
                # 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" level="2" version="1">
  <model id="id_y16y">
    <listOfUnitDefinitions>
      <unitDefinition id="id_0yqd">
        <listOfUnits>
          <unit kind="mole" scale="-6" multiplier="1" offset="0"/>
        </listOfUnits>
      </unitDefinition>
      <unitDefinition id="id_4fev">
        <listOfUnits>
          <unit kind="second" multiplier="60" offset="0"/>
        </listOfUnits>
      </unitDefinition>
    </listOfUnitDefinitions>
    <listOfCompartments>
      <compartment id="id_52il" size="1" outside="id_cyas"/>
      <compartment id="id_rt6a" size="1"/>
      <compartment id="id_cyas" size="1" outside="id_rt6a"/>
    </listOfCompartments>
    <listOfSpecies>
      <species id="id_x589" compartment="id_cyas" initialConcentration="0"/>
      <species id="id_r7xb" compartment="id_52il" initialConcentration="0.36"/>
      <species id="id_22ta" compartment="id_cyas" initialConcentration="0"/>
      <species id="id_plsk" compartment="id_cyas" initialConcentration="0"/>
      <species id="id_0nyd" compartment="id_rt6a" initialConcentration="0"/>
    </listOfSpecies>
  </model>
</sbml>


                ## Max iterations
                20

                Format your response according to the instructions in the system message.
                
## Thoughts
The initial SBML model defines species and compartments but contains no reactions. My first step is to understand the baseline behavior of the system. I will request an "observe" experiment to see how the concentrations of the species change over time from their initial values. This data will provide clues about which species are produced or consumed, and potentially how they interact.

## 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: 10001 rows × 6 columns. 

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

Time   id_x589  id_r7xb  id_22ta   id_plsk   id_0nyd
0         0 3.60e-01        0         0         0
...
1.99e+01  2.05e-01 3.48e-01 3.26e-01 -3.26e-01 -1.93e-01
...
3.98e+01  2.32e-01 3.43e-01 2.71e-01 -2.71e-01 -2.16e-01
...
5.97e+01  2.26e-01 3.84e-01 2.53e-01 -2.53e-01 -2.49e-01
...
7.96e+01  2.30e-01 4.27e-01 2.52e-01 -2.52e-01 -2.97e-01
...
9.95e+01  2.47e-01 4.62e-01 2.28e-01 -2.28e-01 -3.49e-01
...
1.19e+02  2.53e-01 5.11e-01 1.95e-01 -1.95e-01 -4.05e-01
...
1.39e+02  2.39e-01 5.88e-01 1.91e-01 -1.91e-01 -4.67e-01
...
1.59e+02  2.17e-01 6.83e-01 2.31e-01 -2.31e-01 -5.40e-01
...
1.79e+02  2.11e-01 7.73e-01 3.52e-01 -3.52e-01 -6.24e-01
...
1.99e+02  3.15e-01 6.78e-01 6.26e-02 -6.26e-02 -6.33e-01
...
2.19e+02  9.18e-02 8.97e-01 8.83e-01 -8.83e-01 -6.29e-01
...
2.39e+02  7.77e-02 7.50e-01 8.11e-01 -8.11e-01 -4.68e-01
...
2.59e+02  7.23e-02 5.82e-01 6.46e-01 -6.46e-01 -2.94e-01
...
2.78e+02  1.67e-01 3.61e-01 3.71e-01 -3.71e-01 -1.68e-01
...
2.98e+02  2.33e-01 3.08e-01 3.07e-01 -3.07e-01 -1.81e-01
...
3.18e+02  2.15e-01 3.43e-01 2.81e-01 -2.81e-01 -1.98e-01
...
3.38e+02  2.21e-01 3.73e-01 2.83e-01 -2.83e-01 -2.35e-01
...
3.58e+02  2.39e-01 3.93e-01 2.45e-01 -2.45e-01 -2.72e-01
...
3.78e+02  2.37e-01 4.41e-01 2.24e-01 -2.24e-01 -3.18e-01
...
3.98e+02  2.34e-01 5.02e-01 2.27e-01 -2.27e-01 -3.76e-01
...
4.18e+02  2.45e-01 5.57e-01 2.30e-01 -2.30e-01 -4.42e-01
...
4.38e+02  2.77e-01 5.91e-01 1.95e-01 -1.95e-01 -5.09e-01
...
4.57e+02  2.72e-01 6.50e-01 1.29e-01 -1.29e-01 -5.63e-01
...
4.77e+02  1.84e-01 7.88e-01 2.33e-01 -2.33e-01 -6.12e-01
...
4.97e+02  1.20e-01 9.38e-01 8.88e-01 -8.88e-01 -6.98e-01
...
5.17e+02  8.34e-02 8.29e-01 8.56e-01 -8.56e-01 -5.52e-01
...
5.37e+02  7.34e-02 6.70e-01 7.45e-01 -7.45e-01 -3.84e-01
...
5.57e+02  9.93e-02 4.74e-01 5.13e-01 -5.13e-01 -2.13e-01
...
5.77e+02  2.12e-01 3.09e-01 2.84e-01 -2.84e-01 -1.61e-01
...
5.97e+02  2.05e-01 3.48e-01 3.27e-01 -3.27e-01 -1.93e-01
...
6.17e+02  2.32e-01 3.43e-01 2.70e-01 -2.70e-01 -2.15e-01
...
6.36e+02  2.25e-01 3.84e-01 2.54e-01 -2.54e-01 -2.49e-01
...
6.56e+02  2.30e-01 4.26e-01 2.52e-01 -2.52e-01 -2.96e-01
...
6.76e+02  2.47e-01 4.61e-01 2.27e-01 -2.27e-01 -3.48e-01
...
6.96e+02  2.52e-01 5.11e-01 1.96e-01 -1.96e-01 -4.04e-01
...
7.16e+02  2.38e-01 5.88e-01 1.92e-01 -1.92e-01 -4.66e-01
...
7.36e+02  2.17e-01 6.83e-01 2.33e-01 -2.33e-01 -5.40e-01
...
7.56e+02  2.17e-01 7.67e-01 3.51e-01 -3.51e-01 -6.24e-01
...
7.76e+02  3.07e-01 6.84e-01 6.61e-02 -6.61e-02 -6.31e-01
...
7.96e+02  9.22e-02 8.99e-01 8.84e-01 -8.84e-01 -6.31e-01
...
8.15e+02  7.78e-02 7.53e-01 8.13e-01 -8.13e-01 -4.71e-01
...
8.35e+02  7.22e-02 5.85e-01 6.49e-01 -6.49e-01 -2.97e-01
...
8.55e+02  1.66e-01 3.63e-01 3.75e-01 -3.75e-01 -1.68e-01
...
8.75e+02  2.34e-01 3.07e-01 3.05e-01 -3.05e-01 -1.80e-01
...
8.95e+02  2.15e-01 3.43e-01 2.82e-01 -2.82e-01 -1.97e-01
...
9.15e+02  2.22e-01 3.73e-01 2.84e-01 -2.84e-01 -2.34e-01
...
9.35e+02  2.39e-01 3.92e-01 2.44e-01 -2.44e-01 -2.71e-01
...
9.55e+02  2.36e-01 4.41e-01 2.24e-01 -2.24e-01 -3.17e-01
...
9.75e+02  2.20e-01 3.80e-01 2.76e-01 -2.76e-01 -2.39e-01

## 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 initial observation data presents a significant challenge: species `id_plsk` and `id_0nyd` start with initial concentrations of 0 in the provided SBML model, yet their values in the experimental data become negative. Standard biochemical models do not allow for negative concentrations. This suggests a few possibilities:
1.  The data for `id_plsk` and `id_0nyd` might not represent absolute concentrations but rather a change relative to some non-zero baseline that was present in the actual experiment, or some other processed value.
2.  The species `id_plsk` and `id_0nyd` might have non-zero initial concentrations in the "true" biological system, and the input SBML is incomplete in this regard. My final model might need to reflect these true initial concentrations.
3.  The underlying "true" system has properties not captured by standard SBML Level 2 Version 1, or the simulation/observation mechanism is non-standard.

Given the task is to create an SBML model that "accurately represents the biological system" and will be "evaluated by its similarity with the actual system", I need a model whose simulation output matches the experimental data, including these negative values if they are indeed what the system produces. Standard simulators for SBML will not produce negative concentrations for species starting at zero and then being consumed.

Let's analyze the species and their compartments:
-   `id_r7xb`: compartment `id_52il`, initialConcentration="0.36". Observed to decrease. Likely a primary reactant/source.
-   `id_x589`: compartment `id_cyas`, initialConcentration="0". Observed to increase. Likely a product.
-   `id_22ta`: compartment `id_cyas`, initialConcentration="0". Observed to increase. Likely a product.
-   `id_plsk`: compartment `id_cyas`, initialConcentration="0". Observed to decrease (become negative).
-   `id_0nyd`: compartment `id_rt6a`, initialConcentration="0". Observed to decrease (become negative).

The strong inverse correlation between `id_22ta` and `id_plsk` (e.g., `id_22ta` is `X`, `id_plsk` is `-X`) suggests they are tightly coupled, possibly `A + id_plsk -> id_22ta` or `id_plsk -> id_22ta` if `id_plsk` had an initial amount, or `A -> id_22ta` and `A -> (consumes id_plsk)`.

Since species in different compartments cannot directly react unless it's a transport or boundary reaction, `id_r7xb` (in `id_52il`) must somehow influence species in `id_cyas` and `id_rt6a`.

Model Hypothesis (Attempt 1 - Production & Consumption):
If species starting at 0 are consumed, they must first be produced.
1.  `R1: id_r7xb (id_52il) -> id_x589 (id_cyas)` (Produces `id_x589`)
2.  `R2: id_r7xb (id_52il) -> id_plsk (id_cyas)` (Produces `id_plsk`)
3.  `R3: id_plsk (id_cyas) -> id_22ta (id_cyas)` (Consumes `id_plsk` to make `id_22ta`)
4.  `R4: id_r7xb (id_52il) -> id_0nyd (id_rt6a)` (Produces `id_0nyd`)
5.  `R5: id_0nyd (id_rt6a) -> ` (Consumes `id_0nyd`, e.g., degradation)

This model structure (R1-R5) will result in `id_plsk` and `id_0nyd` having positive concentrations. It will not match the negative values in the data. However, it's a starting point to get these species involved.

I will first visualize the experimental data. Then, I will implement this initial hypothesis (R1-R5) into the SBML model, simulate it (with placeholder kinetics for now), and observe the output. This will highlight the discrepancy regarding negative concentrations and guide further refinement or a re-evaluation of how to interpret the data or construct the model.

## Action

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

# Access the experimental data
obs_data = experiment_history['iteration_0']
shared_variables.add("obs_data_iteration_0", obs_data)

# Plot the data
plt.figure(figsize=(12, 8))
for species_id in obs_data.columns:
    if species_id != 'Time':
        plt.plot(obs_data['Time'], obs_data[species_id], label=species_id)
plt.xlabel('Time')
plt.ylabel('Concentration / Value')
plt.title('Observed Species Dynamics (iteration_0)')
plt.legend()
plt.grid(True)
plt.savefig("species_dynamics_plot.png") # Save plot
print("Plot saved as species_dynamics_plot.png")
print(obs_data.describe())

# Modify the input SBML string to add hypothesized reactions
# Hypothesis:
# R1: id_r7xb (id_52il) -> id_x589 (id_cyas) [k1*id_r7xb]
# R2: id_r7xb (id_52il) -> id_plsk (id_cyas) [k2*id_r7xb]
# R3: id_plsk (id_cyas) -> id_22ta (id_cyas) [k3*id_plsk]
# R4: id_r7xb (id_52il) -> id_0nyd (id_rt6a) [k4*id_r7xb]
# R5: id_0nyd (id_rt6a) -> EMPTY [k5*id_0nyd] (degradation)

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

# Helper to add reaction
def add_reaction(model, r_id, reactants_dict, products_dict, formula, reversible=False):
    reaction = model.createReaction()
    reaction.setId(r_id)
    reaction.setReversible(reversible)
    reaction.setFast(False)

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

    for species_id, stoich in products_dict.items():
        product = reaction.createProduct()
        product.setSpecies(species_id)
        product.setStoichiometry(stoich)
        product.setConstant(False) # In SBML L3, this is true by default for products, but L2 might need it. Let's be explicit.

    math_ast = libsbml.parseL3Formula(formula)
    kinetic_law = reaction.createKineticLaw()
    kinetic_law.setMath(math_ast)
    return reaction

# Add parameters (as local parameters for now, or define them globally)
# For simplicity, let's assume k values are 0.1 for now
# These should be global parameters for better practice
param_ids = ["k1", "k2", "k3", "k4", "k5"]
param_values = [0.1, 0.1, 0.1, 0.1, 0.1]

for pid, pval in zip(param_ids, param_values):
    if not model.getParameter(pid): # Add if not exists
        p = model.createParameter()
        p.setId(pid)
        p.setValue(pval)
        p.setConstant(True) # Parameters are usually constant unless specified otherwise

# R1: id_r7xb -> id_x589
add_reaction(model, "R1", {"id_r7xb": 1}, {"id_x589": 1}, "k1 * id_r7xb")

# R2: id_r7xb -> id_plsk
add_reaction(model, "R2", {"id_r7xb": 1}, {"id_plsk": 1}, "k2 * id_r7xb")

# R3: id_plsk -> id_22ta
add_reaction(model, "R3", {"id_plsk": 1}, {"id_22ta": 1}, "k3 * id_plsk")

# R4: id_r7xb -> id_0nyd
add_reaction(model, "R4", {"id_r7xb": 1}, {"id_0nyd": 1}, "k4 * id_r7xb")

# R5: id_0nyd ->
# For degradation, product list is empty.
reaction_r5 = model.createReaction()
reaction_r5.setId("R5")
reaction_r5.setReversible(False)
reaction_r5.setFast(False)
reactant_r5 = reaction_r5.createReactant()
reactant_r5.setSpecies("id_0nyd")
reactant_r5.setStoichiometry(1)
reactant_r5.setConstant(False)
math_ast_r5 = libsbml.parseL3Formula("k5 * id_0nyd")
kinetic_law_r5 = reaction_r5.createKineticLaw()
kinetic_law_r5.setMath(math_ast_r5)


# Write the updated SBML to a string
writer = libsbml.SBMLWriter()
updated_sbml_string_v1 = writer.writeSBMLToString(sbml_doc)
shared_variables.add("hypothetical_sbml_v1", updated_sbml_string_v1)
print("Hypothetical SBML model v1 created and stored.")
# print(updated_sbml_string_v1) # Avoid printing large string

# Simulate this model
sim_data_v1 = simulate(updated_sbml_string_v1)
shared_variables.add("sim_data_v1", sim_data_v1)
print("Simulation of hypothetical_sbml_v1 completed.")

# Plot simulation results
plt.figure(figsize=(12, 8))
for species_id in sim_data_v1.columns:
    if species_id != 'Time':
        plt.plot(sim_data_v1['Time'], sim_data_v1[species_id], label=f"{species_id} (sim v1)")
plt.xlabel('Time')
plt.ylabel('Concentration')
plt.title('Simulated Species Dynamics (Hypothesis v1)')
plt.legend()
plt.grid(True)
plt.savefig("sim_dynamics_plot_v1.png")
print("Plot of v1 simulation saved as sim_dynamics_plot_v1.png")
print(sim_data_v1.describe())

# Compare sim_data_v1 with obs_data focusing on id_plsk and id_0nyd
print("\nComparison of id_plsk (Observed vs Simulated v1):")
print("Observed id_plsk sample:")
print(obs_data[['Time', 'id_plsk']].head())
print("Simulated v1 id_plsk sample:")
print(sim_data_v1[['Time', 'id_plsk']].head())
print(f"Observed id_plsk min: {obs_data['id_plsk'].min()}, max: {obs_data['id_plsk'].max()}")
print(f"Simulated v1 id_plsk min: {sim_data_v1['id_plsk'].min()}, max: {sim_data_v1['id_plsk'].max()}")

print("\nComparison of id_0nyd (Observed vs Simulated v1):")
print("Observed id_0nyd sample:")
print(obs_data[['Time', 'id_0nyd']].head())
print("Simulated v1 id_0nyd sample:")
print(sim_data_v1[['Time', 'id_0nyd']].head())
print(f"Observed id_0nyd min: {obs_data['id_0nyd'].min()}, max: {obs_data['id_0nyd'].max()}")
print(f"Simulated v1 id_0nyd min: {sim_data_v1['id_0nyd'].min()}, max: {sim_data_v1['id_0nyd'].max()}")

```
# 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 primary challenge is the negative concentrations observed for `id_plsk` and `id_0nyd`, and the complex, oscillatory behavior of `id_r7xb`.

**Interpretation of Negative Concentrations:**
I'll assume that `id_plsk` and `id_0nyd` in the SBML model represent "actual" species concentrations (`P_act`, `N_act`) which must be non-negative. The observed values in the data (`P_obs`, `N_obs`) are transformations:
`P_obs(t) = P_act(t) - P_act(0)`
`N_obs(t) = N_act(t) - N_act(0)`
Given `P_obs(0) = 0` and `N_obs(0) = 0` from the data (and initial SBML), this interpretation holds.
This means my SBML model for `id_plsk` and `id_0nyd` must use non-zero initial concentrations (`IC_P`, `IC_N`). For simulation comparison, I will subtract these initial concentrations from the simulated values of `id_plsk` and `id_0nyd`.
Based on observed data, `id_22ta` increases as `id_plsk_obs` decreases (`id_plsk_obs` ≈ `-id_22ta_obs`). This implies `P_act(t) - P_act(0) = -(id_22ta(t) - id_22ta(0))`. Since `id_22ta(0)=0`, `P_act(t) - P_act(0) = -id_22ta(t)`. This is satisfied by a reaction `id_plsk -> id_22ta` (where `id_plsk` is `P_act`).

**Compartment Hierarchy:**
- `compartment id="id_52il" ... outside="id_cyas"` means `id_cyas` is inside `id_52il`.
- `compartment id="id_cyas" ... outside="id_rt6a"` means `id_rt6a` is inside `id_cyas`.
Hierarchy: `id_52il` (outermost) -> `id_cyas` (middle) -> `id_rt6a` (innermost).

Species locations:
- `id_r7xb`: `id_52il`
- `id_x589`, `id_22ta`, `id_plsk`: `id_cyas`
- `id_0nyd`: `id_rt6a`

**Model Hypothesis (v4):**
1.  **`id_r7xb` dynamics (in `id_52il`):**
    -   Production of `id_r7xb` is inhibited by `id_x589` (from `id_cyas`). This creates a negative feedback loop.
        `R_Rprod: -> id_r7xb` (Rate: `Vmax_R / (1 + (id_x589/K_IX)^h_X)`)
    -   Degradation of `id_r7xb`.
        `R_Rdeg: id_r7xb -> ` (Rate: `k_R_deg * id_r7xb`)
2.  **`id_x589` production (in `id_cyas`):**
    -   Catalyzed by `id_r7xb` (from `id_52il`). Assumes an implicit, constant substrate.
        `R_X: -> id_x589` (Rate: `k_X * id_r7xb`)
3.  **`id_plsk` to `id_22ta` conversion (in `id_cyas`):**
    -   `id_plsk` (actual) is converted to `id_22ta`. Catalyzed by `id_r7xb` (from `id_52il`).
        `R_PT: id_plsk -> id_22ta` (Rate: `k_PT * id_plsk * id_r7xb`)
4.  **`id_0nyd` degradation (in `id_rt6a`):**
    -   `id_0nyd` (actual) is degraded. Catalyzed by `id_r7xb` (from `id_52il`).
        `R_N: id_0nyd -> ` (Rate: `k_N * id_0nyd * id_r7xb`)

**Initial Concentrations for SBML Model:**
-   `id_x589(0) = 0`
-   `id_r7xb(0) = 0.36` (from input SBML)
-   `id_22ta(0) = 0`
-   `id_plsk(0) = IC_P` (e.g., 1.0, for `P_act(0)`. Max `id_22ta` is ~0.88, so `IC_P` must be >= 0.88)
-   `id_0nyd(0) = IC_N` (e.g., 1.0, for `N_act(0)`. Max `abs(N_obs)` is ~0.9, so `IC_N` must be >= 0.9)

I will use placeholder kinetic parameter values. The goal is to test this model structure.

## Action

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

# --- Helper function to add reactions (modified for L2V1 compatibility if needed) ---
def add_reaction_l2(model, r_id, reactants_dict, products_dict, formula_str, reversible=False):
    reaction = model.createReaction()
    reaction.setId(r_id)
    reaction.setReversible(reversible)
    # reaction.setFast(False) # Not strictly required for L2, but good practice. Default is false.

    for species_id, stoich in reactants_dict.items():
        reactant = reaction.createReactant()
        reactant.setSpecies(species_id)
        reactant.setStoichiometry(stoich)
        # reactant.setConstant(False) # Not in L2 reactant

    for species_id, stoich in products_dict.items():
        product = reaction.createProduct()
        product.setSpecies(species_id)
        product.setStoichiometry(stoich)
        # product.setConstant(False) # Not in L2 product

    kinetic_law = reaction.createKineticLaw()
    # For L2, setMath requires an ASTNode tree. parseL3Formula might produce L3 math.
    # Simpler to use setFormula for L2 if the simulator supports it, or build AST.
    # Most simulators handle plain formula strings for L2/L3 common subset.
    # Let's assume the simulate() function's backend can parse it.
    # If not, one would need to construct the AST manually.
    kinetic_law.setFormula(formula_str) # Using setFormula for broader L2 compatibility
    return reaction

# --- Load original SBML and experimental data ---
sbml_doc = libsbml.readSBMLFromString(input_sbml_string)
model = sbml_doc.getModel()
obs_data = experiment_history['iteration_0']

# --- Define Initial Concentrations for "actual" species ---
IC_P = 1.5  # Initial concentration for actual id_plsk
IC_N = 1.5  # Initial concentration for actual id_0nyd

# Update initial concentrations in the model for id_plsk and id_0nyd
species_plsk = model.getSpecies('id_plsk')
if species_plsk:
    species_plsk.setInitialConcentration(IC_P)

species_0nyd = model.getSpecies('id_0nyd')
if species_0nyd:
    species_0nyd.setInitialConcentration(IC_N)

# --- Define Parameters ---
param_definitions = {
    "Vmax_R": 0.1, "K_IX": 0.1, "h_X": 2.0, "k_R_deg": 0.1,
    "k_X": 0.1, "k_PT": 0.2, "k_N": 0.1
}

for p_id, p_val in param_definitions.items():
    if not model.getParameter(p_id):
        p = model.createParameter()
        p.setId(p_id)
        p.setValue(p_val)
        p.setConstant(True)
    else: # Update if exists, useful for iterative refinement
        model.getParameter(p_id).setValue(p_val)


# --- Add Reactions ---
# R_Rprod: -> id_r7xb (Rate: Vmax_R / (1 + (id_x589/K_IX)^h_X))
# Compartment: id_r7xb is in 52il. id_x589 is in cyas. This is feedback from child to parent.
formula_Rprod = "Vmax_R / (1 + (id_x589 / K_IX)^h_X)"
add_reaction_l2(model, "R_Rprod", {}, {"id_r7xb": 1}, formula_Rprod)

# R_Rdeg: id_r7xb -> (Rate: k_R_deg * id_r7xb)
formula_Rdeg = "k_R_deg * id_r7xb"
add_reaction_l2(model, "R_Rdeg", {"id_r7xb": 1}, {}, formula_Rdeg)

# R_X: -> id_x589 (Rate: k_X * id_r7xb)
# id_x589 in cyas, id_r7xb in 52il (parent). Catalysis from parent.
formula_X = "k_X * id_r7xb"
add_reaction_l2(model, "R_X", {}, {"id_x589": 1}, formula_X)

# R_PT: id_plsk -> id_22ta (Rate: k_PT * id_plsk * id_r7xb)
# id_plsk, id_22ta in cyas. id_r7xb in 52il (parent). Catalysis from parent.
formula_PT = "k_PT * id_plsk * id_r7xb"
add_reaction_l2(model, "R_PT", {"id_plsk": 1}, {"id_22ta": 1}, formula_PT)

# R_N: id_0nyd -> (Rate: k_N * id_0nyd * id_r7xb)
# id_0nyd in rt6a. id_r7xb in 52il (grandparent). Catalysis from grandparent.
formula_N = "k_N * id_0nyd * id_r7xb"
add_reaction_l2(model, "R_N", {"id_0nyd": 1}, {}, formula_N)


# --- Generate SBML string for the new model (v4) ---
writer = libsbml.SBMLWriter()
sbml_v4_string = writer.writeSBMLToString(sbml_doc)
shared_variables.add("hypothetical_sbml_v4", sbml_v4_string)
# print("SBML Model v4:\n", sbml_v4_string) # Avoid printing large string

# --- Simulate Model v4 ---
sim_data_v4 = simulate(sbml_v4_string)
shared_variables.add("sim_data_v4", sim_data_v4)
print("Simulation of Model v4 completed.")

# --- Analyze Simulation Results ---
print("\nDescriptive statistics for observed data (iteration_0):")
print(obs_data.describe())

print("\nDescriptive statistics for simulated data (Model v4):")
# Adjust id_plsk and id_0nyd for comparison
sim_data_v4_adjusted = sim_data_v4.copy()
sim_data_v4_adjusted['id_plsk'] = sim_data_v4['id_plsk'] - IC_P
sim_data_v4_adjusted['id_0nyd'] = sim_data_v4['id_0nyd'] - IC_N
print(sim_data_v4_adjusted.describe())

# Print some head values for direct comparison
print("\nObserved data (head):")
print(obs_data.head())
print("\nSimulated data v4 (adjusted, head):")
print(sim_data_v4_adjusted.head())

# Check if oscillations are present in id_r7xb_sim
r7xb_sim_stats = sim_data_v4['id_r7xb'].describe()
print(f"\nSimulated id_r7xb stats: min={r7xb_sim_stats['min']:.2e}, max={r7xb_sim_stats['max']:.2e}, std={r7xb_sim_stats['std']:.2e}")
if r7xb_sim_stats['std'] > 1e-3 and (r7xb_sim_stats['max'] - r7xb_sim_stats['min']) / r7xb_sim_stats['mean'] > 0.1 : # Heuristic for oscillation
    print("Simulated id_r7xb shows some dynamic behavior.")
else:
    print("Simulated id_r7xb is relatively stable or decaying, may not be oscillating strongly with these parameters.")

# Check id_plsk vs -id_22ta relationship in simulation
# Using a small tolerance for floating point comparisons
sim_plsk_minus_IC_P = sim_data_v4['id_plsk'] - IC_P
sim_neg_22ta = -sim_data_v4['id_22ta']
diff_plsk_22ta = np.abs(sim_plsk_minus_IC_P - sim_neg_22ta)
print(f"\nMean absolute difference between sim_plsk_adjusted and -sim_id_22ta: {diff_plsk_22ta.mean():.2e}")
if diff_plsk_22ta.mean() < 1e-2: # Arbitrary threshold for "good match"
    print("Simulated data maintains id_plsk_adjusted ≈ -id_22ta relationship.")
else:
    print("Simulated data does NOT strongly maintain id_plsk_adjusted ≈ -id_22ta relationship.")

