
                # 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_2nva">
    <listOfUnitDefinitions>
      <unitDefinition id="id_8ojb">
        <listOfUnits>
          <unit kind="metre" exponent="1" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_kj77">
        <listOfUnits>
          <unit kind="metre" exponent="2" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_xou9">
        <listOfUnits>
          <unit kind="litre" exponent="1" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_idew">
        <listOfUnits>
          <unit kind="second" exponent="1" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_gk1x">
        <listOfUnits>
          <unit kind="mole" exponent="1" scale="0" multiplier="1"/>
        </listOfUnits>
      </unitDefinition>
    </listOfUnitDefinitions>
    <listOfCompartments>
      <compartment id="id_a3j9" spatialDimensions="3" size="1" constant="true"/>
    </listOfCompartments>
    <listOfSpecies>
      <species id="id_xvpm" compartment="id_a3j9" initialConcentration="1e-05" boundaryCondition="false" constant="false"/>
      <species id="id_aa2n" compartment="id_a3j9" initialConcentration="1e-05" boundaryCondition="false" constant="false"/>
      <species id="id_xdgv" compartment="id_a3j9" initialConcentration="1e-05" boundaryCondition="false" constant="false"/>
    </listOfSpecies>
  </model>
</sbml>


                ## Max iterations
                20

                Format your response according to the instructions in the system message.
                
## Thoughts
The initial SBML model defines a system with three species: `id_xvpm`, `id_aa2n`, and `id_xdgv`, all in the same compartment `id_a3j9`. Their initial concentrations are all 1e-05. There are no reactions defined in the model.

My first step is to understand the baseline behavior of the system. I will request an "observe" experiment to see how the concentrations of these species change over time without any perturbations. This data will be crucial for inferring the underlying reactions.

## Action

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

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

Total size: 1001 rows × 4 columns. 

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

Time   id_xvpm  id_aa2n  id_xdgv
0  1.00e-05 1.00e-05 1.00e-05
...
1.00e+01  3.05e-05 5.82e-05 1.70e-04
...
2.00e+01  7.04e-05 2.23e-04 5.40e-04
...
3.00e+01  1.11e-04 5.15e-04 1.16e-03
...
4.00e+01  1.09e-04 8.04e-04 1.78e-03
...
5.00e+01  5.68e-05 8.37e-04 1.95e-03
...
6.00e+01  1.67e-05 5.31e-04 1.63e-03
...
7.00e+01  5.67e-06 2.29e-04 1.21e-03
...
8.00e+01  3.30e-06 1.10e-04 8.63e-04
...
9.00e+01  3.13e-06 7.48e-05 6.19e-04
...
1.00e+02  4.26e-06 6.79e-05 4.55e-04
...
1.10e+02  7.34e-06 7.54e-05 3.60e-04
...
1.20e+02  1.44e-05 9.75e-05 3.37e-04
...
1.30e+02  2.86e-05 1.48e-04 4.10e-04
...
1.40e+02  5.14e-05 2.52e-04 6.20e-04
...
1.50e+02  7.42e-05 4.17e-04 9.69e-04
...
1.60e+02  7.90e-05 5.86e-04 1.34e-03
...
1.70e+02  5.80e-05 6.48e-04 1.52e-03
...
1.80e+02  3.07e-05 5.43e-04 1.42e-03
...
1.90e+02  1.56e-05 3.63e-04 1.16e-03
...
2.00e+02  1.03e-05 2.34e-04 9.04e-04
...
2.10e+02  9.48e-06 1.73e-04 7.03e-04
...
2.20e+02  1.16e-05 1.53e-04 5.71e-04
...
2.30e+02  1.70e-05 1.60e-04 5.11e-04
...
2.40e+02  2.71e-05 1.93e-04 5.37e-04
...
2.50e+02  4.21e-05 2.63e-04 6.62e-04
...
2.60e+02  5.75e-05 3.70e-04 8.83e-04
...
2.70e+02  6.40e-05 4.85e-04 1.14e-03
...
2.80e+02  5.52e-05 5.49e-04 1.30e-03
...
2.90e+02  3.79e-05 5.15e-04 1.29e-03
...
3.00e+02  2.39e-05 4.10e-04 1.14e-03
...
3.10e+02  1.69e-05 3.05e-04 9.56e-04
...
3.20e+02  1.49e-05 2.39e-04 7.89e-04
...
3.30e+02  1.64e-05 2.10e-04 6.74e-04
...
3.40e+02  2.12e-05 2.10e-04 6.21e-04
...
3.50e+02  2.95e-05 2.37e-04 6.41e-04
...
3.60e+02  4.06e-05 2.92e-04 7.38e-04
...
3.70e+02  5.08e-05 3.71e-04 9.00e-04
...
3.80e+02  5.45e-05 4.50e-04 1.07e-03
...
3.90e+02  4.86e-05 4.90e-04 1.18e-03
...
4.00e+02  3.71e-05 4.67e-04 1.18e-03
...
4.10e+02  2.69e-05 3.98e-04 1.08e-03
...
4.20e+02  2.10e-05 3.23e-04 9.43e-04
...
4.30e+02  1.93e-05 2.71e-04 8.17e-04
...
4.40e+02  2.07e-05 2.47e-04 7.30e-04
...
4.50e+02  2.51e-05 2.48e-04 6.95e-04
...
4.60e+02  3.21e-05 2.72e-04 7.20e-04
...
4.70e+02  4.03e-05 3.19e-04 8.04e-04
...
4.80e+02  4.69e-05 3.79e-04 9.27e-04
...
5.00e+02  4.33e-05 4.52e-04 1.11e-03

## 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 shows that all three species (`id_xvpm`, `id_aa2n`, `id_xdgv`) increase from their initial concentrations, peak at different times or magnitudes, and then decrease, with some oscillatory behavior.

- `id_xvpm` peaks first (around t=30-40), then `id_aa2n` and `id_xdgv` peak later (around t=50-60, with `id_xdgv` having a higher peak).
- This sequential peaking suggests a pathway like `id_xvpm` -> `id_aa2n` -> `id_xdgv`.
- The initial increase of `id_xvpm` from a low concentration, without any other species being consumed (as they are all increasing), suggests a zeroth-order production for `id_xvpm` (i.e., `-> id_xvpm`).
- The subsequent decrease of all species suggests degradation or conversion into unobserved products. Since we are told not to add new species, this would be degradation reactions like `S ->`.

Based on this, a plausible initial hypothesis for the reaction network is:
1.  **R1: `-> id_xvpm`** (Production of `id_xvpm`)
2.  **R2: `id_xvpm -> id_aa2n`** (Conversion of `id_xvpm` to `id_aa2n`)
3.  **R3: `id_aa2n -> id_xdgv`** (Conversion of `id_aa2n` to `id_xdgv`)
4.  **R4: `id_xdgv ->`** (Degradation of `id_xdgv`)

To account for the eventual decrease of `id_xvpm` and `id_aa2n` even before `id_xdgv` peaks and degrades, they might also have their own degradation pathways:
5.  **R5: `id_xvpm ->`** (Degradation of `id_xvpm`)
6.  **R6: `id_aa2n ->`** (Degradation of `id_aa2n`)

Let's start by implementing the simpler model (R1-R4) and see how it behaves. The oscillations observed in the data might require more complex mechanisms like feedback loops, which can be explored later if this simpler model is insufficient.

I will now write code to:
1.  Load the experimental data.
2.  Plot the experimental data to visualize the dynamics.
3.  Modify the `input_sbml_string` to include the hypothesized reactions (R1-R4) with some initial guess for parameters.
4.  Simulate the modified SBML model.
5.  Plot the simulation results alongside the experimental data for comparison.

I'll need to choose some initial parameter values for the rate constants. This will be an iterative process.
For R1: `-> S1` (parameter `k1`)
For R2: `S1 -> S2` (parameter `k2`, rate `k2*[S1]`)
For R3: `S2 -> S3` (parameter `k3`, rate `k3*[S2]`)
For R4: `S3 ->` (parameter `k4`, rate `k4*[S3]`)
(S1=`id_xvpm`, S2=`id_aa2n`, S3=`id_xdgv`)

Let's try some initial guesses for parameters.
- `id_xvpm` peaks around 1e-4 at t=30. Production rate should be significant.
- `id_aa2n` peaks around 8e-4 at t=50.
- `id_xdgv` peaks around 2e-3 at t=50.

This is a first attempt, parameter tuning will likely be needed.

## Action

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

# Access experimental data
exp_data_df = experiment_history['iteration_0']

# Plot experimental data
plt.figure(figsize=(10, 6))
plt.plot(exp_data_df['Time'], exp_data_df['id_xvpm'], label='id_xvpm (Experimental)')
plt.plot(exp_data_df['Time'], exp_data_df['id_aa2n'], label='id_aa2n (Experimental)')
plt.plot(exp_data_df['Time'], exp_data_df['id_xdgv'], label='id_xdgv (Experimental)')
plt.xlabel('Time')
plt.ylabel('Concentration')
plt.title('Experimental Data')
plt.legend()
plt.grid(True)
plt.savefig("experimental_data_plot.png") # Save plot to see it
print("Experimental data plot saved as experimental_data_plot.png")
shared_variables.add("experimental_data_plot_iteration_1", plt.gcf())


# Function to add a reaction
def add_reaction(model, r_id, reactants, products, k_val, k_id, reversible=False):
    reaction = model.createReaction()
    reaction.setId(r_id)
    reaction.setReversible(reversible)
    reaction.setFast(False)

    for species_id, stoich in reactants:
        reactant = reaction.createReactant()
        reactant.setSpecies(species_id)
        reactant.setStoichiometry(stoich)
        reactant.setConstant(False)

    for species_id, stoich in products:
        product = reaction.createProduct()
        product.setSpecies(species_id)
        product.setStoichiometry(stoich)
        product.setConstant(False) # In SBML L3V1, product constant is true if it's a boundary species, false otherwise.

    kinetic_law = reaction.createKineticLaw()
    # Create parameter k
    k_param = kinetic_law.createParameter()
    k_param.setId(k_id)
    k_param.setValue(k_val)
    k_param.setConstant(True) # Rate constants are usually constant

    # Create math formula for rate law
    math_ast = libsbml.parseL3Formula(f"{k_id}")
    if reactants:
        formula = f"{k_id}"
        for species_id, stoich in reactants:
            formula += f" * {species_id}" # Assuming mass action for simplicity and stoich=1 for now
            # For stoich > 1, it would be species_id^stoich, but let's keep it simple
        math_ast = libsbml.parseL3Formula(formula)
    
    kinetic_law.setMath(math_ast)
    return reaction

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

# Define species IDs for clarity
s1 = "id_xvpm"
s2 = "id_aa2n"
s3 = "id_xdgv"

# Add reactions to the model
# R1: -> s1 (k1)
# For zeroth order production -> S1, the kinetic law is just k1.
# libsbml requires reactants for mass action, so we model it as X0 -> S1 where X0 is a boundary species.
# However, we are not allowed to add new species.
# A common way to represent -> S1 with rate k1 is to have the kinetic law be just "k1".
# Let's try to implement this directly.

# R1: -> id_xvpm (Production of id_xvpm)
# Kinetic Law: k1
reaction1 = model.createReaction()
reaction1.setId("R1_prod_xvpm")
reaction1.setReversible(False)
reaction1.setFast(False)
prod1 = reaction1.createProduct()
prod1.setSpecies(s1)
prod1.setStoichiometry(1.0)
prod1.setConstant(False)
kl1 = reaction1.createKineticLaw()
p1 = kl1.createParameter()
p1.setId("k1")
p1.setValue(5e-6) # Initial guess: to reach ~1e-4 in 20s, rate ~ 1e-4/20 = 5e-6
kl1.setMath(libsbml.parseL3Formula("k1"))

# R2: id_xvpm -> id_aa2n (Conversion of id_xvpm to id_aa2n)
# Kinetic Law: k2 * [id_xvpm]
reaction2 = model.createReaction()
reaction2.setId("R2_xvpm_to_aa2n")
reaction2.setReversible(False)
reaction2.setFast(False)
reactant2 = reaction2.createReactant()
reactant2.setSpecies(s1)
reactant2.setStoichiometry(1.0)
reactant2.setConstant(False)
product2 = reaction2.createProduct()
product2.setSpecies(s2)
product2.setStoichiometry(1.0)
product2.setConstant(False)
kl2 = reaction2.createKineticLaw()
p2 = kl2.createParameter()
p2.setId("k2")
p2.setValue(0.1) # Initial guess
kl2.setMath(libsbml.parseL3Formula(f"k2 * {s1}"))

# R3: id_aa2n -> id_xdgv (Conversion of id_aa2n to id_xdgv)
# Kinetic Law: k3 * [id_aa2n]
reaction3 = model.createReaction()
reaction3.setId("R3_aa2n_to_xdgv")
reaction3.setReversible(False)
reaction3.setFast(False)
reactant3 = reaction3.createReactant()
reactant3.setSpecies(s2)
reactant3.setStoichiometry(1.0)
reactant3.setConstant(False)
product3 = reaction3.createProduct()
product3.setSpecies(s3)
product3.setStoichiometry(1.0)
product3.setConstant(False)
kl3 = reaction3.createKineticLaw()
p3 = kl3.createParameter()
p3.setId("k3")
p3.setValue(0.1) # Initial guess
kl3.setMath(libsbml.parseL3Formula(f"k3 * {s2}"))

# R4: id_xdgv -> (Degradation of id_xdgv)
# Kinetic Law: k4 * [id_xdgv]
reaction4 = model.createReaction()
reaction4.setId("R4_degrad_xdgv")
reaction4.setReversible(False)
reaction4.setFast(False)
reactant4 = reaction4.createReactant()
reactant4.setSpecies(s3)
reactant4.setStoichiometry(1.0)
reactant4.setConstant(False)
kl4 = reaction4.createKineticLaw()
p4 = kl4.createParameter()
p4.setId("k4")
p4.setValue(0.05) # Initial guess
kl4.setMath(libsbml.parseL3Formula(f"k4 * {s3}"))

# Write the updated SBML to a string
writer = libsbml.SBMLWriter()
current_sbml_model_str = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_sbml_model_str_iteration_1", current_sbml_model_str)

# Simulate the model
sim_data_df = simulate(current_sbml_model_str)

# Plot simulation results vs experimental data
plt.figure(figsize=(12, 8))

plt.plot(exp_data_df['Time'], exp_data_df['id_xvpm'], 'r--', label='id_xvpm (Exp)', alpha=0.7)
plt.plot(sim_data_df['Time'], sim_data_df['id_xvpm'], 'r-', label='id_xvpm (Sim)')

plt.plot(exp_data_df['Time'], exp_data_df['id_aa2n'], 'g--', label='id_aa2n (Exp)', alpha=0.7)
plt.plot(sim_data_df['Time'], sim_data_df['id_aa2n'], 'g-', label='id_aa2n (Sim)')

plt.plot(exp_data_df['Time'], exp_data_df['id_xdgv'], 'b--', label='id_xdgv (Exp)', alpha=0.7)
plt.plot(sim_data_df['Time'], sim_data_df['id_xdgv'], 'b-', label='id_xdgv (Sim)')

plt.xlabel('Time')
plt.ylabel('Concentration')
plt.title('Model Simulation vs Experimental Data (Hypothesis 1)')
plt.legend()
plt.grid(True)
plt.savefig("simulation_plot_hypothesis1.png")
print("Simulation plot saved as simulation_plot_hypothesis1.png")
shared_variables.add("simulation_plot_hypothesis1_iteration_1", plt.gcf())

# Print some info about the simulation
print("Simulation data head:")
print(sim_data_df.head())
print("\nSimulation data tail:")
print(sim_data_df.tail())

# Print parameter values used
print(f"\nParameters used: k1={p1.getValue()}, k2={p2.getValue()}, k3={p3.getValue()}, k4={p4.getValue()}")

```
# Observation

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

## Reminder

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



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

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


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



# Iteration 2
## Thoughts
The previous attempt failed because `matplotlib.pyplot` is not allowed. I need to remove the plotting code. I will proceed with building the SBML model with the hypothesized reactions and parameters, simulate it, and then print the head and tail of the simulation results. This will give me some insight into whether the model is behaving qualitatively as expected, even without a direct visual comparison.

The hypothesized reactions are:
1.  **R1: `-> id_xvpm`** (Production of `id_xvpm`, rate `k1`)
2.  **R2: `id_xvpm -> id_aa2n`** (Conversion of `id_xvpm` to `id_aa2n`, rate `k2*[id_xvpm]`)
3.  **R3: `id_aa2n -> id_xdgv`** (Conversion of `id_aa2n` to `id_xdgv`, rate `k3*[id_aa2n]`)
4.  **R4: `id_xdgv ->`** (Degradation of `id_xdgv`, rate `k4*[id_xdgv]`)

Initial parameter guesses: `k1=5e-6`, `k2=0.1`, `k3=0.1`, `k4=0.05`.

I will:
1.  Load the `input_sbml_string`.
2.  Add the four reactions with their kinetic laws and parameters.
3.  Simulate the modified model.
4.  Print the head and tail of the simulation data.
5.  Save the modified SBML string and the simulation data for future reference.

## Action

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

# Access experimental data for reference if needed (though not plotting here)
exp_data_df = experiment_history['iteration_0']
# print("Experimental data (iteration_0) head:")
# print(exp_data_df.head())
# print("Experimental data (iteration_0) tail:")
# print(exp_data_df.tail())


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

# Define species IDs for clarity
s1 = "id_xvpm"
s2 = "id_aa2n"
s3 = "id_xdgv"

# Parameter values
k1_val = 5e-6
k2_val = 0.1
k3_val = 0.1
k4_val = 0.05

# R1: -> id_xvpm (Production of id_xvpm)
# Kinetic Law: k1
reaction1 = model.createReaction()
reaction1.setId("R1_prod_xvpm")
reaction1.setReversible(False)
reaction1.setFast(False)
prod1 = reaction1.createProduct()
prod1.setSpecies(s1)
prod1.setStoichiometry(1.0)
prod1.setConstant(False)
kl1 = reaction1.createKineticLaw()
p1 = kl1.createParameter()
p1.setId("k1")
p1.setValue(k1_val)
p1.setConstant(True)
kl1.setMath(libsbml.parseL3Formula("k1"))

# R2: id_xvpm -> id_aa2n (Conversion of id_xvpm to id_aa2n)
# Kinetic Law: k2 * [id_xvpm]
reaction2 = model.createReaction()
reaction2.setId("R2_xvpm_to_aa2n")
reaction2.setReversible(False)
reaction2.setFast(False)
reactant2 = reaction2.createReactant()
reactant2.setSpecies(s1)
reactant2.setStoichiometry(1.0)
reactant2.setConstant(False)
product2 = reaction2.createProduct()
product2.setSpecies(s2)
product2.setStoichiometry(1.0)
product2.setConstant(False)
kl2 = reaction2.createKineticLaw()
p2 = kl2.createParameter()
p2.setId("k2")
p2.setValue(k2_val)
p2.setConstant(True)
kl2.setMath(libsbml.parseL3Formula(f"k2 * {s1}"))

# R3: id_aa2n -> id_xdgv (Conversion of id_aa2n to id_xdgv)
# Kinetic Law: k3 * [id_aa2n]
reaction3 = model.createReaction()
reaction3.setId("R3_aa2n_to_xdgv")
reaction3.setReversible(False)
reaction3.setFast(False)
reactant3 = reaction3.createReactant()
reactant3.setSpecies(s2)
reactant3.setStoichiometry(1.0)
reactant3.setConstant(False)
product3 = reaction3.createProduct()
product3.setSpecies(s3)
product3.setStoichiometry(1.0)
product3.setConstant(False)
kl3 = reaction3.createKineticLaw()
p3 = kl3.createParameter()
p3.setId("k3")
p3.setValue(k3_val)
p3.setConstant(True)
kl3.setMath(libsbml.parseL3Formula(f"k3 * {s2}"))

# R4: id_xdgv -> (Degradation of id_xdgv)
# Kinetic Law: k4 * [id_xdgv]
reaction4 = model.createReaction()
reaction4.setId("R4_degrad_xdgv")
reaction4.setReversible(False)
reaction4.setFast(False)
reactant4 = reaction4.createReactant()
reactant4.setSpecies(s3)
reactant4.setStoichiometry(1.0)
reactant4.setConstant(False)
kl4 = reaction4.createKineticLaw()
p4 = kl4.createParameter()
p4.setId("k4")
p4.setValue(k4_val)
p4.setConstant(True)
kl4.setMath(libsbml.parseL3Formula(f"k4 * {s3}"))

# Write the updated SBML to a string
writer = libsbml.SBMLWriter()
current_sbml_model_str = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_sbml_model_str_iteration_2", current_sbml_model_str)

# Simulate the model
sim_data_df = simulate(current_sbml_model_str)
shared_variables.add("sim_data_df_iteration_2", sim_data_df)

# Print simulation results
print("Simulation data (Hypothesis 1) head:")
print(sim_data_df.head())
print("\nSimulation data (Hypothesis 1) tail:")
print(sim_data_df.tail(10)) # Print more of the tail to see later time points

# Print parameter values used
print(f"\nParameters used: k1={k1_val}, k2={k2_val}, k3={k3_val}, k4={k4_val}")

# For comparison, print some key points from experimental data
print("\nExperimental data (iteration_0) for comparison:")
print("Time points around id_xvpm peak (approx t=30-40):")
print(exp_data_df[(exp_data_df['Time'] >= 25) & (exp_data_df['Time'] <= 45)][['Time', 'id_xvpm', 'id_aa2n', 'id_xdgv']])
print("\nTime points around id_aa2n/id_xdgv peak (approx t=50-60):")
print(exp_data_df[(exp_data_df['Time'] >= 45) & (exp_data_df['Time'] <= 65)][['Time', 'id_xvpm', 'id_aa2n', 'id_xdgv']])
print("\nExperimental data at t=500 (end of simulation):")
print(exp_data_df[exp_data_df['Time'] == 500.0])

```
# Observation

## Code Stdout
Simulation data (Hypothesis 1) head:
    id_xvpm   id_aa2n   id_xdgv  Time
0  0.000010  0.000010  0.000010   0.0
1  0.000012  0.000010  0.000010   0.5
2  0.000014  0.000010  0.000010   1.0
3  0.000016  0.000010  0.000011   1.5
4  0.000017  0.000011  0.000011   2.0

Simulation data (Hypothesis 1) tail:
      id_xvpm  id_aa2n  id_xdgv   Time
991   0.00005  0.00005   0.0001  495.5
992   0.00005  0.00005   0.0001  496.0
993   0.00005  0.00005   0.0001  496.5
994   0.00005  0.00005   0.0001  497.0
995   0.00005  0.00005   0.0001  497.5
996   0.00005  0.00005   0.0001  498.0
997   0.00005  0.00005   0.0001  498.5
998   0.00005  0.00005   0.0001  499.0
999   0.00005  0.00005   0.0001  499.5
1000  0.00005  0.00005   0.0001  500.0

Parameters used: k1=5e-06, k2=0.1, k3=0.1, k4=0.05

Experimental data (iteration_0) for comparison:
Time points around id_xvpm peak (approx t=30-40):
    Time   id_xvpm   id_aa2n   id_xdgv
50  25.0  0.000093  0.000357  0.000827
51  25.5  0.000095  0.000372  0.000859
52  26.0  0.000097  0.000387  0.000892
53  26.5  0.000099  0.000403  0.000924
54  27.0  0.000101  0.000418  0.000958
55  27.5  0.000103  0.000434  0.000991
56  28.0  0.000105  0.000450  0.001025
57  28.5  0.000107  0.000466  0.001059
58  29.0  0.000108  0.000482  0.001094
59  29.5  0.000110  0.000499  0.001128
60  30.0  0.000111  0.000515  0.001163
61  30.5  0.000112  0.000532  0.001198
62  31.0  0.000113  0.000548  0.001232
63  31.5  0.000114  0.000564  0.001267
64  32.0  0.000115  0.000581  0.001301
65  32.5  0.000116  0.000597  0.001336
66  33.0  0.000117  0.000613  0.001370
67  33.5  0.000117  0.000629  0.001403
68  34.0  0.000117  0.000645  0.001437
69  34.5  0.000118  0.000660  0.001470
70  35.0  0.000118  0.000675  0.001502
71  35.5  0.000117  0.000690  0.001534
72  36.0  0.000117  0.000705  0.001565
73  36.5  0.000117  0.000719  0.001595
74  37.0  0.000116  0.000733  0.001625
75  37.5  0.000115  0.000746  0.001653
76  38.0  0.000114  0.000759  0.001681
77  38.5  0.000113  0.000771  0.001708
78  39.0  0.000112  0.000783  0.001733
79  39.5  0.000110  0.000794  0.001758
80  40.0  0.000109  0.000804  0.001781
81  40.5  0.000107  0.000814  0.001804
82  41.0  0.000105  0.000823  0.001824
83  41.5  0.000103  0.000832  0.001844
84  42.0  0.000101  0.000839  0.001862
85  42.5  0.000099  0.000846  0.001879
86  43.0  0.000096  0.000852  0.001894
87  43.5  0.000094  0.000857  0.001908
88  44.0  0.000091  0.000861  0.001920
89  44.5  0.000089  0.000865  0.001931
90  45.0  0.000086  0.000867  0.001940

Time points around id_aa2n/id_xdgv peak (approx t=50-60):
     Time   id_xvpm   id_aa2n   id_xdgv
90   45.0  0.000086  0.000867  0.001940
91   45.5  0.000083  0.000868  0.001948
92   46.0  0.000080  0.000869  0.001954
93   46.5  0.000077  0.000868  0.001959
94   47.0  0.000074  0.000867  0.001962
95   47.5  0.000071  0.000864  0.001963
96   48.0  0.000068  0.000861  0.001963
97   48.5  0.000066  0.000856  0.001962
98   49.0  0.000063  0.000851  0.001959
99   49.5  0.000060  0.000845  0.001954
100  50.0  0.000057  0.000837  0.001949
101  50.5  0.000054  0.000829  0.001941
102  51.0  0.000051  0.000819  0.001933
103  51.5  0.000048  0.000809  0.001923
104  52.0  0.000046  0.000798  0.001912
105  52.5  0.000043  0.000786  0.001900
106  53.0  0.000041  0.000773  0.001887
107  53.5  0.000038  0.000759  0.001873
108  54.0  0.000036  0.000745  0.001858
109  54.5  0.000034  0.000729  0.001843
110  55.0  0.000032  0.000713  0.001826
111  55.5  0.000030  0.000697  0.001809
112  56.0  0.000028  0.000680  0.001791
113  56.5  0.000026  0.000662  0.001772
114  57.0  0.000025  0.000644  0.001753
115  57.5  0.000023  0.000626  0.001733
116  58.0  0.000022  0.000607  0.001713
117  58.5  0.000020  0.000588  0.001692
118  59.0  0.000019  0.000569  0.001672
119  59.5  0.000018  0.000550  0.001650
120  60.0  0.000017  0.000531  0.001629
121  60.5  0.000016  0.000512  0.001608
122  61.0  0.000015  0.000493  0.001586
123  61.5  0.000014  0.000475  0.001564
124  62.0  0.000013  0.000456  0.001542
125  62.5  0.000012  0.000438  0.001521
126  63.0  0.000012  0.000421  0.001499
127  63.5  0.000011  0.000404  0.001477
128  64.0  0.000010  0.000387  0.001455
129  64.5  0.000010  0.000371  0.001434
130  65.0  0.000009  0.000355  0.001412

Experimental data at t=500 (end of simulation):
       id_xvpm   id_aa2n   id_xdgv   Time
1000  0.000043  0.000452  0.001113  500.0

## 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_sbml_model_str_iteration_2, sim_data_df_iteration_2)

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


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



# Iteration 3
## Thoughts

The simulation results from Iteration 2, using parameters `k1=5e-6, k2=0.1, k3=0.1, k4=0.05`, show a significant mismatch with the experimental data.

**Comparison:**

*   **`id_xvpm`:**
    *   Experimental peak: ~0.000118 around t=35.
    *   Simulated at t=500: 0.00005. The simulation doesn't show a peak and decline; it seems to reach a steady state or very slow change. The simulated values are much lower than the experimental peak.
*   **`id_aa2n`:**
    *   Experimental peak: ~0.000869 around t=46.
    *   Simulated at t=500: 0.00005. Again, much lower and no clear peak dynamics.
*   **`id_xdgv`:**
    *   Experimental peak: ~0.001963 around t=48.
    *   Simulated at t=500: 0.0001. Much lower and no clear peak dynamics.

**Experimental data at t=500:**
`id_xvpm`: 0.000043, `id_aa2n`: 0.000452, `id_xdgv`: 0.001113

**Simulated data at t=500:**
`id_xvpm`: 0.00005, `id_aa2n`: 0.00005, `id_xdgv`: 0.0001

The model `-> S1 -> S2 -> S3 ->` with the current parameters is not capturing the magnitudes or the dynamic shapes (peaks and declines) seen in the experiment. The simulated species concentrations are too low and they seem to reach a steady state where `k1 = k2*S1 = k3*S2 = k4*S3` (approximately, if S1, S2, S3 are at steady state).
If `S1_ss = k1/k2 = 5e-6 / 0.1 = 5e-5`.
If `S2_ss = k2*S1_ss / k3 = 0.1 * 5e-5 / 0.1 = 5e-5`.
If `S3_ss = k3*S2_ss / k4 = 0.1 * 5e-5 / 0.05 = 1e-4`.
These steady-state values match the simulation output at t=500.

The experimental data shows oscillations and much higher peak values. The simple linear chain with degradation only at the end is insufficient.

**Problems with current model & parameters:**
1.  **Magnitudes are too low:** `k1` (production of `id_xvpm`) is likely too small.
2.  **Dynamics are wrong:** The model quickly reaches a low steady state. The experimental data shows a rise, peak, fall, and then some oscillations.
3.  **Degradation:** The model only has degradation for `id_xdgv`. The experimental data shows `id_xvpm` and `id_aa2n` also decrease after their peaks, suggesting they might also have degradation pathways or are consumed more rapidly than produced. The oscillations suggest more complex regulation, possibly feedback.

**Next Steps - Parameter Tuning & Model Refinement:**

Let's try to adjust parameters first for the current model structure: `-> S1 -> S2 -> S3 ->`.
To get higher peaks:
*   Increase `k1` significantly.
*   The rates `k2`, `k3` control the flow through the pathway.
*   `k4` controls the removal of `S3`.

Let's aim for `id_xvpm` peak of ~1e-4. If `k1` is the main input, and `k2` is the main output for `S1` at peak, then `dS1/dt = k1 - k2*S1 = 0` implies `k1 = k2*S1_peak`.
If `S1_peak` is 1e-4, and `k2` is 0.1, then `k1` should be around `0.1 * 1e-4 = 1e-5`. This is double my previous `k1`.
However, the experimental `id_xvpm` rises from 1e-5 to 1.18e-4 in about 35s. Average rate of increase is (1.18e-4 - 1e-5)/35 = (1.08e-4)/35 approx 3e-6. This is close to my initial `k1` of 5e-6.
The issue might be that `k2` is too high, draining `S1` too quickly, or `k1` is indeed too low for the subsequent species.

The experimental data shows `id_xvpm` decreasing after t=35, `id_aa2n` after t=46, and `id_xdgv` after t=48. This suggests that the consumption/degradation rates become dominant.

The oscillatory behavior is a key feature I'm currently missing. This often implies feedback mechanisms. For example, `id_xdgv` might inhibit the production of `id_xvpm` (R1) or enhance its degradation.

Before adding complexity like feedback, let's try to get the magnitudes and initial peaks somewhat better with the current linear chain model, possibly by adding degradation terms for `S1` and `S2` as well.

Revised model hypothesis (Hypothesis 2):
1.  **R1: `-> id_xvpm`** (k1)
2.  **R2: `id_xvpm -> id_aa2n`** (k2 * [id_xvpm])
3.  **R3: `id_aa2n -> id_xdgv`** (k3 * [id_aa2n])
4.  **R4: `id_xdgv ->`** (k4 * [id_xdgv])
5.  **R5: `id_xvpm ->`** (k5 * [id_xvpm]) (Added degradation for S1)
6.  **R6: `id_aa2n ->`** (k6 * [id_aa2n]) (Added degradation for S2)

Let's try to adjust parameters for this new model.
The net rate of change for `S1` is `k1 - k2*S1 - k5*S1`.
At `S1` peak (around 1.18e-4 at t=35): `k1 - (k2+k5)*S1_peak = 0`. So `k1 = (k2+k5)*S1_peak`.
For `S2` peak (around 8.69e-4 at t=46): `k2*S1 - k3*S2_peak - k6*S2_peak = 0`. So `k2*S1_at_S2_peak = (k3+k6)*S2_peak`.
For `S3` peak (around 1.96e-3 at t=48): `k3*S2 - k4*S3_peak = 0`. So `k3*S2_at_S3_peak = k4*S3_peak`.

This is getting complicated to estimate by hand. Let's try a new set of parameters, focusing on increasing `k1` and adjusting other rates.
Let's try:
*   `k1`: Increase significantly. If `S1` peak is ~1e-4, and it takes ~30s, `k1` could be roughly `1e-4 / 30 = 3.3e-6` if there were no consumption. But there is. Let's try `k1 = 2e-5` (4x previous).
*   `k2`: This rate converts `S1` to `S2`. If `S1` peaks at 1e-4, and `S2` peaks at 8e-4, `k2` should be significant. Let's keep `k2 = 0.1`.
*   `k3`: Converts `S2` to `S3`. If `S2` peaks at 8e-4 and `S3` at 2e-3, `k3` should also be significant. Let's keep `k3 = 0.1`.
*   `k5` (degradation of `S1`): To make `S1` fall after its peak. If `k1 = (k2+k5)*S1_peak`, then `2e-5 = (0.1+k5)*1e-4`. `0.1+k5 = 2e-5/1e-4 = 0.2`. So `k5 = 0.1`.
*   `k6` (degradation of `S2`): To make `S2` fall. `k2*S1_at_S2_peak = (k3+k6)*S2_peak`. At `t=46`, `S1` is ~8e-5, `S2` is ~8.7e-4. `0.1 * 8e-5 = (0.1+k6)*8.7e-4`. `8e-6 = (0.1+k6)*8.7e-4`. `0.1+k6 = 8e-6 / 8.7e-4 = 80e-4 / 8.7e-4 = 80/8.7 ~ 0.009`. This implies `k6` would be negative, which is not possible. This suggests that `k2*S1` is too small compared to `k3*S2` if `S2` is to also degrade.
    This indicates that the simple mass-action degradation might not be the full story, or my parameter estimates are off, or the model structure is still missing something fundamental (like feedback causing the oscillations and sharper peaks/declines).

Let's simplify the parameter estimation for now and try to get somewhat closer to the magnitudes.
New parameter set for Model Hypothesis 2 (with S1 and S2 degradation):
*   `k1 = 2e-5` (Increased production of S1)
*   `k2 = 0.2` (Increased conversion S1->S2)
*   `k3 = 0.2` (Increased conversion S2->S3)
*   `k4 = 0.1` (Degradation of S3)
*   `k5 = 0.05` (Degradation of S1)
*   `k6 = 0.05` (Degradation of S2)

The goal is to see if adding degradation for S1 and S2 helps create the peak-and-fall behavior for them, and if the overall magnitudes are more reasonable. The oscillations are a secondary concern for now, to be addressed if this model fails significantly.

I will implement this new model (Hypothesis 2) and simulate it.

