6et` is low.

Model Structure Attempt 4:
Reaction 1: `id_l6et` -> `id_wnsa` (Rate = `k1 * id_l6et`)
Reaction 2: `id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary` (Rate = `k2 * id_l6et * id_ep2m`)
Reaction 3: `id_ep2m + id_2ary -> ` (Rate = `k3 * id_ep2m * id_2ary / (id_l6et + Km_l6et)`) - Consumption inhibited by `id_l6et`

This introduces a new parameter, `Km_l6et`. Let's implement this and simulate with placeholder parameters.

## Action

### Code
```python
import libsbml
import pandas as pd

# Load the initial SBML model structure (without reactions)
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Clear existing reactions and parameters
while model.getNumReactions() > 0:
    model.removeReaction(0)
while model.getNumParameters() > 0:
    model.removeParameter(0)

# Add parameters for the new reactions
k1 = model.createParameter()
k1.setId("k1")
k1.setValue(0.001) # Placeholder
k1.setConstant(True)

k2 = model.createParameter()
k2.setId("k2")
k2.setValue(0.0001) # Placeholder
k2.setConstant(True)

k3 = model.createParameter()
k3.setId("k3")
k3.setValue(0.1) # Placeholder - might need a larger value for this kinetic form
k3.setConstant(True)

km_l6et = model.createParameter()
km_l6et.setId("Km_l6et")
km_l6et.setValue(100.0) # Placeholder - Km value related to l6et concentration
km_l6et.setConstant(True)


# Reaction 1: id_l6et -> id_wnsa (Mass action)
reaction1 = model.createReaction()
reaction1.setId("reaction1")
reaction1.setReversible(False)
reaction1.setFast(False)
r1_reactant = reaction1.createReactant()
r1_reactant.setSpecies("id_l6et")
r1_reactant.setStoichiometry(1.0)
r1_reactant.setConstant(False)
r1_product = reaction1.createProduct()
r1_product.setSpecies("id_wnsa")
r1_product.setStoichiometry(1.0)
r1_product.setConstant(False)
kl1 = reaction1.createKineticLaw()
kl1.setFormula("k1 * id_l6et")

# Reaction 2: id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary (Autocatalytic production)
reaction2 = model.createReaction()
reaction2.setId("reaction2")
reaction2.setReversible(False)
reaction2.setFast(False)
r2_reactant1 = reaction2.createReactant()
r2_reactant1.setSpecies("id_l6et")
r2_reactant1.setStoichiometry(1.0)
r2_reactant1.setConstant(False)
r2_reactant2 = reaction2.createReactant()
r2_reactant2.setSpecies("id_ep2m")
r2_reactant2.setStoichiometry(1.0)
r2_reactant2.setConstant(False)
r2_product1 = reaction2.createProduct()
r2_product1.setSpecies("id_ep2m")
r2_product1.setStoichiometry(2.0)
r2_product1.setConstant(False)
r2_product2 = reaction2.createProduct()
r2_product2.setSpecies("id_2ary")
r2_product2.setStoichiometry(1.0)
r2_product2.setConstant(False)
kl2 = reaction2.createKineticLaw()
kl2.setFormula("k2 * id_l6et * id_ep2m")

# Reaction 3: id_ep2m + id_2ary -> (Consumption inhibited by id_l6et)
reaction3 = model.createReaction()
reaction3.setId("reaction3")
reaction3.setReversible(False)
reaction3.setFast(False)
r3_reactant1 = reaction3.createReactant()
r3_reactant1.setSpecies("id_ep2m")
r3_reactant1.setStoichiometry(1.0)
r3_reactant1.setConstant(False)
r3_reactant2 = reaction3.createReactant()
r3_reactant2.setSpecies("id_2ary")
r3_reactant2.setStoichiometry(1.0)
r3_reactant2.setConstant(False)
# No products for this reaction

kl3 = reaction3.createKineticLaw()
# Rate = k3 * [ep2m] * [2ary] / ([l6et] + Km_l6et)
kl3.setFormula("k3 * id_ep2m * id_2ary / (id_l6et + Km_l6et)")


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

# Simulate the updated model
sim_df = simulate(updated_sbml)

# Load the experimental data
exp_df = experiment_history['iteration_0']

# Compare simulation and experiment
print("Simulation results (first 10 rows):")
print(sim_df.head(10))

merged_df = pd.merge(sim_df, exp_df, on='Time', suffixes=('_sim', '_exp'))
print("\nDifference between simulation and experiment (first 10 rows):")
print(merged_df.head(10))

# Store the updated SBML for future use
shared_variables.add("current_sbml_inhibited_consumption", updated_sbml)
```
# Observation

## Code Stdout
Simulation results (first 10 rows):
       id_l6et     id_ep2m   id_wnsa     id_2ary  Time
0  1000.000000  200.000000  1.000000    1.000000   0.0
1   982.701791  216.363837  1.793145   17.363837   0.8
2   964.411525  233.439879  2.572057   34.439879   1.6
3   945.145259  251.145173  3.335944   52.145173   2.4
4   924.932612  269.375592  4.084037   70.375592   3.2
5   903.817829  288.005028  4.815596   89.005028   4.0
6   881.859988  306.886244  5.529922  107.886244   4.8
7   859.132947  325.852391  6.226369  126.852391   5.6
8   835.724816  344.719641  6.904355  145.719641   6.4
9   811.736800  363.290866  7.563376  164.290866   7.2

Difference between simulation and experiment (first 10 rows):
   id_l6et_sim  id_ep2m_sim  ...  id_wnsa_exp   id_2ary_exp
0  1000.000000   200.000000  ...     1.000000  1.000000e+00
1   982.701791   216.363837  ...     1.180227  8.719520e+00
2   964.411525   233.439879  ...     3.725243  2.900681e+01
3   945.145259   251.145173  ...     9.300273  2.348123e+00
4   924.932612   269.375592  ...     8.972119  4.980706e-02
5   903.817829   288.005028  ...     8.290540  1.555331e-03
6   881.859988   306.886244  ...     7.653392  7.377733e-05
7   859.132947   325.852391  ...     7.064983  5.178048e-06
8   835.724816   344.719641  ...     6.521802  5.237403e-07
9   811.736800   363.290866  ...     6.020382  7.445900e-08

[10 rows x 9 columns]

## 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)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml, best_sbml_reversible_massaction, current_sbml_autocatalytic, best_sbml_autocatalytic, current_sbml_autocatalytic_consumption, best_sbml_autocatalytic_consumption, current_sbml_autocatalytic_consumption_stoich, current_sbml_inhibited_consumption)

### 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
Introducing the `id_l6et`-inhibited consumption reaction (Rate = `k3 * id_ep2m * id_2ary / (id_l6et + Km_l6et)`) still did not produce the desired fluctuations in the simulation with placeholder parameters. The simulated concentrations remain smooth.

This further reinforces that finding the correct model structure and kinetic laws is challenging. Simple modifications to mass-action kinetics or stoichiometry haven't yielded the fluctuating behavior.

Let's step back and consider the core requirements for oscillations in biochemical systems. They typically involve:
1. Positive feedback (often through autocatalysis).
2. Negative feedback or a limiting resource.
3. Appropriate time scales for the different processes.

Our current model attempts to capture this:
Reaction 2 (`id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary`) provides positive feedback on `id_ep2m`.
Reaction 3 (`id_ep2m + id_2ary -> ` or `id_ep2m + id_2ary -> id_l6et`) provides consumption.
`id_l6et` is a limiting resource that is consumed.

The failure to oscillate might mean:
- The kinetic laws are still incorrect.
- The stoichiometry is incorrect.
- There is a missing reaction or regulatory interaction.
- The parameter values are in a region of parameter space that does not support oscillations for this model structure.

Given the difficulty in guessing the exact kinetic laws, let's try to optimize the parameters for the model structure with `id_l6et`-inhibited consumption (Attempt 4). This model has 4 parameters (`k1`, `k2`, `k3`, `Km_l6et`). If this model structure is capable of oscillations, optimization might find parameters that exhibit this behavior and fit the experimental data better.

I will use the optimization approach again, minimizing the sum of squared errors.

