
                # 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_gvop">
    <listOfCompartments>
      <compartment id="id_mpo9" size="1" units="volume"/>
    </listOfCompartments>
    <listOfSpecies>
      <species id="id_1t80" compartment="id_mpo9" initialConcentration="1" substanceUnits="substance" boundaryCondition="true" constant="true"/>
      <species id="id_ijk0" compartment="id_mpo9" initialConcentration="1" substanceUnits="substance"/>
      <species id="id_9k8m" compartment="id_mpo9" initialConcentration="1" substanceUnits="substance" boundaryCondition="true" constant="true"/>
      <species id="id_bnxy" compartment="id_mpo9" initialConcentration="1" substanceUnits="substance"/>
    </listOfSpecies>
  </model>
</sbml>


                ## Max iterations
                20

                Format your response according to the instructions in the system message.
                
## Thoughts
The initial SBML model defines a system with four species: `id_1t80`, `id_ijk0`, `id_9k8m`, and `id_bnxy`.
- `id_1t80` and `id_9k8m` are boundary species and are constant, meaning their concentrations do not change over time due to reactions within the system. They can act as sources or sinks.
- `id_ijk0` and `id_bnxy` are variable species whose concentrations can change.

Currently, there are no reactions defined in the model. This means if we simulate the current model, the concentrations of `id_ijk0` and `id_bnxy` will remain at their initial values (1.0).

To understand the system's dynamics, I need to perform an experiment. The simplest first step is to observe the system under default conditions. This will show how the concentrations of `id_ijk0` and `id_bnxy` change over time, giving clues about the reactions involving them.

## 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_ijk0  id_bnxy  id_1t80  id_9k8m
0  1.00e+00 1.00e+00 1.00e+00 1.00e+00
...
2.00e-01  4.93e-01 1.86e+00 1.00e+00 1.00e+00
...
4.00e-01  4.57e-01 1.90e+00 1.00e+00 1.00e+00
...
6.00e-01  4.51e-01 1.89e+00 1.00e+00 1.00e+00
...
8.00e-01  4.45e-01 1.88e+00 1.00e+00 1.00e+00
...
1.00e+00  4.39e-01 1.87e+00 1.00e+00 1.00e+00
...
1.20e+00  4.32e-01 1.85e+00 1.00e+00 1.00e+00
...
1.40e+00  4.25e-01 1.83e+00 1.00e+00 1.00e+00
...
1.60e+00  4.16e-01 1.81e+00 1.00e+00 1.00e+00
...
1.80e+00  4.07e-01 1.79e+00 1.00e+00 1.00e+00
...
2.00e+00  3.97e-01 1.77e+00 1.00e+00 1.00e+00
...
2.20e+00  3.87e-01 1.74e+00 1.00e+00 1.00e+00
...
2.40e+00  3.75e-01 1.71e+00 1.00e+00 1.00e+00
...
2.60e+00  3.62e-01 1.68e+00 1.00e+00 1.00e+00
...
2.80e+00  3.48e-01 1.65e+00 1.00e+00 1.00e+00
...
3.00e+00  3.33e-01 1.61e+00 1.00e+00 1.00e+00
...
3.20e+00  3.16e-01 1.56e+00 1.00e+00 1.00e+00
...
3.40e+00  2.99e-01 1.52e+00 1.00e+00 1.00e+00
...
3.60e+00  2.80e-01 1.46e+00 1.00e+00 1.00e+00
...
3.80e+00  2.60e-01 1.41e+00 1.00e+00 1.00e+00
...
4.00e+00  2.40e-01 1.35e+00 1.00e+00 1.00e+00
...
4.20e+00  2.18e-01 1.28e+00 1.00e+00 1.00e+00
...
4.40e+00  1.97e-01 1.21e+00 1.00e+00 1.00e+00
...
4.60e+00  1.74e-01 1.13e+00 1.00e+00 1.00e+00
...
4.80e+00  1.52e-01 1.05e+00 1.00e+00 1.00e+00
...
5.00e+00  1.31e-01 9.69e-01 1.00e+00 1.00e+00
...
5.20e+00  1.10e-01 8.82e-01 1.00e+00 1.00e+00
...
5.40e+00  9.03e-02 7.93e-01 1.00e+00 1.00e+00
...
5.60e+00  7.25e-02 7.04e-01 1.00e+00 1.00e+00
...
5.80e+00  5.67e-02 6.17e-01 1.00e+00 1.00e+00
...
6.00e+00  4.31e-02 5.32e-01 1.00e+00 1.00e+00
...
6.20e+00  3.18e-02 4.52e-01 1.00e+00 1.00e+00
...
6.40e+00  2.28e-02 3.78e-01 1.00e+00 1.00e+00
...
6.60e+00  1.58e-02 3.11e-01 1.00e+00 1.00e+00
...
6.80e+00  1.07e-02 2.52e-01 1.00e+00 1.00e+00
...
7.00e+00  6.99e-03 2.01e-01 1.00e+00 1.00e+00
...
7.20e+00  4.45e-03 1.59e-01 1.00e+00 1.00e+00
...
7.40e+00  2.76e-03 1.24e-01 1.00e+00 1.00e+00
...
7.60e+00  1.68e-03 9.57e-02 1.00e+00 1.00e+00
...
7.80e+00  1.00e-03 7.32e-02 1.00e+00 1.00e+00
...
8.00e+00  5.86e-04 5.57e-02 1.00e+00 1.00e+00
...
8.20e+00  3.38e-04 4.21e-02 1.00e+00 1.00e+00
...
8.40e+00  1.93e-04 3.16e-02 1.00e+00 1.00e+00
...
8.60e+00  1.09e-04 2.37e-02 1.00e+00 1.00e+00
...
8.80e+00  6.15e-05 1.77e-02 1.00e+00 1.00e+00
...
9.00e+00  3.44e-05 1.32e-02 1.00e+00 1.00e+00
...
9.20e+00  1.91e-05 9.83e-03 1.00e+00 1.00e+00
...
9.40e+00  1.06e-05 7.31e-03 1.00e+00 1.00e+00
...
9.60e+00  5.86e-06 5.43e-03 1.00e+00 1.00e+00
...
1.00e+01  1.78e-06 2.99e-03 1.00e+00 1.00e+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 experimental data from `iteration_0` shows that `id_ijk0` decreases over time, while `id_bnxy` first increases, peaks, and then decreases. Both approach low values. The sum `[id_ijk0] + [id_bnxy]` initially increases, suggesting an external source for one or both species, likely from the boundary species `id_1t80` or `id_9k8m`.

Based on this, I hypothesize the following reaction scheme:
1.  **R1: `id_ijk0 -> id_bnxy`** (Conversion of `id_ijk0` to `id_bnxy`)
    *   Rate: `k1 * [id_ijk0]`
2.  **R2: `id_bnxy -> Sink`** (Degradation or conversion of `id_bnxy` to a boundary species, e.g., `id_9k8m`)
    *   Rate: `k2 * [id_bnxy]`
3.  **R3: `Source -> id_bnxy`** (Production of `id_bnxy` from a boundary species, e.g., `id_1t80`)
    *   Rate: `k_source * [id_1t80]`

The differential equations would be:
`d[id_ijk0]/dt = -k1 * [id_ijk0]`
`d[id_bnxy]/dt = k_source * [id_1t80] + k1 * [id_ijk0] - k2 * [id_bnxy]`

Using initial rates and the peak condition for `id_bnxy`:
- `[id_ijk0](0)=1`, `[id_bnxy](0)=1`.
- `d[id_ijk0]/dt (t=0) approx (0.493-1)/0.2 = -2.535`. So, `-k1 * 1 = -2.535` => `k1 = 2.535`.
- `d[id_bnxy]/dt (t=0) approx (1.86-1)/0.2 = 4.3`. So, `k_source * 1 + k1 * 1 - k2 * 1 = 4.3`.
  `k_source + 2.535 - k2 = 4.3` => `k_source - k2 = 1.765`.
- At `id_bnxy` peak (approx t=0.4): `[id_ijk0]=0.457`, `[id_bnxy]=1.90`. `d[id_bnxy]/dt = 0`.
  `k_source * 1 + k1 * 0.457 - k2 * 1.90 = 0`.
  `k_source + 2.535 * 0.457 - 1.90 * k2 = 0` => `k_source + 1.158 - 1.90 * k2 = 0`.

Solving for `k_source` and `k2`:
1. `k_source = 1.765 + k2`
2. `(1.765 + k2) + 1.158 - 1.90 * k2 = 0`
   `2.923 - 0.90 * k2 = 0` => `k2 = 2.923 / 0.90 = 3.248`
   `k_source = 1.765 + 3.248 = 5.013`

So, the estimated parameters are:
- `k1 = 2.535`
- `k2 = 3.248`
- `k_source = 5.013`

I will now add these reactions and parameters to the SBML model and simulate it to compare with the experimental data. I'll use `id_9k8m` as the sink in R2 and `id_1t80` as the source in R3.

## Action

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

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

# Add parameters
params = {
    "param_k1": 2.535,      # For id_ijk0 -> id_bnxy
    "param_k2": 3.248,      # For id_bnxy -> id_9k8m
    "param_k_source": 5.013 # For id_1t80 -> id_bnxy
}

for param_id, param_value in params.items():
    p = model.createParameter()
    p.setId(param_id)
    p.setValue(param_value)
    p.setConstant(True)

# Reaction 1: id_ijk0 -> id_bnxy
r1 = model.createReaction()
r1.setId("R1_ijk0_to_bnxy")
r1.setReversible(False)
r1.setFast(False)

reactant_r1 = r1.createReactant()
reactant_r1.setSpecies("id_ijk0")
reactant_r1.setStoichiometry(1)
reactant_r1.setConstant(False)

product_r1 = r1.createProduct()
product_r1.setSpecies("id_bnxy")
product_r1.setStoichiometry(1)
product_r1.setConstant(False)

kinetic_law_r1 = r1.createKineticLaw()
kinetic_law_r1.setFormula("param_k1 * id_ijk0")

# Reaction 2: id_bnxy -> id_9k8m
r2 = model.createReaction()
r2.setId("R2_bnxy_degradation")
r2.setReversible(False)
r2.setFast(False)

reactant_r2 = r2.createReactant()
reactant_r2.setSpecies("id_bnxy")
reactant_r2.setStoichiometry(1)
reactant_r2.setConstant(False)

product_r2 = r2.createProduct()
product_r2.setSpecies("id_9k8m") # Boundary species, acts as a sink
product_r2.setStoichiometry(1)
product_r2.setConstant(False) # SBML L3 requirement, though species itself is constant

kinetic_law_r2 = r2.createKineticLaw()
kinetic_law_r2.setFormula("param_k2 * id_bnxy")

# Reaction 3: id_1t80 -> id_bnxy
r3 = model.createReaction()
r3.setId("R3_bnxy_production_from_source")
r3.setReversible(False)
r3.setFast(False)

reactant_r3 = r3.createReactant()
reactant_r3.setSpecies("id_1t80") # Boundary species, acts as a source
reactant_r3.setStoichiometry(1)
reactant_r3.setConstant(False) # SBML L3 requirement

product_r3 = r3.createProduct()
product_r3.setSpecies("id_bnxy")
product_r3.setStoichiometry(1)
product_r3.setConstant(False)

kinetic_law_r3 = r3.createKineticLaw()
kinetic_law_r3.setFormula("param_k_source * id_1t80")

# Write the updated SBML to a string
writer = libsbml.SBMLWriter()
updated_sbml_string = writer.writeSBMLToString(sbml_doc)

# Store the new model for potential submission later
shared_variables.add("current_best_model", updated_sbml_string)

# Simulate the updated model
simulated_data = simulate(updated_sbml_string)

# Print some comparison points
exp_data = experiment_history['iteration_0']

print("Comparison of Experimental and Simulated Data:")
print("Species: id_ijk0")
print("Time | Exp_ijk0 | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0, 10.0]:
    exp_val = exp_data[exp_data['Time'] == t_point]['id_ijk0'].values[0] if t_point in exp_data['Time'].values else float('nan')
    sim_val = simulated_data[simulated_data['Time'] == t_point]['id_ijk0'].values[0] if t_point in simulated_data['Time'].values else float('nan')
    print(f"{t_point:.1f} | {exp_val:.3e} | {sim_val:.3e}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 10.0]:
    exp_val = exp_data[exp_data['Time'] == t_point]['id_bnxy'].values[0] if t_point in exp_data['Time'].values else float('nan')
    sim_val = simulated_data[simulated_data['Time'] == t_point]['id_bnxy'].values[0] if t_point in simulated_data['Time'].values else float('nan')
    print(f"{t_point:.1f} | {exp_val:.3e} | {sim_val:.3e}")

# Store data for potential plotting if I had a visual environment
shared_variables.add("exp_data_iter0", exp_data)
shared_variables.add("sim_data_iter1", simulated_data)

# Print the model to see if it's correctly formed (optional, can be long)
# print("\nGenerated SBML Model:")
# print(updated_sbml_string)

```
# Observation

## Code Stdout
Comparison of Experimental and Simulated Data:
Species: id_ijk0
Time | Exp_ijk0 | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 4.933e-01 | 6.023e-01
0.4 | 4.574e-01 | 3.628e-01
1.0 | 4.389e-01 | 7.926e-02
2.0 | 3.975e-01 | 6.282e-03
5.0 | 1.306e-01 | 3.128e-06
10.0 | 1.784e-06 | 1.149e-11

Species: id_bnxy
Time | Exp_bnxy | Sim_bnxy
0.0 | 1.000e+00 | 1.000e+00
0.2 | 1.858e+00 | 1.544e+00
0.4 | 1.900e+00 | 1.715e+00
0.6 | 1.891e+00 | 1.736e+00
1.0 | 1.865e+00 | 1.666e+00
2.0 | 1.769e+00 | 1.560e+00
5.0 | 9.687e-01 | 1.543e+00
10.0 | 2.992e-03 | 1.543e+00

## 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. (current_best_model, exp_data_iter0, sim_data_iter1)

### 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
The previous simulation with the model (R1: `id_ijk0 -> id_bnxy`, R2: `id_bnxy -> id_9k8m`, R3: `id_1t80 -> id_bnxy`) showed that `id_bnxy` reached a non-zero steady state, contradicting experimental data where `id_bnxy` tends towards zero. The key challenge is to find a model where both `id_ijk0` and `id_bnxy` go to zero, yet their sum `[id_ijk0] + [id_bnxy]` initially increases.

The model structure:
R_conv: `id_ijk0 -> 2*id_bnxy` (rate `k_c * [id_ijk0]`)
R_deg: `id_bnxy -> id_9k8m` (rate `k_d * [id_bnxy]`)
This structure allows both species to go to zero and the sum to increase if `k_c * [id_ijk0] > k_d * [id_bnxy]`.
However, the decay profile of `id_ijk0` in the experimental data is complex: its effective first-order rate constant appears to be high initially, then decreases, then increases again. This is not captured by `d[id_ijk0]/dt = -k_c * [id_ijk0]`.

The strange decay pattern of `id_ijk0` (fast-slow-fast) might be influenced by `id_bnxy` (whose concentration goes low-high-low).
If the rate of `id_ijk0` consumption depends on `[id_bnxy]`, for example, `rate = f([id_bnxy]) * [id_ijk0]`, then the effective "constant" `f([id_bnxy])` would need to be High when `[id_bnxy]` is low/initial, Low when `[id_bnxy]` is high (peak), and High again when `[id_bnxy]` becomes very low.
A function like `k_base + k_factor/[id_bnxy]` could produce such a High -> Low -> Very High pattern for the effective rate constant of `id_ijk0` decay.

To test if `[id_bnxy]` influences `id_ijk0`'s decay, I've requested an experiment where `[id_bnxy](0)` is changed to 0.1 (from its default of 1.0), while `[id_ijk0](0)` remains 1.0.
- If `id_ijk0`'s decay profile changes significantly, it confirms a dependency on `[id_bnxy]`.
- If `id_ijk0`'s decay profile remains similar to the original experiment (after accounting for the `exp(-kt)` nature if it were simple), then its decay is largely independent of `[id_bnxy]`.

Let's analyze the new experimental data.

## Action

### Code
```python
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt # Will not display, but useful for local testing if possible

# Access the new experimental data
new_exp_data = experiment_history['iteration_2'] # Assuming current iteration is 2
original_exp_data = experiment_history['iteration_0']

print("New experiment (iteration_2) data with id_bnxy(0)=0.1:")
print(new_exp_data.head())
print("\nOriginal experiment (iteration_0) data with id_bnxy(0)=1.0:")
print(original_exp_data.head())

# Compare id_ijk0 profiles
print("\nComparison of id_ijk0 profiles:")
print("Time | Orig_ijk0 (bnxy0=1) | New_ijk0 (bnxy0=0.1)")

# Use a few common time points for comparison
time_points = [0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 2.0, 5.0] # Add more if needed

for t in time_points:
    orig_val = original_exp_data[np.isclose(original_exp_data['Time'], t)]['id_ijk0'].values
    new_val = new_exp_data[np.isclose(new_exp_data['Time'], t)]['id_ijk0'].values

    orig_str = f"{orig_val[0]:.3e}" if len(orig_val) > 0 else "N/A"
    new_str = f"{new_val[0]:.3e}" if len(new_val) > 0 else "N/A"
    print(f"{t:.1f}  | {orig_str:<20} | {new_str}")

# Calculate effective decay rates for id_ijk0 for both experiments
# k_eff = -ln(C(t)/C(0)) / t
# Or, more locally, k_eff(t_i) approx - (ln(C(t_{i+1})) - ln(C(t_i))) / (t_{i+1} - t_i)

def calculate_local_k_eff(data, species_id):
    times = data['Time'].values
    concentrations = data[species_id].values
    k_eff_values = []
    time_mid_points = []
    for i in range(len(times) - 1):
        if concentrations[i] > 1e-9 and concentrations[i+1] > 1e-9 and times[i+1] > times[i]: # Avoid log(0) or division by zero
            dt = times[i+1] - times[i]
            k = -(np.log(concentrations[i+1]) - np.log(concentrations[i])) / dt
            k_eff_values.append(k)
            time_mid_points.append((times[i] + times[i+1]) / 2)
    return pd.DataFrame({'Time_Mid': time_mid_points, 'k_eff': k_eff_values})

k_eff_orig_ijk0 = calculate_local_k_eff(original_exp_data, 'id_ijk0')
k_eff_new_ijk0 = calculate_local_k_eff(new_exp_data, 'id_ijk0')

shared_variables.add("k_eff_orig_ijk0", k_eff_orig_ijk0)
shared_variables.add("k_eff_new_ijk0", k_eff_new_ijk0)
shared_variables.add("original_exp_data_df", original_exp_data)
shared_variables.add("new_exp_data_df", new_exp_data)


print("\nEffective decay rate k_eff for id_ijk0 (Original Experiment, bnxy(0)=1):")
print(k_eff_orig_ijk0[(k_eff_orig_ijk0['Time_Mid'] <= 2.0)].head(10)) # Show early points

print("\nEffective decay rate k_eff for id_ijk0 (New Experiment, bnxy(0)=0.1):")
print(k_eff_new_ijk0[(k_eff_new_ijk0['Time_Mid'] <= 2.0)].head(10)) # Show early points

# Also look at id_bnxy profiles to see how the perturbation affected it
print("\nComparison of id_bnxy profiles:")
print("Time | Orig_bnxy (bnxy0=1) | New_bnxy (bnxy0=0.1)")
for t in time_points:
    orig_val = original_exp_data[np.isclose(original_exp_data['Time'], t)]['id_bnxy'].values
    new_val = new_exp_data[np.isclose(new_exp_data['Time'], t)]['id_bnxy'].values

    orig_str = f"{orig_val[0]:.3e}" if len(orig_val) > 0 else "N/A"
    new_str = f"{new_val[0]:.3e}" if len(new_val) > 0 else "N/A"
    print(f"{t:.1f}  | {orig_str:<20} | {new_str}")

```
# 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: 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. (current_best_model, exp_data_iter0, sim_data_iter1)

### 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
My previous attempt to analyze data failed because I tried to import `matplotlib` and, more importantly, I hadn't actually requested the experiment I intended to analyze.

The goal is to understand the decay of `id_ijk0` and the production/decay of `id_bnxy`. The initial observation (iteration 0) showed `id_ijk0` decreasing and `id_bnxy` increasing then decreasing, both towards zero. My first model attempt (R1: `id_ijk0 -> id_bnxy`, R2: `id_bnxy -> Sink`, R3: `Source -> id_bnxy`) resulted in `id_bnxy` reaching a non-zero steady state, which was incorrect.

The complex decay pattern of `id_ijk0` (fast-slow-fast when looking at effective rates) and the transient increase of `id_bnxy` are key features.
I hypothesized that the decay of `id_ijk0` might be influenced by the concentration of `id_bnxy`. To test this, I need to perturb the initial concentration of `id_bnxy` and observe the effect on `id_ijk0`'s dynamics.

I will now request the experiment where the initial concentration of `id_bnxy` is set to 0.1 (default is 1.0), while `id_ijk0` remains at its default initial concentration of 1.0.

## Action

### Experiment
```json
{
    "action": "change_initial_concentration",
    "meta_data": {
        "id_bnxy": 0.1
    }
}
```
# Observation

## Experiment Result
Experiment ID: iteration_3
Your requested experiment has finished. The details results are saved as a dataframe in experiment_history[iteration_3]

Total size: 1001 rows × 5 columns. 

Below shows a subset of the table for your convenience. You should access experiment_history[iteration_3] for the full data

Time   id_ijk0  id_bnxy  id_1t80  id_9k8m
0  1.00e+00 1.00e-01 1.00e+00 1.00e+00
...
2.00e-01  3.20e-01 1.29e+00 1.00e+00 1.00e+00
...
4.00e-01  2.41e-01 1.33e+00 1.00e+00 1.00e+00
...
6.00e-01  2.15e-01 1.27e+00 1.00e+00 1.00e+00
...
8.00e-01  1.93e-01 1.20e+00 1.00e+00 1.00e+00
...
1.00e+00  1.71e-01 1.12e+00 1.00e+00 1.00e+00
...
1.20e+00  1.49e-01 1.04e+00 1.00e+00 1.00e+00
...
1.40e+00  1.27e-01 9.55e-01 1.00e+00 1.00e+00
...
1.60e+00  1.07e-01 8.67e-01 1.00e+00 1.00e+00
...
1.80e+00  8.73e-02 7.79e-01 1.00e+00 1.00e+00
...
2.00e+00  6.98e-02 6.90e-01 1.00e+00 1.00e+00
...
2.20e+00  5.43e-02 6.02e-01 1.00e+00 1.00e+00
...
2.40e+00  4.11e-02 5.18e-01 1.00e+00 1.00e+00
...
2.60e+00  3.02e-02 4.39e-01 1.00e+00 1.00e+00
...
2.80e+00  2.15e-02 3.66e-01 1.00e+00 1.00e+00
...
3.00e+00  1.49e-02 3.01e-01 1.00e+00 1.00e+00
...
3.20e+00  9.97e-03 2.43e-01 1.00e+00 1.00e+00
...
3.40e+00  6.50e-03 1.94e-01 1.00e+00 1.00e+00
...
3.60e+00  4.12e-03 1.53e-01 1.00e+00 1.00e+00
...
3.80e+00  2.55e-03 1.19e-01 1.00e+00 1.00e+00
...
4.00e+00  1.54e-03 9.16e-02 1.00e+00 1.00e+00
...
4.20e+00  9.17e-04 7.00e-02 1.00e+00 1.00e+00
...
4.40e+00  5.36e-04 5.32e-02 1.00e+00 1.00e+00
...
4.60e+00  3.09e-04 4.01e-02 1.00e+00 1.00e+00
...
4.80e+00  1.76e-04 3.02e-02 1.00e+00 1.00e+00
...
5.00e+00  9.96e-05 2.26e-02 1.00e+00 1.00e+00
...
5.20e+00  5.59e-05 1.69e-02 1.00e+00 1.00e+00
...
5.40e+00  3.12e-05 1.26e-02 1.00e+00 1.00e+00
...
5.60e+00  1.74e-05 9.37e-03 1.00e+00 1.00e+00
...
5.80e+00  9.62e-06 6.96e-03 1.00e+00 1.00e+00
...
6.00e+00  5.32e-06 5.17e-03 1.00e+00 1.00e+00
...
6.20e+00  2.93e-06 3.84e-03 1.00e+00 1.00e+00
...
6.40e+00  1.62e-06 2.85e-03 1.00e+00 1.00e+00
...
6.60e+00  8.90e-07 2.11e-03 1.00e+00 1.00e+00
...
6.80e+00  4.90e-07 1.57e-03 1.00e+00 1.00e+00
...
7.00e+00  2.69e-07 1.16e-03 1.00e+00 1.00e+00
...
7.20e+00  1.48e-07 8.60e-04 1.00e+00 1.00e+00
...
7.40e+00  8.13e-08 6.38e-04 1.00e+00 1.00e+00
...
7.60e+00  4.46e-08 4.72e-04 1.00e+00 1.00e+00
...
7.80e+00  2.45e-08 3.50e-04 1.00e+00 1.00e+00
...
8.00e+00  1.35e-08 2.59e-04 1.00e+00 1.00e+00
...
8.20e+00  7.39e-09 1.92e-04 1.00e+00 1.00e+00
...
8.40e+00  4.05e-09 1.42e-04 1.00e+00 1.00e+00
...
8.60e+00  2.23e-09 1.05e-04 1.00e+00 1.00e+00
...
8.80e+00  1.22e-09 7.81e-05 1.00e+00 1.00e+00
...
9.00e+00  6.70e-10 5.79e-05 1.00e+00 1.00e+00
...
9.20e+00  3.68e-10 4.29e-05 1.00e+00 1.00e+00
...
9.40e+00  2.02e-10 3.18e-05 1.00e+00 1.00e+00
...
9.60e+00  1.11e-10 2.35e-05 1.00e+00 1.00e+00
...
1.00e+01  3.34e-11 1.29e-05 1.00e+00 1.00e+00

## 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, iteration_3)
- `shared_variables`: all variables you have saved from the previous iterations. (current_best_model, exp_data_iter0, sim_data_iter1)

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


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



# Iteration 4
## Thoughts

I now have two sets of experimental data:
1.  `experiment_history['iteration_0']`: Default initial conditions (`id_ijk0(0)=1`, `id_bnxy(0)=1`).
2.  `experiment_history['iteration_3']`: Perturbed initial condition (`id_ijk0(0)=1`, `id_bnxy(0)=0.1`).

Let's compare the behavior of `id_ijk0` and `id_bnxy` in these two experiments.

**Observations from the new data (iteration_3) vs iteration_0:**

*   **`id_ijk0` decay:**
    *   Original (bnxy(0)=1): At t=0.2, `id_ijk0` = 0.493. At t=0.4, `id_ijk0` = 0.457.
    *   New (bnxy(0)=0.1): At t=0.2, `id_ijk0` = 0.320. At t=0.4, `id_ijk0` = 0.241.
    *   The decay of `id_ijk0` is significantly *faster* in the new experiment where `id_bnxy(0)` was lower. This strongly suggests that `id_bnxy` *inhibits* the decay of `id_ijk0`.

*   **`id_bnxy` profile:**
    *   Original (bnxy(0)=1): Peaks around 1.90 at t=0.4.
    *   New (bnxy(0)=0.1): Peaks around 1.33 at t=0.4.
    *   The peak of `id_bnxy` is lower and reached at a similar time when its initial concentration was lower. This is expected. Both eventually decay towards zero.

**Revised Hypothesis based on `id_bnxy` inhibiting `id_ijk0` decay:**

A possible reaction for `id_ijk0` decay could be:
`R_decay_ijk0: id_ijk0 -> Product(s)` with a rate like `k_decay_ijk0 * [id_ijk0] / (K_i + [id_bnxy])` or `k_decay_ijk0 * [id_ijk0] * (1 / (1 + [id_bnxy]/K_i))`.
This means when `[id_bnxy]` is high, the decay rate of `id_ijk0` is low, and when `[id_bnxy]` is low, the decay rate is high. This matches the observation.

Let's consider the overall system:
1.  **`id_ijk0` consumption:** `id_ijk0 -> ?` (possibly producing `id_bnxy` or something else). The rate is inhibited by `id_bnxy`.
    *   Let's assume `id_ijk0 -> id_bnxy` for now, as `id_bnxy` increases when `id_ijk0` decreases.
    *   `R1: id_ijk0 -> id_bnxy` with rate `k1 * [id_ijk0] / (K_m_bnxy + [id_bnxy])`.
        *   When `[id_bnxy]` is low (e.g., new experiment start), rate is `~k1/K_m_bnxy * [id_ijk0]` (fast).
        *   When `[id_bnxy]` is high (e.g., original experiment peak), rate is `~k1/[id_bnxy] * [id_ijk0]` (slow).

2.  **`id_bnxy` production:**
    *   From `R1`.
    *   Is there another source for `id_bnxy`? In the original experiment, `[id_ijk0]` drops by 0.5, `[id_bnxy]` increases by 0.9. If `id_ijk0 -> id_bnxy` (1:1), this doesn't fully account for the rise.
    *   Perhaps `id_ijk0 -> S * id_bnxy` where S > 1, or there's an independent source like `id_1t80 -> id_bnxy`.
    *   Let's re-examine the sum:
        *   Original: `[ijk0]+[bnxy]` goes from 2.0 to ~2.36 (at t=0.4). Increase of 0.36.
        *   New: `[ijk0]+[bnxy]` goes from 1.1 to ~1.57 (at t=0.4). Increase of 0.47.
    *   The increase in the sum suggests a net production from outside the `ijk0/bnxy` system or a stoichiometry greater than 1 if `ijk0` is the only source.

3.  **`id_bnxy` consumption:** `id_bnxy -> Sink` (e.g., `id_9k8m`).
    *   `R2: id_bnxy -> id_9k8m` with rate `k2 * [id_bnxy]`. This ensures `id_bnxy` goes to zero.

Let's try a model:
*   **R1: `id_1t80 + id_ijk0 -> id_1t80 + 2*id_bnxy`** (Catalyzed conversion with `id_1t80` as a catalyst/enzyme, `id_ijk0` is substrate, produces 2 `id_bnxy`. Rate inhibited by `id_bnxy`).
    *   Rate: `k1 * [id_1t80] * [id_ijk0] / (K_m_bnxy + [id_bnxy])`. Since `[id_1t80]` is constant at 1, this simplifies to `k1_eff * [id_ijk0] / (K_m_bnxy + [id_bnxy])`.
    *   This reaction consumes 1 `id_ijk0` and produces 2 `id_bnxy`. Net change in sum `([ijk0]+[bnxy])` is `(-1 + 2) = +1` for each turnover. This could explain the sum increase.
*   **R2: `id_bnxy -> id_9k8m`** (Degradation of `id_bnxy`).
    *   Rate: `k2 * [id_bnxy]`.

Differential equations:
`d[id_ijk0]/dt = - k1_eff * [id_ijk0] / (K_m_bnxy + [id_bnxy])`
`d[id_bnxy]/dt = 2 * k1_eff * [id_ijk0] / (K_m_bnxy + [id_bnxy]) - k2 * [id_bnxy]`

This model structure seems promising.
- It explains `id_ijk0` decay being faster when `[id_bnxy]` is low.
- It explains `id_bnxy` production and subsequent decay.
- It explains the increase in the sum `[id_ijk0] + [id_bnxy]`.
- Both species go to zero.

I will now try to implement this model and estimate parameters. This is more complex, so I'll start with rough estimates and refine if needed.

Parameter estimation will be tricky.
- `k2`: From the tail of `id_bnxy` decay.
- `k1_eff`, `K_m_bnxy`: From the initial rates and the difference between the two experiments.

Let's analyze the data to get some initial parameter estimates.

## Action

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

# Access experimental data
exp_data_orig = experiment_history['iteration_0']
exp_data_pert = experiment_history['iteration_3'] # bnxy(0)=0.1

shared_variables.add("exp_data_orig_iter4", exp_data_orig)
shared_variables.add("exp_data_pert_iter4", exp_data_pert)

# --- Parameter Estimation ---
# R2: id_bnxy -> id_9k8m (rate k2 * id_bnxy)
# Look at the tail of id_bnxy decay in original experiment when id_ijk0 is very low
# d[bnxy]/dt = -k2 * [bnxy]
# ln([bnxy]_t / [bnxy]_t0) = -k2 * (t - t0)
# k2 = - (ln([bnxy]_t / [bnxy]_t0)) / (t - t0)

# Original data, late time points for id_bnxy decay
# Time 5.0: id_bnxy = 0.1306 (ijk0 is low) -> this is ijk0, bnxy is 0.9687
# Time 6.0: id_bnxy = 0.532 (ijk0 is very low)
# Time 7.0: id_bnxy = 0.201 (ijk0 is very very low)

# Let's use t0=6.0, bnxy0=0.532; t=7.0, bnxy=0.201
if 0.532 > 0 and 0.201 > 0:
    k2_est = - (np.log(0.201 / 0.532)) / (7.0 - 6.0)
else:
    k2_est = 1.0 # fallback
print(f"Estimated k2: {k2_est}") # Expected around 0.97

# R1: id_1t80 + id_ijk0 -> id_1t80 + 2*id_bnxy
# Rate = k1_eff * [id_ijk0] / (K_m_bnxy + [id_bnxy])
# d[id_ijk0]/dt = - Rate
# d[id_bnxy]/dt = 2 * Rate - k2 * [id_bnxy]

# At t=0:
# Exp Orig: id_ijk0(0)=1, id_bnxy(0)=1
# d[ijk0]/dt approx (0.493-1)/0.2 = -2.535
# d[bnxy]/dt approx (1.858-1)/0.2 = 4.29
# So, Rate_orig(0) = 2.535
# 2.535 = k1_eff * 1 / (K_m_bnxy + 1)  => k1_eff = 2.535 * (K_m_bnxy + 1) (Eq 1)
# 4.29 = 2 * 2.535 - k2_est * 1 => 4.29 = 5.07 - k2_est => k2_est = 0.78. This is a bit different from tail estimate. Let's use tail estimate k2=0.973

# Recalculate d[bnxy]/dt with k2_est = 0.973
# d[bnxy]/dt_obs = 4.29
# 2 * Rate_orig(0) - k2_est * [bnxy(0)] = d[bnxy]/dt_obs
# 2 * Rate_orig(0) - 0.973 * 1 = 4.29
# 2 * Rate_orig(0) = 5.263
# Rate_orig(0) = 2.6315
# So, d[ijk0]/dt should be -2.6315. Observed was -2.535. Close enough. Let's use Rate_orig(0) = 2.6315.
# 2.6315 = k1_eff * 1 / (K_m_bnxy + 1) (Eq 1 revised)

# Exp Pert: id_ijk0(0)=1, id_bnxy(0)=0.1
# d[ijk0]/dt approx (0.320-1)/0.2 = -3.4
# d[bnxy]/dt approx (1.29-0.1)/0.2 = 5.95
# So, Rate_pert(0) = 3.4
# 3.4 = k1_eff * 1 / (K_m_bnxy + 0.1) (Eq 2)
# d[bnxy]/dt_obs_pert = 5.95
# 2 * Rate_pert(0) - k2_est * [bnxy(0)] = d[bnxy]/dt_obs_pert
# 2 * Rate_pert(0) - 0.973 * 0.1 = 5.95
# 2 * Rate_pert(0) = 5.95 + 0.0973 = 6.0473
# Rate_pert(0) = 3.02365
# So, d[ijk0]/dt should be -3.02365. Observed was -3.4. Some discrepancy. Let's use Rate_pert(0) = 3.02365.
# 3.02365 = k1_eff * 1 / (K_m_bnxy + 0.1) (Eq 2 revised)

# From Eq 1 revised: k1_eff = 2.6315 * (K_m_bnxy + 1)
# Substitute into Eq 2 revised:
# 3.02365 = 2.6315 * (K_m_bnxy + 1) / (K_m_bnxy + 0.1)
# 3.02365 * (K_m_bnxy + 0.1) = 2.6315 * (K_m_bnxy + 1)
# 3.02365 * K_m_bnxy + 0.302365 = 2.6315 * K_m_bnxy + 2.6315
# (3.02365 - 2.6315) * K_m_bnxy = 2.6315 - 0.302365
# 0.39215 * K_m_bnxy = 2.329135
# K_m_bnxy = 2.329135 / 0.39215
K_m_bnxy_est = 5.939 # Roughly 6

# k1_eff = 2.6315 * (K_m_bnxy_est + 1)
k1_eff_est = 2.6315 * (5.939 + 1) # Roughly 2.6 * 7 = 18.2
k1_eff_est = 2.6315 * 6.939 # = 18.25

param_k1_eff = k1_eff_est
param_K_m_bnxy = K_m_bnxy_est
param_k2 = k2_est # 0.973

print(f"Estimated k1_eff: {param_k1_eff}")
print(f"Estimated K_m_bnxy: {param_K_m_bnxy}")
print(f"Estimated k2: {param_k2}")

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

# Add parameters
p = model.createParameter(); p.setId("k1_eff"); p.setValue(param_k1_eff); p.setConstant(True)
p = model.createParameter(); p.setId("K_m_bnxy"); p.setValue(param_K_m_bnxy); p.setConstant(True)
p = model.createParameter(); p.setId("k2"); p.setValue(param_k2); p.setConstant(True)

# Reaction 1: id_1t80 + id_ijk0 -> id_1t80 + 2*id_bnxy
# Rate: k1_eff * id_1t80 * id_ijk0 / (K_m_bnxy + id_bnxy)
# Since id_1t80 is boundary and constant at 1, its term can be absorbed into k1_eff or explicitly included.
# Let's include it for clarity, so k1_eff is the true rate constant.
r1 = model.createReaction()
r1.setId("R1_ijk0_conv_inhib_bnxy")
r1.setReversible(False)
r1.setFast(False)

# Reactant id_ijk0
reactant1_r1 = r1.createReactant()
reactant1_r1.setSpecies("id_ijk0")
reactant1_r1.setStoichiometry(1)
reactant1_r1.setConstant(False)

# Modifier id_1t80 (as catalyst)
# SBML does not have explicit catalyst role in reactant/product list for mass-action like this.
# It's part of the kinetic law. id_1t80 is already defined as boundaryCondition=true, constant=true.

# Product id_bnxy
product1_r1 = r1.createProduct()
product1_r1.setSpecies("id_bnxy")
product1_r1.setStoichiometry(2)
product1_r1.setConstant(False)

# Kinetic Law for R1
kl_r1 = r1.createKineticLaw()
# Formula: k1_eff * id_1t80 * id_ijk0 / (K_m_bnxy + id_bnxy)
# Since id_1t80 is always 1.0, we can simplify to k1_eff * id_ijk0 / (K_m_bnxy + id_bnxy)
# However, to be more robust if id_1t80 could change in other scenarios (not here), include it.
kl_r1.setFormula("k1_eff * id_1t80 * id_ijk0 / (K_m_bnxy + id_bnxy)")


# Reaction 2: id_bnxy -> id_9k8m
# Rate: k2 * id_bnxy
r2 = model.createReaction()
r2.setId("R2_bnxy_degradation")
r2.setReversible(False)
r2.setFast(False)

# Reactant id_bnxy
reactant1_r2 = r2.createReactant()
reactant1_r2.setSpecies("id_bnxy")
reactant1_r2.setStoichiometry(1)
reactant1_r2.setConstant(False)

# Product id_9k8m (boundary species)
product1_r2 = r2.createProduct()
product1_r2.setSpecies("id_9k8m")
product1_r2.setStoichiometry(1)
product1_r2.setConstant(False) # SBML L3 requirement

# Kinetic Law for R2
kl_r2 = r2.createKineticLaw()
kl_r2.setFormula("k2 * id_bnxy")

# Write the updated SBML
writer = libsbml.SBMLWriter()
updated_sbml_model_iter4 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_best_model_iter4", updated_sbml_model_iter4)

# Simulate with default initial conditions
sim_data_default = simulate(updated_sbml_model_iter4)
shared_variables.add("sim_data_default_iter4", sim_data_default)

# Simulate with perturbed initial conditions (id_bnxy=0.1)
# Need to modify the SBML string for initial concentration for this simulation
temp_doc = libsbml.readSBMLFromString(updated_sbml_model_iter4)
temp_model = temp_doc.getModel()
temp_model.getSpecies("id_bnxy").setInitialConcentration(0.1)
# id_ijk0 remains 1.0 by default from input_sbml_string
# id_1t80 and id_9k8m are boundary, their initial conc is fixed.
sim_sbml_pert = writer.writeSBMLToString(temp_doc)
sim_data_perturbed = simulate(sim_sbml_pert)
shared_variables.add("sim_data_perturbed_iter4", sim_data_perturbed)


# Print comparison for default conditions
print("\nComparison for Default Conditions (Exp vs Sim):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_orig[exp_data_orig['Time'] == t_point]['id_ijk0'].values[0] if t_point in exp_data_orig['Time'].values else float('nan')
    sim_val = sim_data_default[sim_data_default['Time'] == t_point]['id_ijk0'].values[0] if t_point in sim_data_default['Time'].values else float('nan')
    print(f"{t_point:.1f} | {exp_val:.3e} | {sim_val:.3e}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_orig[exp_data_orig['Time'] == t_point]['id_bnxy'].values[0] if t_point in exp_data_orig['Time'].values else float('nan')
    sim_val = sim_data_default[sim_data_default['Time'] == t_point]['id_bnxy'].values[0] if t_point in sim_data_default['Time'].values else float('nan')
    print(f"{t_point:.1f} | {exp_val:.3e} | {sim_val:.3e}")

# Print comparison for perturbed conditions
print("\nComparison for Perturbed Conditions (id_bnxy(0)=0.1) (Exp vs Sim):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_pert[exp_data_pert['Time'] == t_point]['id_ijk0'].values[0] if t_point in exp_data_pert['Time'].values else float('nan')
    sim_val = sim_data_perturbed[sim_data_perturbed['Time'] == t_point]['id_ijk0'].values[0] if t_point in sim_data_perturbed['Time'].values else float('nan')
    print(f"{t_point:.1f} | {exp_val:.3e} | {sim_val:.3e}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_pert[exp_data_pert['Time'] == t_point]['id_bnxy'].values[0] if t_point in exp_data_pert['Time'].values else float('nan')
    sim_val = sim_data_perturbed[sim_data_perturbed['Time'] == t_point]['id_bnxy'].values[0] if t_point in sim_data_perturbed['Time'].values else float('nan')
    print(f"{t_point:.1f} | {exp_val:.3e} | {sim_val:.3e}")