## Action

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

# Access experimental data
exp_data_df = experiment_history['iteration_0']

# Load the initial SBML model string (it's clean, without previous reactions)
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

# Define species IDs
s1 = "id_xvpm"
s2 = "id_aa2n"
s3 = "id_xdgv"

# New parameter values for Hypothesis 2
k1_val = 2.0e-5  # Production of S1
k2_val = 0.2    # S1 -> S2
k3_val = 0.2    # S2 -> S3
k4_val = 0.1    # S3 ->
k5_val = 0.05   # S1 -> (degradation)
k6_val = 0.05   # S2 -> (degradation)

# R1: -> id_xvpm
reaction1 = model.createReaction()
reaction1.setId("R1_prod_xvpm")
reaction1.setReversible(False)
reaction1.setFast(False)
prod1 = reaction1.createProduct()
prod1.setSpecies(s1)
prod1.setStoichiometry(1.0)
prod1.setConstant(False)
kl1 = reaction1.createKineticLaw()
p1 = kl1.createParameter()
p1.setId("k1")
p1.setValue(k1_val)
p1.setConstant(True)
kl1.setMath(libsbml.parseL3Formula("k1"))

# R2: id_xvpm -> id_aa2n
reaction2 = model.createReaction()
reaction2.setId("R2_xvpm_to_aa2n")
reaction2.setReversible(False)
reaction2.setFast(False)
reactant2_s1 = reaction2.createReactant()
reactant2_s1.setSpecies(s1)
reactant2_s1.setStoichiometry(1.0)
reactant2_s1.setConstant(False)
product2_s2 = reaction2.createProduct()
product2_s2.setSpecies(s2)
product2_s2.setStoichiometry(1.0)
product2_s2.setConstant(False)
kl2 = reaction2.createKineticLaw()
p2 = kl2.createParameter()
p2.setId("k2")
p2.setValue(k2_val)
p2.setConstant(True)
kl2.setMath(libsbml.parseL3Formula(f"k2 * {s1}"))

# R3: id_aa2n -> id_xdgv
reaction3 = model.createReaction()
reaction3.setId("R3_aa2n_to_xdgv")
reaction3.setReversible(False)
reaction3.setFast(False)
reactant3_s2 = reaction3.createReactant()
reactant3_s2.setSpecies(s2)
reactant3_s2.setStoichiometry(1.0)
reactant3_s2.setConstant(False)
product3_s3 = reaction3.createProduct()
product3_s3.setSpecies(s3)
product3_s3.setStoichiometry(1.0)
product3_s3.setConstant(False)
kl3 = reaction3.createKineticLaw()
p3 = kl3.createParameter()
p3.setId("k3")
p3.setValue(k3_val)
p3.setConstant(True)
kl3.setMath(libsbml.parseL3Formula(f"k3 * {s2}"))

# R4: id_xdgv ->
reaction4 = model.createReaction()
reaction4.setId("R4_degrad_xdgv")
reaction4.setReversible(False)
reaction4.setFast(False)
reactant4_s3 = reaction4.createReactant()
reactant4_s3.setSpecies(s3)
reactant4_s3.setStoichiometry(1.0)
reactant4_s3.setConstant(False)
kl4 = reaction4.createKineticLaw()
p4 = kl4.createParameter()
p4.setId("k4")
p4.setValue(k4_val)
p4.setConstant(True)
kl4.setMath(libsbml.parseL3Formula(f"k4 * {s3}"))

# R5: id_xvpm -> (degradation of S1)
reaction5 = model.createReaction()
reaction5.setId("R5_degrad_xvpm")
reaction5.setReversible(False)
reaction5.setFast(False)
reactant5_s1 = reaction5.createReactant()
reactant5_s1.setSpecies(s1)
reactant5_s1.setStoichiometry(1.0)
reactant5_s1.setConstant(False)
kl5 = reaction5.createKineticLaw()
p5 = kl5.createParameter()
p5.setId("k5")
p5.setValue(k5_val)
p5.setConstant(True)
kl5.setMath(libsbml.parseL3Formula(f"k5 * {s1}"))

# R6: id_aa2n -> (degradation of S2)
reaction6 = model.createReaction()
reaction6.setId("R6_degrad_aa2n")
reaction6.setReversible(False)
reaction6.setFast(False)
reactant6_s2 = reaction6.createReactant()
reactant6_s2.setSpecies(s2)
reactant6_s2.setStoichiometry(1.0)
reactant6_s2.setConstant(False)
kl6 = reaction6.createKineticLaw()
p6 = kl6.createParameter()
p6.setId("k6")
p6.setValue(k6_val)
p6.setConstant(True)
kl6.setMath(libsbml.parseL3Formula(f"k6 * {s2}"))


# Write the updated SBML to a string
writer = libsbml.SBMLWriter()
current_sbml_model_str_hyp2 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_sbml_model_str_iteration_3", current_sbml_model_str_hyp2)

# Simulate the model
sim_data_df_hyp2 = simulate(current_sbml_model_str_hyp2)
shared_variables.add("sim_data_df_iteration_3", sim_data_df_hyp2)

# Print simulation results
print("Simulation data (Hypothesis 2) head:")
print(sim_data_df_hyp2.head())
print("\nSimulation data (Hypothesis 2) tail:")
print(sim_data_df_hyp2.tail(5))

# Print parameter values used
print(f"\nParameters used for Hypothesis 2: k1={k1_val}, k2={k2_val}, k3={k3_val}, k4={k4_val}, k5={k5_val}, k6={k6_val}")

# For comparison, print some key points from experimental data
print("\nExperimental data (iteration_0) for comparison:")
print("Time points around id_xvpm peak (approx t=35):")
print(exp_data_df[exp_data_df['Time'] == 35.0][['Time', 'id_xvpm', 'id_aa2n', 'id_xdgv']])
print("id_xvpm experimental peak value approx: 1.18e-4")

print("\nTime points around id_aa2n peak (approx t=46):")
print(exp_data_df[exp_data_df['Time'] == 46.0][['Time', 'id_xvpm', 'id_aa2n', 'id_xdgv']])
print("id_aa2n experimental peak value approx: 8.69e-4")

print("\nTime points around id_xdgv peak (approx t=48):")
print(exp_data_df[exp_data_df['Time'] == 48.0][['Time', 'id_xvpm', 'id_aa2n', 'id_xdgv']])
print("id_xdgv experimental peak value approx: 1.96e-3")

print("\nExperimental data at t=500 (end of simulation):")
print(exp_data_df[exp_data_df['Time'] == 500.0][['Time', 'id_xvpm', 'id_aa2n', 'id_xdgv']])

# Print simulated values at times corresponding to experimental peaks for easier comparison
sim_time_points = [35.0, 46.0, 48.0, 500.0]
print("\nSimulated values at specific time points:")
print(sim_data_df_hyp2[sim_data_df_hyp2['Time'].isin(sim_time_points)])

```
# Observation

## Code Stdout
Simulation data (Hypothesis 2) head:
    id_xvpm   id_aa2n   id_xdgv  Time
0  0.000010  0.000010  0.000010   0.0
1  0.000018  0.000010  0.000010   0.5
2  0.000025  0.000011  0.000011   1.0
3  0.000032  0.000012  0.000012   1.5
4  0.000038  0.000014  0.000012   2.0

Simulation data (Hypothesis 2) tail:
      id_xvpm   id_aa2n   id_xdgv   Time
996   0.00008  0.000064  0.000128  498.0
997   0.00008  0.000064  0.000128  498.5
998   0.00008  0.000064  0.000128  499.0
999   0.00008  0.000064  0.000128  499.5
1000  0.00008  0.000064  0.000128  500.0

Parameters used for Hypothesis 2: k1=2e-05, k2=0.2, k3=0.2, k4=0.1, k5=0.05, k6=0.05

Experimental data (iteration_0) for comparison:
Time points around id_xvpm peak (approx t=35):
    Time   id_xvpm   id_aa2n   id_xdgv
70  35.0  0.000118  0.000675  0.001502
id_xvpm experimental peak value approx: 1.18e-4

Time points around id_aa2n peak (approx t=46):
    Time  id_xvpm   id_aa2n   id_xdgv
92  46.0  0.00008  0.000869  0.001954
id_aa2n experimental peak value approx: 8.69e-4

Time points around id_xdgv peak (approx t=48):
    Time   id_xvpm   id_aa2n   id_xdgv
96  48.0  0.000068  0.000861  0.001963
id_xdgv experimental peak value approx: 1.96e-3

Experimental data at t=500 (end of simulation):
       Time   id_xvpm   id_aa2n   id_xdgv
1000  500.0  0.000043  0.000452  0.001113

Simulated values at specific time points:
      id_xvpm   id_aa2n   id_xdgv   Time
70    0.00008  0.000064  0.000119   35.0
92    0.00008  0.000064  0.000125   46.0
96    0.00008  0.000064  0.000125   48.0
1000  0.00008  0.000064  0.000128  500.0

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml_model_str_iteration_2, sim_data_df_iteration_2, current_sbml_model_str_iteration_3, sim_data_df_iteration_3)

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


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



# Iteration 4
## Thoughts

The simulation results for Hypothesis 2 (linear chain with degradation for S1, S2, S3) are still far from the experimental data.

**Model Hypothesis 2 Parameters:**
`k1=2e-5` (Prod S1)
`k2=0.2` (S1->S2)
`k3=0.2` (S2->S3)
`k4=0.1` (S3 deg)
`k5=0.05` (S1 deg)
`k6=0.05` (S2 deg)

**Comparison at Peak Times:**

*   **`id_xvpm` (S1):**
    *   Experimental (t=35): `id_xvpm` = 0.000118
    *   Simulated (t=35): `id_xvpm` = 0.00008
    *   The simulated value is lower. The model reaches a steady state for `id_xvpm` quickly.
        `dS1/dt = k1 - (k2+k5)*S1 = 2e-5 - (0.2+0.05)*S1 = 2e-5 - 0.25*S1`.
        Steady state `S1_ss = 2e-5 / 0.25 = 8e-5`. This is what the simulation shows.
        The experimental data shows a peak and then a decline for `id_xvpm`. My model just goes to a steady state. This means the degradation/consumption terms are not strong enough or not timed correctly to cause a decline after an initial rise.

*   **`id_aa2n` (S2):**
    *   Experimental (t=46): `id_aa2n` = 0.000869
    *   Simulated (t=46): `id_aa2n` = 0.000064
    *   Vastly underestimated.
        `dS2/dt = k2*S1 - (k3+k6)*S2 = 0.2*S1 - (0.2+0.05)*S2 = 0.2*S1 - 0.25*S2`.
        If `S1` is at `8e-5`, then `S2_ss = (0.2 * 8e-5) / 0.25 = 1.6e-5 / 0.25 = 6.4e-5`. This matches the simulation.

*   **`id_xdgv` (S3):**
    *   Experimental (t=48): `id_xdgv` = 0.001963
    *   Simulated (t=48): `id_xdgv` = 0.000125
    *   Vastly underestimated.
        `dS3/dt = k3*S2 - k4*S3 = 0.2*S2 - 0.1*S3`.
        If `S2` is at `6.4e-5`, then `S3_ss = (0.2 * 6.4e-5) / 0.1 = 1.28e-4 / 0.1 = 1.28e-4`. This matches the simulation.

**Comparison at t=500:**
*   Experimental: `id_xvpm`=4.3e-5, `id_aa2n`=4.52e-4, `id_xdgv`=1.113e-3
*   Simulated: `id_xvpm`=8e-5, `id_aa2n`=6.4e-5, `id_xdgv`=1.28e-4

The current model (linear chain with first-order degradations) is not capable of producing the observed peaks and subsequent declines, nor the magnitudes, and certainly not the oscillations. The system quickly settles to a steady state determined by the rate constants.

**Key observations from experimental data not captured:**
1.  **Sharp peaks and declines:** The concentrations rise to a peak and then fall significantly. My model produces flat steady states.
2.  **Magnitudes:** Experimental peaks are much higher.
3.  **Oscillations:** The data snippet shows some up-and-down movement, especially visible in the full dataset's plot (which I can't see now but recall from the first iteration's attempt). This strongly suggests feedback or more complex regulatory mechanisms.

**Revising the strategy:**
A simple linear chain isn't working. The oscillatory behavior is a strong hint. A common motif for oscillations is a negative feedback loop. For example, `id_xdgv` (S3) could inhibit the production of `id_xvpm` (S1).

Let's consider a Goodwin-like oscillator model structure:
*   `-> S1` (production of S1, possibly inhibited by S3)
*   `S1 -> S2`
*   `S2 -> S3`
*   `S1 ->` (degradation of S1)
*   `S2 ->` (degradation of S2)
*   `S3 ->` (degradation of S3)

**Hypothesis 3: Linear chain with S3 inhibiting S1 production, plus individual degradations.**
*   **R1: `-> id_xvpm`** (Rate: `k1 / (1 + (id_xdgv/Ki)^n)`) (Inhibited production of S1 by S3)
*   **R2: `id_xvpm -> id_aa2n`** (k2 * [id_xvpm])
*   **R3: `id_aa2n -> id_xdgv`** (k3 * [id_aa2n])
*   **R4: `id_xdgv ->`** (k4 * [id_xdgv])
*   **R5: `id_xvpm ->`** (k5 * [id_xvpm])
*   **R6: `id_aa2n ->`** (k6 * [id_aa2n])

This introduces new parameters: `Ki` (inhibition constant for S3 on S1 production) and `n` (Hill coefficient for the inhibition). For simplicity, let's start with `n=1` or `n=2`.

Parameter estimation for this will be more challenging.
Let's try to adjust the previous parameters and introduce the feedback.
The maximum uninhibited production rate of S1 is `k1`.
When S3 is high, S1 production should decrease, allowing S1 levels to fall.

Let's try to set parameters to achieve the observed peak heights first, and then see if feedback can induce oscillations and the subsequent fall.
To get `S1_peak` ~ 1e-4: `k1_eff - (k2+k5)*S1_peak = 0`. `k1_eff` is the effective production rate.
If `k1` (max rate) is higher, say `5e-5`.
`k2=0.2`, `k5=0.1`. Then `(k2+k5) = 0.3`.
`S1_peak = k1_eff / 0.3`. If `S1_peak = 1e-4`, then `k1_eff = 0.3 * 1e-4 = 3e-5`.
This `k1_eff` should be `k1 / (1 + (S3_at_S1_peak/Ki)^n)`.
At `S1` peak (t=35), `S3` is `1.5e-3`.
So, `3e-5 = 5e-5 / (1 + (1.5e-3/Ki)^n)`.
`1 + (1.5e-3/Ki)^n = 5e-5 / 3e-5 = 5/3 = 1.667`.
`(1.5e-3/Ki)^n = 0.667`.

This is one equation with two unknowns (`Ki`, `n`). Let's assume `n=2` (a common choice for some cooperativity).
`(1.5e-3/Ki)^2 = 0.667`.
`1.5e-3/Ki = sqrt(0.667) = 0.816`.
`Ki = 1.5e-3 / 0.816 = 1.83e-3`. This value is around the peak concentration of S3.

Let's try these parameters for Hypothesis 3:
*   `k1 = 5e-5` (Max production rate of S1)
*   `Ki = 1.8e-3` (Inhibition constant for S3 on S1 production)
*   `n = 2` (Hill coefficient)
*   `k2 = 0.2` (S1 -> S2)
*   `k3 = 0.2` (S2 -> S3)
*   `k4 = 0.1` (S3 degradation)
*   `k5 = 0.1` (S1 degradation)
*   `k6 = 0.1` (S2 degradation) - increased this from 0.05 to try and get S2 to fall more.

The goal is to see if this feedback can create the initial peaks and subsequent declines, and potentially some oscillatory behavior.

The kinetic law for R1 will be `k1 / (1 + (id_xdgv/Ki_R1)^n_R1)`. I'll need to add `Ki_R1` and `n_R1` as parameters for this reaction.

Let's implement this.

A note on `setConstant(False)` for products: The SBML Level 3 Version 1 Core specification states for `Product`: "The attribute constant is a Boolean value indicating whether the Product references a Species whose quantity is fixed. This attribute is deprecated as of SBML Level 3 Version 1 Core; its value should always be `false`." So my previous usage was correct. Reactants also have `constant="false"`.

The `libsbml.parseL3Formula` should handle the Hill kinetics formula.
The formula will be `k1 / (1 + pow(id_xdgv / Ki_R1, n_R1))`
Parameters in kinetic law: `k1`, `Ki_R1`, `n_R1`.
`n_R1` should be a parameter, but often it's set as a literal number in the formula if it's fixed. For `simulate` to work, it's better if all numbers in the formula are either species or parameters defined in `<listOfParameters>` within the `<kineticLaw>`. Let's make `n_R1` a parameter.

## Action

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

# Access experimental data
exp_data_df = experiment_history['iteration_0']

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

# Define species IDs
s1 = "id_xvpm"
s2 = "id_aa2n"
s3 = "id_xdgv"

# New parameter values for Hypothesis 3 (Feedback Inhibition)
k1_val = 5.0e-5   # Max production rate of S1
Ki_R1_val = 1.8e-3 # Inhibition constant for S3 on S1 production
n_R1_val = 2.0     # Hill coefficient for inhibition
k2_val = 0.2     # S1 -> S2
k3_val = 0.2     # S2 -> S3
k4_val = 0.1     # S3 -> (degradation)
k5_val = 0.1     # S1 -> (degradation)
k6_val = 0.1     # S2 -> (degradation)

# R1: -> id_xvpm (inhibited by id_xdgv)
reaction1 = model.createReaction()
reaction1.setId("R1_prod_xvpm_inhib")
reaction1.setReversible(False)
reaction1.setFast(False)
prod1_s1 = reaction1.createProduct()
prod1_s1.setSpecies(s1)
prod1_s1.setStoichiometry(1.0)
prod1_s1.setConstant(False)
kl1 = reaction1.createKineticLaw()
# Parameter k1 for R1
p_k1_R1 = kl1.createParameter()
p_k1_R1.setId("k1_R1")
p_k1_R1.setValue(k1_val)
p_k1_R1.setConstant(True)
# Parameter Ki for R1
p_Ki_R1 = kl1.createParameter()
p_Ki_R1.setId("Ki_R1")
p_Ki_R1.setValue(Ki_R1_val)
p_Ki_R1.setConstant(True)
# Parameter n for R1
p_n_R1 = kl1.createParameter()
p_n_R1.setId("n_R1")
p_n_R1.setValue(n_R1_val)
p_n_R1.setConstant(True)
# Kinetic formula: k1_R1 / (1 + pow(s3 / Ki_R1, n_R1))
kl1.setMath(libsbml.parseL3Formula(f"k1_R1 / (1 + pow({s3} / Ki_R1, n_R1))"))


# R2: id_xvpm -> id_aa2n
reaction2 = model.createReaction()
reaction2.setId("R2_xvpm_to_aa2n")
reaction2.setReversible(False)
reaction2.setFast(False)
reactant2_s1 = reaction2.createReactant()
reactant2_s1.setSpecies(s1)
reactant2_s1.setStoichiometry(1.0)
reactant2_s1.setConstant(False)
product2_s2 = reaction2.createProduct()
product2_s2.setSpecies(s2)
product2_s2.setStoichiometry(1.0)
product2_s2.setConstant(False)
kl2 = reaction2.createKineticLaw()
p_k2_R2 = kl2.createParameter()
p_k2_R2.setId("k2_R2")
p_k2_R2.setValue(k2_val)
p_k2_R2.setConstant(True)
kl2.setMath(libsbml.parseL3Formula(f"k2_R2 * {s1}"))

# R3: id_aa2n -> id_xdgv
reaction3 = model.createReaction()
reaction3.setId("R3_aa2n_to_xdgv")
reaction3.setReversible(False)
reaction3.setFast(False)
reactant3_s2 = reaction3.createReactant()
reactant3_s2.setSpecies(s2)
reactant3_s2.setStoichiometry(1.0)
reactant3_s2.setConstant(False)
product3_s3 = reaction3.createProduct()
product3_s3.setSpecies(s3)
product3_s3.setStoichiometry(1.0)
product3_s3.setConstant(False)
kl3 = reaction3.createKineticLaw()
p_k3_R3 = kl3.createParameter()
p_k3_R3.setId("k3_R3")
p_k3_R3.setValue(k3_val)
p_k3_R3.setConstant(True)
kl3.setMath(libsbml.parseL3Formula(f"k3_R3 * {s2}"))

# R4: id_xdgv -> (degradation of S3)
reaction4 = model.createReaction()
reaction4.setId("R4_degrad_xdgv")
reaction4.setReversible(False)
reaction4.setFast(False)
reactant4_s3 = reaction4.createReactant()
reactant4_s3.setSpecies(s3)
reactant4_s3.setStoichiometry(1.0)
reactant4_s3.setConstant(False)
kl4 = reaction4.createKineticLaw()
p_k4_R4 = kl4.createParameter()
p_k4_R4.setId("k4_R4")
p_k4_R4.setValue(k4_val)
p_k4_R4.setConstant(True)
kl4.setMath(libsbml.parseL3Formula(f"k4_R4 * {s3}"))

# R5: id_xvpm -> (degradation of S1)
reaction5 = model.createReaction()
reaction5.setId("R5_degrad_xvpm")
reaction5.setReversible(False)
reaction5.setFast(False)
reactant5_s1 = reaction5.createReactant()
reactant5_s1.setSpecies(s1)
reactant5_s1.setStoichiometry(1.0)
reactant5_s1.setConstant(False)
kl5 = reaction5.createKineticLaw()
p_k5_R5 = kl5.createParameter()
p_k5_R5.setId("k5_R5")
p_k5_R5.setValue(k5_val)
p_k5_R5.setConstant(True)
kl5.setMath(libsbml.parseL3Formula(f"k5_R5 * {s1}"))

# R6: id_aa2n -> (degradation of S2)
reaction6 = model.createReaction()
reaction6.setId("R6_degrad_aa2n")
reaction6.setReversible(False)
reaction6.setFast(False)
reactant6_s2 = reaction6.createReactant()
reactant6_s2.setSpecies(s2)
reactant6_s2.setStoichiometry(1.0)
reactant6_s2.setConstant(False)
kl6 = reaction6.createKineticLaw()
p_k6_R6 = kl6.createParameter()
p_k6_R6.setId("k6_R6")
p_k6_R6.setValue(k6_val)
p_k6_R6.setConstant(True)
kl6.setMath(libsbml.parseL3Formula(f"k6_R6 * {s2}"))

# Write the updated SBML to a string
writer = libsbml.SBMLWriter()
current_sbml_model_str_hyp3 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_sbml_model_str_iteration_4", current_sbml_model_str_hyp3)

# Simulate the model
sim_data_df_hyp3 = simulate(current_sbml_model_str_hyp3)
shared_variables.add("sim_data_df_iteration_4", sim_data_df_hyp3)

# Print simulation results
print("Simulation data (Hypothesis 3 - Feedback Inhibition) head:")
print(sim_data_df_hyp3.head())
print("\nSimulation data (Hypothesis 3 - Feedback Inhibition) tail:")
print(sim_data_df_hyp3.tail(5))

# Print parameter values used
print(f"\nParameters used for Hypothesis 3:")
print(f"k1_R1={k1_val}, Ki_R1={Ki_R1_val}, n_R1={n_R1_val}")
print(f"k2_R2={k2_val}, k3_R3={k3_val}, k4_R4={k4_val}, k5_R5={k5_val}, k6_R6={k6_val}")

# Print simulated values at times corresponding to experimental peaks for easier comparison
sim_time_points = [0.0, 35.0, 46.0, 48.0, 100.0, 200.0, 300.0, 400.0, 500.0] # Added more time points
print("\nSimulated values at specific time points (Hypothesis 3):")
# Ensure exact time points exist or interpolate if necessary. Simulator output times are fixed.
# For simplicity, find closest available time points if exact match is not present.
# The simulate function returns specific time points, so direct indexing should work if times are in the output.
# Let's check if these times are in the sim_data_df_hyp3['Time']
actual_sim_times_to_print = []
for t_target in sim_time_points:
    # find the index of the closest time in sim_data_df_hyp3['Time']
    closest_time_idx = (np.abs(sim_data_df_hyp3['Time'] - t_target)).argmin()
    actual_sim_times_to_print.append(sim_data_df_hyp3['Time'].iloc[closest_time_idx])

# Remove duplicates if any from actual_sim_times_to_print
actual_sim_times_to_print = sorted(list(set(actual_sim_times_to_print)))
print(sim_data_df_hyp3[sim_data_df_hyp3['Time'].isin(actual_sim_times_to_print)])


print("\nExperimental data for comparison at these times:")
actual_exp_times_to_print = []
for t_target in sim_time_points:
    closest_time_idx = (np.abs(exp_data_df['Time'] - t_target)).argmin()
    actual_exp_times_to_print.append(exp_data_df['Time'].iloc[closest_time_idx])
actual_exp_times_to_print = sorted(list(set(actual_exp_times_to_print)))
print(exp_data_df[exp_data_df['Time'].isin(actual_exp_times_to_print)][['Time', 'id_xvpm', 'id_aa2n', 'id_xdgv']])

```
# Observation

## Code Stdout
Simulation data (Hypothesis 3 - Feedback Inhibition) head:
    id_xvpm   id_aa2n   id_xdgv  Time
0  0.000010  0.000010  0.000010   0.0
1  0.000032  0.000011  0.000011   0.5
2  0.000051  0.000013  0.000011   1.0
3  0.000067  0.000017  0.000012   1.5
4  0.000081  0.000021  0.000013   2.0

Simulation data (Hypothesis 3 - Feedback Inhibition) tail:
       id_xvpm   id_aa2n   id_xdgv   Time
996   0.000164  0.000109  0.000219  498.0
997   0.000164  0.000109  0.000219  498.5
998   0.000164  0.000109  0.000219  499.0
999   0.000164  0.000109  0.000219  499.5
1000  0.000164  0.000109  0.000219  500.0

Parameters used for Hypothesis 3:
k1_R1=5e-05, Ki_R1=0.0018, n_R1=2.0
k2_R2=0.2, k3_R3=0.2, k4_R4=0.1, k5_R5=0.1, k6_R6=0.1

Simulated values at specific time points (Hypothesis 3):
       id_xvpm   id_aa2n   id_xdgv   Time
0     0.000010  0.000010  0.000010    0.0
70    0.000165  0.000110  0.000206   35.0
92    0.000164  0.000110  0.000215   46.0
96    0.000164  0.000110  0.000216   48.0
200   0.000164  0.000109  0.000219  100.0
400   0.000164  0.000109  0.000219  200.0
600   0.000164  0.000109  0.000219  300.0
800   0.000164  0.000109  0.000219  400.0
1000  0.000164  0.000109  0.000219  500.0

Experimental data for comparison at these times:
       Time   id_xvpm   id_aa2n   id_xdgv
0       0.0  0.000010  0.000010  0.000010
70     35.0  0.000118  0.000675  0.001502
92     46.0  0.000080  0.000869  0.001954
96     48.0  0.000068  0.000861  0.001963
200   100.0  0.000004  0.000068  0.000455
400   200.0  0.000010  0.000234  0.000904
600   300.0  0.000024  0.000410  0.001145
800   400.0  0.000037  0.000467  0.001179
1000  500.0  0.000043  0.000452  0.001113

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml_model_str_iteration_2, sim_data_df_iteration_2, current_sbml_model_str_iteration_3, sim_data_df_iteration_3, current_sbml_model_str_iteration_4, sim_data_df_iteration_4)

### 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 feedback inhibition model (Hypothesis 3) did not produce oscillations and still resulted in a steady state, although at different concentrations than Hypothesis 2.

**Model Hypothesis 3 Parameters:**
`k1_R1=5e-5` (Max prod S1), `Ki_R1=1.8e-3` (Inhib const), `n_R1=2.0` (Hill coeff)
`k2_R2=0.2` (S1->S2), `k3_R3=0.2` (S2->S3)
`k4_R4=0.1` (S3 deg), `k5_R5=0.1` (S1 deg), `k6_R6=0.1` (S2 deg)

**Comparison of Simulated (Hyp3) vs. Experimental:**

*   **`id_xvpm` (S1):**
    *   Sim (t=35): 0.000165. Exp (t=35): 0.000118. (Sim is higher, but in the ballpark of the peak)
    *   Sim (t=500): 0.000164. Exp (t=500): 0.000043. (Sim stays high, Exp drops significantly)
*   **`id_aa2n` (S2):**
    *   Sim (t=46): 0.000110. Exp (t=46): 0.000869. (Sim vastly lower)
    *   Sim (t=500): 0.000109. Exp (t=500): 0.000452. (Sim vastly lower)
*   **`id_xdgv` (S3):**
    *   Sim (t=48): 0.000216. Exp (t=48): 0.001963. (Sim vastly lower)
    *   Sim (t=500): 0.000219. Exp (t=500): 0.001113. (Sim vastly lower)

The model reaches a steady state:
`S1_ss = 0.000164`
`S2_ss = 0.000109`
`S3_ss = 0.000219`

At this steady state, the rate of S1 production is `R1_rate = k1_R1 / (1 + (S3_ss / Ki_R1)^n_R1)`.
`R1_rate = 5e-5 / (1 + (0.000219 / 0.0018)^2) = 5e-5 / (1 + (0.1216)^2) = 5e-5 / (1 + 0.0148) = 5e-5 / 1.0148 = 4.927e-5`.
Rate of S1 consumption = `(k5_R5 + k2_R2) * S1_ss = (0.1 + 0.2) * 0.000164 = 0.3 * 0.000164 = 4.92e-5`.
This confirms the steady state for S1.

For S2: `k2_R2 * S1_ss = 0.2 * 0.000164 = 3.28e-5`.
Consumption of S2: `(k6_R6 + k3_R3) * S2_ss = (0.1 + 0.2) * 0.000109 = 0.3 * 0.000109 = 3.27e-5`. Matches.

For S3: `k3_R3 * S2_ss = 0.2 * 0.000109 = 2.18e-5`.
Consumption of S3: `k4_R4 * S3_ss = 0.1 * 0.000219 = 2.19e-5`. Matches.

The feedback loop, as parameterized, is not strong enough or the system parameters (degradation rates, conversion rates) are not in the correct regime to cause oscillations. For oscillations in a Goodwin-type model, you typically need a sufficiently high Hill coefficient (`n_R1`) and appropriate delays/rate constants through the pathway.

The experimental data shows `id_xvpm` peaking then falling, `id_aa2n` peaking then falling, and `id_xdgv` peaking then falling, followed by what looks like damped oscillations or at least further complex dynamics, not a simple decay to zero or a low steady state. The values at t=500 are still significant.

**Problem:** The model still doesn't capture the peak magnitudes for S2 and S3, nor the post-peak dynamics (the fall and oscillations). The chosen parameters for feedback are not inducing oscillations.

**Further thoughts on oscillations:**
A three-species chain with negative feedback (`S3` inhibits `S1` production) can oscillate if:
1.  The Hill coefficient (`n`) is high enough (typically `n > 8` for a 3-species chain if all degradation rates are equal, though this threshold can be lower with different parameters or additional non-linearities). My `n=2` is likely too low.
2.  The degradation rates and interconversion rates are tuned.

Let's try increasing the Hill coefficient `n_R1` significantly, e.g., to `n_R1 = 8` or `n_R1 = 10`.
Also, the `Ki_R1` value might need adjustment. If `Ki_R1` is too high, the feedback is weak. If it's too low, S1 production might be shut off too strongly and too early. The experimental `S3` peak is ~2e-3. My `Ki_R1` was 1.8e-3, which is reasonable (feedback engages around the peak).

The magnitudes of S2 and S3 are still a major issue.
`S2_peak_exp ~ 8.7e-4`
`S3_peak_exp ~ 1.96e-3`
My simulation gave `S2_ss ~ 1.1e-4` and `S3_ss ~ 2.2e-4`. These are about 8-9 times too low.

To increase S2 and S3 levels, we need either:
-   More S1 flowing into S2 (higher `k2_R2` or higher `S1`).
-   Slower removal of S2 (lower `k6_R6` or `k3_R3` if `k3_R3` is a bottleneck for S3 production but not S2 accumulation).
-   More S2 flowing into S3 (higher `k3_R3` or higher `S2`).
-   Slower removal of S3 (lower `k4_R4`).

Let's try to adjust parameters to boost S2 and S3 levels, and also increase `n_R1`.

**New Parameter Set (Hypothesis 3, attempt 2 - aiming for oscillations and better magnitudes):**
1.  **Feedback parameters (R1: `-> S1` inhib by S3):**
    *   `k1_R1`: Let's keep `5.0e-5`.
    *   `Ki_R1`: Let's keep `1.8e-3`. (S3 concentration at which inhibition is half-maximal)
    *   `n_R1`: Increase to `8`. (Higher Hill coefficient for stronger switch-like behavior)

2.  **Conversion rates:**
    *   `k2_R2` (S1 -> S2): If S1 peak is ~1e-4, and S2 peak needs to be ~8e-4. We need S2 to accumulate more.
        Perhaps `k2_R2` should be higher, or `k6_R6+k3_R3` (S2 removal) should be lower relative to `k2_R2*S1`.
        Let's try increasing `k2_R2` and `k3_R3`.
        `k2_R2 = 0.5` (was 0.2)
        `k3_R3 = 0.5` (was 0.2)

3.  **Degradation rates:**
    *   `k5_R5` (S1 deg): `0.1` (kept)
    *   `k6_R6` (S2 deg): `0.1` (kept)
    *   `k4_R4` (S3 deg): `0.1` (kept)

    Net rate for S1: `R1_prod - (k5_R5 + k2_R2)*S1 = R1_prod - (0.1 + 0.5)*S1 = R1_prod - 0.6*S1`.
    If `S1_peak` is `1e-4`, then `R1_prod_effective` should be `0.6 * 1e-4 = 6e-5`.
    This means my `k1_R1` (max prod rate) of `5e-5` might be too low if S1 is to reach 1e-4 even with some inhibition.
    Let's increase `k1_R1` to `1e-4`.
    Then `R1_prod_effective = 1e-4 / (1 + (S3/Ki)^n)`.
    If `S1_peak` is `1e-4`, then `1e-4 / (1 + (S3_at_S1_peak/Ki)^n) = 0.6 * 1e-4`.
    `1 / (1 + (S3_at_S1_peak/Ki)^n) = 0.6`.
    `1 + (S3_at_S1_peak/Ki)^n = 1/0.6 = 1.667`.
    `(S3_at_S1_peak/Ki)^n = 0.667`.
    If `S3_at_S1_peak` is `1.5e-3` (from exp data), `Ki = 1.8e-3`, `n=8`.
    `(1.5e-3 / 1.8e-3)^8 = (0.833)^8 = (5/6)^8 = 0.23`. This is not 0.667.
    My parameter choices are not entirely consistent, but this is an iterative search. The key is to see if high `n` and adjusted rates can induce oscillations and better magnitudes.

Let's refine the `k1_R1` and `Ki_R1` based on the new `n_R1=8`.
We want `(S3_val/Ki_R1)^8 = 0.667` where `S3_val` is the S3 concentration when S1 is peaking and its production rate is 60% of max.
`S3_val/Ki_R1 = (0.667)^(1/8) = (2/3)^(1/8) = 0.95`.
So, `Ki_R1 = S3_val / 0.95`. If `S3_val` (S3 conc. when S1 peaks) is around `1.5e-3` (experimental value at S1 peak time), then `Ki_R1 = 1.5e-3 / 0.95 = 1.58e-3`.

Revised parameters for Hypothesis 3, attempt 2:
*   `k1_R1 = 1.0e-4` (Max production rate of S1)
*   `Ki_R1 = 1.6e-3` (Inhibition constant for S3 on S1 production)
*   `n_R1 = 8.0` (Hill coefficient)
*   `k2_R2 = 0.5` (S1 -> S2)
*   `k3_R3 = 0.5` (S2 -> S3)
*   `k4_R4 = 0.1` (S3 degradation) - This might be too fast if S3 needs to accumulate to ~2e-3.
    If `S3_peak ~ 2e-3`, then at peak `k3*S2 = k4*S3`. `0.5*S2 = k4*2e-3`.
    If `S2_peak ~ 8e-4`, then `0.5*8e-4 = k4*2e-3` => `4e-4 = k4*2e-3` => `k4 = 4e-4 / 2e-3 = 0.2`.
    Let's try `k4_R4 = 0.2`.
*   `k5_R5 = 0.1` (S1 degradation)
*   `k6_R6 = 0.1` (S2 degradation)
    At `S2_peak ~ 8e-4`, `k2*S1 = (k3+k6)*S2`.
    `0.5*S1_at_S2_peak = (0.5+0.1)*8e-4 = 0.6 * 8e-4 = 4.8e-4`.
    `S1_at_S2_peak = 4.8e-4 / 0.5 = 9.6e-5`. This is plausible as S1 is around this value when S2 peaks (exp data: S1 at S2 peak time t=46 is 8e-5).

This set of parameters seems more self-consistent with the target peak values. The high Hill coefficient is the main new element to try and induce oscillations.

## Action

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

# Access experimental data
exp_data_df = experiment_history['iteration_0']

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

# Define species IDs
s1 = "id_xvpm"
s2 = "id_aa2n"
s3 = "id_xdgv"

# New parameter values for Hypothesis 3, attempt 2 (Feedback Inhibition, high n, adjusted rates)
k1_R1_val = 1.0e-4   # Max production rate of S1
Ki_R1_val = 1.6e-3   # Inhibition constant for S3 on S1 production
n_R1_val = 8.0       # Hill coefficient for inhibition

k2_R2_val = 0.5      # S1 -> S2
k3_R3_val = 0.5      # S2 -> S3

k4_R4_val = 0.2      # S3 -> (degradation)
k5_R5_val = 0.1      # S1 -> (degradation)
k6_R6_val = 0.1      # S2 -> (degradation)

# R1: -> id_xvpm (inhibited by id_xdgv)
reaction1 = model.createReaction()
reaction1.setId("R1_prod_xvpm_inhib")
reaction1.setReversible(False)
reaction1.setFast(False)
prod1_s1 = reaction1.createProduct()
prod1_s1.setSpecies(s1)
prod1_s1.setStoichiometry(1.0)
prod1_s1.setConstant(False)
kl1 = reaction1.createKineticLaw()
p_k1_R1 = kl1.createParameter(); p_k1_R1.setId("k1_R1"); p_k1_R1.setValue(k1_R1_val); p_k1_R1.setConstant(True)
p_Ki_R1 = kl1.createParameter(); p_Ki_R1.setId("Ki_R1"); p_Ki_R1.setValue(Ki_R1_val); p_Ki_R1.setConstant(True)
p_n_R1 = kl1.createParameter(); p_n_R1.setId("n_R1"); p_n_R1.setValue(n_R1_val); p_n_R1.setConstant(True)
kl1.setMath(libsbml.parseL3Formula(f"k1_R1 / (1 + pow({s3} / Ki_R1, n_R1))"))

# R2: id_xvpm -> id_aa2n
reaction2 = model.createReaction()
reaction2.setId("R2_xvpm_to_aa2n")
reaction2.setReversible(False); reaction2.setFast(False)
reactant2_s1 = reaction2.createReactant(); reactant2_s1.setSpecies(s1); reactant2_s1.setStoichiometry(1.0); reactant2_s1.setConstant(False)
product2_s2 = reaction2.createProduct(); product2_s2.setSpecies(s2); product2_s2.setStoichiometry(1.0); product2_s2.setConstant(False)
kl2 = reaction2.createKineticLaw()
p_k2_R2 = kl2.createParameter(); p_k2_R2.setId("k2_R2"); p_k2_R2.setValue(k2_R2_val); p_k2_R2.setConstant(True)
kl2.setMath(libsbml.parseL3Formula(f"k2_R2 * {s1}"))

# R3: id_aa2n -> id_xdgv
reaction3 = model.createReaction()
reaction3.setId("R3_aa2n_to_xdgv")
reaction3.setReversible(False); reaction3.setFast(False)
reactant3_s2 = reaction3.createReactant(); reactant3_s2.setSpecies(s2); reactant3_s2.setStoichiometry(1.0); reactant3_s2.setConstant(False)
product3_s3 = reaction3.createProduct(); product3_s3.setSpecies(s3); product3_s3.setStoichiometry(1.0); product3_s3.setConstant(False)
kl3 = reaction3.createKineticLaw()
p_k3_R3 = kl3.createParameter(); p_k3_R3.setId("k3_R3"); p_k3_R3.setValue(k3_R3_val); p_k3_R3.setConstant(True)
kl3.setMath(libsbml.parseL3Formula(f"k3_R3 * {s2}"))

# R4: id_xdgv -> (degradation of S3)
reaction4 = model.createReaction()
reaction4.setId("R4_degrad_xdgv")
reaction4.setReversible(False); reaction4.setFast(False)
reactant4_s3 = reaction4.createReactant(); reactant4_s3.setSpecies(s3); reactant4_s3.setStoichiometry(1.0); reactant4_s3.setConstant(False)
kl4 = reaction4.createKineticLaw()
p_k4_R4 = kl4.createParameter(); p_k4_R4.setId("k4_R4"); p_k4_R4.setValue(k4_R4_val); p_k4_R4.setConstant(True)
kl4.setMath(libsbml.parseL3Formula(f"k4_R4 * {s3}"))

# R5: id_xvpm -> (degradation of S1)
reaction5 = model.createReaction()
reaction5.setId("R5_degrad_xvpm")
reaction5.setReversible(False); reaction5.setFast(False)
reactant5_s1 = reaction5.createReactant(); reactant5_s1.setSpecies(s1); reactant5_s1.setStoichiometry(1.0); reactant5_s1.setConstant(False)
kl5 = reaction5.createKineticLaw()
p_k5_R5 = kl5.createParameter(); p_k5_R5.setId("k5_R5"); p_k5_R5.setValue(k5_R5_val); p_k5_R5.setConstant(True)
kl5.setMath(libsbml.parseL3Formula(f"k5_R5 * {s1}"))

# R6: id_aa2n -> (degradation of S2)
reaction6 = model.createReaction()
reaction6.setId("R6_degrad_aa2n")
reaction6.setReversible(False); reaction6.setFast(False)
reactant6_s2 = reaction6.createReactant(); reactant6_s2.setSpecies(s2); reactant6_s2.setStoichiometry(1.0); reactant6_s2.setConstant(False)
kl6 = reaction6.createKineticLaw()
p_k6_R6 = kl6.createParameter(); p_k6_R6.setId("k6_R6"); p_k6_R6.setValue(k6_R6_val); p_k6_R6.setConstant(True)
kl6.setMath(libsbml.parseL3Formula(f"k6_R6 * {s2}"))

# Write the updated SBML to a string
writer = libsbml.SBMLWriter()
current_sbml_model_str_hyp3_att2 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_sbml_model_str_iteration_5", current_sbml_model_str_hyp3_att2)

# Simulate the model
sim_data_df_hyp3_att2 = simulate(current_sbml_model_str_hyp3_att2)
shared_variables.add("sim_data_df_iteration_5", sim_data_df_hyp3_att2)

# Print simulation results
print("Simulation data (Hypothesis 3, Attempt 2 - High n) head:")
print(sim_data_df_hyp3_att2.head())
print("\nSimulation data (Hypothesis 3, Attempt 2 - High n) tail:")
print(sim_data_df_hyp3_att2.tail(5))

# Print parameter values used
print(f"\nParameters used for Hypothesis 3, Attempt 2:")
print(f"k1_R1={k1_R1_val}, Ki_R1={Ki_R1_val}, n_R1={n_R1_val}")
print(f"k2_R2={k2_R2_val}, k3_R3={k3_R3_val}")
print(f"k4_R4={k4_R4_val}, k5_R5={k5_R5_val}, k6_R6={k6_R6_val}")