## Action

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

# Load experimental data
exp_df = experiment_history['iteration_0']
exp_time = exp_df['Time'].values
exp_species_data = exp_df[['id_l6et', 'id_ep2m', 'id_wnsa', 'id_2ary']].values

# Get initial concentrations from the experimental data at Time 0
initial_concentrations = exp_species_data[0, :]
species_ids = ['id_l6et', 'id_ep2m', 'id_wnsa', 'id_2ary']

# Function to build SBML model with given parameters (Autocatalytic production, Inhibited Consumption)
def build_sbml_model_inhibited_consumption(k1, k2, k3, Km_l6et):
    sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
    model = sbml_doc.getModel()

    # Clear existing reactions and parameters
    while model.getNumReactions() > 0:
        model.removeReaction(0)
    while model.getNumParameters() > 0:
        model.removeParameter(0)

    # Add parameters
    p_k1 = model.createParameter()
    p_k1.setId("k1")
    p_k1.setValue(k1)
    p_k1.setConstant(True)

    p_k2 = model.createParameter()
    p_k2.setId("k2")
    p_k2.setValue(k2)
    p_k2.setConstant(True)

    p_k3 = model.createParameter()
    p_k3.setId("k3")
    p_k3.setValue(k3)
    p_k3.setConstant(True)

    p_km_l6et = model.createParameter()
    p_km_l6et.setId("Km_l6et")
    p_km_l6et.setValue(Km_l6et)
    p_km_l6et.setConstant(True)


    # Reaction 1: id_l6et -> id_wnsa (Mass action)
    reaction1 = model.createReaction()
    reaction1.setId("reaction1")
    reaction1.setReversible(False)
    reaction1.setFast(False)
    r1_reactant = reaction1.createReactant()
    r1_reactant.setSpecies("id_l6et")
    r1_reactant.setStoichiometry(1.0)
    r1_reactant.setConstant(False)
    r1_product = reaction1.createProduct()
    r1_product.setSpecies("id_wnsa")
    r1_product.setStoichiometry(1.0)
    r1_product.setConstant(False)
    kl1 = reaction1.createKineticLaw()
    kl1.setFormula("k1 * id_l6et")

    # Reaction 2: id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary (Autocatalytic production)
    reaction2 = model.createReaction()
    reaction2.setId("reaction2")
    reaction2.setReversible(False)
    reaction2.setFast(False)
    r2_reactant1 = reaction2.createReactant()
    r2_reactant1.setSpecies("id_l6et")
    r2_reactant1.setStoichiometry(1.0)
    r2_reactant1.setConstant(False)
    r2_reactant2 = reaction2.createReactant()
    r2_reactant2.setSpecies("id_ep2m")
    r2_reactant2.setStoichiometry(1.0)
    r2_reactant2.setConstant(False)
    r2_product1 = reaction2.createProduct()
    r2_product1.setSpecies("id_ep2m")
    r2_product1.setStoichiometry(2.0)
    r2_product1.setConstant(False)
    r2_product2 = reaction2.createProduct()
    r2_product2.setSpecies("id_2ary")
    r2_product2.setStoichiometry(1.0)
    r2_product2.setConstant(False)
    kl2 = reaction2.createKineticLaw()
    kl2.setFormula("k2 * id_l6et * id_ep2m")

    # Reaction 3: id_ep2m + id_2ary -> (Consumption inhibited by id_l6et)
    reaction3 = model.createReaction()
    reaction3.setId("reaction3")
    reaction3.setReversible(False)
    reaction3.setFast(False)
    r3_reactant1 = reaction3.createReactant()
    r3_reactant1.setSpecies("id_ep2m")
    r3_reactant1.setStoichiometry(1.0)
    r3_reactant1.setConstant(False)
    r3_reactant2 = reaction3.createReactant()
    r3_reactant2.setSpecies("id_2ary")
    r3_reactant2.setStoichiometry(1.0)
    r3_reactant2.setConstant(False)
    # No products for this reaction

    kl3 = reaction3.createKineticLaw()
    # Rate = k3 * [ep2m] * [2ary] / ([l6et] + Km_l6et)
    kl3.setFormula("k3 * id_ep2m * id_2ary / (id_l6et + Km_l6et)")


    writer = libsbml.SBMLWriter()
    return writer.writeSBMLToString(sbml_doc)

# Function to simulate the model and return species concentrations
def simulate_model(sbml_string, time_points):
    sim_df = simulate(sbml_string)
    return sim_df

# Objective function to minimize (Sum of Squared Errors)
def objective_function_inhibited_consumption(params):
    k1, k2, k3, Km_l6et = params
    if any(p < 0 for p in params):
        return np.inf # Penalize negative rate constants or Km

    try:
        sbml_string = build_sbml_model_inhibited_consumption(k1, k2, k3, Km_l6et)
        sim_df = simulate_model(sbml_string, exp_time)

        # Merge simulation and experimental data
        merged_df = pd.merge(sim_df, exp_df, on='Time', suffixes=('_sim', '_exp'))

        # Calculate SSE for all species
        sse = 0
        for species_id in species_ids:
             sse += np.sum((merged_df[species_id + '_sim'] - merged_df[species_id + '_exp'])**2)

        return sse
    except Exception as e:
        # print(f"Simulation or merge failed: {e}") # Suppress frequent printing during optimization
        return np.inf # Penalize simulation errors

# Initial guess for parameters (k1, k2, k3, Km_l6et)
# Based on previous k1, k2, k3 values and a guess for Km_l6et
initial_params_guesses = [
    [0.0005, 5e-06, 0.00005, 100.0],
    [0.001, 1e-05, 0.0001, 50.0],
    [0.0001, 1e-07, 0.00001, 200.0]
]

best_sse = np.inf
best_params = None
best_result = None

for initial_params in initial_params_guesses:
    print(f"Attempting optimization with initial guess: {initial_params}")
    # Add bounds to ensure parameters are non-negative
    bounds = [(1e-12, None), (1e-12, None), (1e-12, None), (1e-12, None)] # Small lower bound to avoid zero
    result = minimize(objective_function_inhibited_consumption, initial_params, method='Nelder-Mead', tol=1e-7, bounds=bounds)

    print(f"Result for guess {initial_params}:")
    print(result)

    if result.success and result.fun < best_sse:
        best_sse = result.fun
        best_params = result.x
        best_result = result

print("\nOptimization finished.")
print("Best Result:")
print(best_result)
print(f"\nBest parameters: k1={best_params[0]}, k2={best_params[1]}, k3={best_params[2]}, Km_l6et={best_params[3]}")

# Build the SBML model with the best parameters
best_sbml_string_inhibited_consumption = build_sbml_model_inhibited_consumption(*best_params)

# Simulate with the best parameters and compare visually (by printing data)
sim_df_best = simulate(best_sbml_string_inhibited_consumption)
merged_df_best = pd.merge(sim_df_best, exp_df, on='Time', suffixes=('_sim', '_exp'))

print("\nSimulation with best parameters (inhibited consumption model) vs Experiment (first 10 rows):")
print(merged_df_best.head(10))