```
# Observation

## Code Stdout
Simulation of Model v4 completed.

Descriptive statistics for observed data (iteration_0):
            id_x589       id_r7xb       id_22ta       id_plsk       id_0nyd          Time
count  10001.000000  10001.000000  10001.000000  10001.000000  10001.000000  10001.000000
mean       0.199954      0.547936      0.368897     -0.368897     -0.387890    487.500000
std        0.070800      0.179797      0.235333      0.235333      0.165321    281.500475
min        0.000000      0.297089      0.000000     -0.890884     -0.710021      0.000000
25%        0.165670      0.383741      0.223958     -0.403398     -0.538543    243.750000
50%        0.225494      0.513895      0.270292     -0.270292     -0.368357    487.500000
75%        0.238572      0.679282      0.403398     -0.223958     -0.234269    731.250000
max        0.438452      0.939163      0.890884      0.000000      0.000000    975.000000

Descriptive statistics for simulated data (Model v4):
            id_x589       id_r7xb       id_22ta       id_plsk       id_0nyd          Time
count  10001.000000  10001.000000  10001.000000  10001.000000  10001.000000  10001.000000
mean       1.163887      0.015194      1.331515     -1.331515     -1.016404    487.500000
std        0.241314      0.044519      0.122830      0.122830      0.137344    281.500475
min        0.000000      0.004574      0.000000     -1.422239     -1.158474      0.000000
25%        1.012321      0.005432      1.301938     -1.400909     -1.114466    243.750000
50%        1.210507      0.006857      1.366752     -1.366752     -1.052931    487.500000
75%        1.358592      0.009860      1.400909     -1.301938     -0.954938    731.250000
max        1.479797      0.450818      1.422239      0.000000      0.000000    975.000000

Observed data (head):
    id_x589   id_r7xb   id_22ta   id_plsk   id_0nyd    Time
0  0.000000  0.360000  0.000000  0.000000  0.000000  0.0000
1  0.062929  0.444693  0.384821 -0.384821 -0.147622  0.0975
2  0.068376  0.570143  0.618322 -0.618322 -0.278519  0.1950
3  0.074052  0.690106  0.758522 -0.758522 -0.404158  0.2925
4  0.081118  0.802562  0.840273 -0.840273 -0.523681  0.3900

Simulated data v4 (adjusted, head):
    id_x589   id_r7xb   id_22ta   id_plsk   id_0nyd    Time
0  0.000000  0.360000  0.000000  0.000000  0.000000  0.0000
1  0.003540  0.366206  0.010583 -0.010583 -0.005301  0.0975
2  0.007141  0.372326  0.021270 -0.021270 -0.010673  0.1950
3  0.010800  0.378338  0.032054 -0.032054 -0.016113  0.2925
4  0.014518  0.384216  0.042927 -0.042927 -0.021620  0.3900

Simulated id_r7xb stats: min=4.57e-03, max=4.51e-01, std=4.45e-02
Simulated id_r7xb shows some dynamic behavior.

Mean absolute difference between sim_plsk_adjusted and -sim_id_22ta: 1.17e-15
Simulated data maintains id_plsk_adjusted ≈ -id_22ta relationship.

## 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. (hypothetical_sbml_v4, sim_data_v4)

### 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 previous model (v4) showed some correct relationships (like `id_plsk_adjusted ≈ -id_22ta`) but failed to reproduce the oscillatory behavior of `id_r7xb` and the correct magnitudes for species. The key issues were:
1.  `id_r7xb` did not oscillate correctly; its mean was too low.
2.  `id_x589` lacked a degradation term, leading to uncontrolled accumulation and incorrect feedback dynamics.
3.  Kinetic parameters were not tuned to match observed magnitudes and timescales.

Model v5 aims to address these:
1.  **Introduce degradation for `id_x589`**: `R_Xdeg: id_x589 ->` (Rate: `k_X_deg * id_x589`). This is crucial for `id_x589` to act as a proper negative regulator in a feedback loop.
2.  **Adjust Initial Concentrations**: `IC_P` (for `id_plsk`) and `IC_N` (for `id_0nyd`) will be set closer to the observed maximum change:
    *   `max(obs_data['id_22ta'])` is ~0.89. Let `IC_P = 1.0`.
    *   `max(abs(obs_data['id_0nyd']))` is ~0.71. Let `IC_N = 0.8`.
3.  **Re-tune kinetic parameters** for the `id_r7xb` feedback loop and other reactions, aiming for:
    *   `id_r7xb` oscillations with mean ~0.55, range ~0.3-0.94.
    *   `id_x589` mean ~0.2.
    *   Appropriate rates for `id_plsk -> id_22ta` and `id_0nyd` degradation.

The refined set of reactions for Model v5:
*   `R_Rprod: -> id_r7xb` (Rate: `Vmax_R / (1 + (id_x589/K_IX)^h_X)`)
*   `R_Rdeg: id_r7xb -> ` (Rate: `k_R_deg * id_r7xb`)
*   `R_Xprod: -> id_x589` (Rate: `k_X_prod * id_r7xb`) (Changed from `k_X` to `k_X_prod` for clarity)
*   `R_Xdeg: id_x589 -> ` (Rate: `k_X_deg * id_x589`) (New reaction)
*   `R_PT: id_plsk -> id_22ta` (Rate: `k_PT * id_plsk * id_r7xb`)
*   `R_N: id_0nyd -> ` (Rate: `k_N * id_0nyd * id_r7xb`)

Parameter estimates for v5:
*   `IC_P = 1.0`
*   `IC_N = 0.8`
*   `Vmax_R = 0.25` (Increased slightly to boost id_r7xb mean)
*   `K_IX = 0.1`
*   `h_X = 4` (Increased Hill coefficient for sharper response, promoting oscillation)
*   `k_R_deg = 0.25` (Matched with Vmax_R to aim for a midpoint around 0.5 if inhibition is moderate)
*   `k_X_prod = 0.5`
*   `k_X_deg = 1.0` (This would give `id_x589_ss = (0.5 * id_r7xb) / 1.0 = 0.5 * id_r7xb`. If `id_r7xb` avg is 0.5, `id_x589` avg is 0.25. Close to target 0.2)
*   `k_PT = 0.1` (Increased from 0.05, as previous simulation was too slow for these)
*   `k_N = 0.1` (Increased from 0.05)

I will implement these changes and simulate.

## Action

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

# --- Helper function to add reactions (L2 compatible) ---
def add_reaction_l2(model, r_id, reactants_dict, products_dict, formula_str, reversible=False):
    reaction = model.createReaction()
    reaction.setId(r_id)
    reaction.setReversible(reversible)

    for species_id, stoich in reactants_dict.items():
        reactant = reaction.createReactant()
        reactant.setSpecies(species_id)
        reactant.setStoichiometry(stoich)

    for species_id, stoich in products_dict.items():
        product = reaction.createProduct()
        product.setSpecies(species_id)
        product.setStoichiometry(stoich)

    kinetic_law = reaction.createKineticLaw()
    kinetic_law.setFormula(formula_str)
    return reaction

# --- Load original SBML and experimental data ---
sbml_doc_v5 = libsbml.readSBMLFromString(input_sbml_string) # Start from fresh input_sbml
model_v5 = sbml_doc_v5.getModel()
obs_data = experiment_history['iteration_0']

# --- Define Initial Concentrations for "actual" species ---
IC_P_v5 = 1.0
IC_N_v5 = 0.8

# Update initial concentrations in the model
species_plsk = model_v5.getSpecies('id_plsk')
if species_plsk:
    species_plsk.setInitialConcentration(IC_P_v5)

species_0nyd = model_v5.getSpecies('id_0nyd')
if species_0nyd:
    species_0nyd.setInitialConcentration(IC_N_v5)
    
# Ensure other initial concentrations are as per input_sbml_string (0 for x589, 22ta; 0.36 for r7xb)
model_v5.getSpecies('id_x589').setInitialConcentration(0.0)
model_v5.getSpecies('id_22ta').setInitialConcentration(0.0)
model_v5.getSpecies('id_r7xb').setInitialConcentration(0.36)


# --- Define Parameters for v5 ---
param_definitions_v5 = {
    "Vmax_R": 0.25, "K_IX": 0.1, "h_X": 4.0, "k_R_deg": 0.25,
    "k_X_prod": 0.5, "k_X_deg": 1.0,
    "k_PT": 0.1, "k_N": 0.1
}

for p_id, p_val in param_definitions_v5.items():
    if not model_v5.getParameter(p_id):
        p = model_v5.createParameter()
        p.setId(p_id)
        p.setValue(p_val)
        p.setConstant(True)
    else:
        model_v5.getParameter(p_id).setValue(p_val)

# --- Add Reactions for v5 ---
# R_Rprod: -> id_r7xb (Rate: Vmax_R / (1 + (id_x589/K_IX)^h_X))
formula_Rprod = "Vmax_R / (1 + (id_x589 / K_IX)^h_X)"
add_reaction_l2(model_v5, "R_Rprod", {}, {"id_r7xb": 1}, formula_Rprod)

# R_Rdeg: id_r7xb -> (Rate: k_R_deg * id_r7xb)
formula_Rdeg = "k_R_deg * id_r7xb"
add_reaction_l2(model_v5, "R_Rdeg", {"id_r7xb": 1}, {}, formula_Rdeg)

# R_Xprod: -> id_x589 (Rate: k_X_prod * id_r7xb)
formula_Xprod = "k_X_prod * id_r7xb"
add_reaction_l2(model_v5, "R_Xprod", {}, {"id_x589": 1}, formula_Xprod)

# R_Xdeg: id_x589 -> (Rate: k_X_deg * id_x589) (NEW)
formula_Xdeg = "k_X_deg * id_x589"
add_reaction_l2(model_v5, "R_Xdeg", {"id_x589": 1}, {}, formula_Xdeg)

# R_PT: id_plsk -> id_22ta (Rate: k_PT * id_plsk * id_r7xb)
formula_PT = "k_PT * id_plsk * id_r7xb"
add_reaction_l2(model_v5, "R_PT", {"id_plsk": 1}, {"id_22ta": 1}, formula_PT)

# R_N: id_0nyd -> (Rate: k_N * id_0nyd * id_r7xb)
formula_N = "k_N * id_0nyd * id_r7xb"
add_reaction_l2(model_v5, "R_N", {"id_0nyd": 1}, {}, formula_N)

# --- Generate SBML string for Model v5 ---
writer = libsbml.SBMLWriter()
sbml_v5_string = writer.writeSBMLToString(sbml_doc_v5)
shared_variables.add("hypothetical_sbml_v5", sbml_v5_string)
# print("SBML Model v5 content ready.")

# --- Simulate Model v5 ---
sim_data_v5 = simulate(sbml_v5_string)
shared_variables.add("sim_data_v5", sim_data_v5)
print("Simulation of Model v5 completed.")

# --- Analyze Simulation Results ---
print("\nDescriptive statistics for observed data (iteration_0):")
print(obs_data.describe())

print("\nDescriptive statistics for simulated data (Model v5):")
sim_data_v5_adjusted = sim_data_v5.copy()
sim_data_v5_adjusted['id_plsk'] = sim_data_v5['id_plsk'] - IC_P_v5
sim_data_v5_adjusted['id_0nyd'] = sim_data_v5['id_0nyd'] - IC_N_v5
print(sim_data_v5_adjusted.describe())

print("\nObserved data (head):")
print(obs_data.head())
print("\nSimulated data v5 (adjusted, head):")
print(sim_data_v5_adjusted.head())

r7xb_sim_stats_v5 = sim_data_v5['id_r7xb'].describe()
print(f"\nSimulated id_r7xb (v5) stats: min={r7xb_sim_stats_v5['min']:.3f}, max={r7xb_sim_stats_v5['max']:.3f}, mean={r7xb_sim_stats_v5['mean']:.3f}, std={r7xb_sim_stats_v5['std']:.3f}")
if r7xb_sim_stats_v5['std'] > 0.01 and (r7xb_sim_stats_v5['max'] - r7xb_sim_stats_v5['min']) > 0.1 * r7xb_sim_stats_v5['mean'] :
    print("Simulated id_r7xb (v5) shows dynamic/oscillatory behavior.")
else:
    print("Simulated id_r7xb (v5) is relatively stable or decaying, not strongly oscillating.")

# Check id_plsk vs -id_22ta relationship
sim_plsk_adj_v5 = sim_data_v5['id_plsk'] - IC_P_v5
sim_neg_22ta_v5 = -sim_data_v5['id_22ta']
diff_plsk_22ta_v5 = np.abs(sim_plsk_adj_v5 - sim_neg_22ta_v5)
print(f"\nMean absolute difference between sim_plsk_adjusted (v5) and -sim_id_22ta (v5): {diff_plsk_22ta_v5.mean():.2e}")

```
# Observation

## Code Stdout
Simulation of Model v5 completed.

Descriptive statistics for observed data (iteration_0):
            id_x589       id_r7xb  ...       id_0nyd          Time
count  10001.000000  10001.000000  ...  10001.000000  10001.000000
mean       0.199954      0.547936  ...     -0.387890    487.500000
std        0.070800      0.179797  ...      0.165321    281.500475
min        0.000000      0.297089  ...     -0.710021      0.000000
25%        0.165670      0.383741  ...     -0.538543    243.750000
50%        0.225494      0.513895  ...     -0.368357    487.500000
75%        0.238572      0.679282  ...     -0.234269    731.250000
max        0.438452      0.939163  ...      0.000000    975.000000

[8 rows x 6 columns]

Descriptive statistics for simulated data (Model v5):
            id_x589       id_r7xb  ...       id_0nyd          Time
count  10001.000000  10001.000000  ...  10001.000000  10001.000000
mean       0.129966      0.260217  ...     -0.769503    487.500000
std        0.003079      0.007200  ...      0.104313    281.500475
min        0.000000      0.249920  ...     -0.800000      0.000000
25%        0.129915      0.259831  ...     -0.800000    243.750000
50%        0.129915      0.259831  ...     -0.799998    487.500000
75%        0.129915      0.259831  ...     -0.798631    731.250000
max        0.173696      0.434428  ...      0.000000    975.000000

[8 rows x 6 columns]

Observed data (head):
    id_x589   id_r7xb   id_22ta   id_plsk   id_0nyd    Time
0  0.000000  0.360000  0.000000  0.000000  0.000000  0.0000
1  0.062929  0.444693  0.384821 -0.384821 -0.147622  0.0975
2  0.068376  0.570143  0.618322 -0.618322 -0.278519  0.1950
3  0.074052  0.690106  0.758522 -0.758522 -0.404158  0.2925
4  0.081118  0.802562  0.840273 -0.840273 -0.523681  0.3900

Simulated data v5 (adjusted, head):
    id_x589   id_r7xb   id_22ta   id_plsk   id_0nyd    Time
0  0.000000  0.360000  0.000000  0.000000  0.000000  0.0000
1  0.017087  0.375407  0.003579 -0.003579 -0.002863  0.0975
2  0.033291  0.390327  0.007292 -0.007292 -0.005834  0.1950
3  0.048663  0.404298  0.011131 -0.011131 -0.008905  0.2925
4  0.063217  0.416431  0.015082 -0.015082 -0.012065  0.3900

Simulated id_r7xb (v5) stats: min=0.250, max=0.434, mean=0.260, std=0.007
Simulated id_r7xb (v5) is relatively stable or decaying, not strongly oscillating.

Mean absolute difference between sim_plsk_adjusted (v5) and -sim_id_22ta (v5): 2.66e-16

## 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. (hypothetical_sbml_v4, sim_data_v4, hypothetical_sbml_v5, sim_data_v5)

### 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

Model v5 failed to produce oscillations. The `id_r7xb` concentration quickly settled to a steady state. This indicates that the negative feedback loop involving `id_x589` is either too weak, too slow, or the parameters are not in the oscillatory regime. The mean of `id_r7xb` (0.260) is much lower than observed (0.548), and `id_x589` mean (0.130) is also lower than observed (0.200). The consumption of `id_plsk` and `id_0nyd` is also too slow.

Key observations from v5 simulation:
-   `id_r7xb` reached a steady state around 0.25-0.26.
-   `id_x589` reached a steady state around 0.13.
-   The Hill coefficient `h_X = 4.0` should be sufficient for oscillations if other conditions (delays, relative rates) are met.
-   The formula for `id_r7xb` production is `Vmax_R / (1 + (id_x589/K_IX)^h_X)`. At `id_x589 = 0.13`, `K_IX = 0.1`, `h_X = 4`, the denominator is `1 + (0.13/0.1)^4 = 1 + (1.3)^4 = 1 + 2.8561 = 3.8561`.
    So, production rate = `Vmax_R / 3.8561`.
-   Degradation rate = `k_R_deg * id_r7xb`.
-   At steady state, `Vmax_R / 3.8561 = k_R_deg * id_r7xb_ss`.
    With `Vmax_R = 0.25`, `k_R_deg = 0.25`: `0.25 / 3.8561 = 0.25 * id_r7xb_ss` => `id_r7xb_ss = 1 / 3.8561 = 0.259`. This matches the simulation.

To induce oscillations and match means:
1.  **Increase `Vmax_R`**: This will increase the production drive for `id_r7xb`.
2.  **Adjust `k_R_deg`**: To balance `Vmax_R` for the desired mean.
3.  **Adjust `K_IX`**: This is the sensitivity of `id_r7xb` production to `id_x589`. A `K_IX` value closer to the mean of `id_x589` (target ~0.2) might be better.
4.  **Adjust `k_X_prod` and `k_X_deg`**: To ensure `id_x589` oscillates around its target mean and responds appropriately to `id_r7xb`. The timescale of `id_x589` dynamics relative to `id_r7xb` is crucial for oscillations. If `id_x589` responds too quickly, it can dampen oscillations. If too slowly, it can promote them.
5.  **Increase `k_PT` and `k_N`**: The consumption of `id_plsk` and `id_0nyd` was too slow.

Let's try to make the feedback loop "slower" or more "delayed" implicitly by adjusting rates.
The Goodwin oscillator model, a classic example of a negative feedback oscillator, requires sufficient nonlinearity (Hill coefficient) and a long enough feedback loop (effectively, a delay).

Parameter tuning strategy for Model v6:
-   Target `id_r7xb` mean ~0.55, `id_x589` mean ~0.2.
-   Let `K_IX = 0.2` (closer to target `id_x589` mean).
-   If `id_x589_mean = K_IX = 0.2`, then `(id_x589/K_IX)^h_X = 1`. Denominator = 2.
    Production = `Vmax_R / 2`. Degradation = `k_R_deg * id_r7xb_mean`.
    So, `Vmax_R / 2 = k_R_deg * 0.55`.  `Vmax_R / k_R_deg = 1.1`.
    Let `k_R_deg = 0.5`. Then `Vmax_R = 0.55`.
-   For `id_x589`: `id_x589_ss = (k_X_prod / k_X_deg) * id_r7xb`.
    Target `0.2 = (k_X_prod / k_X_deg) * 0.55`. So `k_X_prod / k_X_deg = 0.2 / 0.55 = 0.36`.
    Let `k_X_deg = 1.0` (as before, for a reasonable timescale for `id_x589`). Then `k_X_prod = 0.36`.
-   `h_X = 4.0` (keep).
-   `k_PT` and `k_N`: Observed `id_22ta` reaches ~0.8 in ~200 time units. `d(id_22ta)/dt = k_PT * id_plsk * id_r7xb`.
    Avg `id_r7xb` ~0.55. `id_plsk` starts at `IC_P_v6 = 1.0`.
    Initial rate ~ `k_PT * 1.0 * 0.36` (initial `id_r7xb`).
    Average rate ~ `k_PT * (IC_P_v6/2) * 0.55`.
    `0.8 / 200 = 0.004`. So, `k_PT * 0.5 * 0.55 = 0.004` => `k_PT * 0.275 = 0.004` => `k_PT = 0.004 / 0.275 = 0.0145`.
    This is much lower than previous. Let's try `k_PT = 0.2` and `k_N = 0.2` to make them faster than v5. The previous values (0.1) were too slow.

Revised parameters for v6:
*   `IC_P_v6 = 1.0`
*   `IC_N_v6 = 0.8`
*   `Vmax_R = 0.55`
*   `K_IX = 0.2`
*   `h_X = 4.0`
*   `k_R_deg = 0.5`
*   `k_X_prod = 0.36`
*   `k_X_deg = 1.0`
*   `k_PT = 0.2`
*   `k_N = 0.2`

The model structure remains the same as v5.

## Action

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

# --- Helper function to add reactions (L2 compatible) ---
def add_reaction_l2(model, r_id, reactants_dict, products_dict, formula_str, reversible=False):
    reaction = model.createReaction()
    reaction.setId(r_id)
    reaction.setReversible(reversible)

    for species_id, stoich in reactants_dict.items():
        reactant = reaction.createReactant()
        reactant.setSpecies(species_id)
        reactant.setStoichiometry(stoich)

    for species_id, stoich in products_dict.items():
        product = reaction.createProduct()
        product.setSpecies(species_id)
        product.setStoichiometry(stoich)

    kinetic_law = reaction.createKineticLaw()
    kinetic_law.setFormula(formula_str)
    return reaction

# --- Load original SBML and experimental data ---
sbml_doc_v6 = libsbml.readSBMLFromString(input_sbml_string) # Start from fresh input_sbml
model_v6 = sbml_doc_v6.getModel()
obs_data = experiment_history['iteration_0']

# --- Define Initial Concentrations for "actual" species ---
IC_P_v6 = 1.0
IC_N_v6 = 0.8

# Update initial concentrations in the model
species_plsk = model_v6.getSpecies('id_plsk')
if species_plsk:
    species_plsk.setInitialConcentration(IC_P_v6)

species_0nyd = model_v6.getSpecies('id_0nyd')
if species_0nyd:
    species_0nyd.setInitialConcentration(IC_N_v6)
    
model_v6.getSpecies('id_x589').setInitialConcentration(0.0)
model_v6.getSpecies('id_22ta').setInitialConcentration(0.0)
model_v6.getSpecies('id_r7xb').setInitialConcentration(0.36) # From original SBML

# --- Define Parameters for v6 ---
param_definitions_v6 = {
    "Vmax_R": 0.55, "K_IX": 0.2, "h_X": 4.0, "k_R_deg": 0.5,
    "k_X_prod": 0.36, "k_X_deg": 1.0,
    "k_PT": 0.2, "k_N": 0.2  # Increased k_PT, k_N
}

for p_id, p_val in param_definitions_v6.items():
    if not model_v6.getParameter(p_id):
        p = model_v6.createParameter()
        p.setId(p_id)
        p.setValue(p_val)
        p.setConstant(True)
    else:
        model_v6.getParameter(p_id).setValue(p_val)

# --- Add Reactions for v6 (same structure as v5) ---
formula_Rprod = "Vmax_R / (1 + (id_x589 / K_IX)^h_X)"
add_reaction_l2(model_v6, "R_Rprod", {}, {"id_r7xb": 1}, formula_Rprod)

formula_Rdeg = "k_R_deg * id_r7xb"
add_reaction_l2(model_v6, "R_Rdeg", {"id_r7xb": 1}, {}, formula_Rdeg)

formula_Xprod = "k_X_prod * id_r7xb"
add_reaction_l2(model_v6, "R_Xprod", {}, {"id_x589": 1}, formula_Xprod)

formula_Xdeg = "k_X_deg * id_x589"
add_reaction_l2(model_v6, "R_Xdeg", {"id_x589": 1}, {}, formula_Xdeg)

formula_PT = "k_PT * id_plsk * id_r7xb"
add_reaction_l2(model_v6, "R_PT", {"id_plsk": 1}, {"id_22ta": 1}, formula_PT)

formula_N = "k_N * id_0nyd * id_r7xb"
add_reaction_l2(model_v6, "R_N", {"id_0nyd": 1}, {}, formula_N)

# --- Generate SBML string for Model v6 ---
writer = libsbml.SBMLWriter()
sbml_v6_string = writer.writeSBMLToString(sbml_doc_v6)
shared_variables.add("hypothetical_sbml_v6", sbml_v6_string)
# print("SBML Model v6 content ready.")

# --- Simulate Model v6 ---
sim_data_v6 = simulate(sbml_v6_string)
shared_variables.add("sim_data_v6", sim_data_v6)
print("Simulation of Model v6 completed.")

# --- Analyze Simulation Results ---
print("\nDescriptive statistics for observed data (iteration_0):")
print(obs_data.describe())

print("\nDescriptive statistics for simulated data (Model v6):")
sim_data_v6_adjusted = sim_data_v6.copy()
sim_data_v6_adjusted['id_plsk'] = sim_data_v6['id_plsk'] - IC_P_v6
sim_data_v6_adjusted['id_0nyd'] = sim_data_v6['id_0nyd'] - IC_N_v6
print(sim_data_v6_adjusted.describe())

print("\nObserved data (head):")
print(obs_data.head())
print("\nSimulated data v6 (adjusted, head):")
print(sim_data_v6_adjusted.head())

r7xb_sim_stats_v6 = sim_data_v6['id_r7xb'].describe()
print(f"\nSimulated id_r7xb (v6) stats: min={r7xb_sim_stats_v6['min']:.3f}, max={r7xb_sim_stats_v6['max']:.3f}, mean={r7xb_sim_stats_v6['mean']:.3f}, std={r7xb_sim_stats_v6['std']:.3f}")
if r7xb_sim_stats_v6['std'] > 0.05 and (r7xb_sim_stats_v6['max'] - r7xb_sim_stats_v6['min']) > 0.2 * r7xb_sim_stats_v6['mean'] : # Adjusted criteria
    print("Simulated id_r7xb (v6) shows significant dynamic/oscillatory behavior.")
else:
    print("Simulated id_r7xb (v6) is not strongly oscillating with these parameters.")

# Check id_plsk vs -id_22ta relationship
sim_plsk_adj_v6 = sim_data_v6['id_plsk'] - IC_P_v6
sim_neg_22ta_v6 = -sim_data_v6['id_22ta']
diff_plsk_22ta_v6 = np.abs(sim_plsk_adj_v6 - sim_neg_22ta_v6)
print(f"\nMean absolute difference between sim_plsk_adjusted (v6) and -sim_id_22ta (v6): {diff_plsk_22ta_v6.mean():.2e}")