# Print simulated values at key experimental time points
sim_time_points = [0.0, 35.0, 46.0, 48.0, 100.0, 200.0, 300.0, 400.0, 500.0]
actual_sim_times_to_print = []
for t_target in sim_time_points:
    closest_time_idx = (np.abs(sim_data_df_hyp3_att2['Time'] - t_target)).argmin()
    actual_sim_times_to_print.append(sim_data_df_hyp3_att2['Time'].iloc[closest_time_idx])
actual_sim_times_to_print = sorted(list(set(actual_sim_times_to_print)))
print("\nSimulated values at specific time points (Hypothesis 3, Attempt 2):")
print(sim_data_df_hyp3_att2[sim_data_df_hyp3_att2['Time'].isin(actual_sim_times_to_print)])

print("\nExperimental data for comparison at these times:")
actual_exp_times_to_print_exp = [] # Renamed to avoid conflict
for t_target in sim_time_points:
    closest_time_idx = (np.abs(exp_data_df['Time'] - t_target)).argmin()
    actual_exp_times_to_print_exp.append(exp_data_df['Time'].iloc[closest_time_idx])
actual_exp_times_to_print_exp = sorted(list(set(actual_exp_times_to_print_exp)))
print(exp_data_df[exp_data_df['Time'].isin(actual_exp_times_to_print_exp)][['Time', 'id_xvpm', 'id_aa2n', 'id_xdgv']])
```
# Observation

## Code Stdout
Simulation data (Hypothesis 3, Attempt 2 - High n) head:
    id_xvpm   id_aa2n   id_xdgv  Time
0  0.000010  0.000010  0.000010   0.0
1  0.000051  0.000014  0.000012   0.5
2  0.000081  0.000025  0.000015   1.0
3  0.000103  0.000039  0.000021   1.5
4  0.000119  0.000053  0.000030   2.0

Simulation data (Hypothesis 3, Attempt 2 - High n) tail:
       id_xvpm   id_aa2n   id_xdgv   Time
996   0.000167  0.000139  0.000347  498.0
997   0.000167  0.000139  0.000347  498.5
998   0.000167  0.000139  0.000347  499.0
999   0.000167  0.000139  0.000347  499.5
1000  0.000167  0.000139  0.000347  500.0

Parameters used for Hypothesis 3, Attempt 2:
k1_R1=0.0001, Ki_R1=0.0016, n_R1=8.0
k2_R2=0.5, k3_R3=0.5
k4_R4=0.2, k5_R5=0.1, k6_R6=0.1

Simulated values at specific time points (Hypothesis 3, Attempt 2):
       id_xvpm   id_aa2n   id_xdgv   Time
0     0.000010  0.000010  0.000010    0.0
70    0.000167  0.000139  0.000347   35.0
92    0.000167  0.000139  0.000347   46.0
96    0.000167  0.000139  0.000347   48.0
200   0.000167  0.000139  0.000347  100.0
400   0.000167  0.000139  0.000347  200.0
600   0.000167  0.000139  0.000347  300.0
800   0.000167  0.000139  0.000347  400.0
1000  0.000167  0.000139  0.000347  500.0

Experimental data for comparison at these times:
       Time   id_xvpm   id_aa2n   id_xdgv
0       0.0  0.000010  0.000010  0.000010
70     35.0  0.000118  0.000675  0.001502
92     46.0  0.000080  0.000869  0.001954
96     48.0  0.000068  0.000861  0.001963
200   100.0  0.000004  0.000068  0.000455
400   200.0  0.000010  0.000234  0.000904
600   300.0  0.000024  0.000410  0.001145
800   400.0  0.000037  0.000467  0.001179
1000  500.0  0.000043  0.000452  0.001113

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml_model_str_iteration_2, sim_data_df_iteration_2, current_sbml_model_str_iteration_3, sim_data_df_iteration_3, current_sbml_model_str_iteration_4, sim_data_df_iteration_4, current_sbml_model_str_iteration_5, sim_data_df_iteration_5)

### 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 second attempt with Hypothesis 3 (feedback inhibition, high Hill coefficient `n_R1=8`, and adjusted rates) still failed to produce oscillations and resulted in a steady state.

**Model Hypothesis 3, Attempt 2 Parameters:**
`k1_R1=1.0e-4` (Max prod S1), `Ki_R1=1.6e-3` (Inhib const), `n_R1=8.0` (Hill coeff)
`k2_R2=0.5` (S1->S2), `k3_R3=0.5` (S2->S3)
`k4_R4=0.2` (S3 deg), `k5_R5=0.1` (S1 deg), `k6_R6=0.1` (S2 deg)

**Comparison of Simulated (Hyp3, Att2) vs. Experimental:**

*   **Steady State Values (Simulated):**
    *   `id_xvpm` (S1): 0.000167
    *   `id_aa2n` (S2): 0.000139
    *   `id_xdgv` (S3): 0.000347

*   **Experimental Values (around peaks/later times):**
    *   `id_xvpm` (S1 peak ~t=35): 0.000118; (t=500): 0.000043
    *   `id_aa2n` (S2 peak ~t=46): 0.000869; (t=500): 0.000452
    *   `id_xdgv` (S3 peak ~t=48): 0.001963; (t=500): 0.001113

**Analysis:**
1.  **No Oscillations:** Even with `n_R1=8`, the system settles to a stable steady state. This implies that the parameters are not in the oscillatory regime. For a 3-stage feedback loop to oscillate, the product of the sensitivities (related to Hill coefficients and gains) around the loop must be high enough, and the delays (related to lifetimes of species) must be appropriate.
2.  **Magnitudes Still Off:**
    *   Simulated S1 (0.000167) is somewhat higher than the experimental peak (0.000118) but doesn't fall.
    *   Simulated S2 (0.000139) is much lower than experimental peak (0.000869). (Factor of ~6x too low)
    *   Simulated S3 (0.000347) is much lower than experimental peak (0.001963). (Factor of ~5.6x too low)

The system is too stable. The degradation rates might be too high relative to production/conversion, or the feedback isn't kicking in correctly to destabilize the steady state.

Let's check the steady state calculations for this attempt:
`S1_ss = 0.000167`, `S2_ss = 0.000139`, `S3_ss = 0.000347`
Rate of S1 production `R1_rate = k1_R1 / (1 + (S3_ss / Ki_R1)^n_R1)`
`R1_rate = 1e-4 / (1 + (0.000347 / 0.0016)^8) = 1e-4 / (1 + (0.216875)^8)`
`(0.216875)^8 = (0.216875^2)^4 = (0.047035)^4 = (0.002212)^2 = 4.89e-6`
`R1_rate = 1e-4 / (1 + 4.89e-6) approx 1e-4`. So inhibition is very weak at this S3_ss.
Rate of S1 consumption = `(k5_R5 + k2_R2) * S1_ss = (0.1 + 0.5) * 0.000167 = 0.6 * 0.000167 = 1.002e-4`. Matches.
Since inhibition is weak, S1 is basically at `k1_R1 / (k5_R5 + k2_R2) = 1e-4 / 0.6 = 1.666e-4`.

This means `S3_ss` is too low to cause significant inhibition. `S3_ss` needs to rise to levels comparable to `Ki_R1` (1.6e-3) for strong feedback.
Current `S3_ss` is `0.000347`.

To get higher `S3_ss`:
-   Increase `k3_R3` (S2->S3 conversion)
-   Decrease `k4_R4` (S3 degradation)
-   Increase `S2_ss`

To get higher `S2_ss`:
-   Increase `k2_R2` (S1->S2 conversion)
-   Decrease `k6_R6` (S2 degradation)
-   Increase `S1_ss` (but S1 is already a bit high compared to exp peak)

The problem might be that the degradation rates `k4, k5, k6` are too fast, preventing accumulation and slowing down the "signal" propagation through the loop, which is necessary for oscillations. Oscillations often require that the species lifetimes are sufficiently long.

Let's try to reduce degradation rates `k4_R4, k5_R5, k6_R6` and see if that helps S2 and S3 build up and potentially create oscillations. If S3 builds up more, the feedback will be stronger.

**New Parameter Set (Hypothesis 3, attempt 3 - Slower Degradation):**
Keep feedback parameters and conversion rates from previous attempt:
*   `k1_R1 = 1.0e-4`
*   `Ki_R1 = 1.6e-3`
*   `n_R1 = 8.0`
*   `k2_R2 = 0.5`
*   `k3_R3 = 0.5`

Modify degradation rates (reduce them by a factor of 5, for example):
*   `k4_R4` (S3 deg): `0.2 / 5 = 0.04` (was 0.2)
*   `k5_R5` (S1 deg): `0.1 / 5 = 0.02` (was 0.1)
*   `k6_R6` (S2 deg): `0.1 / 5 = 0.02` (was 0.1)

Let's analyze the expected steady state with these new degradation rates, assuming weak inhibition first:
`S1_ss_no_inhib = k1_R1 / (k5_R5 + k2_R2) = 1e-4 / (0.02 + 0.5) = 1e-4 / 0.52 = 1.92e-4`. (S1 would be higher)
`S2_ss_no_inhib = k2_R2*S1_ss / (k6_R6 + k3_R3) = 0.5 * 1.92e-4 / (0.02 + 0.5) = 0.5 * 1.92e-4 / 0.52 = 1.85e-4`.
`S3_ss_no_inhib = k3_R3*S2_ss / k4_R4 = 0.5 * 1.85e-4 / 0.04 = 0.5 * 1.85e-4 / 0.04 = 2.31e-3`.

This `S3_ss_no_inhib` of `2.31e-3` is now higher than `Ki_R1` (1.6e-3). So inhibition will be significant.
Let `x = S3_ss / Ki_R1`. The production rate of S1 is `k1_R1 / (1+x^8)`.
Equations for steady state:
1.  `k1_R1 / (1 + (S3/Ki_R1)^8) = (k5_R5 + k2_R2) * S1`
2.  `k2_R2 * S1 = (k6_R6 + k3_R3) * S2`
3.  `k3_R3 * S2 = k4_R4 * S3`

From (2): `S1 = ((k6_R6 + k3_R3) / k2_R2) * S2 = (0.52 / 0.5) * S2 = 1.04 * S2`
From (3): `S2 = (k4_R4 / k3_R3) * S3 = (0.04 / 0.5) * S3 = 0.08 * S3`
So, `S1 = 1.04 * 0.08 * S3 = 0.0832 * S3`.

Substitute S1 into (1):
`k1_R1 / (1 + (S3/Ki_R1)^8) = (k5_R5 + k2_R2) * 0.0832 * S3`
`1e-4 / (1 + (S3/1.6e-3)^8) = 0.52 * 0.0832 * S3 = 0.043264 * S3`
`1e-4 = 0.043264 * S3 * (1 + (S3/1.6e-3)^8)`

This equation needs to be solved for S3.
If S3 is around `Ki_R1 = 1.6e-3`:
LHS = `1e-4`.
RHS = `0.043264 * 1.6e-3 * (1 + 1^8) = 0.043264 * 1.6e-3 * 2 = 6.922e-5 * 2 = 1.38e-4`.
LHS (1e-4) is a bit smaller than RHS (1.38e-4). This means the actual S3_ss will be slightly lower than `Ki_R1` to make the RHS smaller.
If S3_ss is slightly lower than `Ki_R1`, then `(S3/Ki_R1)^8` will be less than 1, making `(1 + (S3/Ki_R1)^8)` closer to 1.
Let's say `S3_ss = 1.5e-3`.
RHS = `0.043264 * 1.5e-3 * (1 + (1.5/1.6)^8) = 0.043264 * 1.5e-3 * (1 + (0.9375)^8) = 0.043264 * 1.5e-3 * (1 + 0.55)`
RHS = `6.49e-5 * 1.55 = 1.005e-4`. This is very close to LHS (1e-4).
So, `S3_ss ~ 1.5e-3`.
Then `S2_ss = 0.08 * 1.5e-3 = 1.2e-4`.
And `S1_ss = 0.0832 * 1.5e-3 = 1.25e-4`.

Predicted steady state with new parameters:
`S1_ss ~ 1.25e-4` (Exp peak ~1.18e-4, then falls to 4.3e-5) - This is a good target for peak.
`S2_ss ~ 1.2e-4` (Exp peak ~8.69e-4) - Still too low.
`S3_ss ~ 1.5e-3` (Exp peak ~1.96e-3) - Closer, good target for peak.

The issue is `S2_ss` is still too low.
`S2_ss = (k2_R2 / (k6_R6 + k3_R3)) * S1_ss`.
`S2_ss = (0.5 / (0.02 + 0.5)) * S1_ss = (0.5 / 0.52) * S1_ss = 0.96 * S1_ss`.
So S2 will track S1 closely, it won't be able to get much higher than S1.
Experimental S2 peak is ~7-8 times S1 peak. This implies `k2_R2` must be much larger than `(k6_R6 + k3_R3)`, or `k6_R6 + k3_R3` must be very small.
If `k3_R3` is small, then S3 won't be produced efficiently.

Perhaps the structure `S1 -> S2 -> S3` is too simple for S2 to accumulate that much more than S1 if S3 also needs to be high.
What if S2 production is not first order in S1, or S2 degradation is slower?
Let's try to make S2 accumulate more by increasing `k2_R2` relative to S2's consumption rates.
Let `k2_R2 = 1.0` (was 0.5).
And keep `k3_R3 = 0.5`.
`k5_R5 = 0.02` (S1 deg)
`k6_R6 = 0.02` (S2 deg)
`k4_R4 = 0.04` (S3 deg)

Recalculate with `k2_R2 = 1.0`:
`S1_ss_no_inhib = k1_R1 / (k5_R5 + k2_R2) = 1e-4 / (0.02 + 1.0) = 1e-4 / 1.02 = 0.98e-4`.
`S2_ss_no_inhib = k2_R2*S1_ss / (k6_R6 + k3_R3) = 1.0 * 0.98e-4 / (0.02 + 0.5) = 0.98e-4 / 0.52 = 1.88e-4`.
`S3_ss_no_inhib = k3_R3*S2_ss / k4_R4 = 0.5 * 1.88e-4 / 0.04 = 2.35e-3`.
This S3 is still high enough to cause inhibition.

Let's re-solve for steady state:
1.  `S1 = ((k6_R6 + k3_R3) / k2_R2) * S2 = (0.52 / 1.0) * S2 = 0.52 * S2`
2.  `S2 = (k4_R4 / k3_R3) * S3 = (0.04 / 0.5) * S3 = 0.08 * S3`
So, `S1 = 0.52 * 0.08 * S3 = 0.0416 * S3`.

Substitute S1 into (1):
`k1_R1 / (1 + (S3/Ki_R1)^8) = (k5_R5 + k2_R2) * 0.0416 * S3`
`1e-4 / (1 + (S3/1.6e-3)^8) = (0.02 + 1.0) * 0.0416 * S3 = 1.02 * 0.0416 * S3 = 0.042432 * S3`
`1e-4 = 0.042432 * S3 * (1 + (S3/1.6e-3)^8)`
This is very similar to the previous equation for S3 (coefficient changed from 0.043264 to 0.042432).
So `S3_ss` will be slightly higher than 1.5e-3. Let's say `S3_ss ~ 1.52e-3`.
Then `S2_ss = 0.08 * 1.52e-3 = 1.216e-4`.
And `S1_ss = 0.0416 * 1.52e-3 = 0.632e-4`.

Predicted steady state:
`S1_ss ~ 0.63e-4` (Exp peak ~1.18e-4, then falls to 4.3e-5) - S1 is now lower, which is good for post-peak.
`S2_ss ~ 1.22e-4` (Exp peak ~8.69e-4) - S2 is still far too low.
`S3_ss ~ 1.52e-3` (Exp peak ~1.96e-3) - S3 is good.

The problem of S2 not accumulating enough persists.
`S2_ss / S1_ss = k2_R2 / (k6_R6 + k3_R3) = 1.0 / (0.02 + 0.5) = 1.0 / 0.52 = 1.92`.
So S2 can be at most ~2x S1 with these parameters. We need it to be ~7-8x S1.
To achieve this, `k2_R2` must be much larger than `k6_R6 + k3_R3`.
This means `k6_R6 + k3_R3` must be small.
If `k3_R3` is small, S3 production is slow.
If `k6_R6` is very small (S2 is very stable), S2 can accumulate.

Let's try making `k6_R6` (S2 degradation) very small.
And `k3_R3` (S2->S3) also relatively small to allow S2 to build up before converting to S3, which might help with delays for oscillation.

**New Parameter Set (Hypothesis 3, attempt 4 - Slower S2 turnover, aiming for S2 accumulation and oscillations):**
Feedback:
*   `k1_R1 = 1.0e-4`
*   `Ki_R1 = 1.6e-3` (S3 value where feedback is strong)
*   `n_R1 = 8.0`

S1 dynamics:
*   `k2_R2 = 0.5` (S1 -> S2 rate, was 1.0, let's reduce it a bit if S1 lifetime is longer)
*   `k5_R5 = 0.02` (S1 degradation)
    S1 lifetime related to `1/(k5_R5+k2_R2) = 1/(0.02+0.5) = 1/0.52 ~ 2 seconds`. (This is very short time, peaks are at 30-50s)
    The rates `k2, k3, k4, k5, k6` are probably too high in general if the timescale of dynamics is tens of seconds.
    The experimental peaks are around 30-60s. This means the rates of change should be slower.
    If `S1_peak ~ 1e-4` and it takes ~30s, then `dS1/dt ~ 1e-4/30 ~ 3e-6`.
    `k1_R1` is `1e-4`. This is much larger. This means S1 should rise very fast.
    The parameters are likely in the wrong order of magnitude for the timescale.

Let's scale down all rate constants (except `n_R1` and `Ki_R1` which is a concentration).
Previous rates were like 0.1, 0.5, etc. These correspond to lifetimes of 2-10 seconds.
If dynamics are on timescale of 100s, rates should be more like 0.01.

Let's try scaling k-values down by a factor of ~10.
Original "good" S3_ss was ~1.5e-3. S1_ss ~0.6e-4. S2_ss ~1.2e-4.
`k1_R1` was `1e-4`. Let's make it `1e-5`.
`Ki_R1 = 1.6e-3` (concentration, keep)
`n_R1 = 8.0` (dimensionless, keep)

Old rates: `k2=0.5, k3=0.5, k4=0.04, k5=0.02, k6=0.02`.
New rates (scaled down by 10):
*   `k2_R2 = 0.05` (S1 -> S2)
*   `k3_R3 = 0.05` (S2 -> S3)
*   `k4_R4 = 0.004` (S3 degradation)
*   `k5_R5 = 0.002` (S1 degradation)
*   `k6_R6 = 0.002` (S2 degradation)

Let's check steady state with these scaled-down rates:
`k1_new = k1_old / 10`. All other `k_new = k_old / 10`.
The steady state concentrations `S_ss` are determined by ratios of ks, or `k_prod / k_deg`.
If `S_ss = k_prod / k_deg`, then `S_ss_new = (k_prod/10) / (k_deg/10) = S_ss_old`.
So the steady state concentrations should be the same as the previous calculation:
`S1_ss ~ 0.63e-4`
`S2_ss ~ 1.22e-4` (still too low)
`S3_ss ~ 1.52e-3`

The main effect of scaling down rates will be to slow down the dynamics. This might help achieve oscillations if the system was previously too fast to establish proper phase relationships.
The problem of S2 being too low (`S2_ss / S1_ss = k2/(k6+k3) = 0.05 / (0.002+0.05) = 0.05 / 0.052 ~ 1`) remains.

To get `S2/S1 ~ 7`: `k2 / (k6+k3) ~ 7`.
Let `k2 = 0.07`. Let `k6+k3 = 0.01`.
If `k6 = 0.002` (very slow S2 degradation), then `k3 = 0.008`.
This makes `k3` (S2->S3) much slower than `k2` (S1->S2). This introduces a delay and could help oscillations.

**Parameter Set (Hypothesis 3, attempt 5 - Scaled rates, S2 accumulation focus):**
*   `k1_R1 = 1.0e-5` (S1 max production)
*   `Ki_R1 = 1.6e-3`
*   `n_R1 = 8.0`

*   `k5_R5 = 0.002` (S1 degradation)
*   `k2_R2 = 0.07` (S1 -> S2)
    *   `S1_eff_lifetime = 1/(0.002+0.07) = 1/0.072 ~ 14s`. (Getting better for timescale)
    *   `S1_ss_no_inhib = 1e-5 / 0.072 = 1.39e-4`. (Target S1 peak ~1.18e-4, so this is good starting point)

*   `k6_R6 = 0.002` (S2 degradation)
*   `k3_R3 = 0.008` (S2 -> S3)
    *   `S2_eff_lifetime = 1/(0.002+0.008) = 1/0.01 = 100s`. (Long lifetime for S2)
    *   `S2_ss_no_inhib / S1_ss_no_inhib = k2/(k6+k3) = 0.07/0.01 = 7`. This is the target ratio!
    *   So `S2_ss_no_inhib = 7 * 1.39e-4 = 9.73e-4`. (Target S2 peak ~8.69e-4, good!)

*   `k4_R4 = ?` (S3 degradation)
    *   `S3_eff_lifetime = 1/k4`.
    *   `S3_ss_no_inhib / S2_ss_no_inhib = k3/k4 = 0.008/k4`.
    *   We want `S3_ss_no_inhib ~ 1.6e-3` (around Ki) to `2e-3` (exp peak).
    *   `S3_ss_no_inhib = (0.008/k4) * 9.73e-4`.
    *   If `S3_ss_no_inhib = 1.8e-3` (midpoint), then `1.8e-3 = (0.008/k4) * 9.73e-4`.
    *   `k4 = 0.008 * 9.73e-4 / 1.8e-3 = 0.008 * 0.54 = 0.0043`.
    *   `S3_eff_lifetime = 1/0.0043 ~ 230s`. (Long lifetime for S3)

This set of parameters looks promising for achieving the correct relative magnitudes and operating on a slower timescale. The long lifetimes and high Hill coefficient might finally produce oscillations.

Final check of S1_ss with feedback:
`S1 = ((k6+k3)/k2) * S2 = (0.01/0.07)*S2 = (1/7)*S2`
`S2 = (k4/k3) * S3 = (0.0043/0.008)*S3 = 0.5375 * S3`
`S1 = (1/7) * 0.5375 * S3 = 0.0768 * S3`

`k1_R1 / (1 + (S3/Ki_R1)^n_R1) = (k5_R5 + k2_R2) * S1`
`1e-5 / (1 + (S3/1.6e-3)^8) = (0.002 + 0.07) * 0.0768 * S3`
`1e-5 / (1 + (S3/1.6e-3)^8) = 0.072 * 0.0768 * S3 = 0.00553 * S3`
`1e-5 = 0.00553 * S3 * (1 + (S3/1.6e-3)^8)`
If `S3_ss = Ki_R1 = 1.6e-3`:
RHS = `0.00553 * 1.6e-3 * (1+1) = 0.00553 * 1.6e-3 * 2 = 8.848e-6 * 2 = 1.77e-5`.
LHS (1e-5) is smaller than RHS (1.77e-5). So S3_ss will be lower than Ki_R1.
Let `S3_ss = 1.3e-3`.
` (1.3/1.6)^8 = (0.8125)^8 = 0.186`.
RHS = `0.00553 * 1.3e-3 * (1 + 0.186) = 0.00553 * 1.3e-3 * 1.186 = 7.189e-6 * 1.186 = 8.52e-6`.
This is now smaller than 1e-5. So S3_ss is between 1.3e-3 and 1.6e-3.
Let's say `S3_ss ~ 1.4e-3`.
Then `S2_ss = 0.5375 * 1.4e-3 = 7.5e-4`. (Exp peak 8.7e-4) - Good.
`S1_ss = 0.0768 * 1.4e-3 = 1.07e-4`. (Exp peak 1.18e-4) - Good.

These steady-state values (if it settles) are very close to experimental peak values. The slower dynamics and high Hill coefficient give the best chance so far for oscillations.

## Action

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

# Access experimental data
exp_data_df = experiment_history['iteration_0']

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

# Define species IDs
s1 = "id_xvpm"
s2 = "id_aa2n"
s3 = "id_xdgv"

# Parameter Set (Hypothesis 3, attempt 5 - Scaled rates, S2 accumulation focus)
k1_R1_val = 1.0e-5   # S1 max production
Ki_R1_val = 1.6e-3   # Inhibition constant for S3 on S1 production
n_R1_val = 8.0       # Hill coefficient for inhibition

k5_R5_val = 0.002    # S1 degradation
k2_R2_val = 0.07     # S1 -> S2

k6_R6_val = 0.002    # S2 degradation
k3_R3_val = 0.008    # S2 -> S3

k4_R4_val = 0.0043   # S3 degradation

# R1: -> id_xvpm (inhibited by id_xdgv)
reaction1 = model.createReaction(); reaction1.setId("R1_prod_xvpm_inhib"); reaction1.setReversible(False); reaction1.setFast(False)
prod1_s1 = reaction1.createProduct(); prod1_s1.setSpecies(s1); prod1_s1.setStoichiometry(1.0); prod1_s1.setConstant(False)
kl1 = reaction1.createKineticLaw()
p_k1_R1 = kl1.createParameter(); p_k1_R1.setId("k1_R1"); p_k1_R1.setValue(k1_R1_val); p_k1_R1.setConstant(True)
p_Ki_R1 = kl1.createParameter(); p_Ki_R1.setId("Ki_R1"); p_Ki_R1.setValue(Ki_R1_val); p_Ki_R1.setConstant(True)
p_n_R1 = kl1.createParameter(); p_n_R1.setId("n_R1"); p_n_R1.setValue(n_R1_val); p_n_R1.setConstant(True)
kl1.setMath(libsbml.parseL3Formula(f"k1_R1 / (1 + pow({s3} / Ki_R1, n_R1))"))

# R2: id_xvpm -> id_aa2n
reaction2 = model.createReaction(); reaction2.setId("R2_xvpm_to_aa2n"); reaction2.setReversible(False); reaction2.setFast(False)
reactant2_s1 = reaction2.createReactant(); reactant2_s1.setSpecies(s1); reactant2_s1.setStoichiometry(1.0); reactant2_s1.setConstant(False)
product2_s2 = reaction2.createProduct(); product2_s2.setSpecies(s2); product2_s2.setStoichiometry(1.0); product2_s2.setConstant(False)
kl2 = reaction2.createKineticLaw()
p_k2_R2 = kl2.createParameter(); p_k2_R2.setId("k2_R2"); p_k2_R2.setValue(k2_R2_val); p_k2_R2.setConstant(True)
kl2.setMath(libsbml.parseL3Formula(f"k2_R2 * {s1}"))

# R3: id_aa2n -> id_xdgv
reaction3 = model.createReaction(); reaction3.setId("R3_aa2n_to_xdgv"); reaction3.setReversible(False); reaction3.setFast(False)
reactant3_s2 = reaction3.createReactant(); reactant3_s2.setSpecies(s2); reactant3_s2.setStoichiometry(1.0); reactant3_s2.setConstant(False)
product3_s3 = reaction3.createProduct(); product3_s3.setSpecies(s3); product3_s3.setStoichiometry(1.0); product3_s3.setConstant(False)
kl3 = reaction3.createKineticLaw()
p_k3_R3 = kl3.createParameter(); p_k3_R3.setId("k3_R3"); p_k3_R3.setValue(k3_R3_val); p_k3_R3.setConstant(True)
kl3.setMath(libsbml.parseL3Formula(f"k3_R3 * {s2}"))

# R4: id_xdgv -> (degradation of S3)
reaction4 = model.createReaction(); reaction4.setId("R4_degrad_xdgv"); reaction4.setReversible(False); reaction4.setFast(False)
reactant4_s3 = reaction4.createReactant(); reactant4_s3.setSpecies(s3); reactant4_s3.setStoichiometry(1.0); reactant4_s3.setConstant(False)
kl4 = reaction4.createKineticLaw()
p_k4_R4 = kl4.createParameter(); p_k4_R4.setId("k4_R4"); p_k4_R4.setValue(k4_R4_val); p_k4_R4.setConstant(True)
kl4.setMath(libsbml.parseL3Formula(f"k4_R4 * {s3}"))

# R5: id_xvpm -> (degradation of S1)
reaction5 = model.createReaction(); reaction5.setId("R5_degrad_xvpm"); reaction5.setReversible(False); reaction5.setFast(False)
reactant5_s1 = reaction5.createReactant(); reactant5_s1.setSpecies(s1); reactant5_s1.setStoichiometry(1.0); reactant5_s1.setConstant(False)
kl5 = reaction5.createKineticLaw()
p_k5_R5 = kl5.createParameter(); p_k5_R5.setId("k5_R5"); p_k5_R5.setValue(k5_R5_val); p_k5_R5.setConstant(True)
kl5.setMath(libsbml.parseL3Formula(f"k5_R5 * {s1}"))

# R6: id_aa2n -> (degradation of S2)
reaction6 = model.createReaction(); reaction6.setId("R6_degrad_aa2n"); reaction6.setReversible(False); reaction6.setFast(False)
reactant6_s2 = reaction6.createReactant(); reactant6_s2.setSpecies(s2); reactant6_s2.setStoichiometry(1.0); reactant6_s2.setConstant(False)
kl6 = reaction6.createKineticLaw()
p_k6_R6 = kl6.createParameter(); p_k6_R6.setId("k6_R6"); p_k6_R6.setValue(k6_R6_val); p_k6_R6.setConstant(True)
kl6.setMath(libsbml.parseL3Formula(f"k6_R6 * {s2}"))

# Write the updated SBML to a string
writer = libsbml.SBMLWriter()
current_sbml_model_str_hyp3_att5 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_sbml_model_str_iteration_6", current_sbml_model_str_hyp3_att5)

# Simulate the model
sim_data_df_hyp3_att5 = simulate(current_sbml_model_str_hyp3_att5)
shared_variables.add("sim_data_df_iteration_6", sim_data_df_hyp3_att5)

# Print simulation results
print(f"Simulation data (Hypothesis 3, Attempt 5) head:\n{sim_data_df_hyp3_att5.head()}")
print(f"\nSimulation data (Hypothesis 3, Attempt 5) tail:\n{sim_data_df_hyp3_att5.tail(5)}")

# Print parameter values used
print(f"\nParameters used for Hypothesis 3, Attempt 5:")
print(f"k1_R1={k1_R1_val}, Ki_R1={Ki_R1_val}, n_R1={n_R1_val}")
print(f"k2_R2={k2_R2_val}, k3_R3={k3_R3_val}")
print(f"k4_R4={k4_R4_val}, k5_R5={k5_R5_val}, k6_R6={k6_R6_val}")

# Print simulated values at key experimental time points
sim_time_points = [0.0, 35.0, 46.0, 48.0, 100.0, 200.0, 300.0, 400.0, 500.0]
actual_sim_times_to_print = []
for t_target in sim_time_points:
    closest_time_idx = (np.abs(sim_data_df_hyp3_att5['Time'] - t_target)).argmin()
    actual_sim_times_to_print.append(sim_data_df_hyp3_att5['Time'].iloc[closest_time_idx])
actual_sim_times_to_print = sorted(list(set(actual_sim_times_to_print))) # Unique sorted times
print("\nSimulated values at specific time points (Hypothesis 3, Attempt 5):")
print(sim_data_df_hyp3_att5[sim_data_df_hyp3_att5['Time'].isin(actual_sim_times_to_print)])

print("\nExperimental data for comparison at these times:")
actual_exp_times_to_print_exp = []
for t_target in sim_time_points:
    closest_time_idx = (np.abs(exp_data_df['Time'] - t_target)).argmin()
    actual_exp_times_to_print_exp.append(exp_data_df['Time'].iloc[closest_time_idx])
actual_exp_times_to_print_exp = sorted(list(set(actual_exp_times_to_print_exp))) # Unique sorted times
print(exp_data_df[exp_data_df['Time'].isin(actual_exp_times_to_print_exp)][['Time', 'id_xvpm', 'id_aa2n', 'id_xdgv']])
```
# Observation

## Code Stdout
Simulation data (Hypothesis 3, Attempt 5) head:
    id_xvpm   id_aa2n  id_xdgv  Time
0  0.000010  0.000010  0.00001   0.0
1  0.000015  0.000010  0.00001   0.5
2  0.000019  0.000011  0.00001   1.0
3  0.000023  0.000012  0.00001   1.5
4  0.000027  0.000012  0.00001   2.0

Simulation data (Hypothesis 3, Attempt 5) tail:
       id_xvpm   id_aa2n   id_xdgv   Time
996   0.000109  0.000852  0.001375  498.0
997   0.000109  0.000852  0.001375  498.5
998   0.000109  0.000851  0.001376  499.0
999   0.000109  0.000851  0.001376  499.5
1000  0.000109  0.000851  0.001377  500.0

Parameters used for Hypothesis 3, Attempt 5:
k1_R1=1e-05, Ki_R1=0.0016, n_R1=8.0
k2_R2=0.07, k3_R3=0.008
k4_R4=0.0043, k5_R5=0.002, k6_R6=0.002

Simulated values at specific time points (Hypothesis 3, Attempt 5):
       id_xvpm   id_aa2n   id_xdgv   Time
0     0.000010  0.000010  0.000010    0.0
70    0.000129  0.000203  0.000033   35.0
92    0.000134  0.000278  0.000052   46.0
96    0.000135  0.000291  0.000056   48.0
200   0.000139  0.000565  0.000210  100.0
400   0.000139  0.000822  0.000607  200.0
600   0.000137  0.000914  0.000967  300.0
800   0.000126  0.000919  0.001230  400.0
1000  0.000109  0.000851  0.001377  500.0

Experimental data for comparison at these times:
       Time   id_xvpm   id_aa2n   id_xdgv
0       0.0  0.000010  0.000010  0.000010
70     35.0  0.000118  0.000675  0.001502
92     46.0  0.000080  0.000869  0.001954
96     48.0  0.000068  0.000861  0.001963
200   100.0  0.000004  0.000068  0.000455
400   200.0  0.000010  0.000234  0.000904
600   300.0  0.000024  0.000410  0.001145
800   400.0  0.000037  0.000467  0.001179
1000  500.0  0.000043  0.000452  0.001113

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml_model_str_iteration_2, sim_data_df_iteration_2, current_sbml_model_str_iteration_3, sim_data_df_iteration_3, current_sbml_model_str_iteration_4, sim_data_df_iteration_4, current_sbml_model_str_iteration_5, sim_data_df_iteration_5, current_sbml_model_str_iteration_6, sim_data_df_iteration_6)

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


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



# Iteration 7
## Thoughts
The previous simulation (Hypothesis 3, Attempt 5) showed promising oscillatory behavior and magnitudes, but the timing of the peaks was too slow compared to the experimental data. Specifically, the simulated S3 rose much slower than in the experiment, delaying the feedback and subsequent dynamics.

To address this, I've re-calculated the rate constants to speed up the S1->S2->S3 pathway while trying to maintain the relative species concentrations and the conditions for oscillation (high Hill coefficient, appropriate lifetimes). This involved increasing `k1_R1`, `k2_R2`, `k3_R3`, and `k4_R4`. The new lifetimes are S1 ~2.8s, S2 ~20s, S3 ~52s. This sequence of increasing lifetimes, combined with faster overall rates, should lead to earlier peaks.

**New Parameter Set (Hypothesis 3, attempt 7 - Faster S2/S3 dynamics):**
*   `k1_R1 = 4.0e-5` (S1 max production, was 1.0e-5)
*   `Ki_R1 = 1.6e-3` (Inhibition constant for S3 on S1 production, kept)
*   `n_R1 = 8.0` (Hill coefficient for inhibition, kept)

*   `k5_R5 = 0.002` (S1 degradation, kept)
*   `k2_R2 = 0.35` (S1 -> S2, was 0.07)

*   `k6_R6 = 0.002` (S2 degradation, kept)
*   `k3_R3 = 0.048` (S2 -> S3, was 0.008)

*   `k4_R4 = 0.019` (S3 degradation, was 0.0043)

The expectation is that these changes will:
1.  Shift the simulated peaks earlier, more in line with experimental observations.
2.  Maintain the oscillatory behavior.
3.  Keep peak magnitudes roughly in the correct range.

The calculated steady-state values (if oscillations damp or it settles) with these parameters are approximately:
`S1_ss ~ 0.82e-4` (Exp peak ~1.18e-4)
`S2_ss ~ 5.74e-4` (Exp peak ~8.69e-4)
`S3_ss ~ 1.45e-3` (Exp peak ~1.96e-3)
These are somewhat lower than the experimental peaks, which is acceptable as the system should oscillate around these values, potentially overshooting them during the initial peaks.

## Action

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

# Access experimental data
exp_data_df = experiment_history['iteration_0']

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

# Define species IDs
s1 = "id_xvpm"
s2 = "id_aa2n"
s3 = "id_xdgv"

# Parameter Set (Hypothesis 3, attempt 7 - Faster S2/S3 dynamics)
k1_R1_val = 4.0e-5   # S1 max production
Ki_R1_val = 1.6e-3   # Inhibition constant for S3 on S1 production
n_R1_val = 8.0       # Hill coefficient for inhibition

k5_R5_val = 0.002    # S1 degradation
k2_R2_val = 0.35     # S1 -> S2

k6_R6_val = 0.002    # S2 degradation
k3_R3_val = 0.048    # S2 -> S3

k4_R4_val = 0.019    # S3 degradation

# R1: -> id_xvpm (inhibited by id_xdgv)
reaction1 = model.createReaction(); reaction1.setId("R1_prod_xvpm_inhib"); reaction1.setReversible(False); reaction1.setFast(False)
prod1_s1 = reaction1.createProduct(); prod1_s1.setSpecies(s1); prod1_s1.setStoichiometry(1.0); prod1_s1.setConstant(False)
kl1 = reaction1.createKineticLaw()
p_k1_R1 = kl1.createParameter(); p_k1_R1.setId("k1_R1"); p_k1_R1.setValue(k1_R1_val); p_k1_R1.setConstant(True)
p_Ki_R1 = kl1.createParameter(); p_Ki_R1.setId("Ki_R1"); p_Ki_R1.setValue(Ki_R1_val); p_Ki_R1.setConstant(True)
p_n_R1 = kl1.createParameter(); p_n_R1.setId("n_R1"); p_n_R1.setValue(n_R1_val); p_n_R1.setConstant(True)
kl1.setMath(libsbml.parseL3Formula(f"k1_R1 / (1 + pow({s3} / Ki_R1, n_R1))"))

# R2: id_xvpm -> id_aa2n
reaction2 = model.createReaction(); reaction2.setId("R2_xvpm_to_aa2n"); reaction2.setReversible(False); reaction2.setFast(False)
reactant2_s1 = reaction2.createReactant(); reactant2_s1.setSpecies(s1); reactant2_s1.setStoichiometry(1.0); reactant2_s1.setConstant(False)
product2_s2 = reaction2.createProduct(); product2_s2.setSpecies(s2); product2_s2.setStoichiometry(1.0); product2_s2.setConstant(False)
kl2 = reaction2.createKineticLaw()
p_k2_R2 = kl2.createParameter(); p_k2_R2.setId("k2_R2"); p_k2_R2.setValue(k2_R2_val); p_k2_R2.setConstant(True)
kl2.setMath(libsbml.parseL3Formula(f"k2_R2 * {s1}"))

# R3: id_aa2n -> id_xdgv
reaction3 = model.createReaction(); reaction3.setId("R3_aa2n_to_xdgv"); reaction3.setReversible(False); reaction3.setFast(False)
reactant3_s2 = reaction3.createReactant(); reactant3_s2.setSpecies(s2); reactant3_s2.setStoichiometry(1.0); reactant3_s2.setConstant(False)
product3_s3 = reaction3.createProduct(); product3_s3.setSpecies(s3); product3_s3.setStoichiometry(1.0); product3_s3.setConstant(False)
kl3 = reaction3.createKineticLaw()
p_k3_R3 = kl3.createParameter(); p_k3_R3.setId("k3_R3"); p_k3_R3.setValue(k3_R3_val); p_k3_R3.setConstant(True)
kl3.setMath(libsbml.parseL3Formula(f"k3_R3 * {s2}"))

# R4: id_xdgv -> (degradation of S3)
reaction4 = model.createReaction(); reaction4.setId("R4_degrad_xdgv"); reaction4.setReversible(False); reaction4.setFast(False)
reactant4_s3 = reaction4.createReactant(); reactant4_s3.setSpecies(s3); reactant4_s3.setStoichiometry(1.0); reactant4_s3.setConstant(False)
kl4 = reaction4.createKineticLaw()
p_k4_R4 = kl4.createParameter(); p_k4_R4.setId("k4_R4"); p_k4_R4.setValue(k4_R4_val); p_k4_R4.setConstant(True)
kl4.setMath(libsbml.parseL3Formula(f"k4_R4 * {s3}"))

# R5: id_xvpm -> (degradation of S1)
reaction5 = model.createReaction(); reaction5.setId("R5_degrad_xvpm"); reaction5.setReversible(False); reaction5.setFast(False)
reactant5_s1 = reaction5.createReactant(); reactant5_s1.setSpecies(s1); reactant5_s1.setStoichiometry(1.0); reactant5_s1.setConstant(False)
kl5 = reaction5.createKineticLaw()
p_k5_R5 = kl5.createParameter(); p_k5_R5.setId("k5_R5"); p_k5_R5.setValue(k5_R5_val); p_k5_R5.setConstant(True)
kl5.setMath(libsbml.parseL3Formula(f"k5_R5 * {s1}"))

# R6: id_aa2n -> (degradation of S2)
reaction6 = model.createReaction(); reaction6.setId("R6_degrad_aa2n"); reaction6.setReversible(False); reaction6.setFast(False)
reactant6_s2 = reaction6.createReactant(); reactant6_s2.setSpecies(s2); reactant6_s2.setStoichiometry(1.0); reactant6_s2.setConstant(False)
kl6 = reaction6.createKineticLaw()
p_k6_R6 = kl6.createParameter(); p_k6_R6.setId("k6_R6"); p_k6_R6.setValue(k6_R6_val); p_k6_R6.setConstant(True)
kl6.setMath(libsbml.parseL3Formula(f"k6_R6 * {s2}"))

# Write the updated SBML to a string
writer = libsbml.SBMLWriter()
current_sbml_model_str_hyp3_att7 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_sbml_model_str_iteration_7", current_sbml_model_str_hyp3_att7)

# Simulate the model
sim_data_df_hyp3_att7 = simulate(current_sbml_model_str_hyp3_att7)
shared_variables.add("sim_data_df_iteration_7", sim_data_df_hyp3_att7)

# Print simulation results
print(f"Simulation data (Hypothesis 3, Attempt 7) head:\n{sim_data_df_hyp3_att7.head()}")
print(f"\nSimulation data (Hypothesis 3, Attempt 7) tail:\n{sim_data_df_hyp3_att7.tail(5)}")

# Print parameter values used
print(f"\nParameters used for Hypothesis 3, Attempt 7:")
print(f"k1_R1={k1_R1_val}, Ki_R1={Ki_R1_val}, n_R1={n_R1_val}")
print(f"k2_R2={k2_R2_val}, k3_R3={k3_R3_val}")
print(f"k4_R4={k4_R4_val}, k5_R5={k5_R5_val}, k6_R6={k6_R6_val}")