# Store the best SBML model
shared_variables.add("best_sbml_inhibited_consumption", best_sbml_string_inhibited_consumption)
```
# Observation

## Code Stdout
Attempting optimization with initial guess: [0.0005, 5e-06, 5e-05, 100.0]
Result for guess [0.0005, 5e-06, 5e-05, 100.0]:
       message: Maximum number of function evaluations has been exceeded.
       success: False
        status: 1
           fun: 234770182.14996058
             x: [ 3.886e-04  2.469e-06  4.238e+08  6.324e+13]
           nit: 498
          nfev: 800
 final_simplex: (array([[ 3.886e-04,  2.469e-06,  4.238e+08,  6.324e+13],
                       [ 3.886e-04,  2.469e-06,  2.542e+08,  3.794e+13],
                       ...,
                       [ 3.886e-04,  2.469e-06,  2.884e+08,  4.304e+13],
                       [ 3.886e-04,  2.469e-06,  2.822e+08,  4.212e+13]],
                      shape=(5, 4)), array([ 2.348e+08,  2.348e+08,  2.348e+08,  2.348e+08,
                        2.348e+08]))
Attempting optimization with initial guess: [0.001, 1e-05, 0.0001, 50.0]
Result for guess [0.001, 1e-05, 0.0001, 50.0]:
       message: Optimization terminated successfully.
       success: True
        status: 0
           fun: 162528533.33563483
             x: [ 1.000e-12  8.569e-06  2.327e-01  1.000e-12]
           nit: 327
          nfev: 677
 final_simplex: (array([[ 1.000e-12,  8.569e-06,  2.327e-01,  1.000e-12],
                       [ 1.000e-12,  8.569e-06,  2.327e-01,  1.000e-12],
                       ...,
                       [ 1.000e-12,  8.569e-06,  2.327e-01,  1.000e-12],
                       [ 1.000e-12,  8.569e-06,  2.327e-01,  1.000e-12]],
                      shape=(5, 4)), array([ 1.625e+08,  1.625e+08,  1.625e+08,  1.625e+08,
                        1.625e+08]))
Attempting optimization with initial guess: [0.0001, 1e-07, 1e-05, 200.0]
Result for guess [0.0001, 1e-07, 1e-05, 200.0]:
       message: Maximum number of function evaluations has been exceeded.
       success: False
        status: 1
           fun: 162528557.97279614
             x: [ 4.030e-09  8.570e-06  2.329e-01  1.000e-12]
           nit: 399
          nfev: 800
 final_simplex: (array([[ 4.030e-09,  8.570e-06,  2.329e-01,  1.000e-12],
                       [ 4.030e-09,  8.570e-06,  2.329e-01,  1.000e-12],
                       ...,
                       [ 4.030e-09,  8.570e-06,  2.329e-01,  1.000e-12],
                       [ 4.030e-09,  8.570e-06,  2.329e-01,  1.000e-12]],
                      shape=(5, 4)), array([ 1.625e+08,  1.625e+08,  1.625e+08,  1.625e+08,
                        1.625e+08]))

Optimization finished.
Best Result:
       message: Optimization terminated successfully.
       success: True
        status: 0
           fun: 162528533.33563483
             x: [ 1.000e-12  8.569e-06  2.327e-01  1.000e-12]
           nit: 327
          nfev: 677
 final_simplex: (array([[ 1.000e-12,  8.569e-06,  2.327e-01,  1.000e-12],
                       [ 1.000e-12,  8.569e-06,  2.327e-01,  1.000e-12],
                       ...,
                       [ 1.000e-12,  8.569e-06,  2.327e-01,  1.000e-12],
                       [ 1.000e-12,  8.569e-06,  2.327e-01,  1.000e-12]],
                      shape=(5, 4)), array([ 1.625e+08,  1.625e+08,  1.625e+08,  1.625e+08,
                        1.625e+08]))

Best parameters: k1=1.0000000000000014e-12, k2=8.568827474718096e-06, k3=0.2327366713368327, Km_l6et=1.0000000000000377e-12

Simulation with best parameters (inhibited consumption model) vs Experiment (first 10 rows):
   id_l6et_sim  id_ep2m_sim  ...  id_wnsa_exp   id_2ary_exp
0  1000.000000   200.000000  ...     1.000000  1.000000e+00
1   998.625414   201.312507  ...     1.180227  8.719520e+00
2   997.243891   202.582817  ...     3.725243  2.900681e+01
3   995.855745   203.811462  ...     9.300273  2.348123e+00
4   994.461294   204.999040  ...     8.972119  4.980706e-02
5   993.060848   206.146184  ...     8.290540  1.555331e-03
6   991.654711   207.253556  ...     7.653392  7.377733e-05
7   990.243183   208.321851  ...     7.064983  5.178048e-06
8   988.826554   209.351797  ...     6.521802  5.237403e-07
9   987.405108   210.344147  ...     6.020382  7.445900e-08

[10 rows x 9 columns]

## 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)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml, best_sbml_reversible_massaction, current_sbml_autocatalytic, best_sbml_autocatalytic, current_sbml_autocatalytic_consumption, best_sbml_autocatalytic_consumption, current_sbml_autocatalytic_consumption_stoich, current_sbml_inhibited_consumption, best_sbml_inhibited_consumption)

### 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
Optimizing the parameters for the model with `id_l6et`-inhibited consumption (Attempt 4) resulted in a slightly lower SSE (around 1.62e+08), but the simulated dynamics still do not show the characteristic fluctuations. The best-fit parameters include `k1` and `Km_l6et` values that are extremely small (close to the lower bound of 1e-12), effectively making Reaction 1 negligible and the `id_l6et` inhibition term in Reaction 3 almost constant (since `id_l6et` is large compared to `Km_l6et`). This suggests the optimization is pushing the model towards a simpler, non-oscillatory behavior because the current structure with mass-action-like kinetics cannot reproduce the fluctuations.

The persistent failure to generate fluctuations with various mass-action and simple non-linear kinetic forms points towards the need for a different type of non-linearity or a different set of reactions.

Let's reconsider the strong positive correlation between `id_ep2m` and `id_2ary` changes. This is a very strong indicator that they are produced and consumed together, likely with 1:1 stoichiometry in the reactions that cause the fluctuations.

The autocatalytic production of `id_ep2m` and `id_2ary` from `id_l6et` seems plausible as the driving force for the increase. The consumption of `id_ep2m` and `id_2ary` is needed to bring the concentrations back down.

What if the autocatalytic reaction is not simply proportional to `id_l6et * id_ep2m`, but has a higher order dependence on `id_ep2m`? For example, `id_l6et + 2*id_ep2m -> 3*id_ep2m + id_2ary` with rate `k2 * id_l6et * id_ep2m^2`. This increases the positive feedback.

Let's try this structure:
Model Structure Attempt 5:
Reaction 1: `id_l6et` -> `id_wnsa` (Rate = `k1 * id_l6et`)
Reaction 2: `id_l6et + 2*id_ep2m -> 3*id_ep2m + id_2ary` (Rate = `k2 * id_l6et * id_ep2m^2`) - Higher-order autocatalysis
Reaction 3: `id_ep2m + id_2ary -> ` (Rate = `k3 * id_ep2m * id_2ary`) - Bimolecular consumption

Let's implement this and simulate with placeholder parameters. Higher-order reactions can sometimes lead to oscillations.

## Action

### Code
```python
import libsbml
import pandas as pd

# Load the initial SBML model structure (without reactions)
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Clear existing reactions and parameters
while model.getNumReactions() > 0:
    model.removeReaction(0)
while model.getNumParameters() > 0:
    model.removeParameter(0)

# Add parameters for the new reactions
k1 = model.createParameter()
k1.setId("k1")
k1.setValue(0.001) # Placeholder
k1.setConstant(True)

k2 = model.createParameter()
k2.setId("k2")
k2.setValue(0.000001) # Placeholder - might need smaller value due to ep2m^2
k2.setConstant(True)

k3 = model.createParameter()
k3.setId("k3")
k3.setValue(0.00001) # Placeholder
k3.setConstant(True)

# Reaction 1: id_l6et -> id_wnsa (Mass action)
reaction1 = model.createReaction()
reaction1.setId("reaction1")
reaction1.setReversible(False)
reaction1.setFast(False)
r1_reactant = reaction1.createReactant()
r1_reactant.setSpecies("id_l6et")
r1_reactant.setStoichiometry(1.0)
r1_reactant.setConstant(False)
r1_product = reaction1.createProduct()
r1_product.setSpecies("id_wnsa")
r1_product.setStoichiometry(1.0)
r1_product.setConstant(False)
kl1 = reaction1.createKineticLaw()
kl1.setFormula("k1 * id_l6et")

# Reaction 2: id_l6et + 2*id_ep2m -> 3*id_ep2m + id_2ary (Higher-order Autocatalytic production)
reaction2 = model.createReaction()
reaction2.setId("reaction2")
reaction2.setReversible(False)
reaction2.setFast(False)
r2_reactant1 = reaction2.createReactant()
r2_reactant1.setSpecies("id_l6et")
r2_reactant1.setStoichiometry(1.0)
r2_reactant1.setConstant(False)
r2_reactant2 = reaction2.createReactant()
r2_reactant2.setSpecies("id_ep2m")
r2_reactant2.setStoichiometry(2.0) # Stoichiometry 2
r2_reactant2.setConstant(False)
r2_product1 = reaction2.createProduct()
r2_product1.setSpecies("id_ep2m")
r2_product1.setStoichiometry(3.0) # Stoichiometry 3 (net +1)
r2_product1.setConstant(False)
r2_product2 = reaction2.createProduct()
r2_product2.setSpecies("id_2ary")
r2_product2.setStoichiometry(1.0)
r2_product2.setConstant(False)
kl2 = reaction2.createKineticLaw()
# Rate = k2 * [l6et] * [ep2m]^2
kl2.setFormula("k2 * id_l6et * id_ep2m * id_ep2m") # SBML requires explicit multiplication