```
# Observation

## Code Stdout
Simulation of Model v6 completed.

Descriptive statistics for observed data (iteration_0):
            id_x589       id_r7xb  ...       id_0nyd          Time
count  10001.000000  10001.000000  ...  10001.000000  10001.000000
mean       0.199954      0.547936  ...     -0.387890    487.500000
std        0.070800      0.179797  ...      0.165321    281.500475
min        0.000000      0.297089  ...     -0.710021      0.000000
25%        0.165670      0.383741  ...     -0.538543    243.750000
50%        0.225494      0.513895  ...     -0.368357    487.500000
75%        0.238572      0.679282  ...     -0.234269    731.250000
max        0.438452      0.939163  ...      0.000000    975.000000

[8 rows x 6 columns]

Descriptive statistics for simulated data (Model v6):
            id_x589       id_r7xb  ...       id_0nyd          Time
count  10001.000000  10001.000000  ...  10001.000000  10001.000000
mean       0.199175      0.553850  ...     -0.792736    487.500000
std        0.004940      0.005750  ...      0.053218    281.500475
min        0.000000      0.360000  ...     -0.800000      0.000000
25%        0.199330      0.553693  ...     -0.800000    243.750000
50%        0.199330      0.553693  ...     -0.800000    487.500000
75%        0.199330      0.553693  ...     -0.800000    731.250000
max        0.216347      0.683743  ...      0.000000    975.000000

[8 rows x 6 columns]

Observed data (head):
    id_x589   id_r7xb   id_22ta   id_plsk   id_0nyd    Time
0  0.000000  0.360000  0.000000  0.000000  0.000000  0.0000
1  0.062929  0.444693  0.384821 -0.384821 -0.147622  0.0975
2  0.068376  0.570143  0.618322 -0.618322 -0.278519  0.1950
3  0.074052  0.690106  0.758522 -0.758522 -0.404158  0.2925
4  0.081118  0.802562  0.840273 -0.840273 -0.523681  0.3900

Simulated data v6 (adjusted, head):
    id_x589   id_r7xb   id_22ta   id_plsk   id_0nyd    Time
0  0.000000  0.360000  0.000000  0.000000  0.000000  0.0000
1  0.012643  0.395210  0.007339 -0.007339 -0.005871  0.0975
2  0.025260  0.428739  0.015284 -0.015284 -0.012227  0.1950
3  0.037798  0.460643  0.023789 -0.023789 -0.019031  0.2925
4  0.050211  0.490936  0.032807 -0.032807 -0.026245  0.3900

Simulated id_r7xb (v6) stats: min=0.360, max=0.684, mean=0.554, std=0.006
Simulated id_r7xb (v6) is not strongly oscillating with these parameters.

Mean absolute difference between sim_plsk_adjusted (v6) and -sim_id_22ta (v6): 8.69e-16

## 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. (hypothetical_sbml_v4, sim_data_v4, hypothetical_sbml_v5, sim_data_v5, hypothetical_sbml_v6, sim_data_v6)

### 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

Model v6 achieved the correct mean values for `id_r7xb` (sim: 0.554, obs: 0.548) and `id_x589` (sim: 0.199, obs: 0.200). However, it *still* failed to produce sustained oscillations for `id_r7xb`. The simulated `id_r7xb` started at 0.36, rose to a peak near 0.68, and then settled to a steady state around 0.55. The standard deviation was very small (0.006), indicating a damped response rather than sustained oscillation. The observed data has `id_r7xb` std of 0.18.

The problem lies in the dynamics of the negative feedback loop. For sustained oscillations in a simple Goodwin-like model (one inhibitor), you typically need:
1.  A sufficiently high Hill coefficient (`h_X = 4.0` should be okay).
2.  A time delay in the feedback. In ODE models, this is often represented by a chain of intermediate species or by making the response of the inhibitor (`id_x589`) slower relative to the regulated species (`id_r7xb`).

In our current model:
-   `id_r7xb` dynamics: `d(id_r7xb)/dt = Vmax_R / (1 + (id_x589/K_IX)^h_X) - k_R_deg * id_r7xb`
-   `id_x589` dynamics: `d(id_x589)/dt = k_X_prod * id_r7xb - k_X_deg * id_x589`

The characteristic time for `id_r7xb` changes is roughly `1/k_R_deg = 1/0.5 = 2` time units.
The characteristic time for `id_x589` changes is `1/k_X_deg = 1/1.0 = 1` time unit.

If the inhibitor (`id_x589`) responds faster than or on a similar timescale to the regulated species (`id_r7xb`), it can quickly counteract changes in `id_r7xb`, leading to a stable steady state. To promote oscillations, `id_x589` should respond more slowly.

Strategy for Model v7:
1.  **Slow down `id_x589` dynamics**: Decrease `k_X_deg`. This will make `id_x589` accumulate and decay more slowly, introducing an effective delay in the feedback.
    -   Keep `k_X_prod / k_X_deg = 0.36` to maintain the `id_x589` mean.
    -   If we decrease `k_X_deg` to, say, `0.2` (making its timescale `1/0.2 = 5` units), then `k_X_prod` must be `0.36 * 0.2 = 0.072`.
2.  **Slightly adjust `k_PT` and `k_N`**: The consumption of `id_plsk` and `id_0nyd` in v6 was still too slow compared to observed data.
    -   Simulated `id_22ta` max was around 0.8 (target ~0.89). `id_0nyd` (adjusted) min was -0.8 (target ~-0.71).
    -   The rates `k_PT = 0.2` and `k_N = 0.2` might need to be increased. Let's try `k_PT = 0.4` and `k_N = 0.3`.

Revised parameters for v7:
*   `IC_P_v7 = 1.0`
*   `IC_N_v7 = 0.8`
*   `Vmax_R = 0.55` (keep)
*   `K_IX = 0.2` (keep)
*   `h_X = 4.0` (keep)
*   `k_R_deg = 0.5` (keep)
*   `k_X_prod = 0.072` (changed from 0.36)
*   `k_X_deg = 0.2` (changed from 1.0, to slow down `id_x589`)
*   `k_PT = 0.4` (increased from 0.2)
*   `k_N = 0.3` (increased from 0.2)

The model structure remains the same. The goal is to see if slowing down `id_x589` can trigger sustained oscillations in `id_r7xb`.

## Action

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

# --- Helper function to add reactions (L2 compatible) ---
def add_reaction_l2(model, r_id, reactants_dict, products_dict, formula_str, reversible=False):
    reaction = model.createReaction()
    reaction.setId(r_id)
    reaction.setReversible(reversible)

    for species_id, stoich in reactants_dict.items():
        reactant = reaction.createReactant()
        reactant.setSpecies(species_id)
        reactant.setStoichiometry(stoich)

    for species_id, stoich in products_dict.items():
        product = reaction.createProduct()
        product.setSpecies(species_id)
        product.setStoichiometry(stoich)

    kinetic_law = reaction.createKineticLaw()
    kinetic_law.setFormula(formula_str)
    return reaction

# --- Load original SBML and experimental data ---
sbml_doc_v7 = libsbml.readSBMLFromString(input_sbml_string)
model_v7 = sbml_doc_v7.getModel()
obs_data = experiment_history['iteration_0']

# --- Define Initial Concentrations ---
IC_P_v7 = 1.0
IC_N_v7 = 0.8

species_plsk = model_v7.getSpecies('id_plsk')
if species_plsk: species_plsk.setInitialConcentration(IC_P_v7)

species_0nyd = model_v7.getSpecies('id_0nyd')
if species_0nyd: species_0nyd.setInitialConcentration(IC_N_v7)
    
model_v7.getSpecies('id_x589').setInitialConcentration(0.0)
model_v7.getSpecies('id_22ta').setInitialConcentration(0.0)
model_v7.getSpecies('id_r7xb').setInitialConcentration(0.36)

# --- Define Parameters for v7 ---
param_definitions_v7 = {
    "Vmax_R": 0.55, "K_IX": 0.2, "h_X": 4.0, "k_R_deg": 0.5,
    "k_X_prod": 0.072, # Changed: 0.36 * (0.2/1.0)
    "k_X_deg": 0.2,    # Changed: Slow down id_x589
    "k_PT": 0.4,       # Increased
    "k_N": 0.3        # Increased
}

for p_id, p_val in param_definitions_v7.items():
    param = model_v7.getParameter(p_id)
    if not param:
        param = model_v7.createParameter()
        param.setId(p_id)
        param.setConstant(True)
    param.setValue(p_val)

# --- Add Reactions for v7 (same structure as v6) ---
formula_Rprod = "Vmax_R / (1 + (id_x589 / K_IX)^h_X)"
add_reaction_l2(model_v7, "R_Rprod", {}, {"id_r7xb": 1}, formula_Rprod)

formula_Rdeg = "k_R_deg * id_r7xb"
add_reaction_l2(model_v7, "R_Rdeg", {"id_r7xb": 1}, {}, formula_Rdeg)

formula_Xprod = "k_X_prod * id_r7xb"
add_reaction_l2(model_v7, "R_Xprod", {}, {"id_x589": 1}, formula_Xprod)

formula_Xdeg = "k_X_deg * id_x589"
add_reaction_l2(model_v7, "R_Xdeg", {"id_x589": 1}, {}, formula_Xdeg)

formula_PT = "k_PT * id_plsk * id_r7xb"
add_reaction_l2(model_v7, "R_PT", {"id_plsk": 1}, {"id_22ta": 1}, formula_PT)

formula_N = "k_N * id_0nyd * id_r7xb"
add_reaction_l2(model_v7, "R_N", {"id_0nyd": 1}, {}, formula_N)

# --- Generate SBML string for Model v7 ---
writer = libsbml.SBMLWriter()
sbml_v7_string = writer.writeSBMLToString(sbml_doc_v7)
shared_variables.add("hypothetical_sbml_v7", sbml_v7_string)

# --- Simulate Model v7 ---
sim_data_v7 = simulate(sbml_v7_string)
shared_variables.add("sim_data_v7", sim_data_v7)
print("Simulation of Model v7 completed.")

# --- Analyze Simulation Results ---
print("\nDescriptive statistics for observed data (iteration_0):")
print(obs_data.describe())

print("\nDescriptive statistics for simulated data (Model v7):")
sim_data_v7_adjusted = sim_data_v7.copy()
sim_data_v7_adjusted['id_plsk'] = sim_data_v7['id_plsk'] - IC_P_v7
sim_data_v7_adjusted['id_0nyd'] = sim_data_v7['id_0nyd'] - IC_N_v7
print(sim_data_v7_adjusted.describe())

r7xb_sim_stats_v7 = sim_data_v7['id_r7xb'].describe()
print(f"\nSimulated id_r7xb (v7) stats: min={r7xb_sim_stats_v7['min']:.3f}, max={r7xb_sim_stats_v7['max']:.3f}, mean={r7xb_sim_stats_v7['mean']:.3f}, std={r7xb_sim_stats_v7['std']:.3f}")
obs_r7xb_std = obs_data['id_r7xb'].std()
if r7xb_sim_stats_v7['std'] > 0.5 * obs_r7xb_std : # Check if std is at least half of observed
    print("Simulated id_r7xb (v7) shows significant dynamic/oscillatory behavior.")
else:
    print("Simulated id_r7xb (v7) is not strongly oscillating with these parameters.")

sim_plsk_adj_v7 = sim_data_v7['id_plsk'] - IC_P_v7
sim_neg_22ta_v7 = -sim_data_v7['id_22ta']
diff_plsk_22ta_v7 = np.abs(sim_plsk_adj_v7 - sim_neg_22ta_v7)
print(f"\nMean absolute difference between sim_plsk_adjusted (v7) and -sim_id_22ta (v7): {diff_plsk_22ta_v7.mean():.2e}")

# Compare key stats
print(f"Observed id_r7xb: mean={obs_data['id_r7xb'].mean():.3f}, std={obs_data['id_r7xb'].std():.3f}, min={obs_data['id_r7xb'].min():.3f}, max={obs_data['id_r7xb'].max():.3f}")
print(f"Simulated id_r7xb (v7): mean={r7xb_sim_stats_v7['mean']:.3f}, std={r7xb_sim_stats_v7['std']:.3f}, min={r7xb_sim_stats_v7['min']:.3f}, max={r7xb_sim_stats_v7['max']:.3f}")

print(f"Observed id_x589: mean={obs_data['id_x589'].mean():.3f}, std={obs_data['id_x589'].std():.3f}")
print(f"Simulated id_x589 (v7): mean={sim_data_v7['id_x589'].mean():.3f}, std={sim_data_v7['id_x589'].std():.3f}")

print(f"Observed id_22ta: max={obs_data['id_22ta'].max():.3f}")
print(f"Simulated id_22ta (v7): max={sim_data_v7['id_22ta'].max():.3f}")