# Print simulated values at key experimental time points
sim_time_points = [0.0, 35.0, 46.0, 48.0, 100.0, 150.0, 200.0, 300.0, 400.0, 500.0] # Added 150s
actual_sim_times_to_print = []
for t_target in sim_time_points:
    closest_time_idx = (np.abs(sim_data_df_hyp3_att7['Time'] - t_target)).argmin()
    actual_sim_times_to_print.append(sim_data_df_hyp3_att7['Time'].iloc[closest_time_idx])
actual_sim_times_to_print = sorted(list(set(actual_sim_times_to_print))) # Unique sorted times
print("\nSimulated values at specific time points (Hypothesis 3, Attempt 7):")
print(sim_data_df_hyp3_att7[sim_data_df_hyp3_att7['Time'].isin(actual_sim_times_to_print)])

print("\nExperimental data for comparison at these times:")
actual_exp_times_to_print_exp = []
for t_target in sim_time_points:
    closest_time_idx = (np.abs(exp_data_df['Time'] - t_target)).argmin()
    actual_exp_times_to_print_exp.append(exp_data_df['Time'].iloc[closest_time_idx])
actual_exp_times_to_print_exp = sorted(list(set(actual_exp_times_to_print_exp))) # Unique sorted times
print(exp_data_df[exp_data_df['Time'].isin(actual_exp_times_to_print_exp)][['Time', 'id_xvpm', 'id_aa2n', 'id_xdgv']])
```
# Observation

## Code Stdout
Simulation data (Hypothesis 3, Attempt 7) head:
    id_xvpm   id_aa2n   id_xdgv  Time
0  0.000010  0.000010  0.000010   0.0
1  0.000027  0.000013  0.000010   0.5
2  0.000041  0.000019  0.000010   1.0
3  0.000053  0.000026  0.000011   1.5
4  0.000062  0.000035  0.000012   2.0

Simulation data (Hypothesis 3, Attempt 7) tail:
       id_xvpm   id_aa2n   id_xdgv   Time
996   0.000081  0.000566  0.001429  498.0
997   0.000081  0.000566  0.001429  498.5
998   0.000081  0.000566  0.001429  499.0
999   0.000081  0.000566  0.001429  499.5
1000  0.000081  0.000566  0.001429  500.0

Parameters used for Hypothesis 3, Attempt 7:
k1_R1=4e-05, Ki_R1=0.0016, n_R1=8.0
k2_R2=0.35, k3_R3=0.048
k4_R4=0.019, k5_R5=0.002, k6_R6=0.002

Simulated values at specific time points (Hypothesis 3, Attempt 7):
       id_xvpm   id_aa2n   id_xdgv   Time
0     0.000010  0.000010  0.000010    0.0
70    0.000114  0.000638  0.000513   35.0
92    0.000113  0.000705  0.000738   46.0
96    0.000113  0.000713  0.000777   48.0
200   0.000081  0.000666  0.001441  100.0
300   0.000078  0.000545  0.001444  150.0
400   0.000082  0.000567  0.001425  200.0
600   0.000081  0.000566  0.001429  300.0
800   0.000081  0.000566  0.001429  400.0
1000  0.000081  0.000566  0.001429  500.0

Experimental data for comparison at these times:
       Time   id_xvpm   id_aa2n   id_xdgv
0       0.0  0.000010  0.000010  0.000010
70     35.0  0.000118  0.000675  0.001502
92     46.0  0.000080  0.000869  0.001954
96     48.0  0.000068  0.000861  0.001963
200   100.0  0.000004  0.000068  0.000455
300   150.0  0.000074  0.000417  0.000969
400   200.0  0.000010  0.000234  0.000904
600   300.0  0.000024  0.000410  0.001145
800   400.0  0.000037  0.000467  0.001179
1000  500.0  0.000043  0.000452  0.001113

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml_model_str_iteration_2, sim_data_df_iteration_2, current_sbml_model_str_iteration_3, sim_data_df_iteration_3, current_sbml_model_str_iteration_4, sim_data_df_iteration_4, current_sbml_model_str_iteration_5, sim_data_df_iteration_5, current_sbml_model_str_iteration_6, sim_data_df_iteration_6, current_sbml_model_str_iteration_7, sim_data_df_iteration_7)

### 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 previous simulation (Hypothesis 3, Attempt 7) showed oscillations but they damped to a steady state that was too high compared to the experimental troughs, and the initial S3 peak was too low and slow.

**Key issues from Attempt 7:**
1.  **S3 initial peak too low/slow:** Experimental S3 at t=35s is ~1.5e-3, simulation was ~0.5e-3.
2.  **Oscillations damp too high:** Simulated steady state values for S1, S2, S3 were much higher than experimental values at later times (e.g., t=100s, t=500s). This means feedback wasn't strong enough to pull concentrations down sufficiently.

**Strategy for Attempt 8:**
1.  **Strengthen feedback:**
    *   Lower `Ki_R1` (inhibition constant for S3 on S1 production) so that inhibition becomes significant at lower S3 concentrations.
    *   Increase `n_R1` (Hill coefficient) for a sharper, more switch-like response.
2.  **Accelerate S3 production:**
    *   Increase `k3_R3` (S2 -> S3 conversion rate).
    *   Adjust `k4_R4` (S3 degradation rate) to maintain an appropriate S3 lifetime and peak, keeping the ratio `k3_R3/k4_R4` suitable for the desired S3/S2 peak ratio.

**Parameter Set (Hypothesis 3, attempt 8 - Lower Ki, higher n, faster S3):**
*   **Feedback & S1 Production (R1):**
    *   `k1_R1 = 4.0e-5` (Max S1 production rate, kept from Att7)
    *   `Ki_R1 = 1.2e-3` (was 1.6e-3 in Att7). Lowered to make feedback engage more strongly at lower S3 levels.
    *   `n_R1 = 10.0` (was 8.0 in Att7). Increased for sharper inhibition.
*   **S1 Dynamics (R2, R5):**
    *   `k5_R5 = 0.002` (S1 degradation, kept)
    *   `k2_R2 = 0.35` (S1 -> S2 conversion, kept)
*   **S2 & S3 Dynamics (R3, R6, R4):**
    *   `k6_R6 = 0.002` (S2 degradation, kept)
    *   `k3_R3 = 0.060` (S2 -> S3 conversion, was 0.048). Increased to speed up S3 formation.
    *   `k4_R4 = 0.024` (S3 degradation, was 0.019). Increased proportionally to `k3_R3` (maintaining `k3/k4` ratio of 2.5, aiming for `S3_peak/S2_peak ~ 2.5`).

**Expected Outcome:**
*   The lower `Ki_R1` and higher `n_R1` should make the negative feedback on S1 production more potent and responsive, leading to deeper troughs in S1, S2, and S3 concentrations after their initial peaks.
*   The increased `k3_R3` (and corresponding `k4_R4`) should result in S3 rising and peaking faster, aligning better with the early experimental data for S3.
*   The combination might lead to more sustained oscillations or oscillations that settle to lower mean values, better matching the experimental data at later time points.

The calculated steady-state values (if oscillations damp) for this parameter set are approximately:
S1_ss ~ 0.78e-4, S2_ss ~ 4.4e-4, S3_ss ~ 1.1e-3.
These are lower than the experimental peaks, which is desirable if the system oscillates with peaks above these values and troughs below.

## Action

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

# Access experimental data
exp_data_df = experiment_history['iteration_0']

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

# Define species IDs
s1 = "id_xvpm"
s2 = "id_aa2n"
s3 = "id_xdgv"

# Parameter Set (Hypothesis 3, attempt 8 - Lower Ki, higher n, faster S3)
k1_R1_val = 4.0e-5   # S1 max production
Ki_R1_val = 1.2e-3   # Inhibition constant for S3 on S1 production (was 1.6e-3)
n_R1_val = 10.0      # Hill coefficient for inhibition (was 8.0)

k5_R5_val = 0.002    # S1 degradation
k2_R2_val = 0.35     # S1 -> S2

k6_R6_val = 0.002    # S2 degradation
k3_R3_val = 0.060    # S2 -> S3 (was 0.048)

k4_R4_val = 0.024    # S3 degradation (was 0.019, k3/k4 = 2.5)

# R1: -> id_xvpm (inhibited by id_xdgv)
reaction1 = model.createReaction(); reaction1.setId("R1_prod_xvpm_inhib"); reaction1.setReversible(False); reaction1.setFast(False)
prod1_s1 = reaction1.createProduct(); prod1_s1.setSpecies(s1); prod1_s1.setStoichiometry(1.0); prod1_s1.setConstant(False)
kl1 = reaction1.createKineticLaw()
p_k1_R1 = kl1.createParameter(); p_k1_R1.setId("k1_R1"); p_k1_R1.setValue(k1_R1_val); p_k1_R1.setConstant(True)
p_Ki_R1 = kl1.createParameter(); p_Ki_R1.setId("Ki_R1"); p_Ki_R1.setValue(Ki_R1_val); p_Ki_R1.setConstant(True)
p_n_R1 = kl1.createParameter(); p_n_R1.setId("n_R1"); p_n_R1.setValue(n_R1_val); p_n_R1.setConstant(True)
kl1.setMath(libsbml.parseL3Formula(f"k1_R1 / (1 + pow({s3} / Ki_R1, n_R1))"))

# R2: id_xvpm -> id_aa2n
reaction2 = model.createReaction(); reaction2.setId("R2_xvpm_to_aa2n"); reaction2.setReversible(False); reaction2.setFast(False)
reactant2_s1 = reaction2.createReactant(); reactant2_s1.setSpecies(s1); reactant2_s1.setStoichiometry(1.0); reactant2_s1.setConstant(False)
product2_s2 = reaction2.createProduct(); product2_s2.setSpecies(s2); product2_s2.setStoichiometry(1.0); product2_s2.setConstant(False)
kl2 = reaction2.createKineticLaw()
p_k2_R2 = kl2.createParameter(); p_k2_R2.setId("k2_R2"); p_k2_R2.setValue(k2_R2_val); p_k2_R2.setConstant(True)
kl2.setMath(libsbml.parseL3Formula(f"k2_R2 * {s1}"))

# R3: id_aa2n -> id_xdgv
reaction3 = model.createReaction(); reaction3.setId("R3_aa2n_to_xdgv"); reaction3.setReversible(False); reaction3.setFast(False)
reactant3_s2 = reaction3.createReactant(); reactant3_s2.setSpecies(s2); reactant3_s2.setStoichiometry(1.0); reactant3_s2.setConstant(False)
product3_s3 = reaction3.createProduct(); product3_s3.setSpecies(s3); product3_s3.setStoichiometry(1.0); product3_s3.setConstant(False)
kl3 = reaction3.createKineticLaw()
p_k3_R3 = kl3.createParameter(); p_k3_R3.setId("k3_R3"); p_k3_R3.setValue(k3_R3_val); p_k3_R3.setConstant(True)
kl3.setMath(libsbml.parseL3Formula(f"k3_R3 * {s2}"))

# R4: id_xdgv -> (degradation of S3)
reaction4 = model.createReaction(); reaction4.setId("R4_degrad_xdgv"); reaction4.setReversible(False); reaction4.setFast(False)
reactant4_s3 = reaction4.createReactant(); reactant4_s3.setSpecies(s3); reactant4_s3.setStoichiometry(1.0); reactant4_s3.setConstant(False)
kl4 = reaction4.createKineticLaw()
p_k4_R4 = kl4.createParameter(); p_k4_R4.setId("k4_R4"); p_k4_R4.setValue(k4_R4_val); p_k4_R4.setConstant(True)
kl4.setMath(libsbml.parseL3Formula(f"k4_R4 * {s3}"))

# R5: id_xvpm -> (degradation of S1)
reaction5 = model.createReaction(); reaction5.setId("R5_degrad_xvpm"); reaction5.setReversible(False); reaction5.setFast(False)
reactant5_s1 = reaction5.createReactant(); reactant5_s1.setSpecies(s1); reactant5_s1.setStoichiometry(1.0); reactant5_s1.setConstant(False)
kl5 = reaction5.createKineticLaw()
p_k5_R5 = kl5.createParameter(); p_k5_R5.setId("k5_R5"); p_k5_R5.setValue(k5_R5_val); p_k5_R5.setConstant(True)
kl5.setMath(libsbml.parseL3Formula(f"k5_R5 * {s1}"))

# R6: id_aa2n -> (degradation of S2)
reaction6 = model.createReaction(); reaction6.setId("R6_degrad_aa2n"); reaction6.setReversible(False); reaction6.setFast(False)
reactant6_s2 = reaction6.createReactant(); reactant6_s2.setSpecies(s2); reactant6_s2.setStoichiometry(1.0); reactant6_s2.setConstant(False)
kl6 = reaction6.createKineticLaw()
p_k6_R6 = kl6.createParameter(); p_k6_R6.setId("k6_R6"); p_k6_R6.setValue(k6_R6_val); p_k6_R6.setConstant(True)
kl6.setMath(libsbml.parseL3Formula(f"k6_R6 * {s2}"))

# Write the updated SBML to a string
writer = libsbml.SBMLWriter()
current_sbml_model_str_hyp3_att8 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_sbml_model_str_iteration_8", current_sbml_model_str_hyp3_att8)

# Simulate the model
sim_data_df_hyp3_att8 = simulate(current_sbml_model_str_hyp3_att8)
shared_variables.add("sim_data_df_iteration_8", sim_data_df_hyp3_att8)

# Print simulation results
print(f"Simulation data (Hypothesis 3, Attempt 8) head:\n{sim_data_df_hyp3_att8.head()}")
print(f"\nSimulation data (Hypothesis 3, Attempt 8) tail:\n{sim_data_df_hyp3_att8.tail(5)}")

# Print parameter values used
print(f"\nParameters used for Hypothesis 3, Attempt 8:")
print(f"k1_R1={k1_R1_val}, Ki_R1={Ki_R1_val}, n_R1={n_R1_val}")
print(f"k2_R2={k2_R2_val}, k3_R3={k3_R3_val}")
print(f"k4_R4={k4_R4_val}, k5_R5={k5_R5_val}, k6_R6={k6_R6_val}")

# Print simulated values at key experimental time points
sim_time_points = [0.0, 35.0, 46.0, 48.0, 60.0, 100.0, 150.0, 200.0, 300.0, 400.0, 500.0] # Added 60s
actual_sim_times_to_print = []
for t_target in sim_time_points:
    # find the index of the closest time in sim_data_df_hyp3_att8['Time']
    if sim_data_df_hyp3_att8.empty: # check if dataframe is empty
        print(f"Simulated data for attempt 8 is empty. Skipping time point comparison.")
        break
    closest_time_idx = (np.abs(sim_data_df_hyp3_att8['Time'] - t_target)).argmin()
    actual_sim_times_to_print.append(sim_data_df_hyp3_att8['Time'].iloc[closest_time_idx])
else: # only execute if loop finished without break
    actual_sim_times_to_print = sorted(list(set(actual_sim_times_to_print))) # Unique sorted times
    print("\nSimulated values at specific time points (Hypothesis 3, Attempt 8):")
    print(sim_data_df_hyp3_att8[sim_data_df_hyp3_att8['Time'].isin(actual_sim_times_to_print)])

print("\nExperimental data for comparison at these times:")
actual_exp_times_to_print_exp = []
for t_target in sim_time_points:
    closest_time_idx = (np.abs(exp_data_df['Time'] - t_target)).argmin()
    actual_exp_times_to_print_exp.append(exp_data_df['Time'].iloc[closest_time_idx])
actual_exp_times_to_print_exp = sorted(list(set(actual_exp_times_to_print_exp))) # Unique sorted times
print(exp_data_df[exp_data_df['Time'].isin(actual_exp_times_to_print_exp)][['Time', 'id_xvpm', 'id_aa2n', 'id_xdgv']])
```
# Observation

## Code Stdout
Simulation data (Hypothesis 3, Attempt 8) head:
    id_xvpm   id_aa2n   id_xdgv  Time
0  0.000010  0.000010  0.000010   0.0
1  0.000027  0.000013  0.000010   0.5
2  0.000041  0.000018  0.000011   1.0
3  0.000053  0.000026  0.000011   1.5
4  0.000062  0.000035  0.000012   2.0

Simulation data (Hypothesis 3, Attempt 8) tail:
       id_xvpm   id_aa2n   id_xdgv   Time
996   0.000078  0.000443  0.001107  498.0
997   0.000078  0.000443  0.001107  498.5
998   0.000078  0.000443  0.001107  499.0
999   0.000078  0.000443  0.001107  499.5
1000  0.000078  0.000443  0.001107  500.0

Parameters used for Hypothesis 3, Attempt 8:
k1_R1=4e-05, Ki_R1=0.0012, n_R1=10.0
k2_R2=0.35, k3_R3=0.06
k4_R4=0.024, k5_R5=0.002, k6_R6=0.002

Simulated values at specific time points (Hypothesis 3, Attempt 8):
       id_xvpm   id_aa2n   id_xdgv   Time
0     0.000010  0.000010  0.000010    0.0
70    0.000114  0.000555  0.000546   35.0
92    0.000113  0.000597  0.000756   46.0
96    0.000113  0.000602  0.000791   48.0
120   0.000105  0.000609  0.000974   60.0
200   0.000069  0.000422  0.001146  100.0
300   0.000080  0.000448  0.001101  150.0
400   0.000078  0.000442  0.001108  200.0
600   0.000078  0.000443  0.001107  300.0
800   0.000078  0.000443  0.001107  400.0
1000  0.000078  0.000443  0.001107  500.0

Experimental data for comparison at these times:
       Time   id_xvpm   id_aa2n   id_xdgv
0       0.0  0.000010  0.000010  0.000010
70     35.0  0.000118  0.000675  0.001502
92     46.0  0.000080  0.000869  0.001954
96     48.0  0.000068  0.000861  0.001963
120    60.0  0.000017  0.000531  0.001629
200   100.0  0.000004  0.000068  0.000455
300   150.0  0.000074  0.000417  0.000969
400   200.0  0.000010  0.000234  0.000904
600   300.0  0.000024  0.000410  0.001145
800   400.0  0.000037  0.000467  0.001179
1000  500.0  0.000043  0.000452  0.001113

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml_model_str_iteration_2, sim_data_df_iteration_2, current_sbml_model_str_iteration_3, sim_data_df_iteration_3, current_sbml_model_str_iteration_4, sim_data_df_iteration_4, current_sbml_model_str_iteration_5, sim_data_df_iteration_5, current_sbml_model_str_iteration_6, sim_data_df_iteration_6, current_sbml_model_str_iteration_7, sim_data_df_iteration_7, current_sbml_model_str_iteration_8, sim_data_df_iteration_8)

### 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 from Attempt 8 showed some improvement:
-   S1 peak (sim: 0.114e-3 at t=35) matches exp (0.118e-3 at t=35) reasonably well.
-   S2 peak (sim: 0.609e-3 at t=60) is lower than exp (0.869e-3 at t=46) and delayed.
-   S3 peak (sim: ~0.001146 at t=100) is much lower than exp (0.001963 at t=48) and significantly delayed.
-   The oscillations damp to a steady state: S1=0.078e-3, S2=0.443e-3, S3=1.107e-3.
    -   Experimental values at t=500: S1=0.043e-3, S2=0.452e-3, S3=1.113e-3.
    -   The simulated S2 and S3 steady states are surprisingly close to the experimental values at t=500. S1 is a bit high.

**Key Issues from Attempt 8:**
1.  **S2 and S3 Peaks Too Low and Too Slow:** The primary issue is that S2 and S3 don't rise fast enough or high enough for their initial peaks.
2.  **Damped Oscillations Settle Too High for S1:** While S2 and S3 steady states are good, S1 settles higher than its experimental counterpart at later times.

**Strategy for Attempt 9:**
The goal is to get the initial peaks of S2 and S3 higher and faster, without losing the good S2/S3 steady state behavior at later times. The S1 steady state also needs to be lower.

1.  **Accelerate S2 and S3 Rise:**
    *   Increase `k2_R2` (S1 -> S2) further to make S2 rise faster from S1.
    *   Increase `k3_R3` (S2 -> S3) further to make S3 rise faster from S2.
    *   To maintain similar peak/steady-state ratios, `k6_R6` (S2_deg) and `k4_R4` (S3_deg) might need adjustment if `k2` and `k3` are changed significantly.
        Recall `S2_ss/S1_ss ~ k2/(k6+k3)` and `S3_ss/S2_ss ~ k3/k4`.

2.  **Refine Feedback for S1:**
    *   The S1 steady state (0.078e-3) is higher than the experimental t=500 value (0.043e-3). This means at the S3 steady state of ~1.1e-3, the inhibition on S1 production is not strong enough.
    *   `Ki_R1` is 1.2e-3. S3_ss (1.1e-3) is slightly below `Ki_R1`. `(S3_ss/Ki_R1)^10 = (1.1/1.2)^10 = (0.916)^10 = 0.41`. So S1 production is `k1_R1 / (1+0.41) = k1_R1 / 1.41`.
    *   `S1_ss = (k1_R1/1.41) / (k5_R5+k2_R2)`.
    *   If we want lower S1_ss, we could slightly decrease `Ki_R1` further or increase `n_R1` (but n=10 is already high). Or decrease `k1_R1`.

Let's try to adjust parameters focusing on the initial peaks first.
Current S2 peak is ~0.6e-3, target ~0.87e-3. Needs ~1.45x increase.
Current S3 peak is ~1.1e-3, target ~1.96e-3. Needs ~1.8x increase.

**Parameter Adjustments (Hypothesis 3, attempt 9):**
*   **Feedback & S1 Production (R1):**
    *   `k1_R1 = 3.5e-5` (was 4.0e-5). Slightly reduce to help lower S1 steady state.
    *   `Ki_R1 = 1.1e-3` (was 1.2e-3). Slightly reduce to strengthen feedback at S3 steady state.
    *   `n_R1 = 10.0` (kept).
*   **S1 Dynamics (R2, R5):**
    *   `k5_R5 = 0.002` (kept).
    *   `k2_R2 = 0.40` (was 0.35). Increase to boost S2 production from S1.
*   **S2 & S3 Dynamics (R3, R6, R4):**
    *   `k6_R6 = 0.002` (kept).
    *   `k3_R3 = 0.075` (was 0.060). Increase significantly to boost S3 production from S2.
    *   `k4_R4 = ?`
        Target `S3_peak/S2_peak ~ 1.96/0.87 ~ 2.25`. So `k3_R3/k4_R4 ~ 2.25`.
        `k4_R4 = k3_R3 / 2.25 = 0.075 / 2.25 = 0.0333`. (was 0.024)

Let's check approximate steady states with these new values:
`S1_eff_deg = k5+k2 = 0.002+0.40 = 0.402`.
`S2_eff_deg = k6+k3 = 0.002+0.075 = 0.077`.
`S3_eff_deg = k4 = 0.0333`.

Ratios:
`S2/S1 ~ k2/(k6+k3) = 0.40 / 0.077 = 5.19`. (Exp peak ratio S2/S1 ~ 0.87/0.118 ~ 7.3. Still a bit low)
`S3/S2 ~ k3/k4 = 0.075 / 0.0333 = 2.25`. (Exp peak ratio S3/S2 ~ 1.96/0.87 ~ 2.25. This matches)

Let's re-evaluate `k2_R2` to get S2/S1 ratio closer to 7.
If `k2/(k6+k3) = 7`, then `k2 = 7 * (0.002+0.075) = 7 * 0.077 = 0.539`.
This is a large increase for `k2_R2`. Let's try `k2_R2 = 0.50`.
Then `S2/S1 ~ 0.50 / 0.077 = 6.49`. Closer.

Revised parameters for Attempt 9:
*   `k1_R1 = 3.0e-5` (Reduced further to compensate for higher S1 drain by increased k2)
*   `Ki_R1 = 1.0e-3` (Reduced further for stronger feedback)
*   `n_R1 = 10.0`
*   `k5_R5 = 0.002`
*   `k2_R2 = 0.50` (S1 -> S2)
*   `k6_R6 = 0.002`
*   `k3_R3 = 0.075` (S2 -> S3)
*   `k4_R4 = 0.0333` (S3 deg)

Approximate steady state calculation:
`S1_eff_deg = 0.002 + 0.50 = 0.502`.
`S2_eff_deg = 0.002 + 0.075 = 0.077`.
`S3_eff_deg = 0.0333`.

`S1 = S2 / (k2/(k6+k3)) = S2 / (0.50/0.077) = S2 / 6.49 = 0.154 * S2`.
`S2 = S3 / (k3/k4) = S3 / (0.075/0.0333) = S3 / 2.25 = 0.444 * S3`.
`S1 = 0.154 * 0.444 * S3 = 0.0684 * S3`.

`k1_R1 / (1 + (S3/Ki_R1)^n_R1) = (k5_R5 + k2_R2) * S1`
`3.0e-5 / (1 + (S3/1.0e-3)^10) = 0.502 * 0.0684 * S3 = 0.03433 * S3`.
`3.0e-5 = 0.03433 * S3 * (1 + (S3/1.0e-3)^10)`.
If `S3_ss = Ki_R1 = 1.0e-3`:
RHS = `0.03433 * 1.0e-3 * (1+1) = 3.433e-5 * 2 = 6.866e-5`.
LHS (3.0e-5) is smaller than RHS. So S3_ss will be lower than `Ki_R1`.
Let `S3_ss = 0.8e-3`. `(0.8/1.0)^10 = (0.8)^10 = 0.107`.
RHS = `0.03433 * 0.8e-3 * (1 + 0.107) = 2.7464e-5 * 1.107 = 3.039e-5`.
This is very close to LHS (3.0e-5). So `S3_ss ~ 0.8e-3`.

Then `S2_ss = 0.444 * 0.8e-3 = 0.355e-3`.
And `S1_ss = 0.0684 * 0.8e-3 = 0.0547e-3`.

Predicted steady state values:
`S1_ss ~ 0.055e-3` (Exp t=500: 0.043e-3) - Good.
`S2_ss ~ 0.355e-3` (Exp t=500: 0.452e-3) - A bit low.
`S3_ss ~ 0.80e-3` (Exp t=500: 1.113e-3) - A bit low.

The steady state values are now generally lower. This might mean the peaks will also be lower.
The main goal is to get the initial peak timing and height right. The increased `k2` and `k3` should help with speed.
The ratios `S2_peak/S1_peak` and `S3_peak/S2_peak` are targeted.
Let's see if this set of parameters improves the initial peak dynamics.

## Action

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

# Access experimental data
exp_data_df = experiment_history['iteration_0']

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

# Define species IDs
s1 = "id_xvpm"
s2 = "id_aa2n"
s3 = "id_xdgv"

# Parameter Set (Hypothesis 3, attempt 9)
k1_R1_val = 3.0e-5
Ki_R1_val = 1.0e-3
n_R1_val = 10.0

k5_R5_val = 0.002
k2_R2_val = 0.50

k6_R6_val = 0.002
k3_R3_val = 0.075

k4_R4_val = 0.0333

# R1: -> id_xvpm (inhibited by id_xdgv)
reaction1 = model.createReaction(); reaction1.setId("R1_prod_xvpm_inhib"); reaction1.setReversible(False); reaction1.setFast(False)
prod1_s1 = reaction1.createProduct(); prod1_s1.setSpecies(s1); prod1_s1.setStoichiometry(1.0); prod1_s1.setConstant(False)
kl1 = reaction1.createKineticLaw()
p_k1_R1 = kl1.createParameter(); p_k1_R1.setId("k1_R1"); p_k1_R1.setValue(k1_R1_val); p_k1_R1.setConstant(True)
p_Ki_R1 = kl1.createParameter(); p_Ki_R1.setId("Ki_R1"); p_Ki_R1.setValue(Ki_R1_val); p_Ki_R1.setConstant(True)
p_n_R1 = kl1.createParameter(); p_n_R1.setId("n_R1"); p_n_R1.setValue(n_R1_val); p_n_R1.setConstant(True)
kl1.setMath(libsbml.parseL3Formula(f"k1_R1 / (1 + pow({s3} / Ki_R1, n_R1))"))

# R2: id_xvpm -> id_aa2n
reaction2 = model.createReaction(); reaction2.setId("R2_xvpm_to_aa2n"); reaction2.setReversible(False); reaction2.setFast(False)
reactant2_s1 = reaction2.createReactant(); reactant2_s1.setSpecies(s1); reactant2_s1.setStoichiometry(1.0); reactant2_s1.setConstant(False)
product2_s2 = reaction2.createProduct(); product2_s2.setSpecies(s2); product2_s2.setStoichiometry(1.0); product2_s2.setConstant(False)
kl2 = reaction2.createKineticLaw()
p_k2_R2 = kl2.createParameter(); p_k2_R2.setId("k2_R2"); p_k2_R2.setValue(k2_R2_val); p_k2_R2.setConstant(True)
kl2.setMath(libsbml.parseL3Formula(f"k2_R2 * {s1}"))

# R3: id_aa2n -> id_xdgv
reaction3 = model.createReaction(); reaction3.setId("R3_aa2n_to_xdgv"); reaction3.setReversible(False); reaction3.setFast(False)
reactant3_s2 = reaction3.createReactant(); reactant3_s2.setSpecies(s2); reactant3_s2.setStoichiometry(1.0); reactant3_s2.setConstant(False)
product3_s3 = reaction3.createProduct(); product3_s3.setSpecies(s3); product3_s3.setStoichiometry(1.0); product3_s3.setConstant(False)
kl3 = reaction3.createKineticLaw()
p_k3_R3 = kl3.createParameter(); p_k3_R3.setId("k3_R3"); p_k3_R3.setValue(k3_R3_val); p_k3_R3.setConstant(True)
kl3.setMath(libsbml.parseL3Formula(f"k3_R3 * {s2}"))

# R4: id_xdgv -> (degradation of S3)
reaction4 = model.createReaction(); reaction4.setId("R4_degrad_xdgv"); reaction4.setReversible(False); reaction4.setFast(False)
reactant4_s3 = reaction4.createReactant(); reactant4_s3.setSpecies(s3); reactant4_s3.setStoichiometry(1.0); reactant4_s3.setConstant(False)
kl4 = reaction4.createKineticLaw()
p_k4_R4 = kl4.createParameter(); p_k4_R4.setId("k4_R4"); p_k4_R4.setValue(k4_R4_val); p_k4_R4.setConstant(True)
kl4.setMath(libsbml.parseL3Formula(f"k4_R4 * {s3}"))

# R5: id_xvpm -> (degradation of S1)
reaction5 = model.createReaction(); reaction5.setId("R5_degrad_xvpm"); reaction5.setReversible(False); reaction5.setFast(False)
reactant5_s1 = reaction5.createReactant(); reactant5_s1.setSpecies(s1); reactant5_s1.setStoichiometry(1.0); reactant5_s1.setConstant(False)
kl5 = reaction5.createKineticLaw()
p_k5_R5 = kl5.createParameter(); p_k5_R5.setId("k5_R5"); p_k5_R5.setValue(k5_R5_val); p_k5_R5.setConstant(True)
kl5.setMath(libsbml.parseL3Formula(f"k5_R5 * {s1}"))

# R6: id_aa2n -> (degradation of S2)
reaction6 = model.createReaction(); reaction6.setId("R6_degrad_aa2n"); reaction6.setReversible(False); reaction6.setFast(False)
reactant6_s2 = reaction6.createReactant(); reactant6_s2.setSpecies(s2); reactant6_s2.setStoichiometry(1.0); reactant6_s2.setConstant(False)
kl6 = reaction6.createKineticLaw()
p_k6_R6 = kl6.createParameter(); p_k6_R6.setId("k6_R6"); p_k6_R6.setValue(k6_R6_val); p_k6_R6.setConstant(True)
kl6.setMath(libsbml.parseL3Formula(f"k6_R6 * {s2}"))

# Write the updated SBML to a string
writer = libsbml.SBMLWriter()
current_sbml_model_str_hyp3_att9 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_sbml_model_str_iteration_9", current_sbml_model_str_hyp3_att9)

# Simulate the model
sim_data_df_hyp3_att9 = simulate(current_sbml_model_str_hyp3_att9)
shared_variables.add("sim_data_df_iteration_9", sim_data_df_hyp3_att9)

# Print simulation results
print(f"Simulation data (Hypothesis 3, Attempt 9) head:\n{sim_data_df_hyp3_att9.head()}")
print(f"\nSimulation data (Hypothesis 3, Attempt 9) tail:\n{sim_data_df_hyp3_att9.tail(5)}")

# Print parameter values used
print(f"\nParameters used for Hypothesis 3, Attempt 9:")
print(f"k1_R1={k1_R1_val}, Ki_R1={Ki_R1_val}, n_R1={n_R1_val}")
print(f"k2_R2={k2_R2_val}, k3_R3={k3_R3_val}")
print(f"k4_R4={k4_R4_val}, k5_R5={k5_R5_val}, k6_R6={k6_R6_val}")

# Print simulated values at key experimental time points
sim_time_points = [0.0, 35.0, 46.0, 48.0, 60.0, 100.0, 150.0, 200.0, 300.0, 400.0, 500.0]
actual_sim_times_to_print = []
if not sim_data_df_hyp3_att9.empty:
    for t_target in sim_time_points:
        closest_time_idx = (np.abs(sim_data_df_hyp3_att9['Time'] - t_target)).argmin()
        actual_sim_times_to_print.append(sim_data_df_hyp3_att9['Time'].iloc[closest_time_idx])
    actual_sim_times_to_print = sorted(list(set(actual_sim_times_to_print)))
    print("\nSimulated values at specific time points (Hypothesis 3, Attempt 9):")
    print(sim_data_df_hyp3_att9[sim_data_df_hyp3_att9['Time'].isin(actual_sim_times_to_print)])
else:
    print("Simulated data for attempt 9 is empty.")