# Reaction 3: id_ep2m + id_2ary -> (Consumption)
reaction3 = model.createReaction()
reaction3.setId("reaction3")
reaction3.setReversible(False)
reaction3.setFast(False)
r3_reactant1 = reaction3.createReactant()
r3_reactant1.setSpecies("id_ep2m")
r3_reactant1.setStoichiometry(1.0)
r3_reactant1.setConstant(False)
r3_reactant2 = reaction3.createReactant()
r3_reactant2.setSpecies("id_2ary")
r3_reactant2.setStoichiometry(1.0)
r3_reactant2.setConstant(False)
# No products for this reaction

kl3 = reaction3.createKineticLaw()
kl3.setFormula("k3 * id_ep2m * id_2ary")


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

# Simulate the updated model
sim_df = simulate(updated_sbml)

# Load the experimental data
exp_df = experiment_history['iteration_0']

# Compare simulation and experiment
print("Simulation results (first 10 rows):")
print(sim_df.head(10))

merged_df = pd.merge(sim_df, exp_df, on='Time', suffixes=('_sim', '_exp'))
print("\nDifference between simulation and experiment (first 10 rows):")
print(merged_df.head(10))

# Store the updated SBML for future use
shared_variables.add("current_sbml_higher_order_autocatalysis", updated_sbml)
```
# Observation

## Code Stdout
Simulation results (first 10 rows):
       id_l6et      id_ep2m   id_wnsa     id_2ary  Time
0  1000.000000   200.000000  1.000000    1.000000   0.0
1   961.940774   237.240232  1.785526   38.240232   0.8
2   909.877906   288.420965  2.535395   89.420965   1.6
3   836.262061   361.014349  3.235618  162.014349   2.4
4   729.752605   466.193625  3.864666  267.193625   3.2
5   578.063503   615.889046  4.391058  416.889046   4.0
6   385.647526   805.006243  4.778237  606.006243   4.8
7   201.627275   983.747659  5.010159  784.747659   5.6
8    83.957924  1094.219191  5.119264  895.219191   6.4
9    30.705574  1139.162578  5.161899  940.162578   7.2

Difference between simulation and experiment (first 10 rows):
   id_l6et_sim  id_ep2m_sim  ...  id_wnsa_exp   id_2ary_exp
0  1000.000000   200.000000  ...     1.000000  1.000000e+00
1   961.940774   237.240232  ...     1.180227  8.719520e+00
2   909.877906   288.420965  ...     3.725243  2.900681e+01
3   836.262061   361.014349  ...     9.300273  2.348123e+00
4   729.752605   466.193625  ...     8.972119  4.980706e-02
5   578.063503   615.889046  ...     8.290540  1.555331e-03
6   385.647526   805.006243  ...     7.653392  7.377733e-05
7   201.627275   983.747659  ...     7.064983  5.178048e-06
8    83.957924  1094.219191  ...     6.521802  5.237403e-07
9    30.705574  1139.162578  ...     6.020382  7.445900e-08

[10 rows x 9 columns]

## 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)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml, best_sbml_reversible_massaction, current_sbml_autocatalytic, best_sbml_autocatalytic, current_sbml_autocatalytic_consumption, best_sbml_autocatalytic_consumption, current_sbml_autocatalytic_consumption_stoich, current_sbml_inhibited_consumption, best_sbml_inhibited_consumption, current_sbml_higher_order_autocatalysis)

### 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
Simulating the model with higher-order autocatalysis (Reaction 2: `id_l6et + 2*id_ep2m -> 3*id_ep2m + id_2ary`) still does not produce sustained oscillations or the sharp fluctuations seen in the experimental data. With the placeholder parameters, the concentrations of `id_ep2m` and `id_2ary` increase rapidly and then plateau as `id_l6et` is depleted. This shows a faster increase than previous models, which is a step in the right direction regarding the speed of the spikes, but it lacks the subsequent sharp decrease.

The absence of the sharp decrease suggests the consumption mechanism (Reaction 3: `id_ep2m + id_2ary -> `) is not strong enough or is not regulated in a way that causes a rapid drop after the peak.

Let's reconsider the consumption reaction. What if the consumption rate is not simply proportional to `id_ep2m * id_2ary`? What if it's proportional to a higher power of `id_ep2m` or `id_2ary`, or both? For example, `id_ep2m + id_2ary -> ` with rate `k3 * id_ep2m^2 * id_2ary^2`. This would make the consumption rate increase much faster as `id_ep2m` and `id_2ary` concentrations rise, potentially leading to a sharper drop after the peak.

Let's try this structure:
Model Structure Attempt 6:
Reaction 1: `id_l6et` -> `id_wnsa` (Rate = `k1 * id_l6et`)
Reaction 2: `id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary` (Rate = `k2 * id_l6et * id_ep2m`) - Autocatalytic production (back to simpler form)
Reaction 3: `id_ep2m + id_2ary -> ` (Rate = `k3 * id_ep2m^2 * id_2ary^2`) - Higher-order consumption

Let's implement this and simulate with placeholder parameters. I'll revert Reaction 2 to the simpler autocatalysis (`id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary`) because the higher-order production didn't immediately yield oscillations and might make parameter fitting harder. The consumption seems more likely to be the missing piece for the sharp drop.

## Action

### Code
```python
import libsbml
import pandas as pd

# Load the initial SBML model structure (without reactions)
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Clear existing reactions and parameters
while model.getNumReactions() > 0:
    model.removeReaction(0)
while model.getNumParameters() > 0:
    model.removeParameter(0)

# Add parameters for the new reactions
k1 = model.createParameter()
k1.setId("k1")
k1.setValue(0.001) # Placeholder
k1.setConstant(True)

k2 = model.createParameter()
k2.setId("k2")
k2.setValue(0.0001) # Placeholder
k2.setConstant(True)

k3 = model.createParameter()
k3.setId("k3")
k3.setValue(1e-9) # Placeholder - very small due to high powers
k3.setConstant(True)

# Reaction 1: id_l6et -> id_wnsa (Mass action)
reaction1 = model.createReaction()
reaction1.setId("reaction1")
reaction1.setReversible(False)
reaction1.setFast(False)
r1_reactant = reaction1.createReactant()
r1_reactant.setSpecies("id_l6et")
r1_reactant.setStoichiometry(1.0)
r1_reactant.setConstant(False)
r1_product = reaction1.createProduct()
r1_product.setSpecies("id_wnsa")
r1_product.setStoichiometry(1.0)
r1_product.setConstant(False)
kl1 = reaction1.createKineticLaw()
kl1.setFormula("k1 * id_l6et")

# Reaction 2: id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary (Autocatalytic production)
reaction2 = model.createReaction()
reaction2.setId("reaction2")
reaction2.setReversible(False)
reaction2.setFast(False)
r2_reactant1 = reaction2.createReactant()
r2_reactant1.setSpecies("id_l6et")
r2_reactant1.setStoichiometry(1.0)
r2_reactant1.setConstant(False)
r2_reactant2 = reaction2.createReactant()
r2_reactant2.setSpecies("id_ep2m")
r2_reactant2.setStoichiometry(1.0)
r2_reactant2.setConstant(False)
r2_product1 = reaction2.createProduct()
r2_product1.setSpecies("id_ep2m")
r2_product1.setStoichiometry(2.0)
r2_product1.setConstant(False)
r2_product2 = reaction2.createProduct()
r2_product2.setSpecies("id_2ary")
r2_product2.setStoichiometry(1.0)
r2_product2.setConstant(False)
kl2 = reaction2.createKineticLaw()
kl2.setFormula("k2 * id_l6et * id_ep2m")

# Reaction 3: id_ep2m + id_2ary -> (Higher-order Consumption)
reaction3 = model.createReaction()
reaction3.setId("reaction3")
reaction3.setReversible(False)
reaction3.setFast(False)
r3_reactant1 = reaction3.createReactant()
r3_reactant1.setSpecies("id_ep2m")
r3_reactant1.setStoichiometry(1.0)
r3_reactant1.setConstant(False)
r3_reactant2 = reaction3.createReactant()
r3_reactant2.setSpecies("id_2ary")
r3_reactant2.setStoichiometry(1.0)
r3_reactant2.setConstant(False)
# No products for this reaction

kl3 = reaction3.createKineticLaw()
# Rate = k3 * [ep2m]^2 * [2ary]^2
kl3.setFormula("k3 * id_ep2m * id_ep2m * id_2ary * id_2ary")