```
# Observation

## Code Stdout
Estimated k2: 0.9733385812825687
Estimated k1_eff: 18.2599785
Estimated K_m_bnxy: 5.939
Estimated k2: 0.9733385812825687

Comparison for Default Conditions (Exp vs Sim):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 4.933e-01 | 6.043e-01
0.4 | 4.574e-01 | 3.730e-01
1.0 | 4.389e-01 | 8.650e-02
2.0 | 3.975e-01 | 6.060e-03
5.0 | 1.306e-01 | 7.982e-07

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e+00 | 1.000e+00
0.2 | 1.858e+00 | 1.536e+00
0.4 | 1.900e+00 | 1.681e+00
1.0 | 1.865e+00 | 1.343e+00
2.0 | 1.769e+00 | 5.922e-01
5.0 | 9.687e-01 | 3.292e-02

Comparison for Perturbed Conditions (id_bnxy(0)=0.1) (Exp vs Sim):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 3.200e-01 | 5.687e-01
0.4 | 2.413e-01 | 3.364e-01
1.0 | 1.707e-01 | 7.125e-02
2.0 | 6.978e-02 | 4.555e-03
5.0 | 9.958e-05 | 5.606e-07

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e-01 | 1.000e-01
0.2 | 1.287e+00 | 8.575e-01
0.4 | 1.327e+00 | 1.124e+00
1.0 | 1.120e+00 | 1.000e+00
2.0 | 6.897e-01 | 4.476e-01
5.0 | 2.260e-02 | 2.488e-02

## 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, iteration_3)
- `shared_variables`: all variables you have saved from the previous iterations. (current_best_model, exp_data_iter0, sim_data_iter1, exp_data_orig_iter4, exp_data_pert_iter4, current_best_model_iter4, sim_data_default_iter4, sim_data_perturbed_iter4)

### 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 current model (R1: `id_1t80 + id_ijk0 -> id_1t80 + 2*id_bnxy` with rate `k1_eff * id_1t80 * id_ijk0 / (K_m_bnxy + id_bnxy)` and R2: `id_bnxy -> id_9k8m` with rate `k2 * id_bnxy`) captures the qualitative behavior:
- `id_ijk0` decay is inhibited by `id_bnxy`.
- `id_bnxy` shows a rise and fall.
- Both species go to zero.
- The sum `[id_ijk0] + [id_bnxy]` increases initially due to the 2:1 stoichiometry in R1.

However, the quantitative fit is not perfect.
**Default conditions (bnxy(0)=1):**
- `id_ijk0`: Simulated decay is initially slower (0.604 vs 0.493 at t=0.2) then faster. The experimental data shows a more pronounced "slowing down" of `id_ijk0` decay around its peak `id_bnxy` concentration.
- `id_bnxy`: Simulated peak is lower (1.68 vs 1.90) and occurs slightly later. Decay is faster in simulation.

**Perturbed conditions (bnxy(0)=0.1):**
- `id_ijk0`: Simulated decay is significantly slower than experimental (0.568 vs 0.320 at t=0.2). This is the opposite of what I expected from the inhibition term. My parameter estimation for `K_m_bnxy` or `k1_eff` might be off, or the form of inhibition is not quite right.
- `id_bnxy`: Simulated rise is slower and peak is lower (1.12 vs 1.33).

The discrepancy in the perturbed case for `id_ijk0` (simulated decay is *slower* than experimental, while `id_bnxy` is low) is particularly concerning. The rate law `k1_eff * [id_ijk0] / (K_m_bnxy + [id_bnxy])` means that when `[id_bnxy]` is small (0.1), the denominator `(K_m_bnxy + 0.1)` is smaller than when `[id_bnxy]` is 1 `(K_m_bnxy + 1)`, leading to a *faster* rate for `id_ijk0` decay in the perturbed case. This is consistent with the experimental observation that `id_ijk0` decays faster when `id_bnxy(0)` is low.
Original: `Rate_ijk0_decay(0) = k1_eff / (K_m_bnxy + 1)`
Perturbed: `Rate_ijk0_decay(0) = k1_eff / (K_m_bnxy + 0.1)`
Since `K_m_bnxy` (est. ~6) is positive, the perturbed rate should indeed be higher.

Let's re-check the initial rate calculations from the simulation output vs. experimental:
**Default (bnxy(0)=1):**
- Exp `d[ijk0]/dt(0)` ~ -2.535. Sim `d[ijk0]/dt(0)` from model: `-18.26 * 1 / (5.939 + 1)` = `-18.26 / 6.939` = -2.63. (Matches my calculation).
- Exp `id_ijk0(0.2)` = 0.493. Sim `id_ijk0(0.2)` = 0.604. Simulated decay is slower.

**Perturbed (bnxy(0)=0.1):**
- Exp `d[ijk0]/dt(0)` ~ -3.4. Sim `d[ijk0]/dt(0)` from model: `-18.26 * 1 / (5.939 + 0.1)` = `-18.26 / 6.039` = -3.02. (Matches my calculation).
- Exp `id_ijk0(0.2)` = 0.320. Sim `id_ijk0(0.2)` = 0.568. Simulated decay is much slower.

The issue is that my estimated initial rates from the simulation (based on parameters derived from experimental initial slopes) are leading to simulated concentrations at t=0.2 that are higher (slower decay) than observed. This suggests the model, even with these parameters, doesn't capture the initial phase correctly or the numerical integration over the interval [0, 0.2] with these rates results in this.

The fact that the simulated `id_ijk0` decay is *slower* than experimental in *both* cases for the first 0.2s is a problem.
Perhaps the inhibition by `id_bnxy` is stronger, or the basal rate (when `id_bnxy` is low) is higher.

Let's re-evaluate the parameters `k1_eff` and `K_m_bnxy`.
The equations were:
1. `Rate_orig(0) = k1_eff / (K_m_bnxy + 1)` (Observed `d[ijk0]/dt` ~ -2.535, so `Rate_orig(0)` ~ 2.535)
2. `Rate_pert(0) = k1_eff / (K_m_bnxy + 0.1)` (Observed `d[ijk0]/dt` ~ -3.4, so `Rate_pert(0)` ~ 3.4)

Let `R_o = 2.535` and `R_p = 3.4`.
`k1_eff = R_o * (K_m_bnxy + 1)`
`k1_eff = R_p * (K_m_bnxy + 0.1)`
`R_o * (K_m_bnxy + 1) = R_p * (K_m_bnxy + 0.1)`
`2.535 * K_m_bnxy + 2.535 = 3.4 * K_m_bnxy + 0.34`
`2.535 - 0.34 = (3.4 - 2.535) * K_m_bnxy`
`2.195 = 0.865 * K_m_bnxy`
`K_m_bnxy = 2.195 / 0.865 = 2.5375`

`k1_eff = 2.535 * (2.5375 + 1) = 2.535 * 3.5375 = 8.967`
Or using `R_p`: `k1_eff = 3.4 * (2.5375 + 0.1) = 3.4 * 2.6375 = 8.9675`

So, new estimates:
`param_k1_eff = 8.9675`
`param_K_m_bnxy = 2.5375`
`param_k2` remains `0.9733` (from tail of `id_bnxy` decay).

These new parameters give a lower `K_m_bnxy` (stronger inhibition at lower `id_bnxy` concentrations) and a lower `k1_eff` (lower maximal rate).
Let's try simulating with these parameters.

The previous `k1_eff` was ~18.26, `K_m_bnxy` ~5.94.
New `k1_eff` ~8.97, `K_m_bnxy` ~2.54.

The ratio `k1_eff / K_m_bnxy` (rate when `id_bnxy` is very small) changed from `18.26/5.94 = 3.07` to `8.97/2.54 = 3.53`. This suggests a slightly faster rate when `id_bnxy` is very low, which is good for the perturbed case.
The rate when `id_bnxy = 1` is `k1_eff / (K_m_bnxy + 1)` which changed from `18.26 / (5.94+1) = 2.63` to `8.97 / (2.54+1) = 2.53`. This is very close to the target `R_o = 2.535`.

This re-estimation seems more consistent with the initial slopes of `id_ijk0`.
I will update the model with these new parameters and re-simulate.

## Action

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

# Access experimental data
exp_data_orig = experiment_history['iteration_0']
exp_data_pert = experiment_history['iteration_3'] # bnxy(0)=0.1

# New parameter estimates
param_k1_eff_new = 8.9675
param_K_m_bnxy_new = 2.5375
param_k2_new = 0.9733385812825687 # Unchanged

print(f"New Estimated k1_eff: {param_k1_eff_new}")
print(f"New Estimated K_m_bnxy: {param_K_m_bnxy_new}")
print(f"New Estimated k2: {param_k2_new}")

# Build the SBML model using the existing structure but new parameters
# Retrieve the model structure from the previous best model if stored, or rebuild
# For safety, rebuild from input_sbml_string to ensure clean state
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Add parameters
p = model.createParameter(); p.setId("k1_eff"); p.setValue(param_k1_eff_new); p.setConstant(True)
p = model.createParameter(); p.setId("K_m_bnxy"); p.setValue(param_K_m_bnxy_new); p.setConstant(True)
p = model.createParameter(); p.setId("k2"); p.setValue(param_k2_new); p.setConstant(True)

# Reaction 1: id_1t80 + id_ijk0 -> id_1t80 + 2*id_bnxy
r1 = model.createReaction()
r1.setId("R1_ijk0_conv_inhib_bnxy")
r1.setReversible(False)
r1.setFast(False)
reactant1_r1 = r1.createReactant()
reactant1_r1.setSpecies("id_ijk0")
reactant1_r1.setStoichiometry(1)
reactant1_r1.setConstant(False)
product1_r1 = r1.createProduct()
product1_r1.setSpecies("id_bnxy")
product1_r1.setStoichiometry(2)
product1_r1.setConstant(False)
kl_r1 = r1.createKineticLaw()
kl_r1.setFormula("k1_eff * id_1t80 * id_ijk0 / (K_m_bnxy + id_bnxy)")

# Reaction 2: id_bnxy -> id_9k8m
r2 = model.createReaction()
r2.setId("R2_bnxy_degradation")
r2.setReversible(False)
r2.setFast(False)
reactant1_r2 = r2.createReactant()
reactant1_r2.setSpecies("id_bnxy")
reactant1_r2.setStoichiometry(1)
reactant1_r2.setConstant(False)
product1_r2 = r2.createProduct()
product1_r2.setSpecies("id_9k8m")
product1_r2.setStoichiometry(1)
product1_r2.setConstant(False)
kl_r2 = r2.createKineticLaw()
kl_r2.setFormula("k2 * id_bnxy")

writer = libsbml.SBMLWriter()
updated_sbml_model_iter5 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_best_model_iter5", updated_sbml_model_iter5)

# Simulate with default initial conditions
# Ensure species initial concentrations are default for this simulation
# The input_sbml_string has id_ijk0=1, id_bnxy=1.
# When we read from input_sbml_string, these are set.
# If we were reusing a modified doc, we'd need to reset them.
sim_doc_default = libsbml.readSBMLFromString(updated_sbml_model_iter5)
# Ensure default initial conditions from problem spec (ijk0=1, bnxy=1)
sim_model_default = sim_doc_default.getModel()
sim_model_default.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_default.getSpecies("id_bnxy").setInitialConcentration(1.0)
sbml_for_sim_default = writer.writeSBMLToString(sim_doc_default)
sim_data_default = simulate(sbml_for_sim_default)


# Simulate with perturbed initial conditions (id_bnxy=0.1)
sim_doc_perturbed = libsbml.readSBMLFromString(updated_sbml_model_iter5)
sim_model_perturbed = sim_doc_perturbed.getModel()
sim_model_perturbed.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_perturbed.getSpecies("id_bnxy").setInitialConcentration(0.1)
sbml_for_sim_perturbed = writer.writeSBMLToString(sim_doc_perturbed)
sim_data_perturbed = simulate(sbml_for_sim_perturbed)

# Store for inspection
shared_variables.add("sim_data_default_iter5", sim_data_default)
shared_variables.add("sim_data_perturbed_iter5", sim_data_perturbed)

# Print comparison for default conditions
print("\nComparison for Default Conditions (Exp vs Sim with new params):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_orig[exp_data_orig['Time'] == t_point]['id_ijk0'].values[0] if t_point in exp_data_orig['Time'].values else float('nan')
    sim_val = sim_data_default[sim_data_default['Time'] == t_point]['id_ijk0'].values[0] if t_point in sim_data_default['Time'].values else float('nan')
    print(f"{t_point:.1f} | {exp_val:.3e} | {sim_val:.3e}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_orig[exp_data_orig['Time'] == t_point]['id_bnxy'].values[0] if t_point in exp_data_orig['Time'].values else float('nan')
    sim_val = sim_data_default[sim_data_default['Time'] == t_point]['id_bnxy'].values[0] if t_point in sim_data_default['Time'].values else float('nan')
    print(f"{t_point:.1f} | {exp_val:.3e} | {sim_val:.3e}")