print("\nExperimental data for comparison at these times:")
actual_exp_times_to_print_exp = []
for t_target in sim_time_points:
    closest_time_idx = (np.abs(exp_data_df['Time'] - t_target)).argmin()
    actual_exp_times_to_print_exp.append(exp_data_df['Time'].iloc[closest_time_idx])
actual_exp_times_to_print_exp = sorted(list(set(actual_exp_times_to_print_exp)))
print(exp_data_df[exp_data_df['Time'].isin(actual_exp_times_to_print_exp)][['Time', 'id_xvpm', 'id_aa2n', 'id_xdgv']])
```
# Observation

## Code Stdout
Simulation data (Hypothesis 3, Attempt 9) head:
    id_xvpm   id_aa2n   id_xdgv  Time
0  0.000010  0.000010  0.000010   0.0
1  0.000021  0.000013  0.000010   0.5
2  0.000030  0.000019  0.000011   1.0
3  0.000036  0.000027  0.000011   1.5
4  0.000042  0.000035  0.000012   2.0

Simulation data (Hypothesis 3, Attempt 9) tail:
       id_xvpm   id_aa2n   id_xdgv   Time
996   0.000054  0.000353  0.000794  498.0
997   0.000054  0.000353  0.000794  498.5
998   0.000054  0.000353  0.000794  499.0
999   0.000054  0.000353  0.000794  499.5
1000  0.000054  0.000353  0.000794  500.0

Parameters used for Hypothesis 3, Attempt 9:
k1_R1=3e-05, Ki_R1=0.001, n_R1=10.0
k2_R2=0.5, k3_R3=0.075
k4_R4=0.0333, k5_R5=0.002, k6_R6=0.002

Simulated values at specific time points (Hypothesis 3, Attempt 9):
       id_xvpm   id_aa2n   id_xdgv   Time
0     0.000010  0.000010  0.000010    0.0
70    0.000060  0.000359  0.000425   35.0
92    0.000060  0.000375  0.000549   46.0
96    0.000060  0.000377  0.000568   48.0
120   0.000059  0.000382  0.000663   60.0
200   0.000055  0.000362  0.000792  100.0
300   0.000054  0.000352  0.000795  150.0
400   0.000054  0.000353  0.000794  200.0
600   0.000054  0.000353  0.000794  300.0
800   0.000054  0.000353  0.000794  400.0
1000  0.000054  0.000353  0.000794  500.0

Experimental data for comparison at these times:
       Time   id_xvpm   id_aa2n   id_xdgv
0       0.0  0.000010  0.000010  0.000010
70     35.0  0.000118  0.000675  0.001502
92     46.0  0.000080  0.000869  0.001954
96     48.0  0.000068  0.000861  0.001963
120    60.0  0.000017  0.000531  0.001629
200   100.0  0.000004  0.000068  0.000455
300   150.0  0.000074  0.000417  0.000969
400   200.0  0.000010  0.000234  0.000904
600   300.0  0.000024  0.000410  0.001145
800   400.0  0.000037  0.000467  0.001179
1000  500.0  0.000043  0.000452  0.001113

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml_model_str_iteration_2, sim_data_df_iteration_2, current_sbml_model_str_iteration_3, sim_data_df_iteration_3, current_sbml_model_str_iteration_4, sim_data_df_iteration_4, current_sbml_model_str_iteration_5, sim_data_df_iteration_5, current_sbml_model_str_iteration_6, sim_data_df_iteration_6, current_sbml_model_str_iteration_7, sim_data_df_iteration_7, current_sbml_model_str_iteration_8, sim_data_df_iteration_8, current_sbml_model_str_iteration_9, sim_data_df_iteration_9)

### 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 simulation from Attempt 9 did not achieve the desired peaks.
-   S1 peak (sim: 0.060e-3 at t=35-48) is much lower than exp (0.118e-3 at t=35).
-   S2 peak (sim: 0.382e-3 at t=60) is much lower than exp (0.869e-3 at t=46).
-   S3 peak (sim: ~0.000795 at t=150) is much lower than exp (0.001963 at t=48) and very delayed.
-   The system damps to a steady state: S1=0.054e-3, S2=0.353e-3, S3=0.794e-3.
    -   Experimental values at t=500: S1=0.043e-3, S2=0.452e-3, S3=1.113e-3.
    -   Simulated S1 steady state is good. S2 and S3 steady states are lower than experimental.

The parameter changes in Attempt 9 (reducing `k1_R1` and `Ki_R1`, increasing `k2_R2` and `k3_R3`) resulted in overall lower concentrations and did not improve the peak heights or timings sufficiently. The system became too damped.

**Key Issues from Attempt 9:**
1.  **All Peaks Too Low:** The primary problem is that none of the species reach their experimental peak heights.
2.  **S3 Peak Still Too Slow:** S3 dynamics are sluggish.
3.  **Oscillations are Overdamped:** The system settles to a steady state too quickly without prominent oscillations.

**Strategy for Attempt 10:**
The system needs more "energy" or "activity" to reach higher peaks and oscillate more vigorously. This primarily means increasing the production rate of the first species (`k1_R1`) and ensuring the downstream conversion rates (`k2_R2`, `k3_R3`) are fast enough to propagate this. The feedback parameters (`Ki_R1`, `n_R1`) and degradation rates will then shape the oscillations.

1.  **Increase S1 Production:**
    *   Significantly increase `k1_R1` to drive higher concentrations throughout the pathway.
2.  **Adjust Downstream Rates:**
    *   `k2_R2` and `k3_R3` need to be fast enough. The ratios `k2/(k6+k3)` and `k3/k4` determine relative concentrations.
3.  **Re-evaluate Feedback and Degradation:**
    *   With higher overall flux, `Ki_R1` might need to be adjusted (likely increased) if S3 peaks become much higher.
    *   Degradation rates (`k4, k5, k6`) determine the lifetimes and influence oscillatory period and damping.

Let's go back to a `k1_R1` similar to Attempt 7 or 8 (e.g., 4e-5) or even higher, as the peaks in Attempt 9 were too low across the board.
The S2/S1 and S3/S2 peak ratios are roughly S2/S1 ~7 and S3/S2 ~2.25.

**Revised Parameters (Hypothesis 3, attempt 10):**
*   **Feedback & S1 Production (R1):**
    *   `k1_R1 = 6.0e-5` (was 3.0e-5 in Att9, 4.0e-5 in Att8). Increase significantly.
    *   `Ki_R1 = 1.5e-3` (was 1.0e-3 in Att9, 1.2e-3 in Att8). Increase as S3 peak will be higher.
    *   `n_R1 = 10.0` (kept).
*   **S1 Dynamics (R2, R5):**
    *   `k5_R5 = 0.002` (kept).
    *   `k2_R2 = 0.50` (kept from Att9, this gave S2/S1 ratio ~6.5).
        *   `S1_eff_deg = k5+k2 = 0.002+0.50 = 0.502`.
        *   `S1_peak_no_inhib ~ k1_R1 / S1_eff_deg = 6.0e-5 / 0.502 = 1.195e-4`. This matches exp S1 peak.
*   **S2 & S3 Dynamics (R3, R6, R4):**
    *   `k6_R6 = 0.002` (kept).
    *   `k3_R3 = 0.075` (kept from Att9).
        *   `S2_eff_deg = k6+k3 = 0.002+0.075 = 0.077`.
        *   `S2_peak_no_inhib ~ (k2/S2_eff_deg)*S1_peak = (0.50/0.077)*1.195e-4 = 6.49 * 1.195e-4 = 7.75e-4`. (Exp S2 peak ~8.69e-4. Closer).
    *   `k4_R4 = ?`
        *   Target `S3_peak/S2_peak ~ 2.25`. So `k3_R3/k4_R4 ~ 2.25`.
        *   `k4_R4 = k3_R3 / 2.25 = 0.075 / 2.25 = 0.0333`. (Same as Att9).
        *   `S3_peak_no_inhib ~ (k3/k4)*S2_peak = 2.25 * 7.75e-4 = 1.74e-3`. (Exp S3 peak ~1.96e-3. Closer).

This set of parameters aims to restore peak heights. The `Ki_R1` is set around the expected S3 peak. The increased `k1_R1` should provide the necessary drive. The key will be whether this also produces good oscillatory dynamics and settles to appropriate lower values.

Approximate steady state calculation with these parameters:
`S1 = 0.0684 * S3` (from Att9, as k ratios are similar for this part)
`k1_R1 / (1 + (S3/Ki_R1)^n_R1) = (k5_R5 + k2_R2) * S1`
`6.0e-5 / (1 + (S3/1.5e-3)^10) = 0.502 * 0.0684 * S3 = 0.03433 * S3`.
`6.0e-5 = 0.03433 * S3 * (1 + (S3/1.5e-3)^10)`.
If `S3_ss = Ki_R1 = 1.5e-3`:
RHS = `0.03433 * 1.5e-3 * (1+1) = 5.1495e-5 * 2 = 10.299e-5`.
LHS (6.0e-5) is smaller than RHS. So S3_ss will be lower than `Ki_R1`.
Let `S3_ss = 1.2e-3`. `(1.2/1.5)^10 = (0.8)^10 = 0.107`.
RHS = `0.03433 * 1.2e-3 * (1 + 0.107) = 4.1196e-5 * 1.107 = 4.56e-5`.
LHS (6.0e-5) is now larger than RHS. So S3_ss is between 1.2e-3 and 1.5e-3.
Let's estimate `S3_ss ~ 1.3e-3`.
RHS for `S3=1.3e-3`: `(1.3/1.5)^10 = (0.866)^10 = 0.25`.
RHS = `0.03433 * 1.3e-3 * (1 + 0.25) = 4.4629e-5 * 1.25 = 5.57e-5`.
This is close to LHS (6.0e-5). So `S3_ss ~ 1.3e-3`.

Then `S2_ss = 0.444 * 1.3e-3 = 0.577e-3`.
And `S1_ss = 0.0684 * 1.3e-3 = 0.0889e-3`.

Predicted steady state values:
`S1_ss ~ 0.089e-3` (Exp t=500: 0.043e-3) - Higher than desired.
`S2_ss ~ 0.577e-3` (Exp t=500: 0.452e-3) - Higher than desired.
`S3_ss ~ 1.3e-3` (Exp t=500: 1.113e-3) - Higher than desired.

The steady state values are predicted to be higher than the experimental t=500 values. This might mean the oscillations won't dip low enough. However, getting the initial peaks correct is the priority, then fine-tuning the feedback strength (`Ki_R1`, `n_R1`) and possibly degradation rates to control the trough depth and steady state.

## Action

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

# Access experimental data
exp_data_df = experiment_history['iteration_0']

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

# Define species IDs
s1 = "id_xvpm"
s2 = "id_aa2n"
s3 = "id_xdgv"

# Parameter Set (Hypothesis 3, attempt 10)
k1_R1_val = 6.0e-5
Ki_R1_val = 1.5e-3
n_R1_val = 10.0

k5_R5_val = 0.002
k2_R2_val = 0.50

k6_R6_val = 0.002
k3_R3_val = 0.075

k4_R4_val = 0.0333

# R1: -> id_xvpm (inhibited by id_xdgv)
reaction1 = model.createReaction(); reaction1.setId("R1_prod_xvpm_inhib"); reaction1.setReversible(False); reaction1.setFast(False)
prod1_s1 = reaction1.createProduct(); prod1_s1.setSpecies(s1); prod1_s1.setStoichiometry(1.0); prod1_s1.setConstant(False)
kl1 = reaction1.createKineticLaw()
p_k1_R1 = kl1.createParameter(); p_k1_R1.setId("k1_R1"); p_k1_R1.setValue(k1_R1_val); p_k1_R1.setConstant(True)
p_Ki_R1 = kl1.createParameter(); p_Ki_R1.setId("Ki_R1"); p_Ki_R1.setValue(Ki_R1_val); p_Ki_R1.setConstant(True)
p_n_R1 = kl1.createParameter(); p_n_R1.setId("n_R1"); p_n_R1.setValue(n_R1_val); p_n_R1.setConstant(True)
kl1.setMath(libsbml.parseL3Formula(f"k1_R1 / (1 + pow({s3} / Ki_R1, n_R1))"))

# R2: id_xvpm -> id_aa2n
reaction2 = model.createReaction(); reaction2.setId("R2_xvpm_to_aa2n"); reaction2.setReversible(False); reaction2.setFast(False)
reactant2_s1 = reaction2.createReactant(); reactant2_s1.setSpecies(s1); reactant2_s1.setStoichiometry(1.0); reactant2_s1.setConstant(False)
product2_s2 = reaction2.createProduct(); product2_s2.setSpecies(s2); product2_s2.setStoichiometry(1.0); product2_s2.setConstant(False)
kl2 = reaction2.createKineticLaw()
p_k2_R2 = kl2.createParameter(); p_k2_R2.setId("k2_R2"); p_k2_R2.setValue(k2_R2_val); p_k2_R2.setConstant(True)
kl2.setMath(libsbml.parseL3Formula(f"k2_R2 * {s1}"))

# R3: id_aa2n -> id_xdgv
reaction3 = model.createReaction(); reaction3.setId("R3_aa2n_to_xdgv"); reaction3.setReversible(False); reaction3.setFast(False)
reactant3_s2 = reaction3.createReactant(); reactant3_s2.setSpecies(s2); reactant3_s2.setStoichiometry(1.0); reactant3_s2.setConstant(False)
product3_s3 = reaction3.createProduct(); product3_s3.setSpecies(s3); product3_s3.setStoichiometry(1.0); product3_s3.setConstant(False)
kl3 = reaction3.createKineticLaw()
p_k3_R3 = kl3.createParameter(); p_k3_R3.setId("k3_R3"); p_k3_R3.setValue(k3_R3_val); p_k3_R3.setConstant(True)
kl3.setMath(libsbml.parseL3Formula(f"k3_R3 * {s2}"))

# R4: id_xdgv -> (degradation of S3)
reaction4 = model.createReaction(); reaction4.setId("R4_degrad_xdgv"); reaction4.setReversible(False); reaction4.setFast(False)
reactant4_s3 = reaction4.createReactant(); reactant4_s3.setSpecies(s3); reactant4_s3.setStoichiometry(1.0); reactant4_s3.setConstant(False)
kl4 = reaction4.createKineticLaw()
p_k4_R4 = kl4.createParameter(); p_k4_R4.setId("k4_R4"); p_k4_R4.setValue(k4_R4_val); p_k4_R4.setConstant(True)
kl4.setMath(libsbml.parseL3Formula(f"k4_R4 * {s3}"))

# R5: id_xvpm -> (degradation of S1)
reaction5 = model.createReaction(); reaction5.setId("R5_degrad_xvpm"); reaction5.setReversible(False); reaction5.setFast(False)
reactant5_s1 = reaction5.createReactant(); reactant5_s1.setSpecies(s1); reactant5_s1.setStoichiometry(1.0); reactant5_s1.setConstant(False)
kl5 = reaction5.createKineticLaw()
p_k5_R5 = kl5.createParameter(); p_k5_R5.setId("k5_R5"); p_k5_R5.setValue(k5_R5_val); p_k5_R5.setConstant(True)
kl5.setMath(libsbml.parseL3Formula(f"k5_R5 * {s1}"))

# R6: id_aa2n -> (degradation of S2)
reaction6 = model.createReaction(); reaction6.setId("R6_degrad_aa2n"); reaction6.setReversible(False); reaction6.setFast(False)
reactant6_s2 = reaction6.createReactant(); reactant6_s2.setSpecies(s2); reactant6_s2.setStoichiometry(1.0); reactant6_s2.setConstant(False)
kl6 = reaction6.createKineticLaw()
p_k6_R6 = kl6.createParameter(); p_k6_R6.setId("k6_R6"); p_k6_R6.setValue(k6_R6_val); p_k6_R6.setConstant(True)
kl6.setMath(libsbml.parseL3Formula(f"k6_R6 * {s2}"))

# Write the updated SBML to a string
writer = libsbml.SBMLWriter()
current_sbml_model_str_hyp3_att10 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_sbml_model_str_iteration_10", current_sbml_model_str_hyp3_att10)

# Simulate the model
sim_data_df_hyp3_att10 = simulate(current_sbml_model_str_hyp3_att10)
shared_variables.add("sim_data_df_iteration_10", sim_data_df_hyp3_att10)

# Print simulation results
print(f"Simulation data (Hypothesis 3, Attempt 10) head:\n{sim_data_df_hyp3_att10.head()}")
print(f"\nSimulation data (Hypothesis 3, Attempt 10) tail:\n{sim_data_df_hyp3_att10.tail(5)}")

# Print parameter values used
print(f"\nParameters used for Hypothesis 3, Attempt 10:")
print(f"k1_R1={k1_R1_val}, Ki_R1={Ki_R1_val}, n_R1={n_R1_val}")
print(f"k2_R2={k2_R2_val}, k3_R3={k3_R3_val}")
print(f"k4_R4={k4_R4_val}, k5_R5={k5_R5_val}, k6_R6={k6_R6_val}")

# Print simulated values at key experimental time points
sim_time_points = [0.0, 35.0, 46.0, 48.0, 60.0, 100.0, 150.0, 200.0, 300.0, 400.0, 500.0]
actual_sim_times_to_print = []
if not sim_data_df_hyp3_att10.empty:
    for t_target in sim_time_points:
        closest_time_idx = (np.abs(sim_data_df_hyp3_att10['Time'] - t_target)).argmin()
        actual_sim_times_to_print.append(sim_data_df_hyp3_att10['Time'].iloc[closest_time_idx])
    actual_sim_times_to_print = sorted(list(set(actual_sim_times_to_print)))
    print("\nSimulated values at specific time points (Hypothesis 3, Attempt 10):")
    print(sim_data_df_hyp3_att10[sim_data_df_hyp3_att10['Time'].isin(actual_sim_times_to_print)])
else:
    print("Simulated data for attempt 10 is empty.")

print("\nExperimental data for comparison at these times:")
actual_exp_times_to_print_exp = []
for t_target in sim_time_points:
    closest_time_idx = (np.abs(exp_data_df['Time'] - t_target)).argmin()
    actual_exp_times_to_print_exp.append(exp_data_df['Time'].iloc[closest_time_idx])
actual_exp_times_to_print_exp = sorted(list(set(actual_exp_times_to_print_exp)))
print(exp_data_df[exp_data_df['Time'].isin(actual_exp_times_to_print_exp)][['Time', 'id_xvpm', 'id_aa2n', 'id_xdgv']])
```
# Observation

## Code Stdout
Simulation data (Hypothesis 3, Attempt 10) head:
    id_xvpm   id_aa2n   id_xdgv  Time
0  0.000010  0.000010  0.000010   0.0
1  0.000034  0.000015  0.000010   0.5
2  0.000053  0.000025  0.000011   1.0
3  0.000068  0.000039  0.000012   1.5
4  0.000079  0.000056  0.000013   2.0

Simulation data (Hypothesis 3, Attempt 10) tail:
       id_xvpm   id_aa2n   id_xdgv   Time
996   0.000091  0.000592  0.001334  498.0
997   0.000091  0.000592  0.001334  498.5
998   0.000091  0.000592  0.001334  499.0
999   0.000091  0.000592  0.001334  499.5
1000  0.000091  0.000592  0.001334  500.0

Parameters used for Hypothesis 3, Attempt 10:
k1_R1=6e-05, Ki_R1=0.0015, n_R1=10.0
k2_R2=0.5, k3_R3=0.075
k4_R4=0.0333, k5_R5=0.002, k6_R6=0.002

Simulated values at specific time points (Hypothesis 3, Attempt 10):
       id_xvpm   id_aa2n   id_xdgv   Time
0     0.000010  0.000010  0.000010    0.0
70    0.000119  0.000715  0.000837   35.0
92    0.000116  0.000745  0.001088   46.0
96    0.000115  0.000745  0.001126   48.0
120   0.000100  0.000714  0.001299   60.0
200   0.000090  0.000575  0.001338  100.0
300   0.000091  0.000594  0.001335  150.0
400   0.000091  0.000592  0.001334  200.0
600   0.000091  0.000592  0.001334  300.0
800   0.000091  0.000592  0.001334  400.0
1000  0.000091  0.000592  0.001334  500.0

Experimental data for comparison at these times:
       Time   id_xvpm   id_aa2n   id_xdgv
0       0.0  0.000010  0.000010  0.000010
70     35.0  0.000118  0.000675  0.001502
92     46.0  0.000080  0.000869  0.001954
96     48.0  0.000068  0.000861  0.001963
120    60.0  0.000017  0.000531  0.001629
200   100.0  0.000004  0.000068  0.000455
300   150.0  0.000074  0.000417  0.000969
400   200.0  0.000010  0.000234  0.000904
600   300.0  0.000024  0.000410  0.001145
800   400.0  0.000037  0.000467  0.001179
1000  500.0  0.000043  0.000452  0.001113

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml_model_str_iteration_2, sim_data_df_iteration_2, current_sbml_model_str_iteration_3, sim_data_df_iteration_3, current_sbml_model_str_iteration_4, sim_data_df_iteration_4, current_sbml_model_str_iteration_5, sim_data_df_iteration_5, current_sbml_model_str_iteration_6, sim_data_df_iteration_6, current_sbml_model_str_iteration_7, sim_data_df_iteration_7, current_sbml_model_str_iteration_8, sim_data_df_iteration_8, current_sbml_model_str_iteration_9, sim_data_df_iteration_9, current_sbml_model_str_iteration_10, sim_data_df_iteration_10)

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


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



# Iteration 11
## Action

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

# Access experimental data
exp_data_df = experiment_history['iteration_0']

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

# Define species IDs
s1 = "id_xvpm"
s2 = "id_aa2n"
s3 = "id_xdgv"

# Parameter Set (Hypothesis 3, attempt 11b - very fast cascade)
k1_R1_val = 1.3e-4
Ki_R1_val = 1.2e-3 
n_R1_val = 10.0

k5_R5_val = 0.002
k2_R2_val = 1.10  # Was 0.50

k6_R6_val = 0.002
k3_R3_val = 0.15  # Was 0.075

k4_R4_val = 0.0667 # Was 0.0333, k3/k4 = 2.25

# R1: -> id_xvpm (inhibited by id_xdgv)
reaction1 = model.createReaction(); reaction1.setId("R1_prod_xvpm_inhib"); reaction1.setReversible(False); reaction1.setFast(False)
prod1_s1 = reaction1.createProduct(); prod1_s1.setSpecies(s1); prod1_s1.setStoichiometry(1.0); prod1_s1.setConstant(False)
kl1 = reaction1.createKineticLaw()
p_k1_R1 = kl1.createParameter(); p_k1_R1.setId("k1_R1"); p_k1_R1.setValue(k1_R1_val); p_k1_R1.setConstant(True)
p_Ki_R1 = kl1.createParameter(); p_Ki_R1.setId("Ki_R1"); p_Ki_R1.setValue(Ki_R1_val); p_Ki_R1.setConstant(True)
p_n_R1 = kl1.createParameter(); p_n_R1.setId("n_R1"); p_n_R1.setValue(n_R1_val); p_n_R1.setConstant(True)
kl1.setMath(libsbml.parseL3Formula(f"k1_R1 / (1 + pow({s3} / Ki_R1, n_R1))"))

# R2: id_xvpm -> id_aa2n
reaction2 = model.createReaction(); reaction2.setId("R2_xvpm_to_aa2n"); reaction2.setReversible(False); reaction2.setFast(False)
reactant2_s1 = reaction2.createReactant(); reactant2_s1.setSpecies(s1); reactant2_s1.setStoichiometry(1.0); reactant2_s1.setConstant(False)
product2_s2 = reaction2.createProduct(); product2_s2.setSpecies(s2); product2_s2.setStoichiometry(1.0); product2_s2.setConstant(False)
kl2 = reaction2.createKineticLaw()
p_k2_R2 = kl2.createParameter(); p_k2_R2.setId("k2_R2"); p_k2_R2.setValue(k2_R2_val); p_k2_R2.setConstant(True)
kl2.setMath(libsbml.parseL3Formula(f"k2_R2 * {s1}"))

# R3: id_aa2n -> id_xdgv
reaction3 = model.createReaction(); reaction3.setId("R3_aa2n_to_xdgv"); reaction3.setReversible(False); reaction3.setFast(False)
reactant3_s2 = reaction3.createReactant(); reactant3_s2.setSpecies(s2); reactant3_s2.setStoichiometry(1.0); reactant3_s2.setConstant(False)
product3_s3 = reaction3.createProduct(); product3_s3.setSpecies(s3); product3_s3.setStoichiometry(1.0); product3_s3.setConstant(False)
kl3 = reaction3.createKineticLaw()
p_k3_R3 = kl3.createParameter(); p_k3_R3.setId("k3_R3"); p_k3_R3.setValue(k3_R3_val); p_k3_R3.setConstant(True)
kl3.setMath(libsbml.parseL3Formula(f"k3_R3 * {s2}"))

# R4: id_xdgv -> (degradation of S3)
reaction4 = model.createReaction(); reaction4.setId("R4_degrad_xdgv"); reaction4.setReversible(False); reaction4.setFast(False)
reactant4_s3 = reaction4.createReactant(); reactant4_s3.setSpecies(s3); reactant4_s3.setStoichiometry(1.0); reactant4_s3.setConstant(False)
kl4 = reaction4.createKineticLaw()
p_k4_R4 = kl4.createParameter(); p_k4_R4.setId("k4_R4"); p_k4_R4.setValue(k4_R4_val); p_k4_R4.setConstant(True)
kl4.setMath(libsbml.parseL3Formula(f"k4_R4 * {s3}"))

# R5: id_xvpm -> (degradation of S1)
reaction5 = model.createReaction(); reaction5.setId("R5_degrad_xvpm"); reaction5.setReversible(False); reaction5.setFast(False)
reactant5_s1 = reaction5.createReactant(); reactant5_s1.setSpecies(s1); reactant5_s1.setStoichiometry(1.0); reactant5_s1.setConstant(False)
kl5 = reaction5.createKineticLaw()
p_k5_R5 = kl5.createParameter(); p_k5_R5.setId("k5_R5"); p_k5_R5.setValue(k5_R5_val); p_k5_R5.setConstant(True)
kl5.setMath(libsbml.parseL3Formula(f"k5_R5 * {s1}"))

# R6: id_aa2n -> (degradation of S2)
reaction6 = model.createReaction(); reaction6.setId("R6_degrad_aa2n"); reaction6.setReversible(False); reaction6.setFast(False)
reactant6_s2 = reaction6.createReactant(); reactant6_s2.setSpecies(s2); reactant6_s2.setStoichiometry(1.0); reactant6_s2.setConstant(False)
kl6 = reaction6.createKineticLaw()
p_k6_R6 = kl6.createParameter(); p_k6_R6.setId("k6_R6"); p_k6_R6.setValue(k6_R6_val); p_k6_R6.setConstant(True)
kl6.setMath(libsbml.parseL3Formula(f"k6_R6 * {s2}"))

# Write the updated SBML to a string
writer = libsbml.SBMLWriter()
current_sbml_model_str_hyp3_att11b = writer.writeSBMLToString(sbml_doc)
# Using iteration_11 for shared_variables key as per thought process
shared_variables.add("current_sbml_model_str_iteration_11", current_sbml_model_str_hyp3_att11b) 

# Simulate the model
sim_data_df_hyp3_att11b = simulate(current_sbml_model_str_hyp3_att11b)
shared_variables.add("sim_data_df_iteration_11", sim_data_df_hyp3_att11b)

# Print simulation results
print(f"Simulation data (Hypothesis 3, Attempt 11b) head:\n{sim_data_df_hyp3_att11b.head()}")
print(f"\nSimulation data (Hypothesis 3, Attempt 11b) tail:\n{sim_data_df_hyp3_att11b.tail(5)}")

# Print parameter values used
print(f"\nParameters used for Hypothesis 3, Attempt 11b:")
print(f"k1_R1={k1_R1_val}, Ki_R1={Ki_R1_val}, n_R1={n_R1_val}")
print(f"k2_R2={k2_R2_val}, k3_R3={k3_R3_val}")
print(f"k4_R4={k4_R4_val}, k5_R5={k5_R5_val}, k6_R6={k6_R6_val}")

# Print simulated values at key experimental time points
sim_time_points = [0.0, 35.0, 46.0, 48.0, 60.0, 100.0, 150.0, 200.0, 300.0, 400.0, 500.0]
actual_sim_times_to_print = []
if not sim_data_df_hyp3_att11b.empty:
    for t_target in sim_time_points:
        closest_time_idx = (np.abs(sim_data_df_hyp3_att11b['Time'] - t_target)).argmin()
        actual_sim_times_to_print.append(sim_data_df_hyp3_att11b['Time'].iloc[closest_time_idx])
    actual_sim_times_to_print = sorted(list(set(actual_sim_times_to_print)))
    print("\nSimulated values at specific time points (Hypothesis 3, Attempt 11b):")
    print(sim_data_df_hyp3_att11b[sim_data_df_hyp3_att11b['Time'].isin(actual_sim_times_to_print)])
else:
    print("Simulated data for attempt 11b is empty.")

print("\nExperimental data for comparison at these times:")
actual_exp_times_to_print_exp = []
for t_target in sim_time_points:
    closest_time_idx = (np.abs(exp_data_df['Time'] - t_target)).argmin()
    actual_exp_times_to_print_exp.append(exp_data_df['Time'].iloc[closest_time_idx])
actual_exp_times_to_print_exp = sorted(list(set(actual_exp_times_to_print_exp)))
print(exp_data_df[exp_data_df['Time'].isin(actual_exp_times_to_print_exp)][['Time', 'id_xvpm', 'id_aa2n', 'id_xdgv']])
```
# Observation

## Code Stdout
Simulation data (Hypothesis 3, Attempt 11b) head:
    id_xvpm   id_aa2n   id_xdgv  Time
0  0.000010  0.000010  0.000010   0.0
1  0.000056  0.000028  0.000011   0.5
2  0.000082  0.000063  0.000014   1.0
3  0.000097  0.000106  0.000020   1.5
4  0.000106  0.000153  0.000029   2.0

Simulation data (Hypothesis 3, Attempt 11b) tail:
       id_xvpm   id_aa2n   id_xdgv   Time
996   0.000071  0.000512  0.001152  498.0
997   0.000071  0.000512  0.001152  498.5
998   0.000071  0.000512  0.001152  499.0
999   0.000071  0.000512  0.001152  499.5
1000  0.000071  0.000512  0.001152  500.0

Parameters used for Hypothesis 3, Attempt 11b:
k1_R1=0.00013, Ki_R1=0.0012, n_R1=10.0
k2_R2=1.1, k3_R3=0.15
k4_R4=0.0667, k5_R5=0.002, k6_R6=0.002

Simulated values at specific time points (Hypothesis 3, Attempt 11b):
       id_xvpm   id_aa2n   id_xdgv   Time
0     0.000010  0.000010  0.000010    0.0
70    0.000060  0.000448  0.001185   35.0
92    0.000076  0.000516  0.001131   46.0
96    0.000075  0.000524  0.001136   48.0
120   0.000069  0.000514  0.001158   60.0
200   0.000071  0.000512  0.001152  100.0
300   0.000071  0.000512  0.001152  150.0
400   0.000071  0.000512  0.001152  200.0
600   0.000071  0.000512  0.001152  300.0
800   0.000071  0.000512  0.001152  400.0
1000  0.000071  0.000512  0.001152  500.0

Experimental data for comparison at these times:
       Time   id_xvpm   id_aa2n   id_xdgv
0       0.0  0.000010  0.000010  0.000010
70     35.0  0.000118  0.000675  0.001502
92     46.0  0.000080  0.000869  0.001954
96     48.0  0.000068  0.000861  0.001963
120    60.0  0.000017  0.000531  0.001629
200   100.0  0.000004  0.000068  0.000455
300   150.0  0.000074  0.000417  0.000969
400   200.0  0.000010  0.000234  0.000904
600   300.0  0.000024  0.000410  0.001145
800   400.0  0.000037  0.000467  0.001179
1000  500.0  0.000043  0.000452  0.001113

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml_model_str_iteration_2, sim_data_df_iteration_2, current_sbml_model_str_iteration_3, sim_data_df_iteration_3, current_sbml_model_str_iteration_4, sim_data_df_iteration_4, current_sbml_model_str_iteration_5, sim_data_df_iteration_5, current_sbml_model_str_iteration_6, sim_data_df_iteration_6, current_sbml_model_str_iteration_7, sim_data_df_iteration_7, current_sbml_model_str_iteration_8, sim_data_df_iteration_8, current_sbml_model_str_iteration_9, sim_data_df_iteration_9, current_sbml_model_str_iteration_10, sim_data_df_iteration_10, current_sbml_model_str_iteration_11, sim_data_df_iteration_11)

### 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 previous attempt (11b) showed some promise with a good S3 steady state, but the initial peaks for all species were too low, and S1/S2 steady states were a bit high. The system also damped too quickly.

The strategy for Attempt 12 is to:
1.  Adjust `k1_R1` (S1 production) and downstream conversion rates (`k2_R2`, `k3_R3`) to better match the *initial peak heights* calculated without considering inhibition. The idea is to provide enough "input" for the peaks.
2.  Slightly adjust `Ki_R1` (feedback threshold) to be around the target S3 peak.
3.  Maintain a high Hill coefficient (`n_R1=10`) for sharp feedback.
4.  Keep degradation rates (`k5_R5`, `k6_R6`, `k4_R4`) relatively slow to support oscillations and allow accumulation. The ratios of conversion to degradation rates are crucial for relative concentrations.

The new parameter set for Attempt 12 aims to achieve:
*   S1 peak (uninhibited) ~1.14e-4
*   S2 peak (uninhibited) ~7.82e-4
*   S3 peak (uninhibited) ~1.76e-3

These uninhibited targets are close to the experimental peak values. The actual simulated peaks will be affected by the feedback kicking in. The calculated steady-state values for this new set are: S1 ~0.078e-3, S2 ~0.533e-3, S3 ~1.2e-3. The S3 steady state is very close to the experimental t=500 value. S1 and S2 are a bit higher than their t=500 experimental values, suggesting the troughs might not be deep enough if the system settles there.

The key is to see if these parameters can generate more pronounced oscillations and better initial peak matching.

## Action

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

# Access experimental data
exp_data_df = experiment_history['iteration_0']

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

# Define species IDs
s1 = "id_xvpm"
s2 = "id_aa2n"
s3 = "id_xdgv"

# Parameter Set (Hypothesis 3, attempt 12)
k1_R1_val = 0.8e-4   # S1 max production
Ki_R1_val = 1.3e-3   # Inhibition constant
n_R1_val = 10.0      # Hill coefficient

k5_R5_val = 0.002    # S1 degradation
k2_R2_val = 0.70     # S1 -> S2

k6_R6_val = 0.002    # S2 degradation
k3_R3_val = 0.10     # S2 -> S3

k4_R4_val = 0.0444   # S3 degradation (k3/k4 = 0.10/0.0444 ~ 2.25)

# R1: -> id_xvpm (inhibited by id_xdgv)
reaction1 = model.createReaction(); reaction1.setId("R1_prod_xvpm_inhib"); reaction1.setReversible(False); reaction1.setFast(False)
prod1_s1 = reaction1.createProduct(); prod1_s1.setSpecies(s1); prod1_s1.setStoichiometry(1.0); prod1_s1.setConstant(False)
kl1 = reaction1.createKineticLaw()
p_k1_R1 = kl1.createParameter(); p_k1_R1.setId("k1_R1"); p_k1_R1.setValue(k1_R1_val); p_k1_R1.setConstant(True)
p_Ki_R1 = kl1.createParameter(); p_Ki_R1.setId("Ki_R1"); p_Ki_R1.setValue(Ki_R1_val); p_Ki_R1.setConstant(True)
p_n_R1 = kl1.createParameter(); p_n_R1.setId("n_R1"); p_n_R1.setValue(n_R1_val); p_n_R1.setConstant(True)
kl1.setMath(libsbml.parseL3Formula(f"k1_R1 / (1 + pow({s3} / Ki_R1, n_R1))"))

# R2: id_xvpm -> id_aa2n
reaction2 = model.createReaction(); reaction2.setId("R2_xvpm_to_aa2n"); reaction2.setReversible(False); reaction2.setFast(False)
reactant2_s1 = reaction2.createReactant(); reactant2_s1.setSpecies(s1); reactant2_s1.setStoichiometry(1.0); reactant2_s1.setConstant(False)
product2_s2 = reaction2.createProduct(); product2_s2.setSpecies(s2); product2_s2.setStoichiometry(1.0); product2_s2.setConstant(False)
kl2 = reaction2.createKineticLaw()
p_k2_R2 = kl2.createParameter(); p_k2_R2.setId("k2_R2"); p_k2_R2.setValue(k2_R2_val); p_k2_R2.setConstant(True)
kl2.setMath(libsbml.parseL3Formula(f"k2_R2 * {s1}"))

# R3: id_aa2n -> id_xdgv
reaction3 = model.createReaction(); reaction3.setId("R3_aa2n_to_xdgv"); reaction3.setReversible(False); reaction3.setFast(False)
reactant3_s2 = reaction3.createReactant(); reactant3_s2.setSpecies(s2); reactant3_s2.setStoichiometry(1.0); reactant3_s2.setConstant(False)
product3_s3 = reaction3.createProduct(); product3_s3.setSpecies(s3); product3_s3.setStoichiometry(1.0); product3_s3.setConstant(False)
kl3 = reaction3.createKineticLaw()
p_k3_R3 = kl3.createParameter(); p_k3_R3.setId("k3_R3"); p_k3_R3.setValue(k3_R3_val); p_k3_R3.setConstant(True)
kl3.setMath(libsbml.parseL3Formula(f"k3_R3 * {s2}"))

# R4: id_xdgv -> (degradation of S3)
reaction4 = model.createReaction(); reaction4.setId("R4_degrad_xdgv"); reaction4.setReversible(False); reaction4.setFast(False)
reactant4_s3 = reaction4.createReactant(); reactant4_s3.setSpecies(s3); reactant4_s3.setStoichiometry(1.0); reactant4_s3.setConstant(False)
kl4 = reaction4.createKineticLaw()
p_k4_R4 = kl4.createParameter(); p_k4_R4.setId("k4_R4"); p_k4_R4.setValue(k4_R4_val); p_k4_R4.setConstant(True)
kl4.setMath(libsbml.parseL3Formula(f"k4_R4 * {s3}"))

# R5: id_xvpm -> (degradation of S1)
reaction5 = model.createReaction(); reaction5.setId("R5_degrad_xvpm"); reaction5.setReversible(False); reaction5.setFast(False)
reactant5_s1 = reaction5.createReactant(); reactant5_s1.setSpecies(s1); reactant5_s1.setStoichiometry(1.0); reactant5_s1.setConstant(False)
kl5 = reaction5.createKineticLaw()
p_k5_R5 = kl5.createParameter(); p_k5_R5.setId("k5_R5"); p_k5_R5.setValue(k5_R5_val); p_k5_R5.setConstant(True)
kl5.setMath(libsbml.parseL3Formula(f"k5_R5 * {s1}"))

# R6: id_aa2n -> (degradation of S2)
reaction6 = model.createReaction(); reaction6.setId("R6_degrad_aa2n"); reaction6.setReversible(False); reaction6.setFast(False)
reactant6_s2 = reaction6.createReactant(); reactant6_s2.setSpecies(s2); reactant6_s2.setStoichiometry(1.0); reactant6_s2.setConstant(False)
kl6 = reaction6.createKineticLaw()
p_k6_R6 = kl6.createParameter(); p_k6_R6.setId("k6_R6"); p_k6_R6.setValue(k6_R6_val); p_k6_R6.setConstant(True)
kl6.setMath(libsbml.parseL3Formula(f"k6_R6 * {s2}"))

# Write the updated SBML to a string
writer = libsbml.SBMLWriter()
current_sbml_model_str_hyp3_att12 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_sbml_model_str_iteration_12", current_sbml_model_str_hyp3_att12)

# Simulate the model
sim_data_df_hyp3_att12 = simulate(current_sbml_model_str_hyp3_att12)
shared_variables.add("sim_data_df_iteration_12", sim_data_df_hyp3_att12)

# Print simulation results
print(f"Simulation data (Hypothesis 3, Attempt 12) head:\n{sim_data_df_hyp3_att12.head()}")
print(f"\nSimulation data (Hypothesis 3, Attempt 12) tail:\n{sim_data_df_hyp3_att12.tail(5)}")

# Print parameter values used
print(f"\nParameters used for Hypothesis 3, Attempt 12:")
print(f"k1_R1={k1_R1_val}, Ki_R1={Ki_R1_val}, n_R1={n_R1_val}")
print(f"k2_R2={k2_R2_val}, k3_R3={k3_R3_val}")
print(f"k4_R4={k4_R4_val}, k5_R5={k5_R5_val}, k6_R6={k6_R6_val}")

# Print simulated values at key experimental time points
sim_time_points = [0.0, 35.0, 46.0, 48.0, 60.0, 100.0, 150.0, 200.0, 300.0, 400.0, 500.0]
actual_sim_times_to_print = []
if not sim_data_df_hyp3_att12.empty:
    for t_target in sim_time_points:
        closest_time_idx = (np.abs(sim_data_df_hyp3_att12['Time'] - t_target)).argmin()
        actual_sim_times_to_print.append(sim_data_df_hyp3_att12['Time'].iloc[closest_time_idx])
    actual_sim_times_to_print = sorted(list(set(actual_sim_times_to_print)))
    print("\nSimulated values at specific time points (Hypothesis 3, Attempt 12):")
    print(sim_data_df_hyp3_att12[sim_data_df_hyp3_att12['Time'].isin(actual_sim_times_to_print)])
else:
    print("Simulated data for attempt 12 is empty.")

print("\nExperimental data for comparison at these times:")
actual_exp_times_to_print_exp = []
for t_target in sim_time_points:
    closest_time_idx = (np.abs(exp_data_df['Time'] - t_target)).argmin()
    actual_exp_times_to_print_exp.append(exp_data_df['Time'].iloc[closest_time_idx])
actual_exp_times_to_print_exp = sorted(list(set(actual_exp_times_to_print_exp)))
print(exp_data_df[exp_data_df['Time'].isin(actual_exp_times_to_print_exp)][['Time', 'id_xvpm', 'id_aa2n', 'id_xdgv']])
```
# Observation

## Code Stdout
Simulation data (Hypothesis 3, Attempt 12) head:
    id_xvpm   id_aa2n   id_xdgv  Time
0  0.000010  0.000010  0.000010   0.0
1  0.000041  0.000019  0.000010   0.5
2  0.000062  0.000035  0.000012   1.0
3  0.000078  0.000058  0.000014   1.5
4  0.000088  0.000083  0.000017   2.0

Simulation data (Hypothesis 3, Attempt 12) tail:
       id_xvpm   id_aa2n   id_xdgv   Time
996   0.000078  0.000534  0.001204  498.0
997   0.000078  0.000534  0.001204  498.5
998   0.000078  0.000534  0.001204  499.0
999   0.000078  0.000534  0.001204  499.5
1000  0.000078  0.000534  0.001204  500.0

Parameters used for Hypothesis 3, Attempt 12:
k1_R1=8e-05, Ki_R1=0.0013, n_R1=10.0
k2_R2=0.7, k3_R3=0.1
k4_R4=0.0444, k5_R5=0.002, k6_R6=0.002

Simulated values at specific time points (Hypothesis 3, Attempt 12):
       id_xvpm   id_aa2n   id_xdgv   Time
0     0.000010  0.000010  0.000010    0.0
70    0.000100  0.000730  0.001103   35.0
92    0.000068  0.000604  0.001261   46.0
96    0.000066  0.000577  0.001266   48.0
120   0.000072  0.000496  0.001227   60.0
200   0.000077  0.000536  0.001206  100.0
300   0.000078  0.000535  0.001204  150.0
400   0.000078  0.000534  0.001204  200.0
600   0.000078  0.000534  0.001204  300.0
800   0.000078  0.000534  0.001204  400.0
1000  0.000078  0.000534  0.001204  500.0

Experimental data for comparison at these times:
       Time   id_xvpm   id_aa2n   id_xdgv
0       0.0  0.000010  0.000010  0.000010
70     35.0  0.000118  0.000675  0.001502
92     46.0  0.000080  0.000869  0.001954
96     48.0  0.000068  0.000861  0.001963
120    60.0  0.000017  0.000531  0.001629
200   100.0  0.000004  0.000068  0.000455
300   150.0  0.000074  0.000417  0.000969
400   200.0  0.000010  0.000234  0.000904
600   300.0  0.000024  0.000410  0.001145
800   400.0  0.000037  0.000467  0.001179
1000  500.0  0.000043  0.000452  0.001113

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml_model_str_iteration_2, sim_data_df_iteration_2, current_sbml_model_str_iteration_3, sim_data_df_iteration_3, current_sbml_model_str_iteration_4, sim_data_df_iteration_4, current_sbml_model_str_iteration_5, sim_data_df_iteration_5, current_sbml_model_str_iteration_6, sim_data_df_iteration_6, current_sbml_model_str_iteration_7, sim_data_df_iteration_7, current_sbml_model_str_iteration_8, sim_data_df_iteration_8, current_sbml_model_str_iteration_9, sim_data_df_iteration_9, current_sbml_model_str_iteration_10, sim_data_df_iteration_10, current_sbml_model_str_iteration_11, sim_data_df_iteration_11, current_sbml_model_str_iteration_12, sim_data_df_iteration_12)

### 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

Attempt 12 showed some progress, but also highlighted persistent issues:
-   **S1 peak (sim: 0.100e-3 at t=35)** is getting closer to exp (0.118e-3 at t=35).
-   **S2 peak (sim: 0.730e-3 at t=35)** is now earlier, but still a bit lower than exp (0.869e-3 at t=46). The timing is off.
-   **S3 peak (sim: ~0.001266 at t=48)** is much lower than exp (0.001963 at t=48). The timing is better.
-   **Oscillations:** The system still damps to a steady state: S1=0.078e-3, S2=0.534e-3, S3=1.204e-3.
    -   Experimental values at t=500: S1=0.043e-3, S2=0.452e-3, S3=1.113e-3.
    -   Simulated S3 steady state is good. S1 and S2 steady states are too high. This means the troughs of the oscillations are not deep enough, especially for S1 and S2.

**Key Issues from Attempt 12:**
1.  **S2 and S3 Peak Magnitudes Still Too Low:** S2 needs to be higher, and S3 significantly higher.
2.  **S1 and S2 Troughs/Steady State Too High:** The feedback isn't pulling S1 and S2 down enough after the initial peaks.
3.  **Oscillations are Damped:** The system doesn't show sustained oscillations like the experimental data suggests (especially the later time points showing rises after initial falls).

**Strategy for Attempt 13:**
The model needs to be more "excitable" to get higher peaks and more sustained oscillations. The feedback also needs to be more effective at creating lower troughs for S1 and S2.

1.  **Increase Overall Production/Conversion Rates:**
    *   Increase `k1_R1` (S1 production) further to drive higher peaks.
    *   Increase `k2_R2` (S1->S2) and `k3_R3` (S2->S3) to ensure rapid conversion and build-up for S2 and S3 peaks.
2.  **Adjust Feedback Parameters (`Ki_R1`, `n_R1`):**
    *   `Ki_R1` should be set relative to the *target S3 peak*. If we aim for a higher S3 peak, `Ki_R1` might need to be slightly higher too, so inhibition doesn't clamp down too early.
    *   `n_R1` is already high (10), which is good for sharp response.
3.  **Review Degradation Rates (`k4, k5, k6`):**
    *   These control lifetimes and influence oscillatory period and damping. If they are too fast, peaks might not be reached, and oscillations might damp quickly. If too slow, the system might respond sluggishly or overshoot excessively. The current slow degradation rates (0.002 for S1/S2) are likely good for promoting oscillations.