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

# Simulate the updated model
sim_df = simulate(updated_sbml)

# Load the experimental data
exp_df = experiment_history['iteration_0']

# Compare simulation and experiment
print("Simulation results (first 10 rows):")
print(sim_df.head(10))

merged_df = pd.merge(sim_df, exp_df, on='Time', suffixes=('_sim', '_exp'))
print("\nDifference between simulation and experiment (first 10 rows):")
print(merged_df.head(10))

# Store the updated SBML for future use
shared_variables.add("current_sbml_higher_order_consumption", updated_sbml)
```
# Observation

## Code Stdout
Simulation results (first 10 rows):
       id_l6et     id_ep2m   id_wnsa     id_2ary  Time
0  1000.000000  200.000000  1.000000    1.000000   0.0
1   982.697935  216.505044  1.793145   17.505044   0.8
2   964.382211  234.012526  2.572045   35.012526   1.6
3   945.046286  252.489904  3.335885   53.489904   2.4
4   924.696208  271.869574  4.083849   72.869574   3.2
5   903.353337  292.039279  4.815134   93.039279   4.0
6   881.057103  312.832991  5.528961  113.832991   4.8
7   857.867741  334.023391  6.224588  135.023391   5.6
8   833.868558  355.318857  6.901333  156.318857   6.4
9   809.166970  376.369014  7.558590  177.369014   7.2

Difference between simulation and experiment (first 10 rows):
   id_l6et_sim  id_ep2m_sim  ...  id_wnsa_exp   id_2ary_exp
0  1000.000000   200.000000  ...     1.000000  1.000000e+00
1   982.697935   216.505044  ...     1.180227  8.719520e+00
2   964.382211   234.012526  ...     3.725243  2.900681e+01
3   945.046286   252.489904  ...     9.300273  2.348123e+00
4   924.696208   271.869574  ...     8.972119  4.980706e-02
5   903.353337   292.039279  ...     8.290540  1.555331e-03
6   881.057103   312.832991  ...     7.653392  7.377733e-05
7   857.867741   334.023391  ...     7.064983  5.178048e-06
8   833.868558   355.318857  ...     6.521802  5.237403e-07
9   809.166970   376.369014  ...     6.020382  7.445900e-08

[10 rows x 9 columns]

## 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)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml, best_sbml_reversible_massaction, current_sbml_autocatalytic, best_sbml_autocatalytic, current_sbml_autocatalytic_consumption, best_sbml_autocatalytic_consumption, current_sbml_autocatalytic_consumption_stoich, current_sbml_inhibited_consumption, best_sbml_inhibited_consumption, current_sbml_higher_order_autocatalysis, current_sbml_higher_order_consumption)

### 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
Simulating the model with higher-order consumption (Reaction 3: `id_ep2m + id_2ary -> ` with rate `k3 * id_ep2m^2 * id_2ary^2`) still does not produce the sharp fluctuations. The concentrations increase smoothly, similar to previous attempts. The placeholder parameters are likely not optimal, but the lack of any oscillatory behavior suggests this structure with these kinetics is still not correct.

I have tried several model structures involving autocatalysis and consumption, with variations in stoichiometry and simple non-linear kinetics (inhibition by `id_l6et`, higher-order mass action), but none have reproduced the characteristic fluctuations.

The strong positive correlation between `id_ep2m` and `id_2ary` changes, and their coupled spiking behavior, remains the most significant clue. This suggests a core oscillatory module involving these two species, likely fueled by `id_l6et`.

Let's consider a classic oscillatory system structure that involves a precursor, an activator, and an inhibitor.
Precursor -> Activator
Activator + Precursor -> 2 * Activator + Inhibitor
Inhibitor + Activator -> Precursor
Inhibitor -> Degraded

Mapping this to our species:
`id_l6et` could be the Precursor.
`id_ep2m` could be the Activator (autocatalytic production).
`id_2ary` could be the Inhibitor (produced with the activator, inhibits the activator or promotes its consumption).

Let's try a model structure based on this idea:
Reaction 1: `id_l6et` -> `id_wnsa` (Rate = `k1 * id_l6et`) - This seems consistent and separate.
Reaction 2: `id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary` (Rate = `k2 * id_l6et * id_ep2m`) - Autocatalytic production of Activator (`id_ep2m`) and Inhibitor (`id_2ary`) from Precursor (`id_l6et`).
Reaction 3: `id_ep2m + id_2ary -> id_l6et` (Rate = `k3 * id_ep2m * id_2ary`) - Inhibitor (`id_2ary`) and Activator (`id_ep2m`) react to regenerate Precursor (`id_l6et`). This provides negative feedback by removing the activator and inhibitor and regenerating the resource.

This is the same structure as Attempt 2 (reversible reaction `id_l6et <=> id_ep2m + id_2ary` with mass action in both directions, but written as two irreversible reactions). I already optimized parameters for this and it didn't oscillate.

Perhaps the kinetic laws for the activator-inhibitor module are not simple mass action. The rate of Reaction 3 (`id_ep2m + id_2ary -> id_l6et`) might need to be non-linear to cause oscillations. What if the rate is proportional to `id_ep2m * id_2ary^2`? This would mean the inhibitor (`id_2ary`) has a stronger effect on the consumption rate.

Model Structure Attempt 7:
Reaction 1: `id_l6et` -> `id_wnsa` (Rate = `k1 * id_l6et`)
Reaction 2: `id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary` (Rate = `k2 * id_l6et * id_ep2m`)
Reaction 3: `id_ep2m + 2*id_2ary -> id_l6et + id_2ary` (Net: `id_ep2m + id_2ary -> id_l6et`) (Rate = `k3 * id_ep2m * id_2ary^2`) - Consumption rate depends quadratically on Inhibitor (`id_2ary`).

Let's implement this and simulate with placeholder parameters.

## Action

### Code
```python
import libsbml
import pandas as pd

# Load the initial SBML model structure (without reactions)
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Clear existing reactions and parameters
while model.getNumReactions() > 0:
    model.removeReaction(0)
while model.getNumParameters() > 0:
    model.removeParameter(0)

# Add parameters for the new reactions
k1 = model.createParameter()
k1.setId("k1")
k1.setValue(0.001) # Placeholder
k1.setConstant(True)

k2 = model.createParameter()
k2.setId("k2")
k2.setValue(0.0001) # Placeholder
k2.setConstant(True)

k3 = model.createParameter()
k3.setId("k3")
k3.setValue(1e-7) # Placeholder - smaller due to id_2ary^2
k3.setConstant(True)

# Reaction 1: id_l6et -> id_wnsa (Mass action)
reaction1 = model.createReaction()
reaction1.setId("reaction1")
reaction1.setReversible(False)
reaction1.setFast(False)
r1_reactant = reaction1.createReactant()
r1_reactant.setSpecies("id_l6et")
r1_reactant.setStoichiometry(1.0)
r1_reactant.setConstant(False)
r1_product = reaction1.createProduct()
r1_product.setSpecies("id_wnsa")
r1_product.setStoichiometry(1.0)
r1_product.setConstant(False)
kl1 = reaction1.createKineticLaw()
kl1.setFormula("k1 * id_l6et")

# Reaction 2: id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary (Autocatalytic production)
reaction2 = model.createReaction()
reaction2.setId("reaction2")
reaction2.setReversible(False)
reaction2.setFast(False)
r2_reactant1 = reaction2.createReactant()
r2_reactant1.setSpecies("id_l6et")
r2_reactant1.setStoichiometry(1.0)
r2_reactant1.setConstant(False)
r2_reactant2 = reaction2.createReactant()
r2_reactant2.setSpecies("id_ep2m")
r2_reactant2.setStoichiometry(1.0)
r2_reactant2.setConstant(False)
r2_product1 = reaction2.createProduct()
r2_product1.setSpecies("id_ep2m")
r2_product1.setStoichiometry(2.0)
r2_product1.setConstant(False)
r2_product2 = reaction2.createProduct()
r2_product2.setSpecies("id_2ary")
r2_product2.setStoichiometry(1.0)
r2_product2.setConstant(False)
kl2 = reaction2.createKineticLaw()
kl2.setFormula("k2 * id_l6et * id_ep2m")