print(f"Observed id_0nyd (adj): min={obs_data['id_0nyd'].min():.3f}") # This is already "adjusted" in data
print(f"Simulated id_0nyd (adj, v7): min={sim_data_v7_adjusted['id_0nyd'].min():.3f}")

```
# Observation

## Code Stdout
Simulation of Model v7 completed.

Descriptive statistics for observed data (iteration_0):
            id_x589       id_r7xb  ...       id_0nyd          Time
count  10001.000000  10001.000000  ...  10001.000000  10001.000000
mean       0.199954      0.547936  ...     -0.387890    487.500000
std        0.070800      0.179797  ...      0.165321    281.500475
min        0.000000      0.297089  ...     -0.710021      0.000000
25%        0.165670      0.383741  ...     -0.538543    243.750000
50%        0.225494      0.513895  ...     -0.368357    487.500000
75%        0.238572      0.679282  ...     -0.234269    731.250000
max        0.438452      0.939163  ...      0.000000    975.000000

[8 rows x 6 columns]

Descriptive statistics for simulated data (Model v7):
            id_x589       id_r7xb  ...       id_0nyd          Time
count  10001.000000  10001.000000  ...  10001.000000  10001.000000
mean       0.198833      0.555170  ...     -0.796004    487.500000
std        0.008579      0.021270  ...      0.040047    281.500475
min        0.000000      0.360000  ...     -0.800000      0.000000
25%        0.199330      0.553693  ...     -0.800000    243.750000
50%        0.199330      0.553693  ...     -0.800000    487.500000
75%        0.199330      0.553693  ...     -0.800000    731.250000
max        0.211360      0.911463  ...      0.000000    975.000000

[8 rows x 6 columns]

Simulated id_r7xb (v7) stats: min=0.360, max=0.911, mean=0.555, std=0.021
Simulated id_r7xb (v7) is not strongly oscillating with these parameters.

Mean absolute difference between sim_plsk_adjusted (v7) and -sim_id_22ta (v7): 5.42e-16
Observed id_r7xb: mean=0.548, std=0.180, min=0.297, max=0.939
Simulated id_r7xb (v7): mean=0.555, std=0.021, min=0.360, max=0.911
Observed id_x589: mean=0.200, std=0.071
Simulated id_x589 (v7): mean=0.199, std=0.009
Observed id_22ta: max=0.891
Simulated id_22ta (v7): max=1.000
Observed id_0nyd (adj): min=-0.710
Simulated id_0nyd (adj, v7): min=-0.800

## 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. (hypothetical_sbml_v4, sim_data_v4, hypothetical_sbml_v5, sim_data_v5, hypothetical_sbml_v6, sim_data_v6, hypothetical_sbml_v7, sim_data_v7)

### 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

Model v7 still failed to produce sustained oscillations, even after slowing down `id_x589` dynamics. The standard deviation of `id_r7xb` (0.021) is far from the observed (0.180). The system settles to a steady state. The means are good, but the dynamics are wrong.
-   Simulated `id_r7xb`: mean=0.555 (obs=0.548), std=0.021 (obs=0.180), min=0.360 (obs=0.297), max=0.911 (obs=0.939). Max is good, min is too high.
-   Simulated `id_x589`: mean=0.199 (obs=0.200), std=0.009 (obs=0.071). Std is too low.
-   Simulated `id_22ta` max=1.000 (obs=0.891). This means `IC_P_v7 = 1.0` is fully consumed. This is good.
-   Simulated `id_0nyd` (adj) min=-0.800 (obs=-0.710). `IC_N_v7 = 0.8` is fully consumed. This is good.

The lack of oscillation suggests that the simple negative feedback loop with one intermediate (`id_x589`) might not be sufficient, or the parameter space for oscillation is very narrow and hard to hit with manual tuning. Oscillations in biochemical systems often arise from:
1.  **Time delays**: Explicit or implicit (e.g., a longer chain of reactions in the feedback loop).
2.  **Multiple feedback loops**: Positive and negative feedback interacting.
3.  **Zero-order ultrasensitivity / cooperativity**: High Hill coefficients are a form of this.
4.  **Substrate depletion**: If a reactant for `id_r7xb` production was also part of the loop.

We've tried slowing `id_x589` (implicit delay) and have a high Hill coefficient.
Perhaps the issue is that `id_x589` is too directly coupled. A longer chain for the negative feedback might be needed. However, the problem states "You do not need to add new species." This implies the existing species must form the oscillatory circuit.

Let's reconsider the parameters for the `id_r7xb` <-> `id_x589` loop.
The timescale for `id_x589` (controlled by `k_X_deg`) needs to be "just right" relative to `id_r7xb` (controlled by `k_R_deg`).
In v7, `1/k_R_deg = 1/0.5 = 2`. `1/k_X_deg = 1/0.2 = 5`. `id_x589` is slower.

What if `id_x589` needs to be even slower to create a more pronounced phase lag?
Or, what if the Hill coefficient `h_X` needs to be even higher? (Though 4 is already quite high).

Let's try to make `id_x589` significantly slower.
If `k_X_deg` is very small, `id_x589` will integrate `id_r7xb` over a long time.
Current parameters for `id_r7xb` and `id_x589`:
*   `Vmax_R = 0.55`, `K_IX = 0.2`, `h_X = 4.0`, `k_R_deg = 0.5`
*   `k_X_prod = 0.072`, `k_X_deg = 0.2`

Let's try `k_X_deg = 0.05` (timescale = 20).
Then `k_X_prod = 0.36 * 0.05 = 0.018`.
This makes `id_x589` respond very slowly.

The other species (`id_plsk`, `id_0nyd`) seem to be behaving reasonably once `id_r7xb` is correct, as their consumption is driven by `id_r7xb`. The `k_PT` and `k_N` values from v7 (`0.4`, `0.3`) resulted in full consumption, which matches the data where `id_22ta` reaches a plateau and `id_0nyd` (adj) reaches a minimum plateau. The exact values of these plateaus depend on `IC_P` and `IC_N`.
Observed `id_22ta` max ~0.89. So `IC_P` should be ~0.89.
Observed `id_0nyd` (adj) min ~-0.71. So `IC_N` should be ~0.71.
Let's adjust `IC_P` and `IC_N` to these values.

Revised parameters for Model v8:
*   `IC_P_v8 = 0.89` (refined)
*   `IC_N_v8 = 0.71` (refined)
*   `Vmax_R = 0.55`
*   `K_IX = 0.2`
*   `h_X = 4.0` (Let's try increasing this to 5 or 6 to see if it helps push into oscillation) -> Let's try `h_X = 6.0`
*   `k_R_deg = 0.5`
*   `k_X_prod = 0.018` (changed: `0.36 * (0.05/1.0)`)
*   `k_X_deg = 0.05` (changed: to make `id_x589` much slower)
*   `k_PT = 0.4` (keep from v7, might need slight tuning if oscillation period changes)
*   `k_N = 0.3` (keep from v7)

The main changes are significantly slowing `id_x589` and increasing `h_X`, plus refining `IC_P` and `IC_N`.

## Action

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

# --- Helper function to add reactions (L2 compatible) ---
def add_reaction_l2(model, r_id, reactants_dict, products_dict, formula_str, reversible=False):
    reaction = model.createReaction()
    reaction.setId(r_id)
    reaction.setReversible(reversible)

    for species_id, stoich in reactants_dict.items():
        reactant = reaction.createReactant()
        reactant.setSpecies(species_id)
        reactant.setStoichiometry(stoich)

    for species_id, stoich in products_dict.items():
        product = reaction.createProduct()
        product.setSpecies(species_id)
        product.setStoichiometry(stoich)

    kinetic_law = reaction.createKineticLaw()
    kinetic_law.setFormula(formula_str)
    return reaction

# --- Load original SBML and experimental data ---
sbml_doc_v8 = libsbml.readSBMLFromString(input_sbml_string)
model_v8 = sbml_doc_v8.getModel()
obs_data = experiment_history['iteration_0']

# --- Define Initial Concentrations ---
IC_P_v8 = 0.89 # Refined based on obs max(id_22ta)
IC_N_v8 = 0.71 # Refined based on obs min(abs(id_0nyd))

species_plsk = model_v8.getSpecies('id_plsk')
if species_plsk: species_plsk.setInitialConcentration(IC_P_v8)

species_0nyd = model_v8.getSpecies('id_0nyd')
if species_0nyd: species_0nyd.setInitialConcentration(IC_N_v8)
    
model_v8.getSpecies('id_x589').setInitialConcentration(0.0)
model_v8.getSpecies('id_22ta').setInitialConcentration(0.0)
model_v8.getSpecies('id_r7xb').setInitialConcentration(0.36)

# --- Define Parameters for v8 ---
param_definitions_v8 = {
    "Vmax_R": 0.55, "K_IX": 0.2, "h_X": 6.0,    # Increased h_X
    "k_R_deg": 0.5,
    "k_X_prod": 0.018, # Changed: 0.36 * (0.05/1.0) where 0.36 was kXp/kXd ratio
    "k_X_deg": 0.05,   # Changed: Slow down id_x589 significantly
    "k_PT": 0.4,
    "k_N": 0.3
}
# Recalculate k_X_prod based on target mean for id_x589 (0.2) and id_r7xb (0.55)
# Target: id_x589_mean = (k_X_prod / k_X_deg) * id_r7xb_mean
# 0.2 = (k_X_prod / 0.05) * 0.55  => k_X_prod = (0.2 * 0.05) / 0.55 = 0.01 / 0.55 = 0.01818
param_definitions_v8["k_X_prod"] = 0.018181818


for p_id, p_val in param_definitions_v8.items():
    param = model_v8.getParameter(p_id)
    if not param:
        param = model_v8.createParameter()
        param.setId(p_id)
        param.setConstant(True)
    param.setValue(p_val)

# --- Add Reactions for v8 (same structure) ---
formula_Rprod = "Vmax_R / (1 + (id_x589 / K_IX)^h_X)"
add_reaction_l2(model_v8, "R_Rprod", {}, {"id_r7xb": 1}, formula_Rprod)

formula_Rdeg = "k_R_deg * id_r7xb"
add_reaction_l2(model_v8, "R_Rdeg", {"id_r7xb": 1}, {}, formula_Rdeg)

formula_Xprod = "k_X_prod * id_r7xb"
add_reaction_l2(model_v8, "R_Xprod", {}, {"id_x589": 1}, formula_Xprod)

formula_Xdeg = "k_X_deg * id_x589"
add_reaction_l2(model_v8, "R_Xdeg", {"id_x589": 1}, {}, formula_Xdeg)

formula_PT = "k_PT * id_plsk * id_r7xb"
add_reaction_l2(model_v8, "R_PT", {"id_plsk": 1}, {"id_22ta": 1}, formula_PT)

formula_N = "k_N * id_0nyd * id_r7xb"
add_reaction_l2(model_v8, "R_N", {"id_0nyd": 1}, {}, formula_N)

# --- Generate SBML string for Model v8 ---
writer = libsbml.SBMLWriter()
sbml_v8_string = writer.writeSBMLToString(sbml_doc_v8)
shared_variables.add("hypothetical_sbml_v8", sbml_v8_string)

# --- Simulate Model v8 ---
sim_data_v8 = simulate(sbml_v8_string)
shared_variables.add("sim_data_v8", sim_data_v8)
print("Simulation of Model v8 completed.")

# --- Analyze Simulation Results ---
sim_data_v8_adjusted = sim_data_v8.copy()
sim_data_v8_adjusted['id_plsk'] = sim_data_v8['id_plsk'] - IC_P_v8
sim_data_v8_adjusted['id_0nyd'] = sim_data_v8['id_0nyd'] - IC_N_v8

r7xb_sim_stats_v8 = sim_data_v8['id_r7xb'].describe()
print(f"\nSimulated id_r7xb (v8) stats: min={r7xb_sim_stats_v8['min']:.3f}, max={r7xb_sim_stats_v8['max']:.3f}, mean={r7xb_sim_stats_v8['mean']:.3f}, std={r7xb_sim_stats_v8['std']:.3f}")
obs_r7xb_std = obs_data['id_r7xb'].std()
if r7xb_sim_stats_v8['std'] > 0.7 * obs_r7xb_std : # Stricter criteria for oscillation std
    print("Simulated id_r7xb (v8) shows significant dynamic/oscillatory behavior.")
else:
    print("Simulated id_r7xb (v8) is not strongly oscillating with these parameters.")

print(f"Observed id_r7xb: mean={obs_data['id_r7xb'].mean():.3f}, std={obs_data['id_r7xb'].std():.3f}, min={obs_data['id_r7xb'].min():.3f}, max={obs_data['id_r7xb'].max():.3f}")
print(f"Simulated id_r7xb (v8): mean={r7xb_sim_stats_v8['mean']:.3f}, std={r7xb_sim_stats_v8['std']:.3f}, min={r7xb_sim_stats_v8['min']:.3f}, max={r7xb_sim_stats_v8['max']:.3f}")

print(f"Observed id_x589: mean={obs_data['id_x589'].mean():.3f}, std={obs_data['id_x589'].std():.3f}, min={obs_data['id_x589'].min():.3f}, max={obs_data['id_x589'].max():.3f}")
print(f"Simulated id_x589 (v8): mean={sim_data_v8['id_x589'].mean():.3f}, std={sim_data_v8['id_x589'].std():.3f}, min={sim_data_v8['id_x589'].min():.3f}, max={sim_data_v8['id_x589'].max():.3f}")

print(f"Observed id_22ta: max={obs_data['id_22ta'].max():.3f}")
print(f"Simulated id_22ta (v8): max={sim_data_v8['id_22ta'].max():.3f}")

print(f"Observed id_0nyd (adj): min={obs_data['id_0nyd'].min():.3f}")
print(f"Simulated id_0nyd (adj, v8): min={sim_data_v8_adjusted['id_0nyd'].min():.3f}")

```
# Observation

## Code Stdout
Simulation of Model v8 completed.

Simulated id_r7xb (v8) stats: min=0.360, max=1.075, mean=0.557, std=0.053
Simulated id_r7xb (v8) is not strongly oscillating with these parameters.
Observed id_r7xb: mean=0.548, std=0.180, min=0.297, max=0.939
Simulated id_r7xb (v8): mean=0.557, std=0.053, min=0.360, max=1.075
Observed id_x589: mean=0.200, std=0.071, min=0.000, max=0.438
Simulated id_x589 (v8): mean=0.198, std=0.014, min=0.000, max=0.200
Observed id_22ta: max=0.891
Simulated id_22ta (v8): max=0.890
Observed id_0nyd (adj): min=-0.710
Simulated id_0nyd (adj, v8): min=-0.710

## 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. (hypothetical_sbml_v4, sim_data_v4, hypothetical_sbml_v5, sim_data_v5, hypothetical_sbml_v6, sim_data_v6, hypothetical_sbml_v7, sim_data_v7, hypothetical_sbml_v8, sim_data_v8)

### 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

Model v8 is getting closer but still no sustained oscillations.
-   `id_r7xb`: mean is good (0.557 vs 0.548 obs). Max is a bit high (1.075 vs 0.939 obs). Min is too high (0.360 vs 0.297 obs). Critically, std is still too low (0.053 vs 0.180 obs).
-   `id_x589`: mean is good (0.198 vs 0.200 obs). Std is too low (0.014 vs 0.071 obs). Max is too low (0.200 vs 0.438 obs). Min is okay (0.000 vs 0.000 obs).
-   `id_22ta` max is perfect (0.890 vs 0.891 obs).
-   `id_0nyd` (adj) min is perfect (-0.710 vs -0.710 obs).
This means `IC_P_v8 = 0.89` and `IC_N_v8 = 0.71` are correct. Also, `k_PT=0.4` and `k_N=0.3` seem to be working well given the `id_r7xb` profile.

The core issue remains the `id_r7xb` <-> `id_x589` oscillator.
We have:
-   `Vmax_R = 0.55`
-   `K_IX = 0.2` (This is the concentration of `id_x589` at which inhibition is half-maximal for the `(id_x589/K_IX)^h_X` term, effectively where `id_x589` starts to strongly inhibit)
-   `h_X = 6.0` (Hill coefficient)
-   `k_R_deg = 0.5` (timescale for `id_r7xb` is ~2 time units)
-   `k_X_prod = 0.01818`
-   `k_X_deg = 0.05` (timescale for `id_x589` is ~20 time units)

`id_x589` is now much slower than `id_r7xb`. This should promote oscillations. The high Hill coefficient (`h_X=6`) should also help.
Why is it still not oscillating with sufficient amplitude?

Possible reasons:
1.  **Parameter balance**: The balance between production (`Vmax_R`) and degradation (`k_R_deg`) of `id_r7xb`, and the inhibition strength (`K_IX`, `h_X`), might be such that it settles into a stable focus rather than a limit cycle.
2.  **`K_IX` value**: The `K_IX` is set to the mean of `id_x589`. If `id_x589` needs to swing more to cause `id_r7xb` to swing, perhaps `K_IX` needs to be lower, making the inhibition kick in more sharply at lower `id_x589` levels. Or, if `id_x589` doesn't rise enough, `K_IX` might be too high.
    -   Observed `id_x589` max is 0.438. Simulated `id_x589` max is 0.200. This is a major discrepancy. `id_x589` is not rising enough.
    -   This means `k_X_prod * id_r7xb` is not high enough on average, or `k_X_deg` is too high relative to `k_X_prod` for the peak.
    -   The mean of `id_x589` is correct, so the ratio `k_X_prod / k_X_deg` is correct (0.01818 / 0.05 = 0.3636).
    -   If `id_x589` needs to reach a higher max, its production rate needs to be higher when `id_r7xb` is high, or its degradation needs to be slower (already very slow).

Let's analyze `id_x589` dynamics: `d(id_x589)/dt = k_X_prod * id_r7xb - k_X_deg * id_x589`.
If `id_r7xb` peaks at ~1.0 (as in sim v8, or obs 0.939), then `id_x589` production rate at peak `id_r7xb` is `0.01818 * 1.0 = 0.01818`.
If `id_x589` needs to reach 0.438, and its degradation rate constant is `k_X_deg = 0.05`, then at steady state `id_x589_ss = (k_X_prod/k_X_deg) * id_r7xb_avg = 0.3636 * 0.55 = 0.2`. This is the mean.
To get a higher peak for `id_x589`, we need `id_r7xb` to stay high for longer, or `k_X_prod` to be larger (which would also increase the mean of `id_x589` unless `k_X_deg` also increases, which would speed it up again).

The problem might be that `id_r7xb`'s production is shut down too effectively or too quickly by `id_x589` reaching its mean of 0.2.
If `K_IX = 0.2`, then when `id_x589` is 0.2, the term `(id_x589/K_IX)^h_X = (0.2/0.2)^6 = 1^6 = 1`.
The production rate of `id_r7xb` becomes `Vmax_R / (1+1) = Vmax_R / 2`.
This is `0.55 / 2 = 0.275`.
The degradation rate of `id_r7xb` when `id_r7xb` is at its mean (0.55) is `k_R_deg * id_r7xb = 0.5 * 0.55 = 0.275`.
This is why it settles at the mean.

For oscillations, `id_x589` needs to "overshoot" its mean significantly to drive `id_r7xb` down, and then "undershoot" to allow `id_r7xb` to rise high.
The simulated `id_x589` max is 0.2, which is its mean. It's not overshooting at all. This is the key problem. `id_x589` is acting like a perfect proportional controller with no lag, despite `k_X_deg` being small.

This implies that the timescale separation (20 for `id_x589` vs 2 for `id_r7xb`) is still not enough, or the system is overly damped.

What if `K_IX` is actually lower than the mean of `id_x589`?
Say `K_IX = 0.1`.
Then when `id_x589` reaches 0.1, inhibition starts. When `id_x589` reaches 0.2 (its mean), `(0.2/0.1)^6 = 2^6 = 64`.
Production of `id_r7xb` would be `Vmax_R / (1+64) = Vmax_R / 65`. This is very strong inhibition.
This would make `id_r7xb` drop sharply once `id_x589` crosses 0.1.
This might make `id_x589` also drop, leading to an undershoot.

Let's try Model v9 with:
-   `K_IX = 0.1` (making inhibition more sensitive at lower `id_x589` levels).
-   To maintain `id_r7xb` mean around 0.55:
    If `id_x589_mean = 0.2`, then `(id_x589_mean/K_IX)^h_X = (0.2/0.1)^6 = 2^6 = 64`.
    `Vmax_R / (1+64) = k_R_deg * id_r7xb_mean`
    `Vmax_R / 65 = 0.5 * 0.55 = 0.275`
    `Vmax_R = 0.275 * 65 = 17.875`. This seems very high for `Vmax_R`.
    This approach of setting `K_IX` far from `id_x589_mean` and compensating with `Vmax_R` might be too extreme.

Alternative: The system might require a delay that is not simply achieved by slowing one species.
However, "no new species".

Let's re-evaluate the timescales.
`id_r7xb` timescale `1/k_R_deg = 1/0.5 = 2`.
`id_x589` timescale `1/k_X_deg = 1/0.05 = 20`.
This is a 10x separation. Usually, this is enough for some oscillatory behavior in a 2-species negative feedback if the gain is high. The gain is high due to `h_X=6`.

Perhaps the initial condition of `id_r7xb = 0.36` is too close to the eventual trough of the oscillation, preventing it from "kicking off" properly if the basin of attraction for the limit cycle is small or requires a larger perturbation. This is unlikely to be the sole cause for sustained non-oscillation.

Let's reconsider the `id_x589` max value. Observed `id_x589` max is 0.438. Simulated is 0.2.
If `id_x589` needs to reach 0.438, and its mean is 0.2, it needs to swing significantly.
If `id_x589` reaches 0.438, and `K_IX = 0.2`, then `(0.438/0.2)^6 = (2.19)^6 = 100.8`.
Inhibition term `1 / (1 + 100.8) = 1/101.8`. This would shut down `id_r7xb` production.
This part seems okay. The issue is `id_x589` *not reaching* that high.

To make `id_x589` go higher, given `d(id_x589)/dt = k_X_prod * id_r7xb - k_X_deg * id_x589`:
-   `id_r7xb` needs to be higher for longer.
-   `k_X_prod` needs to be larger (but this changes the mean unless `k_X_deg` also changes).
-   `k_X_deg` needs to be smaller (already very small).

What if the `Vmax_R` is too low, so `id_r7xb` cannot climb high enough to drive `id_x589` high enough?
Simulated `id_r7xb` max was 1.075. Observed is 0.939. So `id_r7xb` *can* go high.
When `id_r7xb` is 1.0, `d(id_x589)/dt` (initially, if `id_x589` is low) = `0.01818 * 1.0 = 0.01818`.
If `id_x589` is at its mean of 0.2, `k_X_deg * id_x589 = 0.05 * 0.2 = 0.01`.
So, when `id_r7xb` is high (1.0) and `id_x589` is at its mean (0.2), `d(id_x589)/dt = 0.01818 - 0.01 = 0.00818 > 0`. So `id_x589` will rise.
It rises until `k_X_prod * id_r7xb = k_X_deg * id_x589_peak`.
`id_x589_peak = (k_X_prod / k_X_deg) * id_r7xb_at_that_moment`.
`id_x589_peak = 0.3636 * id_r7xb_at_that_moment`.
If `id_r7xb_at_that_moment` is around its peak of 1.0, then `id_x589_peak` could be ~0.36. This is better than 0.2, but still not 0.438.
This suggests that `id_r7xb` doesn't stay at its peak long enough for `id_x589` to catch up, because `id_x589` (even at 0.2) is already strongly inhibiting `id_r7xb` production.

This points back to `K_IX`. If `K_IX` is higher, say `K_IX = 0.3`, then `id_x589` needs to rise higher before significant inhibition occurs.
Let's try `K_IX = 0.3`.
-   Target `id_x589_mean = 0.2`.
-   `h_X = 6.0`.
-   `(id_x589_mean/K_IX)^h_X = (0.2/0.3)^6 = (2/3)^6 = 64/729 = 0.0878`.
-   `Vmax_R / (1 + 0.0878) = k_R_deg * id_r7xb_mean`
    `Vmax_R / 1.0878 = 0.5 * 0.55 = 0.275`
    `Vmax_R = 0.275 * 1.0878 = 0.299`. This is a more reasonable `Vmax_R`.

Parameters for Model v9:
*   `IC_P_v9 = 0.89`
*   `IC_N_v9 = 0.71`
*   `Vmax_R = 0.299` (changed)
*   `K_IX = 0.3` (changed)
*   `h_X = 6.0` (keep)
*   `k_R_deg = 0.5` (keep)
*   `k_X_prod` and `k_X_deg` need to ensure `id_x589_mean = 0.2` and `id_x589_timescale = 20`.
    `k_X_deg = 0.05` (keep for timescale = 20).
    `k_X_prod = (id_x589_mean / id_r7xb_mean) * k_X_deg = (0.2 / 0.55) * 0.05 = 0.3636 * 0.05 = 0.01818` (this calculation is independent of `K_IX` and `Vmax_R` for the mean, so it stays the same).
*   `k_PT = 0.4`
*   `k_N = 0.3`

The main idea is that by increasing `K_IX`, `id_x589` has to rise to higher levels before it effectively inhibits `id_r7xb` production. This might allow `id_x589` to achieve a higher peak, closer to the observed 0.438, and induce larger swings.

## Action

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

# --- Helper function to add reactions (L2 compatible) ---
def add_reaction_l2(model, r_id, reactants_dict, products_dict, formula_str, reversible=False):
    reaction = model.createReaction()
    reaction.setId(r_id)
    reaction.setReversible(reversible)

    for species_id, stoich in reactants_dict.items():
        reactant = reaction.createReactant()
        reactant.setSpecies(species_id)
        reactant.setStoichiometry(stoich)

    for species_id, stoich in products_dict.items():
        product = reaction.createProduct()
        product.setSpecies(species_id)
        product.setStoichiometry(stoich)

    kinetic_law = reaction.createKineticLaw()
    kinetic_law.setFormula(formula_str)
    return reaction

# --- Load original SBML and experimental data ---
sbml_doc_v9 = libsbml.readSBMLFromString(input_sbml_string)
model_v9 = sbml_doc_v9.getModel()
obs_data = experiment_history['iteration_0']

# --- Define Initial Concentrations ---
IC_P_v9 = 0.89
IC_N_v9 = 0.71

species_plsk = model_v9.getSpecies('id_plsk')
if species_plsk: species_plsk.setInitialConcentration(IC_P_v9)

species_0nyd = model_v9.getSpecies('id_0nyd')
if species_0nyd: species_0nyd.setInitialConcentration(IC_N_v9)
    
model_v9.getSpecies('id_x589').setInitialConcentration(0.0)
model_v9.getSpecies('id_22ta').setInitialConcentration(0.0)
model_v9.getSpecies('id_r7xb').setInitialConcentration(0.36)

# --- Define Parameters for v9 ---
param_definitions_v9 = {
    "Vmax_R": 0.299145, # Changed: 0.275 * (1 + (0.2/0.3)^6)
    "K_IX": 0.3,       # Changed: Higher K_IX
    "h_X": 6.0,
    "k_R_deg": 0.5,
    "k_X_prod": 0.018181818, # (0.2 / 0.55) * 0.05
    "k_X_deg": 0.05,
    "k_PT": 0.4,
    "k_N": 0.3
}

for p_id, p_val in param_definitions_v9.items():
    param = model_v9.getParameter(p_id)
    if not param:
        param = model_v9.createParameter()
        param.setId(p_id)
        param.setConstant(True)
    param.setValue(p_val)

# --- Add Reactions for v9 (same structure) ---
formula_Rprod = "Vmax_R / (1 + (id_x589 / K_IX)^h_X)"
add_reaction_l2(model_v9, "R_Rprod", {}, {"id_r7xb": 1}, formula_Rprod)

formula_Rdeg = "k_R_deg * id_r7xb"
add_reaction_l2(model_v9, "R_Rdeg", {"id_r7xb": 1}, {}, formula_Rdeg)

formula_Xprod = "k_X_prod * id_r7xb"
add_reaction_l2(model_v9, "R_Xprod", {}, {"id_x589": 1}, formula_Xprod)

formula_Xdeg = "k_X_deg * id_x589"
add_reaction_l2(model_v9, "R_Xdeg", {"id_x589": 1}, {}, formula_Xdeg)

formula_PT = "k_PT * id_plsk * id_r7xb"
add_reaction_l2(model_v9, "R_PT", {"id_plsk": 1}, {"id_22ta": 1}, formula_PT)

formula_N = "k_N * id_0nyd * id_r7xb"
add_reaction_l2(model_v9, "R_N", {"id_0nyd": 1}, {}, formula_N)

# --- Generate SBML string for Model v9 ---
writer = libsbml.SBMLWriter()
sbml_v9_string = writer.writeSBMLToString(sbml_doc_v9)
shared_variables.add("hypothetical_sbml_v9", sbml_v9_string)

# --- Simulate Model v9 ---
sim_data_v9 = simulate(sbml_v9_string)
shared_variables.add("sim_data_v9", sim_data_v9)
print("Simulation of Model v9 completed.")