**Revised Parameters (Hypothesis 3, attempt 13):**
Let's aim for higher uninhibited peaks:
*   Target S1 peak ~ 1.2e-4
*   Target S2 peak ~ 0.9e-3 (S2/S1 ~ 7.5)
*   Target S3 peak ~ 2.0e-3 (S3/S2 ~ 2.22)

*   **Feedback & S1 Production (R1):**
    *   `k1_R1 = 0.9e-4` (was 0.8e-4). Increased.
    *   `Ki_R1 = 1.6e-3` (was 1.3e-3). Increased slightly, to be around the new target S3 peak.
    *   `n_R1 = 10.0` (kept).
*   **S1 Dynamics (R2, R5):**
    *   `k5_R5 = 0.002` (kept).
    *   `k2_R2 = 0.75` (was 0.70). Increased.
        *   `S1_eff_deg = k5+k2 = 0.002+0.75 = 0.752`.
        *   `S1_peak_no_inhib ~ k1_R1 / S1_eff_deg = 0.9e-4 / 0.752 = 1.196e-4`. (Matches target S1 peak).
*   **S2 & S3 Dynamics (R3, R6, R4):**
    *   `k6_R6 = 0.002` (kept).
    *   `k3_R3 = 0.10` (kept from Att12, as S2/S1 ratio was good).
        *   `S2_eff_deg = k6+k3 = 0.002+0.10 = 0.102`.
        *   `S2_peak_no_inhib ~ (k2/S2_eff_deg)*S1_peak = (0.75/0.102)*1.196e-4 = 7.35 * 1.196e-4 = 8.79e-4`. (Matches target S2 peak).
    *   `k4_R4 = ?`
        *   Target `S3_peak/S2_peak ~ 2.22`. So `k3_R3/k4_R4 ~ 2.22`.
        *   `k4_R4 = k3_R3 / 2.22 = 0.10 / 2.22 = 0.045`. (was 0.0444).
        *   `S3_peak_no_inhib ~ (k3/k4)*S2_peak = 2.22 * 8.79e-4 = 1.95e-3`. (Matches target S3 peak).

This set of parameters is designed to hit the experimental peak values more closely *before* strong feedback kicks in. The `Ki_R1` is set near the target S3 peak. The hope is that this stronger initial pulse, combined with the sharp feedback, will lead to more pronounced and sustained oscillations, and also help in achieving lower trough values due to stronger feedback engagement at higher S3 levels.

Approximate steady state calculation:
`S1_eff_deg = 0.752`.
`S2_eff_deg = 0.102`.
`S3_eff_deg = 0.045`.

`S1 = S2 / (k2/(k6+k3)) = S2 / (0.75/0.102) = S2 / 7.35 = 0.136 * S2`.
`S2 = S3 / (k3/k4) = S3 / (0.10/0.045) = S3 / 2.22 = 0.450 * S3`.
`S1 = 0.136 * 0.450 * S3 = 0.0612 * S3`.

`k1_R1 / (1 + (S3/Ki_R1)^n_R1) = (k5_R5 + k2_R2) * S1`
`0.9e-4 / (1 + (S3/1.6e-3)^10) = 0.752 * 0.0612 * S3 = 0.0460 * S3`.
`0.9e-4 = 0.0460 * S3 * (1 + (S3/1.6e-3)^10)`.
If `S3_ss = Ki_R1 = 1.6e-3`:
RHS = `0.0460 * 1.6e-3 * (1+1) = 7.36e-5 * 2 = 14.72e-5 = 1.472e-4`.
LHS (0.9e-4) is smaller than RHS. So S3_ss will be lower than `Ki_R1`.
Let `S3_ss = 1.3e-3`. `(1.3/1.6)^10 = (0.8125)^10 = 0.125`. (approx, (0.8)^10=0.107, (0.85)^10=0.196)
RHS = `0.0460 * 1.3e-3 * (1 + 0.125) = 5.98e-5 * 1.125 = 6.73e-5`.
LHS (0.9e-4) is larger than RHS. So S3_ss is between 1.3e-3 and 1.6e-3.
Let's try `S3_ss = 1.4e-3`. `(1.4/1.6)^10 = (0.875)^10 = 0.263`.
RHS = `0.0460 * 1.4e-3 * (1 + 0.263) = 6.44e-5 * 1.263 = 8.13e-5`.
Closer to 0.9e-4. Let's estimate `S3_ss ~ 1.45e-3`.

Then `S2_ss = 0.450 * 1.45e-3 = 0.65e-3`.
And `S1_ss = 0.0612 * 1.45e-3 = 0.0887e-3`.

Predicted steady state values:
`S1_ss ~ 0.089e-3` (Exp t=500: 0.043e-3) - Still high.
`S2_ss ~ 0.65e-3` (Exp t=500: 0.452e-3) - Still high.
`S3_ss ~ 1.45e-3` (Exp t=500: 1.113e-3) - Still high.

The predicted steady states are higher than desired. This suggests that even if oscillations occur, they might not dip low enough. The `Ki_R1` might need to be lower to enforce a lower S3 steady state, which would then pull S1 and S2 down. However, a lower `Ki_R1` might also clamp the initial S3 peak too much. This is a delicate balance.
Let's proceed with these parameters to see the peak behavior first. If peaks are good but troughs are too high, then `Ki_R1` can be tuned down in the next iteration.

## Action

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

# Access experimental data
exp_data_df = experiment_history['iteration_0']

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

# Define species IDs
s1 = "id_xvpm"
s2 = "id_aa2n"
s3 = "id_xdgv"

# Parameter Set (Hypothesis 3, attempt 13)
k1_R1_val = 0.9e-4
Ki_R1_val = 1.6e-3
n_R1_val = 10.0

k5_R5_val = 0.002
k2_R2_val = 0.75

k6_R6_val = 0.002
k3_R3_val = 0.10

k4_R4_val = 0.045

# R1: -> id_xvpm (inhibited by id_xdgv)
reaction1 = model.createReaction(); reaction1.setId("R1_prod_xvpm_inhib"); reaction1.setReversible(False); reaction1.setFast(False)
prod1_s1 = reaction1.createProduct(); prod1_s1.setSpecies(s1); prod1_s1.setStoichiometry(1.0); prod1_s1.setConstant(False)
kl1 = reaction1.createKineticLaw()
p_k1_R1 = kl1.createParameter(); p_k1_R1.setId("k1_R1"); p_k1_R1.setValue(k1_R1_val); p_k1_R1.setConstant(True)
p_Ki_R1 = kl1.createParameter(); p_Ki_R1.setId("Ki_R1"); p_Ki_R1.setValue(Ki_R1_val); p_Ki_R1.setConstant(True)
p_n_R1 = kl1.createParameter(); p_n_R1.setId("n_R1"); p_n_R1.setValue(n_R1_val); p_n_R1.setConstant(True)
kl1.setMath(libsbml.parseL3Formula(f"k1_R1 / (1 + pow({s3} / Ki_R1, n_R1))"))

# R2: id_xvpm -> id_aa2n
reaction2 = model.createReaction(); reaction2.setId("R2_xvpm_to_aa2n"); reaction2.setReversible(False); reaction2.setFast(False)
reactant2_s1 = reaction2.createReactant(); reactant2_s1.setSpecies(s1); reactant2_s1.setStoichiometry(1.0); reactant2_s1.setConstant(False)
product2_s2 = reaction2.createProduct(); product2_s2.setSpecies(s2); product2_s2.setStoichiometry(1.0); product2_s2.setConstant(False)
kl2 = reaction2.createKineticLaw()
p_k2_R2 = kl2.createParameter(); p_k2_R2.setId("k2_R2"); p_k2_R2.setValue(k2_R2_val); p_k2_R2.setConstant(True)
kl2.setMath(libsbml.parseL3Formula(f"k2_R2 * {s1}"))

# R3: id_aa2n -> id_xdgv
reaction3 = model.createReaction(); reaction3.setId("R3_aa2n_to_xdgv"); reaction3.setReversible(False); reaction3.setFast(False)
reactant3_s2 = reaction3.createReactant(); reactant3_s2.setSpecies(s2); reactant3_s2.setStoichiometry(1.0); reactant3_s2.setConstant(False)
product3_s3 = reaction3.createProduct(); product3_s3.setSpecies(s3); product3_s3.setStoichiometry(1.0); product3_s3.setConstant(False)
kl3 = reaction3.createKineticLaw()
p_k3_R3 = kl3.createParameter(); p_k3_R3.setId("k3_R3"); p_k3_R3.setValue(k3_R3_val); p_k3_R3.setConstant(True)
kl3.setMath(libsbml.parseL3Formula(f"k3_R3 * {s2}"))

# R4: id_xdgv -> (degradation of S3)
reaction4 = model.createReaction(); reaction4.setId("R4_degrad_xdgv"); reaction4.setReversible(False); reaction4.setFast(False)
reactant4_s3 = reaction4.createReactant(); reactant4_s3.setSpecies(s3); reactant4_s3.setStoichiometry(1.0); reactant4_s3.setConstant(False)
kl4 = reaction4.createKineticLaw()
p_k4_R4 = kl4.createParameter(); p_k4_R4.setId("k4_R4"); p_k4_R4.setValue(k4_R4_val); p_k4_R4.setConstant(True)
kl4.setMath(libsbml.parseL3Formula(f"k4_R4 * {s3}"))

# R5: id_xvpm -> (degradation of S1)
reaction5 = model.createReaction(); reaction5.setId("R5_degrad_xvpm"); reaction5.setReversible(False); reaction5.setFast(False)
reactant5_s1 = reaction5.createReactant(); reactant5_s1.setSpecies(s1); reactant5_s1.setStoichiometry(1.0); reactant5_s1.setConstant(False)
kl5 = reaction5.createKineticLaw()
p_k5_R5 = kl5.createParameter(); p_k5_R5.setId("k5_R5"); p_k5_R5.setValue(k5_R5_val); p_k5_R5.setConstant(True)
kl5.setMath(libsbml.parseL3Formula(f"k5_R5 * {s1}"))

# R6: id_aa2n -> (degradation of S2)
reaction6 = model.createReaction(); reaction6.setId("R6_degrad_aa2n"); reaction6.setReversible(False); reaction6.setFast(False)
reactant6_s2 = reaction6.createReactant(); reactant6_s2.setSpecies(s2); reactant6_s2.setStoichiometry(1.0); reactant6_s2.setConstant(False)
kl6 = reaction6.createKineticLaw()
p_k6_R6 = kl6.createParameter(); p_k6_R6.setId("k6_R6"); p_k6_R6.setValue(k6_R6_val); p_k6_R6.setConstant(True)
kl6.setMath(libsbml.parseL3Formula(f"k6_R6 * {s2}"))

# Write the updated SBML to a string
writer = libsbml.SBMLWriter()
current_sbml_model_str_hyp3_att13 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_sbml_model_str_iteration_13", current_sbml_model_str_hyp3_att13)

# Simulate the model
sim_data_df_hyp3_att13 = simulate(current_sbml_model_str_hyp3_att13)
shared_variables.add("sim_data_df_iteration_13", sim_data_df_hyp3_att13)

# Print simulation results
print(f"Simulation data (Hypothesis 3, Attempt 13) head:\n{sim_data_df_hyp3_att13.head()}")
print(f"\nSimulation data (Hypothesis 3, Attempt 13) tail:\n{sim_data_df_hyp3_att13.tail(5)}")

# Print parameter values used
print(f"\nParameters used for Hypothesis 3, Attempt 13:")
print(f"k1_R1={k1_R1_val}, Ki_R1={Ki_R1_val}, n_R1={n_R1_val}")
print(f"k2_R2={k2_R2_val}, k3_R3={k3_R3_val}")
print(f"k4_R4={k4_R4_val}, k5_R5={k5_R5_val}, k6_R6={k6_R6_val}")

# Print simulated values at key experimental time points
sim_time_points = [0.0, 35.0, 46.0, 48.0, 60.0, 100.0, 150.0, 200.0, 300.0, 400.0, 500.0]
actual_sim_times_to_print = []
if not sim_data_df_hyp3_att13.empty:
    for t_target in sim_time_points:
        closest_time_idx = (np.abs(sim_data_df_hyp3_att13['Time'] - t_target)).argmin()
        actual_sim_times_to_print.append(sim_data_df_hyp3_att13['Time'].iloc[closest_time_idx])
    actual_sim_times_to_print = sorted(list(set(actual_sim_times_to_print)))
    print("\nSimulated values at specific time points (Hypothesis 3, Attempt 13):")
    print(sim_data_df_hyp3_att13[sim_data_df_hyp3_att13['Time'].isin(actual_sim_times_to_print)])
else:
    print("Simulated data for attempt 13 is empty.")

print("\nExperimental data for comparison at these times:")
actual_exp_times_to_print_exp = []
for t_target in sim_time_points:
    closest_time_idx = (np.abs(exp_data_df['Time'] - t_target)).argmin()
    actual_exp_times_to_print_exp.append(exp_data_df['Time'].iloc[closest_time_idx])
actual_exp_times_to_print_exp = sorted(list(set(actual_exp_times_to_print_exp)))
print(exp_data_df[exp_data_df['Time'].isin(actual_exp_times_to_print_exp)][['Time', 'id_xvpm', 'id_aa2n', 'id_xdgv']])
```
# Observation

## Code Stdout
Simulation data (Hypothesis 3, Attempt 13) head:
    id_xvpm   id_aa2n   id_xdgv  Time
0  0.000010  0.000010  0.000010   0.0
1  0.000044  0.000020  0.000010   0.5
2  0.000068  0.000040  0.000012   1.0
3  0.000084  0.000066  0.000014   1.5
4  0.000095  0.000095  0.000018   2.0

Simulation data (Hypothesis 3, Attempt 13) tail:
       id_xvpm   id_aa2n   id_xdgv   Time
996   0.000088  0.000649  0.001443  498.0
997   0.000088  0.000649  0.001443  498.5
998   0.000088  0.000649  0.001443  499.0
999   0.000088  0.000649  0.001443  499.5
1000  0.000088  0.000649  0.001443  500.0

Parameters used for Hypothesis 3, Attempt 13:
k1_R1=9e-05, Ki_R1=0.0016, n_R1=10.0
k2_R2=0.75, k3_R3=0.1
k4_R4=0.045, k5_R5=0.002, k6_R6=0.002

Simulated values at specific time points (Hypothesis 3, Attempt 13):
       id_xvpm   id_aa2n   id_xdgv   Time
0     0.000010  0.000010  0.000010    0.0
70    0.000113  0.000839  0.001240   35.0
92    0.000089  0.000764  0.001458   46.0
96    0.000085  0.000741  0.001476   48.0
120   0.000080  0.000634  0.001485   60.0
200   0.000088  0.000653  0.001442  100.0
300   0.000088  0.000649  0.001443  150.0
400   0.000088  0.000649  0.001443  200.0
600   0.000088  0.000649  0.001443  300.0
800   0.000088  0.000649  0.001443  400.0
1000  0.000088  0.000649  0.001443  500.0

Experimental data for comparison at these times:
       Time   id_xvpm   id_aa2n   id_xdgv
0       0.0  0.000010  0.000010  0.000010
70     35.0  0.000118  0.000675  0.001502
92     46.0  0.000080  0.000869  0.001954
96     48.0  0.000068  0.000861  0.001963
120    60.0  0.000017  0.000531  0.001629
200   100.0  0.000004  0.000068  0.000455
300   150.0  0.000074  0.000417  0.000969
400   200.0  0.000010  0.000234  0.000904
600   300.0  0.000024  0.000410  0.001145
800   400.0  0.000037  0.000467  0.001179
1000  500.0  0.000043  0.000452  0.001113

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml_model_str_iteration_2, sim_data_df_iteration_2, current_sbml_model_str_iteration_3, sim_data_df_iteration_3, current_sbml_model_str_iteration_4, sim_data_df_iteration_4, current_sbml_model_str_iteration_5, sim_data_df_iteration_5, current_sbml_model_str_iteration_6, sim_data_df_iteration_6, current_sbml_model_str_iteration_7, sim_data_df_iteration_7, current_sbml_model_str_iteration_8, sim_data_df_iteration_8, current_sbml_model_str_iteration_9, sim_data_df_iteration_9, current_sbml_model_str_iteration_10, sim_data_df_iteration_10, current_sbml_model_str_iteration_11, sim_data_df_iteration_11, current_sbml_model_str_iteration_12, sim_data_df_iteration_12, current_sbml_model_str_iteration_13, sim_data_df_iteration_13)

### 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

Attempt 13 showed some improvements in peak heights but still has issues:
-   **S1 peak (sim: 0.113e-3 at t=35)** is very close to exp (0.118e-3 at t=35). Good.
-   **S2 peak (sim: 0.839e-3 at t=35)** is now close to exp (0.869e-3 at t=46), but the timing is too early (S2 peaks with S1).
-   **S3 peak (sim: ~0.001485 at t=60)** is still lower than exp (0.001963 at t=48). Timing is a bit slow.
-   **Oscillations:** The system still damps to a steady state: S1=0.088e-3, S2=0.649e-3, S3=1.443e-3.
    -   Experimental values at t=500: S1=0.043e-3, S2=0.452e-3, S3=1.113e-3.
    -   All simulated steady states are too high. The system doesn't dip low enough.

**Key Issues from Attempt 13:**
1.  **S2 Peak Timing:** S2 peaks too early, concurrently with S1, instead of after S1. This suggests S2 production from S1 (`k2_R2`) might be too fast relative to S2's own consumption/conversion (`k6_R6 + k3_R3`).
2.  **S3 Peak Magnitude Still Too Low:** Needs to be higher.
3.  **All Troughs/Steady States Too High:** Feedback isn't strong enough or isn't timed right to pull concentrations down to experimental trough levels.
4.  **Oscillations are Damped:** Still not getting sustained oscillations that match the later experimental data points (e.g., the rise in S1 and S2 after t=100, then fall, then another rise).

**Strategy for Attempt 14:**
1.  **Delay S2 Peak Relative to S1:**
    *   Reduce `k2_R2` (S1->S2 conversion) to slow down S2's rise relative to S1. This should create a lag.
    *   To maintain S2 peak height, `k1_R1` might need a slight increase if S1 peak is affected by slower `k2_R2`.
2.  **Increase S3 Peak Magnitude:**
    *   Increase `k3_R3` (S2->S3 conversion) to drive more S2 into S3.
    *   Adjust `k4_R4` (S3 degradation) to maintain the S3/S2 peak ratio.
3.  **Improve Trough Depths (Strengthen Feedback Effect at Lower Concentrations):**
    *   Decrease `Ki_R1` (feedback threshold). If S3 steady state is around 1.1e-3 (exp), `Ki_R1` should be around this value or slightly lower to ensure feedback is active.
    *   The current `Ki_R1 = 1.6e-3` is too high if the target S3 trough/steady state is ~1.1e-3.
4.  **Promote Sustained Oscillations:**
    *   The combination of appropriate delays (S1->S2->S3), strong feedback (high `n_R1`, well-tuned `Ki_R1`), and appropriate species lifetimes (set by degradation rates) is key. The degradation rates `k5, k6, k4` are already quite low, which is generally good for oscillations.

**Revised Parameters (Hypothesis 3, attempt 14):**
*   **Feedback & S1 Production (R1):**
    *   `k1_R1 = 1.0e-4` (was 0.9e-4). Increase slightly to ensure S1 peak is robust.
    *   `Ki_R1 = 1.2e-3` (was 1.6e-3). Lowered significantly to make feedback stronger at lower S3 levels, aiming for lower troughs.
    *   `n_R1 = 10.0` (kept).
*   **S1 Dynamics (R2, R5):**
    *   `k5_R5 = 0.002` (kept).
    *   `k2_R2 = 0.60` (was 0.75). Reduced to delay S2 peak.
        *   `S1_eff_deg = k5+k2 = 0.002+0.60 = 0.602`.
        *   `S1_peak_no_inhib ~ k1_R1 / S1_eff_deg = 1.0e-4 / 0.602 = 1.66e-4`. This is now higher than exp. S1 peak, but feedback will reduce it.
*   **S2 & S3 Dynamics (R3, R6, R4):**
    *   `k6_R6 = 0.002` (kept).
    *   `k3_R3 = 0.12` (was 0.10). Increased to boost S3 production.
        *   `S2_eff_deg = k6+k3 = 0.002+0.12 = 0.122`.
        *   Target S2 peak ~0.9e-3. `S2_peak ~ (k2/S2_eff_deg)*S1_actual_peak`.
        *   `k2/S2_eff_deg = 0.60/0.122 = 4.91`. If S1 actual peak is ~1.2e-4, S2 peak ~ 4.91 * 1.2e-4 = 5.9e-4. This is too low.
            We need a higher `k2/S2_eff_deg` ratio.
            Let's keep `k2_R2 = 0.60`. To get S2/S1 ratio ~7.5 (for peaks): `0.60 / S2_eff_deg = 7.5` => `S2_eff_deg = 0.60/7.5 = 0.08`.
            So `k6+k3 = 0.08`. Since `k6=0.002`, `k3_R3 = 0.078`. (This is lower than previous 0.10).
            Let's try `k3_R3 = 0.08`. Then `S2_eff_deg = 0.082`. `k2/S2_eff_deg = 0.60/0.082 = 7.31`. Good.
    *   `k3_R3 = 0.08` (revised from above).
    *   `k4_R4 = ?`
        *   Target `S3_peak/S2_peak ~ 2.22`. So `k3_R3/k4_R4 ~ 2.22`.
        *   `k4_R4 = k3_R3 / 2.22 = 0.08 / 2.22 = 0.036`.
        *   With these, `S1_peak_no_inhib ~ 1.66e-4`. `S2_peak_no_inhib ~ 7.31 * S1_peak_actual`. `S3_peak_no_inhib ~ 2.22 * S2_peak_actual`.

This iterative adjustment is tricky. The main changes are:
-   `k1_R1` increased.
-   `Ki_R1` significantly decreased to strengthen feedback at lower S3.
-   `k2_R2` decreased to try and lag S2 peak.
-   `k3_R3` adjusted based on `k2_R2` to maintain S2/S1 ratio.
-   `k4_R4` adjusted based on `k3_R3` to maintain S3/S2 ratio.

Let's re-evaluate the target "uninhibited" peaks with the new `k1_R1=1.0e-4`, `k2_R2=0.60`, `k3_R3=0.08`, `k4_R4=0.036`.
`S1_peak_no_inhib = 1.0e-4 / (0.002+0.60) = 1.0e-4 / 0.602 = 1.66e-4`.
`S2_peak_no_inhib = (0.60 / (0.002+0.08)) * 1.66e-4 = (0.60/0.082) * 1.66e-4 = 7.317 * 1.66e-4 = 12.15e-4 = 1.215e-3`. (This is higher than exp S2 peak 0.87e-3).
`S3_peak_no_inhib = (0.08 / 0.036) * 1.215e-3 = 2.22 * 1.215e-3 = 2.69e-3`. (This is higher than exp S3 peak 1.96e-3).

The uninhibited peaks are now higher than experimental ones. This is good, as feedback will bring them down. The `Ki_R1=1.2e-3` is now well below the uninhibited S3 peak, so feedback will be strong.

Final parameter set for Attempt 14:
*   `k1_R1 = 1.0e-4`
*   `Ki_R1 = 1.2e-3`
*   `n_R1 = 10.0`
*   `k5_R5 = 0.002`
*   `k2_R2 = 0.60`
*   `k6_R6 = 0.002`
*   `k3_R3 = 0.08`
*   `k4_R4 = 0.036`

This set aims for strong initial pulses that get curtailed by potent feedback, hopefully leading to sustained oscillations and lower troughs.

## Action

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

# Access experimental data
exp_data_df = experiment_history['iteration_0']

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

# Define species IDs
s1 = "id_xvpm"
s2 = "id_aa2n"
s3 = "id_xdgv"

# Parameter Set (Hypothesis 3, attempt 14)
k1_R1_val = 1.0e-4
Ki_R1_val = 1.2e-3
n_R1_val = 10.0

k5_R5_val = 0.002
k2_R2_val = 0.60

k6_R6_val = 0.002
k3_R3_val = 0.08

k4_R4_val = 0.036

# R1: -> id_xvpm (inhibited by id_xdgv)
reaction1 = model.createReaction(); reaction1.setId("R1_prod_xvpm_inhib"); reaction1.setReversible(False); reaction1.setFast(False)
prod1_s1 = reaction1.createProduct(); prod1_s1.setSpecies(s1); prod1_s1.setStoichiometry(1.0); prod1_s1.setConstant(False)
kl1 = reaction1.createKineticLaw()
p_k1_R1 = kl1.createParameter(); p_k1_R1.setId("k1_R1"); p_k1_R1.setValue(k1_R1_val); p_k1_R1.setConstant(True)
p_Ki_R1 = kl1.createParameter(); p_Ki_R1.setId("Ki_R1"); p_Ki_R1.setValue(Ki_R1_val); p_Ki_R1.setConstant(True)
p_n_R1 = kl1.createParameter(); p_n_R1.setId("n_R1"); p_n_R1.setValue(n_R1_val); p_n_R1.setConstant(True)
kl1.setMath(libsbml.parseL3Formula(f"k1_R1 / (1 + pow({s3} / Ki_R1, n_R1))"))

# R2: id_xvpm -> id_aa2n
reaction2 = model.createReaction(); reaction2.setId("R2_xvpm_to_aa2n"); reaction2.setReversible(False); reaction2.setFast(False)
reactant2_s1 = reaction2.createReactant(); reactant2_s1.setSpecies(s1); reactant2_s1.setStoichiometry(1.0); reactant2_s1.setConstant(False)
product2_s2 = reaction2.createProduct(); product2_s2.setSpecies(s2); product2_s2.setStoichiometry(1.0); product2_s2.setConstant(False)
kl2 = reaction2.createKineticLaw()
p_k2_R2 = kl2.createParameter(); p_k2_R2.setId("k2_R2"); p_k2_R2.setValue(k2_R2_val); p_k2_R2.setConstant(True)
kl2.setMath(libsbml.parseL3Formula(f"k2_R2 * {s1}"))

# R3: id_aa2n -> id_xdgv
reaction3 = model.createReaction(); reaction3.setId("R3_aa2n_to_xdgv"); reaction3.setReversible(False); reaction3.setFast(False)
reactant3_s2 = reaction3.createReactant(); reactant3_s2.setSpecies(s2); reactant3_s2.setStoichiometry(1.0); reactant3_s2.setConstant(False)
product3_s3 = reaction3.createProduct(); product3_s3.setSpecies(s3); product3_s3.setStoichiometry(1.0); product3_s3.setConstant(False)
kl3 = reaction3.createKineticLaw()
p_k3_R3 = kl3.createParameter(); p_k3_R3.setId("k3_R3"); p_k3_R3.setValue(k3_R3_val); p_k3_R3.setConstant(True)
kl3.setMath(libsbml.parseL3Formula(f"k3_R3 * {s2}"))

# R4: id_xdgv -> (degradation of S3)
reaction4 = model.createReaction(); reaction4.setId("R4_degrad_xdgv"); reaction4.setReversible(False); reaction4.setFast(False)
reactant4_s3 = reaction4.createReactant(); reactant4_s3.setSpecies(s3); reactant4_s3.setStoichiometry(1.0); reactant4_s3.setConstant(False)
kl4 = reaction4.createKineticLaw()
p_k4_R4 = kl4.createParameter(); p_k4_R4.setId("k4_R4"); p_k4_R4.setValue(k4_R4_val); p_k4_R4.setConstant(True)
kl4.setMath(libsbml.parseL3Formula(f"k4_R4 * {s3}"))

# R5: id_xvpm -> (degradation of S1)
reaction5 = model.createReaction(); reaction5.setId("R5_degrad_xvpm"); reaction5.setReversible(False); reaction5.setFast(False)
reactant5_s1 = reaction5.createReactant(); reactant5_s1.setSpecies(s1); reactant5_s1.setStoichiometry(1.0); reactant5_s1.setConstant(False)
kl5 = reaction5.createKineticLaw()
p_k5_R5 = kl5.createParameter(); p_k5_R5.setId("k5_R5"); p_k5_R5.setValue(k5_R5_val); p_k5_R5.setConstant(True)
kl5.setMath(libsbml.parseL3Formula(f"k5_R5 * {s1}"))

# R6: id_aa2n -> (degradation of S2)
reaction6 = model.createReaction(); reaction6.setId("R6_degrad_aa2n"); reaction6.setReversible(False); reaction6.setFast(False)
reactant6_s2 = reaction6.createReactant(); reactant6_s2.setSpecies(s2); reactant6_s2.setStoichiometry(1.0); reactant6_s2.setConstant(False)
kl6 = reaction6.createKineticLaw()
p_k6_R6 = kl6.createParameter(); p_k6_R6.setId("k6_R6"); p_k6_R6.setValue(k6_R6_val); p_k6_R6.setConstant(True)
kl6.setMath(libsbml.parseL3Formula(f"k6_R6 * {s2}"))

# Write the updated SBML to a string
writer = libsbml.SBMLWriter()
current_sbml_model_str_hyp3_att14 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_sbml_model_str_iteration_14", current_sbml_model_str_hyp3_att14)

# Simulate the model
sim_data_df_hyp3_att14 = simulate(current_sbml_model_str_hyp3_att14)
shared_variables.add("sim_data_df_iteration_14", sim_data_df_hyp3_att14)

# Print simulation results
print(f"Simulation data (Hypothesis 3, Attempt 14) head:\n{sim_data_df_hyp3_att14.head()}")
print(f"\nSimulation data (Hypothesis 3, Attempt 14) tail:\n{sim_data_df_hyp3_att14.tail(5)}")

# Print parameter values used
print(f"\nParameters used for Hypothesis 3, Attempt 14:")
print(f"k1_R1={k1_R1_val}, Ki_R1={Ki_R1_val}, n_R1={n_R1_val}")
print(f"k2_R2={k2_R2_val}, k3_R3={k3_R3_val}")
print(f"k4_R4={k4_R4_val}, k5_R5={k5_R5_val}, k6_R6={k6_R6_val}")

# Print simulated values at key experimental time points
sim_time_points = [0.0, 35.0, 46.0, 48.0, 60.0, 100.0, 150.0, 200.0, 300.0, 400.0, 500.0]
actual_sim_times_to_print = []
if not sim_data_df_hyp3_att14.empty:
    for t_target in sim_time_points:
        closest_time_idx = (np.abs(sim_data_df_hyp3_att14['Time'] - t_target)).argmin()
        actual_sim_times_to_print.append(sim_data_df_hyp3_att14['Time'].iloc[closest_time_idx])
    actual_sim_times_to_print = sorted(list(set(actual_sim_times_to_print)))
    print("\nSimulated values at specific time points (Hypothesis 3, Attempt 14):")
    print(sim_data_df_hyp3_att14[sim_data_df_hyp3_att14['Time'].isin(actual_sim_times_to_print)])
else:
    print("Simulated data for attempt 14 is empty.")

print("\nExperimental data for comparison at these times:")
actual_exp_times_to_print_exp = []
for t_target in sim_time_points:
    closest_time_idx = (np.abs(exp_data_df['Time'] - t_target)).argmin()
    actual_exp_times_to_print_exp.append(exp_data_df['Time'].iloc[closest_time_idx])
actual_exp_times_to_print_exp = sorted(list(set(actual_exp_times_to_print_exp)))
print(exp_data_df[exp_data_df['Time'].isin(actual_exp_times_to_print_exp)][['Time', 'id_xvpm', 'id_aa2n', 'id_xdgv']])
```
# Observation

## Code Stdout
Simulation data (Hypothesis 3, Attempt 14) head:
    id_xvpm   id_aa2n   id_xdgv  Time
0  0.000010  0.000010  0.000010   0.0
1  0.000051  0.000019  0.000010   0.5
2  0.000081  0.000038  0.000011   1.0
3  0.000103  0.000063  0.000013   1.5
4  0.000119  0.000093  0.000016   2.0

Simulation data (Hypothesis 3, Attempt 14) tail:
       id_xvpm  id_aa2n   id_xdgv   Time
996   0.000075  0.00055  0.001223  498.0
997   0.000075  0.00055  0.001223  498.5
998   0.000075  0.00055  0.001223  499.0
999   0.000075  0.00055  0.001223  499.5
1000  0.000075  0.00055  0.001223  500.0

Parameters used for Hypothesis 3, Attempt 14:
k1_R1=0.0001, Ki_R1=0.0012, n_R1=10.0
k2_R2=0.6, k3_R3=0.08
k4_R4=0.036, k5_R5=0.002, k6_R6=0.002

Simulated values at specific time points (Hypothesis 3, Attempt 14):
       id_xvpm   id_aa2n   id_xdgv   Time
0     0.000010  0.000010  0.000010    0.0
70    0.000057  0.000882  0.001330   35.0
92    0.000031  0.000508  0.001380   46.0
96    0.000034  0.000467  0.001359   48.0
120   0.000074  0.000429  0.001209   60.0
200   0.000071  0.000537  0.001234  100.0
300   0.000075  0.000549  0.001224  150.0
400   0.000075  0.000550  0.001223  200.0
600   0.000075  0.000550  0.001223  300.0
800   0.000075  0.000550  0.001223  400.0
1000  0.000075  0.000550  0.001223  500.0

Experimental data for comparison at these times:
       Time   id_xvpm   id_aa2n   id_xdgv
0       0.0  0.000010  0.000010  0.000010
70     35.0  0.000118  0.000675  0.001502
92     46.0  0.000080  0.000869  0.001954
96     48.0  0.000068  0.000861  0.001963
120    60.0  0.000017  0.000531  0.001629
200   100.0  0.000004  0.000068  0.000455
300   150.0  0.000074  0.000417  0.000969
400   200.0  0.000010  0.000234  0.000904
600   300.0  0.000024  0.000410  0.001145
800   400.0  0.000037  0.000467  0.001179
1000  500.0  0.000043  0.000452  0.001113

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml_model_str_iteration_2, sim_data_df_iteration_2, current_sbml_model_str_iteration_3, sim_data_df_iteration_3, current_sbml_model_str_iteration_4, sim_data_df_iteration_4, current_sbml_model_str_iteration_5, sim_data_df_iteration_5, current_sbml_model_str_iteration_6, sim_data_df_iteration_6, current_sbml_model_str_iteration_7, sim_data_df_iteration_7, current_sbml_model_str_iteration_8, sim_data_df_iteration_8, current_sbml_model_str_iteration_9, sim_data_df_iteration_9, current_sbml_model_str_iteration_10, sim_data_df_iteration_10, current_sbml_model_str_iteration_11, sim_data_df_iteration_11, current_sbml_model_str_iteration_12, sim_data_df_iteration_12, current_sbml_model_str_iteration_13, sim_data_df_iteration_13, current_sbml_model_str_iteration_14, sim_data_df_iteration_14)

### 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

Attempt 14 showed some interesting dynamics:
-   **S1 peak (sim: ~0.12e-4 at t<5s, then drops to 0.057e-3 at t=35s)**. The initial S1 peak is too fast and too high (uninhibited calculation was 1.66e-4). The value at t=35s (0.057e-3) is much lower than exp (0.118e-3).
-   **S2 peak (sim: 0.882e-3 at t=35s)** is very close to exp (0.869e-3 at t=46s). Good magnitude, but timing is a bit early.
-   **S3 peak (sim: ~0.001380 at t=46s)** is lower than exp (0.001963 at t=48s). Magnitude is off, timing is good.
-   **Oscillations:** The system shows an initial overshoot then settles to a steady state: S1=0.075e-3, S2=0.550e-3, S3=1.223e-3.
    -   Experimental values at t=500: S1=0.043e-3, S2=0.452e-3, S3=1.113e-3.
    -   Simulated S3 steady state is good. S1 and S2 steady states are too high.

**Key Issues from Attempt 14:**
1.  **S1 Dynamics:** The initial S1 peak is too high and too fast, then it drops too low by t=35s. This suggests `k1_R1` might be too high, or `k2_R2` (consumption of S1) is too fast initially, or feedback kicks in too hard on S1.
2.  **S3 Peak Magnitude Still Too Low:** Needs to be higher.
3.  **S1 and S2 Troughs/Steady State Too High:** Feedback isn't pulling S1 and S2 down enough.
4.  **Oscillations are Damped:** Still not getting sustained oscillations that match the later experimental data points.

The fact that S2 peak magnitude is good at t=35s, while S1 is already low, suggests S1 is being converted to S2 very efficiently. The S3 peak is forming around the right time but not high enough.

**Strategy for Attempt 15:**
The goal is to get a more controlled S1 peak, a higher S3 peak, and better trough behavior for S1/S2, hopefully leading to more sustained oscillations.

1.  **Refine S1 Dynamics:**
    *   Reduce `k1_R1` slightly to prevent S1 from overshooting too much initially.
    *   Perhaps slightly reduce `k2_R2` to slow S1 consumption, allowing S1 to peak a bit later and not fall so drastically before S2 peaks.
2.  **Boost S3 Peak:**
    *   Increase `k3_R3` (S2->S3 conversion) to drive more S2 into S3.
    *   Adjust `k4_R4` (S3 degradation) to maintain the S3/S2 peak ratio.
3.  **Improve Trough Depths:**
    *   `Ki_R1` is currently 1.2e-3. The experimental S3 at t=500 is 1.113e-3. This `Ki_R1` seems appropriate for the steady-state S3 level. The issue might be that S3 itself isn't falling low enough during oscillations to de-repress S1 production sufficiently for another strong pulse.
    *   The degradation rates `k5, k6, k4` are critical for how low the species can fall.

Let's try to make the cascade S1->S2->S3 slightly less aggressive initially for S1, but ensure S3 can reach its peak.

**Revised Parameters (Hypothesis 3, attempt 15):**
*   **Feedback & S1 Production (R1):**
    *   `k1_R1 = 0.85e-4` (was 1.0e-4). Reduced slightly.
    *   `Ki_R1 = 1.2e-3` (kept).
    *   `n_R1 = 10.0` (kept).
*   **S1 Dynamics (R2, R5):**
    *   `k5_R5 = 0.002` (kept).
    *   `k2_R2 = 0.50` (was 0.60). Reduced to slow S1 consumption and potentially delay S2 peak slightly.
        *   `S1_eff_deg = k5+k2 = 0.002+0.50 = 0.502`.
        *   `S1_peak_no_inhib ~ k1_R1 / S1_eff_deg = 0.85e-4 / 0.502 = 1.69e-4`. (Still high, feedback will reduce).
*   **S2 & S3 Dynamics (R3, R6, R4):**
    *   `k6_R6 = 0.002` (kept).
    *   `k3_R3 = 0.10` (was 0.08). Increased to boost S3 production.
        *   `S2_eff_deg = k6+k3 = 0.002+0.10 = 0.102`.
        *   Target S2 peak ~0.9e-3. `S2_peak ~ (k2/S2_eff_deg)*S1_actual_peak`.
        *   `k2/S2_eff_deg = 0.50/0.102 = 4.90`. If S1 actual peak is ~1.2e-4 (exp), S2 peak ~ 4.90 * 1.2e-4 = 5.88e-4. This is now too low.
            This shows the sensitivity. If `k2` is reduced, and `k3` increased, the S2/S1 ratio `k2/(k6+k3)` decreases.

Let's rethink. We need S2 peak ~0.87e-3 and S1 peak ~1.18e-3. S2/S1 ~ 7.3.
And S3 peak ~1.96e-3 and S2 peak ~0.87e-3. S3/S2 ~ 2.25.

Revised parameters for Attempt 15, focusing on these ratios and magnitudes:
*   `k1_R1 = 0.85e-4`
*   `Ki_R1 = 1.4e-3` (Slightly increased as target S3 peak is high)
*   `n_R1 = 10.0`
*   `k5_R5 = 0.002`
*   `k2_R2 = 0.65` (S1->S2)
    *   `S1_eff_deg = 0.002 + 0.65 = 0.652`.
    *   `S1_peak_no_inhib = 0.85e-4 / 0.652 = 1.30e-4`. (Good target for S1 peak).
*   `k6_R6 = 0.002`
*   `k3_R3 = ?`
    *   We need `k2/(k6+k3) ~ 7.3`. So `0.65 / (0.002+k3) = 7.3`.
    *   `0.65 = 7.3 * 0.002 + 7.3 * k3 = 0.0146 + 7.3 * k3`.
    *   `7.3 * k3 = 0.65 - 0.0146 = 0.6354`.
    *   `k3 = 0.6354 / 7.3 = 0.087`.
    *   So, `k3_R3 = 0.087`.
    *   `S2_peak_no_inhib ~ 7.3 * 1.30e-4 = 9.49e-4`. (Good target for S2 peak).
*   `k4_R4 = ?`
    *   We need `k3/k4 ~ 2.25`. So `0.087 / k4 = 2.25`.
    *   `k4 = 0.087 / 2.25 = 0.0386`.
    *   `S3_peak_no_inhib ~ 2.25 * 9.49e-4 = 2.135e-3`. (Good target for S3 peak).

This set seems more consistent with achieving the target uninhibited peak heights and ratios. The `Ki_R1=1.4e-3` is below the uninhibited S3 peak, so feedback will engage.

Final parameter set for Attempt 15:
*   `k1_R1 = 0.85e-4`
*   `Ki_R1 = 1.4e-3`
*   `n_R1 = 10.0`
*   `k5_R5 = 0.002`
*   `k2_R2 = 0.65`
*   `k6_R6 = 0.002`
*   `k3_R3 = 0.087`
*   `k4_R4 = 0.0386`

This is a careful retuning. The goal is to get the initial pulse right, hoping the feedback and slow degradations will then lead to more sustained oscillations and appropriate trough levels.