# Print comparison for perturbed conditions
print("\nComparison for Perturbed Conditions (id_bnxy(0)=0.1) (Exp vs Sim with new params):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_pert[exp_data_pert['Time'] == t_point]['id_ijk0'].values[0] if t_point in exp_data_pert['Time'].values else float('nan')
    sim_val = sim_data_perturbed[sim_data_perturbed['Time'] == t_point]['id_ijk0'].values[0] if t_point in sim_data_perturbed['Time'].values else float('nan')
    print(f"{t_point:.1f} | {exp_val:.3e} | {sim_val:.3e}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_pert[exp_data_pert['Time'] == t_point]['id_bnxy'].values[0] if t_point in exp_data_pert['Time'].values else float('nan')
    sim_val = sim_data_perturbed[sim_data_perturbed['Time'] == t_point]['id_bnxy'].values[0] if t_point in sim_data_perturbed['Time'].values else float('nan')
    print(f"{t_point:.1f} | {exp_val:.3e} | {sim_val:.3e}")
```
# Observation

## Code Stdout
New Estimated k1_eff: 8.9675
New Estimated K_m_bnxy: 2.5375
New Estimated k2: 0.9733385812825687

Comparison for Default Conditions (Exp vs Sim with new params):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 4.933e-01 | 6.257e-01
0.4 | 4.574e-01 | 4.049e-01
1.0 | 4.389e-01 | 1.077e-01
2.0 | 3.975e-01 | 8.137e-03
5.0 | 1.306e-01 | 4.298e-07

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e+00 | 1.000e+00
0.2 | 1.858e+00 | 1.497e+00
0.4 | 1.900e+00 | 1.630e+00
1.0 | 1.865e+00 | 1.333e+00
2.0 | 1.769e+00 | 6.099e-01
5.0 | 9.687e-01 | 3.419e-02

Comparison for Perturbed Conditions (id_bnxy(0)=0.1) (Exp vs Sim with new params):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 3.200e-01 | 5.587e-01
0.4 | 2.413e-01 | 3.374e-01
1.0 | 1.707e-01 | 7.709e-02
2.0 | 6.978e-02 | 4.817e-03
5.0 | 9.958e-05 | 2.118e-07

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e-01 | 1.000e-01
0.2 | 1.287e+00 | 8.735e-01
0.4 | 1.327e+00 | 1.117e+00
1.0 | 1.120e+00 | 9.914e-01
2.0 | 6.897e-01 | 4.505e-01
5.0 | 2.260e-02 | 2.505e-02

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_3)
- `shared_variables`: all variables you have saved from the previous iterations. (current_best_model, exp_data_iter0, sim_data_iter1, exp_data_orig_iter4, exp_data_pert_iter4, current_best_model_iter4, sim_data_default_iter4, sim_data_perturbed_iter4, current_best_model_iter5, sim_data_default_iter5, sim_data_perturbed_iter5)

### 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

The re-estimated parameters (`k1_eff = 8.9675`, `K_m_bnxy = 2.5375`, `k2 = 0.9733`) improved the fit for the initial decay rate of `id_ijk0` conceptually, but the simulation results still show discrepancies:

**Default conditions (bnxy(0)=1):**
- `id_ijk0`: Sim(t=0.2) = 0.626 vs Exp(t=0.2) = 0.493. Simulated decay is still too slow initially.
- `id_bnxy`: Sim peak ~1.63 vs Exp peak ~1.90. Simulated peak is too low.

**Perturbed conditions (bnxy(0)=0.1):**
- `id_ijk0`: Sim(t=0.2) = 0.559 vs Exp(t=0.2) = 0.320. Simulated decay is significantly too slow.
- `id_bnxy`: Sim peak ~1.12 vs Exp peak ~1.33. Simulated peak is too low.

The model structure `R1: id_1t80 + id_ijk0 -> id_1t80 + 2*id_bnxy` (rate `k1_eff * id_1t80 * id_ijk0 / (K_m_bnxy + id_bnxy)`) and `R2: id_bnxy -> id_9k8m` (rate `k2 * id_bnxy`) seems to capture the qualitative aspects, but the quantitative fit, especially for the initial phase and peak heights, is off.

The fact that `id_ijk0` decay is consistently simulated as slower than observed in the initial phase (0-0.2s) for both conditions, despite parameters being derived from these initial slopes, suggests a few possibilities:
1.  The chosen kinetic law for R1, while directionally correct for inhibition, might not be the exact form.
2.  The stoichiometry of `id_bnxy` production (currently 2) might be different.
3.  There might be other reactions missing.
4.  Parameter estimation based solely on initial slopes might be too simplistic for this coupled system. A more global optimization approach might yield better parameters for the current model structure.

Let's reconsider the initial rates and the `id_bnxy` production.
In the default case: `id_ijk0` drops by ~0.5 (from 1 to 0.493) by t=0.2. `id_bnxy` rises by ~0.86 (from 1 to 1.858).
If `id_ijk0 -> 2*id_bnxy`, then 0.5 `id_ijk0` consumed should produce 1.0 `id_bnxy`.
Net change in `id_bnxy` = `produced - k2*integral(id_bnxy*dt)`.
`0.86 = 1.0 - k2*integral(id_bnxy*dt)`.
`k2*integral(id_bnxy*dt) = 0.14`.
Average `id_bnxy` over [0, 0.2] is roughly `(1+1.858)/2 = 1.429`.
So, `0.9733 * 1.429 * 0.2 = 0.9733 * 0.2858 = 0.278`. This is higher than 0.14.
This suggests that either the stoichiometry of 2 is too high, or `k2` is too high, or the production of `id_bnxy` from `id_ijk0` is less than 2 for each `id_ijk0`.

If the stoichiometry was `id_ijk0 -> S * id_bnxy`:
`d[bnxy]/dt = S * Rate_R1 - k2 * [bnxy]`
`d[ijk0]/dt = -Rate_R1`
Change in `bnxy` = `S * (change in ijk0) - k2 * integral(bnxy*dt)`
`0.86 = S * 0.507 - 0.278` (using `delta_ijk0 = 1-0.493 = 0.507`)
`1.138 = S * 0.507`
`S = 1.138 / 0.507 = 2.24`
This suggests stoichiometry might be slightly higher than 2, or my `k2` estimate is off, or the simple averaging for the integral is too crude.

Let's try to adjust `k1_eff` to better match the `id_ijk0` decay at t=0.2, and see what happens to `id_bnxy`.
If `id_ijk0(0.2)` needs to be `0.493` (default) and `0.320` (perturbed), the current model with `K_m_bnxy=2.5375` gives:
Default: `Rate_R1(0) = k1_eff / (2.5375 + 1) = k1_eff / 3.5375`. Target `id_ijk0(0.2)=0.493`.
Perturbed: `Rate_R1(0) = k1_eff / (2.5375 + 0.1) = k1_eff / 2.6375`. Target `id_ijk0(0.2)=0.320`.

The simulated `id_ijk0` values at t=0.2 were 0.626 (default) and 0.559 (perturbed).
To get lower `id_ijk0` values (faster decay), `k1_eff` needs to be larger.
Let's try to increase `k1_eff` by a factor.
Default: Exp/Sim = 0.493/0.626 ~ 0.787.
Perturbed: Exp/Sim = 0.320/0.559 ~ 0.572.
This suggests a simple scaling of `k1_eff` won't fix both simultaneously, indicating `K_m_bnxy` might also need adjustment or the model form is still not quite right.

The discrepancy is larger for the perturbed case (needs much faster decay).
Current `k1_eff / (K_m_bnxy + 0.1)` is too small.
If `K_m_bnxy` is made smaller, the inhibition kicks in more strongly at lower `[bnxy]`, and the rate `k1_eff / (K_m_bnxy + [bnxy])` becomes more sensitive to `[bnxy]`.
When `[bnxy]` is very small (like 0.1), a smaller `K_m_bnxy` would make the denominator smaller, increasing the rate.
When `[bnxy]` is larger (like 1), a smaller `K_m_bnxy` would also make the denominator smaller, increasing the rate, but the relative effect of `[bnxy]` in the denominator becomes larger.

Let's try to manually tune `k1_eff` and `K_m_bnxy`.
Suppose `K_m_bnxy` is smaller, e.g., `K_m_bnxy = 1.0`.
Then from `Rate_orig(0) = 2.535 = k1_eff / (1.0 + 1)` => `k1_eff = 2.535 * 2 = 5.07`.
And `Rate_pert(0) = 3.4 = k1_eff / (1.0 + 0.1)` => `k1_eff = 3.4 * 1.1 = 3.74`.
These `k1_eff` values are inconsistent, meaning `K_m_bnxy=1.0` is not right if we strictly fit initial slopes.

The problem might be that the simple Michaelis-Menten type inhibition `1/(K_m + S)` is not steep enough or shaped correctly.
What if the stoichiometry `S` for `id_ijk0 -> S * id_bnxy` is a parameter to fit?
Let's assume `S=2` is correct for now as it's a common biological pattern (e.g. dimerization then modification).

Consider the possibility of an uncompetitive or non-competitive inhibition pattern if we think of `id_bnxy` as an inhibitor of an "enzyme" that converts `id_ijk0`.
The current form `rate = Vm * [S] / (Km_s + [S]) * (1 / (1 + [I]/Ki))` is not what I have.
I have `rate = k_cat * [E_total] * [S] / (Km_s + [S])` where the "enzyme" part is `id_1t80` and the rate `k1_eff * [id_ijk0]` is essentially `Vmax_apparent * [id_ijk0] / K_m_for_ijk0` if `id_ijk0` itself had Michaelis-Menten kinetics, but it's simpler here, `k1_eff * [id_ijk0]`. The inhibition is `1 / (K_m_bnxy + [id_bnxy])`. This is not standard form.
It should be `k1_eff * [id_ijk0] * (K_m_bnxy / (K_m_bnxy + [id_bnxy]))` or `k1_eff * [id_ijk0] / (1 + [id_bnxy]/K_i_bnxy)`.
My current form `k1_eff * [id_ijk0] / (K_m_bnxy + [id_bnxy])` is like `(k1_eff/K_m_bnxy) * [id_ijk0] / (1 + [id_bnxy]/K_m_bnxy)`.
So `k_prime = k1_eff / K_m_bnxy` is the effective rate constant when `[bnxy]` is low, and `K_i = K_m_bnxy` is the inhibition constant.
Let's use this formulation: `Rate = k_prime * [id_ijk0] / (1 + [id_bnxy]/K_i_bnxy)`.
`k_prime = k1_eff / K_m_bnxy`
`K_i_bnxy = K_m_bnxy`

Using my previous re-estimated parameters:
`k_prime = 8.9675 / 2.5375 = 3.534`
`K_i_bnxy = 2.5375`
So the rate law for R1 becomes: `(3.534 * id_1t80 * id_ijk0) / (1 + id_bnxy / 2.5375)`

Let's test this formulation with these parameters. It's mathematically equivalent to what I had, just a re-parameterization. So it won't change the simulation outcome.

The discrepancy in `id_bnxy` peak height (simulated is lower) suggests that either `id_bnxy` production is underestimated or its degradation `k2` is overestimated.
My `k2 = 0.9733` was from `ln(0.201 / 0.532) / (7.0 - 6.0)`.
At t=6, `id_ijk0` is `4.31e-2`. At t=7, `id_ijk0` is `6.99e-3`.
The contribution to `d[bnxy]/dt` from R1 is `2 * Rate_R1`.
`Rate_R1(t=6) = k_prime * [ijk0] / (1 + [bnxy]/K_i) = 3.534 * 0.0431 / (1 + 0.532/2.5375) = 0.152 / (1 + 0.209) = 0.152 / 1.209 = 0.1257`.
So `2 * Rate_R1(t=6) = 0.2514`.
`d[bnxy]/dt (t=6) = 0.2514 - k2 * 0.532`.
Experimental `d[bnxy]/dt` around t=6: `(bnxy(6.2)-bnxy(5.8))/(6.2-5.8) = (0.452 - 0.617)/0.4 = -0.165/0.4 = -0.4125`.
So, `-0.4125 = 0.2514 - k2 * 0.532`.
`k2 * 0.532 = 0.2514 + 0.4125 = 0.6639`.
`k2 = 0.6639 / 0.532 = 1.248`.
This new `k2` is higher than before (0.9733). A higher `k2` would make `id_bnxy` decay faster and peak lower, which is the opposite of what's needed for the peak.

This suggests my assumption that `id_ijk0` contribution is negligible at late times for `k2` estimation might be flawed, or the model structure/parameters are still off.

Let's try to improve the fit by focusing on the initial phase and peak.
If `id_bnxy` peak is too low, we need more net production. This means either:
- Increase stoichiometry `S` (currently 2).
- Increase `k1_eff` (or `k_prime`).
- Decrease `K_m_bnxy` (or `K_i_bnxy`) if `[bnxy]` is above `K_m_bnxy` at peak.
- Decrease `k2`.

Let's try decreasing `k2`. If `k2` is smaller, `id_bnxy` will accumulate more.
Original `k2` was ~0.97. What if `k2` is, say, 0.7?
This would affect the initial slope calculation for `K_m_bnxy` and `k1_eff` if I had used the `d[bnxy]/dt` equations. I primarily used `d[ijk0]/dt` for those, so they are less dependent on `k2`.

Let's try the model with `k1_eff = 8.9675`, `K_m_bnxy = 2.5375` and a smaller `k2`, e.g., `k2 = 0.7`.

If this doesn't significantly improve things, I might need to consider a different kinetic law for the inhibition or an additional reaction.
One common alternative for inhibition is `Vmax / (1 + ([I]/Ki)^h)`, a Hill-type inhibition, if the cooperativity of inhibition is important.

Let's try the parameter adjustment first.

## Action

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

# Access experimental data
exp_data_orig = experiment_history['iteration_0']
exp_data_pert = experiment_history['iteration_3'] # bnxy(0)=0.1

# Parameters from last iteration
param_k1_eff_prev = 8.9675
param_K_m_bnxy_prev = 2.5375
# param_k2_prev = 0.9733385812825687

# Try a new k2
param_k1_eff_adj = param_k1_eff_prev
param_K_m_bnxy_adj = param_K_m_bnxy_prev
param_k2_adj = 0.7  # Reduced k2

print(f"Adjusted k1_eff: {param_k1_eff_adj}")
print(f"Adjusted K_m_bnxy: {param_K_m_bnxy_adj}")
print(f"Adjusted k2: {param_k2_adj}")

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

# Add parameters
p = model.createParameter(); p.setId("k1_eff"); p.setValue(param_k1_eff_adj); p.setConstant(True)
p = model.createParameter(); p.setId("K_m_bnxy"); p.setValue(param_K_m_bnxy_adj); p.setConstant(True)
p = model.createParameter(); p.setId("k2"); p.setValue(param_k2_adj); p.setConstant(True)

# Reaction 1: id_1t80 + id_ijk0 -> id_1t80 + 2*id_bnxy
r1 = model.createReaction()
r1.setId("R1_ijk0_conv_inhib_bnxy")
r1.setReversible(False)
r1.setFast(False)
reactant1_r1 = r1.createReactant()
reactant1_r1.setSpecies("id_ijk0")
reactant1_r1.setStoichiometry(1)
reactant1_r1.setConstant(False)
product1_r1 = r1.createProduct()
product1_r1.setSpecies("id_bnxy")
product1_r1.setStoichiometry(2)
product1_r1.setConstant(False)
kl_r1 = r1.createKineticLaw()
kl_r1.setFormula("k1_eff * id_1t80 * id_ijk0 / (K_m_bnxy + id_bnxy)")

# Reaction 2: id_bnxy -> id_9k8m
r2 = model.createReaction()
r2.setId("R2_bnxy_degradation")
r2.setReversible(False)
r2.setFast(False)
reactant1_r2 = r2.createReactant()
reactant1_r2.setSpecies("id_bnxy")
reactant1_r2.setStoichiometry(1)
reactant1_r2.setConstant(False)
product1_r2 = r2.createProduct()
product1_r2.setSpecies("id_9k8m")
product1_r2.setStoichiometry(1)
product1_r2.setConstant(False)
kl_r2 = r2.createKineticLaw()
kl_r2.setFormula("k2 * id_bnxy")

writer = libsbml.SBMLWriter()
updated_sbml_model_iter6 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_best_model_iter6", updated_sbml_model_iter6)

# Simulate with default initial conditions
sim_doc_default = libsbml.readSBMLFromString(updated_sbml_model_iter6)
sim_model_default = sim_doc_default.getModel()
sim_model_default.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_default.getSpecies("id_bnxy").setInitialConcentration(1.0)
sbml_for_sim_default = writer.writeSBMLToString(sim_doc_default)
sim_data_default = simulate(sbml_for_sim_default)

# Simulate with perturbed initial conditions (id_bnxy=0.1)
sim_doc_perturbed = libsbml.readSBMLFromString(updated_sbml_model_iter6)
sim_model_perturbed = sim_doc_perturbed.getModel()
sim_model_perturbed.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_perturbed.getSpecies("id_bnxy").setInitialConcentration(0.1)
sbml_for_sim_perturbed = writer.writeSBMLToString(sim_doc_perturbed)
sim_data_perturbed = simulate(sbml_for_sim_perturbed)

shared_variables.add("sim_data_default_iter6", sim_data_default)
shared_variables.add("sim_data_perturbed_iter6", sim_data_perturbed)

# Print comparison for default conditions
print("\nComparison for Default Conditions (Exp vs Sim with adjusted k2):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_orig[exp_data_orig['Time'] == t_point]['id_ijk0'].values[0] if t_point in exp_data_orig['Time'].values else float('nan')
    sim_val = sim_data_default[sim_data_default['Time'] == t_point]['id_ijk0'].values[0] if t_point in sim_data_default['Time'].values else float('nan')
    print(f"{t_point:.1f} | {exp_val:.3e} | {sim_val:.3e}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0]: # Added 0.6 for bnxy peak
    exp_val = exp_data_orig[exp_data_orig['Time'] == t_point]['id_bnxy'].values[0] if t_point in exp_data_orig['Time'].values else float('nan')
    sim_val = sim_data_default[sim_data_default['Time'] == t_point]['id_bnxy'].values[0] if t_point in sim_data_default['Time'].values else float('nan')
    print(f"{t_point:.1f} | {exp_val:.3e} | {sim_val:.3e}")

# Print comparison for perturbed conditions
print("\nComparison for Perturbed Conditions (id_bnxy(0)=0.1) (Exp vs Sim with adjusted k2):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_pert[exp_data_pert['Time'] == t_point]['id_ijk0'].values[0] if t_point in exp_data_pert['Time'].values else float('nan')
    sim_val = sim_data_perturbed[sim_data_perturbed['Time'] == t_point]['id_ijk0'].values[0] if t_point in sim_data_perturbed['Time'].values else float('nan')
    print(f"{t_point:.1f} | {exp_val:.3e} | {sim_val:.3e}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0]: # Added 0.6 for bnxy peak
    exp_val = exp_data_pert[exp_data_pert['Time'] == t_point]['id_bnxy'].values[0] if t_point in exp_data_pert['Time'].values else float('nan')
    sim_val = sim_data_perturbed[sim_data_perturbed['Time'] == t_point]['id_bnxy'].values[0] if t_point in sim_data_perturbed['Time'].values else float('nan')
    print(f"{t_point:.1f} | {exp_val:.3e} | {sim_val:.3e}")
```
# Observation

## Code Stdout
Adjusted k1_eff: 8.9675
Adjusted K_m_bnxy: 2.5375
Adjusted k2: 0.7

Comparison for Default Conditions (Exp vs Sim with adjusted k2):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 4.933e-01 | 6.279e-01
0.4 | 4.574e-01 | 4.103e-01
1.0 | 4.389e-01 | 1.166e-01
2.0 | 3.975e-01 | 1.100e-02
5.0 | 1.306e-01 | 1.099e-06

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e+00 | 1.000e+00
0.2 | 1.858e+00 | 1.559e+00
0.4 | 1.900e+00 | 1.759e+00
0.6 | 1.891e+00 | 1.788e+00
1.0 | 1.865e+00 | 1.616e+00
2.0 | 1.769e+00 | 9.371e-01
5.0 | 9.687e-01 | 1.184e-01

Comparison for Perturbed Conditions (id_bnxy(0)=0.1) (Exp vs Sim with adjusted k2):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 3.200e-01 | 5.598e-01
0.4 | 2.413e-01 | 3.404e-01
1.0 | 1.707e-01 | 8.205e-02
2.0 | 6.978e-02 | 6.157e-03
5.0 | 9.958e-05 | 4.403e-07

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e-01 | 1.000e-01
0.2 | 1.287e+00 | 9.006e-01
0.4 | 1.327e+00 | 1.190e+00
0.6 | 1.268e+00 | 1.273e+00
1.0 | 1.120e+00 | 1.184e+00
2.0 | 6.897e-01 | 6.833e-01
5.0 | 2.260e-02 | 8.565e-02

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_3)
- `shared_variables`: all variables you have saved from the previous iterations. (current_best_model, exp_data_iter0, sim_data_iter1, exp_data_orig_iter4, exp_data_pert_iter4, current_best_model_iter4, sim_data_default_iter4, sim_data_perturbed_iter4, current_best_model_iter5, sim_data_default_iter5, sim_data_perturbed_iter5, current_best_model_iter6, sim_data_default_iter6, sim_data_perturbed_iter6)

### 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

Reducing `k2` from ~0.97 to 0.7 helped improve the `id_bnxy` peak height and persistence, making it closer to the experimental data, especially for the default condition.
- Default `id_bnxy`: Sim peak ~1.79 (at t=0.6) vs Exp peak ~1.90 (at t=0.4-0.6). This is better.
- Perturbed `id_bnxy`: Sim peak ~1.27 (at t=0.6) vs Exp peak ~1.33 (at t=0.4). Also better.
The tail of `id_bnxy` decay is now slower in simulation, which might be a trade-off. (e.g. Default: Sim(t=5)=0.118 vs Exp(t=5)=0.9687 - wait, Exp(t=5) for bnxy is 0.9687, not 0.13. My previous comparison point was wrong. Exp(t=5) id_bnxy is 0.9687. Sim(t=5) id_bnxy is 0.118. So simulated decay is much faster than experimental. This means k2=0.7 is too high, not too low. My previous k2=0.97 was even higher. This is confusing. Let me recheck experimental data for id_bnxy at t=5.0. `experiment_history['iteration_0']` at t=5.0, `id_bnxy` is 0.9687. `experiment_history['iteration_3']` at t=5.0, `id_bnxy` is 0.0226.

Okay, there's a significant difference in `id_bnxy` tail behavior between the two experiments.
- Default (bnxy(0)=1): `id_bnxy` is still high at t=5 (0.9687).
- Perturbed (bnxy(0)=0.1): `id_bnxy` is low at t=5 (0.0226).

My current model with `k2=0.7` gives:
- Default sim `id_bnxy(t=5)` = 0.118. This is much lower than experimental 0.9687. So `k2=0.7` is too *high* for the default case tail.
- Perturbed sim `id_bnxy(t=5)` = 0.085. This is higher than experimental 0.0226, but closer.

This implies that the decay rate of `id_bnxy` might not be a simple first-order `k2 * [id_bnxy]`. Or, the production of `id_bnxy` (via R1) is still significant at t=5 in the default case but not in the perturbed case.
Let's check `id_ijk0` at t=5:
- Default: Exp `id_ijk0(t=5)` = 0.1306. Sim `id_ijk0(t=5)` = 1.1e-6. Simulated `id_ijk0` is near zero.
- Perturbed: Exp `id_ijk0(t=5)` = 9.9e-5. Sim `id_ijk0(t=5)` = 4.4e-7. Simulated `id_ijk0` is near zero.

If simulated `id_ijk0` is near zero by t=5, then R1 (`...id_ijk0 / (... + id_bnxy)`) is producing very little `id_bnxy`. So the `id_bnxy` dynamics at t=5 are mostly governed by `-k2 * [id_bnxy]`.
If default exp `id_bnxy(t=5)` is 0.9687 and it's still decaying, then `k2` must be small enough to allow this persistence.
If perturbed exp `id_bnxy(t=5)` is 0.0226, this implies a faster relative decay or it started lower and decayed.

This is a strong indication that `k2` might not be constant, or there's another source/sink for `id_bnxy` that behaves differently in the two conditions.
However, the problem statement implies discovering missing *reactions*, not necessarily complex rate laws for existing species if the species themselves are part of the reaction.

Let's re-evaluate `k2` based on the default experiment's tail, assuming `id_ijk0` contribution is small.
Exp Default: `id_bnxy(t=5)=0.9687`, `id_bnxy(t=7)=0.201`.
`k2_default_tail = -ln(0.201/0.9687) / (7-5) = -ln(0.2075) / 2 = -(-1.572) / 2 = 0.786`.
This `k2 = 0.786` is higher than the `k2=0.7` I just tried.

Exp Perturbed: `id_bnxy(t=4)=0.0916`, `id_bnxy(t=5)=0.0226`.
`k2_perturbed_tail = -ln(0.0226/0.0916) / (5-4) = -ln(0.2467) = -(-1.399) = 1.399`.
This `k2` is much higher.

This is a significant finding: the effective first-order decay constant for `id_bnxy` appears to be different in the two conditions, or at different concentration ranges of `id_bnxy`.
- When `id_bnxy` is high (default tail), `k2_eff ~ 0.79`.
- When `id_bnxy` is low (perturbed tail), `k2_eff ~ 1.4`.

This could mean the degradation of `id_bnxy` is not simple first order. For example, it could be Michaelis-Menten like: `Vm_deg * [bnxy] / (Km_deg + [bnxy])`.
- If `[bnxy]` is high (>> Km_deg), rate ~ `Vm_deg` (zero order). This doesn't fit `d[bnxy]/dt = -k[bnxy]`.
- If `[bnxy]` is low (<< Km_deg), rate ~ `(Vm_deg/Km_deg) * [bnxy]` (first order).
This would imply `Vm_deg/Km_deg` is different at high vs low `[bnxy]`, which doesn't make sense for this form.

What if `id_ijk0` influences `id_bnxy` degradation?
E.g., `id_bnxy -> Sink`, rate `(k2_base + k2_ijk0_factor * [id_ijk0]) * [id_bnxy]`.
At late times, `[id_ijk0]` is low in both experiments. So this wouldn't explain the difference in `k2_eff`.

What if one of the source species, `id_1t80` or `id_9k8m`, is involved in `id_bnxy` degradation in a non-trivial way? They are constant.

Let's reconsider the `id_ijk0` decay. It's still simulated too slowly in the initial phase for both conditions.
Default: Sim(0.2)=0.628 vs Exp(0.2)=0.493. Needs to be faster.
Perturbed: Sim(0.2)=0.560 vs Exp(0.2)=0.320. Needs to be much faster.

To make `id_ijk0` decay faster, `k1_eff` needs to be larger, or `(K_m_bnxy + [id_bnxy])` needs to be smaller.
Current params: `k1_eff = 8.9675`, `K_m_bnxy = 2.5375`.
Rate for default at t=0: `8.9675 / (2.5375 + 1) = 8.9675 / 3.5375 = 2.535`.
Rate for perturbed at t=0: `8.9675 / (2.5375 + 0.1) = 8.9675 / 2.6375 = 3.40`.
These rates were derived from experimental `d[ijk0]/dt(0)`. The simulation integrates these rates. If the simulated `[ijk0]` at t=0.2 is too high, it means the *average* rate over [0, 0.2] was too low. This can happen if `[bnxy]` rises too quickly and inhibits R1 too soon/strongly.

The `id_bnxy` simulated peaks are:
Default: Sim peak ~1.79 (k2=0.7). Exp peak ~1.90. (Sim still a bit low)
Perturbed: Sim peak ~1.27 (k2=0.7). Exp peak ~1.33. (Sim a bit low)

If `k2` is actually smaller for the default case (e.g., `k2_def = 0.4-0.5` to match `id_bnxy(5)=0.96`), and larger for perturbed (`k2_pert = 1.4`). This is complex.

Let's assume `k2` *is* constant for now, and try to improve the initial `id_ijk0` fit and `id_bnxy` peak.
To get faster `id_ijk0` decay: increase `k1_eff`.
If `k1_eff` is increased, `id_bnxy` production also increases, potentially leading to higher `id_bnxy` peaks. This might be good.
Let's try `k1_eff` = 12 (was 8.9675).
And keep `K_m_bnxy = 2.5375`.
And what `k2`? The `k2_default_tail = 0.786` and `k2_perturbed_tail = 1.399` are problematic.
Let's use an intermediate `k2`, perhaps the `k2=0.7` was a reasonable compromise if we have to pick one constant value, or the previous `k2=0.9733`.
The simulation with `k2=0.7` gave `id_bnxy(5)` for default as 0.118 (exp 0.9687) - too fast decay.
The simulation with `k2=0.9733` (iter 5) gave `id_bnxy(5)` for default as 0.034 (exp 0.9687) - even faster decay.
This means `k2` needs to be much smaller if `id_ijk0` is truly depleted by t=5.

What if `id_ijk0` is *not* depleted by t=5 in the default experiment?
Exp default `id_ijk0(t=5) = 0.1306`. This is not negligible.
So R1 is still active: `Rate_R1(t=5, default) = k1_eff * 0.1306 / (K_m_bnxy + 0.9687)`.
Using `k1_eff=8.9675, K_m_bnxy=2.5375`:
`Rate_R1(t=5, default) = 8.9675 * 0.1306 / (2.5375 + 0.9687) = 1.171 / 3.5062 = 0.334`.
So, `d[bnxy]/dt (t=5, default) = 2 * 0.334 - k2 * 0.9687 = 0.668 - k2 * 0.9687`.
Experimental `d[bnxy]/dt` from t=5 to t=6 (default): `(0.532 - 0.9687)/1 = -0.4367`.
So, `-0.4367 = 0.668 - k2 * 0.9687`.
`k2 * 0.9687 = 0.668 + 0.4367 = 1.1047`.
`k2 = 1.1047 / 0.9687 = 1.140`.

Now for perturbed case at t=5:
Exp perturbed `id_ijk0(t=5) = 9.958e-5`. This is very low.
Exp perturbed `id_bnxy(t=5) = 0.0226`.
`Rate_R1(t=5, perturbed) = k1_eff * 9.958e-5 / (K_m_bnxy + 0.0226)`.
`Rate_R1(t=5, perturbed) = 8.9675 * 9.958e-5 / (2.5375 + 0.0226) = 8.929e-4 / 2.5601 = 3.488e-4`. (Very small).
`d[bnxy]/dt (t=5, perturbed) = 2 * 3.488e-4 - k2 * 0.0226 = 6.976e-4 - k2 * 0.0226`.
Experimental `d[bnxy]/dt` from t=5 to t=5.2 (perturbed): `(0.0169 - 0.0226)/0.2 = -0.0057/0.2 = -0.0285`.
So, `-0.0285 = 6.976e-4 - k2 * 0.0226`.
`k2 * 0.0226 = 6.976e-4 + 0.0285 = 0.0291976`.
`k2 = 0.0291976 / 0.0226 = 1.292`.

The two `k2` estimates (1.140 and 1.292) are now much closer! This is encouraging. It suggests that the persistence of `id_ijk0` in the default case at t=5 is important.
Let's use an average `k2 = (1.140 + 1.292)/2 = 1.216`.

So, new parameter set to try:
`k1_eff`: Needs to be higher to speed up initial `id_ijk0` decay. Let's try `k1_eff = 15`. (Original estimate was ~9).
`K_m_bnxy = 2.5375` (This was from initial `id_ijk0` slopes, might need re-evaluation if `k1_eff` changes significantly, but let's try one change at a time for `k1_eff` then `k2`).
`k2 = 1.216`.

Let's re-evaluate `k1_eff` and `K_m_bnxy` using the initial `d[ijk0]/dt` slopes:
`R_o = 2.535 = k1_eff / (K_m_bnxy + 1)`
`R_p = 3.4   = k1_eff / (K_m_bnxy + 0.1)`
This system yielded `K_m_bnxy = 2.5375` and `k1_eff = 8.9675`.
These values are tied. If I want a faster initial decay, I need to scale both `R_o` and `R_p` effectively.
The simulation with `k1_eff=8.9675` gave `id_ijk0(0.2)` as 0.628 (default) and 0.560 (perturbed).
Target `id_ijk0(0.2)` is 0.493 (default) and 0.320 (perturbed).
The required increase in average rate over [0,0.2] is larger for perturbed.

This suggests the sensitivity to `[bnxy]` (i.e. `K_m_bnxy`) might be off.
If `K_m_bnxy` is smaller, the term `(K_m_bnxy + [bnxy])` changes more percent-wise when `[bnxy]` goes from 0.1 to 1.
Let `K_m_bnxy = 1.0`.
Then `2.535 = k1_eff / 2` => `k1_eff = 5.07`.
And `3.4 = k1_eff / 1.1` => `k1_eff = 3.74`.
Still inconsistent. The ratio `R_p/R_o = 3.4/2.535 = 1.341`.
This should be equal to `(K_m_bnxy + 1) / (K_m_bnxy + 0.1)`.
`1.341 * (K_m_bnxy + 0.1) = K_m_bnxy + 1`
`1.341 * K_m_bnxy + 0.1341 = K_m_bnxy + 1`
`0.341 * K_m_bnxy = 1 - 0.1341 = 0.8659`
`K_m_bnxy = 0.8659 / 0.341 = 2.539`. This is the same `K_m_bnxy` as before. My algebra was correct.

The issue is that fitting to initial point slopes `d[S]/dt` doesn't guarantee matching `S(t)` at a later time point like `t=0.2` if the rate changes significantly over that interval.
The simulated `id_ijk0` decay is too slow. This means the actual inhibition by `id_bnxy` (which grows from t=0) is kicking in too hard or too fast in the model, or the initial rate without inhibition is not high enough.
The rate is `k1_eff * [id_ijk0] / (K_m_bnxy + [id_bnxy])`.
To make it faster overall, `k1_eff` should be higher.
To make it less inhibited by early `[id_bnxy]` rise, `K_m_bnxy` should be higher (makes denominator less sensitive to small `[id_bnxy]`).

Let's try:
- Increase `k1_eff` significantly. Say, `k1_eff = 20`.
- To maintain the ratio `R_p/R_o = 1.341`, `K_m_bnxy` must remain `2.539`.
  If `k1_eff = 20`, then `K_m_bnxy + 1 = 20 / 2.535 = 7.89` => `K_m_bnxy = 6.89`.
  And `K_m_bnxy + 0.1 = 20 / 3.4 = 5.88` => `K_m_bnxy = 5.78`.
  These `K_m_bnxy` are not equal. This means the initial slopes `R_o` and `R_p` are not perfectly fit by this model structure. One or both of these slope estimates, or the model form, must be imperfect.

Let's assume the model form is correct. The experimental data has noise.
Let's try `k1_eff = 15`, `K_m_bnxy = 4.0` (higher than 2.54, so less sensitive initially).
And `k2 = 1.216`.

Parameters:
`param_k1_eff = 15.0`
`param_K_m_bnxy = 4.0`
`param_k2 = 1.216`

This is another round of parameter tuning. If this doesn't work, I might need to consider a different reaction for `id_bnxy` degradation or production, or a different inhibition mechanism.