# Reaction 3: id_ep2m + 2*id_2ary -> id_l6et + id_2ary (Consumption rate depends quadratically on id_2ary)
reaction3 = model.createReaction()
reaction3.setId("reaction3")
reaction3.setReversible(False)
reaction3.setFast(False)
r3_reactant1 = reaction3.createReactant()
r3_reactant1.setSpecies("id_ep2m")
r3_reactant1.setStoichiometry(1.0)
r3_reactant1.setConstant(False)
r3_reactant2 = reaction3.createReactant()
r3_reactant2.setSpecies("id_2ary")
r3_reactant2.setStoichiometry(2.0) # Stoichiometry 2
r3_reactant2.setConstant(False)
r3_product1 = reaction3.createProduct()
r3_product1.setSpecies("id_l6et")
r3_product1.setStoichiometry(1.0)
r3_product1.setConstant(False)
r3_product2 = reaction3.createProduct()
r3_product2.setSpecies("id_2ary")
r3_product2.setStoichiometry(1.0) # Stoichiometry 1 (net consumption of 1 id_2ary)
r3_product2.setConstant(False)

kl3 = reaction3.createKineticLaw()
# Rate = k3 * [ep2m] * [2ary]^2
kl3.setFormula("k3 * id_ep2m * id_2ary * id_2ary")


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

# Simulate the updated model
sim_df = simulate(updated_sbml)

# Load the experimental data
exp_df = experiment_history['iteration_0']

# Compare simulation and experiment
print("Simulation results (first 10 rows):")
print(sim_df.head(10))

merged_df = pd.merge(sim_df, exp_df, on='Time', suffixes=('_sim', '_exp'))
print("\nDifference between simulation and experiment (first 10 rows):")
print(merged_df.head(10))

# Store the updated SBML for future use
shared_variables.add("current_sbml_inhibitor_squared", updated_sbml)
```
# Observation

## Code Stdout
Simulation results (first 10 rows):
       id_l6et     id_ep2m   id_wnsa     id_2ary  Time
0  1000.000000  200.000000  1.000000    1.000000   0.0
1   982.699702  216.507154  1.793144   17.507154   0.8
2   964.396155  234.031795  2.572050   35.031795   1.6
3   945.095108  252.568979  3.335912   53.568979   2.4
4   924.817425  272.098634  4.083942   73.098634   3.2
5   903.601224  292.583405  4.815371   93.583405   4.0
6   881.503599  313.966932  5.529470  114.966932   4.8
7   858.602294  336.172143  6.225563  137.172143   5.6
8   834.996549  359.100405  6.903047  160.100405   6.4
9   810.807097  382.631498  7.561405  183.631498   7.2

Difference between simulation and experiment (first 10 rows):
   id_l6et_sim  id_ep2m_sim  ...  id_wnsa_exp   id_2ary_exp
0  1000.000000   200.000000  ...     1.000000  1.000000e+00
1   982.699702   216.507154  ...     1.180227  8.719520e+00
2   964.396155   234.031795  ...     3.725243  2.900681e+01
3   945.095108   252.568979  ...     9.300273  2.348123e+00
4   924.817425   272.098634  ...     8.972119  4.980706e-02
5   903.601224   292.583405  ...     8.290540  1.555331e-03
6   881.503599   313.966932  ...     7.653392  7.377733e-05
7   858.602294   336.172143  ...     7.064983  5.178048e-06
8   834.996549   359.100405  ...     6.521802  5.237403e-07
9   810.807097   382.631498  ...     6.020382  7.445900e-08

[10 rows x 9 columns]

## 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)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml, best_sbml_reversible_massaction, current_sbml_autocatalytic, best_sbml_autocatalytic, current_sbml_autocatalytic_consumption, best_sbml_autocatalytic_consumption, current_sbml_autocatalytic_consumption_stoich, current_sbml_inhibited_consumption, best_sbml_inhibited_consumption, current_sbml_higher_order_autocatalysis, current_sbml_higher_order_consumption, current_sbml_inhibitor_squared)

### 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
Simulating the model with the consumption rate proportional to `id_ep2m * id_2ary^2` (Attempt 7) still did not produce the desired fluctuations. The simulated concentrations remain smooth.

I have now tried several model structures and kinetic laws, focusing on autocatalysis and consumption/feedback mechanisms, but none have reproduced the sharp, coupled fluctuations of `id_ep2m` and `id_2ary`. This is the key missing dynamic behavior.

The consistent failure suggests that either:
1. The fundamental reaction network structure is different from what I've hypothesized.
2. The kinetic laws are more complex or involve different types of regulation (e.g., thresholds, cooperativity, or regulation by species not directly involved as reactants/products in that step).
3. There is a missing species that acts as a key regulator or intermediate in the oscillatory module (though I was instructed not to add new species).

Given the limited iterations remaining and the difficulty in guessing complex kinetic laws or entirely different reaction networks, my best approach is to try and optimize the parameters for the model structure that seems most plausible based on the correlations and the need for positive and negative feedback. The structure with autocatalytic production and a consumption reaction that regenerates the precursor (`id_l6et`) (Attempt 2, or Reactions 2 and 3 from Attempt 7 without the squared term) is a common motif for oscillations.

Let's go back to the structure:
Reaction 1: `id_l6et` -> `id_wnsa` (Rate = `k1 * id_l6et`)
Reaction 2: `id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary` (Rate = `k2 * id_l6et * id_ep2m`)
Reaction 3: `id_ep2m + id_2ary -> id_l6et` (Rate = `k3 * id_ep2m * id_2ary`)

This is the same as the model I optimized in Iteration 8, which didn't produce oscillations with the found parameters. However, it's possible that with different initial guesses or a more thorough optimization, parameters that *do* produce oscillations for this structure could be found, if they exist. Oscillatory systems often have specific regions in parameter space where oscillations occur.

Let's try optimizing this model structure again, perhaps with a wider range of initial guesses or focusing the optimization on parameters that might drive oscillations (like `k2` and `k3`). I will reuse the optimization code from Iteration 8, but maybe try a few more diverse initial guesses.

## Action

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

# Load experimental data
exp_df = experiment_history['iteration_0']
exp_time = exp_df['Time'].values
exp_species_data = exp_df[['id_l6et', 'id_ep2m', 'id_wnsa', 'id_2ary']].values

# Get initial concentrations from the experimental data at Time 0
initial_concentrations = exp_species_data[0, :]
species_ids = ['id_l6et', 'id_ep2m', 'id_wnsa', 'id_2ary']

# Function to build SBML model with given parameters (Autocatalytic production, Regeneration consumption)
def build_sbml_model_autocatalytic_regeneration(k1, k2, k3):
    sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
    model = sbml_doc.getModel()

    # Clear existing reactions and parameters
    while model.getNumReactions() > 0:
        model.removeReaction(0)
    while model.getNumParameters() > 0:
        model.removeParameter(0)

    # Add parameters
    p_k1 = model.createParameter()
    p_k1.setId("k1")
    p_k1.setValue(k1)
    p_k1.setConstant(True)

    p_k2 = model.createParameter()
    p_k2.setId("k2")
    p_k2.setValue(k2)
    p_k2.setConstant(True)

    p_k3 = model.createParameter()
    p_k3.setId("k3")
    p_k3.setValue(k3)
    p_k3.setConstant(True)

    # Reaction 1: id_l6et -> id_wnsa (Mass action)
    reaction1 = model.createReaction()
    reaction1.setId("reaction1")
    reaction1.setReversible(False)
    reaction1.setFast(False)
    r1_reactant = reaction1.createReactant()
    r1_reactant.setSpecies("id_l6et")
    r1_reactant.setStoichiometry(1.0)
    r1_reactant.setConstant(False)
    r1_product = reaction1.createProduct()
    r1_product.setSpecies("id_wnsa")
    r1_product.setStoichiometry(1.0)
    r1_product.setConstant(False)
    kl1 = reaction1.createKineticLaw()
    kl1.setFormula("k1 * id_l6et")

    # Reaction 2: id_l6et + id_ep2m -> 2 * id_ep2m + id_2ary (Autocatalytic production)
    reaction2 = model.createReaction()
    reaction2.setId("reaction2")
    reaction2.setReversible(False)
    reaction2.setFast(False)
    r2_reactant1 = reaction2.createReactant()
    r2_reactant1.setSpecies("id_l6et")
    r2_reactant1.setStoichiometry(1.0)
    r2_reactant1.setConstant(False)
    r2_reactant2 = reaction2.createReactant()
    r2_reactant2.setSpecies("id_ep2m")
    r2_reactant2.setStoichiometry(1.0)
    r2_reactant2.setConstant(False)
    r2_product1 = reaction2.createProduct()
    r2_product1.setSpecies("id_ep2m")
    r2_product1.setStoichiometry(2.0)
    r2_product1.setConstant(False)
    r2_product2 = reaction2.createProduct()
    r2_product2.setSpecies("id_2ary")
    r2_product2.setStoichiometry(1.0)
    r2_product2.setConstant(False)
    kl2 = reaction2.createKineticLaw()
    kl2.setFormula("k2 * id_l6et * id_ep2m")

    # Reaction 3: id_ep2m + id_2ary -> id_l6et (Consumption/Regeneration)
    reaction3 = model.createReaction()
    reaction3.setId("reaction3")
    reaction3.setReversible(False)
    reaction3.setFast(False)
    r3_reactant1 = reaction3.createReactant()
    r3_reactant1.setSpecies("id_ep2m")
    r3_reactant1.setStoichiometry(1.0)
    r3_reactant1.setConstant(False)
    r3_reactant2 = reaction3.createReactant()
    r3_reactant2.setSpecies("id_2ary")
    r3_reactant2.setStoichiometry(1.0)
    r3_reactant2.setConstant(False)
    r3_product = reaction3.createProduct()
    r3_product.setSpecies("id_l6et")
    r3_product.setStoichiometry(1.0)
    r3_product.setConstant(False)
    kl3 = reaction3.createKineticLaw()
    kl3.setFormula("k3 * id_ep2m * id_2ary")


    writer = libsbml.SBMLWriter()
    return writer.writeSBMLToString(sbml_doc)

# Function to simulate the model and return species concentrations
def simulate_model(sbml_string, time_points):
    sim_df = simulate(sbml_string)
    return sim_df

# Objective function to minimize (Sum of Squared Errors)
def objective_function_autocatalytic_regeneration(params):
    k1, k2, k3 = params
    if any(p < 0 for p in params):
        return np.inf # Penalize negative rate constants

    try:
        sbml_string = build_sbml_model_autocatalytic_regeneration(k1, k2, k3)
        sim_df = simulate_model(sbml_string, exp_time)

        # Merge simulation and experimental data
        merged_df = pd.merge(sim_df, exp_df, on='Time', suffixes=('_sim', '_exp'))

        # Calculate SSE for all species
        sse = 0
        for species_id in species_ids:
             sse += np.sum((merged_df[species_id + '_sim'] - merged_df[species_id + '_exp'])**2)

        return sse
    except Exception as e:
        # print(f"Simulation or merge failed: {e}") # Suppress frequent printing during optimization
        return np.inf # Penalize simulation errors

# Initial guess for parameters (k1, k2, k3)
# Trying a few more diverse guesses, including some higher values for k2/k3
initial_params_guesses = [
    [0.0005, 5e-06, 2.7e-05], # Best from Iteration 8
    [0.001, 1e-05, 5e-05],
    [0.0001, 1e-07, 1e-06],
    [0.0008, 0.00008, 0.00008], # Higher k2, k3
    [0.0003, 0.00003, 0.00003]
]

best_sse = np.inf
best_params = None
best_result = None

for initial_params in initial_params_guesses:
    print(f"Attempting optimization with initial guess: {initial_params}")
    # Add bounds to ensure parameters are non-negative
    bounds = [(1e-12, None), (1e-12, None), (1e-12, None)] # Small lower bound to avoid zero
    result = minimize(objective_function_autocatalytic_regeneration, initial_params, method='Nelder-Mead', tol=1e-7, bounds=bounds)

    print(f"Result for guess {initial_params}:")
    print(result)

    if result.success and result.fun < best_sse:
        best_sse = result.fun
        best_params = result.x
        best_result = result

print("\nOptimization finished.")
print("Best Result:")
print(best_result)
print(f"\nBest parameters: k1={best_params[0]}, k2={best_params[1]}, k3={best_params[2]}")

# Build the SBML model with the best parameters
best_sbml_string_autocatalytic_regeneration = build_sbml_model_autocatalytic_regeneration(*best_params)

# Simulate with the best parameters and compare visually (by printing data)
sim_df_best = simulate(best_sbml_string_autocatalytic_regeneration)
merged_df_best = pd.merge(sim_df_best, exp_df, on='Time', suffixes=('_sim', '_exp'))

print("\nSimulation with best parameters (autocatalytic regeneration model) vs Experiment (first 10 rows):")
print(merged_df_best.head(10))

# Store the best SBML model
shared_variables.add("best_sbml_autocatalytic_regeneration", best_sbml_string_autocatalytic_regeneration)
```
# Observation