## Action

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

# Access experimental data
exp_data_df = experiment_history['iteration_0']

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

# Define species IDs
s1 = "id_xvpm"
s2 = "id_aa2n"
s3 = "id_xdgv"

# Parameter Set (Hypothesis 3, attempt 15)
k1_R1_val = 0.85e-4
Ki_R1_val = 1.4e-3
n_R1_val = 10.0

k5_R5_val = 0.002
k2_R2_val = 0.65

k6_R6_val = 0.002
k3_R3_val = 0.087

k4_R4_val = 0.0386

# R1: -> id_xvpm (inhibited by id_xdgv)
reaction1 = model.createReaction(); reaction1.setId("R1_prod_xvpm_inhib"); reaction1.setReversible(False); reaction1.setFast(False)
prod1_s1 = reaction1.createProduct(); prod1_s1.setSpecies(s1); prod1_s1.setStoichiometry(1.0); prod1_s1.setConstant(False)
kl1 = reaction1.createKineticLaw()
p_k1_R1 = kl1.createParameter(); p_k1_R1.setId("k1_R1"); p_k1_R1.setValue(k1_R1_val); p_k1_R1.setConstant(True)
p_Ki_R1 = kl1.createParameter(); p_Ki_R1.setId("Ki_R1"); p_Ki_R1.setValue(Ki_R1_val); p_Ki_R1.setConstant(True)
p_n_R1 = kl1.createParameter(); p_n_R1.setId("n_R1"); p_n_R1.setValue(n_R1_val); p_n_R1.setConstant(True)
kl1.setMath(libsbml.parseL3Formula(f"k1_R1 / (1 + pow({s3} / Ki_R1, n_R1))"))

# R2: id_xvpm -> id_aa2n
reaction2 = model.createReaction(); reaction2.setId("R2_xvpm_to_aa2n"); reaction2.setReversible(False); reaction2.setFast(False)
reactant2_s1 = reaction2.createReactant(); reactant2_s1.setSpecies(s1); reactant2_s1.setStoichiometry(1.0); reactant2_s1.setConstant(False)
product2_s2 = reaction2.createProduct(); product2_s2.setSpecies(s2); product2_s2.setStoichiometry(1.0); product2_s2.setConstant(False)
kl2 = reaction2.createKineticLaw()
p_k2_R2 = kl2.createParameter(); p_k2_R2.setId("k2_R2"); p_k2_R2.setValue(k2_R2_val); p_k2_R2.setConstant(True)
kl2.setMath(libsbml.parseL3Formula(f"k2_R2 * {s1}"))

# R3: id_aa2n -> id_xdgv
reaction3 = model.createReaction(); reaction3.setId("R3_aa2n_to_xdgv"); reaction3.setReversible(False); reaction3.setFast(False)
reactant3_s2 = reaction3.createReactant(); reactant3_s2.setSpecies(s2); reactant3_s2.setStoichiometry(1.0); reactant3_s2.setConstant(False)
product3_s3 = reaction3.createProduct(); product3_s3.setSpecies(s3); product3_s3.setStoichiometry(1.0); product3_s3.setConstant(False)
kl3 = reaction3.createKineticLaw()
p_k3_R3 = kl3.createParameter(); p_k3_R3.setId("k3_R3"); p_k3_R3.setValue(k3_R3_val); p_k3_R3.setConstant(True)
kl3.setMath(libsbml.parseL3Formula(f"k3_R3 * {s2}"))

# R4: id_xdgv -> (degradation of S3)
reaction4 = model.createReaction(); reaction4.setId("R4_degrad_xdgv"); reaction4.setReversible(False); reaction4.setFast(False)
reactant4_s3 = reaction4.createReactant(); reactant4_s3.setSpecies(s3); reactant4_s3.setStoichiometry(1.0); reactant4_s3.setConstant(False)
kl4 = reaction4.createKineticLaw()
p_k4_R4 = kl4.createParameter(); p_k4_R4.setId("k4_R4"); p_k4_R4.setValue(k4_R4_val); p_k4_R4.setConstant(True)
kl4.setMath(libsbml.parseL3Formula(f"k4_R4 * {s3}"))

# R5: id_xvpm -> (degradation of S1)
reaction5 = model.createReaction(); reaction5.setId("R5_degrad_xvpm"); reaction5.setReversible(False); reaction5.setFast(False)
reactant5_s1 = reaction5.createReactant(); reactant5_s1.setSpecies(s1); reactant5_s1.setStoichiometry(1.0); reactant5_s1.setConstant(False)
kl5 = reaction5.createKineticLaw()
p_k5_R5 = kl5.createParameter(); p_k5_R5.setId("k5_R5"); p_k5_R5.setValue(k5_R5_val); p_k5_R5.setConstant(True)
kl5.setMath(libsbml.parseL3Formula(f"k5_R5 * {s1}"))

# R6: id_aa2n -> (degradation of S2)
reaction6 = model.createReaction(); reaction6.setId("R6_degrad_aa2n"); reaction6.setReversible(False); reaction6.setFast(False)
reactant6_s2 = reaction6.createReactant(); reactant6_s2.setSpecies(s2); reactant6_s2.setStoichiometry(1.0); reactant6_s2.setConstant(False)
kl6 = reaction6.createKineticLaw()
p_k6_R6 = kl6.createParameter(); p_k6_R6.setId("k6_R6"); p_k6_R6.setValue(k6_R6_val); p_k6_R6.setConstant(True)
kl6.setMath(libsbml.parseL3Formula(f"k6_R6 * {s2}"))

# Write the updated SBML to a string
writer = libsbml.SBMLWriter()
current_sbml_model_str_hyp3_att15 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_sbml_model_str_iteration_15", current_sbml_model_str_hyp3_att15)

# Simulate the model
sim_data_df_hyp3_att15 = simulate(current_sbml_model_str_hyp3_att15)
shared_variables.add("sim_data_df_iteration_15", sim_data_df_hyp3_att15)

# Print simulation results
print(f"Simulation data (Hypothesis 3, Attempt 15) head:\n{sim_data_df_hyp3_att15.head()}")
print(f"\nSimulation data (Hypothesis 3, Attempt 15) tail:\n{sim_data_df_hyp3_att15.tail(5)}")

# Print parameter values used
print(f"\nParameters used for Hypothesis 3, Attempt 15:")
print(f"k1_R1={k1_R1_val}, Ki_R1={Ki_R1_val}, n_R1={n_R1_val}")
print(f"k2_R2={k2_R2_val}, k3_R3={k3_R3_val}")
print(f"k4_R4={k4_R4_val}, k5_R5={k5_R5_val}, k6_R6={k6_R6_val}")

# Print simulated values at key experimental time points
sim_time_points = [0.0, 35.0, 46.0, 48.0, 60.0, 100.0, 150.0, 200.0, 300.0, 400.0, 500.0]
actual_sim_times_to_print = []
if not sim_data_df_hyp3_att15.empty:
    for t_target in sim_time_points:
        closest_time_idx = (np.abs(sim_data_df_hyp3_att15['Time'] - t_target)).argmin()
        actual_sim_times_to_print.append(sim_data_df_hyp3_att15['Time'].iloc[closest_time_idx])
    actual_sim_times_to_print = sorted(list(set(actual_sim_times_to_print)))
    print("\nSimulated values at specific time points (Hypothesis 3, Attempt 15):")
    print(sim_data_df_hyp3_att15[sim_data_df_hyp3_att15['Time'].isin(actual_sim_times_to_print)])
else:
    print("Simulated data for attempt 15 is empty.")

print("\nExperimental data for comparison at these times:")
actual_exp_times_to_print_exp = []
for t_target in sim_time_points:
    closest_time_idx = (np.abs(exp_data_df['Time'] - t_target)).argmin()
    actual_exp_times_to_print_exp.append(exp_data_df['Time'].iloc[closest_time_idx])
actual_exp_times_to_print_exp = sorted(list(set(actual_exp_times_to_print_exp)))
print(exp_data_df[exp_data_df['Time'].isin(actual_exp_times_to_print_exp)][['Time', 'id_xvpm', 'id_aa2n', 'id_xdgv']])
```
# Observation

## Code Stdout
Simulation data (Hypothesis 3, Attempt 15) head:
    id_xvpm   id_aa2n   id_xdgv  Time
0  0.000010  0.000010  0.000010   0.0
1  0.000043  0.000018  0.000010   0.5
2  0.000068  0.000035  0.000011   1.0
3  0.000085  0.000058  0.000013   1.5
4  0.000098  0.000085  0.000016   2.0

Simulation data (Hypothesis 3, Attempt 15) tail:
       id_xvpm   id_aa2n   id_xdgv   Time
996   0.000081  0.000591  0.001333  498.0
997   0.000081  0.000591  0.001333  498.5
998   0.000081  0.000591  0.001333  499.0
999   0.000081  0.000591  0.001333  499.5
1000  0.000081  0.000591  0.001333  500.0

Parameters used for Hypothesis 3, Attempt 15:
k1_R1=8.5e-05, Ki_R1=0.0014, n_R1=10.0
k2_R2=0.65, k3_R3=0.087
k4_R4=0.0386, k5_R5=0.002, k6_R6=0.002

Simulated values at specific time points (Hypothesis 3, Attempt 15):
       id_xvpm   id_aa2n   id_xdgv   Time
0     0.000010  0.000010  0.000010    0.0
70    0.000115  0.000878  0.001192   35.0
92    0.000067  0.000719  0.001410   46.0
96    0.000063  0.000679  0.001422   48.0
120   0.000067  0.000531  0.001382   60.0
200   0.000080  0.000599  0.001337  100.0
300   0.000081  0.000592  0.001333  150.0
400   0.000081  0.000591  0.001333  200.0
600   0.000081  0.000591  0.001333  300.0
800   0.000081  0.000591  0.001333  400.0
1000  0.000081  0.000591  0.001333  500.0

Experimental data for comparison at these times:
       Time   id_xvpm   id_aa2n   id_xdgv
0       0.0  0.000010  0.000010  0.000010
70     35.0  0.000118  0.000675  0.001502
92     46.0  0.000080  0.000869  0.001954
96     48.0  0.000068  0.000861  0.001963
120    60.0  0.000017  0.000531  0.001629
200   100.0  0.000004  0.000068  0.000455
300   150.0  0.000074  0.000417  0.000969
400   200.0  0.000010  0.000234  0.000904
600   300.0  0.000024  0.000410  0.001145
800   400.0  0.000037  0.000467  0.001179
1000  500.0  0.000043  0.000452  0.001113

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml_model_str_iteration_2, sim_data_df_iteration_2, current_sbml_model_str_iteration_3, sim_data_df_iteration_3, current_sbml_model_str_iteration_4, sim_data_df_iteration_4, current_sbml_model_str_iteration_5, sim_data_df_iteration_5, current_sbml_model_str_iteration_6, sim_data_df_iteration_6, current_sbml_model_str_iteration_7, sim_data_df_iteration_7, current_sbml_model_str_iteration_8, sim_data_df_iteration_8, current_sbml_model_str_iteration_9, sim_data_df_iteration_9, current_sbml_model_str_iteration_10, sim_data_df_iteration_10, current_sbml_model_str_iteration_11, sim_data_df_iteration_11, current_sbml_model_str_iteration_12, sim_data_df_iteration_12, current_sbml_model_str_iteration_13, sim_data_df_iteration_13, current_sbml_model_str_iteration_14, sim_data_df_iteration_14, current_sbml_model_str_iteration_15, sim_data_df_iteration_15)

### 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

Attempt 15 yielded the following results:
-   **S1 peak (sim: 0.115e-3 at t=35s)** is very close to exp (0.118e-3 at t=35s). Excellent.
-   **S2 peak (sim: 0.878e-3 at t=35s)** magnitude is excellent compared to exp (0.869e-3 at t=46s), but S2 still peaks too early with S1.
-   **S3 peak (sim: ~0.001422 at t=48s)** is still lower than exp (0.001963 at t=48s). Magnitude is off, but timing is good.
-   **Oscillations:** The system damps to a steady state: S1=0.081e-3, S2=0.591e-3, S3=1.333e-3.
    -   Experimental values at t=500: S1=0.043e-3, S2=0.452e-3, S3=1.113e-3.
    -   All simulated steady states are too high. The system doesn't dip low enough.

**Key Issues from Attempt 15:**
1.  **S2 Peak Timing:** S2 peaks too early. This has been a persistent issue. It means S1 is converted to S2 too rapidly, or S2 is consumed/converted too slowly *relative to its production from S1*.
2.  **S3 Peak Magnitude Still Too Low:** Needs to be higher.
3.  **All Troughs/Steady States Too High:** Feedback isn't strong enough or effective enough to bring concentrations down to experimental trough levels.
4.  **Oscillations are Damped:** Still not getting sustained oscillations that match the later experimental data points.

**Strategy for Attempt 16:**
The model is getting closer on initial peak magnitudes for S1 and S2, but the dynamics (S2 lag, S3 peak height, sustained oscillations, trough depths) are still off.

1.  **Delay S2 Peak Relative to S1:**
    *   To make S2 peak *after* S1, the rate of S2 accumulation must be such that it continues to rise even as S1 starts to fall. This means `k2_R2 * S1` must be greater than `(k6_R6 + k3_R3) * S2` for a period after S1 peaks.
    *   Reducing `k2_R2` (S1->S2 conversion) is the primary way to introduce this lag. If `k2_R2` is reduced, `k1_R1` might need to be adjusted to maintain S1 peak.
2.  **Increase S3 Peak Magnitude:**
    *   Increase `k3_R3` (S2->S3 conversion).
    *   Adjust `k4_R4` (S3 degradation) to maintain the S3/S2 peak ratio.
3.  **Improve Trough Depths & Promote Oscillations:**
    *   Decrease `Ki_R1` (feedback threshold) further. If the experimental S3 trough goes significantly below 1.1e-3, then `Ki_R1` needs to be lower to keep feedback active and suppress S1 production during troughs.
    *   The Hill coefficient `n_R1=10` is already high.
    *   The degradation rates `k5, k6, k4` are crucial. If they are too high, oscillations damp. If too low, the system might be slow to recover or respond. The current values (0.002 for S1/S2 deg) are very low, which should favor oscillations but might make troughs shallower if production isn't shut off hard enough.

Let's try a more significant reduction in `k2_R2` to force a lag in S2.
If `k2_R2` is reduced, S1 will accumulate more (or be degraded by `k5_R5`).
Target S1 peak ~1.18e-4. Target S2 peak ~0.87e-3. Target S3 peak ~1.96e-3.

**Revised Parameters (Hypothesis 3, attempt 16):**
*   **Feedback & S1 Production (R1):**
    *   `k1_R1 = 0.75e-4` (was 0.85e-4). Reduced slightly, as S1 consumption will be slower.
    *   `Ki_R1 = 1.0e-3` (was 1.4e-3). Lowered significantly to make feedback very strong when S3 is around 1e-3 or higher. This should help with troughs.
    *   `n_R1 = 10.0` (kept).
*   **S1 Dynamics (R2, R5):**
    *   `k5_R5 = 0.002` (kept).
    *   `k2_R2 = 0.40` (was 0.65). Significantly reduced to delay S2 peak.
        *   `S1_eff_deg = k5+k2 = 0.002+0.40 = 0.402`.
        *   `S1_peak_no_inhib ~ k1_R1 / S1_eff_deg = 0.75e-4 / 0.402 = 1.86e-4`. (This is high, but strong feedback from low Ki should bring it down to ~1.2e-4).
*   **S2 & S3 Dynamics (R3, R6, R4):**
    *   `k6_R6 = 0.002` (kept).
    *   `k3_R3 = ?`
        *   We need S2/S1 peak ratio ~ 7.3. `k2/(k6+k3) ~ 7.3`.
        *   `0.40 / (0.002+k3) = 7.3`.
        *   `0.40 = 7.3 * 0.002 + 7.3 * k3 = 0.0146 + 7.3 * k3`.
        *   `7.3 * k3 = 0.40 - 0.0146 = 0.3854`.
        *   `k3 = 0.3854 / 7.3 = 0.0528`.
        *   So, `k3_R3 = 0.053`.
        *   `S2_peak_no_inhib ~ 7.3 * S1_actual_peak`. If S1 actual peak is ~1.2e-4, S2 peak ~ 8.76e-4. (Good).
    *   `k4_R4 = ?`
        *   We need S3/S2 peak ratio ~ 2.25. `k3/k4 ~ 2.25`.
        *   `0.053 / k4 = 2.25`.
        *   `k4 = 0.053 / 2.25 = 0.0235`.
        *   `S3_peak_no_inhib ~ 2.25 * 8.76e-4 = 1.97e-3`. (Good).

This set of parameters:
*   `k1_R1 = 0.75e-4`
*   `Ki_R1 = 1.0e-3`
*   `n_R1 = 10.0`
*   `k5_R5 = 0.002`
*   `k2_R2 = 0.40`
*   `k6_R6 = 0.002`
*   `k3_R3 = 0.053`
*   `k4_R4 = 0.0235`

This parameter set has a significantly lower `k2_R2` which should introduce the S2 lag. `Ki_R1` is lowered to improve trough depth. The uninhibited peak calculations suggest magnitudes could be in the right ballpark. The main uncertainty is whether this will produce sustained oscillations and the correct S2 lag.

## Action

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

# Access experimental data
exp_data_df = experiment_history['iteration_0']

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

# Define species IDs
s1 = "id_xvpm"
s2 = "id_aa2n"
s3 = "id_xdgv"

# Parameter Set (Hypothesis 3, attempt 16)
k1_R1_val = 0.75e-4
Ki_R1_val = 1.0e-3
n_R1_val = 10.0

k5_R5_val = 0.002
k2_R2_val = 0.40

k6_R6_val = 0.002
k3_R3_val = 0.053

k4_R4_val = 0.0235

# R1: -> id_xvpm (inhibited by id_xdgv)
reaction1 = model.createReaction(); reaction1.setId("R1_prod_xvpm_inhib"); reaction1.setReversible(False); reaction1.setFast(False)
prod1_s1 = reaction1.createProduct(); prod1_s1.setSpecies(s1); prod1_s1.setStoichiometry(1.0); prod1_s1.setConstant(False)
kl1 = reaction1.createKineticLaw()
p_k1_R1 = kl1.createParameter(); p_k1_R1.setId("k1_R1"); p_k1_R1.setValue(k1_R1_val); p_k1_R1.setConstant(True)
p_Ki_R1 = kl1.createParameter(); p_Ki_R1.setId("Ki_R1"); p_Ki_R1.setValue(Ki_R1_val); p_Ki_R1.setConstant(True)
p_n_R1 = kl1.createParameter(); p_n_R1.setId("n_R1"); p_n_R1.setValue(n_R1_val); p_n_R1.setConstant(True)
kl1.setMath(libsbml.parseL3Formula(f"k1_R1 / (1 + pow({s3} / Ki_R1, n_R1))"))

# R2: id_xvpm -> id_aa2n
reaction2 = model.createReaction(); reaction2.setId("R2_xvpm_to_aa2n"); reaction2.setReversible(False); reaction2.setFast(False)
reactant2_s1 = reaction2.createReactant(); reactant2_s1.setSpecies(s1); reactant2_s1.setStoichiometry(1.0); reactant2_s1.setConstant(False)
product2_s2 = reaction2.createProduct(); product2_s2.setSpecies(s2); product2_s2.setStoichiometry(1.0); product2_s2.setConstant(False)
kl2 = reaction2.createKineticLaw()
p_k2_R2 = kl2.createParameter(); p_k2_R2.setId("k2_R2"); p_k2_R2.setValue(k2_R2_val); p_k2_R2.setConstant(True)
kl2.setMath(libsbml.parseL3Formula(f"k2_R2 * {s1}"))

# R3: id_aa2n -> id_xdgv
reaction3 = model.createReaction(); reaction3.setId("R3_aa2n_to_xdgv"); reaction3.setReversible(False); reaction3.setFast(False)
reactant3_s2 = reaction3.createReactant(); reactant3_s2.setSpecies(s2); reactant3_s2.setStoichiometry(1.0); reactant3_s2.setConstant(False)
product3_s3 = reaction3.createProduct(); product3_s3.setSpecies(s3); product3_s3.setStoichiometry(1.0); product3_s3.setConstant(False)
kl3 = reaction3.createKineticLaw()
p_k3_R3 = kl3.createParameter(); p_k3_R3.setId("k3_R3"); p_k3_R3.setValue(k3_R3_val); p_k3_R3.setConstant(True)
kl3.setMath(libsbml.parseL3Formula(f"k3_R3 * {s2}"))

# R4: id_xdgv -> (degradation of S3)
reaction4 = model.createReaction(); reaction4.setId("R4_degrad_xdgv"); reaction4.setReversible(False); reaction4.setFast(False)
reactant4_s3 = reaction4.createReactant(); reactant4_s3.setSpecies(s3); reactant4_s3.setStoichiometry(1.0); reactant4_s3.setConstant(False)
kl4 = reaction4.createKineticLaw()
p_k4_R4 = kl4.createParameter(); p_k4_R4.setId("k4_R4"); p_k4_R4.setValue(k4_R4_val); p_k4_R4.setConstant(True)
kl4.setMath(libsbml.parseL3Formula(f"k4_R4 * {s3}"))

# R5: id_xvpm -> (degradation of S1)
reaction5 = model.createReaction(); reaction5.setId("R5_degrad_xvpm"); reaction5.setReversible(False); reaction5.setFast(False)
reactant5_s1 = reaction5.createReactant(); reactant5_s1.setSpecies(s1); reactant5_s1.setStoichiometry(1.0); reactant5_s1.setConstant(False)
kl5 = reaction5.createKineticLaw()
p_k5_R5 = kl5.createParameter(); p_k5_R5.setId("k5_R5"); p_k5_R5.setValue(k5_R5_val); p_k5_R5.setConstant(True)
kl5.setMath(libsbml.parseL3Formula(f"k5_R5 * {s1}"))

# R6: id_aa2n -> (degradation of S2)
reaction6 = model.createReaction(); reaction6.setId("R6_degrad_aa2n"); reaction6.setReversible(False); reaction6.setFast(False)
reactant6_s2 = reaction6.createReactant(); reactant6_s2.setSpecies(s2); reactant6_s2.setStoichiometry(1.0); reactant6_s2.setConstant(False)
kl6 = reaction6.createKineticLaw()
p_k6_R6 = kl6.createParameter(); p_k6_R6.setId("k6_R6"); p_k6_R6.setValue(k6_R6_val); p_k6_R6.setConstant(True)
kl6.setMath(libsbml.parseL3Formula(f"k6_R6 * {s2}"))

# Write the updated SBML to a string
writer = libsbml.SBMLWriter()
current_sbml_model_str_hyp3_att16 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_sbml_model_str_iteration_16", current_sbml_model_str_hyp3_att16)

# Simulate the model
sim_data_df_hyp3_att16 = simulate(current_sbml_model_str_hyp3_att16)
shared_variables.add("sim_data_df_iteration_16", sim_data_df_hyp3_att16)

# Print simulation results
print(f"Simulation data (Hypothesis 3, Attempt 16) head:\n{sim_data_df_hyp3_att16.head()}")
print(f"\nSimulation data (Hypothesis 3, Attempt 16) tail:\n{sim_data_df_hyp3_att16.tail(5)}")

# Print parameter values used
print(f"\nParameters used for Hypothesis 3, Attempt 16:")
print(f"k1_R1={k1_R1_val}, Ki_R1={Ki_R1_val}, n_R1={n_R1_val}")
print(f"k2_R2={k2_R2_val}, k3_R3={k3_R3_val}")
print(f"k4_R4={k4_R4_val}, k5_R5={k5_R5_val}, k6_R6={k6_R6_val}")

# Print simulated values at key experimental time points
sim_time_points = [0.0, 35.0, 46.0, 48.0, 60.0, 100.0, 150.0, 200.0, 300.0, 400.0, 500.0]
actual_sim_times_to_print = []
if not sim_data_df_hyp3_att16.empty:
    for t_target in sim_time_points:
        closest_time_idx = (np.abs(sim_data_df_hyp3_att16['Time'] - t_target)).argmin()
        actual_sim_times_to_print.append(sim_data_df_hyp3_att16['Time'].iloc[closest_time_idx])
    actual_sim_times_to_print = sorted(list(set(actual_sim_times_to_print)))
    print("\nSimulated values at specific time points (Hypothesis 3, Attempt 16):")
    print(sim_data_df_hyp3_att16[sim_data_df_hyp3_att16['Time'].isin(actual_sim_times_to_print)])
else:
    print("Simulated data for attempt 16 is empty.")

print("\nExperimental data for comparison at these times:")
actual_exp_times_to_print_exp = []
for t_target in sim_time_points:
    closest_time_idx = (np.abs(exp_data_df['Time'] - t_target)).argmin()
    actual_exp_times_to_print_exp.append(exp_data_df['Time'].iloc[closest_time_idx])
actual_exp_times_to_print_exp = sorted(list(set(actual_exp_times_to_print_exp)))
print(exp_data_df[exp_data_df['Time'].isin(actual_exp_times_to_print_exp)][['Time', 'id_xvpm', 'id_aa2n', 'id_xdgv']])
```
# Observation

## Code Stdout
Simulation data (Hypothesis 3, Attempt 16) head:
    id_xvpm   id_aa2n   id_xdgv  Time
0  0.000010  0.000010  0.000010   0.0
1  0.000042  0.000015  0.000010   0.5
2  0.000068  0.000026  0.000011   1.0
3  0.000090  0.000041  0.000011   1.5
4  0.000108  0.000059  0.000013   2.0

Simulation data (Hypothesis 3, Attempt 16) tail:
       id_xvpm   id_aa2n   id_xdgv   Time
996   0.000065  0.000472  0.001065  498.0
997   0.000065  0.000472  0.001065  498.5
998   0.000065  0.000472  0.001065  499.0
999   0.000065  0.000472  0.001065  499.5
1000  0.000065  0.000472  0.001065  500.0

Parameters used for Hypothesis 3, Attempt 16:
k1_R1=7.5e-05, Ki_R1=0.001, n_R1=10.0
k2_R2=0.4, k3_R3=0.053
k4_R4=0.0235, k5_R5=0.002, k6_R6=0.002

Simulated values at specific time points (Hypothesis 3, Attempt 16):
       id_xvpm   id_aa2n   id_xdgv   Time
0     0.000010  0.000010  0.000010    0.0
70    0.000142  0.001089  0.000962   35.0
92    0.000029  0.000823  0.001246   46.0
96    0.000022  0.000757  0.001271   48.0
120   0.000014  0.000443  0.001279   60.0
200   0.000090  0.000499  0.001010  100.0
300   0.000064  0.000447  0.001063  150.0
400   0.000063  0.000472  0.001068  200.0
600   0.000065  0.000472  0.001065  300.0
800   0.000065  0.000472  0.001065  400.0
1000  0.000065  0.000472  0.001065  500.0

Experimental data for comparison at these times:
       Time   id_xvpm   id_aa2n   id_xdgv
0       0.0  0.000010  0.000010  0.000010
70     35.0  0.000118  0.000675  0.001502
92     46.0  0.000080  0.000869  0.001954
96     48.0  0.000068  0.000861  0.001963
120    60.0  0.000017  0.000531  0.001629
200   100.0  0.000004  0.000068  0.000455
300   150.0  0.000074  0.000417  0.000969
400   200.0  0.000010  0.000234  0.000904
600   300.0  0.000024  0.000410  0.001145
800   400.0  0.000037  0.000467  0.001179
1000  500.0  0.000043  0.000452  0.001113

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml_model_str_iteration_2, sim_data_df_iteration_2, current_sbml_model_str_iteration_3, sim_data_df_iteration_3, current_sbml_model_str_iteration_4, sim_data_df_iteration_4, current_sbml_model_str_iteration_5, sim_data_df_iteration_5, current_sbml_model_str_iteration_6, sim_data_df_iteration_6, current_sbml_model_str_iteration_7, sim_data_df_iteration_7, current_sbml_model_str_iteration_8, sim_data_df_iteration_8, current_sbml_model_str_iteration_9, sim_data_df_iteration_9, current_sbml_model_str_iteration_10, sim_data_df_iteration_10, current_sbml_model_str_iteration_11, sim_data_df_iteration_11, current_sbml_model_str_iteration_12, sim_data_df_iteration_12, current_sbml_model_str_iteration_13, sim_data_df_iteration_13, current_sbml_model_str_iteration_14, sim_data_df_iteration_14, current_sbml_model_str_iteration_15, sim_data_df_iteration_15, current_sbml_model_str_iteration_16, sim_data_df_iteration_16)

### 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

Attempt 16 showed some interesting dynamics, including oscillations, but still has discrepancies:
-   **S1 peak (sim: 0.142e-3 at t=35s)** is higher than exp (0.118e-3 at t=35s). Then it drops very sharply.
-   **S2 peak (sim: 1.089e-3 at t=35s)** is higher than exp (0.869e-3 at t=46s) and peaks too early with S1.
-   **S3 peak (sim: ~0.001279 at t=60s)** is lower than exp (0.001963 at t=48s). Magnitude is off, timing a bit slow.
-   **Oscillations:** The system does oscillate, but the troughs for S1 (e.g., 0.014e-3 at t=60s) are very low, potentially too low compared to exp. S2 also has deep troughs. The system settles to S1=0.065e-3, S2=0.472e-3, S3=1.065e-3.
    -   Experimental values at t=500: S1=0.043e-3, S2=0.452e-3, S3=1.113e-3.
    -   Simulated S2 and S3 steady states are very good. S1 steady state is a bit high.

**Key Issues from Attempt 16:**
1.  **S1/S2 Initial Peaks Too High & S2 Timing:** S1 and S2 overshoot their experimental peaks, and S2 still peaks with S1. This suggests `k1_R1` is too high, and/or `k2_R2` is still too fast relative to S1's own dynamics or S2's consumption.
2.  **S3 Peak Magnitude Still Too Low.**
3.  **S1 Troughs Might Be Too Deep Initially:** While we want lower troughs, the first S1 trough at t=60s (0.014e-3) is very close to the experimental S1 value at t=100s (0.004e-3) which is a later, deeper trough.
4.  **Sustained Oscillations:** The model *is* oscillating, which is a major step forward. The period and damping need refinement.

**Strategy for Attempt 17:**
The main goal is to temper the initial S1/S2 overshoot, get the S2 peak lag, improve S3 peak height, and refine the oscillatory behavior (amplitude and mean).

1.  **Control S1/S2 Initial Peaks and S2 Lag:**
    *   Reduce `k1_R1` to lower the initial S1 pulse.
    *   Further reduce `k2_R2` (S1->S2 conversion) to ensure S1 peaks and starts falling *before* S2 peaks. This is crucial for the lag.
2.  **Boost S3 Peak:**
    *   Increase `k3_R3` (S2->S3 conversion).
    *   Adjust `k4_R4` (S3 degradation) to maintain the S3/S2 peak ratio.
3.  **Refine Feedback for Troughs/Steady State:**
    *   `Ki_R1 = 1.0e-3` was effective in creating oscillations. The S3 steady state (1.065e-3) is close to this `Ki_R1` and also close to the experimental S3 at t=500 (1.113e-3). This seems like a good `Ki_R1` value.
    *   The S1 steady state (0.065e-3) is still higher than exp (0.043e-3). This implies that at S3_ss ~1.0e-3, S1 production is not suppressed enough. This could mean `k1_R1` is still a bit too high for the steady state, or `k5_R5` (S1 degradation) is too low.

Let's try to get the S2 lag. This requires `k2_R2` to be slow enough that S1 can peak and decline while `k2_R2*S1` is still building up S2.

**Revised Parameters (Hypothesis 3, attempt 17):**
*   **Feedback & S1 Production (R1):**
    *   `k1_R1 = 0.60e-4` (was 0.75e-4). Reduced significantly to lower S1 initial peak.
    *   `Ki_R1 = 1.0e-3` (kept).
    *   `n_R1 = 10.0` (kept).
*   **S1 Dynamics (R2, R5):**
    *   `k5_R5 = 0.002` (kept).
    *   `k2_R2 = 0.25` (was 0.40). Significantly reduced to delay S2 peak.
        *   `S1_eff_deg = k5+k2 = 0.002+0.25 = 0.252`.
        *   `S1_peak_no_inhib ~ k1_R1 / S1_eff_deg = 0.60e-4 / 0.252 = 2.38e-4`. (This is very high, strong feedback is essential).
*   **S2 & S3 Dynamics (R3, R6, R4):**
    *   `k6_R6 = 0.002` (kept).
    *   `k3_R3 = ?`
        *   Target S2/S1 peak ratio ~ 7.3. `k2/(k6+k3) ~ 7.3`.
        *   `0.25 / (0.002+k3) = 7.3`.
        *   `0.25 = 7.3 * 0.002 + 7.3 * k3 = 0.0146 + 7.3 * k3`.
        *   `7.3 * k3 = 0.25 - 0.0146 = 0.2354`.
        *   `k3 = 0.2354 / 7.3 = 0.0322`.
        *   So, `k3_R3 = 0.032`.
        *   `S2_peak_no_inhib ~ 7.3 * S1_actual_peak`. If S1 actual peak (with feedback) is ~1.2e-4, S2 peak ~ 8.76e-4.
    *   `k4_R4 = ?`
        *   Target S3/S2 peak ratio ~ 2.25. `k3/k4 ~ 2.25`.
        *   `0.032 / k4 = 2.25`.
        *   `k4 = 0.032 / 2.25 = 0.0142`.
        *   `S3_peak_no_inhib ~ 2.25 * 8.76e-4 = 1.97e-3`.

This set of parameters:
*   `k1_R1 = 0.60e-4`
*   `Ki_R1 = 1.0e-3`
*   `n_R1 = 10.0`
*   `k5_R5 = 0.002`
*   `k2_R2 = 0.25`
*   `k6_R6 = 0.002`
*   `k3_R3 = 0.032`
*   `k4_R4 = 0.0142`

The uninhibited S1 peak is calculated high (2.38e-4), but `Ki_R1` is low (1.0e-3). If S3 rises quickly to ~1.0e-3, it will strongly inhibit S1 production. This might give a sharp S1 peak around the experimental value, then a fall. The much slower `k2_R2` (0.25) should create a significant lag for S2. The downstream rates `k3, k4` are also slower now. This might slow down S3 too much.

The lifetimes are now:
S1: `1/0.252 ~ 4s` (very short)
S2: `1/(0.002+0.032) = 1/0.034 ~ 29s`
S3: `1/0.0142 ~ 70s`
This sequence of S1 < S2 < S3 lifetimes is generally good for oscillations in a 3-stage loop.

Let's check the steady state for S1 with these parameters, assuming S3_ss ~ Ki_R1 = 1.0e-3.
At S3=1.0e-3, S1 prod rate = `k1_R1 / (1 + (1.0e-3/1.0e-3)^10) = k1_R1 / 2 = 0.30e-4`.
`S1_ss = Prod_rate / (k5+k2) = 0.30e-4 / 0.252 = 1.19e-4`. This is too high for S1 steady state (target ~0.04e-3).
This means `Ki_R1=1.0e-3` is too high if we want S1_ss to be low when S3_ss is around 1.0e-3.
To get S1_ss lower, S1 production at S3_ss must be lower. This means `k1_R1` must be lower, or feedback stronger (lower `Ki_R1` or higher `n`).

If `Ki_R1` is set to, say, `0.8e-3`, and S3_ss is `1.0e-3`.
Then `(S3/Ki)^n = (1.0/0.8)^10 = (1.25)^10 = 9.3`.
S1 prod rate = `k1_R1 / (1+9.3) = k1_R1 / 10.3`.
`S1_ss = (k1_R1/10.3) / (k5+k2) = (0.60e-4/10.3) / 0.252 = (0.058e-4) / 0.252 = 0.23e-4`.
This S1_ss is now lower than exp (0.043e-3). This might be too strong.

Let's stick with `Ki_R1 = 1.0e-3` for now and see the dynamic response. The steady state calculation assumes it *settles*. If it oscillates with wide amplitude, the average might be different, and troughs could be low.

The main focus of this attempt is to achieve the S2 lag and see if oscillations become more sustained with these slower cascade rates (especially k2, k3, k4 being slower than in Att16).