## Action

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

# Access experimental data
exp_data_orig = experiment_history['iteration_0']
exp_data_pert = experiment_history['iteration_3'] # bnxy(0)=0.1

# New parameters based on refined analysis
param_k1_eff_new = 15.0
param_K_m_bnxy_new = 4.0
param_k2_new = 1.216

print(f"New trial k1_eff: {param_k1_eff_new}")
print(f"New trial K_m_bnxy: {param_K_m_bnxy_new}")
print(f"New trial k2: {param_k2_new}")

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

# Add parameters
p = model.createParameter(); p.setId("k1_eff"); p.setValue(param_k1_eff_new); p.setConstant(True)
p = model.createParameter(); p.setId("K_m_bnxy"); p.setValue(param_K_m_bnxy_new); p.setConstant(True)
p = model.createParameter(); p.setId("k2"); p.setValue(param_k2_new); p.setConstant(True)

# Reaction 1: id_1t80 + id_ijk0 -> id_1t80 + 2*id_bnxy
r1 = model.createReaction()
r1.setId("R1_ijk0_conv_inhib_bnxy")
r1.setReversible(False)
r1.setFast(False)
reactant1_r1 = r1.createReactant()
reactant1_r1.setSpecies("id_ijk0")
reactant1_r1.setStoichiometry(1)
reactant1_r1.setConstant(False)
product1_r1 = r1.createProduct()
product1_r1.setSpecies("id_bnxy")
product1_r1.setStoichiometry(2)
product1_r1.setConstant(False)
kl_r1 = r1.createKineticLaw()
kl_r1.setFormula("k1_eff * id_1t80 * id_ijk0 / (K_m_bnxy + id_bnxy)")

# Reaction 2: id_bnxy -> id_9k8m
r2 = model.createReaction()
r2.setId("R2_bnxy_degradation")
r2.setReversible(False)
r2.setFast(False)
reactant1_r2 = r2.createReactant()
reactant1_r2.setSpecies("id_bnxy")
reactant1_r2.setStoichiometry(1)
reactant1_r2.setConstant(False)
product1_r2 = r2.createProduct()
product1_r2.setSpecies("id_9k8m")
product1_r2.setStoichiometry(1)
product1_r2.setConstant(False)
kl_r2 = r2.createKineticLaw()
kl_r2.setFormula("k2 * id_bnxy")

writer = libsbml.SBMLWriter()
updated_sbml_model_iter7 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_best_model_iter7", updated_sbml_model_iter7)

# Simulate with default initial conditions
sim_doc_default = libsbml.readSBMLFromString(updated_sbml_model_iter7)
sim_model_default = sim_doc_default.getModel()
sim_model_default.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_default.getSpecies("id_bnxy").setInitialConcentration(1.0)
sbml_for_sim_default = writer.writeSBMLToString(sim_doc_default)
sim_data_default = simulate(sbml_for_sim_default)

# Simulate with perturbed initial conditions (id_bnxy=0.1)
sim_doc_perturbed = libsbml.readSBMLFromString(updated_sbml_model_iter7)
sim_model_perturbed = sim_doc_perturbed.getModel()
sim_model_perturbed.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_perturbed.getSpecies("id_bnxy").setInitialConcentration(0.1)
sbml_for_sim_perturbed = writer.writeSBMLToString(sim_doc_perturbed)
sim_data_perturbed = simulate(sbml_for_sim_perturbed)

shared_variables.add("sim_data_default_iter7", sim_data_default)
shared_variables.add("sim_data_perturbed_iter7", sim_data_perturbed)

# Print comparison for default conditions
print("\nComparison for Default Conditions (Exp vs Sim with new params iter7):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")


print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

# Print comparison for perturbed conditions
print("\nComparison for Perturbed Conditions (id_bnxy(0)=0.1) (Exp vs Sim with new params iter7):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")
```
# Observation

## Code Stdout
New trial k1_eff: 15.0
New trial K_m_bnxy: 4.0
New trial k2: 1.216

Comparison for Default Conditions (Exp vs Sim with new params iter7):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 4.933e-01 | 5.692e-01
0.4 | 4.574e-01 | 3.332e-01
1.0 | 4.389e-01 | 6.309e-02
2.0 | 3.975e-01 | 2.595e-03
5.0 | 1.306e-01 | 4.472e-08

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e+00 | 1.000e+00
0.2 | 1.858e+00 | 1.539e+00
0.4 | 1.900e+00 | 1.621e+00
0.6 | 1.891e+00 | 1.515e+00
1.0 | 1.865e+00 | 1.131e+00
2.0 | 1.769e+00 | 3.880e-01
5.0 | 9.687e-01 | 1.031e-02

Comparison for Perturbed Conditions (id_bnxy(0)=0.1) (Exp vs Sim with new params iter7):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 3.200e-01 | 5.194e-01
0.4 | 2.413e-01 | 2.867e-01
1.0 | 1.707e-01 | 4.797e-02
2.0 | 6.978e-02 | 1.748e-03
5.0 | 9.958e-05 | 2.816e-08

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e-01 | 1.000e-01
0.2 | 1.287e+00 | 9.175e-01
0.4 | 1.327e+00 | 1.127e+00
0.6 | 1.268e+00 | 1.107e+00
1.0 | 1.120e+00 | 8.495e-01
2.0 | 6.897e-01 | 2.917e-01
5.0 | 2.260e-02 | 7.734e-03

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_3)
- `shared_variables`: all variables you have saved from the previous iterations. (current_best_model, exp_data_iter0, sim_data_iter1, exp_data_orig_iter4, exp_data_pert_iter4, current_best_model_iter4, sim_data_default_iter4, sim_data_perturbed_iter4, current_best_model_iter5, sim_data_default_iter5, sim_data_perturbed_iter5, current_best_model_iter6, sim_data_default_iter6, sim_data_perturbed_iter6, current_best_model_iter7, sim_data_default_iter7, sim_data_perturbed_iter7)

### 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 parameter set `k1_eff = 15.0`, `K_m_bnxy = 4.0`, `k2 = 1.216` yielded:

**Default condition (bnxy(0)=1):**
- `id_ijk0`: Sim(0.2)=0.569 vs Exp(0.2)=0.493. Better, but still a bit slow. Sim(0.4)=0.333 vs Exp(0.4)=0.457. Now sim is faster than exp after the initial phase. This means the inhibition by `bnxy` might be kicking in too strongly or the shape of `id_ijk0` decay is not quite right.
- `id_bnxy`: Sim peak ~1.62 (t=0.4) vs Exp peak ~1.90. Still too low. Sim(t=5)=0.010 vs Exp(t=5)=0.9687. `id_bnxy` decay is far too rapid in simulation. This `k2=1.216` is too high for the default condition's tail.

**Perturbed condition (bnxy(0)=0.1):**
- `id_ijk0`: Sim(0.2)=0.519 vs Exp(0.2)=0.320. Still significantly too slow.
- `id_bnxy`: Sim peak ~1.13 (t=0.4) vs Exp peak ~1.33. Too low. Sim(t=5)=0.0077 vs Exp(t=5)=0.0226. Decay is too rapid here as well, but closer than default.

The major issue is that `k2` cannot be a single constant to match both tails:
- Default tail needs small `k2` (to keep `id_bnxy` high at t=5, like `0.9687`).
- Perturbed tail needs larger `k2` (to get `id_bnxy` low at t=5, like `0.0226`).

This strongly suggests that the degradation of `id_bnxy` (Reaction R2) is not `k2 * [id_bnxy]`.
It needs to be slower when `[id_bnxy]` is high (default case) and faster when `[id_bnxy]` is low (perturbed case, relative to its concentration). This is the opposite of simple Michaelis-Menten saturation for degradation `Vm*S/(Km+S)`.

Possibilities for R2 (`id_bnxy` degradation):
1.  **Zero-order component + First-order component**: `k_zero + k_first * [id_bnxy]`.
    If `k_zero` is significant, it would make decay faster at low `[id_bnxy]`.
    At high `[id_bnxy]`, `k_first` dominates. If `k_first` is small, decay is slow.
    This could work:
    - Default (high `[bnxy]`): `d[bnxy]/dt ~ -k_first * [bnxy]`. If `k_first` is small, `bnxy` persists.
    - Perturbed (low `[bnxy]`): `d[bnxy]/dt ~ -k_zero`. If `k_zero` is appropriately sized, it can drive `bnxy` down.
    Let's test this idea for R2: `id_bnxy -> id_9k8m` with rate `k2_zero + k2_first * id_bnxy`.

Let's try to estimate `k2_zero` and `k2_first`.
**Perturbed tail (low `[bnxy]`):**
`d[bnxy]/dt ~ -k2_zero` (if `k2_first * [bnxy]` is small).
`id_bnxy` goes from `0.0916` (t=4) to `0.0226` (t=5). Change of `-0.069` in 1s. So `k2_zero ~ 0.069`.
At `t=5`, `[bnxy]=0.0226`. `k2_first * 0.0226` should be small compared to `0.069`.

**Default tail (high `[bnxy]`):**
`d[bnxy]/dt = -(k2_zero + k2_first * [bnxy])`.
We also have production from R1: `2 * Rate_R1`.
`d[bnxy]/dt (exp) = Rate_prod - (k2_zero + k2_first * [bnxy])`.
At t=5 (default): `d[bnxy]/dt ~ -0.4367`. `[bnxy]=0.9687`. `Rate_prod = 0.668` (from previous calculation with `k1_eff=8.9675`, `K_m_bnxy=2.5375`).
If I use the new `k1_eff=15, K_m_bnxy=4`:
`Rate_R1(t=5, default) = 15 * 0.1306 / (4 + 0.9687) = 1.959 / 4.9687 = 0.394`.
`Rate_prod = 2 * 0.394 = 0.788`.
So, `-0.4367 = 0.788 - (k2_zero + k2_first * 0.9687)`.
`k2_zero + k2_first * 0.9687 = 0.788 + 0.4367 = 1.2247`.
Using `k2_zero = 0.069`:
`0.069 + k2_first * 0.9687 = 1.2247`.
`k2_first * 0.9687 = 1.2247 - 0.069 = 1.1557`.
`k2_first = 1.1557 / 0.9687 = 1.193`.

So, proposed parameters for R2: `k2_zero = 0.069`, `k2_first = 1.193`.
Kinetic law for R2: `k2_zero + k2_first * id_bnxy`.

Now for R1 parameters (`k1_eff`, `K_m_bnxy`):
The initial `id_ijk0` decay is still too slow.
Default: Sim(0.2)=0.569 (target 0.493)
Perturbed: Sim(0.2)=0.519 (target 0.320)
The current `k1_eff=15, K_m_bnxy=4` gives initial rates:
`Rate_R1_def(0) = 15 * 1 / (4+1) = 15/5 = 3.0`. (Exp slope ~2.535)
`Rate_R1_pert(0) = 15 * 1 / (4+0.1) = 15/4.1 = 3.65`. (Exp slope ~3.4)
These initial rates are higher than the experimental slopes. This means the inhibition by `id_bnxy` as it rises from t=0 to t=0.2 must be stronger in the simulation than in reality, or the integration over the interval with these rates leads to this.

If `K_m_bnxy` is higher, inhibition is weaker.
Let's try to match `id_ijk0(t=0.2)` more closely. This is hard without optimization.
Let's stick with `k1_eff=15, K_m_bnxy=4` for R1 for now, and see how the new R2 law affects things. The `id_bnxy` dynamics will change, which will feed back into R1.

New parameter set:
R1: `k1_eff = 15.0`, `K_m_bnxy = 4.0`
R2: `k2_zero = 0.069`, `k2_first = 1.193` (New kinetic law: `k2_zero + k2_first * id_bnxy`)

This is a significant model structure change for R2.

## Action

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

# Access experimental data
exp_data_orig = experiment_history['iteration_0']
exp_data_pert = experiment_history['iteration_3'] # bnxy(0)=0.1

# Parameters for R1
param_k1_eff = 15.0
param_K_m_bnxy = 4.0

# Parameters for R2 (new kinetic law)
param_k2_zero = 0.069
param_k2_first = 1.193

print(f"R1 params: k1_eff={param_k1_eff}, K_m_bnxy={param_K_m_bnxy}")
print(f"R2 params: k2_zero={param_k2_zero}, k2_first={param_k2_first}")

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

# Add parameters for R1
p = model.createParameter(); p.setId("k1_eff"); p.setValue(param_k1_eff); p.setConstant(True)
p = model.createParameter(); p.setId("K_m_bnxy"); p.setValue(param_K_m_bnxy); p.setConstant(True)

# Add parameters for R2
p = model.createParameter(); p.setId("k2_zero"); p.setValue(param_k2_zero); p.setConstant(True)
p = model.createParameter(); p.setId("k2_first"); p.setValue(param_k2_first); p.setConstant(True)


# Reaction 1: id_1t80 + id_ijk0 -> id_1t80 + 2*id_bnxy
r1 = model.createReaction()
r1.setId("R1_ijk0_conv_inhib_bnxy")
r1.setReversible(False)
r1.setFast(False)
reactant1_r1 = r1.createReactant()
reactant1_r1.setSpecies("id_ijk0")
reactant1_r1.setStoichiometry(1)
reactant1_r1.setConstant(False)
product1_r1 = r1.createProduct()
product1_r1.setSpecies("id_bnxy")
product1_r1.setStoichiometry(2)
product1_r1.setConstant(False)
kl_r1 = r1.createKineticLaw()
kl_r1.setFormula("k1_eff * id_1t80 * id_ijk0 / (K_m_bnxy + id_bnxy)")

# Reaction 2: id_bnxy -> id_9k8m (New kinetic law)
r2 = model.createReaction()
r2.setId("R2_bnxy_degradation_mixed_order")
r2.setReversible(False)
r2.setFast(False)
reactant1_r2 = r2.createReactant()
reactant1_r2.setSpecies("id_bnxy")
reactant1_r2.setStoichiometry(1) # Stoichiometry of reactant consumption
reactant1_r2.setConstant(False)
product1_r2 = r2.createProduct()
product1_r2.setSpecies("id_9k8m")
product1_r2.setStoichiometry(1) # Stoichiometry of product formation
product1_r2.setConstant(False)
kl_r2 = r2.createKineticLaw()
kl_r2.setFormula("k2_zero + k2_first * id_bnxy") # Check: this rate is per unit volume

writer = libsbml.SBMLWriter()
updated_sbml_model_iter8 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_best_model_iter8", updated_sbml_model_iter8)

# Simulate with default initial conditions
sim_doc_default = libsbml.readSBMLFromString(updated_sbml_model_iter8)
sim_model_default = sim_doc_default.getModel()
sim_model_default.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_default.getSpecies("id_bnxy").setInitialConcentration(1.0)
sbml_for_sim_default = writer.writeSBMLToString(sim_doc_default)
sim_data_default = simulate(sbml_for_sim_default)

# Simulate with perturbed initial conditions (id_bnxy=0.1)
sim_doc_perturbed = libsbml.readSBMLFromString(updated_sbml_model_iter8)
sim_model_perturbed = sim_doc_perturbed.getModel()
sim_model_perturbed.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_perturbed.getSpecies("id_bnxy").setInitialConcentration(0.1)
sbml_for_sim_perturbed = writer.writeSBMLToString(sim_doc_perturbed)
sim_data_perturbed = simulate(sbml_for_sim_perturbed)

shared_variables.add("sim_data_default_iter8", sim_data_default)
shared_variables.add("sim_data_perturbed_iter8", sim_data_perturbed)

# Print comparison for default conditions
print("\nComparison for Default Conditions (Exp vs Sim with new R2 law):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]: # Added more points for bnxy
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

# Print comparison for perturbed conditions
print("\nComparison for Perturbed Conditions (id_bnxy(0)=0.1) (Exp vs Sim with new R2 law):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]: # Added more points for bnxy
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")
```
# Observation

## Code Stdout
R1 params: k1_eff=15.0, K_m_bnxy=4.0
R2 params: k2_zero=0.069, k2_first=1.193

Comparison for Default Conditions (Exp vs Sim with new R2 law):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 4.933e-01 | 5.690e-01
0.4 | 4.574e-01 | 3.328e-01
1.0 | 4.389e-01 | 6.270e-02
2.0 | 3.975e-01 | 2.525e-03
5.0 | 1.306e-01 | 3.790e-08

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e+00 | 1.000e+00
0.2 | 1.858e+00 | 1.532e+00
0.4 | 1.900e+00 | 1.610e+00
0.6 | 1.891e+00 | 1.501e+00
1.0 | 1.865e+00 | 1.110e+00
2.0 | 1.769e+00 | 3.496e-01
5.0 | 9.687e-01 | -4.626e-02
7.0 | 2.015e-01 | -5.677e-02

Comparison for Perturbed Conditions (id_bnxy(0)=0.1) (Exp vs Sim with new R2 law):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 3.200e-01 | 5.190e-01
0.4 | 2.413e-01 | 2.861e-01
1.0 | 1.707e-01 | 4.751e-02
2.0 | 6.978e-02 | 1.686e-03
5.0 | 9.958e-05 | 2.352e-08

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e-01 | 1.000e-01
0.2 | 1.287e+00 | 9.083e-01
0.4 | 1.327e+00 | 1.112e+00
0.6 | 1.268e+00 | 1.087e+00
1.0 | 1.120e+00 | 8.225e-01
2.0 | 6.897e-01 | 2.492e-01
5.0 | 2.260e-02 | -4.913e-02
7.0 | 1.161e-03 | -5.704e-02

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_3)
- `shared_variables`: all variables you have saved from the previous iterations. (current_best_model, exp_data_iter0, sim_data_iter1, exp_data_orig_iter4, exp_data_pert_iter4, current_best_model_iter4, sim_data_default_iter4, sim_data_perturbed_iter4, current_best_model_iter5, sim_data_default_iter5, sim_data_perturbed_iter5, current_best_model_iter6, sim_data_default_iter6, sim_data_perturbed_iter6, current_best_model_iter7, sim_data_default_iter7, sim_data_perturbed_iter7, current_best_model_iter8, sim_data_default_iter8, sim_data_perturbed_iter8)

### 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

The simulation results with the new R2 law (`k2_zero + k2_first * id_bnxy`) are problematic: `id_bnxy` becomes negative. This indicates that the rate law `k2_zero + k2_first * id_bnxy` can lead to a net production if `id_bnxy` is very low and the production term from R1 is smaller than `k2_zero`. More importantly, it means that the degradation rate itself does not go to zero when `id_bnxy` goes to zero, which is physically unrealistic for a degradation pathway unless `id_bnxy` is a precursor to something that is then actively removed. The `k2_zero` term implies a constant rate of removal, irrespective of `id_bnxy` concentration. This is only valid if `id_bnxy` is always present at a level that can sustain this rate. When `id_bnxy` approaches zero, a constant removal term `k2_zero` will drive it negative.

The kinetic law for degradation must ensure the rate becomes zero if the species concentration is zero.
A common form for mixed-order kinetics that does this is Michaelis-Menten: `Vm * S / (Km + S)`.
- At low S: `(Vm/Km) * S` (first order).
- At high S: `Vm` (zero order).

This means:
- If `[bnxy]` is high (default tail), degradation rate `~Vm`. This would be the "slower" phase if `Vm` is not too large.
- If `[bnxy]` is low (perturbed tail), degradation rate `~ (Vm/Km) * [bnxy]`. This is first order.

Let's re-evaluate the tails:
- Default tail (high `[bnxy]`): `id_bnxy` from 0.9687 (t=5) to 0.2015 (t=7). `d[bnxy]/dt` (ignoring R1 for a moment) `~ (0.2015-0.9687)/2 = -0.7672/2 = -0.3836`. If this is `~Vm_deg`, then `Vm_deg = 0.3836`.
- Perturbed tail (low `[bnxy]`): `id_bnxy` from 0.0916 (t=4) to 0.0226 (t=5). `d[bnxy]/dt` (ignoring R1) `~ (0.0226-0.0916)/1 = -0.069`. If this is `~ (Vm_deg/Km_deg) * [bnxy_avg]`. Average `[bnxy]` is `(0.0916+0.0226)/2 = 0.0571`.
  So, `-0.069 = - (Vm_deg/Km_deg) * 0.0571`.
  `(Vm_deg/Km_deg) = 0.069 / 0.0571 = 1.208`.
  If `Vm_deg = 0.3836`, then `Km_deg = Vm_deg / 1.208 = 0.3836 / 1.208 = 0.3175`.

So, for R2: `id_bnxy -> id_9k8m`, rate `Vm_deg * id_bnxy / (Km_deg + id_bnxy)`
with `Vm_deg = 0.3836` and `Km_deg = 0.3175`.

Let's check the R1 contribution again with these new R2 parameters.
**Default, t=5:** `[ijk0]=0.1306`, `[bnxy]=0.9687`.
R1 params: `k1_eff=15`, `K_m_bnxy=4`.
`Rate_R1_prod = 2 * 15 * 0.1306 / (4 + 0.9687) = 2 * 0.394 = 0.788`.
Rate R2_deg = `0.3836 * 0.9687 / (0.3175 + 0.9687) = 0.3836 * 0.9687 / 1.2862 = 0.3716 / 1.2862 = 0.289`.
Net `d[bnxy]/dt = 0.788 - 0.289 = 0.499`.
Experimental `d[bnxy]/dt` (t=5 to t=7 average) was `-0.3836`.
This is a mismatch. The production from R1 is too high, or degradation R2 is too low with these Vm/Km.

The problem is that `Vm_deg` was estimated assuming R1 is zero.
Let `R2_actual_rate(t)` be the true rate of degradation.
Default, t=5: `d[bnxy]/dt_exp = Rate_R1_prod - R2_actual_rate(t=5)`.
`-0.3836 = 0.788 - R2_actual_rate(t=5)`.
`R2_actual_rate(t=5) = 0.788 + 0.3836 = 1.1716`.
This is the degradation rate needed at `[bnxy]=0.9687`.
So, `1.1716 = Vm_deg * 0.9687 / (Km_deg + 0.9687)`. (Eq A)

Perturbed, t=4.5 (midpoint of 4-5): `[ijk0]` is very low, assume `Rate_R1_prod ~ 0`.
`[bnxy]` avg is `0.0571`. `d[bnxy]/dt_exp ~ -0.069`.
So, `-0.069 = 0 - R2_actual_rate(t=4.5)`.
`R2_actual_rate(t=4.5) = 0.069`.
This is the degradation rate needed at `[bnxy]=0.0571`.
So, `0.069 = Vm_deg * 0.0571 / (Km_deg + 0.0571)`. (Eq B)

We have two equations for `Vm_deg` and `Km_deg`.
From B: `0.069 * (Km_deg + 0.0571) = Vm_deg * 0.0571`
`Vm_deg = (0.069/0.0571) * (Km_deg + 0.0571) = 1.2084 * (Km_deg + 0.0571)`.
Substitute into A:
`1.1716 = 1.2084 * (Km_deg + 0.0571) * 0.9687 / (Km_deg + 0.9687)`
`1.1716 * (Km_deg + 0.9687) = 1.2084 * 0.9687 * (Km_deg + 0.0571)`
`1.1716 * Km_deg + 1.1716 * 0.9687 = 1.1705 * (Km_deg + 0.0571)`
`1.1716 * Km_deg + 1.1349 = 1.1705 * Km_deg + 1.1705 * 0.0571`
`1.1716 * Km_deg + 1.1349 = 1.1705 * Km_deg + 0.06684`
`(1.1716 - 1.1705) * Km_deg = 0.06684 - 1.1349`
`0.0011 * Km_deg = -1.06806`.
This gives a negative `Km_deg`, which is not possible.

This means my assumption that R1 production is `0.788` at t=5 (default) using `k1_eff=15, K_m_bnxy=4` might be too high, or the Michaelis-Menten form for R2 is not fitting these two points simultaneously.
The `id_ijk0` values are experimental. The `k1_eff, K_m_bnxy` are model parameters.
If `id_ijk0` (sim) at t=5 is much lower than experimental `0.1306` (which it was in iter 7: `4.4e-8`), then `Rate_R1_prod` would be near zero.
Let's re-evaluate `Vm_deg, Km_deg` assuming `Rate_R1_prod` is negligible at late times for *both* experiments.
This brings back:
`Vm_deg = 0.3836` (from default tail, `d[bnxy]/dt ~ -Vm if [bnxy] >> Km`)
`(Vm_deg/Km_deg) = 1.208` (from perturbed tail, `d[bnxy]/dt ~ -(Vm/Km)[bnxy] if [bnxy] << Km`)
`Km_deg = Vm_deg / 1.208 = 0.3836 / 1.208 = 0.3175`.
This assumes the high `[bnxy]` in default tail is saturating (zero order) and low `[bnxy]` in perturbed tail is linear (first order).
Is `0.9687 >> 0.3175`? Yes, by a factor of 3. So, somewhat saturating.
Is `0.0571 << 0.3175`? Yes, by a factor of 5.5. So, somewhat linear.
This set of `Vm_deg=0.3836, Km_deg=0.3175` seems like a reasonable starting point for Michaelis-Menten R2.

Now, the R1 parameters: `k1_eff`, `K_m_bnxy`.
The initial `id_ijk0` decay is still too slow with `k1_eff=15, K_m_bnxy=4`.
Default: Sim(0.2)=0.569 (target 0.493)
Perturbed: Sim(0.2)=0.519 (target 0.320)
To make decay faster, especially for perturbed, `k1_eff` needs to be higher, or `K_m_bnxy` needs to be lower (to make `1/(K_m+[bnxy])` larger when `[bnxy]` is small).
Let's try `k1_eff = 25` and `K_m_bnxy = 3.0`.
Initial rates:
`Rate_R1_def(0) = 25 / (3+1) = 25/4 = 6.25`. (Exp slope ~2.5) -> Too high.
`Rate_R1_pert(0) = 25 / (3+0.1) = 25/3.1 = 8.06`. (Exp slope ~3.4) -> Too high.

The initial slope fitting `K_m_bnxy = 2.539` and `k1_eff = 8.9675` was based on `d[ijk0]/dt`.
Let's use these R1 parameters and the new R2 (Michaelis-Menten).
R1: `k1_eff = 8.9675`, `K_m_bnxy = 2.539`
R2: `Vm_deg = 0.3836`, `Km_deg = 0.3175`. Law: `Vm_deg * id_bnxy / (Km_deg + id_bnxy)`

This combination should hopefully fix the `id_bnxy` tails and negative values, and R1 is based on initial `id_ijk0` slopes.

## Action

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

# Access experimental data
exp_data_orig = experiment_history['iteration_0']
exp_data_pert = experiment_history['iteration_3'] # bnxy(0)=0.1

# Parameters for R1 (from initial slope fitting)
param_k1_eff = 8.9675
param_K_m_bnxy = 2.539

# Parameters for R2 (Michaelis-Menten degradation)
param_Vm_deg = 0.3836
param_Km_deg = 0.3175

print(f"R1 params: k1_eff={param_k1_eff}, K_m_bnxy={param_K_m_bnxy}")
print(f"R2 params (MM): Vm_deg={param_Vm_deg}, Km_deg={param_Km_deg}")

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

# Add parameters for R1
p = model.createParameter(); p.setId("k1_eff"); p.setValue(param_k1_eff); p.setConstant(True)
p = model.createParameter(); p.setId("K_m_bnxy"); p.setValue(param_K_m_bnxy); p.setConstant(True)

# Add parameters for R2
p = model.createParameter(); p.setId("Vm_deg"); p.setValue(param_Vm_deg); p.setConstant(True)
p = model.createParameter(); p.setId("Km_deg"); p.setValue(param_Km_deg); p.setConstant(True)

# Reaction 1: id_1t80 + id_ijk0 -> id_1t80 + 2*id_bnxy
r1 = model.createReaction()
r1.setId("R1_ijk0_conv_inhib_bnxy")
r1.setReversible(False)
r1.setFast(False)
reactant1_r1 = r1.createReactant()
reactant1_r1.setSpecies("id_ijk0")
reactant1_r1.setStoichiometry(1)
reactant1_r1.setConstant(False)
product1_r1 = r1.createProduct()
product1_r1.setSpecies("id_bnxy")
product1_r1.setStoichiometry(2)
product1_r1.setConstant(False)
kl_r1 = r1.createKineticLaw()
kl_r1.setFormula("k1_eff * id_1t80 * id_ijk0 / (K_m_bnxy + id_bnxy)")

# Reaction 2: id_bnxy -> id_9k8m (Michaelis-Menten kinetic law)
r2 = model.createReaction()
r2.setId("R2_bnxy_degradation_MM")
r2.setReversible(False)
r2.setFast(False)
reactant1_r2 = r2.createReactant()
reactant1_r2.setSpecies("id_bnxy")
reactant1_r2.setStoichiometry(1)
reactant1_r2.setConstant(False)
product1_r2 = r2.createProduct()
product1_r2.setSpecies("id_9k8m")
product1_r2.setStoichiometry(1)
product1_r2.setConstant(False)
kl_r2 = r2.createKineticLaw()
kl_r2.setFormula("Vm_deg * id_bnxy / (Km_deg + id_bnxy)")

writer = libsbml.SBMLWriter()
updated_sbml_model_iter9 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_best_model_iter9", updated_sbml_model_iter9)

# Simulate with default initial conditions
sim_doc_default = libsbml.readSBMLFromString(updated_sbml_model_iter9)
sim_model_default = sim_doc_default.getModel()
sim_model_default.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_default.getSpecies("id_bnxy").setInitialConcentration(1.0)
sbml_for_sim_default = writer.writeSBMLToString(sim_doc_default)
sim_data_default = simulate(sbml_for_sim_default)

# Simulate with perturbed initial conditions (id_bnxy=0.1)
sim_doc_perturbed = libsbml.readSBMLFromString(updated_sbml_model_iter9)
sim_model_perturbed = sim_doc_perturbed.getModel()
sim_model_perturbed.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_perturbed.getSpecies("id_bnxy").setInitialConcentration(0.1)
sbml_for_sim_perturbed = writer.writeSBMLToString(sim_doc_perturbed)
sim_data_perturbed = simulate(sbml_for_sim_perturbed)

shared_variables.add("sim_data_default_iter9", sim_data_default)
shared_variables.add("sim_data_perturbed_iter9", sim_data_perturbed)

# Print comparison for default conditions
print("\nComparison for Default Conditions (Exp vs Sim with MM R2):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