## Code Stdout
Attempting optimization with initial guess: [0.0005, 5e-06, 2.7e-05]
Result for guess [0.0005, 5e-06, 2.7e-05]:
       message: Optimization terminated successfully.
       success: True
        status: 0
           fun: 256189774.76280296
             x: [ 5.524e-04  5.274e-06  2.718e-05]
           nit: 176
          nfev: 396
 final_simplex: (array([[ 5.524e-04,  5.274e-06,  2.718e-05],
                       [ 5.524e-04,  5.274e-06,  2.718e-05],
                       [ 5.524e-04,  5.274e-06,  2.718e-05],
                       [ 5.524e-04,  5.274e-06,  2.718e-05]]), array([ 2.562e+08,  2.562e+08,  2.562e+08,  2.562e+08]))
Attempting optimization with initial guess: [0.001, 1e-05, 5e-05]
Result for guess [0.001, 1e-05, 5e-05]:
       message: Optimization terminated successfully.
       success: True
        status: 0
           fun: 256189774.42126322
             x: [ 5.522e-04  5.272e-06  2.717e-05]
           nit: 189
          nfev: 432
 final_simplex: (array([[ 5.522e-04,  5.272e-06,  2.717e-05],
                       [ 5.522e-04,  5.272e-06,  2.717e-05],
                       [ 5.522e-04,  5.272e-06,  2.717e-05],
                       [ 5.522e-04,  5.272e-06,  2.717e-05]]), array([ 2.562e+08,  2.562e+08,  2.562e+08,  2.562e+08]))
Attempting optimization with initial guess: [0.0001, 1e-07, 1e-06]
Result for guess [0.0001, 1e-07, 1e-06]:
       message: Optimization terminated successfully.
       success: True
        status: 0
           fun: 264933853.49016804
             x: [ 5.448e-04  1.092e-06  1.000e-12]
           nit: 162
          nfev: 350
 final_simplex: (array([[ 5.448e-04,  1.092e-06,  1.000e-12],
                       [ 5.448e-04,  1.092e-06,  1.000e-12],
                       [ 5.448e-04,  1.092e-06,  1.000e-12],
                       [ 5.448e-04,  1.092e-06,  1.000e-12]]), array([ 2.649e+08,  2.649e+08,  2.649e+08,  2.649e+08]))
Attempting optimization with initial guess: [0.0008, 8e-05, 8e-05]
Result for guess [0.0008, 8e-05, 8e-05]:
       message: Optimization terminated successfully.
       success: True
        status: 0
           fun: 256189773.65751734
             x: [ 5.523e-04  5.276e-06  2.718e-05]
           nit: 233
          nfev: 501
 final_simplex: (array([[ 5.523e-04,  5.276e-06,  2.718e-05],
                       [ 5.523e-04,  5.276e-06,  2.718e-05],
                       [ 5.523e-04,  5.276e-06,  2.718e-05],
                       [ 5.523e-04,  5.276e-06,  2.718e-05]]), array([ 2.562e+08,  2.562e+08,  2.562e+08,  2.562e+08]))
Attempting optimization with initial guess: [0.0003, 3e-05, 3e-05]
Result for guess [0.0003, 3e-05, 3e-05]:
       message: Optimization terminated successfully.
       success: True
        status: 0
           fun: 256189774.4058951
             x: [ 5.523e-04  5.276e-06  2.719e-05]
           nit: 202
          nfev: 440
 final_simplex: (array([[ 5.523e-04,  5.276e-06,  2.719e-05],
                       [ 5.523e-04,  5.276e-06,  2.719e-05],
                       [ 5.523e-04,  5.276e-06,  2.719e-05],
                       [ 5.523e-04,  5.276e-06,  2.719e-05]]), array([ 2.562e+08,  2.562e+08,  2.562e+08,  2.562e+08]))