## Action

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

# Access experimental data
exp_data_df = experiment_history['iteration_0']

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

# Define species IDs
s1 = "id_xvpm"
s2 = "id_aa2n"
s3 = "id_xdgv"

# Parameter Set (Hypothesis 3, attempt 17)
k1_R1_val = 0.60e-4
Ki_R1_val = 1.0e-3
n_R1_val = 10.0

k5_R5_val = 0.002
k2_R2_val = 0.25

k6_R6_val = 0.002
k3_R3_val = 0.032

k4_R4_val = 0.0142

# R1: -> id_xvpm (inhibited by id_xdgv)
reaction1 = model.createReaction(); reaction1.setId("R1_prod_xvpm_inhib"); reaction1.setReversible(False); reaction1.setFast(False)
prod1_s1 = reaction1.createProduct(); prod1_s1.setSpecies(s1); prod1_s1.setStoichiometry(1.0); prod1_s1.setConstant(False)
kl1 = reaction1.createKineticLaw()
p_k1_R1 = kl1.createParameter(); p_k1_R1.setId("k1_R1"); p_k1_R1.setValue(k1_R1_val); p_k1_R1.setConstant(True)
p_Ki_R1 = kl1.createParameter(); p_Ki_R1.setId("Ki_R1"); p_Ki_R1.setValue(Ki_R1_val); p_Ki_R1.setConstant(True)
p_n_R1 = kl1.createParameter(); p_n_R1.setId("n_R1"); p_n_R1.setValue(n_R1_val); p_n_R1.setConstant(True)
kl1.setMath(libsbml.parseL3Formula(f"k1_R1 / (1 + pow({s3} / Ki_R1, n_R1))"))

# R2: id_xvpm -> id_aa2n
reaction2 = model.createReaction(); reaction2.setId("R2_xvpm_to_aa2n"); reaction2.setReversible(False); reaction2.setFast(False)
reactant2_s1 = reaction2.createReactant(); reactant2_s1.setSpecies(s1); reactant2_s1.setStoichiometry(1.0); reactant2_s1.setConstant(False)
product2_s2 = reaction2.createProduct(); product2_s2.setSpecies(s2); product2_s2.setStoichiometry(1.0); product2_s2.setConstant(False)
kl2 = reaction2.createKineticLaw()
p_k2_R2 = kl2.createParameter(); p_k2_R2.setId("k2_R2"); p_k2_R2.setValue(k2_R2_val); p_k2_R2.setConstant(True)
kl2.setMath(libsbml.parseL3Formula(f"k2_R2 * {s1}"))

# R3: id_aa2n -> id_xdgv
reaction3 = model.createReaction(); reaction3.setId("R3_aa2n_to_xdgv"); reaction3.setReversible(False); reaction3.setFast(False)
reactant3_s2 = reaction3.createReactant(); reactant3_s2.setSpecies(s2); reactant3_s2.setStoichiometry(1.0); reactant3_s2.setConstant(False)
product3_s3 = reaction3.createProduct(); product3_s3.setSpecies(s3); product3_s3.setStoichiometry(1.0); product3_s3.setConstant(False)
kl3 = reaction3.createKineticLaw()
p_k3_R3 = kl3.createParameter(); p_k3_R3.setId("k3_R3"); p_k3_R3.setValue(k3_R3_val); p_k3_R3.setConstant(True)
kl3.setMath(libsbml.parseL3Formula(f"k3_R3 * {s2}"))

# R4: id_xdgv -> (degradation of S3)
reaction4 = model.createReaction(); reaction4.setId("R4_degrad_xdgv"); reaction4.setReversible(False); reaction4.setFast(False)
reactant4_s3 = reaction4.createReactant(); reactant4_s3.setSpecies(s3); reactant4_s3.setStoichiometry(1.0); reactant4_s3.setConstant(False)
kl4 = reaction4.createKineticLaw()
p_k4_R4 = kl4.createParameter(); p_k4_R4.setId("k4_R4"); p_k4_R4.setValue(k4_R4_val); p_k4_R4.setConstant(True)
kl4.setMath(libsbml.parseL3Formula(f"k4_R4 * {s3}"))

# R5: id_xvpm -> (degradation of S1)
reaction5 = model.createReaction(); reaction5.setId("R5_degrad_xvpm"); reaction5.setReversible(False); reaction5.setFast(False)
reactant5_s1 = reaction5.createReactant(); reactant5_s1.setSpecies(s1); reactant5_s1.setStoichiometry(1.0); reactant5_s1.setConstant(False)
kl5 = reaction5.createKineticLaw()
p_k5_R5 = kl5.createParameter(); p_k5_R5.setId("k5_R5"); p_k5_R5.setValue(k5_R5_val); p_k5_R5.setConstant(True)
kl5.setMath(libsbml.parseL3Formula(f"k5_R5 * {s1}"))

# R6: id_aa2n -> (degradation of S2)
reaction6 = model.createReaction(); reaction6.setId("R6_degrad_aa2n"); reaction6.setReversible(False); reaction6.setFast(False)
reactant6_s2 = reaction6.createReactant(); reactant6_s2.setSpecies(s2); reactant6_s2.setStoichiometry(1.0); reactant6_s2.setConstant(False)
kl6 = reaction6.createKineticLaw()
p_k6_R6 = kl6.createParameter(); p_k6_R6.setId("k6_R6"); p_k6_R6.setValue(k6_R6_val); p_k6_R6.setConstant(True)
kl6.setMath(libsbml.parseL3Formula(f"k6_R6 * {s2}"))

# Write the updated SBML to a string
writer = libsbml.SBMLWriter()
current_sbml_model_str_hyp3_att17 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_sbml_model_str_iteration_17", current_sbml_model_str_hyp3_att17)

# Simulate the model
sim_data_df_hyp3_att17 = simulate(current_sbml_model_str_hyp3_att17)
shared_variables.add("sim_data_df_iteration_17", sim_data_df_hyp3_att17)

# Print simulation results
print(f"Simulation data (Hypothesis 3, Attempt 17) head:\n{sim_data_df_hyp3_att17.head()}")
print(f"\nSimulation data (Hypothesis 3, Attempt 17) tail:\n{sim_data_df_hyp3_att17.tail(5)}")

# Print parameter values used
print(f"\nParameters used for Hypothesis 3, Attempt 17:")
print(f"k1_R1={k1_R1_val}, Ki_R1={Ki_R1_val}, n_R1={n_R1_val}")
print(f"k2_R2={k2_R2_val}, k3_R3={k3_R3_val}")
print(f"k4_R4={k4_R4_val}, k5_R5={k5_R5_val}, k6_R6={k6_R6_val}")

# Print simulated values at key experimental time points
sim_time_points = [0.0, 35.0, 46.0, 48.0, 60.0, 100.0, 150.0, 200.0, 300.0, 400.0, 500.0]
actual_sim_times_to_print = []
if not sim_data_df_hyp3_att17.empty:
    for t_target in sim_time_points:
        closest_time_idx = (np.abs(sim_data_df_hyp3_att17['Time'] - t_target)).argmin()
        actual_sim_times_to_print.append(sim_data_df_hyp3_att17['Time'].iloc[closest_time_idx])
    actual_sim_times_to_print = sorted(list(set(actual_sim_times_to_print)))
    print("\nSimulated values at specific time points (Hypothesis 3, Attempt 17):")
    print(sim_data_df_hyp3_att17[sim_data_df_hyp3_att17['Time'].isin(actual_sim_times_to_print)])
else:
    print("Simulated data for attempt 17 is empty.")

print("\nExperimental data for comparison at these times:")
actual_exp_times_to_print_exp = []
for t_target in sim_time_points:
    closest_time_idx = (np.abs(exp_data_df['Time'] - t_target)).argmin()
    actual_exp_times_to_print_exp.append(exp_data_df['Time'].iloc[closest_time_idx])
actual_exp_times_to_print_exp = sorted(list(set(actual_exp_times_to_print_exp)))
print(exp_data_df[exp_data_df['Time'].isin(actual_exp_times_to_print_exp)][['Time', 'id_xvpm', 'id_aa2n', 'id_xdgv']])
```
# Observation

## Code Stdout
Simulation data (Hypothesis 3, Attempt 17) head:
    id_xvpm   id_aa2n   id_xdgv  Time
0  0.000010  0.000010  0.000010   0.0
1  0.000037  0.000013  0.000010   0.5
2  0.000061  0.000019  0.000010   1.0
3  0.000082  0.000027  0.000011   1.5
4  0.000100  0.000038  0.000011   2.0

Simulation data (Hypothesis 3, Attempt 17) tail:
       id_xvpm   id_aa2n  id_xdgv   Time
996   0.000066  0.000488   0.0011  498.0
997   0.000066  0.000488   0.0011  498.5
998   0.000066  0.000488   0.0011  499.0
999   0.000066  0.000488   0.0011  499.5
1000  0.000066  0.000488   0.0011  500.0

Parameters used for Hypothesis 3, Attempt 17:
k1_R1=6e-05, Ki_R1=0.001, n_R1=10.0
k2_R2=0.25, k3_R3=0.032
k4_R4=0.0142, k5_R5=0.002, k6_R6=0.002

Simulated values at specific time points (Hypothesis 3, Attempt 17):
       id_xvpm   id_aa2n   id_xdgv   Time
0     0.000010  0.000010  0.000010    0.0
70    0.000238  0.001141  0.000588   35.0
92    0.000207  0.001311  0.000907   46.0
96    0.000187  0.001320  0.000965   48.0
120   0.000048  0.001138  0.001257   60.0
200   0.000011  0.000352  0.001335  100.0
300   0.000102  0.000460  0.001024  150.0
400   0.000053  0.000507  0.001135  200.0
600   0.000065  0.000496  0.001105  300.0
800   0.000066  0.000490  0.001100  400.0
1000  0.000066  0.000488  0.001100  500.0

Experimental data for comparison at these times:
       Time   id_xvpm   id_aa2n   id_xdgv
0       0.0  0.000010  0.000010  0.000010
70     35.0  0.000118  0.000675  0.001502
92     46.0  0.000080  0.000869  0.001954
96     48.0  0.000068  0.000861  0.001963
120    60.0  0.000017  0.000531  0.001629
200   100.0  0.000004  0.000068  0.000455
300   150.0  0.000074  0.000417  0.000969
400   200.0  0.000010  0.000234  0.000904
600   300.0  0.000024  0.000410  0.001145
800   400.0  0.000037  0.000467  0.001179
1000  500.0  0.000043  0.000452  0.001113

## Reminder

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



### Available Global Variables
- `input_sbml_string`: Original incomplete model
- `experiment_history`: Results from all previous experiments (iteration_0)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml_model_str_iteration_2, sim_data_df_iteration_2, current_sbml_model_str_iteration_3, sim_data_df_iteration_3, current_sbml_model_str_iteration_4, sim_data_df_iteration_4, current_sbml_model_str_iteration_5, sim_data_df_iteration_5, current_sbml_model_str_iteration_6, sim_data_df_iteration_6, current_sbml_model_str_iteration_7, sim_data_df_iteration_7, current_sbml_model_str_iteration_8, sim_data_df_iteration_8, current_sbml_model_str_iteration_9, sim_data_df_iteration_9, current_sbml_model_str_iteration_10, sim_data_df_iteration_10, current_sbml_model_str_iteration_11, sim_data_df_iteration_11, current_sbml_model_str_iteration_12, sim_data_df_iteration_12, current_sbml_model_str_iteration_13, sim_data_df_iteration_13, current_sbml_model_str_iteration_14, sim_data_df_iteration_14, current_sbml_model_str_iteration_15, sim_data_df_iteration_15, current_sbml_model_str_iteration_16, sim_data_df_iteration_16, current_sbml_model_str_iteration_17, sim_data_df_iteration_17)

### 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

Attempt 17 showed significant improvement in oscillatory behavior and some peak characteristics:
-   **S1 peak (sim: 0.238e-3 at t=35s)** is too high compared to exp (0.118e-3 at t=35s).
-   **S2 peak (sim: 1.320e-3 at t=48s)** is too high compared to exp (0.869e-3 at t=46s), but the lag is now present (S2 peaks after S1 starts to fall from its max).
-   **S3 peak (sim: ~0.001335 at t=100s)** is lower than exp (0.001963 at t=48s) and significantly delayed.
-   **Oscillations:** The system oscillates! The troughs for S1 (e.g., 0.011e-3 at t=100s) are very low, matching the later experimental S1 values. S2 also has deep troughs. The system settles to S1=0.066e-3, S2=0.488e-3, S3=1.100e-3.
    -   Experimental values at t=500: S1=0.043e-3, S2=0.452e-3, S3=1.113e-3.
    -   Simulated S2 and S3 steady states are excellent. S1 steady state is a bit high.

**Key Issues from Attempt 17:**
1.  **S1/S2 Initial Peaks Too High:** The first peaks of S1 and S2 overshoot.
2.  **S3 Peak Magnitude Too Low and Too Slow.**
3.  **S1 Steady State a Bit High.**

**Strategy for Attempt 18 (Final Tuning):**
The model is now oscillating and capturing many features. This will be about fine-tuning.
1.  **Reduce S1/S2 Initial Peaks:**
    *   Slightly reduce `k1_R1` (S1 production) to lower the initial S1 pulse. This should also lower the S2 peak.
2.  **Improve S3 Peak (Height and Timing):**
    *   Increase `k3_R3` (S2->S3 conversion) to make S3 rise faster and potentially higher from S2.
    *   Adjust `k4_R4` (S3 degradation) to ensure S3 peak isn't too short-lived and to maintain the S3/S2 ratio. A slightly slower `k4_R4` might allow S3 to accumulate more.
3.  **Refine S1 Steady State:**
    *   If S1 steady state is still high after adjusting `k1_R1`, a very slight increase in `k5_R5` (S1 degradation) could be considered, but this might damp oscillations. Alternatively, ensuring `Ki_R1` is perfectly tuned for the S3 steady state is key. `Ki_R1 = 1.0e-3` seems to give good S3 steady state, so the issue might be the S1 production rate when S3 is at that level.

Let's try to primarily adjust `k1_R1` for S1/S2 peaks, and `k3_R3`/`k4_R4` for S3 peak.

**Revised Parameters (Hypothesis 3, attempt 18 - Fine-tuning):**
*   **Feedback & S1 Production (R1):**
    *   `k1_R1 = 0.45e-4` (was 0.60e-4). Reduced to lower S1/S2 initial peaks.
    *   `Ki_R1 = 1.0e-3` (kept).
    *   `n_R1 = 10.0` (kept).
*   **S1 Dynamics (R2, R5):**
    *   `k5_R5 = 0.002` (kept).
    *   `k2_R2 = 0.25` (kept from Att17, as it produced S2 lag).
        *   `S1_eff_deg = 0.252`.
        *   `S1_peak_no_inhib ~ k1_R1 / S1_eff_deg = 0.45e-4 / 0.252 = 1.78e-4`. (Still a bit high, but feedback will act).
*   **S2 & S3 Dynamics (R3, R6, R4):**
    *   `k6_R6 = 0.002` (kept).
    *   `k3_R3 = 0.040` (was 0.032). Increased to boost S3 production and speed.
        *   `S2_eff_deg = k6+k3 = 0.002+0.040 = 0.042`.
        *   S2/S1 ratio `k2/(k6+k3) = 0.25/0.042 = 5.95`. (Was ~7.3. This will lower S2 peak relative to S1).
            If S1 actual peak is ~1.2e-4, S2 peak ~ 5.95 * 1.2e-4 = 7.14e-4. (Target S2 peak ~8.7e-4. This is now lower).
            This is a trade-off. Increasing `k3` to help S3 also increases S2 consumption.
    *   Let's re-calculate `k3` to maintain S2/S1 ratio while `k2` is fixed.
        If `k2/(k6+k3) = 7.3` (target ratio), and `k2=0.25`, `k6=0.002`.
        `0.25 / (0.002+k3) = 7.3` => `0.25 = 0.0146 + 7.3*k3` => `7.3*k3 = 0.2354` => `k3 = 0.0322`. This was the Att17 value.
        To increase S3 production without lowering S2 peak too much, we need to increase `k3` but also ensure S2 itself is high enough.
        Perhaps the S2/S1 ratio target was too aggressive if S2 needs to feed S3 more.

Let's try a moderate increase in `k3` and adjust `k4` accordingly.
*   `k3_R3 = 0.045` (was 0.032 in Att17, a significant increase).
*   `k4_R4 = ?`
    *   Target S3/S2 peak ratio ~ 2.25. `k3/k4 ~ 2.25`.
    *   `0.045 / k4 = 2.25` => `k4 = 0.045 / 2.25 = 0.020`. (was 0.0142 in Att17).
    *   S2_eff_deg = `0.002 + 0.045 = 0.047`.
    *   S2/S1 ratio = `0.25 / 0.047 = 5.32`. (Lower than target 7.3).
    *   If S1 actual peak ~1.0e-4 (due to lower k1), S2 peak ~ 5.3e-4. This is too low.

This shows that simply increasing `k3` to boost S3 will hurt S2 if S2 production isn't also increased.
The problem is coupled. The previous attempt (Att17) had good S2 peak height (1.3e-3, too high) but S3 was too low (0.0013).

Let's use Att17 as a base and try to:
1.  Reduce `k1_R1` to lower S1 and S2 initial overshoots.
2.  Increase `k3_R3` moderately to boost S3, and adjust `k4_R4`.
3.  Keep `Ki_R1` low for good troughs.

Parameters from Att17:
`k1=0.60e-4, Ki=1.0e-3, n=10, k5=0.002, k2=0.25, k6=0.002, k3=0.032, k4=0.0142`
Simulated S1 peak ~0.24e-3 (target 0.12e-3) -> Too high
Simulated S2 peak ~1.3e-3 (target 0.87e-3) -> Too high
Simulated S3 peak ~1.3e-3 (target 1.96e-3) -> Too low

New approach for Attempt 18:
*   `k1_R1 = 0.40e-4` (Drastically reduce k1 to control S1/S2 overshoot)
*   `Ki_R1 = 1.0e-3` (Keep for good troughs)
*   `n_R1 = 10.0`
*   `k5_R5 = 0.002`
*   `k2_R2 = 0.25` (Keep, as S2 lag was appearing)
    *   `S1_peak_no_inhib = 0.40e-4 / (0.002+0.25) = 0.40e-4 / 0.252 = 1.58e-4`. (Still high, but lower than Att17's 2.38e-4. Feedback should bring it to ~1.2e-4)
*   `k6_R6 = 0.002`
*   `k3_R3 = 0.050` (was 0.032. Increase significantly to boost S3)
    *   S2/S1 ratio `k2/(k6+k3) = 0.25 / (0.002+0.050) = 0.25/0.052 = 4.8`. (This will make S2 peak lower relative to S1 peak).
        If S1 actual peak ~1.2e-4, S2 peak ~ 4.8 * 1.2e-4 = 5.76e-4. (Target 8.7e-4. This is too low).
This is the core difficulty.

If the model structure is correct, there should be a parameter set. The oscillations in Att17 were the most promising so far in terms of shape and trough depth. The main issue was S1/S2 overshooting and S3 being too low/slow.

Let's try to preserve the oscillatory nature of Att17 but scale down the initial pulse and boost S3.
Att17 params: `k1=0.6e-4, Ki=1e-3, n=10, k5=0.002, k2=0.25, k6=0.002, k3=0.032, k4=0.0142`
S1_ss ~0.066e-3, S2_ss ~0.488e-3, S3_ss ~1.1e-3 (S2, S3 steady states are good)

To reduce S1 overshoot from Att17 (peak 0.238e-3) to ~0.12e-3:
Reduce `k1_R1`. Let `k1_R1 = 0.35e-4` (was 0.60e-4).
This will roughly halve the S1 peak if feedback scales similarly.
If S1 peak is halved, S2 peak will also be halved (from 1.3e-3 to ~0.65e-3). This is too low for S2.

This suggests that `k2_R2` in Att17 (0.25) might be too high if `k1_R1` needs to be low to control S1 peak.
A slower `k2_R2` would mean S1 is consumed less rapidly, so `k1_R1` wouldn't need to be as high for a given S1 peak.

Let's try to get the S1 steady state right first.
Exp S1_ss ~0.043e-3. Sim S1_ss (Att17) ~0.066e-3.
To reduce S1_ss from 0.066 to 0.043 (factor of 0.65), assuming S3_ss and feedback strength are similar, `k1_R1` should be reduced by this factor.
New `k1_R1 = 0.6e-4 * 0.65 = 0.39e-4`. Let's use `0.4e-4`.
Keep other rates from Att17 for now, and see how this `k1_R1` affects peaks and oscillations.
This primarily targets the steady state of S1.

Parameters for Attempt 18 (based on Att17, reduced k1):
*   `k1_R1 = 0.40e-4` (was 0.60e-4 in Att17)
*   `Ki_R1 = 1.0e-3` (kept from Att17)
*   `n_R1 = 10.0` (kept from Att17)
*   `k5_R5 = 0.002` (kept from Att17)
*   `k2_R2 = 0.25` (kept from Att17)
*   `k6_R6 = 0.002` (kept from Att17)
*   `k3_R3 = 0.032` (kept from Att17)
*   `k4_R4 = 0.0142` (kept from Att17)

This is a direct test of reducing `k1_R1` from the Att17 set.
Expected: S1/S2 peaks will be lower. S3 peak will be lower. Oscillations might be less vigorous but troughs might be better relative to peaks. S1 steady state should be closer to experimental.

## Action

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

# Access experimental data
exp_data_df = experiment_history['iteration_0']

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

# Define species IDs
s1 = "id_xvpm"
s2 = "id_aa2n"
s3 = "id_xdgv"

# Parameter Set (Hypothesis 3, attempt 18 - Based on Att17, reduced k1)
k1_R1_val = 0.40e-4  # Reduced from Att17's 0.60e-4
Ki_R1_val = 1.0e-3   # Kept from Att17
n_R1_val = 10.0      # Kept from Att17

k5_R5_val = 0.002    # Kept from Att17
k2_R2_val = 0.25     # Kept from Att17

k6_R6_val = 0.002    # Kept from Att17
k3_R3_val = 0.032    # Kept from Att17

k4_R4_val = 0.0142   # Kept from Att17

# R1: -> id_xvpm (inhibited by id_xdgv)
reaction1 = model.createReaction(); reaction1.setId("R1_prod_xvpm_inhib"); reaction1.setReversible(False); reaction1.setFast(False)
prod1_s1 = reaction1.createProduct(); prod1_s1.setSpecies(s1); prod1_s1.setStoichiometry(1.0); prod1_s1.setConstant(False)
kl1 = reaction1.createKineticLaw()
p_k1_R1 = kl1.createParameter(); p_k1_R1.setId("k1_R1"); p_k1_R1.setValue(k1_R1_val); p_k1_R1.setConstant(True)
p_Ki_R1 = kl1.createParameter(); p_Ki_R1.setId("Ki_R1"); p_Ki_R1.setValue(Ki_R1_val); p_Ki_R1.setConstant(True)
p_n_R1 = kl1.createParameter(); p_n_R1.setId("n_R1"); p_n_R1.setValue(n_R1_val); p_n_R1.setConstant(True)
kl1.setMath(libsbml.parseL3Formula(f"k1_R1 / (1 + pow({s3} / Ki_R1, n_R1))"))

# R2: id_xvpm -> id_aa2n
reaction2 = model.createReaction(); reaction2.setId("R2_xvpm_to_aa2n"); reaction2.setReversible(False); reaction2.setFast(False)
reactant2_s1 = reaction2.createReactant(); reactant2_s1.setSpecies(s1); reactant2_s1.setStoichiometry(1.0); reactant2_s1.setConstant(False)
product2_s2 = reaction2.createProduct(); product2_s2.setSpecies(s2); product2_s2.setStoichiometry(1.0); product2_s2.setConstant(False)
kl2 = reaction2.createKineticLaw()
p_k2_R2 = kl2.createParameter(); p_k2_R2.setId("k2_R2"); p_k2_R2.setValue(k2_R2_val); p_k2_R2.setConstant(True)
kl2.setMath(libsbml.parseL3Formula(f"k2_R2 * {s1}"))

# R3: id_aa2n -> id_xdgv
reaction3 = model.createReaction(); reaction3.setId("R3_aa2n_to_xdgv"); reaction3.setReversible(False); reaction3.setFast(False)
reactant3_s2 = reaction3.createReactant(); reactant3_s2.setSpecies(s2); reactant3_s2.setStoichiometry(1.0); reactant3_s2.setConstant(False)
product3_s3 = reaction3.createProduct(); product3_s3.setSpecies(s3); product3_s3.setStoichiometry(1.0); product3_s3.setConstant(False)
kl3 = reaction3.createKineticLaw()
p_k3_R3 = kl3.createParameter(); p_k3_R3.setId("k3_R3"); p_k3_R3.setValue(k3_R3_val); p_k3_R3.setConstant(True)
kl3.setMath(libsbml.parseL3Formula(f"k3_R3 * {s2}"))

# R4: id_xdgv -> (degradation of S3)
reaction4 = model.createReaction(); reaction4.setId("R4_degrad_xdgv"); reaction4.setReversible(False); reaction4.setFast(False)
reactant4_s3 = reaction4.createReactant(); reactant4_s3.setSpecies(s3); reactant4_s3.setStoichiometry(1.0); reactant4_s3.setConstant(False)
kl4 = reaction4.createKineticLaw()
p_k4_R4 = kl4.createParameter(); p_k4_R4.setId("k4_R4"); p_k4_R4.setValue(k4_R4_val); p_k4_R4.setConstant(True)
kl4.setMath(libsbml.parseL3Formula(f"k4_R4 * {s3}"))

# R5: id_xvpm -> (degradation of S1)
reaction5 = model.createReaction(); reaction5.setId("R5_degrad_xvpm"); reaction5.setReversible(False); reaction5.setFast(False)
reactant5_s1 = reaction5.createReactant(); reactant5_s1.setSpecies(s1); reactant5_s1.setStoichiometry(1.0); reactant5_s1.setConstant(False)
kl5 = reaction5.createKineticLaw()
p_k5_R5 = kl5.createParameter(); p_k5_R5.setId("k5_R5"); p_k5_R5.setValue(k5_R5_val); p_k5_R5.setConstant(True)
kl5.setMath(libsbml.parseL3Formula(f"k5_R5 * {s1}"))

# R6: id_aa2n -> (degradation of S2)
reaction6 = model.createReaction(); reaction6.setId("R6_degrad_aa2n"); reaction6.setReversible(False); reaction6.setFast(False)
reactant6_s2 = reaction6.createReactant(); reactant6_s2.setSpecies(s2); reactant6_s2.setStoichiometry(1.0); reactant6_s2.setConstant(False)
kl6 = reaction6.createKineticLaw()
p_k6_R6 = kl6.createParameter(); p_k6_R6.setId("k6_R6"); p_k6_R6.setValue(k6_R6_val); p_k6_R6.setConstant(True)
kl6.setMath(libsbml.parseL3Formula(f"k6_R6 * {s2}"))

# Write the updated SBML to a string
writer = libsbml.SBMLWriter()
current_sbml_model_str_hyp3_att18 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("current_sbml_model_str_iteration_18", current_sbml_model_str_hyp3_att18)

# Simulate the model
sim_data_df_hyp3_att18 = simulate(current_sbml_model_str_hyp3_att18)
shared_variables.add("sim_data_df_iteration_18", sim_data_df_hyp3_att18)

# Print simulation results
print(f"Simulation data (Hypothesis 3, Attempt 18) head:\n{sim_data_df_hyp3_att18.head()}")
print(f"\nSimulation data (Hypothesis 3, Attempt 18) tail:\n{sim_data_df_hyp3_att18.tail(5)}")

# Print parameter values used
print(f"\nParameters used for Hypothesis 3, Attempt 18:")
print(f"k1_R1={k1_R1_val}, Ki_R1={Ki_R1_val}, n_R1={n_R1_val}")
print(f"k2_R2={k2_R2_val}, k3_R3={k3_R3_val}")
print(f"k4_R4={k4_R4_val}, k5_R5={k5_R5_val}, k6_R6={k6_R6_val}")

# Print simulated values at key experimental time points
sim_time_points = [0.0, 35.0, 46.0, 48.0, 60.0, 100.0, 150.0, 200.0, 300.0, 400.0, 500.0]
actual_sim_times_to_print = []
if not sim_data_df_hyp3_att18.empty:
    for t_target in sim_time_points:
        closest_time_idx = (np.abs(sim_data_df_hyp3_att18['Time'] - t_target)).argmin()
        actual_sim_times_to_print.append(sim_data_df_hyp3_att18['Time'].iloc[closest_time_idx])
    actual_sim_times_to_print = sorted(list(set(actual_sim_times_to_print)))
    print("\nSimulated values at specific time points (Hypothesis 3, Attempt 18):")
    print(sim_data_df_hyp3_att18[sim_data_df_hyp3_att18['Time'].isin(actual_sim_times_to_print)])
else:
    print("Simulated data for attempt 18 is empty.")

print("\nExperimental data for comparison at these times:")
actual_exp_times_to_print_exp = []
for t_target in sim_time_points:
    closest_time_idx = (np.abs(exp_data_df['Time'] - t_target)).argmin()
    actual_exp_times_to_print_exp.append(exp_data_df['Time'].iloc[closest_time_idx])
actual_exp_times_to_print_exp = sorted(list(set(actual_exp_times_to_print_exp)))
print(exp_data_df[exp_data_df['Time'].isin(actual_exp_times_to_print_exp)][['Time', 'id_xvpm', 'id_aa2n', 'id_xdgv']])
```
# Observation

## Code Stdout
Simulation data (Hypothesis 3, Attempt 18) head:
    id_xvpm   id_aa2n   id_xdgv  Time
0  0.000010  0.000010  0.000010   0.0
1  0.000028  0.000012  0.000010   0.5
2  0.000043  0.000016  0.000010   1.0
3  0.000057  0.000022  0.000010   1.5
4  0.000069  0.000030  0.000011   2.0

Simulation data (Hypothesis 3, Attempt 18) tail:
       id_xvpm   id_aa2n   id_xdgv   Time
996   0.000063  0.000463  0.001043  498.0
997   0.000063  0.000463  0.001043  498.5
998   0.000063  0.000463  0.001043  499.0
999   0.000063  0.000463  0.001043  499.5
1000  0.000063  0.000463  0.001043  500.0

Parameters used for Hypothesis 3, Attempt 18:
k1_R1=4e-05, Ki_R1=0.001, n_R1=10.0
k2_R2=0.25, k3_R3=0.032
k4_R4=0.0142, k5_R5=0.002, k6_R6=0.002

Simulated values at specific time points (Hypothesis 3, Attempt 18):
       id_xvpm   id_aa2n   id_xdgv   Time
0     0.000010  0.000010  0.000010    0.0
70    0.000159  0.000763  0.000398   35.0
92    0.000158  0.000889  0.000611   46.0
96    0.000158  0.000907  0.000651   48.0
120   0.000138  0.000975  0.000884   60.0
200   0.000019  0.000479  0.001223  100.0
300   0.000074  0.000382  0.001002  150.0
400   0.000062  0.000505  0.001051  200.0
600   0.000064  0.000469  0.001040  300.0
800   0.000063  0.000463  0.001042  400.0
1000  0.000063  0.000463  0.001043  500.0

Experimental data for comparison at these times:
       Time   id_xvpm   id_aa2n   id_xdgv
0       0.0  0.000010  0.000010  0.000010
70     35.0  0.000118  0.000675  0.001502
92     46.0  0.000080  0.000869  0.001954
96     48.0  0.000068  0.000861  0.001963
120    60.0  0.000017  0.000531  0.001629
200   100.0  0.000004  0.000068  0.000455
300   150.0  0.000074  0.000417  0.000969
400   200.0  0.000010  0.000234  0.000904
600   300.0  0.000024  0.000410  0.001145
800   400.0  0.000037  0.000467  0.001179
1000  500.0  0.000043  0.000452  0.001113

## 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)
- `shared_variables`: all variables you have saved from the previous iterations. (current_sbml_model_str_iteration_2, sim_data_df_iteration_2, current_sbml_model_str_iteration_3, sim_data_df_iteration_3, current_sbml_model_str_iteration_4, sim_data_df_iteration_4, current_sbml_model_str_iteration_5, sim_data_df_iteration_5, current_sbml_model_str_iteration_6, sim_data_df_iteration_6, current_sbml_model_str_iteration_7, sim_data_df_iteration_7, current_sbml_model_str_iteration_8, sim_data_df_iteration_8, current_sbml_model_str_iteration_9, sim_data_df_iteration_9, current_sbml_model_str_iteration_10, sim_data_df_iteration_10, current_sbml_model_str_iteration_11, sim_data_df_iteration_11, current_sbml_model_str_iteration_12, sim_data_df_iteration_12, current_sbml_model_str_iteration_13, sim_data_df_iteration_13, current_sbml_model_str_iteration_14, sim_data_df_iteration_14, current_sbml_model_str_iteration_15, sim_data_df_iteration_15, current_sbml_model_str_iteration_16, sim_data_df_iteration_16, current_sbml_model_str_iteration_17, sim_data_df_iteration_17, current_sbml_model_str_iteration_18, sim_data_df_iteration_18)

### 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

Attempt 18 (which was a simple reduction of `k1_R1` from Att17's parameters) showed:
-   **S1 peak (sim: 0.159e-3 at t=35s)** is higher than exp (0.118e-3 at t=35s).
-   **S2 peak (sim: 0.907e-3 at t=48s)** magnitude is good compared to exp (0.869e-3 at t=46s), and there's a lag! S2 peaks after S1. This is a key improvement.
-   **S3 peak (sim: ~0.001223 at t=100s)** is lower than exp (0.001963 at t=48s) and significantly delayed.
-   **Oscillations:** The system oscillates. The S1 steady state (0.063e-3) is still a bit high (exp 0.043e-3). S2 steady state (0.463e-3) is excellent (exp 0.452e-3). S3 steady state (1.043e-3) is excellent (exp 1.113e-3).

**Key Issues from Attempt 18:**
1.  **S1 Initial Peak Too High.**
2.  **S3 Peak Magnitude Too Low and Too Slow.**
3.  **S1 Steady State a Bit High.** (Though S2/S3 steady states are good).

The S2 lag is a major breakthrough. The steady-state behavior for S2 and S3 is also very good. The main remaining problems are the S1 initial peak, S3 initial peak, and S1 steady state.

**Strategy for Attempt 19 (Final Model Candidate):**
This is likely the last iteration for parameter tuning. The focus is on getting the S1 peak down, S3 peak up and faster, and S1 steady state a bit lower, while preserving the good S2/S3 steady states and oscillations.

1.  **Reduce S1 Initial Peak:**
    *   Slightly reduce `k1_R1` again.
2.  **Improve S3 Peak (Height and Timing):**
    *   Increase `k3_R3` (S2->S3 conversion) to make S3 rise faster and higher.
    *   Adjust `k4_R4` (S3 degradation) to ensure S3 peak isn't too short-lived and to maintain the S3/S2 ratio. A slightly slower `k4_R4` relative to the new `k3_R3` might help S3 accumulate more for its peak, but this could also slow its fall.
3.  **Refine S1 Steady State:**
    *   The `Ki_R1 = 1.0e-3` seems to be working well for S3 steady state. If S1 steady state is still high after `k1_R1` adjustment, it's tricky. Increasing `k5_R5` (S1 degradation) is an option but can damp oscillations.

Let's use parameters from Attempt 18 as the base:
Att18: `k1=0.4e-4, Ki=1e-3, n=10, k5=0.002, k2=0.25, k6=0.002, k3=0.032, k4=0.0142`

**Revised Parameters (Hypothesis 3, attempt 19):**
*   **Feedback & S1 Production (R1):**
    *   `k1_R1 = 0.35e-4` (was 0.40e-4). Further slight reduction.
    *   `Ki_R1 = 1.0e-3` (kept).
    *   `n_R1 = 10.0` (kept).
*   **S1 Dynamics (R2, R5):**
    *   `k5_R5 = 0.002` (kept).
    *   `k2_R2 = 0.25` (kept, as S2 lag was good).
        *   `S1_eff_deg = 0.252`.
        *   `S1_peak_no_inhib ~ k1_R1 / S1_eff_deg = 0.35e-4 / 0.252 = 1.39e-4`. (Target S1 peak ~1.2e-4. This should be good with feedback).
*   **S2 & S3 Dynamics (R3, R6, R4):**
    *   `k6_R6 = 0.002` (kept).
    *   `k3_R3 = 0.045` (was 0.032. Increased significantly to boost S3).
        *   S2/S1 ratio `k2/(k6+k3) = 0.25 / (0.002+0.045) = 0.25/0.047 = 5.32`.
        *   If S1 actual peak ~1.2e-4, S2 peak ~ 5.32 * 1.2e-4 = 6.38e-4. (Target S2 peak ~8.7e-4. This is now too low due to increased k3).
            This highlights the coupling. Increasing `k3` to help S3 hurts S2's peak height if S2 production (`k2*S1`) doesn't compensate.

We need S2 peak to be around 0.87e-3.
If S1 peak is ~1.2e-4, then `k2/(k6+k3)` needs to be `0.87e-3 / 1.2e-4 = 7.25`.
Current `k2=0.25`, `k6=0.002`. So `0.25 / (0.002+k3) = 7.25`.
`0.25 = 7.25 * 0.002 + 7.25 * k3 = 0.0145 + 7.25 * k3`.
`7.25 * k3 = 0.25 - 0.0145 = 0.2355`.
`k3 = 0.2355 / 7.25 = 0.0325`.
This brings `k3` back to the Att18 value (0.032). So, just increasing `k3` from Att18 values will lower S2 peak.

This means if we want a higher S3 peak, and S2 peak is already good (from Att18 sim: 0.9e-3), we should primarily adjust `k3` and `k4` to get S3 higher, without changing `k2` too much from Att18.
Att18 S2 peak was 0.907e-3 at t=48s. S1 peak was 0.159e-3 at t=35s. S2/S1 ratio was ~5.7.
Exp S2/S1 ratio ~7.3.

Let's try to get the S2/S1 ratio closer to 7.3, and S3/S2 ratio to 2.25.
Use `k1_R1 = 0.35e-4`.
`S1_peak_no_inhib = 1.39e-4`. Assume actual S1 peak with feedback is `S1_act_pk ~ 1.1e-4`.
Target `S2_act_pk = 7.3 * S1_act_pk = 7.3 * 1.1e-4 = 8.03e-4`.
`k2/(k6+k3) = 7.3`. Let `k2 = 0.25` (from Att18). `0.25 / (0.002+k3) = 7.3` => `k3 = 0.0322` (same as Att18).
This means the S2/S1 ratio was already implicitly set by these `k2, k3, k6` values in Att18, but the actual S1 peak was higher.

The issue might be that the *uninhibited* S1 peak calculation is misleading when feedback is very strong.
The simulation from Att18: S1 peak 0.159e-3, S2 peak 0.907e-3. S2/S1 = 5.7.
Experimental: S1 peak 0.118e-3, S2 peak 0.869e-3. S2/S1 = 7.36.

To increase S2/S1 ratio from 5.7 to 7.3 (factor of 1.28):
Either increase `k2` or decrease `(k6+k3)`.
Let's try increasing `k2` slightly from Att18's 0.25.
And increase `k3` significantly to boost S3.

Final attempt parameters for submission (Attempt 19):
*   `k1_R1 = 0.38e-4` (Fine-tuned from 0.40e-4 of Att18, aiming for S1 peak ~0.12-0.14e-3)
*   `Ki_R1 = 0.95e-3` (Slightly lower than Att18's 1.0e-3, to get lower S1 steady state)
*   `n_R1 = 10.0`
*   `k5_R5 = 0.002`
*   `k2_R2 = 0.28` (was 0.25 in Att18. Increased to boost S2/S1 ratio)
*   `k6_R6 = 0.002`
*   `k3_R3 = 0.045` (was 0.032 in Att18. Increased significantly to boost S3 height & speed)
*   `k4_R4 = 0.020` (k3/k4 = 2.25. Was 0.0142 in Att18. Increased due to k3 increase)

This set tries to:
- Lower S1 peak slightly with `k1_R1`.
- Increase S2 peak relative to S1 with `k2_R2`.
- Significantly boost S3 production with `k3_R3`.
- Maintain S3/S2 ratio with `k4_R4`.
- Sharpen feedback for lower S1 steady state with `Ki_R1`.

This is the best educated guess based on trends. The system is highly non-linear.

## Action

### Submit
```python
import libsbml

# Parameter Set (Hypothesis 3, attempt 19 - Final Candidate)
k1_R1_val = 0.38e-4
Ki_R1_val = 0.95e-3
n_R1_val = 10.0

k5_R5_val = 0.002    # S1 degradation
k2_R2_val = 0.28     # S1 -> S2

k6_R6_val = 0.002    # S2 degradation
k3_R3_val = 0.045    # S2 -> S3

k4_R4_val = 0.020    # S3 degradation (k3/k4 = 0.045/0.020 = 2.25)

sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()

s1 = "id_xvpm"
s2 = "id_aa2n"
s3 = "id_xdgv"

# R1: -> id_xvpm (inhibited by id_xdgv)
reaction1 = model.createReaction(); reaction1.setId("R1_prod_xvpm_inhib"); reaction1.setReversible(False); reaction1.setFast(False)
prod1_s1 = reaction1.createProduct(); prod1_s1.setSpecies(s1); prod1_s1.setStoichiometry(1.0); prod1_s1.setConstant(False)
kl1 = reaction1.createKineticLaw()
p_k1_R1 = kl1.createParameter(); p_k1_R1.setId("k1_R1"); p_k1_R1.setValue(k1_R1_val); p_k1_R1.setConstant(True)
p_Ki_R1 = kl1.createParameter(); p_Ki_R1.setId("Ki_R1"); p_Ki_R1.setValue(Ki_R1_val); p_Ki_R1.setConstant(True)
p_n_R1 = kl1.createParameter(); p_n_R1.setId("n_R1"); p_n_R1.setValue(n_R1_val); p_n_R1.setConstant(True)
kl1.setMath(libsbml.parseL3Formula(f"k1_R1 / (1 + pow({s3} / Ki_R1, n_R1))"))

# R2: id_xvpm -> id_aa2n
reaction2 = model.createReaction(); reaction2.setId("R2_xvpm_to_aa2n"); reaction2.setReversible(False); reaction2.setFast(False)
reactant2_s1 = reaction2.createReactant(); reactant2_s1.setSpecies(s1); reactant2_s1.setStoichiometry(1.0); reactant2_s1.setConstant(False)
product2_s2 = reaction2.createProduct(); product2_s2.setSpecies(s2); product2_s2.setStoichiometry(1.0); product2_s2.setConstant(False)
kl2 = reaction2.createKineticLaw()
p_k2_R2 = kl2.createParameter(); p_k2_R2.setId("k2_R2"); p_k2_R2.setValue(k2_R2_val); p_k2_R2.setConstant(True)
kl2.setMath(libsbml.parseL3Formula(f"k2_R2 * {s1}"))

# R3: id_aa2n -> id_xdgv
reaction3 = model.createReaction(); reaction3.setId("R3_aa2n_to_xdgv"); reaction3.setReversible(False); reaction3.setFast(False)
reactant3_s2 = reaction3.createReactant(); reactant3_s2.setSpecies(s2); reactant3_s2.setStoichiometry(1.0); reactant3_s2.setConstant(False)
product3_s3 = reaction3.createProduct(); product3_s3.setSpecies(s3); product3_s3.setStoichiometry(1.0); product3_s3.setConstant(False)
kl3 = reaction3.createKineticLaw()
p_k3_R3 = kl3.createParameter(); p_k3_R3.setId("k3_R3"); p_k3_R3.setValue(k3_R3_val); p_k3_R3.setConstant(True)
kl3.setMath(libsbml.parseL3Formula(f"k3_R3 * {s2}"))

# R4: id_xdgv -> (degradation of S3)
reaction4 = model.createReaction(); reaction4.setId("R4_degrad_xdgv"); reaction4.setReversible(False); reaction4.setFast(False)
reactant4_s3 = reaction4.createReactant(); reactant4_s3.setSpecies(s3); reactant4_s3.setStoichiometry(1.0); reactant4_s3.setConstant(False)
kl4 = reaction4.createKineticLaw()
p_k4_R4 = kl4.createParameter(); p_k4_R4.setId("k4_R4"); p_k4_R4.setValue(k4_R4_val); p_k4_R4.setConstant(True)
kl4.setMath(libsbml.parseL3Formula(f"k4_R4 * {s3}"))

# R5: id_xvpm -> (degradation of S1)
reaction5 = model.createReaction(); reaction5.setId("R5_degrad_xvpm"); reaction5.setReversible(False); reaction5.setFast(False)
reactant5_s1 = reaction5.createReactant(); reactant5_s1.setSpecies(s1); reactant5_s1.setStoichiometry(1.0); reactant5_s1.setConstant(False)
kl5 = reaction5.createKineticLaw()
p_k5_R5 = kl5.createParameter(); p_k5_R5.setId("k5_R5"); p_k5_R5.setValue(k5_R5_val); p_k5_R5.setConstant(True)
kl5.setMath(libsbml.parseL3Formula(f"k5_R5 * {s1}"))

# R6: id_aa2n -> (degradation of S2)
reaction6 = model.createReaction(); reaction6.setId("R6_degrad_aa2n"); reaction6.setReversible(False); reaction6.setFast(False)
reactant6_s2 = reaction6.createReactant(); reactant6_s2.setSpecies(s2); reactant6_s2.setStoichiometry(1.0); reactant6_s2.setConstant(False)
kl6 = reaction6.createKineticLaw()
p_k6_R6 = kl6.createParameter(); p_k6_R6.setId("k6_R6"); p_k6_R6.setValue(k6_R6_val); p_k6_R6.setConstant(True)
kl6.setMath(libsbml.parseL3Formula(f"k6_R6 * {s2}"))

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

print("Final SBML model generated with parameters from Attempt 19.")
# The actual simulation and comparison would happen after submission.
# This is the model I believe is the best fit.
```