# Print comparison for perturbed conditions
print("\nComparison for Perturbed Conditions (id_bnxy(0)=0.1) (Exp vs Sim with MM R2):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")
```
# Observation

## Code Stdout
R1 params: k1_eff=8.9675, K_m_bnxy=2.539
R2 params (MM): Vm_deg=0.3836, Km_deg=0.3175

Comparison for Default Conditions (Exp vs Sim with MM R2):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 4.933e-01 | 6.318e-01
0.4 | 4.574e-01 | 4.205e-01
1.0 | 4.389e-01 | 1.373e-01
2.0 | 3.975e-01 | 2.213e-02
5.0 | 1.306e-01 | 4.566e-05

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e+00 | 1.000e+00
0.2 | 1.858e+00 | 1.674e+00
0.4 | 1.900e+00 | 2.031e+00
0.6 | 1.891e+00 | 2.232e+00
1.0 | 1.865e+00 | 2.396e+00
2.0 | 1.769e+00 | 2.288e+00
5.0 | 9.687e-01 | 1.354e+00
7.0 | 2.015e-01 | 7.675e-01

Comparison for Perturbed Conditions (id_bnxy(0)=0.1) (Exp vs Sim with MM R2):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 3.200e-01 | 5.608e-01
0.4 | 2.413e-01 | 3.442e-01
1.0 | 1.707e-01 | 9.139e-02
2.0 | 6.978e-02 | 1.029e-02
5.0 | 9.958e-05 | 5.288e-06

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e-01 | 1.000e-01
0.2 | 1.287e+00 | 9.313e-01
0.4 | 1.327e+00 | 1.304e+00
0.6 | 1.268e+00 | 1.493e+00
1.0 | 1.120e+00 | 1.620e+00
2.0 | 6.897e-01 | 1.463e+00
5.0 | 2.260e-02 | 6.137e-01
7.0 | 1.161e-03 | 2.009e-01

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_3)
- `shared_variables`: all variables you have saved from the previous iterations. (current_best_model, exp_data_iter0, sim_data_iter1, exp_data_orig_iter4, exp_data_pert_iter4, current_best_model_iter4, sim_data_default_iter4, sim_data_perturbed_iter4, current_best_model_iter5, sim_data_default_iter5, sim_data_perturbed_iter5, current_best_model_iter6, sim_data_default_iter6, sim_data_perturbed_iter6, current_best_model_iter7, sim_data_default_iter7, sim_data_perturbed_iter7, current_best_model_iter8, sim_data_default_iter8, sim_data_perturbed_iter8, current_best_model_iter9, sim_data_default_iter9, sim_data_perturbed_iter9)

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


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



# Iteration 10
## Thoughts

The Michaelis-Menten degradation for R2 (`Vm_deg * id_bnxy / (Km_deg + id_bnxy)`) with `Vm_deg=0.3836, Km_deg=0.3175` and R1 (`k1_eff * id_ijk0 / (K_m_bnxy + id_bnxy)`) with `k1_eff=8.9675, K_m_bnxy=2.539` has produced interesting results:

**`id_ijk0` dynamics:**
- Default: Sim(0.2)=0.632 vs Exp(0.2)=0.493. Still too slow.
- Perturbed: Sim(0.2)=0.561 vs Exp(0.2)=0.320. Still too slow.
The `id_ijk0` decay remains a problem. The initial decay is not captured well. The parameters `k1_eff, K_m_bnxy` were derived from initial slopes, but the integrated behavior over 0.2s is off. This suggests the inhibition by `id_bnxy` (which is rising) is too effective in the simulation during this early phase.

**`id_bnxy` dynamics:**
- Default:
    - Peak: Sim peak ~2.40 (t=1.0) vs Exp peak ~1.90 (t=0.4-0.6). Simulated peak is too high and too late.
    - Tail: Sim(t=5)=1.35 vs Exp(t=5)=0.9687. Sim(t=7)=0.76 vs Exp(t=7)=0.20. The simulated decay is too slow in the tail.
- Perturbed:
    - Peak: Sim peak ~1.62 (t=1.0) vs Exp peak ~1.33 (t=0.4). Simulated peak is too high and too late.
    - Tail: Sim(t=5)=0.61 vs Exp(t=5)=0.0226. Sim(t=7)=0.20 vs Exp(t=7)=0.001. Simulated decay is far too slow.

The Michaelis-Menten R2 parameters (`Vm_deg=0.3836, Km_deg=0.3175`) were estimated assuming R1 production is negligible in the tails.
- For default tail (high `[bnxy]`): `R2_rate ~ Vm_deg = 0.3836`.
- For perturbed tail (low `[bnxy]`): `R2_rate ~ (Vm_deg/Km_deg)*[bnxy] = (0.3836/0.3175)*[bnxy] = 1.208*[bnxy]`.

The simulated `id_bnxy` is now too high and persists too long in both cases. This means the degradation (R2) is too slow overall with these MM parameters.
This implies `Vm_deg` should be higher, and/or `Km_deg` should be adjusted.

If `Vm_deg` is increased, the max degradation rate increases.
If `Km_deg` is increased, it takes higher `[bnxy]` to reach saturation, making it behave more like first-order for longer.

Let's re-evaluate the R2 parameters based on the latest simulation's `id_ijk0` values, which are more realistic for R1 production during the tail.
Simulated `id_ijk0` at t=5 (default) is `4.5e-5` (very low). So R1 prod is negligible.
Simulated `id_ijk0` at t=5 (perturbed) is `5.2e-6` (very low). So R1 prod is negligible.
My previous assumption for negligible R1 in tails for MM parameter estimation was okay based on *simulated* `ijk0`.

The issue is that the experimental `id_ijk0(t=5, default)` is `0.1306`, which is *not* negligible.
This means the R1 parameters (`k1_eff, K_m_bnxy`) are causing `id_ijk0` to deplete too quickly in the simulation compared to reality.

This points back to the R1 term: `k1_eff * id_ijk0 / (K_m_bnxy + id_bnxy)`.
If simulated `id_ijk0` goes to zero too fast:
- `k1_eff` is too high relative to the inhibition.
- Or, `K_m_bnxy` is too low, making the inhibition `(K_m_bnxy + id_bnxy)` term not grow fast enough to slow down `id_ijk0` decay appropriately.

Let's reconsider the `id_ijk0` decay profile.
Original experiment: `id_ijk0` starts at 1.0.
t=0.2, 0.493 (-0.507)
t=0.4, 0.457 (-0.036) -> decay slows significantly
t=1.0, 0.439 (-0.018 over 0.6s) -> very slow
t=2.0, 0.397 (-0.042 over 1.0s) -> still slow
t=5.0, 0.131 (-0.266 over 3.0s) -> speeds up again?

This "slow-down then speed-up" of `id_ijk0` decay is complex.
The inhibition term `1/(K_m_bnxy + id_bnxy)`:
`id_bnxy` goes low -> high -> low.
So inhibition factor `1/(K_m+bnxy)` goes high -> low -> high.
This means `id_ijk0` decay rate should be fast -> slow -> fast. This matches the general experimental trend.

The problem is quantitative fitting.
The simulated `id_ijk0` values at t=0.2 are too high (decay too slow).
Default: Sim(0.2)=0.632 vs Exp(0.2)=0.493.
Perturbed: Sim(0.2)=0.561 vs Exp(0.2)=0.320.

To make initial `id_ijk0` decay faster:
- Increase `k1_eff`.
- Decrease `K_m_bnxy` (this makes the denominator smaller when `id_bnxy` is small, e.g. at t=0 or for perturbed case).

Let's try to adjust R1 parameters first, then revisit R2 if `id_bnxy` is still off.
Target: `id_ijk0(0.2, default) ~ 0.493`, `id_ijk0(0.2, perturbed) ~ 0.320`.
Current R1: `k1_eff=8.9675, K_m_bnxy=2.539`.
Sim default `id_ijk0(0)`: rate `8.9675/(2.539+1) = 2.535`.
Sim perturbed `id_ijk0(0)`: rate `8.9675/(2.539+0.1) = 3.40`.

If `K_m_bnxy` is reduced, say to `1.0`.
To keep `Rate_def(0) ~ 2.535`, `k1_eff = 2.535 * (1.0+1) = 5.07`.
Then `Rate_pert(0) = 5.07 / (1.0+0.1) = 5.07 / 1.1 = 4.61`. (Exp rate ~3.4). This is now too high for perturbed.
This shows that `K_m_bnxy` cannot be arbitrarily changed without affecting the ratio of initial rates. The value `K_m_bnxy=2.539` was derived from this ratio.

Perhaps the stoichiometry `S=2` for `id_ijk0 -> S * id_bnxy` is incorrect.
If `S` is higher, `id_bnxy` rises faster. This would inhibit R1 sooner/stronger, making `id_ijk0` decay slower. This is the opposite of what's needed for `id_ijk0`.
If `S` is lower (e.g., 1), `id_bnxy` rises slower. R1 inhibition is less, so `id_ijk0` decays faster. This might help `id_ijk0`.
But if `S=1`, the sum `[ijk0]+[bnxy]` would not increase, it would be constant (ignoring R2). Experimentally, the sum increases.
Original: `[ijk0]+[bnxy]` from 2.0 to ~2.36. Increase of 0.36.
If `S=1.5`: `d(sum)/dt = -Rate1 + 1.5*Rate1 - Rate2 = 0.5*Rate1 - Rate2`. This could still give an increase.

Let's try `S=1.5` and re-evaluate R1 parameters.
The `d[ijk0]/dt` equations don't change. So `k1_eff=8.9675, K_m_bnxy=2.539` are still the best fit for initial `id_ijk0` slopes.
The `id_bnxy` equation changes: `d[bnxy]/dt = 1.5 * Rate_R1 - Rate_R2`.
This will affect `id_bnxy` levels, which feeds back to R1.

Let's try:
R1: `k1_eff = 8.9675`, `K_m_bnxy = 2.539`. Stoichiometry `id_ijk0 -> 1.5 id_bnxy`.
R2: `Vm_deg = 0.3836`, `Km_deg = 0.3175`. (Michaelis-Menten).

This is a model structure change (stoichiometry).

## Action

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

# Access experimental data
exp_data_orig = experiment_history['iteration_0']
exp_data_pert = experiment_history['iteration_3'] # bnxy(0)=0.1

# Parameters for R1
param_k1_eff = 8.9675
param_K_m_bnxy = 2.539
# New stoichiometry for id_bnxy production in R1
stoich_S_bnxy_R1 = 1.5

# Parameters for R2 (Michaelis-Menten degradation)
param_Vm_deg = 0.3836
param_Km_deg = 0.3175

print(f"R1 params: k1_eff={param_k1_eff}, K_m_bnxy={param_K_m_bnxy}, Stoich_bnxy={stoich_S_bnxy_R1}")
print(f"R2 params (MM): Vm_deg={param_Vm_deg}, Km_deg={param_Km_deg}")

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

# Add parameters for R1
p = model.createParameter(); p.setId("k1_eff"); p.setValue(param_k1_eff); p.setConstant(True)
p = model.createParameter(); p.setId("K_m_bnxy"); p.setValue(param_K_m_bnxy); p.setConstant(True)

# Add parameters for R2
p = model.createParameter(); p.setId("Vm_deg"); p.setValue(param_Vm_deg); p.setConstant(True)
p = model.createParameter(); p.setId("Km_deg"); p.setValue(param_Km_deg); p.setConstant(True)

# Reaction 1: id_1t80 + id_ijk0 -> id_1t80 + S*id_bnxy
r1 = model.createReaction()
r1.setId("R1_ijk0_conv_inhib_bnxy")
r1.setReversible(False)
r1.setFast(False)
reactant1_r1 = r1.createReactant()
reactant1_r1.setSpecies("id_ijk0")
reactant1_r1.setStoichiometry(1) # Consumes 1 id_ijk0
reactant1_r1.setConstant(False)
product1_r1 = r1.createProduct()
product1_r1.setSpecies("id_bnxy")
product1_r1.setStoichiometry(stoich_S_bnxy_R1) # Produces S id_bnxy
product1_r1.setConstant(False)
kl_r1 = r1.createKineticLaw()
kl_r1.setFormula("k1_eff * id_1t80 * id_ijk0 / (K_m_bnxy + id_bnxy)")

# Reaction 2: id_bnxy -> id_9k8m (Michaelis-Menten kinetic law)
r2 = model.createReaction()
r2.setId("R2_bnxy_degradation_MM")
r2.setReversible(False)
r2.setFast(False)
reactant1_r2 = r2.createReactant()
reactant1_r2.setSpecies("id_bnxy")
reactant1_r2.setStoichiometry(1)
reactant1_r2.setConstant(False)
product1_r2 = r2.createProduct()
product1_r2.setSpecies("id_9k8m")
product1_r2.setStoichiometry(1)
product1_r2.setConstant(False)
kl_r2 = r2.createKineticLaw()
kl_r2.setFormula("Vm_deg * id_bnxy / (Km_deg + id_bnxy)")

writer = libsbml.SBMLWriter()
updated_sbml_model_iter10 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_best_model_iter10", updated_sbml_model_iter10)

# Simulate with default initial conditions
sim_doc_default = libsbml.readSBMLFromString(updated_sbml_model_iter10)
sim_model_default = sim_doc_default.getModel()
sim_model_default.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_default.getSpecies("id_bnxy").setInitialConcentration(1.0)
sbml_for_sim_default = writer.writeSBMLToString(sim_doc_default)
sim_data_default = simulate(sbml_for_sim_default)

# Simulate with perturbed initial conditions (id_bnxy=0.1)
sim_doc_perturbed = libsbml.readSBMLFromString(updated_sbml_model_iter10)
sim_model_perturbed = sim_doc_perturbed.getModel()
sim_model_perturbed.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_perturbed.getSpecies("id_bnxy").setInitialConcentration(0.1)
sbml_for_sim_perturbed = writer.writeSBMLToString(sim_doc_perturbed)
sim_data_perturbed = simulate(sbml_for_sim_perturbed)

shared_variables.add("sim_data_default_iter10", sim_data_default)
shared_variables.add("sim_data_perturbed_iter10", sim_data_perturbed)

# Print comparison for default conditions
print("\nComparison for Default Conditions (Exp vs Sim with S=1.5 and MM R2):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

# Print comparison for perturbed conditions
print("\nComparison for Perturbed Conditions (id_bnxy(0)=0.1) (Exp vs Sim with S=1.5 and MM R2):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")
```
# Observation

## Code Stdout
R1 params: k1_eff=8.9675, K_m_bnxy=2.539, Stoich_bnxy=1.5
R2 params (MM): Vm_deg=0.3836, Km_deg=0.3175

Comparison for Default Conditions (Exp vs Sim with S=1.5 and MM R2):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 4.933e-01 | 6.248e-01
0.4 | 4.574e-01 | 4.070e-01
1.0 | 4.389e-01 | 1.218e-01
2.0 | 3.975e-01 | 1.639e-02
5.0 | 1.306e-01 | 1.627e-05

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e+00 | 1.000e+00
0.2 | 1.858e+00 | 1.502e+00
0.4 | 1.900e+00 | 1.764e+00
0.6 | 1.891e+00 | 1.904e+00
1.0 | 1.865e+00 | 1.994e+00
2.0 | 1.769e+00 | 1.823e+00
5.0 | 9.687e-01 | 9.188e-01
7.0 | 2.015e-01 | 4.088e-01

Comparison for Perturbed Conditions (id_bnxy(0)=0.1) (Exp vs Sim with S=1.5 and MM R2):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 3.200e-01 | 5.491e-01
0.4 | 2.413e-01 | 3.254e-01
1.0 | 1.707e-01 | 7.580e-02
2.0 | 6.978e-02 | 6.558e-03
5.0 | 9.958e-05 | 1.268e-06

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e-01 | 1.000e-01
0.2 | 1.287e+00 | 7.327e-01
0.4 | 1.327e+00 | 1.012e+00
0.6 | 1.268e+00 | 1.143e+00
1.0 | 1.120e+00 | 1.206e+00
2.0 | 6.897e-01 | 1.011e+00
5.0 | 2.260e-02 | 2.803e-01
7.0 | 1.161e-03 | 5.144e-02

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_3)
- `shared_variables`: all variables you have saved from the previous iterations. (current_best_model, exp_data_iter0, sim_data_iter1, exp_data_orig_iter4, exp_data_pert_iter4, current_best_model_iter4, sim_data_default_iter4, sim_data_perturbed_iter4, current_best_model_iter5, sim_data_default_iter5, sim_data_perturbed_iter5, current_best_model_iter6, sim_data_default_iter6, sim_data_perturbed_iter6, current_best_model_iter7, sim_data_default_iter7, sim_data_perturbed_iter7, current_best_model_iter8, sim_data_default_iter8, sim_data_perturbed_iter8, current_best_model_iter9, sim_data_default_iter9, sim_data_perturbed_iter9, current_best_model_iter10, sim_data_default_iter10, sim_data_perturbed_iter10)

### 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

Changing R1 stoichiometry from `S=2` to `S=1.5` (`id_ijk0 -> 1.5 id_bnxy`) had the following effects, with R1 params (`k1_eff=8.9675, K_m_bnxy=2.539`) and R2 params (MM: `Vm_deg=0.3836, Km_deg=0.3175`):

**`id_ijk0` dynamics:**
- Default: Sim(0.2)=0.625 vs Exp(0.2)=0.493. Still too slow.
- Perturbed: Sim(0.2)=0.549 vs Exp(0.2)=0.320. Still too slow.
Changing `S` to 1.5 made `id_bnxy` rise slightly slower, which means R1 inhibition was slightly less, leading to slightly faster `id_ijk0` decay compared to S=2 (e.g. default Sim(0.2) was 0.632 with S=2, now 0.625 with S=1.5). This is a tiny improvement in the right direction for `id_ijk0`, but not nearly enough.

**`id_bnxy` dynamics:**
- Default:
    - Peak: Sim peak ~1.99 (t=1.0) vs Exp peak ~1.90 (t=0.4-0.6). Peak height is good, but too late.
    - Tail: Sim(t=5)=0.919 vs Exp(t=5)=0.9687. Good match! Sim(t=7)=0.409 vs Exp(t=7)=0.20. Simulated decay now a bit too slow in the far tail.
- Perturbed:
    - Peak: Sim peak ~1.21 (t=1.0) vs Exp peak ~1.33 (t=0.4). Peak height a bit low, and too late.
    - Tail: Sim(t=5)=0.280 vs Exp(t=5)=0.0226. Far too slow decay. Sim(t=7)=0.051 vs Exp(t=7)=0.001. Far too slow.

The change to `S=1.5` significantly improved the `id_bnxy` level at t=5 for the default case. However, it made the perturbed case tail much worse (too slow decay).
This indicates that the R2 Michaelis-Menten parameters (`Vm_deg=0.3836, Km_deg=0.3175`) are not robust to changes in `id_bnxy` production rate from R1. They were estimated assuming R1 production was negligible in the tails. With `S=1.5`, the `id_bnxy` levels are different, so R1 production might not be negligible, or the MM parameters themselves need to be different.

The primary remaining issue is the slow initial decay of `id_ijk0`.
Sim `id_ijk0(0.2)` needs to be much lower.
This requires the *average* rate of R1 over [0, 0.2] to be higher.
The rate is `k1_eff * [id_ijk0] / (K_m_bnxy + [id_bnxy])`.
At t=0, `[id_bnxy]` is fixed by initial conditions. As `t` increases, `[id_bnxy]` increases, inhibiting R1.
If the simulated `id_ijk0` decay is too slow, it means either:
1.  `k1_eff` is too small.
2.  The term `(K_m_bnxy + [id_bnxy])` grows too quickly or is too large, over-inhibiting R1.
    - If `K_m_bnxy` is larger, inhibition is weaker (denominator larger). This would speed up `id_ijk0` decay if `[id_bnxy]` is not zero.
    - If `[id_bnxy]` (simulated) rises faster than reality, it over-inhibits.

The parameters `k1_eff=8.9675, K_m_bnxy=2.539` were derived to fit the *instantaneous* slopes `d[ijk0]/dt` at `t=0`.
Default: `Rate(0) = 8.9675 / (2.539+1) = 2.535`.
Perturbed: `Rate(0) = 8.9675 / (2.539+0.1) = 3.40`.
These match the experimental slopes.
If the simulation over [0, 0.2] with these initial rates results in `id_ijk0(0.2)` being too high, it implies that the *simulated* `[id_bnxy]` rises in a way that dampens the R1 rate more than what happens experimentally.

What if the inhibition by `id_bnxy` is less potent? This means `K_m_bnxy` should be larger.
If `K_m_bnxy` is larger, then for a given `k1_eff`, the rate `k1_eff / (K_m_bnxy + [bnxy])` would be smaller.
This is tricky. Let's try to increase `k1_eff` substantially, and also increase `K_m_bnxy` to try and maintain the initial rate ratio but make inhibition less sensitive.

Let `R_o = 2.535` (default initial rate for `id_ijk0` decay)
Let `R_p = 3.40` (perturbed initial rate for `id_ijk0` decay)
We need `k1_eff / (K_m_bnxy + 1) = R_o`
And `k1_eff / (K_m_bnxy + 0.1) = R_p`
Dividing: `(K_m_bnxy + 0.1) / (K_m_bnxy + 1) = R_o / R_p = 2.535 / 3.40 = 0.7456`
`K_m_bnxy + 0.1 = 0.7456 * K_m_bnxy + 0.7456`
`(1 - 0.7456) * K_m_bnxy = 0.7456 - 0.1`
`0.2544 * K_m_bnxy = 0.6456`
`K_m_bnxy = 0.6456 / 0.2544 = 2.538` (This is consistent, just re-deriving).

To get `id_ijk0(0.2)` lower, the average rate of R1 must be higher.
If `k1_eff` is increased, say to 12 (from 8.9675).
Then `K_m_bnxy + 1 = 12 / 2.535 = 4.733` => `K_m_bnxy = 3.733`.
And `K_m_bnxy + 0.1 = 12 / 3.40 = 3.529` => `K_m_bnxy = 3.429`.
These `K_m_bnxy` values are not identical but are closer than some previous attempts. Let's average them: `K_m_bnxy_avg = (3.733 + 3.429)/2 = 3.581`.
So, new R1 trial: `k1_eff = 12`, `K_m_bnxy = 3.581`.
Stoichiometry `S=1.5`.
R2 MM params: `Vm_deg=0.3836, Km_deg=0.3175`.

This attempts to increase the overall rate of R1 while trying to maintain the observed ratio of initial rates under default vs perturbed conditions. A higher `K_m_bnxy` means `id_bnxy` needs to rise to higher levels before it strongly inhibits R1.

## Action

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

# Access experimental data
exp_data_orig = experiment_history['iteration_0']
exp_data_pert = experiment_history['iteration_3'] # bnxy(0)=0.1

# Parameters for R1 - new trial
param_k1_eff = 12.0
param_K_m_bnxy = 3.581
stoich_S_bnxy_R1 = 1.5

# Parameters for R2 (Michaelis-Menten degradation) - kept from previous
param_Vm_deg = 0.3836
param_Km_deg = 0.3175

print(f"R1 params: k1_eff={param_k1_eff}, K_m_bnxy={param_K_m_bnxy}, Stoich_bnxy={stoich_S_bnxy_R1}")
print(f"R2 params (MM): Vm_deg={param_Vm_deg}, Km_deg={param_Km_deg}")

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

# Add parameters for R1
p = model.createParameter(); p.setId("k1_eff"); p.setValue(param_k1_eff); p.setConstant(True)
p = model.createParameter(); p.setId("K_m_bnxy"); p.setValue(param_K_m_bnxy); p.setConstant(True)

# Add parameters for R2
p = model.createParameter(); p.setId("Vm_deg"); p.setValue(param_Vm_deg); p.setConstant(True)
p = model.createParameter(); p.setId("Km_deg"); p.setValue(param_Km_deg); p.setConstant(True)

# Reaction 1: id_1t80 + id_ijk0 -> id_1t80 + S*id_bnxy
r1 = model.createReaction()
r1.setId("R1_ijk0_conv_inhib_bnxy")
r1.setReversible(False)
r1.setFast(False)
reactant1_r1 = r1.createReactant()
reactant1_r1.setSpecies("id_ijk0")
reactant1_r1.setStoichiometry(1)
reactant1_r1.setConstant(False)
product1_r1 = r1.createProduct()
product1_r1.setSpecies("id_bnxy")
product1_r1.setStoichiometry(stoich_S_bnxy_R1)
product1_r1.setConstant(False)
kl_r1 = r1.createKineticLaw()
kl_r1.setFormula("k1_eff * id_1t80 * id_ijk0 / (K_m_bnxy + id_bnxy)")

# Reaction 2: id_bnxy -> id_9k8m (Michaelis-Menten kinetic law)
r2 = model.createReaction()
r2.setId("R2_bnxy_degradation_MM")
r2.setReversible(False)
r2.setFast(False)
reactant1_r2 = r2.createReactant()
reactant1_r2.setSpecies("id_bnxy")
reactant1_r2.setStoichiometry(1)
reactant1_r2.setConstant(False)
product1_r2 = r2.createProduct()
product1_r2.setSpecies("id_9k8m")
product1_r2.setStoichiometry(1)
product1_r2.setConstant(False)
kl_r2 = r2.createKineticLaw()
kl_r2.setFormula("Vm_deg * id_bnxy / (Km_deg + id_bnxy)")

writer = libsbml.SBMLWriter()
updated_sbml_model_iter11 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_best_model_iter11", updated_sbml_model_iter11)

# Simulate with default initial conditions
sim_doc_default = libsbml.readSBMLFromString(updated_sbml_model_iter11)
sim_model_default = sim_doc_default.getModel()
sim_model_default.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_default.getSpecies("id_bnxy").setInitialConcentration(1.0)
sbml_for_sim_default = writer.writeSBMLToString(sim_doc_default)
sim_data_default = simulate(sbml_for_sim_default)

# Simulate with perturbed initial conditions (id_bnxy=0.1)
sim_doc_perturbed = libsbml.readSBMLFromString(updated_sbml_model_iter11)
sim_model_perturbed = sim_doc_perturbed.getModel()
sim_model_perturbed.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_perturbed.getSpecies("id_bnxy").setInitialConcentration(0.1)
sbml_for_sim_perturbed = writer.writeSBMLToString(sim_doc_perturbed)
sim_data_perturbed = simulate(sbml_for_sim_perturbed)

shared_variables.add("sim_data_default_iter11", sim_data_default)
shared_variables.add("sim_data_perturbed_iter11", sim_data_perturbed)

# Print comparison for default conditions
print("\nComparison for Default Conditions (Exp vs Sim with new R1 params):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

# Print comparison for perturbed conditions
print("\nComparison for Perturbed Conditions (id_bnxy(0)=0.1) (Exp vs Sim with new R1 params):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")
```
# Observation

## Code Stdout
R1 params: k1_eff=12.0, K_m_bnxy=3.581, Stoich_bnxy=1.5
R2 params (MM): Vm_deg=0.3836, Km_deg=0.3175

Comparison for Default Conditions (Exp vs Sim with new R1 params):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 4.933e-01 | 6.107e-01
0.4 | 4.574e-01 | 3.867e-01
1.0 | 4.389e-01 | 1.053e-01
2.0 | 3.975e-01 | 1.200e-02
5.0 | 1.306e-01 | 8.161e-06

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e+00 | 1.000e+00
0.2 | 1.858e+00 | 1.522e+00
0.4 | 1.900e+00 | 1.794e+00
0.6 | 1.891e+00 | 1.935e+00
1.0 | 1.865e+00 | 2.018e+00
2.0 | 1.769e+00 | 1.828e+00
5.0 | 9.687e-01 | 9.178e-01
7.0 | 2.015e-01 | 4.080e-01

Comparison for Perturbed Conditions (id_bnxy(0)=0.1) (Exp vs Sim with new R1 params):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 3.200e-01 | 5.514e-01
0.4 | 2.413e-01 | 3.223e-01
1.0 | 1.707e-01 | 7.065e-02
2.0 | 6.978e-02 | 5.530e-03
5.0 | 9.958e-05 | 1.030e-06

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e-01 | 1.000e-01
0.2 | 1.287e+00 | 7.295e-01
0.4 | 1.327e+00 | 1.017e+00
0.6 | 1.268e+00 | 1.151e+00
1.0 | 1.120e+00 | 1.214e+00
2.0 | 6.897e-01 | 1.012e+00
5.0 | 2.260e-02 | 2.801e-01
7.0 | 1.161e-03 | 5.138e-02

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_3)
- `shared_variables`: all variables you have saved from the previous iterations. (current_best_model, exp_data_iter0, sim_data_iter1, exp_data_orig_iter4, exp_data_pert_iter4, current_best_model_iter4, sim_data_default_iter4, sim_data_perturbed_iter4, current_best_model_iter5, sim_data_default_iter5, sim_data_perturbed_iter5, current_best_model_iter6, sim_data_default_iter6, sim_data_perturbed_iter6, current_best_model_iter7, sim_data_default_iter7, sim_data_perturbed_iter7, current_best_model_iter8, sim_data_default_iter8, sim_data_perturbed_iter8, current_best_model_iter9, sim_data_default_iter9, sim_data_perturbed_iter9, current_best_model_iter10, sim_data_default_iter10, sim_data_perturbed_iter10, current_best_model_iter11, sim_data_default_iter11, sim_data_perturbed_iter11)

### 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

The model from Iteration 11 showed some promise, particularly for the default `id_bnxy` tail at t=5s when using `S=1.5` for R1's stoichiometry. However, several key issues persist:
1.  **Initial `id_ijk0` decay (0-0.2s) is too slow** in simulations for both default and perturbed conditions, even though the R1 parameters (`k1_eff`, `K_m_bnxy`) are chosen to match the instantaneous experimental slopes at t=0. This suggests that the simulated `id_bnxy` rises too quickly or its inhibitory effect becomes too strong over this initial interval compared to reality.
2.  **Simulated `id_ijk0` depletes too rapidly after the initial phase** (e.g., at t >= 1.0s) compared to experimental data, especially in the default case.
3.  **`id_bnxy` peaks in simulations are generally too late.**
4.  **The tail of `id_bnxy` decay for the perturbed case is far too slow** in simulation, indicating the R2 Michaelis-Menten parameters are not capturing this condition well.

The contradiction in `id_ijk0` behavior (simulated R1 too slow initially, then too fast later) is a major challenge for the current R1 kinetic form `k1_eff * [id_ijk0] / (K_m_bnxy + [id_bnxy])` with constant parameters.

Let's try to address point (1) by making the R2 degradation of `id_bnxy` faster, especially at moderate `id_bnxy` concentrations. This might keep `id_bnxy` levels lower during the initial phase [0-0.2s], thus reducing the inhibition on R1 and allowing `id_ijk0` to decay faster.

The R2 Michaelis-Menten parameters from Iteration 11 were `Vm_deg=0.3836, Km_deg=0.3175`.
The effective first-order rate for low `[bnxy]` (perturbed tail) is `Vm_deg/Km_deg = 1.208`. The experimental rate was ~1.399.
The saturated rate for high `[bnxy]` (default tail, ignoring R1 production) was `Vm_deg = 0.3836`. The experimental rate was ~0.38.

Let's try to increase `Vm_deg` to make degradation faster when `[bnxy]` is higher (like during its peak and initial rise). To keep the perturbed tail somewhat consistent, `Km_deg` would also need to increase if `Vm_deg/Km_deg` is to be around 1.2-1.4.
Suppose we target `Vm_deg/Km_deg = 1.3`.
If we increase `Vm_deg` to, say, `0.6`. Then `Km_deg = 0.6 / 1.3 = 0.46`.
New R2 trial parameters: `Vm_deg = 0.6`, `Km_deg = 0.46`.
This will make R2 degradation faster across a range of `id_bnxy` concentrations compared to the previous set.

R1 parameters will be kept as in Iteration 11: `k1_eff = 12.0`, `K_m_bnxy = 3.581`, and stoichiometry `S=1.5`. These R1 parameters were an attempt to improve initial `id_ijk0` decay by increasing `k1_eff` and `K_m_bnxy` (weaker inhibition sensitivity).

Revised parameter set for this iteration:
R1: `k1_eff = 12.0`, `K_m_bnxy = 3.581`, `S=1.5`.
R2: `Vm_deg = 0.6`, `Km_deg = 0.46` (Michaelis-Menten law).

## Action

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

# Access experimental data
exp_data_orig = experiment_history['iteration_0']
exp_data_pert = experiment_history['iteration_3'] # bnxy(0)=0.1

# Parameters for R1 (kept from Iteration 11)
param_k1_eff = 12.0
param_K_m_bnxy = 3.581
stoich_S_bnxy_R1 = 1.5

# Parameters for R2 (Michaelis-Menten degradation) - new trial
param_Vm_deg = 0.6
param_Km_deg = 0.46

print(f"R1 params: k1_eff={param_k1_eff}, K_m_bnxy={param_K_m_bnxy}, Stoich_bnxy={stoich_S_bnxy_R1}")
print(f"R2 params (MM): Vm_deg={param_Vm_deg}, Km_deg={param_Km_deg}")

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

# Add parameters for R1
p = model.createParameter(); p.setId("k1_eff"); p.setValue(param_k1_eff); p.setConstant(True)
p = model.createParameter(); p.setId("K_m_bnxy"); p.setValue(param_K_m_bnxy); p.setConstant(True)

# Add parameters for R2
p = model.createParameter(); p.setId("Vm_deg"); p.setValue(param_Vm_deg); p.setConstant(True)
p = model.createParameter(); p.setId("Km_deg"); p.setValue(param_Km_deg); p.setConstant(True)

# Reaction 1: id_1t80 + id_ijk0 -> id_1t80 + S*id_bnxy
r1 = model.createReaction()
r1.setId("R1_ijk0_conv_inhib_bnxy")
r1.setReversible(False)
r1.setFast(False)
reactant1_r1 = r1.createReactant()
reactant1_r1.setSpecies("id_ijk0")
reactant1_r1.setStoichiometry(1)
reactant1_r1.setConstant(False)
product1_r1 = r1.createProduct()
product1_r1.setSpecies("id_bnxy")
product1_r1.setStoichiometry(stoich_S_bnxy_R1)
product1_r1.setConstant(False)
kl_r1 = r1.createKineticLaw()
kl_r1.setFormula("k1_eff * id_1t80 * id_ijk0 / (K_m_bnxy + id_bnxy)")

# Reaction 2: id_bnxy -> id_9k8m (Michaelis-Menten kinetic law)
r2 = model.createReaction()
r2.setId("R2_bnxy_degradation_MM")
r2.setReversible(False)
r2.setFast(False)
reactant1_r2 = r2.createReactant()
reactant1_r2.setSpecies("id_bnxy")
reactant1_r2.setStoichiometry(1)
reactant1_r2.setConstant(False)
product1_r2 = r2.createProduct()
product1_r2.setSpecies("id_9k8m")
product1_r2.setStoichiometry(1)
product1_r2.setConstant(False)
kl_r2 = r2.createKineticLaw()
kl_r2.setFormula("Vm_deg * id_bnxy / (Km_deg + id_bnxy)")

writer = libsbml.SBMLWriter()
updated_sbml_model_iter12 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_best_model_iter12", updated_sbml_model_iter12)

# Simulate with default initial conditions
sim_doc_default = libsbml.readSBMLFromString(updated_sbml_model_iter12)
sim_model_default = sim_doc_default.getModel()
sim_model_default.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_default.getSpecies("id_bnxy").setInitialConcentration(1.0)
sbml_for_sim_default = writer.writeSBMLToString(sim_doc_default)
sim_data_default = simulate(sbml_for_sim_default)

# Simulate with perturbed initial conditions (id_bnxy=0.1)
sim_doc_perturbed = libsbml.readSBMLFromString(updated_sbml_model_iter12)
sim_model_perturbed = sim_doc_perturbed.getModel()
sim_model_perturbed.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_perturbed.getSpecies("id_bnxy").setInitialConcentration(0.1)
sbml_for_sim_perturbed = writer.writeSBMLToString(sim_doc_perturbed)
sim_data_perturbed = simulate(sbml_for_sim_perturbed)

shared_variables.add("sim_data_default_iter12", sim_data_default)
shared_variables.add("sim_data_perturbed_iter12", sim_data_perturbed)

# Print comparison for default conditions
print("\nComparison for Default Conditions (Exp vs Sim with new R2 MM params):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

# Print comparison for perturbed conditions
print("\nComparison for Perturbed Conditions (id_bnxy(0)=0.1) (Exp vs Sim with new R2 MM params):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")
```
# Observation

## Code Stdout
R1 params: k1_eff=12.0, K_m_bnxy=3.581, Stoich_bnxy=1.5
R2 params (MM): Vm_deg=0.6, Km_deg=0.46

Comparison for Default Conditions (Exp vs Sim with new R2 MM params):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 4.933e-01 | 6.100e-01
0.4 | 4.574e-01 | 3.849e-01
1.0 | 4.389e-01 | 1.024e-01
2.0 | 3.975e-01 | 1.068e-02
5.0 | 1.306e-01 | 3.470e-06

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e+00 | 1.000e+00
0.2 | 1.858e+00 | 1.497e+00
0.4 | 1.900e+00 | 1.741e+00
0.6 | 1.891e+00 | 1.853e+00
1.0 | 1.865e+00 | 1.876e+00
2.0 | 1.769e+00 | 1.540e+00
5.0 | 9.687e-01 | 3.916e-01
7.0 | 2.015e-01 | 5.938e-02

Comparison for Perturbed Conditions (id_bnxy(0)=0.1) (Exp vs Sim with new R2 MM params):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 3.200e-01 | 5.510e-01
0.4 | 2.413e-01 | 3.212e-01
1.0 | 1.707e-01 | 6.884e-02
2.0 | 6.978e-02 | 4.917e-03
5.0 | 9.958e-05 | 5.212e-07

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e-01 | 1.000e-01
0.2 | 1.287e+00 | 7.165e-01
0.4 | 1.327e+00 | 9.829e-01
0.6 | 1.268e+00 | 1.094e+00
1.0 | 1.120e+00 | 1.108e+00
2.0 | 6.897e-01 | 7.981e-01
5.0 | 2.260e-02 | 7.836e-02
7.0 | 1.161e-03 | 6.741e-03

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_3)
- `shared_variables`: all variables you have saved from the previous iterations. (current_best_model, exp_data_iter0, sim_data_iter1, exp_data_orig_iter4, exp_data_pert_iter4, current_best_model_iter4, sim_data_default_iter4, sim_data_perturbed_iter4, current_best_model_iter5, sim_data_default_iter5, sim_data_perturbed_iter5, current_best_model_iter6, sim_data_default_iter6, sim_data_perturbed_iter6, current_best_model_iter7, sim_data_default_iter7, sim_data_perturbed_iter7, current_best_model_iter8, sim_data_default_iter8, sim_data_perturbed_iter8, current_best_model_iter9, sim_data_default_iter9, sim_data_perturbed_iter9, current_best_model_iter10, sim_data_default_iter10, sim_data_perturbed_iter10, current_best_model_iter11, sim_data_default_iter11, sim_data_perturbed_iter11, current_best_model_iter12, sim_data_default_iter12, sim_data_perturbed_iter12)

### 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

The previous iterations have struggled to simultaneously fit:
1.  The initial rapid decay of `id_ijk0`, especially in the perturbed case.
2.  The subsequent slowing and then slight re-acceleration of `id_ijk0` decay.
3.  The peak height and timing of `id_bnxy`.
4.  The tail behavior of `id_bnxy` under both default (slow decay from a high value) and perturbed (faster relative decay from a low value) conditions.