# --- Analyze Simulation Results ---
sim_data_v9_adjusted = sim_data_v9.copy()
sim_data_v9_adjusted['id_plsk'] = sim_data_v9['id_plsk'] - IC_P_v9
sim_data_v9_adjusted['id_0nyd'] = sim_data_v9['id_0nyd'] - IC_N_v9

r7xb_sim_stats_v9 = sim_data_v9['id_r7xb'].describe()
x589_sim_stats_v9 = sim_data_v9['id_x589'].describe()

print(f"\nObserved id_r7xb: mean={obs_data['id_r7xb'].mean():.3f}, std={obs_data['id_r7xb'].std():.3f}, min={obs_data['id_r7xb'].min():.3f}, max={obs_data['id_r7xb'].max():.3f}")
print(f"Simulated id_r7xb (v9): mean={r7xb_sim_stats_v9['mean']:.3f}, std={r7xb_sim_stats_v9['std']:.3f}, min={r7xb_sim_stats_v9['min']:.3f}, max={r7xb_sim_stats_v9['max']:.3f}")

print(f"Observed id_x589: mean={obs_data['id_x589'].mean():.3f}, std={obs_data['id_x589'].std():.3f}, min={obs_data['id_x589'].min():.3f}, max={obs_data['id_x589'].max():.3f}")
print(f"Simulated id_x589 (v9): mean={x589_sim_stats_v9['mean']:.3f}, std={x589_sim_stats_v9['std']:.3f}, min={x589_sim_stats_v9['min']:.3f}, max={x589_sim_stats_v9['max']:.3f}")

if r7xb_sim_stats_v9['std'] > 0.1: # Threshold for significant oscillation
    print("Simulated id_r7xb (v9) shows significant dynamic/oscillatory behavior.")
else:
    print("Simulated id_r7xb (v9) is not strongly oscillating with these parameters.")

print(f"Observed id_22ta: max={obs_data['id_22ta'].max():.3f}")
print(f"Simulated id_22ta (v9): max={sim_data_v9['id_22ta'].max():.3f}")
print(f"Observed id_0nyd (adj): min={obs_data['id_0nyd'].min():.3f}")
print(f"Simulated id_0nyd (adj, v9): min={sim_data_v9_adjusted['id_0nyd'].min():.3f}")

```
# Observation

## Code Stdout
Simulation of Model v9 completed.

Observed id_r7xb: mean=0.548, std=0.180, min=0.297, max=0.939
Simulated id_r7xb (v9): mean=0.552, std=0.009, min=0.360, max=0.598
Observed id_x589: mean=0.200, std=0.071, min=0.000, max=0.438
Simulated id_x589 (v9): mean=0.196, std=0.019, min=0.000, max=0.200
Simulated id_r7xb (v9) is not strongly oscillating with these parameters.
Observed id_22ta: max=0.891
Simulated id_22ta (v9): max=0.890
Observed id_0nyd (adj): min=-0.710
Simulated id_0nyd (adj, v9): min=-0.710

## 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. (hypothetical_sbml_v4, sim_data_v4, hypothetical_sbml_v5, sim_data_v5, hypothetical_sbml_v6, sim_data_v6, hypothetical_sbml_v7, sim_data_v7, hypothetical_sbml_v8, sim_data_v8, hypothetical_sbml_v9, sim_data_v9)

### 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

Model v9 was a step backward for oscillations. `id_r7xb` stddev dropped to 0.009. Max `id_r7xb` dropped to 0.598. Max `id_x589` remained at 0.200.
The attempt to raise `K_IX` to allow `id_x589` to grow larger before inhibiting `id_r7xb` did not work as intended; it seems to have stabilized the system further. The `Vmax_R` was adjusted to maintain the mean, but the dynamics worsened.

This is becoming a difficult parameter tuning problem for the oscillator. The core structure `R_Rprod`, `R_Rdeg`, `R_Xprod`, `R_Xdeg` is standard for a negative feedback loop. The issue is finding the parameter region for sustained oscillations that match the observed amplitude and period.

Let's review the conditions for oscillation in a two-variable system like the `id_r7xb` (activator/regulated) and `id_x589` (inhibitor) loop:
1.  **Negative feedback**: Present.
2.  **Sufficient time delay (phase lag)**: We've tried to achieve this by making `id_x589` kinetics slower (`1/k_X_deg` large).
3.  **Sufficient nonlinearity (steepness of response)**: `h_X = 6.0` is high.
4.  **Unstable steady state**: The parameters must lead to an unstable focus or node that spirals out to a limit cycle.

The fact that `id_x589` max is stuck at ~0.2 (its mean) is the most telling sign. It means `id_x589` is not "overshooting".
If `id_x589` is slow (timescale 20) and `id_r7xb` is faster (timescale 2), when `id_r7xb` rises, `id_x589` should slowly follow. By the time `id_x589` rises enough to inhibit `id_r7xb`, `id_r7xb` should have already peaked and started to fall. Then `id_x589` continues to rise for a bit (due to its slow response and memory of high `id_r7xb`), strongly inhibiting `id_r7xb` and driving it low. Then `id_x589` starts to fall slowly, eventually allowing `id_r7xb` to rise again. This is the classic phase lag leading to oscillation.

Why is this not happening?
-   Perhaps `k_R_deg` (rate of `id_r7xb` decay) is too fast, so `id_r7xb` falls too quickly before `id_x589` can build up significantly past its mean.
-   Or `Vmax_R` (max production of `id_r7xb`) is not high enough to let `id_r7xb` "escape" to a high peak.

Let's go back to Model v8 parameters as a base, since it had a slightly better `id_r7xb` std (0.053) and max (1.075), even if not oscillating properly.
Model v8 params:
*   `IC_P = 0.89`, `IC_N = 0.71`
*   `Vmax_R = 0.55`, `K_IX = 0.2`, `h_X = 6.0`, `k_R_deg = 0.5`
*   `k_X_prod = 0.01818`, `k_X_deg = 0.05`
*   `k_PT = 0.4`, `k_N = 0.3`

Observed `id_r7xb`: mean=0.548, std=0.180, min=0.297, max=0.939
Sim v8 `id_r7xb`: mean=0.557, std=0.053, min=0.360, max=1.075
Observed `id_x589`: mean=0.200, std=0.071, min=0.000, max=0.438
Sim v8 `id_x589`: mean=0.198, std=0.014, min=0.000, max=0.200

The simulated `id_r7xb` max (1.075) is higher than observed (0.939).
The simulated `id_r7xb` min (0.360) is higher than observed (0.297).
The simulated `id_x589` max (0.200) is much lower than observed (0.438).

To get `id_x589` to have a higher max, it needs to be driven more strongly by `id_r7xb`.
This means `k_X_prod` should be larger.
If `k_X_prod` increases, to maintain `id_x589_mean = 0.2`, `k_X_deg` must also increase proportionally.
`id_x589_mean = (k_X_prod / k_X_deg) * id_r7xb_mean`.
Ratio `k_X_prod / k_X_deg = id_x589_mean / id_r7xb_mean = 0.2 / 0.55 = 0.3636`.
Current: `k_X_prod = 0.01818`, `k_X_deg = 0.05`. Ratio = 0.3636. Timescale `1/k_X_deg = 20`.

If we make `id_x589` respond faster, e.g., timescale 10 (`k_X_deg = 0.1`):
Then `k_X_prod = 0.3636 * 0.1 = 0.03636`.
This would make `id_x589` less "delayed". This is counter to the usual strategy for promoting oscillations.

What if the problem is that `id_r7xb` degradation (`k_R_deg`) is too fast?
If `k_R_deg` is smaller, `id_r7xb` is "slower".
Let `k_R_deg = 0.25` (timescale for `id_r7xb` = 4).
Keep `id_x589` timescale at 20 (`k_X_deg = 0.05`). Now separation is 5x.
To maintain `id_r7xb_mean = 0.55` with `K_IX = 0.2`, `h_X = 6`:
Inhibition term at `id_x589_mean = 0.2` is `(0.2/0.2)^6 = 1`. Denominator `1+1=2`.
`Vmax_R / 2 = k_R_deg * id_r7xb_mean`
`Vmax_R / 2 = 0.25 * 0.55 = 0.1375`
`Vmax_R = 0.275`. (This was the `Vmax_R` from v8 divided by 2).

To maintain `id_x589_mean = 0.2` with `id_r7xb_mean = 0.55` and `k_X_deg = 0.05`:
`k_X_prod = (0.2 / 0.55) * 0.05 = 0.01818` (this remains unchanged as it depends on means and `k_X_deg`).

Parameters for Model v10:
*   `IC_P_v10 = 0.89`, `IC_N_v10 = 0.71`
*   `Vmax_R = 0.275` (changed)
*   `K_IX = 0.2` (back to v8 value)
*   `h_X = 6.0`
*   `k_R_deg = 0.25` (changed, `id_r7xb` is slower)
*   `k_X_prod = 0.018181818`
*   `k_X_deg = 0.05` (`id_x589` timescale 20, `id_r7xb` timescale 4. Ratio 5x)
*   `k_PT = 0.4`, `k_N = 0.3`

The strategy here is to slow down `id_r7xb` itself, while keeping `id_x589` slow. This changes the relative speeds.
If `id_r7xb` changes more slowly, `id_x589` has more time to react to changes in `id_r7xb` before `id_r7xb` changes again. This might not be good.
The usual requirement is that the inhibitor (`id_x589`) should be slower than the species it inhibits (`id_r7xb`).
In v8: `id_r7xb` timescale 2, `id_x589` timescale 20. (Inhibitor 10x slower).
In v10: `id_r7xb` timescale 4, `id_x589` timescale 20. (Inhibitor 5x slower).
This reduces the timescale separation. This might also be counterproductive.

Let's try one more attempt at making `id_x589` even slower, keeping `id_r7xb` as in v8.
Model v8 params:
*   `Vmax_R = 0.55`, `K_IX = 0.2`, `h_X = 6.0`, `k_R_deg = 0.5` (timescale 2)
*   `k_X_prod = 0.01818`, `k_X_deg = 0.05` (timescale 20)

Make `id_x589` timescale 40: `k_X_deg = 0.025`.
`k_X_prod = (0.2 / 0.55) * 0.025 = 0.3636 * 0.025 = 0.0090909`.
Separation is now 20x.

Parameters for Model v10 (revised strategy):
*   `IC_P_v10 = 0.89`, `IC_N_v10 = 0.71`
*   `Vmax_R = 0.55` (from v8)
*   `K_IX = 0.2` (from v8)
*   `h_X = 6.0` (from v8)
*   `k_R_deg = 0.5` (from v8, `id_r7xb` timescale 2)
*   `k_X_prod = 0.009090909` (changed)
*   `k_X_deg = 0.025` (changed, `id_x589` timescale 40)
*   `k_PT = 0.4`, `k_N = 0.3`

This is a pure timescale separation increase. If this doesn't work, the simple 2-variable feedback might be missing something more complex from the true system that cannot be captured without altering species interactions or adding species (which is disallowed).

One final check: the period of oscillation. Observed data seems to have a period of roughly 200 time units (peaks around t=200, 400, 600, etc. for `id_r7xb`).
The period of a Goodwin-type oscillator is often related to the sum of the effective delays, roughly `1/k_X_deg`.
If `1/k_X_deg = 40`, this is much shorter than 200.
This suggests that the rates might be too fast overall for the observed period.
If `1/k_X_deg` needs to be, say, 100 (half period), then `k_X_deg = 0.01`.
And `1/k_R_deg` might need to be, say, 10. `k_R_deg = 0.1`.
Let's try to scale down all rates in the oscillator loop.

Model v10 (Strategy 3 - Slower overall dynamics for longer period):
Target `id_r7xb` timescale ~10 => `k_R_deg = 0.1`.
Target `id_x589` timescale ~100 => `k_X_deg = 0.01`. (Separation 10x)

Recalculate parameters:
*   `IC_P_v10 = 0.89`, `IC_N_v10 = 0.71`
*   `k_R_deg = 0.1`
*   `k_X_deg = 0.01`
*   `K_IX = 0.2`, `h_X = 6.0` (keep these from v8 which had some amplitude)
*   `Vmax_R / (1 + (0.2/0.2)^6) = k_R_deg * id_r7xb_mean`
    `Vmax_R / 2 = 0.1 * 0.55 = 0.055`
    `Vmax_R = 0.11`.
*   `k_X_prod = (id_x589_mean / id_r7xb_mean) * k_X_deg = (0.2 / 0.55) * 0.01 = 0.363636 * 0.01 = 0.00363636`.
*   `k_PT` and `k_N` also need to be scaled down if `id_r7xb` is active for longer periods.
    The integral of `id_r7xb` over one cycle should be similar. If period is 5x longer, rates might need to be 5x smaller.
    Old `k_R_deg=0.5`, new `k_R_deg=0.1` (5x slower).
    So, try `k_PT = 0.4 / 5 = 0.08`.
    `k_N = 0.3 / 5 = 0.06`.

This is a more drastic change.

## Action

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

# --- Helper function to add reactions (L2 compatible) ---
def add_reaction_l2(model, r_id, reactants_dict, products_dict, formula_str, reversible=False):
    reaction = model.createReaction()
    reaction.setId(r_id)
    reaction.setReversible(reversible)

    for species_id, stoich in reactants_dict.items():
        reactant = reaction.createReactant()
        reactant.setSpecies(species_id)
        reactant.setStoichiometry(stoich)

    for species_id, stoich in products_dict.items():
        product = reaction.createProduct()
        product.setSpecies(species_id)
        product.setStoichiometry(stoich)

    kinetic_law = reaction.createKineticLaw()
    kinetic_law.setFormula(formula_str)
    return reaction

# --- Load original SBML and experimental data ---
sbml_doc_v10 = libsbml.readSBMLFromString(input_sbml_string)
model_v10 = sbml_doc_v10.getModel()
obs_data = experiment_history['iteration_0']

# --- Define Initial Concentrations ---
IC_P_v10 = 0.89
IC_N_v10 = 0.71

species_plsk = model_v10.getSpecies('id_plsk')
if species_plsk: species_plsk.setInitialConcentration(IC_P_v10)

species_0nyd = model_v10.getSpecies('id_0nyd')
if species_0nyd: species_0nyd.setInitialConcentration(IC_N_v10)
    
model_v10.getSpecies('id_x589').setInitialConcentration(0.0)
model_v10.getSpecies('id_22ta').setInitialConcentration(0.0)
model_v10.getSpecies('id_r7xb').setInitialConcentration(0.36) # True initial value

# --- Define Parameters for v10 (Strategy 3 - Slower overall dynamics) ---
param_definitions_v10 = {
    "Vmax_R": 0.11,         # Changed: Vmax_R / 2 = k_R_deg * mean_r7xb
    "K_IX": 0.2,
    "h_X": 6.0,
    "k_R_deg": 0.1,         # Changed: Slower id_r7xb dynamics (timescale ~10)
    "k_X_prod": 0.00363636, # Changed: (0.2/0.55)*k_X_deg
    "k_X_deg": 0.01,        # Changed: Slower id_x589 dynamics (timescale ~100)
    "k_PT": 0.08,           # Changed: Scaled down
    "k_N": 0.06            # Changed: Scaled down
}

for p_id, p_val in param_definitions_v10.items():
    param = model_v10.getParameter(p_id)
    if not param:
        param = model_v10.createParameter()
        param.setId(p_id)
        param.setConstant(True)
    param.setValue(p_val)

# --- Add Reactions for v10 (same structure) ---
formula_Rprod = "Vmax_R / (1 + (id_x589 / K_IX)^h_X)"
add_reaction_l2(model_v10, "R_Rprod", {}, {"id_r7xb": 1}, formula_Rprod)

formula_Rdeg = "k_R_deg * id_r7xb"
add_reaction_l2(model_v10, "R_Rdeg", {"id_r7xb": 1}, {}, formula_Rdeg)

formula_Xprod = "k_X_prod * id_r7xb"
add_reaction_l2(model_v10, "R_Xprod", {}, {"id_x589": 1}, formula_Xprod)

formula_Xdeg = "k_X_deg * id_x589"
add_reaction_l2(model_v10, "R_Xdeg", {"id_x589": 1}, {}, formula_Xdeg)

formula_PT = "k_PT * id_plsk * id_r7xb"
add_reaction_l2(model_v10, "R_PT", {"id_plsk": 1}, {"id_22ta": 1}, formula_PT)

formula_N = "k_N * id_0nyd * id_r7xb"
add_reaction_l2(model_v10, "R_N", {"id_0nyd": 1}, {}, formula_N)

# --- Generate SBML string for Model v10 ---
writer = libsbml.SBMLWriter()
sbml_v10_string = writer.writeSBMLToString(sbml_doc_v10)
shared_variables.add("hypothetical_sbml_v10", sbml_v10_string)

# --- Simulate Model v10 ---
sim_data_v10 = simulate(sbml_v10_string)
shared_variables.add("sim_data_v10", sim_data_v10)
print("Simulation of Model v10 completed.")

# --- Analyze Simulation Results ---
sim_data_v10_adjusted = sim_data_v10.copy()
sim_data_v10_adjusted['id_plsk'] = sim_data_v10['id_plsk'] - IC_P_v10
sim_data_v10_adjusted['id_0nyd'] = sim_data_v10['id_0nyd'] - IC_N_v10

r7xb_sim_stats_v10 = sim_data_v10['id_r7xb'].describe()
x589_sim_stats_v10 = sim_data_v10['id_x589'].describe()

print(f"\nObserved id_r7xb: mean={obs_data['id_r7xb'].mean():.3f}, std={obs_data['id_r7xb'].std():.3f}, min={obs_data['id_r7xb'].min():.3f}, max={obs_data['id_r7xb'].max():.3f}")
print(f"Simulated id_r7xb (v10): mean={r7xb_sim_stats_v10['mean']:.3f}, std={r7xb_sim_stats_v10['std']:.3f}, min={r7xb_sim_stats_v10['min']:.3f}, max={r7xb_sim_stats_v10['max']:.3f}")

print(f"Observed id_x589: mean={obs_data['id_x589'].mean():.3f}, std={obs_data['id_x589'].std():.3f}, min={obs_data['id_x589'].min():.3f}, max={obs_data['id_x589'].max():.3f}")
print(f"Simulated id_x589 (v10): mean={x589_sim_stats_v10['mean']:.3f}, std={x589_sim_stats_v10['std']:.3f}, min={x589_sim_stats_v10['min']:.3f}, max={x589_sim_stats_v10['max']:.3f}")

if r7xb_sim_stats_v10['std'] > 0.15 : # Looking for std close to observed 0.18
    print("Simulated id_r7xb (v10) shows significant dynamic/oscillatory behavior.")
else:
    print("Simulated id_r7xb (v10) is not strongly oscillating with these parameters.")

# Check period by finding peaks (simple method)
time_points = sim_data_v10['Time'].to_numpy()
r7xb_values = sim_data_v10['id_r7xb'].to_numpy()
from scipy.signal import find_peaks
peaks, _ = find_peaks(r7xb_values, height=r7xb_sim_stats_v10['mean'], distance=50) # distance in samples
if len(peaks) > 2:
    peak_times = time_points[peaks]
    periods = np.diff(peak_times)
    print(f"Simulated peak times for id_r7xb: {peak_times}")
    print(f"Simulated periods for id_r7xb: {periods}, mean period: {np.mean(periods):.2f}")
else:
    print("Not enough peaks found to estimate period reliably.")

```
# Observation

## Code Stdout
Simulation of Model v10 completed.

Observed id_r7xb: mean=0.548, std=0.180, min=0.297, max=0.939
Simulated id_r7xb (v10): mean=0.585, std=0.114, min=0.360, max=1.075
Observed id_x589: mean=0.200, std=0.071, min=0.000, max=0.438
Simulated id_x589 (v10): mean=0.192, std=0.031, min=0.000, max=0.200
Simulated id_r7xb (v10) is not strongly oscillating with these parameters.
Not enough peaks found to estimate period reliably.

## 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. (hypothetical_sbml_v4, sim_data_v4, hypothetical_sbml_v5, sim_data_v5, hypothetical_sbml_v6, sim_data_v6, hypothetical_sbml_v7, sim_data_v7, hypothetical_sbml_v8, sim_data_v8, hypothetical_sbml_v9, sim_data_v9, hypothetical_sbml_v10, sim_data_v10)

### 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

Model v10 showed some improvement in `id_r7xb` std (0.114 vs obs 0.180), but `id_x589` was still not swinging enough (max stuck at 0.2, std 0.031 vs obs 0.071 and max 0.438). The system did not produce clear oscillations. The primary strategy is to get `id_x589` to oscillate with a larger amplitude, which should, in turn, drive `id_r7xb` oscillations more strongly.

The main hypothesis for Model v11 is to make `id_x589` dynamics even slower, thus increasing the phase lag and promoting larger overshoots.
-   `id_r7xb` timescale: `1/k_R_deg = 1/0.1 = 10` time units.
-   `id_x589` timescale: `1/k_X_deg = 1/0.005 = 200` time units.
This creates a 20x timescale separation. The period of `id_x589` response is now very long, which might align better with the observed slow oscillations.

We will also fine-tune `Vmax_R` and `k_X_prod` to precisely target the observed means of `id_r7xb` (0.548) and `id_x589` (0.200) with the new rate constants.
-   Target `id_r7xb_mean = 0.548`. With `K_IX = 0.2`, `h_X = 6.0`, `k_R_deg = 0.1`, and assuming `id_x589_mean = 0.2`, the inhibition term `(id_x589/K_IX)^h_X = 1`. So, `Vmax_R / (1+1) = k_R_deg * id_r7xb_mean`.
    `Vmax_R = 2 * 0.1 * 0.548 = 0.1096`.
-   Target `id_x589_mean = 0.2`. With `id_r7xb_mean = 0.548` and `k_X_deg = 0.005`:
    `k_X_prod = (id_x589_mean / id_r7xb_mean) * k_X_deg = (0.2 / 0.548) * 0.005 = 0.3649635 * 0.005 = 0.0018248175`.

The parameters for `k_PT` and `k_N` from v10 (`0.08`, `0.06`) were scaled based on `id_r7xb` becoming 5x slower. This logic should hold.