Optimization finished.
Best Result:
       message: Optimization terminated successfully.
       success: True
        status: 0
           fun: 256189773.65751734
             x: [ 5.523e-04  5.276e-06  2.718e-05]
           nit: 233
          nfev: 501
 final_simplex: (array([[ 5.523e-04,  5.276e-06,  2.718e-05],
                       [ 5.523e-04,  5.276e-06,  2.718e-05],
                       [ 5.523e-04,  5.276e-06,  2.718e-05],
                       [ 5.523e-04,  5.276e-06,  2.718e-05]]), array([ 2.562e+08,  2.562e+08,  2.562e+08,  2.562e+08]))

Best parameters: k1=0.0005522562663375448, k2=5.275803506691436e-06, k3=2.7178660945818187e-05

Simulation with best parameters (autocatalytic regeneration model) vs Experiment (first 10 rows):
   id_l6et_sim  id_ep2m_sim  ...  id_wnsa_exp   id_2ary_exp
0  1000.000000   200.000000  ...     1.000000  1.000000e+00
1   998.719313   200.839167  ...     1.180227  8.719520e+00
2   997.440433   201.677090  ...     3.725243  2.900681e+01
3   996.163400   202.513731  ...     9.300273  2.348123e+00
4   994.888251   203.349051  ...     8.972119  4.980706e-02
5   993.615023   204.183014  ...     8.290540  1.555331e-03
6   992.343752   205.015582  ...     7.653392  7.377733e-05
7   991.074473   205.846719  ...     7.064983  5.178048e-06
8   989.807224   206.676386  ...     6.521802  5.237403e-07
9   988.542038   207.504550  ...     6.020382  7.445900e-08

[10 rows x 9 columns]

## 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)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml, best_sbml_reversible_massaction, current_sbml_autocatalytic, best_sbml_autocatalytic, current_sbml_autocatalytic_consumption, best_sbml_autocatalytic_consumption, current_sbml_autocatalytic_consumption_stoich, current_sbml_inhibited_consumption, best_sbml_inhibited_consumption, current_sbml_higher_order_autocatalysis, current_sbml_higher_order_consumption, current_sbml_inhibitor_squared, best_sbml_autocatalytic_regeneration)

### 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
Optimizing the parameters for the autocatalytic production and regeneration consumption model (Attempt 2/7) again did not yield parameters that produce the observed fluctuations. The SSE remains high, and the simulated dynamics are smooth.

With only 3 iterations left, it's unlikely I can discover a completely new reaction network or complex kinetic law and successfully optimize its parameters. The current model structures based on autocatalysis and consumption/regeneration with mass-action or simple non-linear kinetics have failed to reproduce the key dynamic feature (fluctuations).

The strong correlation between `id_ep2m` and `id_2ary` changes, and their coupled spiking behavior, is the most defining characteristic of the experimental data that my models fail to capture.

Given the constraints and remaining iterations, I have two main options:
1. Make a final attempt to refine the most plausible model structure (autocatalysis + regeneration) by trying to force it towards oscillatory behavior, perhaps by manually adjusting parameters or trying a different optimization strategy if available (though Nelder-Mead is generally suitable for non-smooth landscapes that might arise from stiff ODEs near oscillatory regions).
2. Submit the best model found so far, acknowledging its limitations in capturing the fluctuations, but representing the general trends observed in the data.

Attempting to force oscillations in a model that doesn't naturally exhibit them in the found parameter space is risky and might lead to unstable simulations or parameters that don't generalize well. The optimization results consistently point to a non-oscillatory regime for the tested mass-action based models.

The best model found so far in terms of SSE was the autocatalytic consumption model (Attempt 3) with SSE ~1.62e+08, although it didn't oscillate. The autocatalytic regeneration model (Attempt 2/7) had a slightly higher SSE ~2.56e+08. Neither captures the fluctuations.

Let's look at the best parameters found for the autocatalytic consumption model (Attempt 3) from Iteration 11: k1=4.466e-07, k2=8.693e-06, k3=0.000434. And for the autocatalytic regeneration model (Attempt 2/7) from Iteration 18: k1=0.000552, k2=5.276e-06, k3=2.718e-05. The parameter values are quite different, yet both models fail to oscillate and have high SSE.

The most consistent observation is the strong coupling of `id_ep2m` and `id_2ary` dynamics. The simplest explanation for this coupled production and consumption, along with the decrease in `id_l6et` and increase in `id_wnsa`, is the set of reactions:
Reaction 1: `id_l6et` -> `id_wnsa`
Reaction 2: `id_l6et` -> `id_ep2m` + `id_2ary`
Reaction 3: `id_ep2m` + `id_2ary` -> `id_l6et`

This is the reversible reaction `id_l6et <=> id_ep2m + id_2ary` combined with `id_l6et` -> `id_wnsa`. The failure to reproduce fluctuations with mass-action kinetics for the reversible part suggests the kinetics are more complex. However, without more experimental data (e.g., perturbations of specific species) or the ability to test more complex kinetic laws easily, identifying the exact non-linear mechanism is very difficult.

Given the remaining iterations, I will submit the model structure that seems most plausible based on the overall species changes and correlations, even if it doesn't perfectly capture the fluctuations with simple kinetics. The model with Reaction 1 (`id_l6et` -> `id_wnsa`) and the reversible Reaction 2 (`id_l6et <=> id_ep2m + id_2ary`) seems the most parsimonious explanation for the observed species relationships, assuming the complex dynamics arise from non-mass-action kinetics in the reversible step.

I will use the SBML structure from Iteration 5 (reversible mass action) and include the best parameters found for that structure in Iteration 7 (k1=0.00057, k2=0.0679, k_reverse2=0.00135). While this model didn't oscillate, it represents a plausible network structure. It's possible the true kinetics are non-mass-action, but I cannot confidently determine them with the current information and tools.

I will prepare the final SBML model based on the reversible reaction structure with the best parameters from Iteration 7.

## Action

### Submit
```python
import libsbml

# Load the initial SBML model structure
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Clear existing reactions and parameters
while model.getNumReactions() > 0:
    model.removeReaction(0)
while model.getNumParameters() > 0:
    model.removeParameter(0)

# Add parameters from the best fit of the reversible mass-action model (Iteration 7)
k1_best = 0.0005703295852496197
k2_best = 0.06791860618156326
k_reverse2_best = 0.0013551623348530858

p_k1 = model.createParameter()
p_k1.setId("k1")
p_k1.setValue(k1_best)
p_k1.setConstant(True)

p_k2 = model.createParameter()
p_k2.setId("k2")
p_k2.setValue(k2_best)
p_k2.setConstant(True)

p_k_reverse2 = model.createParameter()
p_k_reverse2.setId("k_reverse2")
p_k_reverse2.setValue(k_reverse2_best)
p_k_reverse2.setConstant(True)

# Reaction 1: id_l6et -> id_wnsa (Mass action)
reaction1 = model.createReaction()
reaction1.setId("reaction1")
reaction1.setReversible(False)
reaction1.setFast(False)
r1_reactant = reaction1.createReactant()
r1_reactant.setSpecies("id_l6et")
r1_reactant.setStoichiometry(1.0)
r1_reactant.setConstant(False)
r1_product = reaction1.createProduct()
r1_product.setSpecies("id_wnsa")
r1_product.setStoichiometry(1.0)
r1_product.setConstant(False)
kl1 = reaction1.createKineticLaw()
kl1.setFormula("k1 * id_l6et")

# Reaction 2: id_l6et <=> id_ep2m + id_2ary (Reversible Mass action)
reaction2 = model.createReaction()
reaction2.setId("reaction2")
reaction2.setReversible(True)
reaction2.setFast(False)
r2_reactant = reaction2.createReactant()
r2_reactant.setSpecies("id_l6et")
r2_reactant.setStoichiometry(1.0)
r2_reactant.setConstant(False)
r2_product1 = reaction2.createProduct()
r2_product1.setSpecies("id_ep2m")
r2_product1.setStoichiometry(1.0)
r2_product1.setConstant(False)
r2_product2 = reaction2.createProduct()
r2_product2.setSpecies("id_2ary")
r2_product2.setStoichiometry(1.0)
r2_product2.setConstant(False)
kl2 = reaction2.createKineticLaw()
kl2.setFormula("k2 * id_l6et - k_reverse2 * id_ep2m * id_2ary")

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