The model structure so far is:
R1: `id_ijk0 -> S * id_bnxy`, rate `k1_eff * id_1t80 * id_ijk0 / (K_m_bnxy + id_bnxy)` (where `id_1t80=1`)
R2: `id_bnxy -> id_9k8m`, rate `Vm_deg * id_bnxy / (Km_deg + id_bnxy)`

The persistent issue with slow initial `id_ijk0` decay, even when instantaneous `t=0` slopes are matched, suggests that the inhibition by the rising `id_bnxy` is too strong or kicks in too effectively in the model during the [0, 0.2s] interval.

I propose adding a new, uninhibited, first-order decay pathway for `id_ijk0`. This will provide a baseline decay rate and contribute to faster initial decay, especially if the main pathway (R1) is significantly inhibited.
Let this new reaction be:
**R0: `id_ijk0 -> id_9k8m`** (decay to a sink, not producing `id_bnxy`)
Rate: `k_r0 * id_ijk0`

The differential equation for `id_ijk0` becomes:
`d[id_ijk0]/dt = - (k_r1_eff * [id_1t80] / (Km_r1_bnxy + [id_bnxy]) + k_r0) * [id_ijk0]`

Let's try to estimate parameters:
Experimental initial slopes for `id_ijk0` decay: `Rate_def(0) = 2.535`, `Rate_pert(0) = 3.40`.
So,
1.  `2.535 = k_r1_eff / (Km_r1_bnxy + 1.0) + k_r0` (since `[id_bnxy](0)=1.0` for default)
2.  `3.40  = k_r1_eff / (Km_r1_bnxy + 0.1) + k_r0` (since `[id_bnxy](0)=0.1` for perturbed)

Subtracting (1) from (2):
`0.865 = k_r1_eff * [1/(Km_r1_bnxy + 0.1) - 1/(Km_r1_bnxy + 1.0)]`
`0.865 = k_r1_eff * [0.9 / ((Km_r1_bnxy + 0.1)(Km_r1_bnxy + 1.0))]`

Let's assume a value for `k_r0`. If `k_r0 = 0.5 s^-1`.
Then from (1): `k_r1_eff / (Km_r1_bnxy + 1.0) = 2.535 - 0.5 = 2.035`.
And from (2): `k_r1_eff / (Km_r1_bnxy + 0.1) = 3.40 - 0.5 = 2.90`.
Dividing these two new equations: `(Km_r1_bnxy + 0.1) / (Km_r1_bnxy + 1.0) = 2.035 / 2.90 = 0.7017`.
`Km_r1_bnxy + 0.1 = 0.7017 * Km_r1_bnxy + 0.7017`.
`0.2983 * Km_r1_bnxy = 0.6017`.
`Km_r1_bnxy = 0.6017 / 0.2983 = 2.017`.
Then `k_r1_eff = 2.035 * (2.017 + 1.0) = 2.035 * 3.017 = 6.140`.

So, for R0 and R1:
`k_r0 = 0.5`
`k_r1_eff = 6.140`
`Km_r1_bnxy = 2.017`
Stoichiometry for R1: `S=1.5` (kept from Iteration 11/12 as it helped `id_bnxy` default tail).

For R2 (Michaelis-Menten degradation of `id_bnxy`):
`Vm_r2 * id_bnxy / (Km_r2_bnxy + id_bnxy)`
Re-estimating `Vm_r2, Km_r2_bnxy` assuming R0 and R1 production of `id_bnxy` is negligible in far tails:
Perturbed tail: `d[bnxy]/dt ~ -0.069` at `[bnxy]~0.0571`. So `Vm_r2/Km_r2_bnxy * 0.0571 = 0.069` => `Vm_r2/Km_r2_bnxy = 1.2084`.
Default tail: `d[bnxy]/dt ~ -0.3836` at `[bnxy]~0.9687` (average over t=5-7, ignoring R1). If this is near saturation, `Vm_r2 ~ 0.3836`.
Then `Km_r2_bnxy = Vm_r2 / 1.2084 = 0.3836 / 1.2084 = 0.3175`.
These are the same R2 parameters as Iteration 10 and 11 (`Vm_deg=0.3836, Km_deg=0.3175`). Iteration 12 used different ones that worsened tails.

Final proposed parameter set for this iteration:
R0: `k_r0 = 0.5`
R1: `k_r1_eff = 6.140`, `Km_r1_bnxy = 2.017`, `S=1.5`
R2: `Vm_r2 = 0.3836`, `Km_r2_bnxy = 0.3175`

## Action

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

# Access experimental data
exp_data_orig = experiment_history['iteration_0']
exp_data_pert = experiment_history['iteration_3'] # bnxy(0)=0.1

# Parameters for R0 (New reaction: id_ijk0 -> id_9k8m)
param_k_r0 = 0.5

# Parameters for R1 (id_ijk0 -> 1.5 id_bnxy, inhibited by id_bnxy)
param_k_r1_eff = 6.140
param_Km_r1_bnxy = 2.017
stoich_S_bnxy_R1 = 1.5

# Parameters for R2 (id_bnxy -> id_9k8m, Michaelis-Menten)
param_Vm_r2 = 0.3836
param_Km_r2_bnxy = 0.3175

print(f"R0 param: k_r0={param_k_r0}")
print(f"R1 params: k_r1_eff={param_k_r1_eff}, Km_r1_bnxy={param_Km_r1_bnxy}, Stoich_bnxy={stoich_S_bnxy_R1}")
print(f"R2 params (MM): Vm_r2={param_Vm_r2}, Km_r2_bnxy={param_Km_r2_bnxy}")

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

# Add parameters
p = model.createParameter(); p.setId("k_r0"); p.setValue(param_k_r0); p.setConstant(True)
p = model.createParameter(); p.setId("k_r1_eff"); p.setValue(param_k_r1_eff); p.setConstant(True)
p = model.createParameter(); p.setId("Km_r1_bnxy"); p.setValue(param_Km_r1_bnxy); p.setConstant(True)
p = model.createParameter(); p.setId("Vm_r2"); p.setValue(param_Vm_r2); p.setConstant(True)
p = model.createParameter(); p.setId("Km_r2_bnxy"); p.setValue(param_Km_r2_bnxy); p.setConstant(True)

# Reaction R0: id_ijk0 -> id_9k8m
r0 = model.createReaction()
r0.setId("R0_ijk0_decay_uninhibited")
r0.setReversible(False)
r0.setFast(False)
reactant_r0 = r0.createReactant()
reactant_r0.setSpecies("id_ijk0")
reactant_r0.setStoichiometry(1)
reactant_r0.setConstant(False)
product_r0 = r0.createProduct() # Product is a sink
product_r0.setSpecies("id_9k8m")
product_r0.setStoichiometry(1)
product_r0.setConstant(False)
kl_r0 = r0.createKineticLaw()
kl_r0.setFormula("k_r0 * id_ijk0")

# Reaction R1: id_1t80 + id_ijk0 -> id_1t80 + S*id_bnxy
r1 = model.createReaction()
r1.setId("R1_ijk0_to_bnxy_inhib_bnxy")
r1.setReversible(False)
r1.setFast(False)
reactant1_r1 = r1.createReactant()
reactant1_r1.setSpecies("id_ijk0")
reactant1_r1.setStoichiometry(1)
reactant1_r1.setConstant(False)
product1_r1 = r1.createProduct()
product1_r1.setSpecies("id_bnxy")
product1_r1.setStoichiometry(stoich_S_bnxy_R1)
product1_r1.setConstant(False)
kl_r1 = r1.createKineticLaw()
# Assuming id_1t80 is constant at 1.0 and its effect is in k_r1_eff
kl_r1.setFormula("k_r1_eff * id_1t80 * id_ijk0 / (Km_r1_bnxy + id_bnxy)")


# Reaction R2: id_bnxy -> id_9k8m (Michaelis-Menten kinetic law)
r2 = model.createReaction()
r2.setId("R2_bnxy_degradation_MM")
r2.setReversible(False)
r2.setFast(False)
reactant1_r2 = r2.createReactant()
reactant1_r2.setSpecies("id_bnxy")
reactant1_r2.setStoichiometry(1)
reactant1_r2.setConstant(False)
product1_r2 = r2.createProduct()
product1_r2.setSpecies("id_9k8m")
product1_r2.setStoichiometry(1)
product1_r2.setConstant(False)
kl_r2 = r2.createKineticLaw()
kl_r2.setFormula("Vm_r2 * id_bnxy / (Km_r2_bnxy + id_bnxy)")

writer = libsbml.SBMLWriter()
updated_sbml_model_iter13 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_best_model_iter13", updated_sbml_model_iter13)

# Simulate with default initial conditions
sim_doc_default = libsbml.readSBMLFromString(updated_sbml_model_iter13)
sim_model_default = sim_doc_default.getModel()
sim_model_default.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_default.getSpecies("id_bnxy").setInitialConcentration(1.0)
sbml_for_sim_default = writer.writeSBMLToString(sim_doc_default)
sim_data_default = simulate(sbml_for_sim_default)

# Simulate with perturbed initial conditions (id_bnxy=0.1)
sim_doc_perturbed = libsbml.readSBMLFromString(updated_sbml_model_iter13)
sim_model_perturbed = sim_doc_perturbed.getModel()
sim_model_perturbed.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_perturbed.getSpecies("id_bnxy").setInitialConcentration(0.1)
sbml_for_sim_perturbed = writer.writeSBMLToString(sim_doc_perturbed)
sim_data_perturbed = simulate(sbml_for_sim_perturbed)

shared_variables.add("sim_data_default_iter13", sim_data_default)
shared_variables.add("sim_data_perturbed_iter13", sim_data_perturbed)

# Print comparison for default conditions
print("\nComparison for Default Conditions (Exp vs Sim with R0, new R1, MM R2):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

# Print comparison for perturbed conditions
print("\nComparison for Perturbed Conditions (Exp vs Sim with R0, new R1, MM R2):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"
    sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")
```
# Observation

## Code Stdout
R0 param: k_r0=0.5
R1 params: k_r1_eff=6.14, Km_r1_bnxy=2.017, Stoich_bnxy=1.5
R2 params (MM): Vm_r2=0.3836, Km_r2_bnxy=0.3175

Comparison for Default Conditions (Exp vs Sim with R0, new R1, MM R2):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 4.933e-01 | 6.188e-01
0.4 | 4.574e-01 | 3.950e-01
1.0 | 4.389e-01 | 1.084e-01
2.0 | 3.975e-01 | 1.226e-02
5.0 | 1.306e-01 | 6.791e-06

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e+00 | 1.000e+00
0.2 | 1.858e+00 | 1.392e+00
0.4 | 1.900e+00 | 1.590e+00
0.6 | 1.891e+00 | 1.687e+00
1.0 | 1.865e+00 | 1.727e+00
2.0 | 1.769e+00 | 1.516e+00
5.0 | 9.687e-01 | 6.514e-01
7.0 | 2.015e-01 | 2.237e-01

Comparison for Perturbed Conditions (Exp vs Sim with R0, new R1, MM R2):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 3.200e-01 | 5.441e-01
0.4 | 2.413e-01 | 3.162e-01
1.0 | 1.707e-01 | 6.790e-02
2.0 | 6.978e-02 | 4.989e-03
5.0 | 9.958e-05 | 5.738e-07

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e-01 | 1.000e-01
0.2 | 1.287e+00 | 6.311e-01
0.4 | 1.327e+00 | 8.562e-01
0.6 | 1.268e+00 | 9.544e-01
1.0 | 1.120e+00 | 9.835e-01
2.0 | 6.897e-01 | 7.768e-01
5.0 | 2.260e-02 | 1.523e-01
7.0 | 1.161e-03 | 2.059e-02

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_3)
- `shared_variables`: all variables you have saved from the previous iterations. (current_best_model, exp_data_iter0, sim_data_iter1, exp_data_orig_iter4, exp_data_pert_iter4, current_best_model_iter4, sim_data_default_iter4, sim_data_perturbed_iter4, current_best_model_iter5, sim_data_default_iter5, sim_data_perturbed_iter5, current_best_model_iter6, sim_data_default_iter6, sim_data_perturbed_iter6, current_best_model_iter7, sim_data_default_iter7, sim_data_perturbed_iter7, current_best_model_iter8, sim_data_default_iter8, sim_data_perturbed_iter8, current_best_model_iter9, sim_data_default_iter9, sim_data_perturbed_iter9, current_best_model_iter10, sim_data_default_iter10, sim_data_perturbed_iter10, current_best_model_iter11, sim_data_default_iter11, sim_data_perturbed_iter11, current_best_model_iter12, sim_data_default_iter12, sim_data_perturbed_iter12, current_best_model_iter13, sim_data_default_iter13, sim_data_perturbed_iter13)

### 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

The addition of R0 (`id_ijk0 -> Sink`, rate `k_r0 * id_ijk0`) was intended to improve the initial decay of `id_ijk0`.
Current model:
R0: `id_ijk0 -> Sink`, `k_r0 = 0.5`
R1: `id_ijk0 -> 1.5 id_bnxy`, rate `k_r1_eff * id_ijk0 / (Km_r1_bnxy + id_bnxy)` with `k_r1_eff = 6.140`, `Km_r1_bnxy = 2.017`
R2: `id_bnxy -> Sink`, rate `Vm_r2 * id_bnxy / (Km_r2_bnxy + id_bnxy)` with `Vm_r2 = 0.3836`, `Km_r2_bnxy = 0.3175`

**`id_ijk0` dynamics:**
- Default: Sim(0.2)=0.619 vs Exp(0.2)=0.493. Still too slow.
- Perturbed: Sim(0.2)=0.544 vs Exp(0.2)=0.320. Still too slow.
The R0 term did not sufficiently accelerate the initial decay. The values of `id_ijk0` at t=0.2 are very similar to Iteration 12 (which didn't have R0 but had different R1 params). This means the R1 pathway is still dominant and its inhibition is the main factor.

**`id_bnxy` dynamics:**
- Default:
    - Peak: Sim peak ~1.73 (t=1.0) vs Exp peak ~1.90 (t=0.4-0.6). Peak height is better, but still too late.
    - Tail: Sim(t=5)=0.651 vs Exp(t=5)=0.9687. Decay too fast. Sim(t=7)=0.224 vs Exp(t=7)=0.20. Close at t=7.
- Perturbed:
    - Peak: Sim peak ~0.98 (t=1.0) vs Exp peak ~1.33 (t=0.4). Peak height too low, and too late.
    - Tail: Sim(t=5)=0.152 vs Exp(t=5)=0.0226. Too slow decay. Sim(t=7)=0.021 vs Exp(t=7)=0.001. Too slow.

The `id_bnxy` tails are now problematic again, especially for the perturbed case (too slow decay). This suggests the R2 MM parameters are not quite right, or the interplay with R1 (which produces `id_bnxy`) is complex.

The core issue remains the `id_ijk0` initial decay. The combined rate from R0 and R1 at t=0 was designed to match experimental slopes.
Total initial rate for `id_ijk0` decay:
Default: `k_r0 + k_r1_eff / (Km_r1_bnxy + 1.0) = 0.5 + 6.140 / (2.017 + 1) = 0.5 + 6.140 / 3.017 = 0.5 + 2.035 = 2.535`. (Matches Exp)
Perturbed: `k_r0 + k_r1_eff / (Km_r1_bnxy + 0.1) = 0.5 + 6.140 / (2.017 + 0.1) = 0.5 + 6.140 / 2.117 = 0.5 + 2.900 = 3.400`. (Matches Exp)

If the instantaneous rates at t=0 are correct, but `id_ijk0(0.2)` is too high, it means the *average* rate of `id_ijk0` consumption over [0, 0.2s] is lower in simulation than in reality. This happens because `id_bnxy` (simulated) rises and inhibits R1.
This implies:
1.  Simulated `id_bnxy` rises too fast/high initially, OR
2.  The inhibitory effect of `id_bnxy` (via `Km_r1_bnxy`) is too strong for small `id_bnxy`.

Let's look at simulated `id_bnxy` at t=0.2:
Default: Sim(0.2)=1.392 vs Exp(0.2)=1.858. Simulated `id_bnxy` is *lower*. This would mean *less* inhibition on R1, so `id_ijk0` should decay *faster*, not slower. This contradicts the observation for `id_ijk0`.

This is a fundamental conflict.
- If `id_bnxy` (sim) is lower than exp at t=0.2, R1 should be less inhibited, `id_ijk0` (sim) should be lower.
- But `id_ijk0` (sim) is higher than exp at t=0.2.

This could mean:
- The `k_r0` term is too small. If `k_r0` is larger, it provides more uninhibited decay.
- The `k_r1_eff` term (max rate of R1 when uninhibited by `id_bnxy`) is too small.
- The `Km_r1_bnxy` (inhibition constant for R1) is too small, making R1 too sensitive to `id_bnxy`.

Let's try to increase `k_r0` significantly.
If `k_r0 = 1.5` (was 0.5).
1.  `k_r1_eff / (Km_r1_bnxy + 1.0) = 2.535 - 1.5 = 1.035`.
2.  `k_r1_eff / (Km_r1_bnxy + 0.1) = 3.40 - 1.5 = 1.90`.
Dividing: `(Km_r1_bnxy + 0.1) / (Km_r1_bnxy + 1.0) = 1.035 / 1.90 = 0.5447`.
`Km_r1_bnxy + 0.1 = 0.5447 * Km_r1_bnxy + 0.5447`.
`0.4553 * Km_r1_bnxy = 0.4447`.
`Km_r1_bnxy = 0.4447 / 0.4553 = 0.9767`.
Then `k_r1_eff = 1.035 * (0.9767 + 1.0) = 1.035 * 1.9767 = 2.046`.

New R0/R1 parameters:
`k_r0 = 1.5`
`k_r1_eff = 2.046`
`Km_r1_bnxy = 0.9767`
Stoichiometry `S=1.5` for R1.
R2 MM params: `Vm_r2=0.3836, Km_r2_bnxy=0.3175` (revert to these as they were better for default tail previously).

This set has a much larger contribution from the uninhibited R0 pathway. The R1 pathway now has a smaller `k_r1_eff` and a smaller `Km_r1_bnxy` (meaning R1 is more sensitive to `id_bnxy` but its max rate is lower).

## Action

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

# Access experimental data
exp_data_orig = experiment_history['iteration_0']
exp_data_pert = experiment_history['iteration_3'] # bnxy(0)=0.1

# Parameters for R0 (New reaction: id_ijk0 -> id_9k8m) - Increased k_r0
param_k_r0 = 1.5

# Parameters for R1 (id_ijk0 -> 1.5 id_bnxy, inhibited by id_bnxy) - Recalculated
param_k_r1_eff = 2.046
param_Km_r1_bnxy = 0.9767
stoich_S_bnxy_R1 = 1.5

# Parameters for R2 (id_bnxy -> id_9k8m, Michaelis-Menten) - Reverted to Iteration 10/11 values
param_Vm_r2 = 0.3836
param_Km_r2_bnxy = 0.3175

print(f"R0 param: k_r0={param_k_r0}")
print(f"R1 params: k_r1_eff={param_k_r1_eff}, Km_r1_bnxy={param_Km_r1_bnxy}, Stoich_bnxy={stoich_S_bnxy_R1}")
print(f"R2 params (MM): Vm_r2={param_Vm_r2}, Km_r2_bnxy={param_Km_r2_bnxy}")

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

# Add parameters
p = model.createParameter(); p.setId("k_r0"); p.setValue(param_k_r0); p.setConstant(True)
p = model.createParameter(); p.setId("k_r1_eff"); p.setValue(param_k_r1_eff); p.setConstant(True)
p = model.createParameter(); p.setId("Km_r1_bnxy"); p.setValue(param_Km_r1_bnxy); p.setConstant(True)
p = model.createParameter(); p.setId("Vm_r2"); p.setValue(param_Vm_r2); p.setConstant(True)
p = model.createParameter(); p.setId("Km_r2_bnxy"); p.setValue(param_Km_r2_bnxy); p.setConstant(True)

# Reaction R0: id_ijk0 -> id_9k8m
r0 = model.createReaction()
r0.setId("R0_ijk0_decay_uninhibited")
r0.setReversible(False)
r0.setFast(False)
reactant_r0 = r0.createReactant()
reactant_r0.setSpecies("id_ijk0")
reactant_r0.setStoichiometry(1)
reactant_r0.setConstant(False)
product_r0 = r0.createProduct()
product_r0.setSpecies("id_9k8m")
product_r0.setStoichiometry(1)
product_r0.setConstant(False)
kl_r0 = r0.createKineticLaw()
kl_r0.setFormula("k_r0 * id_ijk0")

# Reaction R1: id_1t80 + id_ijk0 -> id_1t80 + S*id_bnxy
r1 = model.createReaction()
r1.setId("R1_ijk0_to_bnxy_inhib_bnxy")
r1.setReversible(False)
r1.setFast(False)
reactant1_r1 = r1.createReactant()
reactant1_r1.setSpecies("id_ijk0")
reactant1_r1.setStoichiometry(1)
reactant1_r1.setConstant(False)
product1_r1 = r1.createProduct()
product1_r1.setSpecies("id_bnxy")
product1_r1.setStoichiometry(stoich_S_bnxy_R1)
product1_r1.setConstant(False)
kl_r1 = r1.createKineticLaw()
kl_r1.setFormula("k_r1_eff * id_1t80 * id_ijk0 / (Km_r1_bnxy + id_bnxy)")

# Reaction R2: id_bnxy -> id_9k8m (Michaelis-Menten kinetic law)
r2 = model.createReaction()
r2.setId("R2_bnxy_degradation_MM")
r2.setReversible(False)
r2.setFast(False)
reactant1_r2 = r2.createReactant()
reactant1_r2.setSpecies("id_bnxy")
reactant1_r2.setStoichiometry(1)
reactant1_r2.setConstant(False)
product1_r2 = r2.createProduct()
product1_r2.setSpecies("id_9k8m")
product1_r2.setStoichiometry(1)
product1_r2.setConstant(False)
kl_r2 = r2.createKineticLaw()
kl_r2.setFormula("Vm_r2 * id_bnxy / (Km_r2_bnxy + id_bnxy)")

writer = libsbml.SBMLWriter()
updated_sbml_model_iter14 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_best_model_iter14", updated_sbml_model_iter14)

# Simulate
sim_doc_default = libsbml.readSBMLFromString(updated_sbml_model_iter14)
sim_model_default = sim_doc_default.getModel()
sim_model_default.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_default.getSpecies("id_bnxy").setInitialConcentration(1.0)
sbml_for_sim_default = writer.writeSBMLToString(sim_doc_default)
sim_data_default = simulate(sbml_for_sim_default)

sim_doc_perturbed = libsbml.readSBMLFromString(updated_sbml_model_iter14)
sim_model_perturbed = sim_doc_perturbed.getModel()
sim_model_perturbed.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_perturbed.getSpecies("id_bnxy").setInitialConcentration(0.1)
sbml_for_sim_perturbed = writer.writeSBMLToString(sim_doc_perturbed)
sim_data_perturbed = simulate(sbml_for_sim_perturbed)

shared_variables.add("sim_data_default_iter14", sim_data_default)
shared_variables.add("sim_data_perturbed_iter14", sim_data_perturbed)

# Print comparison
print("\nComparison for Default Conditions (Exp vs Sim with new R0/R1 params):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"; sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"; sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nComparison for Perturbed Conditions (Exp vs Sim with new R0/R1 params):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"; sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"; sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")
```
# Observation

## Code Stdout
R0 param: k_r0=1.5
R1 params: k_r1_eff=2.046, Km_r1_bnxy=0.9767, Stoich_bnxy=1.5
R2 params (MM): Vm_r2=0.3836, Km_r2_bnxy=0.3175

Comparison for Default Conditions (Exp vs Sim with new R0/R1 params):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 4.933e-01 | 6.082e-01
0.4 | 4.574e-01 | 3.739e-01
1.0 | 4.389e-01 | 8.772e-02
2.0 | 3.975e-01 | 7.330e-03
5.0 | 1.306e-01 | 1.503e-06

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e+00 | 1.000e+00
0.2 | 1.858e+00 | 1.174e+00
0.4 | 1.900e+00 | 1.248e+00
0.6 | 1.891e+00 | 1.268e+00
1.0 | 1.865e+00 | 1.227e+00
2.0 | 1.769e+00 | 9.762e-01
5.0 | 9.687e-01 | 2.563e-01
7.0 | 2.015e-01 | 4.457e-02

Comparison for Perturbed Conditions (Exp vs Sim with new R0/R1 params):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 3.200e-01 | 5.357e-01
0.4 | 2.413e-01 | 3.007e-01
1.0 | 1.707e-01 | 5.572e-02
2.0 | 6.978e-02 | 3.050e-03
5.0 | 9.958e-05 | 1.469e-07

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e-01 | 1.000e-01
0.2 | 1.287e+00 | 4.283e-01
0.4 | 1.327e+00 | 5.511e-01
0.6 | 1.268e+00 | 5.926e-01
1.0 | 1.120e+00 | 5.734e-01
2.0 | 6.897e-01 | 3.806e-01
5.0 | 2.260e-02 | 3.099e-02
7.0 | 1.161e-03 | 3.020e-03

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_3)
- `shared_variables`: all variables you have saved from the previous iterations. (current_best_model, exp_data_iter0, sim_data_iter1, exp_data_orig_iter4, exp_data_pert_iter4, current_best_model_iter4, sim_data_default_iter4, sim_data_perturbed_iter4, current_best_model_iter5, sim_data_default_iter5, sim_data_perturbed_iter5, current_best_model_iter6, sim_data_default_iter6, sim_data_perturbed_iter6, current_best_model_iter7, sim_data_default_iter7, sim_data_perturbed_iter7, current_best_model_iter8, sim_data_default_iter8, sim_data_perturbed_iter8, current_best_model_iter9, sim_data_default_iter9, sim_data_perturbed_iter9, current_best_model_iter10, sim_data_default_iter10, sim_data_perturbed_iter10, current_best_model_iter11, sim_data_default_iter11, sim_data_perturbed_iter11, current_best_model_iter12, sim_data_default_iter12, sim_data_perturbed_iter12, current_best_model_iter13, sim_data_default_iter13, sim_data_perturbed_iter13, current_best_model_iter14, sim_data_default_iter14, sim_data_perturbed_iter14)

### 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

The model from Iteration 14 with a significant R0 pathway (`k_r0=1.5`) and adjusted R1 parameters:
R0: `id_ijk0 -> Sink`, `k_r0 = 1.5`
R1: `id_ijk0 -> 1.5 id_bnxy`, rate `k_r1_eff * id_ijk0 / (Km_r1_bnxy + id_bnxy)` with `k_r1_eff = 2.046`, `Km_r1_bnxy = 0.9767`
R2: `id_bnxy -> Sink`, rate `Vm_r2 * id_bnxy / (Km_r2_bnxy + id_bnxy)` with `Vm_r2 = 0.3836`, `Km_r2_bnxy = 0.3175`

**`id_ijk0` dynamics:**
- Default: Sim(0.2)=0.608 vs Exp(0.2)=0.493. Still too slow.
- Perturbed: Sim(0.2)=0.536 vs Exp(0.2)=0.320. Still too slow.
Even with `k_r0=1.5`, the initial decay is not fast enough. The `id_ijk0` values at t=0.2 are still higher than experimental.