Parameters for Model v11:
*   `IC_P_v11 = 0.89`, `IC_N_v11 = 0.71`
*   `Vmax_R = 0.1096`
*   `K_IX = 0.2`
*   `h_X = 6.0`
*   `k_R_deg = 0.1`
*   `k_X_prod = 0.0018248175`
*   `k_X_deg = 0.005`
*   `k_PT = 0.08`
*   `k_N = 0.06`

## Action

### Code
```python
import libsbml
import pandas as pd
import numpy as np
from scipy.signal import find_peaks # Import for peak analysis

# --- Helper function to add reactions (L2 compatible) ---
def add_reaction_l2(model, r_id, reactants_dict, products_dict, formula_str, reversible=False):
    reaction = model.createReaction()
    reaction.setId(r_id)
    reaction.setReversible(reversible)

    for species_id, stoich in reactants_dict.items():
        reactant = reaction.createReactant()
        reactant.setSpecies(species_id)
        reactant.setStoichiometry(stoich)

    for species_id, stoich in products_dict.items():
        product = reaction.createProduct()
        product.setSpecies(species_id)
        product.setStoichiometry(stoich)

    kinetic_law = reaction.createKineticLaw()
    kinetic_law.setFormula(formula_str)
    return reaction

# --- Load original SBML and experimental data ---
sbml_doc_v11 = libsbml.readSBMLFromString(input_sbml_string)
model_v11 = sbml_doc_v11.getModel()
obs_data = experiment_history['iteration_0']

# --- Define Initial Concentrations ---
IC_P_v11 = 0.89
IC_N_v11 = 0.71

species_plsk = model_v11.getSpecies('id_plsk')
if species_plsk: species_plsk.setInitialConcentration(IC_P_v11)

species_0nyd = model_v11.getSpecies('id_0nyd')
if species_0nyd: species_0nyd.setInitialConcentration(IC_N_v11)
    
model_v11.getSpecies('id_x589').setInitialConcentration(0.0)
model_v11.getSpecies('id_22ta').setInitialConcentration(0.0)
model_v11.getSpecies('id_r7xb').setInitialConcentration(0.36)

# --- Define Parameters for v11 ---
param_definitions_v11 = {
    "Vmax_R": 0.1096,         # Adjusted for target id_r7xb_mean = 0.548
    "K_IX": 0.2,
    "h_X": 6.0,
    "k_R_deg": 0.1,           # id_r7xb timescale ~10
    "k_X_prod": 0.0018248175, # Adjusted for target id_x589_mean = 0.2 and new k_X_deg
    "k_X_deg": 0.005,         # id_x589 timescale ~200
    "k_PT": 0.08,
    "k_N": 0.06
}

for p_id, p_val in param_definitions_v11.items():
    param = model_v11.getParameter(p_id)
    if not param:
        param = model_v11.createParameter()
        param.setId(p_id)
        param.setConstant(True)
    param.setValue(p_val)

# --- Add Reactions for v11 (same structure) ---
formula_Rprod = "Vmax_R / (1 + (id_x589 / K_IX)^h_X)"
add_reaction_l2(model_v11, "R_Rprod", {}, {"id_r7xb": 1}, formula_Rprod)

formula_Rdeg = "k_R_deg * id_r7xb"
add_reaction_l2(model_v11, "R_Rdeg", {"id_r7xb": 1}, {}, formula_Rdeg)

formula_Xprod = "k_X_prod * id_r7xb"
add_reaction_l2(model_v11, "R_Xprod", {}, {"id_x589": 1}, formula_Xprod)

formula_Xdeg = "k_X_deg * id_x589"
add_reaction_l2(model_v11, "R_Xdeg", {"id_x589": 1}, {}, formula_Xdeg)

formula_PT = "k_PT * id_plsk * id_r7xb"
add_reaction_l2(model_v11, "R_PT", {"id_plsk": 1}, {"id_22ta": 1}, formula_PT)

formula_N = "k_N * id_0nyd * id_r7xb"
add_reaction_l2(model_v11, "R_N", {"id_0nyd": 1}, {}, formula_N)

# --- Generate SBML string for Model v11 ---
writer = libsbml.SBMLWriter()
sbml_v11_string = writer.writeSBMLToString(sbml_doc_v11)
shared_variables.add("hypothetical_sbml_v11", sbml_v11_string)

# --- Simulate Model v11 ---
sim_data_v11 = simulate(sbml_v11_string)
shared_variables.add("sim_data_v11", sim_data_v11)
print("Simulation of Model v11 completed.")

# --- Analyze Simulation Results ---
sim_data_v11_adjusted = sim_data_v11.copy()
sim_data_v11_adjusted['id_plsk'] = sim_data_v11['id_plsk'] - IC_P_v11
sim_data_v11_adjusted['id_0nyd'] = sim_data_v11['id_0nyd'] - IC_N_v11

r7xb_sim_stats_v11 = sim_data_v11['id_r7xb'].describe()
x589_sim_stats_v11 = sim_data_v11['id_x589'].describe()

print(f"\nObserved id_r7xb: mean={obs_data['id_r7xb'].mean():.3f}, std={obs_data['id_r7xb'].std():.3f}, min={obs_data['id_r7xb'].min():.3f}, max={obs_data['id_r7xb'].max():.3f}")
print(f"Simulated id_r7xb (v11): mean={r7xb_sim_stats_v11['mean']:.3f}, std={r7xb_sim_stats_v11['std']:.3f}, min={r7xb_sim_stats_v11['min']:.3f}, max={r7xb_sim_stats_v11['max']:.3f}")

print(f"Observed id_x589: mean={obs_data['id_x589'].mean():.3f}, std={obs_data['id_x589'].std():.3f}, min={obs_data['id_x589'].min():.3f}, max={obs_data['id_x589'].max():.3f}")
print(f"Simulated id_x589 (v11): mean={x589_sim_stats_v11['mean']:.3f}, std={x589_sim_stats_v11['std']:.3f}, min={x589_sim_stats_v11['min']:.3f}, max={x589_sim_stats_v11['max']:.3f}")

if r7xb_sim_stats_v11['std'] > 0.15 and x589_sim_stats_v11['std'] > 0.05: # Check std dev for both
    print("Simulated id_r7xb and id_x589 (v11) show potentially significant dynamic/oscillatory behavior.")
    # Peak analysis
    time_points = sim_data_v11['Time'].to_numpy()
    r7xb_values = sim_data_v11['id_r7xb'].to_numpy()
    # Adjust find_peaks parameters: distance might need to be larger if period is long
    # Simulation time is ~1000. If period is ~200, expect ~5 peaks.
    # distance = period_samples / 2.  period_samples = period_time / time_step.
    # time_step = 0.0975. period_time ~ 200. period_samples ~ 200 / 0.0975 ~ 2050 samples.
    # This means default simulation length might only capture a few cycles.
    # Let's use a distance based on expected number of points in a half-period.
    # If period ~200 time units, half period ~100 time units.
    # Number of samples in 100 time units = 100 / 0.0975 ~ 1025 samples.
    peaks_r7xb, _ = find_peaks(r7xb_values, height=r7xb_sim_stats_v11['mean'], distance=500) # distance in samples (500 samples ~ 50 time units)
    
    if len(peaks_r7xb) > 1: # Need at least 2 peaks to calculate one period
        peak_times_r7xb = time_points[peaks_r7xb]
        periods_r7xb = np.diff(peak_times_r7xb)
        print(f"Simulated peak times for id_r7xb: {peak_times_r7xb}")
        print(f"Simulated periods for id_r7xb: {periods_r7xb}, mean period: {np.mean(periods_r7xb):.2f}")
    else:
        print("Not enough peaks found for id_r7xb to estimate period reliably.")
else:
    print("Simulated id_r7xb or id_x589 (v11) is not strongly oscillating with these parameters.")

print(f"Observed id_22ta: max={obs_data['id_22ta'].max():.3f}")
print(f"Simulated id_22ta (v11): max={sim_data_v11['id_22ta'].max():.3f}")
print(f"Observed id_0nyd (adj): min={obs_data['id_0nyd'].min():.3f}")
print(f"Simulated id_0nyd (adj, v11): min={sim_data_v11_adjusted['id_0nyd'].min():.3f}")

```
# Observation

## Code Stdout
Simulation of Model v11 completed.

Observed id_r7xb: mean=0.548, std=0.180, min=0.297, max=0.939
Simulated id_r7xb (v11): mean=0.619, std=0.157, min=0.360, max=1.090
Observed id_x589: mean=0.200, std=0.071, min=0.000, max=0.438
Simulated id_x589 (v11): mean=0.185, std=0.040, min=0.000, max=0.200
Simulated id_r7xb or id_x589 (v11) is not strongly oscillating with these parameters.
Observed id_22ta: max=0.891
Simulated id_22ta (v11): max=0.890
Observed id_0nyd (adj): min=-0.710
Simulated id_0nyd (adj, v11): min=-0.710

## 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. (hypothetical_sbml_v4, sim_data_v4, hypothetical_sbml_v5, sim_data_v5, hypothetical_sbml_v6, sim_data_v6, hypothetical_sbml_v7, sim_data_v7, hypothetical_sbml_v8, sim_data_v8, hypothetical_sbml_v9, sim_data_v9, hypothetical_sbml_v10, sim_data_v10, hypothetical_sbml_v11, sim_data_v11)

### 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

Model v11 showed progress:
-   `id_r7xb` stddev increased to 0.157 (obs: 0.180). This is the closest yet.
-   `id_r7xb` mean is 0.619 (obs: 0.548) - a bit high.
-   `id_r7xb` min is 0.360 (obs: 0.297) - still too high.
-   `id_r7xb` max is 1.090 (obs: 0.939) - a bit high.
-   `id_x589` stddev is 0.040 (obs: 0.071) - still too low.
-   `id_x589` mean is 0.185 (obs: 0.200) - a bit low.
-   `id_x589` max is 0.200 (obs: 0.438) - **critically, still stuck near its mean, not swinging high enough.**

The key issue is that `id_x589` is not achieving a high enough peak. If `id_x589` (the inhibitor) doesn't rise sufficiently, it cannot drive `id_r7xb` down to its observed minimum, nor will `id_x589` itself show a large enough amplitude.
The previous models suggest that with a high Hill coefficient (`h_X=6`), `id_x589` tends to be "clamped" around `K_IX` because inhibition becomes extremely strong once `id_x589` surpasses `K_IX`.
Therefore, to allow `id_x589` to reach a peak near the observed 0.438, `K_IX` should be set near this value.

Let's try `K_IX = 0.4`.
We need to recalculate `Vmax_R` to maintain the target mean of `id_r7xb` (0.548), assuming `id_x589` mean is 0.200.
The other rates (`k_R_deg`, `k_X_prod`, `k_X_deg`, `k_PT`, `k_N`) from v11 seemed to provide a good timescale and consumption profile for `id_plsk`/`id_0nyd`.

Recalculation for Model v12:
-   Target `id_r7xb_mean = 0.548`.
-   Target `id_x589_mean = 0.200`.
-   `K_IX = 0.4`.
-   `h_X = 6.0`.
-   `k_R_deg = 0.1` (from v11, `id_r7xb` timescale ~10).
-   `k_X_deg = 0.005` (from v11, `id_x589` timescale ~200).

1.  Calculate `Vmax_R`:
    At steady state (mean levels), `d(id_r7xb)/dt = 0`:
    `Vmax_R / (1 + (id_x589_mean / K_IX)^h_X) = k_R_deg * id_r7xb_mean`
    The inhibition term: `(id_x589_mean / K_IX)^h_X = (0.200 / 0.4)^6 = (0.5)^6 = 1/64 = 0.015625`.
    So, `Vmax_R / (1 + 0.015625) = 0.1 * 0.548`.
    `Vmax_R / 1.015625 = 0.0548`.
    `Vmax_R = 0.0548 * 1.015625 = 0.055636`.

2.  Calculate `k_X_prod`:
    At steady state (mean levels), `d(id_x589)/dt = 0`:
    `k_X_prod * id_r7xb_mean = k_X_deg * id_x589_mean`.
    `k_X_prod = (k_X_deg * id_x589_mean) / id_r7xb_mean = (0.005 * 0.200) / 0.548 = 0.001 / 0.548 = 0.0018248175`.
    This value for `k_X_prod` is the same as in v11, as it depends on the target means and `k_X_deg`.

Parameters for Model v12:
*   `IC_P_v12 = 0.89`, `IC_N_v12 = 0.71`
*   `Vmax_R = 0.055636` (changed)
*   `K_IX = 0.4` (changed)
*   `h_X = 6.0` (kept)
*   `k_R_deg = 0.1` (kept from v11)
*   `k_X_prod = 0.0018248175` (kept from v11)
*   `k_X_deg = 0.005` (kept from v11)
*   `k_PT = 0.08` (kept from v11)
*   `k_N = 0.06` (kept from v11)

The hypothesis is that increasing `K_IX` will allow `id_x589` to rise to higher values before strong inhibition kicks in, potentially leading to a larger amplitude for `id_x589` and consequently for `id_r7xb`.

## Action

### Code
```python
import libsbml
import pandas as pd
import numpy as np
from scipy.signal import find_peaks

# --- Helper function to add reactions (L2 compatible) ---
def add_reaction_l2(model, r_id, reactants_dict, products_dict, formula_str, reversible=False):
    reaction = model.createReaction()
    reaction.setId(r_id)
    reaction.setReversible(reversible)

    for species_id, stoich in reactants_dict.items():
        reactant = reaction.createReactant()
        reactant.setSpecies(species_id)
        reactant.setStoichiometry(stoich)

    for species_id, stoich in products_dict.items():
        product = reaction.createProduct()
        product.setSpecies(species_id)
        product.setStoichiometry(stoich)

    kinetic_law = reaction.createKineticLaw()
    kinetic_law.setFormula(formula_str)
    return reaction

# --- Load original SBML and experimental data ---
sbml_doc_v12 = libsbml.readSBMLFromString(input_sbml_string)
model_v12 = sbml_doc_v12.getModel()
obs_data = experiment_history['iteration_0']

# --- Define Initial Concentrations ---
IC_P_v12 = 0.89
IC_N_v12 = 0.71

species_plsk = model_v12.getSpecies('id_plsk')
if species_plsk: species_plsk.setInitialConcentration(IC_P_v12)

species_0nyd = model_v12.getSpecies('id_0nyd')
if species_0nyd: species_0nyd.setInitialConcentration(IC_N_v12)
    
model_v12.getSpecies('id_x589').setInitialConcentration(0.0)
model_v12.getSpecies('id_22ta').setInitialConcentration(0.0)
model_v12.getSpecies('id_r7xb').setInitialConcentration(0.36)

# --- Define Parameters for v12 ---
param_definitions_v12 = {
    "Vmax_R": 0.055636,
    "K_IX": 0.4,            # Increased K_IX
    "h_X": 6.0,
    "k_R_deg": 0.1,
    "k_X_prod": 0.0018248175,
    "k_X_deg": 0.005,
    "k_PT": 0.08,
    "k_N": 0.06
}

for p_id, p_val in param_definitions_v12.items():
    param = model_v12.getParameter(p_id)
    if not param:
        param = model_v12.createParameter()
        param.setId(p_id)
        param.setConstant(True)
    param.setValue(p_val)

# --- Add Reactions for v12 (same structure) ---
formula_Rprod = "Vmax_R / (1 + (id_x589 / K_IX)^h_X)"
add_reaction_l2(model_v12, "R_Rprod", {}, {"id_r7xb": 1}, formula_Rprod)

formula_Rdeg = "k_R_deg * id_r7xb"
add_reaction_l2(model_v12, "R_Rdeg", {"id_r7xb": 1}, {}, formula_Rdeg)

formula_Xprod = "k_X_prod * id_r7xb"
add_reaction_l2(model_v12, "R_Xprod", {}, {"id_x589": 1}, formula_Xprod)

formula_Xdeg = "k_X_deg * id_x589"
add_reaction_l2(model_v12, "R_Xdeg", {"id_x589": 1}, {}, formula_Xdeg)

formula_PT = "k_PT * id_plsk * id_r7xb"
add_reaction_l2(model_v12, "R_PT", {"id_plsk": 1}, {"id_22ta": 1}, formula_PT)

formula_N = "k_N * id_0nyd * id_r7xb"
add_reaction_l2(model_v12, "R_N", {"id_0nyd": 1}, {}, formula_N)

# --- Generate SBML string for Model v12 ---
writer = libsbml.SBMLWriter()
sbml_v12_string = writer.writeSBMLToString(sbml_doc_v12)
shared_variables.add("hypothetical_sbml_v12", sbml_v12_string)

# --- Simulate Model v12 ---
sim_data_v12 = simulate(sbml_v12_string)
shared_variables.add("sim_data_v12", sim_data_v12)
print("Simulation of Model v12 completed.")

# --- Analyze Simulation Results ---
sim_data_v12_adjusted = sim_data_v12.copy()
sim_data_v12_adjusted['id_plsk'] = sim_data_v12['id_plsk'] - IC_P_v12
sim_data_v12_adjusted['id_0nyd'] = sim_data_v12['id_0nyd'] - IC_N_v12

r7xb_sim_stats_v12 = sim_data_v12['id_r7xb'].describe()
x589_sim_stats_v12 = sim_data_v12['id_x589'].describe()

print(f"\nObserved id_r7xb: mean={obs_data['id_r7xb'].mean():.3f}, std={obs_data['id_r7xb'].std():.3f}, min={obs_data['id_r7xb'].min():.3f}, max={obs_data['id_r7xb'].max():.3f}")
print(f"Simulated id_r7xb (v12): mean={r7xb_sim_stats_v12['mean']:.3f}, std={r7xb_sim_stats_v12['std']:.3f}, min={r7xb_sim_stats_v12['min']:.3f}, max={r7xb_sim_stats_v12['max']:.3f}")

print(f"Observed id_x589: mean={obs_data['id_x589'].mean():.3f}, std={obs_data['id_x589'].std():.3f}, min={obs_data['id_x589'].min():.3f}, max={obs_data['id_x589'].max():.3f}")
print(f"Simulated id_x589 (v12): mean={x589_sim_stats_v12['mean']:.3f}, std={x589_sim_stats_v12['std']:.3f}, min={x589_sim_stats_v12['min']:.3f}, max={x589_sim_stats_v12['max']:.3f}")

if r7xb_sim_stats_v12['std'] > 0.15 and x589_sim_stats_v12['std'] > 0.06: # Target std_obs for x589 is 0.071
    print("Simulated id_r7xb and id_x589 (v12) show significant dynamic/oscillatory behavior.")
    time_points = sim_data_v12['Time'].to_numpy()
    r7xb_values = sim_data_v12['id_r7xb'].to_numpy()
    # distance for find_peaks: if period ~200 time units, half period ~100 time units.
    # samples in 100 time units = 100 / (time_points[1]-time_points[0]) if time_points[0] is 0.
    # Assuming time step is roughly 0.1, distance ~ 1000 samples.
    # A safer, more general distance might be related to total number of points / expected number of cycles.
    # Total points = 10001. Expected cycles ~ 1000/200 = 5.
    # Distance ~ 10001 / (5*2) = 1000 samples.
    peaks_r7xb, _ = find_peaks(r7xb_values, height=r7xb_sim_stats_v12['mean'], distance=800) # distance in samples
    
    if len(peaks_r7xb) > 1:
        peak_times_r7xb = time_points[peaks_r7xb]
        periods_r7xb = np.diff(peak_times_r7xb)
        print(f"Simulated peak times for id_r7xb: {peak_times_r7xb}")
        print(f"Simulated periods for id_r7xb: {periods_r7xb}, mean period: {np.mean(periods_r7xb):.2f}")
    else:
        print("Not enough peaks found for id_r7xb to estimate period reliably.")
else:
    print("Simulated id_r7xb or id_x589 (v12) is not strongly oscillating with these parameters.")

print(f"Observed id_22ta: max={obs_data['id_22ta'].max():.3f}")
print(f"Simulated id_22ta (v12): max={sim_data_v12['id_22ta'].max():.3f}")
print(f"Observed id_0nyd (adj): min={obs_data['id_0nyd'].min():.3f}")
print(f"Simulated id_0nyd (adj, v12): min={sim_data_v12_adjusted['id_0nyd'].min():.3f}")

```
# Observation

## Code Stdout
Simulation of Model v12 completed.

Observed id_r7xb: mean=0.548, std=0.180, min=0.297, max=0.939
Simulated id_r7xb (v12): mean=0.550, std=0.014, min=0.360, max=0.556
Observed id_x589: mean=0.200, std=0.071, min=0.000, max=0.438
Simulated id_x589 (v12): mean=0.160, std=0.050, min=0.000, max=0.199
Simulated id_r7xb or id_x589 (v12) is not strongly oscillating with these parameters.
Observed id_22ta: max=0.891
Simulated id_22ta (v12): max=0.890
Observed id_0nyd (adj): min=-0.710
Simulated id_0nyd (adj, v12): min=-0.710

## 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. (hypothetical_sbml_v4, sim_data_v4, hypothetical_sbml_v5, sim_data_v5, hypothetical_sbml_v6, sim_data_v6, hypothetical_sbml_v7, sim_data_v7, hypothetical_sbml_v8, sim_data_v8, hypothetical_sbml_v9, sim_data_v9, hypothetical_sbml_v10, sim_data_v10, hypothetical_sbml_v11, sim_data_v11, hypothetical_sbml_v12, sim_data_v12)

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


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



# Iteration 11
## Thoughts

Model v12 was a significant step backward for `id_r7xb` oscillations (std dev 0.014). Raising `K_IX` to 0.4, while intended to allow `id_x589` to rise higher, seems to have overly stabilized the system by making the inhibition too weak until `id_x589` is very high. The `id_x589` mean also dropped to 0.160 (target 0.200), and its max was still stuck at 0.199.

This suggests that the interplay between `K_IX` and `h_X` is critical and sensitive.
Model v11 had the best `id_r7xb` std so far (0.157), with `K_IX = 0.2` and `h_X = 6.0`.
Its parameters were:
*   `Vmax_R = 0.1096`, `K_IX = 0.2`, `h_X = 6.0`
*   `k_R_deg = 0.1` (timescale 10)
*   `k_X_prod = 0.0018248`, `k_X_deg = 0.005` (timescale 200)
*   `k_PT = 0.08`, `k_N = 0.06`

The main issues with v11 were:
1.  `id_r7xb` mean was a bit high (0.619 vs 0.548).
2.  `id_r7xb` min was too high (0.360 vs 0.297).
3.  `id_x589` std was too low (0.040 vs 0.071), and max too low (0.200 vs 0.438).
4.  `id_x589` mean was a bit low (0.185 vs 0.200).

Let's try to refine Model v11.
To get `id_x589` to swing higher (target max 0.438) and have a higher std:
-   Its production (`k_X_prod * id_r7xb`) must be stronger when `id_r7xb` is high.
-   Its degradation (`k_X_deg * id_x589`) must allow it to accumulate.

If `id_x589` needs to reach 0.438, and its mean is 0.2, this is a large swing.
The current `k_X_deg = 0.005` (timescale 200) is already very slow. Making it slower might make the period too long or dampen oscillations if `id_r7xb` changes faster.

Consider the Hill coefficient `h_X`. A very high `h_X` (like 6) makes the inhibition very switch-like around `K_IX`.
If `K_IX = 0.2`, then `id_x589` values above 0.2 very strongly inhibit `id_r7xb` production. This might be preventing `id_r7xb` from staying high long enough to drive `id_x589` significantly above 0.2.
What if `h_X` is slightly lower, e.g., `h_X = 4`? This makes the inhibition less steep.
If `h_X = 4`:
-   Target `id_r7xb_mean = 0.548`.
-   Target `id_x589_mean = 0.200`.
-   `K_IX = 0.2`.
-   `k_R_deg = 0.1`.
-   `k_X_deg = 0.005`.

1.  Recalculate `Vmax_R` with `h_X = 4`:
    Inhibition term: `(id_x589_mean / K_IX)^h_X = (0.200 / 0.2)^4 = (1)^4 = 1`.
    `Vmax_R / (1 + 1) = k_R_deg * id_r7xb_mean`.
    `Vmax_R / 2 = 0.1 * 0.548 = 0.0548`.
    `Vmax_R = 0.1096`. (This is the same as v11 because the inhibition term at mean `id_x589` is 1 for any `h_X` if `id_x589_mean = K_IX`).

2.  `k_X_prod` remains `0.0018248175` as it depends on target means and `k_X_deg`.

So, the only change from v11 would be `h_X` from 6 to 4.
This might make the feedback "softer", potentially allowing `id_x589` to achieve a wider range if `id_r7xb` can sustain higher levels for longer before inhibition fully clamps down.

Parameters for Model v13:
*   `IC_P_v13 = 0.89`, `IC_N_v13 = 0.71`
*   `Vmax_R = 0.1096` (same as v11)
*   `K_IX = 0.2` (same as v11)
*   `h_X = 4.0` (changed from 6.0)
*   `k_R_deg = 0.1` (same as v11)
*   `k_X_prod = 0.0018248175` (same as v11)
*   `k_X_deg = 0.005` (same as v11)
*   `k_PT = 0.08` (same as v11)
*   `k_N = 0.06` (same as v11)

This is a targeted change to see if a less steep Hill slope helps `id_x589` escape being clamped at its mean.
The `id_r7xb` mean in v11 (0.619) was higher than target (0.548). `Vmax_R` was calculated for target 0.548. The simulation result being higher suggests the system dynamics don't perfectly settle at the analytically derived mean when oscillating. We might need to slightly adjust `Vmax_R` downwards if the mean remains high. For now, let's test `h_X=4`.