**`id_bnxy` dynamics:**
- Default:
    - Peak: Sim peak ~1.27 (t=0.6-1.0) vs Exp peak ~1.90. Peak is now too low and still a bit late.
    - Tail: Sim(t=5)=0.256 vs Exp(t=5)=0.9687. Decay too fast.
- Perturbed:
    - Peak: Sim peak ~0.59 (t=0.6) vs Exp peak ~1.33. Peak far too low.
    - Tail: Sim(t=5)=0.031 vs Exp(t=5)=0.0226. This is actually a good match! Sim(t=7)=0.003 vs Exp(t=7)=0.001. Also good.

The `id_bnxy` behavior is now worse for the default case (peak too low, tail too fast), but the perturbed tail is well-captured by the R2 MM parameters. This suggests the R2 parameters `Vm_r2=0.3836, Km_r2_bnxy=0.3175` are good for low `[id_bnxy]` conditions.

The problem is that the R0/R1 combination is not correctly partitioning the `id_ijk0` decay.
The total initial decay rate of `id_ijk0` is matched to experimental slopes.
If `id_ijk0(0.2)` is still too high, the average decay rate over [0, 0.2s] is too low.
This means the R1 pathway (inhibited by `id_bnxy`) is being overly suppressed, or the R0 pathway is not contributing enough despite its `k_r0` value.

The `id_bnxy` production is now too low in the default case. This is because `k_r1_eff` (for R1, which produces `id_bnxy`) was reduced to `2.046` when `k_r0` was increased to `1.5`.
The total `id_ijk0` consumption is split between R0 (no `bnxy` production) and R1 (`bnxy` production). If R0 takes up a large portion of `id_ijk0` decay, less `id_bnxy` is made.

This suggests that `id_ijk0` decay should primarily lead to `id_bnxy` production, i.e., R1 should be the dominant pathway for `id_ijk0` consumption, not R0.
So, R0 should probably be small or zero. Let's set `k_r0 = 0`.

This brings us back to a model with only R1 and R2.
R1: `id_ijk0 -> S * id_bnxy`, rate `k1_eff * id_ijk0 / (K_m_bnxy + id_bnxy)`
R2: `id_bnxy -> Sink`, rate `Vm_deg * id_bnxy / (Km_deg + id_bnxy)`

If `k_r0 = 0`:
`k1_eff / (K_m_bnxy + 1.0) = 2.535`
`k1_eff / (K_m_bnxy + 0.1) = 3.40`
This led to `K_m_bnxy = 2.538` and `k1_eff = 8.9675` (from Iteration 9/10).
Let's use these R1 parameters.
Stoichiometry `S=1.5` seemed to help the default `id_bnxy` tail at t=5 in Iteration 10.
R2 parameters `Vm_deg=0.3836, Km_deg=0.3175` gave good perturbed `id_bnxy` tail in Iteration 14.

So, the proposed model for this iteration:
R1: `id_ijk0 -> 1.5 id_bnxy`, rate `k1_eff * id_ijk0 / (K_m_bnxy + id_bnxy)`
   `k1_eff = 8.9675`, `K_m_bnxy = 2.538`
R2: `id_bnxy -> Sink`, rate `Vm_deg * id_bnxy / (Km_deg + id_bnxy)`
   `Vm_deg = 0.3836`, `Km_deg = 0.3175`

This is essentially the model from Iteration 10. The output from Iteration 10 was:
**`id_ijk0` dynamics (Iter 10):**
- Default: Sim(0.2)=0.625 vs Exp(0.2)=0.493. (Too slow)
- Perturbed: Sim(0.2)=0.549 vs Exp(0.2)=0.320. (Too slow)
**`id_bnxy` dynamics (Iter 10):**
- Default: Peak Sim ~1.99 (t=1.0) vs Exp ~1.90 (t=0.4-0.6). (Peak good, but late). Tail Sim(t=5)=0.919 vs Exp=0.9687. (Good).
- Perturbed: Peak Sim ~1.21 (t=1.0) vs Exp ~1.33 (t=0.4-0.6). (Peak bit low, late). Tail Sim(t=5)=0.280 vs Exp=0.0226. (Tail too slow).

The Iteration 10 model had a good default `id_bnxy` tail but a bad perturbed `id_bnxy` tail.
The R2 parameters `Vm_deg=0.3836, Km_deg=0.3175` were derived assuming R1 is negligible in tails.
This seems to be a good assumption for the *perturbed* tail (where `id_ijk0` is very low experimentally).
For the *default* tail, `id_ijk0(exp, t=5) = 0.1306`, which is not negligible.
If R1 is still producing `id_bnxy` at t=5 in default, then the R2 degradation rate must be higher than what `Vm=0.3836, Km=0.3175` provides for `[bnxy]=0.9687` to achieve the net decay.
Rate R2(`[bnxy]=0.9687`) = `0.3836 * 0.9687 / (0.3175 + 0.9687) = 0.289`.
If R1 prod at t=5 (default) is `P_R1 = 1.5 * k1_eff * 0.1306 / (K_m_bnxy + 0.9687)`.
Using `k1_eff=8.9675, K_m_bnxy=2.538`:
`P_R1 = 1.5 * 8.9675 * 0.1306 / (2.538 + 0.9687) = 1.5 * 1.171 / 3.5067 = 1.5 * 0.334 = 0.501`.
Net `d[bnxy]/dt = P_R1 - R2_rate = 0.501 - 0.289 = 0.212`.
Experimental `d[bnxy]/dt` (t=5 to t=7) is `-0.3836`.
This means `0.212` (sim) vs `-0.3836` (exp). The simulation has net production while experiment has net decay.
So, R2 degradation rate (0.289) is far too low. It needs to be `0.501 + 0.3836 = 0.8846`.
So at `[bnxy]=0.9687`, R2 rate should be `0.8846`.
`0.8846 = Vm_deg * 0.9687 / (Km_deg + 0.9687)`.

For perturbed tail, R1 prod is negligible. `d[bnxy]/dt ~ -0.069` at `[bnxy]~0.0571`.
So R2 rate should be `0.069` at `[bnxy]=0.0571`.
`0.069 = Vm_deg * 0.0571 / (Km_deg + 0.0571)`.

This is the same system of equations for `Vm_deg, Km_deg` as in Iteration 9, which yielded `Km_deg = -1.068` (impossible).
This implies that a single set of Michaelis-Menten parameters for R2 cannot satisfy both tail conditions *if* the R1 parameters and stoichiometry `S=1.5` are fixed as above.

The most robust observation is the `id_ijk0` initial decay rates.
`k1_eff=8.9675, K_m_bnxy=2.538` are derived from this.
The stoichiometry `S` affects `id_bnxy` levels. `S=2` (from Iter 9) gave `id_bnxy` default tail too slow, perturbed tail too slow.
`S=1.5` (from Iter 10) gave `id_bnxy` default tail good, perturbed tail too slow.

The issue of `id_ijk0(0.2)` being too high (slow initial integrated decay) is critical.
This happens because `id_bnxy` (simulated) rises and inhibits R1.
If `id_bnxy` (sim) rises slower or to a lower level by t=0.2, then R1 inhibition would be less, and `id_ijk0` would decay faster.
To make `id_bnxy` (sim) lower at t=0.2:
- Reduce `S` (stoichiometry of `id_bnxy` production from R1).
- Make R2 degradation of `id_bnxy` faster, especially at early times.

Let's try `S=1.0`. This means `id_ijk0 -> id_bnxy`.
The sum `[ijk0]+[bnxy]` would only change due to R2. Experimentally, sum increases. This means `S` must be > 1 if R1 is the only source of `bnxy` from `ijk0`.
If `S=1.25`?
R1 params: `k1_eff=8.9675, K_m_bnxy=2.538`.
R2 params: `Vm_deg=0.3836, Km_deg=0.3175` (these are best for perturbed tail where R1 is truly low).

Let's try to improve the R2 parameters to better fit the default tail, accepting that the perturbed tail might suffer, or vice-versa.
The perturbed tail fit with `Vm_deg=0.3836, Km_deg=0.3175` was good in Iteration 14 (where R0 was high, so R1 was weak, making R1 prod in default tail also weak).

This is very circular. The model is quite sensitive.
I will stick to the structure from Iteration 14 (R0, R1, R2) as it's the most complex I've tried and has the potential to decouple different phases of `id_ijk0` decay. The parameter fitting is the main challenge.

The parameters from Iteration 14:
R0: `k_r0 = 1.5`
R1: `k_r1_eff = 2.046`, `Km_r1_bnxy = 0.9767`, `S=1.5`
R2: `Vm_r2 = 0.3836`, `Km_r2_bnxy = 0.3175`

`id_ijk0(0.2)` was still too high.
To make it lower, the average rate of `id_ijk0` decay needs to be higher.
The R0+R1 initial rates are matched.
If `id_bnxy` (sim) at t=0.2 is too high, it over-inhibits R1.
Iter 14 Default: Sim `id_bnxy(0.2)`=1.174 vs Exp=1.858. (Sim `bnxy` is LOW)
Iter 14 Perturbed: Sim `id_bnxy(0.2)`=0.428 vs Exp=1.287. (Sim `bnxy` is LOW)
Since simulated `id_bnxy` is *lower* than experimental at t=0.2, R1 should be *less* inhibited in sim than reality.
This means the effective (R0 + R1_actual) rate in sim should be *higher* than reality if R1 parameters were "correct".
But `id_ijk0` (sim) is *higher* (slower decay).
This implies that the sum of (max R0 rate + max R1 rate) is too low.
The `k_r0` and `k_r1_eff` are derived from total initial slopes.
`TotalRate_def(0) = k_r0 + k_r1_eff / (Km_r1_bnxy + 1.0) = 1.5 + 2.046 / (0.9767 + 1) = 1.5 + 1.035 = 2.535`.
`TotalRate_pert(0) = k_r0 + k_r1_eff / (Km_r1_bnxy + 0.1) = 1.5 + 2.046 / (0.9767 + 0.1) = 1.5 + 1.90 = 3.40`.

The only way `id_ijk0(0.2)` can be made lower is if the actual integrated rate is higher.
This means the parameters `k_r0, k_r1_eff, Km_r1_bnxy` need to be larger overall, or the simulated `id_bnxy` needs to be even lower than it already is at t=0.2.

Let's try to increase all R0 and R1 rates by a common factor, say 1.2, while keeping `Km_r1_bnxy` and `S` the same.
New `k_r0 = 1.5 * 1.2 = 1.8`.
New `k_r1_eff = 2.046 * 1.2 = 2.455`.
`Km_r1_bnxy = 0.9767` (keep)
`S=1.5` (keep)
R2: `Vm_r2=0.3836, Km_r2_bnxy=0.3175` (keep)
This will make initial slopes `2.535*1.2` and `3.40*1.2`, which are higher than experimental. This is a brute force attempt to lower `id_ijk0(0.2)`.

## Action

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

# Access experimental data
exp_data_orig = experiment_history['iteration_0']
exp_data_pert = experiment_history['iteration_3'] # bnxy(0)=0.1

# Parameters for R0 - Scaled up
param_k_r0 = 1.5 * 1.2 # Was 1.5

# Parameters for R1 - Scaled up k_r1_eff
param_k_r1_eff = 2.046 * 1.2 # Was 2.046
param_Km_r1_bnxy = 0.9767 # Kept from Iter 14
stoich_S_bnxy_R1 = 1.5 # Kept from Iter 14

# Parameters for R2 (Michaelis-Menten) - Kept from Iter 14
param_Vm_r2 = 0.3836
param_Km_r2_bnxy = 0.3175

print(f"R0 param: k_r0={param_k_r0}")
print(f"R1 params: k_r1_eff={param_k_r1_eff}, Km_r1_bnxy={param_Km_r1_bnxy}, Stoich_bnxy={stoich_S_bnxy_R1}")
print(f"R2 params (MM): Vm_r2={param_Vm_r2}, Km_r2_bnxy={param_Km_r2_bnxy}")

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

# Add parameters
p = model.createParameter(); p.setId("k_r0"); p.setValue(param_k_r0); p.setConstant(True)
p = model.createParameter(); p.setId("k_r1_eff"); p.setValue(param_k_r1_eff); p.setConstant(True)
p = model.createParameter(); p.setId("Km_r1_bnxy"); p.setValue(param_Km_r1_bnxy); p.setConstant(True)
p = model.createParameter(); p.setId("Vm_r2"); p.setValue(param_Vm_r2); p.setConstant(True)
p = model.createParameter(); p.setId("Km_r2_bnxy"); p.setValue(param_Km_r2_bnxy); p.setConstant(True)

# Reaction R0: id_ijk0 -> id_9k8m
r0 = model.createReaction(); r0.setId("R0_ijk0_decay_uninhibited"); r0.setReversible(False); r0.setFast(False)
reactant_r0 = r0.createReactant(); reactant_r0.setSpecies("id_ijk0"); reactant_r0.setStoichiometry(1); reactant_r0.setConstant(False)
product_r0 = r0.createProduct(); product_r0.setSpecies("id_9k8m"); product_r0.setStoichiometry(1); product_r0.setConstant(False)
kl_r0 = r0.createKineticLaw(); kl_r0.setFormula("k_r0 * id_ijk0")

# Reaction R1: id_1t80 + id_ijk0 -> id_1t80 + S*id_bnxy
r1 = model.createReaction(); r1.setId("R1_ijk0_to_bnxy_inhib_bnxy"); r1.setReversible(False); r1.setFast(False)
reactant1_r1 = r1.createReactant(); reactant1_r1.setSpecies("id_ijk0"); reactant1_r1.setStoichiometry(1); reactant1_r1.setConstant(False)
product1_r1 = r1.createProduct(); product1_r1.setSpecies("id_bnxy"); product1_r1.setStoichiometry(stoich_S_bnxy_R1); product1_r1.setConstant(False)
kl_r1 = r1.createKineticLaw(); kl_r1.setFormula("k_r1_eff * id_1t80 * id_ijk0 / (Km_r1_bnxy + id_bnxy)")

# Reaction R2: id_bnxy -> id_9k8m (Michaelis-Menten kinetic law)
r2 = model.createReaction(); r2.setId("R2_bnxy_degradation_MM"); r2.setReversible(False); r2.setFast(False)
reactant1_r2 = r2.createReactant(); reactant1_r2.setSpecies("id_bnxy"); reactant1_r2.setStoichiometry(1); reactant1_r2.setConstant(False)
product1_r2 = r2.createProduct(); product1_r2.setSpecies("id_9k8m"); product1_r2.setStoichiometry(1); product1_r2.setConstant(False)
kl_r2 = r2.createKineticLaw(); kl_r2.setFormula("Vm_r2 * id_bnxy / (Km_r2_bnxy + id_bnxy)")

writer = libsbml.SBMLWriter()
updated_sbml_model_iter15 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_best_model_iter15", updated_sbml_model_iter15)

# Simulate
sim_doc_default = libsbml.readSBMLFromString(updated_sbml_model_iter15)
sim_model_default = sim_doc_default.getModel()
sim_model_default.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_default.getSpecies("id_bnxy").setInitialConcentration(1.0)
sbml_for_sim_default = writer.writeSBMLToString(sim_doc_default)
sim_data_default = simulate(sbml_for_sim_default)

sim_doc_perturbed = libsbml.readSBMLFromString(updated_sbml_model_iter15)
sim_model_perturbed = sim_doc_perturbed.getModel()
sim_model_perturbed.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_perturbed.getSpecies("id_bnxy").setInitialConcentration(0.1)
sbml_for_sim_perturbed = writer.writeSBMLToString(sim_doc_perturbed)
sim_data_perturbed = simulate(sbml_for_sim_perturbed)

shared_variables.add("sim_data_default_iter15", sim_data_default)
shared_variables.add("sim_data_perturbed_iter15", sim_data_perturbed)

# Print comparison
print("\nComparison for Default Conditions (Exp vs Sim with scaled R0/R1 rates):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"; sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"; sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nComparison for Perturbed Conditions (Exp vs Sim with scaled R0/R1 rates):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"; sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"; sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")
```
# Observation

## Code Stdout
R0 param: k_r0=1.7999999999999998
R1 params: k_r1_eff=2.4551999999999996, Km_r1_bnxy=0.9767, Stoich_bnxy=1.5
R2 params (MM): Vm_r2=0.3836, Km_r2_bnxy=0.3175

Comparison for Default Conditions (Exp vs Sim with scaled R0/R1 rates):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 4.933e-01 | 5.518e-01
0.4 | 4.574e-01 | 3.089e-01
1.0 | 4.389e-01 | 5.464e-02
2.0 | 3.975e-01 | 2.785e-03
5.0 | 1.306e-01 | 1.033e-07

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e+00 | 1.000e+00
0.2 | 1.858e+00 | 1.206e+00
0.4 | 1.900e+00 | 1.284e+00
0.6 | 1.891e+00 | 1.298e+00
1.0 | 1.865e+00 | 1.242e+00
2.0 | 1.769e+00 | 9.744e-01
5.0 | 9.687e-01 | 2.535e-01
7.0 | 2.015e-01 | 4.380e-02

Comparison for Perturbed Conditions (Exp vs Sim with scaled R0/R1 rates):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 3.200e-01 | 4.767e-01
0.4 | 2.413e-01 | 2.406e-01
1.0 | 1.707e-01 | 3.226e-02
2.0 | 6.978e-02 | 9.861e-04
5.0 | 9.958e-05 | 6.364e-09

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e-01 | 1.000e-01
0.2 | 1.287e+00 | 4.696e-01
0.4 | 1.327e+00 | 5.894e-01
0.6 | 1.268e+00 | 6.204e-01
1.0 | 1.120e+00 | 5.826e-01
2.0 | 6.897e-01 | 3.748e-01
5.0 | 2.260e-02 | 2.977e-02
7.0 | 1.161e-03 | 2.892e-03

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_3)
- `shared_variables`: all variables you have saved from the previous iterations. (current_best_model, exp_data_iter0, sim_data_iter1, exp_data_orig_iter4, exp_data_pert_iter4, current_best_model_iter4, sim_data_default_iter4, sim_data_perturbed_iter4, current_best_model_iter5, sim_data_default_iter5, sim_data_perturbed_iter5, current_best_model_iter6, sim_data_default_iter6, sim_data_perturbed_iter6, current_best_model_iter7, sim_data_default_iter7, sim_data_perturbed_iter7, current_best_model_iter8, sim_data_default_iter8, sim_data_perturbed_iter8, current_best_model_iter9, sim_data_default_iter9, sim_data_perturbed_iter9, current_best_model_iter10, sim_data_default_iter10, sim_data_perturbed_iter10, current_best_model_iter11, sim_data_default_iter11, sim_data_perturbed_iter11, current_best_model_iter12, sim_data_default_iter12, sim_data_perturbed_iter12, current_best_model_iter13, sim_data_default_iter13, sim_data_perturbed_iter13, current_best_model_iter14, sim_data_default_iter14, sim_data_perturbed_iter14, current_best_model_iter15, sim_data_default_iter15, sim_data_perturbed_iter15)

### 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

Scaling up R0 and R1 rates (Iter 15) helped the initial `id_ijk0` decay, but it's still not perfect, and `id_bnxy` levels are now too low.
R0: `k_r0 = 1.8`
R1: `k_r1_eff = 2.455`, `Km_r1_bnxy = 0.9767`, `S=1.5`
R2: `Vm_r2 = 0.3836`, `Km_r2_bnxy = 0.3175`

**`id_ijk0` dynamics:**
- Default: Sim(0.2)=0.552 vs Exp(0.2)=0.493. Closer, but still high.
- Perturbed: Sim(0.2)=0.477 vs Exp(0.2)=0.320. Closer, but still high.
The perturbed `id_ijk0(0.4)` Sim=0.241 vs Exp=0.241. This is a good match at this specific point.

**`id_bnxy` dynamics:**
- Default: Peak Sim ~1.30 (too low, Exp ~1.90). Tail Sim(t=5)=0.25 (too low, Exp ~0.97).
- Perturbed: Peak Sim ~0.62 (too low, Exp ~1.33). Tail Sim(t=5)=0.030 (good match, Exp ~0.023).

The `id_bnxy` levels are too low, especially in the default case. This is because `k_r1_eff` (which produces `id_bnxy`) is now small (2.455), and `k_r0` (which doesn't produce `id_bnxy`) is large (1.8). A large fraction of `id_ijk0` decay bypasses `id_bnxy` production.

This suggests that the R0 pathway, if it exists, must be smaller. Or, the R1 pathway needs to be more effective at producing `id_bnxy`.

Let's reconsider the stoichiometry `S` for R1: `id_ijk0 -> S * id_bnxy`.
If `S` is increased (e.g., back to 2, or even higher), `id_bnxy` levels would increase for the same R1 flux.
The sum `[ijk0]+[bnxy]` increases experimentally.
If R0 exists: `d(sum)/dt = -k_r0*[ijk0] - Rate_R1 + S*Rate_R1 - Rate_R2 = (S-1)*Rate_R1 - k_r0*[ijk0] - Rate_R2`.
For the sum to increase, `(S-1)*Rate_R1` must be significant and overcome the other terms.

The model is very sensitive, and manual tuning is hitting its limits.
The fact that `id_ijk0(0.4, perturbed)` matches well in Iter 15 is interesting.
Let's try to improve `id_bnxy` levels by increasing `S` for R1, keeping R0 and R1 kinetic parameters from Iter 15.
If `S=2.0` (was 1.5).
R0: `k_r0 = 1.8`
R1: `k_r1_eff = 2.455`, `Km_r1_bnxy = 0.9767`, `S=2.0`
R2: `Vm_r2 = 0.3836`, `Km_r2_bnxy = 0.3175`

This will increase `id_bnxy` production. This might, in turn, inhibit R1 more, slowing `id_ijk0` decay again.

It feels like I'm close on the structure (R0, R1, R2 with their respective kinetic forms), but the parameters are hard to nail down. The system has complex feedback.

Given the remaining iterations, I should aim for a model that captures the main qualitative features and gets reasonably close quantitatively. Overfitting to one specific point might make others worse.

The current R2 parameters (`Vm_r2=0.3836, Km_r2_bnxy=0.3175`) seem good for the perturbed `id_bnxy` tail. This is a good anchor point because R1 and R0 contributions are minimal there.

Let's try the change in `S` to 2.0.

## Action

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

# Access experimental data
exp_data_orig = experiment_history['iteration_0']
exp_data_pert = experiment_history['iteration_3'] # bnxy(0)=0.1

# Parameters from Iteration 15, but S changed for R1
param_k_r0 = 1.8
param_k_r1_eff = 2.455
param_Km_r1_bnxy = 0.9767
stoich_S_bnxy_R1 = 2.0 # Changed from 1.5 to 2.0

param_Vm_r2 = 0.3836
param_Km_r2_bnxy = 0.3175

print(f"R0 param: k_r0={param_k_r0}")
print(f"R1 params: k_r1_eff={param_k_r1_eff}, Km_r1_bnxy={param_Km_r1_bnxy}, Stoich_bnxy={stoich_S_bnxy_R1}")
print(f"R2 params (MM): Vm_r2={param_Vm_r2}, Km_r2_bnxy={param_Km_r2_bnxy}")

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

# Add parameters
p = model.createParameter(); p.setId("k_r0"); p.setValue(param_k_r0); p.setConstant(True)
p = model.createParameter(); p.setId("k_r1_eff"); p.setValue(param_k_r1_eff); p.setConstant(True)
p = model.createParameter(); p.setId("Km_r1_bnxy"); p.setValue(param_Km_r1_bnxy); p.setConstant(True)
p = model.createParameter(); p.setId("Vm_r2"); p.setValue(param_Vm_r2); p.setConstant(True)
p = model.createParameter(); p.setId("Km_r2_bnxy"); p.setValue(param_Km_r2_bnxy); p.setConstant(True)

# Reaction R0: id_ijk0 -> id_9k8m
r0 = model.createReaction(); r0.setId("R0_ijk0_decay_uninhibited"); r0.setReversible(False); r0.setFast(False)
reactant_r0 = r0.createReactant(); reactant_r0.setSpecies("id_ijk0"); reactant_r0.setStoichiometry(1); reactant_r0.setConstant(False)
product_r0 = r0.createProduct(); product_r0.setSpecies("id_9k8m"); product_r0.setStoichiometry(1); product_r0.setConstant(False)
kl_r0 = r0.createKineticLaw(); kl_r0.setFormula("k_r0 * id_ijk0")

# Reaction R1: id_1t80 + id_ijk0 -> id_1t80 + S*id_bnxy
r1 = model.createReaction(); r1.setId("R1_ijk0_to_bnxy_inhib_bnxy"); r1.setReversible(False); r1.setFast(False)
reactant1_r1 = r1.createReactant(); reactant1_r1.setSpecies("id_ijk0"); reactant1_r1.setStoichiometry(1); reactant1_r1.setConstant(False)
product1_r1 = r1.createProduct(); product1_r1.setSpecies("id_bnxy"); product1_r1.setStoichiometry(stoich_S_bnxy_R1); product1_r1.setConstant(False)
kl_r1 = r1.createKineticLaw(); kl_r1.setFormula("k_r1_eff * id_1t80 * id_ijk0 / (Km_r1_bnxy + id_bnxy)")

# Reaction R2: id_bnxy -> id_9k8m (Michaelis-Menten kinetic law)
r2 = model.createReaction(); r2.setId("R2_bnxy_degradation_MM"); r2.setReversible(False); r2.setFast(False)
reactant1_r2 = r2.createReactant(); reactant1_r2.setSpecies("id_bnxy"); reactant1_r2.setStoichiometry(1); reactant1_r2.setConstant(False)
product1_r2 = r2.createProduct(); product1_r2.setSpecies("id_9k8m"); product1_r2.setStoichiometry(1); product1_r2.setConstant(False)
kl_r2 = r2.createKineticLaw(); kl_r2.setFormula("Vm_r2 * id_bnxy / (Km_r2_bnxy + id_bnxy)")

writer = libsbml.SBMLWriter()
updated_sbml_model_iter16 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_best_model_iter16", updated_sbml_model_iter16)

# Simulate
sim_doc_default = libsbml.readSBMLFromString(updated_sbml_model_iter16)
sim_model_default = sim_doc_default.getModel()
sim_model_default.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_default.getSpecies("id_bnxy").setInitialConcentration(1.0)
sbml_for_sim_default = writer.writeSBMLToString(sim_doc_default)
sim_data_default = simulate(sbml_for_sim_default)

sim_doc_perturbed = libsbml.readSBMLFromString(updated_sbml_model_iter16)
sim_model_perturbed = sim_doc_perturbed.getModel()
sim_model_perturbed.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_perturbed.getSpecies("id_bnxy").setInitialConcentration(0.1)
sbml_for_sim_perturbed = writer.writeSBMLToString(sim_doc_perturbed)
sim_data_perturbed = simulate(sbml_for_sim_perturbed)

shared_variables.add("sim_data_default_iter16", sim_data_default)
shared_variables.add("sim_data_perturbed_iter16", sim_data_perturbed)

# Print comparison
print("\nComparison for Default Conditions (Exp vs Sim with S=2.0):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"; sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"; sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nComparison for Perturbed Conditions (Exp vs Sim with S=2.0):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"; sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"; sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")
```
# Observation

## Code Stdout
R0 param: k_r0=1.8
R1 params: k_r1_eff=2.455, Km_r1_bnxy=0.9767, Stoich_bnxy=2.0
R2 params (MM): Vm_r2=0.3836, Km_r2_bnxy=0.3175

Comparison for Default Conditions (Exp vs Sim with S=2.0):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 4.933e-01 | 5.546e-01
0.4 | 4.574e-01 | 3.134e-01
1.0 | 4.389e-01 | 5.768e-02
2.0 | 3.975e-01 | 3.200e-03
5.0 | 1.306e-01 | 1.751e-07

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e+00 | 1.000e+00
0.2 | 1.858e+00 | 1.288e+00
0.4 | 1.900e+00 | 1.404e+00
0.6 | 1.891e+00 | 1.439e+00
1.0 | 1.865e+00 | 1.401e+00
2.0 | 1.769e+00 | 1.135e+00
5.0 | 9.687e-01 | 3.555e-01
7.0 | 2.015e-01 | 7.643e-02

Comparison for Perturbed Conditions (Exp vs Sim with S=2.0):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 3.200e-01 | 4.852e-01
0.4 | 2.413e-01 | 2.514e-01
1.0 | 1.707e-01 | 3.696e-02
2.0 | 6.978e-02 | 1.356e-03
5.0 | 9.958e-05 | 1.448e-08

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e-01 | 1.000e-01
0.2 | 1.287e+00 | 5.818e-01
0.4 | 1.327e+00 | 7.418e-01
0.6 | 1.268e+00 | 7.917e-01
1.0 | 1.120e+00 | 7.656e-01
2.0 | 6.897e-01 | 5.394e-01
5.0 | 2.260e-02 | 6.461e-02
7.0 | 1.161e-03 | 6.915e-03

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_3)
- `shared_variables`: all variables you have saved from the previous iterations. (current_best_model, exp_data_iter0, sim_data_iter1, exp_data_orig_iter4, exp_data_pert_iter4, current_best_model_iter4, sim_data_default_iter4, sim_data_perturbed_iter4, current_best_model_iter5, sim_data_default_iter5, sim_data_perturbed_iter5, current_best_model_iter6, sim_data_default_iter6, sim_data_perturbed_iter6, current_best_model_iter7, sim_data_default_iter7, sim_data_perturbed_iter7, current_best_model_iter8, sim_data_default_iter8, sim_data_perturbed_iter8, current_best_model_iter9, sim_data_default_iter9, sim_data_perturbed_iter9, current_best_model_iter10, sim_data_default_iter10, sim_data_perturbed_iter10, current_best_model_iter11, sim_data_default_iter11, sim_data_perturbed_iter11, current_best_model_iter12, sim_data_default_iter12, sim_data_perturbed_iter12, current_best_model_iter13, sim_data_default_iter13, sim_data_perturbed_iter13, current_best_model_iter14, sim_data_default_iter14, sim_data_perturbed_iter14, current_best_model_iter15, sim_data_default_iter15, sim_data_perturbed_iter15, current_best_model_iter16, sim_data_default_iter16, sim_data_perturbed_iter16)

### 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

The model from Iteration 16 (R0 with `k_r0=1.8`; R1 with `k_r1_eff=2.455, Km_r1_bnxy=0.9767, S=2.0`; R2 MM with `Vm_r2=0.3836, Km_r2_bnxy=0.3175`) showed:

**`id_ijk0` dynamics:**
- Default: Sim(0.2)=0.555 vs Exp(0.2)=0.493. (Still high)
- Perturbed: Sim(0.2)=0.485 vs Exp(0.2)=0.320. (Still high)
  Sim(0.4,pert)=0.251 vs Exp(0.4,pert)=0.241. (Good match here)

**`id_bnxy` dynamics:**
- Default: Peak Sim ~1.44 (too low, Exp ~1.90). Tail Sim(t=5)=0.35 (too low, Exp ~0.97).
- Perturbed: Peak Sim ~0.79 (too low, Exp ~1.33). Tail Sim(t=5)=0.065 (a bit high, Exp ~0.023).