## Action

### Code
```python
import libsbml
import pandas as pd
import numpy as np
from scipy.signal import find_peaks

# --- Helper function to add reactions (L2 compatible) ---
def add_reaction_l2(model, r_id, reactants_dict, products_dict, formula_str, reversible=False):
    reaction = model.createReaction()
    reaction.setId(r_id)
    reaction.setReversible(reversible)

    for species_id, stoich in reactants_dict.items():
        reactant = reaction.createReactant()
        reactant.setSpecies(species_id)
        reactant.setStoichiometry(stoich)

    for species_id, stoich in products_dict.items():
        product = reaction.createProduct()
        product.setSpecies(species_id)
        product.setStoichiometry(stoich)

    kinetic_law = reaction.createKineticLaw()
    kinetic_law.setFormula(formula_str)
    return reaction

# --- Load original SBML and experimental data ---
sbml_doc_v13 = libsbml.readSBMLFromString(input_sbml_string)
model_v13 = sbml_doc_v13.getModel()
obs_data = experiment_history['iteration_0']

# --- Define Initial Concentrations ---
IC_P_v13 = 0.89
IC_N_v13 = 0.71

species_plsk = model_v13.getSpecies('id_plsk')
if species_plsk: species_plsk.setInitialConcentration(IC_P_v13)

species_0nyd = model_v13.getSpecies('id_0nyd')
if species_0nyd: species_0nyd.setInitialConcentration(IC_N_v13)
    
model_v13.getSpecies('id_x589').setInitialConcentration(0.0)
model_v13.getSpecies('id_22ta').setInitialConcentration(0.0)
model_v13.getSpecies('id_r7xb').setInitialConcentration(0.36)

# --- Define Parameters for v13 ---
# Based on v11, but h_X changed from 6 to 4.
# Vmax_R and k_X_prod are calculated for target means id_r7xb=0.548, id_x589=0.200
param_definitions_v13 = {
    "Vmax_R": 0.1096,         # 2 * k_R_deg * target_r7xb_mean (since K_IX = target_x589_mean)
    "K_IX": 0.2,
    "h_X": 4.0,               # Changed from 6.0
    "k_R_deg": 0.1,
    "k_X_prod": 0.0018248175, # (target_x589_mean / target_r7xb_mean) * k_X_deg
    "k_X_deg": 0.005,
    "k_PT": 0.08,
    "k_N": 0.06
}

for p_id, p_val in param_definitions_v13.items():
    param = model_v13.getParameter(p_id)
    if not param:
        param = model_v13.createParameter()
        param.setId(p_id)
        param.setConstant(True)
    param.setValue(p_val)

# --- Add Reactions for v13 (same structure) ---
formula_Rprod = "Vmax_R / (1 + (id_x589 / K_IX)^h_X)"
add_reaction_l2(model_v13, "R_Rprod", {}, {"id_r7xb": 1}, formula_Rprod)

formula_Rdeg = "k_R_deg * id_r7xb"
add_reaction_l2(model_v13, "R_Rdeg", {"id_r7xb": 1}, {}, formula_Rdeg)

formula_Xprod = "k_X_prod * id_r7xb"
add_reaction_l2(model_v13, "R_Xprod", {}, {"id_x589": 1}, formula_Xprod)

formula_Xdeg = "k_X_deg * id_x589"
add_reaction_l2(model_v13, "R_Xdeg", {"id_x589": 1}, {}, formula_Xdeg)

formula_PT = "k_PT * id_plsk * id_r7xb"
add_reaction_l2(model_v13, "R_PT", {"id_plsk": 1}, {"id_22ta": 1}, formula_PT)

formula_N = "k_N * id_0nyd * id_r7xb"
add_reaction_l2(model_v13, "R_N", {"id_0nyd": 1}, {}, formula_N)

# --- Generate SBML string for Model v13 ---
writer = libsbml.SBMLWriter()
sbml_v13_string = writer.writeSBMLToString(sbml_doc_v13)
shared_variables.add("hypothetical_sbml_v13", sbml_v13_string)

# --- Simulate Model v13 ---
sim_data_v13 = simulate(sbml_v13_string)
shared_variables.add("sim_data_v13", sim_data_v13)
print("Simulation of Model v13 completed.")

# --- Analyze Simulation Results ---
sim_data_v13_adjusted = sim_data_v13.copy()
sim_data_v13_adjusted['id_plsk'] = sim_data_v13['id_plsk'] - IC_P_v13
sim_data_v13_adjusted['id_0nyd'] = sim_data_v13['id_0nyd'] - IC_N_v13

r7xb_sim_stats_v13 = sim_data_v13['id_r7xb'].describe()
x589_sim_stats_v13 = sim_data_v13['id_x589'].describe()

print(f"\nObserved id_r7xb: mean={obs_data['id_r7xb'].mean():.3f}, std={obs_data['id_r7xb'].std():.3f}, min={obs_data['id_r7xb'].min():.3f}, max={obs_data['id_r7xb'].max():.3f}")
print(f"Simulated id_r7xb (v13): mean={r7xb_sim_stats_v13['mean']:.3f}, std={r7xb_sim_stats_v13['std']:.3f}, min={r7xb_sim_stats_v13['min']:.3f}, max={r7xb_sim_stats_v13['max']:.3f}")

print(f"Observed id_x589: mean={obs_data['id_x589'].mean():.3f}, std={obs_data['id_x589'].std():.3f}, min={obs_data['id_x589'].min():.3f}, max={obs_data['id_x589'].max():.3f}")
print(f"Simulated id_x589 (v13): mean={x589_sim_stats_v13['mean']:.3f}, std={x589_sim_stats_v13['std']:.3f}, min={x589_sim_stats_v13['min']:.3f}, max={x589_sim_stats_v13['max']:.3f}")

if r7xb_sim_stats_v13['std'] > 0.15 and x589_sim_stats_v13['std'] > 0.06:
    print("Simulated id_r7xb and id_x589 (v13) show significant dynamic/oscillatory behavior.")
    time_points = sim_data_v13['Time'].to_numpy()
    r7xb_values = sim_data_v13['id_r7xb'].to_numpy()
    peaks_r7xb, _ = find_peaks(r7xb_values, height=r7xb_sim_stats_v13['mean'], distance=800) 
    
    if len(peaks_r7xb) > 1:
        peak_times_r7xb = time_points[peaks_r7xb]
        periods_r7xb = np.diff(peak_times_r7xb)
        print(f"Simulated peak times for id_r7xb: {peak_times_r7xb}")
        print(f"Simulated periods for id_r7xb: {periods_r7xb}, mean period: {np.mean(periods_r7xb):.2f}")
    else:
        print("Not enough peaks found for id_r7xb to estimate period reliably.")
else:
    print("Simulated id_r7xb or id_x589 (v13) is not strongly oscillating with these parameters.")

```
# Observation

## Code Stdout
Simulation of Model v13 completed.

Observed id_r7xb: mean=0.548, std=0.180, min=0.297, max=0.939
Simulated id_r7xb (v13): mean=0.616, std=0.146, min=0.360, max=1.080
Observed id_x589: mean=0.200, std=0.071, min=0.000, max=0.438
Simulated id_x589 (v13): mean=0.184, std=0.041, min=0.000, max=0.200
Simulated id_r7xb or id_x589 (v13) is not strongly oscillating with these parameters.

## 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. (hypothetical_sbml_v4, sim_data_v4, hypothetical_sbml_v5, sim_data_v5, hypothetical_sbml_v6, sim_data_v6, hypothetical_sbml_v7, sim_data_v7, hypothetical_sbml_v8, sim_data_v8, hypothetical_sbml_v9, sim_data_v9, hypothetical_sbml_v10, sim_data_v10, hypothetical_sbml_v11, sim_data_v11, hypothetical_sbml_v12, sim_data_v12, hypothetical_sbml_v13, sim_data_v13)

### 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

Model v13 (with `h_X=4`) did not significantly improve upon v11 (with `h_X=6`).
-   `id_r7xb` std: 0.146 (v13) vs 0.157 (v11). Obs: 0.180. (Slightly worse)
-   `id_r7xb` mean: 0.616 (v13) vs 0.619 (v11). Obs: 0.548. (Still high)
-   `id_x589` std: 0.041 (v13) vs 0.040 (v11). Obs: 0.071. (Still low)
-   `id_x589` max: 0.200 (v13) vs 0.200 (v11). Obs: 0.438. (Still stuck at mean)

The critical issue remains: `id_x589` does not swing high enough. Its maximum value in simulations is consistently around its mean (0.2), far from the observed maximum of 0.438. This prevents strong feedback to drive `id_r7xb` down sufficiently, thus limiting the amplitude of oscillations for both species.

Model v11 parameters (which gave the best `id_r7xb` std so far):
*   `Vmax_R = 0.1096`, `K_IX = 0.2`, `h_X = 6.0`
*   `k_R_deg = 0.1` (timescale 10)
*   `k_X_prod = 0.0018248`, `k_X_deg = 0.005` (timescale 200)
*   `k_PT = 0.08`, `k_N = 0.06`

Let's focus on making `id_x589` achieve a higher peak.
`d(id_x589)/dt = k_X_prod * id_r7xb - k_X_deg * id_x589`.
At the peak of `id_x589`, `d(id_x589)/dt` would be zero if `id_r7xb` was constant.
`id_x589_peak ≈ (k_X_prod / k_X_deg) * id_r7xb_at_peak_of_x589`.
The ratio `k_X_prod / k_X_deg = (0.2 / 0.548) = 0.36496`.
So, `id_x589_peak ≈ 0.36496 * id_r7xb_value_when_x589_peaks`.
If `id_r7xb` can reach a max of ~1.0 (as seen in v11 sim: 1.090; obs: 0.939), then `id_x589` could theoretically peak around `0.36496 * 1.0 = 0.365`. This is closer to the observed 0.438.
Why isn't it reaching this?
Perhaps `id_r7xb` falls too quickly from its peak due to the inhibition from `id_x589` (even when `id_x589` is still rising towards its own peak).

The `id_r7xb` mean in v11 (0.619) was higher than the target (0.548). This suggests that `Vmax_R = 0.1096` might be a bit too high for the oscillating system.
Let's try to lower `Vmax_R` slightly to bring the mean of `id_r7xb` down. This might also affect the amplitude and min/max values.
If `id_r7xb` mean decreases, and `k_X_prod` is tied to this mean (to keep `id_x589` mean at 0.2), then `k_X_prod` would increase if `id_r7xb` mean decreases. This could help `id_x589` rise.

Revised strategy for Model v14 (based on v11):
1.  Slightly decrease `Vmax_R` to lower `id_r7xb` mean closer to 0.548.
    Let's try reducing `Vmax_R` by ~10% from v11's value: `0.1096 * 0.9 = 0.09864`.
2.  Recalculate `k_X_prod` based on this new assumption for `id_r7xb` mean (target 0.548) and `id_x589` mean (target 0.2).
    `k_X_prod = (target_id_x589_mean / target_id_r7xb_mean) * k_X_deg`
    `k_X_prod = (0.200 / 0.548) * 0.005 = 0.0018248175`. This calculation actually doesn't change if we use the *target* means. The previous `Vmax_R` was already set for the target mean. The discrepancy is that the *simulated* mean was higher.
    So, if we lower `Vmax_R`, the simulated `id_r7xb` mean should drop. If it drops below 0.548, then `id_x589` mean might also drop below 0.2 if `k_X_prod` is not adjusted.

Let's assume the new `Vmax_R = 0.09864` will result in `id_r7xb_sim_mean` closer to 0.548.
The other parameters from v11 remain:
*   `K_IX = 0.2`, `h_X = 6.0`
*   `k_R_deg = 0.1`
*   `k_X_deg = 0.005`
*   `k_PT = 0.08`, `k_N = 0.06`

The key is that `k_X_prod` should be `(0.2 / id_r7xb_actual_sim_mean) * k_X_deg`.
If `id_r7xb_actual_sim_mean` from v11 was 0.619, and we want `id_x589_mean` to be 0.2:
`k_X_prod_for_v11_sim_mean = (0.2 / 0.619) * 0.005 = 0.0016155`.
The `k_X_prod` used in v11 (0.0018248) was based on target `id_r7xb_mean=0.548`. This might have contributed to `id_x589_mean` being slightly low (0.185) because `id_r7xb_mean` was actually higher.

Let's stick to parameters aimed at target means first.
Model v14 parameters:
*   `IC_P_v14 = 0.89`, `IC_N_v14 = 0.71`
*   `Vmax_R = 0.09864` (Reduced from v11's 0.1096, aiming to lower `id_r7xb` mean)
*   `K_IX = 0.2`
*   `h_X = 6.0`
*   `k_R_deg = 0.1`
*   `k_X_prod = 0.0018248175` (Calculated for target means `id_r7xb_mean=0.548`, `id_x589_mean=0.2`)
*   `k_X_deg = 0.005`
*   `k_PT = 0.08`
*   `k_N = 0.06`

This is a fine-tuning of `Vmax_R` from v11. The hope is that a lower `id_r7xb` mean (and perhaps a slightly different waveform due to lower `Vmax_R`) might allow `id_x589` to exhibit larger swings if the system is sensitive to the overall level of `id_r7xb`.

If `id_x589` max is still stuck at 0.2, then the problem is more fundamental. It could be that `K_IX=0.2` combined with `h_X=6` creates such a potent "wall" of inhibition at `id_x589=0.2` that `id_r7xb` production is almost completely shut off, preventing `id_r7xb` from staying high long enough to drive `id_x589` further up.
If this attempt fails, the next step might be to slightly increase `K_IX` (e.g., to 0.22 or 0.25) while keeping `h_X=6`, to shift that "wall" a bit higher, and readjust `Vmax_R`.

## Action

### Code
```python
import libsbml
import pandas as pd
import numpy as np
from scipy.signal import find_peaks

# --- Helper function to add reactions (L2 compatible) ---
def add_reaction_l2(model, r_id, reactants_dict, products_dict, formula_str, reversible=False):
    reaction = model.createReaction()
    reaction.setId(r_id)
    reaction.setReversible(reversible)

    for species_id, stoich in reactants_dict.items():
        reactant = reaction.createReactant()
        reactant.setSpecies(species_id)
        reactant.setStoichiometry(stoich)

    for species_id, stoich in products_dict.items():
        product = reaction.createProduct()
        product.setSpecies(species_id)
        product.setStoichiometry(stoich)

    kinetic_law = reaction.createKineticLaw()
    kinetic_law.setFormula(formula_str)
    return reaction

# --- Load original SBML and experimental data ---
sbml_doc_v14 = libsbml.readSBMLFromString(input_sbml_string)
model_v14 = sbml_doc_v14.getModel()
obs_data = experiment_history['iteration_0']

# --- Define Initial Concentrations ---
IC_P_v14 = 0.89
IC_N_v14 = 0.71

species_plsk = model_v14.getSpecies('id_plsk')
if species_plsk: species_plsk.setInitialConcentration(IC_P_v14)

species_0nyd = model_v14.getSpecies('id_0nyd')
if species_0nyd: species_0nyd.setInitialConcentration(IC_N_v14)
    
model_v14.getSpecies('id_x589').setInitialConcentration(0.0)
model_v14.getSpecies('id_22ta').setInitialConcentration(0.0)
model_v14.getSpecies('id_r7xb').setInitialConcentration(0.36)

# --- Define Parameters for v14 ---
# Based on v11, but Vmax_R reduced to lower id_r7xb mean.
# k_X_prod is calculated for target means id_r7xb=0.548, id_x589=0.200
param_definitions_v14 = {
    "Vmax_R": 0.09864,        # Reduced from v11's 0.1096 (approx 10% reduction)
    "K_IX": 0.2,
    "h_X": 6.0,
    "k_R_deg": 0.1,
    "k_X_prod": 0.0018248175, # (0.200 / 0.548) * 0.005
    "k_X_deg": 0.005,
    "k_PT": 0.08,
    "k_N": 0.06
}

for p_id, p_val in param_definitions_v14.items():
    param = model_v14.getParameter(p_id)
    if not param:
        param = model_v14.createParameter()
        param.setId(p_id)
        param.setConstant(True)
    param.setValue(p_val)

# --- Add Reactions for v14 (same structure) ---
formula_Rprod = "Vmax_R / (1 + (id_x589 / K_IX)^h_X)"
add_reaction_l2(model_v14, "R_Rprod", {}, {"id_r7xb": 1}, formula_Rprod)

formula_Rdeg = "k_R_deg * id_r7xb"
add_reaction_l2(model_v14, "R_Rdeg", {"id_r7xb": 1}, {}, formula_Rdeg)

formula_Xprod = "k_X_prod * id_r7xb"
add_reaction_l2(model_v14, "R_Xprod", {}, {"id_x589": 1}, formula_Xprod)

formula_Xdeg = "k_X_deg * id_x589"
add_reaction_l2(model_v14, "R_Xdeg", {"id_x589": 1}, {}, formula_Xdeg)

formula_PT = "k_PT * id_plsk * id_r7xb"
add_reaction_l2(model_v14, "R_PT", {"id_plsk": 1}, {"id_22ta": 1}, formula_PT)

formula_N = "k_N * id_0nyd * id_r7xb"
add_reaction_l2(model_v14, "R_N", {"id_0nyd": 1}, {}, formula_N)

# --- Generate SBML string for Model v14 ---
writer = libsbml.SBMLWriter()
sbml_v14_string = writer.writeSBMLToString(sbml_doc_v14)
shared_variables.add("hypothetical_sbml_v14", sbml_v14_string)

# --- Simulate Model v14 ---
sim_data_v14 = simulate(sbml_v14_string)
shared_variables.add("sim_data_v14", sim_data_v14)
print("Simulation of Model v14 completed.")

# --- Analyze Simulation Results ---
sim_data_v14_adjusted = sim_data_v14.copy()
sim_data_v14_adjusted['id_plsk'] = sim_data_v14['id_plsk'] - IC_P_v14
sim_data_v14_adjusted['id_0nyd'] = sim_data_v14['id_0nyd'] - IC_N_v14

r7xb_sim_stats_v14 = sim_data_v14['id_r7xb'].describe()
x589_sim_stats_v14 = sim_data_v14['id_x589'].describe()

print(f"\nObserved id_r7xb: mean={obs_data['id_r7xb'].mean():.3f}, std={obs_data['id_r7xb'].std():.3f}, min={obs_data['id_r7xb'].min():.3f}, max={obs_data['id_r7xb'].max():.3f}")
print(f"Simulated id_r7xb (v14): mean={r7xb_sim_stats_v14['mean']:.3f}, std={r7xb_sim_stats_v14['std']:.3f}, min={r7xb_sim_stats_v14['min']:.3f}, max={r7xb_sim_stats_v14['max']:.3f}")

print(f"Observed id_x589: mean={obs_data['id_x589'].mean():.3f}, std={obs_data['id_x589'].std():.3f}, min={obs_data['id_x589'].min():.3f}, max={obs_data['id_x589'].max():.3f}")
print(f"Simulated id_x589 (v14): mean={x589_sim_stats_v14['mean']:.3f}, std={x589_sim_stats_v14['std']:.3f}, min={x589_sim_stats_v14['min']:.3f}, max={x589_sim_stats_v14['max']:.3f}")

if r7xb_sim_stats_v14['std'] > 0.15 and x589_sim_stats_v14['std'] > 0.06:
    print("Simulated id_r7xb and id_x589 (v14) show significant dynamic/oscillatory behavior.")
    time_points = sim_data_v14['Time'].to_numpy()
    r7xb_values = sim_data_v14['id_r7xb'].to_numpy()
    peaks_r7xb, _ = find_peaks(r7xb_values, height=r7xb_sim_stats_v14['mean'], distance=800) 
    
    if len(peaks_r7xb) > 1:
        peak_times_r7xb = time_points[peaks_r7xb]
        periods_r7xb = np.diff(peak_times_r7xb)
        print(f"Simulated peak times for id_r7xb: {peak_times_r7xb}")
        print(f"Simulated periods for id_r7xb: {periods_r7xb}, mean period: {np.mean(periods_r7xb):.2f}")
    else:
        print("Not enough peaks found for id_r7xb to estimate period reliably.")
else:
    print("Simulated id_r7xb or id_x589 (v14) is not strongly oscillating with these parameters.")

```
# Observation

## Code Stdout
Simulation of Model v14 completed.

Observed id_r7xb: mean=0.548, std=0.180, min=0.297, max=0.939
Simulated id_r7xb (v14): mean=0.599, std=0.136, min=0.360, max=0.982
Observed id_x589: mean=0.200, std=0.071, min=0.000, max=0.438
Simulated id_x589 (v14): mean=0.179, std=0.041, min=0.000, max=0.195
Simulated id_r7xb or id_x589 (v14) is not strongly oscillating with these parameters.

## 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. (hypothetical_sbml_v4, sim_data_v4, hypothetical_sbml_v5, sim_data_v5, hypothetical_sbml_v6, sim_data_v6, hypothetical_sbml_v7, sim_data_v7, hypothetical_sbml_v8, sim_data_v8, hypothetical_sbml_v9, sim_data_v9, hypothetical_sbml_v10, sim_data_v10, hypothetical_sbml_v11, sim_data_v11, hypothetical_sbml_v12, sim_data_v12, hypothetical_sbml_v13, sim_data_v13, hypothetical_sbml_v14, sim_data_v14)

### 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

Model v14 (reducing `Vmax_R` from v11) did not improve things significantly.
-   `id_r7xb` std: 0.136 (v14) vs 0.157 (v11). Obs: 0.180. (Worse)
-   `id_r7xb` mean: 0.599 (v14) vs 0.619 (v11). Obs: 0.548. (Still high, but closer)
-   `id_x589` std: 0.041 (v14) vs 0.040 (v11). Obs: 0.071. (Still low)
-   `id_x589` max: 0.195 (v14) vs 0.200 (v11). Obs: 0.438. (Still stuck near mean)
-   `id_x589` mean: 0.179 (v14) vs 0.185 (v11). Obs: 0.200. (Still low)

The core problem persists: `id_x589` does not achieve a high enough peak (observed 0.438, simulated ~0.2). This limits its ability to strongly inhibit `id_r7xb` and drive `id_r7xb` to its observed minimum (0.297, simulated ~0.360). Consequently, the amplitudes of both oscillations are too small.

Let's reconsider the parameters of Model v11, which had the best `id_r7xb` std (0.157):
*   `Vmax_R = 0.1096`, `K_IX = 0.2`, `h_X = 6.0`
*   `k_R_deg = 0.1` (timescale 10)
*   `k_X_prod = 0.0018248`, `k_X_deg = 0.005` (timescale 200)

The fact that `id_x589_max` is stuck near `K_IX` (0.2) when `h_X` is high (6.0) suggests that the inhibition by `id_x589` on `id_r7xb` production becomes overwhelmingly strong as soon as `id_x589` slightly exceeds `K_IX`. This prevents `id_r7xb` from staying high long enough to drive `id_x589` much further up.

Strategy for Model v15:
1.  **Increase `K_IX`**: Shift the point of strong inhibition to a higher `id_x589` concentration. If `id_x589` needs to peak around 0.4, perhaps `K_IX` should be somewhere between its mean (0.2) and its peak (0.4). Let's try `K_IX = 0.25`. This is a compromise between v11 (`K_IX=0.2`) and v12 (`K_IX=0.4`, which was too high).
2.  **Keep `h_X = 6.0`**: Maintain the steepness of inhibition.
3.  **Keep `k_R_deg = 0.1` and `k_X_deg = 0.005`**: These timescales from v11 seemed promising.
4.  **Recalculate `Vmax_R`** to target `id_r7xb_mean = 0.548` with the new `K_IX`, assuming `id_x589_mean = 0.200`.
    Inhibition term: `(id_x589_mean / K_IX)^h_X = (0.200 / 0.25)^6 = (0.8)^6 = 0.262144`.
    `Vmax_R / (1 + 0.262144) = k_R_deg * id_r7xb_mean`.
    `Vmax_R / 1.262144 = 0.1 * 0.548 = 0.0548`.
    `Vmax_R = 0.0548 * 1.262144 = 0.069165`.
5.  **Recalculate `k_X_prod`** to target `id_x589_mean = 0.200` with `id_r7xb_mean = 0.548` and `k_X_deg = 0.005`.
    `k_X_prod = (0.200 / 0.548) * 0.005 = 0.0018248175`. (This remains the same as it's based on target means and `k_X_deg`).
6.  Keep `k_PT = 0.08`, `k_N = 0.06`.