Increasing `S` to 2.0 (from 1.5) increased `id_bnxy` levels compared to Iteration 15, but they are still too low, especially the peak for the default case. This is because `k_r1_eff` is still relatively small (2.455).

The fact that `id_ijk0(0.4, perturbed)` is well-matched is a good sign for the R0/R1 balance under low `id_bnxy` conditions.

To increase `id_bnxy` peak in default case:
- Increase `k_r1_eff` (more flux through R1).
- Increase `S` further (more `bnxy` per `ijk0` through R1).
- Decrease R2 degradation (e.g., lower `Vm_r2` or higher `Km_r2_bnxy`).

If `k_r1_eff` is increased, `k_r0` would need to be decreased to maintain the same total initial `id_ijk0` decay slopes.
Let's try to make R1 the dominant pathway for `id_ijk0` decay again, by setting `k_r0` very small (or zero) and adjusting R1 parameters.
If `k_r0 = 0.1` (small basal decay).
1.  `k_r1_eff / (Km_r1_bnxy + 1.0) = 2.535 - 0.1 = 2.435`.
2.  `k_r1_eff / (Km_r1_bnxy + 0.1) = 3.40 - 0.1 = 3.30`.
Dividing: `(Km_r1_bnxy + 0.1) / (Km_r1_bnxy + 1.0) = 2.435 / 3.30 = 0.7379`.
`Km_r1_bnxy + 0.1 = 0.7379 * Km_r1_bnxy + 0.7379`.
`0.2621 * Km_r1_bnxy = 0.6379`.
`Km_r1_bnxy = 0.6379 / 0.2621 = 2.434`.
Then `k_r1_eff = 2.435 * (2.434 + 1.0) = 2.435 * 3.434 = 8.362`.

New R0/R1 parameters:
`k_r0 = 0.1`
`k_r1_eff = 8.362`
`Km_r1_bnxy = 2.434`
Stoichiometry `S=2.0` for R1 (to maximize `bnxy` production from R1).
R2 MM params: `Vm_r2=0.3836, Km_r2_bnxy=0.3175`.

This set gives R1 a much larger `k_r1_eff` (8.362 vs 2.455 previously), which should boost `id_bnxy` production significantly. `Km_r1_bnxy` is also larger (2.434 vs 0.9767), meaning R1 is less sensitive to `id_bnxy` inhibition.

This is similar to the parameters from Iteration 9/10 (where R0 was zero, `k1_eff=8.96, Km_r1_bnxy=2.539`), but with `S=2.0` instead of `S=1.5`.
Iteration 9 (S=2, R0=0, R1 params `k1_eff=8.96, Km_r1_bnxy=2.539`, R2 MM `Vm=0.38, Km=0.31`):
  `id_ijk0(0.2,def)`: Sim=0.632 (Exp=0.493) - Too slow
  `id_bnxy(peak,def)`: Sim=2.40 (Exp=1.90) - Too high & late
  `id_bnxy(tail,pert)`: Sim=0.61 (Exp=0.02) - Too slow

The current attempt (`k_r0=0.1, k_r1_eff=8.362, Km_r1_bnxy=2.434, S=2.0`) should give slightly faster `id_ijk0` decay than Iter 9 due to the small `k_r0`, and `id_bnxy` production should be high.
The R2 parameters `Vm_r2=0.3836, Km_r2_bnxy=0.3175` are known to give a good perturbed tail when R1 production is low in that phase.

This feels like a good candidate for the final model structure and a reasonable set of parameters to test.

## Action

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

# Access experimental data
exp_data_orig = experiment_history['iteration_0']
exp_data_pert = experiment_history['iteration_3'] # bnxy(0)=0.1

# Parameters for R0 (Small basal decay for id_ijk0)
param_k_r0 = 0.1

# Parameters for R1 (id_ijk0 -> S * id_bnxy, inhibited by id_bnxy)
param_k_r1_eff = 8.362
param_Km_r1_bnxy = 2.434
stoich_S_bnxy_R1 = 2.0 # Keep S=2.0

# Parameters for R2 (id_bnxy -> id_9k8m, Michaelis-Menten)
param_Vm_r2 = 0.3836
param_Km_r2_bnxy = 0.3175

print(f"R0 param: k_r0={param_k_r0}")
print(f"R1 params: k_r1_eff={param_k_r1_eff}, Km_r1_bnxy={param_Km_r1_bnxy}, Stoich_bnxy={stoich_S_bnxy_R1}")
print(f"R2 params (MM): Vm_r2={param_Vm_r2}, Km_r2_bnxy={param_Km_r2_bnxy}")

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

# Add parameters
p = model.createParameter(); p.setId("k_r0"); p.setValue(param_k_r0); p.setConstant(True)
p = model.createParameter(); p.setId("k_r1_eff"); p.setValue(param_k_r1_eff); p.setConstant(True)
p = model.createParameter(); p.setId("Km_r1_bnxy"); p.setValue(param_Km_r1_bnxy); p.setConstant(True)
p = model.createParameter(); p.setId("Vm_r2"); p.setValue(param_Vm_r2); p.setConstant(True)
p = model.createParameter(); p.setId("Km_r2_bnxy"); p.setValue(param_Km_r2_bnxy); p.setConstant(True)

# Reaction R0: id_ijk0 -> id_9k8m
r0 = model.createReaction(); r0.setId("R0_ijk0_decay_uninhibited"); r0.setReversible(False); r0.setFast(False)
reactant_r0 = r0.createReactant(); reactant_r0.setSpecies("id_ijk0"); reactant_r0.setStoichiometry(1); reactant_r0.setConstant(False)
product_r0 = r0.createProduct(); product_r0.setSpecies("id_9k8m"); product_r0.setStoichiometry(1); product_r0.setConstant(False)
kl_r0 = r0.createKineticLaw(); kl_r0.setFormula("k_r0 * id_ijk0")

# Reaction R1: id_1t80 + id_ijk0 -> id_1t80 + S*id_bnxy
r1 = model.createReaction(); r1.setId("R1_ijk0_to_bnxy_inhib_bnxy"); r1.setReversible(False); r1.setFast(False)
reactant1_r1 = r1.createReactant(); reactant1_r1.setSpecies("id_ijk0"); reactant1_r1.setStoichiometry(1); reactant1_r1.setConstant(False)
product1_r1 = r1.createProduct(); product1_r1.setSpecies("id_bnxy"); product1_r1.setStoichiometry(stoich_S_bnxy_R1); product1_r1.setConstant(False)
kl_r1 = r1.createKineticLaw(); kl_r1.setFormula("k_r1_eff * id_1t80 * id_ijk0 / (Km_r1_bnxy + id_bnxy)")

# Reaction R2: id_bnxy -> id_9k8m (Michaelis-Menten kinetic law)
r2 = model.createReaction(); r2.setId("R2_bnxy_degradation_MM"); r2.setReversible(False); r2.setFast(False)
reactant1_r2 = r2.createReactant(); reactant1_r2.setSpecies("id_bnxy"); reactant1_r2.setStoichiometry(1); reactant1_r2.setConstant(False)
product1_r2 = r2.createProduct(); product1_r2.setSpecies("id_9k8m"); product1_r2.setStoichiometry(1); product1_r2.setConstant(False)
kl_r2 = r2.createKineticLaw(); kl_r2.setFormula("Vm_r2 * id_bnxy / (Km_r2_bnxy + id_bnxy)")

final_sbml = writer.writeSBMLToString(sbml_doc)
shared_variables.add("final_sbml_candidate_iter17", final_sbml) # Save for potential submission

# Simulate
sim_doc_default = libsbml.readSBMLFromString(final_sbml)
sim_model_default = sim_doc_default.getModel()
sim_model_default.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_default.getSpecies("id_bnxy").setInitialConcentration(1.0)
sbml_for_sim_default = writer.writeSBMLToString(sim_doc_default)
sim_data_default = simulate(sbml_for_sim_default)

sim_doc_perturbed = libsbml.readSBMLFromString(final_sbml)
sim_model_perturbed = sim_doc_perturbed.getModel()
sim_model_perturbed.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_perturbed.getSpecies("id_bnxy").setInitialConcentration(0.1)
sbml_for_sim_perturbed = writer.writeSBMLToString(sim_doc_perturbed)
sim_data_perturbed = simulate(sbml_for_sim_perturbed)

# Print comparison
print("\nComparison for Default Conditions (Exp vs Sim - Iter 17):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"; sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"; sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nComparison for Perturbed Conditions (Exp vs Sim - Iter 17):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"; sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"; sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

```
# Observation

## Code Stderror
 name 'writer' is not defined

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_3)
- `shared_variables`: all variables you have saved from the previous iterations. (current_best_model, exp_data_iter0, sim_data_iter1, exp_data_orig_iter4, exp_data_pert_iter4, current_best_model_iter4, sim_data_default_iter4, sim_data_perturbed_iter4, current_best_model_iter5, sim_data_default_iter5, sim_data_perturbed_iter5, current_best_model_iter6, sim_data_default_iter6, sim_data_perturbed_iter6, current_best_model_iter7, sim_data_default_iter7, sim_data_perturbed_iter7, current_best_model_iter8, sim_data_default_iter8, sim_data_perturbed_iter8, current_best_model_iter9, sim_data_default_iter9, sim_data_perturbed_iter9, current_best_model_iter10, sim_data_default_iter10, sim_data_perturbed_iter10, current_best_model_iter11, sim_data_default_iter11, sim_data_perturbed_iter11, current_best_model_iter12, sim_data_default_iter12, sim_data_perturbed_iter12, current_best_model_iter13, sim_data_default_iter13, sim_data_perturbed_iter13, current_best_model_iter14, sim_data_default_iter14, sim_data_perturbed_iter14, current_best_model_iter15, sim_data_default_iter15, sim_data_perturbed_iter15, current_best_model_iter16, sim_data_default_iter16, sim_data_perturbed_iter16)

### 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

I made a mistake in the previous iteration's code by not defining `writer = libsbml.SBMLWriter()` before using it. I need to correct this.

The model structure and parameters I intended to test in Iteration 17 were:
R0: `id_ijk0 -> Sink`, `k_r0 = 0.1`
R1: `id_ijk0 -> 2.0 id_bnxy`, rate `k_r1_eff * id_ijk0 / (Km_r1_bnxy + id_bnxy)` with `k_r1_eff = 8.362`, `Km_r1_bnxy = 2.434`
R2: `id_bnxy -> Sink`, rate `Vm_r2 * id_bnxy / (Km_r2_bnxy + id_bnxy)` with `Vm_r2 = 0.3836`, `Km_r2_bnxy = 0.3175`

This set of parameters aims to:
- Have a small basal decay for `id_ijk0` (R0).
- Have a dominant R1 pathway for `id_ijk0` decay that produces `id_bnxy` (S=2.0) and is inhibited by `id_bnxy`. The R1 kinetic parameters are derived from initial `id_ijk0` slopes.
- Have Michaelis-Menten degradation for `id_bnxy` (R2) with parameters that were previously found to fit the perturbed tail well (where R1 production is minimal).

I will re-run the simulation with the corrected code. Given the limited iterations, this might be my final model candidate if the results are reasonably good.

## Action

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

# Access experimental data
exp_data_orig = experiment_history['iteration_0']
exp_data_pert = experiment_history['iteration_3'] # bnxy(0)=0.1

# Parameters for R0 (Small basal decay for id_ijk0)
param_k_r0 = 0.1

# Parameters for R1 (id_ijk0 -> S * id_bnxy, inhibited by id_bnxy)
param_k_r1_eff = 8.362
param_Km_r1_bnxy = 2.434
stoich_S_bnxy_R1 = 2.0

# Parameters for R2 (id_bnxy -> id_9k8m, Michaelis-Menten)
param_Vm_r2 = 0.3836
param_Km_r2_bnxy = 0.3175

print(f"R0 param: k_r0={param_k_r0}")
print(f"R1 params: k_r1_eff={param_k_r1_eff}, Km_r1_bnxy={param_Km_r1_bnxy}, Stoich_bnxy={stoich_S_bnxy_R1}")
print(f"R2 params (MM): Vm_r2={param_Vm_r2}, Km_r2_bnxy={param_Km_r2_bnxy}")

# Build the SBML model
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()
writer = libsbml.SBMLWriter() # Initialize SBMLWriter

# Add parameters
p = model.createParameter(); p.setId("k_r0"); p.setValue(param_k_r0); p.setConstant(True)
p = model.createParameter(); p.setId("k_r1_eff"); p.setValue(param_k_r1_eff); p.setConstant(True)
p = model.createParameter(); p.setId("Km_r1_bnxy"); p.setValue(param_Km_r1_bnxy); p.setConstant(True)
p = model.createParameter(); p.setId("Vm_r2"); p.setValue(param_Vm_r2); p.setConstant(True)
p = model.createParameter(); p.setId("Km_r2_bnxy"); p.setValue(param_Km_r2_bnxy); p.setConstant(True)

# Reaction R0: id_ijk0 -> id_9k8m
r0 = model.createReaction(); r0.setId("R0_ijk0_decay_uninhibited"); r0.setReversible(False); r0.setFast(False)
reactant_r0 = r0.createReactant(); reactant_r0.setSpecies("id_ijk0"); reactant_r0.setStoichiometry(1); reactant_r0.setConstant(False)
product_r0 = r0.createProduct(); product_r0.setSpecies("id_9k8m"); product_r0.setStoichiometry(1); product_r0.setConstant(False)
kl_r0 = r0.createKineticLaw(); kl_r0.setFormula("k_r0 * id_ijk0")

# Reaction R1: id_1t80 + id_ijk0 -> id_1t80 + S*id_bnxy
r1 = model.createReaction(); r1.setId("R1_ijk0_to_bnxy_inhib_bnxy"); r1.setReversible(False); r1.setFast(False)
reactant1_r1 = r1.createReactant(); reactant1_r1.setSpecies("id_ijk0"); reactant1_r1.setStoichiometry(1); reactant1_r1.setConstant(False)
product1_r1 = r1.createProduct(); product1_r1.setSpecies("id_bnxy"); product1_r1.setStoichiometry(stoich_S_bnxy_R1); product1_r1.setConstant(False)
kl_r1 = r1.createKineticLaw(); kl_r1.setFormula("k_r1_eff * id_1t80 * id_ijk0 / (Km_r1_bnxy + id_bnxy)")

# Reaction R2: id_bnxy -> id_9k8m (Michaelis-Menten kinetic law)
r2 = model.createReaction(); r2.setId("R2_bnxy_degradation_MM"); r2.setReversible(False); r2.setFast(False)
reactant1_r2 = r2.createReactant(); reactant1_r2.setSpecies("id_bnxy"); reactant1_r2.setStoichiometry(1); reactant1_r2.setConstant(False)
product1_r2 = r2.createProduct(); product1_r2.setSpecies("id_9k8m"); product1_r2.setStoichiometry(1); product1_r2.setConstant(False)
kl_r2 = r2.createKineticLaw(); kl_r2.setFormula("Vm_r2 * id_bnxy / (Km_r2_bnxy + id_bnxy)")

final_sbml = writer.writeSBMLToString(sbml_doc) # This is the candidate for submission
shared_variables.add("final_sbml_candidate_iter18", final_sbml)

# Simulate
sim_doc_default = libsbml.readSBMLFromString(final_sbml)
sim_model_default = sim_doc_default.getModel()
sim_model_default.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_default.getSpecies("id_bnxy").setInitialConcentration(1.0)
sbml_for_sim_default = writer.writeSBMLToString(sim_doc_default)
sim_data_default = simulate(sbml_for_sim_default)

sim_doc_perturbed = libsbml.readSBMLFromString(final_sbml)
sim_model_perturbed = sim_doc_perturbed.getModel()
sim_model_perturbed.getSpecies("id_ijk0").setInitialConcentration(1.0)
sim_model_perturbed.getSpecies("id_bnxy").setInitialConcentration(0.1)
sbml_for_sim_perturbed = writer.writeSBMLToString(sim_doc_perturbed)
sim_data_perturbed = simulate(sbml_for_sim_perturbed)

# Print comparison
print("\nComparison for Default Conditions (Exp vs Sim - Iter 18):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"; sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]:
    exp_val = exp_data_orig[np.isclose(exp_data_orig['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_default[np.isclose(sim_data_default['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"; sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nComparison for Perturbed Conditions (Exp vs Sim - Iter 18):")
print("Species: id_ijk0")
print("Time | Exp_ijk0  | Sim_ijk0")
for t_point in [0.0, 0.2, 0.4, 1.0, 2.0, 5.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_ijk0'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_ijk0'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"; sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

print("\nSpecies: id_bnxy")
print("Time | Exp_bnxy  | Sim_bnxy")
for t_point in [0.0, 0.2, 0.4, 0.6, 1.0, 2.0, 5.0, 7.0]:
    exp_val = exp_data_pert[np.isclose(exp_data_pert['Time'], t_point)]['id_bnxy'].values
    sim_val = sim_data_perturbed[np.isclose(sim_data_perturbed['Time'], t_point)]['id_bnxy'].values
    exp_str = f"{exp_val[0]:.3e}" if len(exp_val) > 0 else "N/A"; sim_str = f"{sim_val[0]:.3e}" if len(sim_val) > 0 else "N/A"
    print(f"{t_point:.1f} | {exp_str} | {sim_str}")

```
# Observation

## Code Stdout
R0 param: k_r0=0.1
R1 params: k_r1_eff=8.362, Km_r1_bnxy=2.434, Stoich_bnxy=2.0
R2 params (MM): Vm_r2=0.3836, Km_r2_bnxy=0.3175

Comparison for Default Conditions (Exp vs Sim - Iter 18):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 4.933e-01 | 6.302e-01
0.4 | 4.574e-01 | 4.173e-01
1.0 | 4.389e-01 | 1.333e-01
2.0 | 3.975e-01 | 2.055e-02
5.0 | 1.306e-01 | 3.638e-05

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e+00 | 1.000e+00
0.2 | 1.858e+00 | 1.646e+00
0.4 | 1.900e+00 | 1.986e+00
0.6 | 1.891e+00 | 2.175e+00
1.0 | 1.865e+00 | 2.323e+00
2.0 | 1.769e+00 | 2.199e+00
5.0 | 9.687e-01 | 1.268e+00
7.0 | 2.015e-01 | 6.930e-01

Comparison for Perturbed Conditions (Exp vs Sim - Iter 18):
Species: id_ijk0
Time | Exp_ijk0  | Sim_ijk0
0.0 | 1.000e+00 | 1.000e+00
0.2 | 3.200e-01 | 5.593e-01
0.4 | 2.413e-01 | 3.416e-01
1.0 | 1.707e-01 | 8.883e-02
2.0 | 6.978e-02 | 9.588e-03
5.0 | 9.958e-05 | 4.248e-06

Species: id_bnxy
Time | Exp_bnxy  | Sim_bnxy
0.0 | 1.000e-01 | 1.000e-01
0.2 | 1.287e+00 | 9.047e-01
0.4 | 1.327e+00 | 1.263e+00
0.6 | 1.268e+00 | 1.442e+00
1.0 | 1.120e+00 | 1.557e+00
2.0 | 6.897e-01 | 1.392e+00
5.0 | 2.260e-02 | 5.556e-01
7.0 | 1.161e-03 | 1.681e-01

## Reminder

 ### Remaining Iterations for Interactions:
- Iterations: 2/20



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0, iteration_3)
- `shared_variables`: all variables you have saved from the previous iterations. (current_best_model, exp_data_iter0, sim_data_iter1, exp_data_orig_iter4, exp_data_pert_iter4, current_best_model_iter4, sim_data_default_iter4, sim_data_perturbed_iter4, current_best_model_iter5, sim_data_default_iter5, sim_data_perturbed_iter5, current_best_model_iter6, sim_data_default_iter6, sim_data_perturbed_iter6, current_best_model_iter7, sim_data_default_iter7, sim_data_perturbed_iter7, current_best_model_iter8, sim_data_default_iter8, sim_data_perturbed_iter8, current_best_model_iter9, sim_data_default_iter9, sim_data_perturbed_iter9, current_best_model_iter10, sim_data_default_iter10, sim_data_perturbed_iter10, current_best_model_iter11, sim_data_default_iter11, sim_data_perturbed_iter11, current_best_model_iter12, sim_data_default_iter12, sim_data_perturbed_iter12, current_best_model_iter13, sim_data_default_iter13, sim_data_perturbed_iter13, current_best_model_iter14, sim_data_default_iter14, sim_data_perturbed_iter14, current_best_model_iter15, sim_data_default_iter15, sim_data_perturbed_iter15, current_best_model_iter16, sim_data_default_iter16, sim_data_perturbed_iter16, final_sbml_candidate_iter18)

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


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



# Iteration 19
## Thoughts

The model from Iteration 18 (R0 with `k_r0=0.1`; R1 with `k_r1_eff=8.362, Km_r1_bnxy=2.434, S=2.0`; R2 MM with `Vm_r2=0.3836, Km_r2_bnxy=0.3175`) showed:

**`id_ijk0` dynamics:**
- Default: Sim(0.2)=0.630 vs Exp(0.2)=0.493. (Still high/slow)
- Perturbed: Sim(0.2)=0.559 vs Exp(0.2)=0.320. (Still high/slow)
The initial `id_ijk0` decay is the most persistent issue. The parameters for R0 and R1 were derived to match instantaneous t=0 slopes, but the integrated decay over [0, 0.2s] is consistently too slow.

**`id_bnxy` dynamics:**
- Default: Peak Sim ~2.32 (t=1.0) vs Exp peak ~1.90 (t=0.4-0.6). Peak is now too high and late. Tail Sim(t=5)=1.27 vs Exp(t=5)=0.9687 (reasonable). Sim(t=7)=0.69 vs Exp(t=7)=0.20 (too slow decay).
- Perturbed: Peak Sim ~1.56 (t=1.0) vs Exp peak ~1.33 (t=0.4-0.6). Peak a bit high and late. Tail Sim(t=5)=0.55 vs Exp(t=5)=0.0226 (far too slow decay).

The `id_bnxy` tails, especially for the perturbed case, are now much worse with these R1 parameters. The R2 parameters (`Vm_r2=0.3836, Km_r2_bnxy=0.3175`) were good for the perturbed tail when R1 production was low in that phase. The current R1 (`k_r1_eff=8.362`) is strong, so `id_bnxy` production is high, leading to `id_bnxy` persisting too long.

This suggests that `Vm_r2` needs to be higher or `Km_r2_bnxy` lower to increase `id_bnxy` degradation.
Let's try to adjust R2 parameters.
If perturbed tail `d[bnxy]/dt ~ -0.069` at `[bnxy]~0.0571`.
If R1 production is truly negligible there, then `Vm_r2 * 0.0571 / (Km_r2_bnxy + 0.0571) = 0.069`.
If default tail `d[bnxy]/dt ~ -0.3836` at `[bnxy]~0.9687`.
If R1 production at default t=5 (using current R1 params and exp `id_ijk0(t=5)=0.1306`):
`P_R1 = 2.0 * 8.362 * 0.1306 / (2.434 + 0.9687) = 2.0 * 2.181 / 3.4027 = 2.0 * 0.641 = 1.282`.
Then R2 rate at default t=5 needs to be `1.282 + 0.3836 = 1.6656`.
So, `Vm_r2 * 0.9687 / (Km_r2_bnxy + 0.9687) = 1.6656`.

This is the same system of equations for `Vm_r2, Km_r2_bnxy` that previously led to negative `Km_r2_bnxy`.
This indicates that the assumption about R1 production (based on experimental `id_ijk0`) and the MM form for R2 are not simultaneously compatible across both conditions.

Given the few remaining iterations, I need to choose a model that provides the best overall compromise.
The model from Iteration 16 had some decent features for `id_ijk0(0.4, perturbed)` and the perturbed `id_bnxy` tail was okay.
Iteration 16 params:
R0: `k_r0 = 1.8`
R1: `k_r1_eff = 2.455`, `Km_r1_bnxy = 0.9767`, `S=2.0`
R2: `Vm_r2 = 0.3836`, `Km_r2_bnxy = 0.3175`

Let's re-examine Iteration 16 results:
**`id_ijk0` (Iter 16):**
- Default: Sim(0.2)=0.555 (Exp=0.493) - High
- Perturbed: Sim(0.2)=0.485 (Exp=0.320) - High. Sim(0.4)=0.251 (Exp=0.241) - Good.
**`id_bnxy` (Iter 16):**
- Default: Peak Sim ~1.44 (Exp ~1.90) - Low. Tail Sim(t=5)=0.35 (Exp ~0.97) - Low.
- Perturbed: Peak Sim ~0.79 (Exp ~1.33) - Low. Tail Sim(t=5)=0.065 (Exp ~0.023) - A bit high.

The `id_bnxy` levels in Iteration 16 were generally too low. This was due to a high `k_r0` and low `k_r1_eff`.
To increase `id_bnxy` levels from Iteration 16, we need more flux through R1.
This means increasing `k_r1_eff` and, to compensate for initial `id_ijk0` slopes, decreasing `k_r0`.

This line of reasoning led to Iteration 18's parameters, which made `id_bnxy` too high/persistent.
The system is highly coupled.

Let's try a slight modification of Iteration 18 parameters.
R0: `k_r0=0.1`
R1: `k_r1_eff=8.362`, `Km_r1_bnxy=2.434`, `S=2.0`
R2: `Vm_r2`, `Km_r2_bnxy`.
The `id_bnxy` tails were too slow in Iter 18. This means R2 degradation was too slow.
To speed up R2: Increase `Vm_r2` or decrease `Km_r2_bnxy`.
The old R2 params `Vm_r2=0.3836, Km_r2_bnxy=0.3175` gave `Vm/Km = 1.208`.
Perturbed tail needs `d[bnxy]/dt ~ -0.069` at `[bnxy]~0.0571`. Rate = `1.208 * 0.0571 = 0.069`. This is good.
Default tail needs `d[bnxy]/dt ~ -0.38` at `[bnxy]~0.97` (ignoring R1). Rate = `0.3836 * 0.97 / (0.3175+0.97) = 0.3836 * 0.97/1.2875 = 0.3836 * 0.753 = 0.289`. This is lower than 0.38.
So `Vm_r2` might need to be higher.
Let's try `Vm_r2 = 0.5`. Then `Km_r2_bnxy = Vm_r2 / 1.208 = 0.5 / 1.208 = 0.414`.
New R2 trial: `Vm_r2 = 0.5`, `Km_r2_bnxy = 0.414`.

Final attempt with this structure:
R0: `k_r0 = 0.1`
R1: `k_r1_eff = 8.362`, `Km_r1_bnxy = 2.434`, `S=2.0`
R2: `Vm_r2 = 0.5`, `Km_r2_bnxy = 0.414`

This is my last chance to tune. I will use this model for submission.

## Action

### Submit
```python
import pandas as pd
import numpy as np
import libsbml

# Parameters chosen for the final model submission
# Based on Iteration 18 structure, with adjusted R2 parameters

# Parameters for R0 (Small basal decay for id_ijk0)
param_k_r0 = 0.1

# Parameters for R1 (id_ijk0 -> S * id_bnxy, inhibited by id_bnxy)
param_k_r1_eff = 8.362
param_Km_r1_bnxy = 2.434
stoich_S_bnxy_R1 = 2.0

# Parameters for R2 (id_bnxy -> id_9k8m, Michaelis-Menten) - Adjusted
param_Vm_r2 = 0.5 # Was 0.3836
param_Km_r2_bnxy = 0.414 # Was 0.3175, Vm/Km ratio is ~1.208

print(f"Final Model Parameters:")
print(f"R0 param: k_r0={param_k_r0}")
print(f"R1 params: k_r1_eff={param_k_r1_eff}, Km_r1_bnxy={param_Km_r1_bnxy}, Stoich_bnxy={stoich_S_bnxy_R1}")
print(f"R2 params (MM): Vm_r2={param_Vm_r2}, Km_r2_bnxy={param_Km_r2_bnxy}")

# Build the SBML model
sbml_doc = libsbml.readSBMLFromString(input_sbml_string) # Start from the initial incomplete model
model = sbml_doc.getModel()
writer = libsbml.SBMLWriter()

# Add parameters
p = model.createParameter(); p.setId("k_r0"); p.setValue(param_k_r0); p.setConstant(True)
p = model.createParameter(); p.setId("k_r1_eff"); p.setValue(param_k_r1_eff); p.setConstant(True)
p = model.createParameter(); p.setId("Km_r1_bnxy"); p.setValue(param_Km_r1_bnxy); p.setConstant(True)
p = model.createParameter(); p.setId("Vm_r2"); p.setValue(param_Vm_r2); p.setConstant(True)
p = model.createParameter(); p.setId("Km_r2_bnxy"); p.setValue(param_Km_r2_bnxy); p.setConstant(True)

# Reaction R0: id_ijk0 -> id_9k8m
r0 = model.createReaction(); r0.setId("R0_ijk0_decay_uninhibited"); r0.setReversible(False); r0.setFast(False)
reactant_r0 = r0.createReactant(); reactant_r0.setSpecies("id_ijk0"); reactant_r0.setStoichiometry(1); reactant_r0.setConstant(False)
product_r0 = r0.createProduct(); product_r0.setSpecies("id_9k8m"); product_r0.setStoichiometry(1); product_r0.setConstant(False)
kl_r0 = r0.createKineticLaw(); kl_r0.setFormula("k_r0 * id_ijk0")

# Reaction R1: id_1t80 + id_ijk0 -> id_1t80 + S*id_bnxy
# Species id_1t80 is boundaryCondition=true, constant=true, initialConcentration=1. It acts as a catalyst/enzyme.
r1 = model.createReaction(); r1.setId("R1_ijk0_to_bnxy_inhib_bnxy"); r1.setReversible(False); r1.setFast(False)
reactant1_r1 = r1.createReactant(); reactant1_r1.setSpecies("id_ijk0"); reactant1_r1.setStoichiometry(1); reactant1_r1.setConstant(False)
# id_1t80 is a modifier, its concentration appears in the kinetic law
modifier_r1 = r1.createModifier()
modifier_r1.setSpecies("id_1t80")

product1_r1 = r1.createProduct(); product1_r1.setSpecies("id_bnxy"); product1_r1.setStoichiometry(stoich_S_bnxy_R1); product1_r1.setConstant(False)
kl_r1 = r1.createKineticLaw(); kl_r1.setFormula("k_r1_eff * id_1t80 * id_ijk0 / (Km_r1_bnxy + id_bnxy)")

# Reaction R2: id_bnxy -> id_9k8m (Michaelis-Menten kinetic law)
r2 = model.createReaction(); r2.setId("R2_bnxy_degradation_MM"); r2.setReversible(False); r2.setFast(False)
reactant1_r2 = r2.createReactant(); reactant1_r2.setSpecies("id_bnxy"); reactant1_r2.setStoichiometry(1); reactant1_r2.setConstant(False)
product1_r2 = r2.createProduct(); product1_r2.setSpecies("id_9k8m"); product1_r2.setStoichiometry(1); product1_r2.setConstant(False)
kl_r2 = r2.createKineticLaw(); kl_r2.setFormula("Vm_r2 * id_bnxy / (Km_r2_bnxy + id_bnxy)")

final_sbml = writer.writeSBMLToString(sbml_doc)
# print(final_sbml) # For verification if needed

# The variable 'final_sbml' will be picked up for submission.
```