Parameters for Model v15:
*   `IC_P_v15 = 0.89`, `IC_N_v15 = 0.71`
*   `Vmax_R = 0.069165` (changed)
*   `K_IX = 0.25` (changed)
*   `h_X = 6.0`
*   `k_R_deg = 0.1`
*   `k_X_prod = 0.0018248175`
*   `k_X_deg = 0.005`
*   `k_PT = 0.08`
*   `k_N = 0.06`

This approach attempts to "allow" `id_x589` to rise higher by making the inhibition less severe at `id_x589 = 0.2`. If `id_x589` can now reach higher values, it might then be able to drive `id_r7xb` lower and create larger oscillations.
The `Vmax_R` is lower than in v11, which might help bring the `id_r7xb` mean down from the 0.6+ values seen in v11/v13/v14.

## Action

### Code
```python
import libsbml
import pandas as pd
import numpy as np
from scipy.signal import find_peaks

# --- Helper function to add reactions (L2 compatible) ---
def add_reaction_l2(model, r_id, reactants_dict, products_dict, formula_str, reversible=False):
    reaction = model.createReaction()
    reaction.setId(r_id)
    reaction.setReversible(reversible)

    for species_id, stoich in reactants_dict.items():
        reactant = reaction.createReactant()
        reactant.setSpecies(species_id)
        reactant.setStoichiometry(stoich)

    for species_id, stoich in products_dict.items():
        product = reaction.createProduct()
        product.setSpecies(species_id)
        product.setStoichiometry(stoich)

    kinetic_law = reaction.createKineticLaw()
    kinetic_law.setFormula(formula_str)
    return reaction

# --- Load original SBML and experimental data ---
sbml_doc_v15 = libsbml.readSBMLFromString(input_sbml_string)
model_v15 = sbml_doc_v15.getModel()
obs_data = experiment_history['iteration_0']

# --- Define Initial Concentrations ---
IC_P_v15 = 0.89
IC_N_v15 = 0.71

species_plsk = model_v15.getSpecies('id_plsk')
if species_plsk: species_plsk.setInitialConcentration(IC_P_v15)

species_0nyd = model_v15.getSpecies('id_0nyd')
if species_0nyd: species_0nyd.setInitialConcentration(IC_N_v15)
    
model_v15.getSpecies('id_x589').setInitialConcentration(0.0)
model_v15.getSpecies('id_22ta').setInitialConcentration(0.0)
model_v15.getSpecies('id_r7xb').setInitialConcentration(0.36)

# --- Define Parameters for v15 ---
param_definitions_v15 = {
    "Vmax_R": 0.069165472,    # Recalculated: (0.1 * 0.548) * (1 + (0.2/0.25)^6)
    "K_IX": 0.25,             # Changed from 0.2
    "h_X": 6.0,
    "k_R_deg": 0.1,
    "k_X_prod": 0.0018248175, # (0.200 / 0.548) * 0.005
    "k_X_deg": 0.005,
    "k_PT": 0.08,
    "k_N": 0.06
}

for p_id, p_val in param_definitions_v15.items():
    param = model_v15.getParameter(p_id)
    if not param:
        param = model_v15.createParameter()
        param.setId(p_id)
        param.setConstant(True)
    param.setValue(p_val)

# --- Add Reactions for v15 (same structure) ---
formula_Rprod = "Vmax_R / (1 + (id_x589 / K_IX)^h_X)"
add_reaction_l2(model_v15, "R_Rprod", {}, {"id_r7xb": 1}, formula_Rprod)

formula_Rdeg = "k_R_deg * id_r7xb"
add_reaction_l2(model_v15, "R_Rdeg", {"id_r7xb": 1}, {}, formula_Rdeg)

formula_Xprod = "k_X_prod * id_r7xb"
add_reaction_l2(model_v15, "R_Xprod", {}, {"id_x589": 1}, formula_Xprod)

formula_Xdeg = "k_X_deg * id_x589"
add_reaction_l2(model_v15, "R_Xdeg", {"id_x589": 1}, {}, formula_Xdeg)

formula_PT = "k_PT * id_plsk * id_r7xb"
add_reaction_l2(model_v15, "R_PT", {"id_plsk": 1}, {"id_22ta": 1}, formula_PT)

formula_N = "k_N * id_0nyd * id_r7xb"
add_reaction_l2(model_v15, "R_N", {"id_0nyd": 1}, {}, formula_N)

# --- Generate SBML string for Model v15 ---
writer = libsbml.SBMLWriter()
sbml_v15_string = writer.writeSBMLToString(sbml_doc_v15)
shared_variables.add("hypothetical_sbml_v15", sbml_v15_string)

# --- Simulate Model v15 ---
sim_data_v15 = simulate(sbml_v15_string)
shared_variables.add("sim_data_v15", sim_data_v15)
print("Simulation of Model v15 completed.")

# --- Analyze Simulation Results ---
sim_data_v15_adjusted = sim_data_v15.copy()
sim_data_v15_adjusted['id_plsk'] = sim_data_v15['id_plsk'] - IC_P_v15
sim_data_v15_adjusted['id_0nyd'] = sim_data_v15['id_0nyd'] - IC_N_v15

r7xb_sim_stats_v15 = sim_data_v15['id_r7xb'].describe()
x589_sim_stats_v15 = sim_data_v15['id_x589'].describe()

print(f"\nObserved id_r7xb: mean={obs_data['id_r7xb'].mean():.3f}, std={obs_data['id_r7xb'].std():.3f}, min={obs_data['id_r7xb'].min():.3f}, max={obs_data['id_r7xb'].max():.3f}")
print(f"Simulated id_r7xb (v15): mean={r7xb_sim_stats_v15['mean']:.3f}, std={r7xb_sim_stats_v15['std']:.3f}, min={r7xb_sim_stats_v15['min']:.3f}, max={r7xb_sim_stats_v15['max']:.3f}")

print(f"Observed id_x589: mean={obs_data['id_x589'].mean():.3f}, std={obs_data['id_x589'].std():.3f}, min={obs_data['id_x589'].min():.3f}, max={obs_data['id_x589'].max():.3f}")
print(f"Simulated id_x589 (v15): mean={x589_sim_stats_v15['mean']:.3f}, std={x589_sim_stats_v15['std']:.3f}, min={x589_sim_stats_v15['min']:.3f}, max={x589_sim_stats_v15['max']:.3f}")

if r7xb_sim_stats_v15['std'] > 0.15 and x589_sim_stats_v15['std'] > 0.065: # Target std_obs for x589 is 0.071
    print("Simulated id_r7xb and id_x589 (v15) show significant dynamic/oscillatory behavior.")
    time_points = sim_data_v15['Time'].to_numpy()
    r7xb_values = sim_data_v15['id_r7xb'].to_numpy()
    peaks_r7xb, _ = find_peaks(r7xb_values, height=r7xb_sim_stats_v15['mean'], distance=800) 
    
    if len(peaks_r7xb) > 1:
        peak_times_r7xb = time_points[peaks_r7xb]
        periods_r7xb = np.diff(peak_times_r7xb)
        print(f"Simulated peak times for id_r7xb: {peak_times_r7xb}")
        print(f"Simulated periods for id_r7xb: {periods_r7xb}, mean period: {np.mean(periods_r7xb):.2f}")
    else:
        print("Not enough peaks found for id_r7xb to estimate period reliably.")
else:
    print("Simulated id_r7xb or id_x589 (v15) is not strongly oscillating with these parameters.")

```
# Observation

## Code Stdout
Simulation of Model v15 completed.

Observed id_r7xb: mean=0.548, std=0.180, min=0.297, max=0.939
Simulated id_r7xb (v15): mean=0.586, std=0.054, min=0.360, max=0.691
Observed id_x589: mean=0.200, std=0.071, min=0.000, max=0.438
Simulated id_x589 (v15): mean=0.173, std=0.048, min=0.000, max=0.200
Simulated id_r7xb or id_x589 (v15) is not strongly oscillating with these parameters.

## 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. (hypothetical_sbml_v4, sim_data_v4, hypothetical_sbml_v5, sim_data_v5, hypothetical_sbml_v6, sim_data_v6, hypothetical_sbml_v7, sim_data_v7, hypothetical_sbml_v8, sim_data_v8, hypothetical_sbml_v9, sim_data_v9, hypothetical_sbml_v10, sim_data_v10, hypothetical_sbml_v11, sim_data_v11, hypothetical_sbml_v12, sim_data_v12, hypothetical_sbml_v13, sim_data_v13, hypothetical_sbml_v14, sim_data_v14, hypothetical_sbml_v15, sim_data_v15)

### 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

Model v15 (with `K_IX = 0.25`) was also a step backward for `id_r7xb` oscillations (std dev 0.054). `id_x589` max was still stuck at 0.2.
It seems that `K_IX` values around `id_x589_mean` (0.2) combined with high `h_X` (6.0) are very effective at clamping `id_x589` near `K_IX`, preventing it from overshooting and driving oscillations.

The best `id_r7xb` std (0.157) was achieved with Model v11:
*   `Vmax_R = 0.1096`, `K_IX = 0.2`, `h_X = 6.0`
*   `k_R_deg = 0.1` (timescale 10)
*   `k_X_prod = 0.0018248`, `k_X_deg = 0.005` (timescale 200)
*   `k_PT = 0.08`, `k_N = 0.06`

Key issues with v11:
1.  `id_r7xb` mean: 0.619 (obs: 0.548) - Too high.
2.  `id_r7xb` min: 0.360 (obs: 0.297) - Too high.
3.  `id_r7xb` max: 1.090 (obs: 0.939) - Too high.
4.  `id_x589` std: 0.040 (obs: 0.071) - Too low.
5.  `id_x589` max: 0.200 (obs: 0.438) - Too low, stuck at `K_IX`.
6.  `id_x589` mean: 0.185 (obs: 0.200) - Slightly low.

The fact that `id_x589_max` is pinned at `K_IX` when `h_X` is high is a strong indicator.
If `id_x589` cannot rise above `K_IX`, it cannot cause `id_r7xb` to drop significantly.
For `id_x589` to reach 0.438, `K_IX` must be higher than 0.2 if `h_X` is high.
However, Model v12 with `K_IX = 0.4` and `h_X = 6` also failed (std 0.014). This suggests that if `K_IX` is too far from `id_x589_mean`, the inhibition is too weak for too long, and the system stabilizes.

What if the Hill coefficient `h_X` is the main issue? A very high `h_X` makes the system very sensitive to `id_x589` crossing `K_IX`.
Perhaps a lower `h_X` (e.g., 2 or 3) would create a "softer" negative feedback, allowing for wider swings.
Model v13 used `h_X=4` with `K_IX=0.2` and got `id_r7xb` std 0.146, `id_x589` std 0.041, `id_x589` max 0.2. Still not good enough.

Let's reconsider the period. Observed period is ~200.
In v11, `id_r7xb` timescale (`1/k_R_deg`) = 10. `id_x589` timescale (`1/k_X_deg`) = 200.
The `id_x589` timescale matches the observed period, which is a good sign for phase lag.

The problem might be the "gain" of the `id_x589` response.
`id_x589_peak ≈ (k_X_prod / k_X_deg) * id_r7xb_at_peak_of_x589`.
Ratio `k_X_prod / k_X_deg = (target_id_x589_mean / target_id_r7xb_mean) = 0.2 / 0.548 = 0.36496`.
If `id_r7xb` peaks at ~1.0, then `id_x589` should try to peak at `0.365`.
This is not happening because `id_r7xb` production is shut down too effectively when `id_x589` approaches `K_IX=0.2`.

This implies `K_IX` *must* be higher than 0.2 to allow `id_x589` to rise above 0.2.
But if `K_IX` is too high (like 0.4 in v12), the system damps.
This suggests a narrow window for `K_IX` or that `h_X=6` is too aggressive.

Let's try `h_X = 3` (a moderate Hill coefficient) and set `K_IX` to be slightly above `id_x589_mean`, e.g., `K_IX = 0.22`.
This means inhibition starts to get strong as `id_x589` passes 0.22.
We need `id_x589` to reach ~0.4.

Parameters for Model v16:
*   `IC_P_v16 = 0.89`, `IC_N_v16 = 0.71`
*   `k_R_deg = 0.1` (timescale 10)
*   `k_X_deg = 0.005` (timescale 200)
*   `K_IX = 0.22` (slightly higher than `id_x589_mean`)
*   `h_X = 3.0` (softer feedback)

1.  Recalculate `Vmax_R` for `id_r7xb_mean = 0.548`, `id_x589_mean = 0.200`:
    Inhibition term: `(id_x589_mean / K_IX)^h_X = (0.200 / 0.22)^3 = (0.90909)^3 = 0.7513`.
    `Vmax_R / (1 + 0.7513) = k_R_deg * id_r7xb_mean`.
    `Vmax_R / 1.7513 = 0.1 * 0.548 = 0.0548`.
    `Vmax_R = 0.0548 * 1.7513 = 0.09597`.

2.  `k_X_prod = (0.200 / 0.548) * 0.005 = 0.0018248175`.

3.  `k_PT = 0.08`, `k_N = 0.06`.

This model attempts a "gentler" inhibition (`h_X=3`) with `K_IX` set just above the target mean of `id_x589`, hoping this allows `id_x589` to build up to higher levels without `id_r7xb` production being completely shut off too early.
The `Vmax_R` is now lower than v11 (0.09597 vs 0.1096), which should help bring `id_r7xb_mean` down.

## Action

### Code
```python
import libsbml
import pandas as pd
import numpy as np
from scipy.signal import find_peaks

# --- Helper function to add reactions (L2 compatible) ---
def add_reaction_l2(model, r_id, reactants_dict, products_dict, formula_str, reversible=False):
    reaction = model.createReaction()
    reaction.setId(r_id)
    reaction.setReversible(reversible)

    for species_id, stoich in reactants_dict.items():
        reactant = reaction.createReactant()
        reactant.setSpecies(species_id)
        reactant.setStoichiometry(stoich)

    for species_id, stoich in products_dict.items():
        product = reaction.createProduct()
        product.setSpecies(species_id)
        product.setStoichiometry(stoich)

    kinetic_law = reaction.createKineticLaw()
    kinetic_law.setFormula(formula_str)
    return reaction

# --- Load original SBML and experimental data ---
sbml_doc_v16 = libsbml.readSBMLFromString(input_sbml_string)
model_v16 = sbml_doc_v16.getModel()
obs_data = experiment_history['iteration_0']

# --- Define Initial Concentrations ---
IC_P_v16 = 0.89
IC_N_v16 = 0.71

species_plsk = model_v16.getSpecies('id_plsk')
if species_plsk: species_plsk.setInitialConcentration(IC_P_v16)

species_0nyd = model_v16.getSpecies('id_0nyd')
if species_0nyd: species_0nyd.setInitialConcentration(IC_N_v16)
    
model_v16.getSpecies('id_x589').setInitialConcentration(0.0)
model_v16.getSpecies('id_22ta').setInitialConcentration(0.0)
model_v16.getSpecies('id_r7xb').setInitialConcentration(0.36)

# --- Define Parameters for v16 ---
param_definitions_v16 = {
    "Vmax_R": 0.09597124,     # Recalculated: (0.1 * 0.548) * (1 + (0.2/0.22)^3)
    "K_IX": 0.22,             # Changed
    "h_X": 3.0,               # Changed
    "k_R_deg": 0.1,
    "k_X_prod": 0.0018248175, # (0.200 / 0.548) * 0.005
    "k_X_deg": 0.005,
    "k_PT": 0.08,
    "k_N": 0.06
}

for p_id, p_val in param_definitions_v16.items():
    param = model_v16.getParameter(p_id)
    if not param:
        param = model_v16.createParameter()
        param.setId(p_id)
        param.setConstant(True)
    param.setValue(p_val)

# --- Add Reactions for v16 (same structure) ---
formula_Rprod = "Vmax_R / (1 + (id_x589 / K_IX)^h_X)"
add_reaction_l2(model_v16, "R_Rprod", {}, {"id_r7xb": 1}, formula_Rprod)

formula_Rdeg = "k_R_deg * id_r7xb"
add_reaction_l2(model_v16, "R_Rdeg", {"id_r7xb": 1}, {}, formula_Rdeg)

formula_Xprod = "k_X_prod * id_r7xb"
add_reaction_l2(model_v16, "R_Xprod", {}, {"id_x589": 1}, formula_Xprod)

formula_Xdeg = "k_X_deg * id_x589"
add_reaction_l2(model_v16, "R_Xdeg", {"id_x589": 1}, {}, formula_Xdeg)

formula_PT = "k_PT * id_plsk * id_r7xb"
add_reaction_l2(model_v16, "R_PT", {"id_plsk": 1}, {"id_22ta": 1}, formula_PT)

formula_N = "k_N * id_0nyd * id_r7xb"
add_reaction_l2(model_v16, "R_N", {"id_0nyd": 1}, {}, formula_N)

# --- Generate SBML string for Model v16 ---
writer = libsbml.SBMLWriter()
sbml_v16_string = writer.writeSBMLToString(sbml_doc_v16)
shared_variables.add("hypothetical_sbml_v16", sbml_v16_string)

# --- Simulate Model v16 ---
sim_data_v16 = simulate(sbml_v16_string)
shared_variables.add("sim_data_v16", sim_data_v16)
print("Simulation of Model v16 completed.")

# --- Analyze Simulation Results ---
sim_data_v16_adjusted = sim_data_v16.copy()
sim_data_v16_adjusted['id_plsk'] = sim_data_v16['id_plsk'] - IC_P_v16
sim_data_v16_adjusted['id_0nyd'] = sim_data_v16['id_0nyd'] - IC_N_v16

r7xb_sim_stats_v16 = sim_data_v16['id_r7xb'].describe()
x589_sim_stats_v16 = sim_data_v16['id_x589'].describe()

print(f"\nObserved id_r7xb: mean={obs_data['id_r7xb'].mean():.3f}, std={obs_data['id_r7xb'].std():.3f}, min={obs_data['id_r7xb'].min():.3f}, max={obs_data['id_r7xb'].max():.3f}")
print(f"Simulated id_r7xb (v16): mean={r7xb_sim_stats_v16['mean']:.3f}, std={r7xb_sim_stats_v16['std']:.3f}, min={r7xb_sim_stats_v16['min']:.3f}, max={r7xb_sim_stats_v16['max']:.3f}")

print(f"Observed id_x589: mean={obs_data['id_x589'].mean():.3f}, std={obs_data['id_x589'].std():.3f}, min={obs_data['id_x589'].min():.3f}, max={obs_data['id_x589'].max():.3f}")
print(f"Simulated id_x589 (v16): mean={x589_sim_stats_v16['mean']:.3f}, std={x589_sim_stats_v16['std']:.3f}, min={x589_sim_stats_v16['min']:.3f}, max={x589_sim_stats_v16['max']:.3f}")

if r7xb_sim_stats_v16['std'] > 0.15 and x589_sim_stats_v16['std'] > 0.065:
    print("Simulated id_r7xb and id_x589 (v16) show significant dynamic/oscillatory behavior.")
    time_points = sim_data_v16['Time'].to_numpy()
    r7xb_values = sim_data_v16['id_r7xb'].to_numpy()
    peaks_r7xb, _ = find_peaks(r7xb_values, height=r7xb_sim_stats_v16['mean'], distance=800) 
    
    if len(peaks_r7xb) > 1:
        peak_times_r7xb = time_points[peaks_r7xb]
        periods_r7xb = np.diff(peak_times_r7xb)
        print(f"Simulated peak times for id_r7xb: {peak_times_r7xb}")
        print(f"Simulated periods for id_r7xb: {periods_r7xb}, mean period: {np.mean(periods_r7xb):.2f}")
    else:
        print("Not enough peaks found for id_r7xb to estimate period reliably.")
else:
    print("Simulated id_r7xb or id_x589 (v16) is not strongly oscillating with these parameters.")

```
# Observation

## Code Stdout
Simulation of Model v16 completed.

Observed id_r7xb: mean=0.548, std=0.180, min=0.297, max=0.939
Simulated id_r7xb (v16): mean=0.606, std=0.110, min=0.360, max=0.942
Observed id_x589: mean=0.200, std=0.071, min=0.000, max=0.438
Simulated id_x589 (v16): mean=0.180, std=0.043, min=0.000, max=0.200
Simulated id_r7xb or id_x589 (v16) is not strongly oscillating with these parameters.

## 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. (hypothetical_sbml_v4, sim_data_v4, hypothetical_sbml_v5, sim_data_v5, hypothetical_sbml_v6, sim_data_v6, hypothetical_sbml_v7, sim_data_v7, hypothetical_sbml_v8, sim_data_v8, hypothetical_sbml_v9, sim_data_v9, hypothetical_sbml_v10, sim_data_v10, hypothetical_sbml_v11, sim_data_v11, hypothetical_sbml_v12, sim_data_v12, hypothetical_sbml_v13, sim_data_v13, hypothetical_sbml_v14, sim_data_v14, hypothetical_sbml_v15, sim_data_v15, hypothetical_sbml_v16, sim_data_v16)

### 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

Model v16 (with `K_IX=0.22`, `h_X=3`) did not yield the desired breakthrough.
-   `id_r7xb` std: 0.110 (Obs: 0.180).
-   `id_x589` std: 0.043 (Obs: 0.071), max: 0.200 (Obs: 0.438). Still stuck.

The consistent failure of `id_x589` to rise significantly above `K_IX` (when `K_IX` is near 0.2) or above its mean, regardless of `h_X` or timescale adjustments, is the primary obstacle.

Let's go back to Model v11 parameters, which gave the highest `id_r7xb` std (0.157), and try a more aggressive approach to make `id_x589` peak higher.
Model v11:
*   `Vmax_R = 0.1096`, `K_IX = 0.2`, `h_X = 6.0`
*   `k_R_deg = 0.1` (timescale 10)
*   `k_X_prod = 0.0018248`, `k_X_deg = 0.005` (timescale 200)

If `id_x589_max` is stuck at `K_IX=0.2`, it means that as soon as `id_x589` reaches 0.2, `id_r7xb` production is so strongly inhibited that `id_r7xb` cannot sustain levels high enough to push `id_x589` further.
To get `id_x589` to peak at ~0.4, `id_r7xb` must be high while `id_x589` is rising from ~0.2 to ~0.4.
This implies that even when `id_x589` is, say, 0.3, `id_r7xb` production must still be substantial.
This requires `K_IX` to be higher than 0.3, perhaps around the target peak of `id_x589` (0.4).

Let's try `K_IX = 0.35` (midway to 0.4, target peak of `id_x589`).
Keep `h_X = 6.0` for strong switch-like behavior once `id_x589` approaches this higher `K_IX`.
Keep timescales from v11: `k_R_deg = 0.1`, `k_X_deg = 0.005`.

Recalculate `Vmax_R` and `k_X_prod` for Model v17:
-   Target `id_r7xb_mean = 0.548`, `id_x589_mean = 0.200`.
-   `K_IX = 0.35`.
-   `h_X = 6.0`.
-   `k_R_deg = 0.1`.
-   `k_X_deg = 0.005`.

1.  `Vmax_R`:
    Inhibition term: `(id_x589_mean / K_IX)^h_X = (0.200 / 0.35)^6 = (0.5714)^6 = 0.0340`.
    `Vmax_R / (1 + 0.0340) = k_R_deg * id_r7xb_mean`.
    `Vmax_R / 1.0340 = 0.1 * 0.548 = 0.0548`.
    `Vmax_R = 0.0548 * 1.0340 = 0.05666`.

2.  `k_X_prod = (0.200 / 0.548) * 0.005 = 0.0018248175`.

Parameters for Model v17:
*   `IC_P_v17 = 0.89`, `IC_N_v17 = 0.71`
*   `Vmax_R = 0.05666`
*   `K_IX = 0.35`
*   `h_X = 6.0`
*   `k_R_deg = 0.1`
*   `k_X_prod = 0.0018248175`
*   `k_X_deg = 0.005`
*   `k_PT = 0.08`, `k_N = 0.06` (from v11)

This is a significant increase in `K_IX` compared to v11 (`0.35` vs `0.2`). The `Vmax_R` is substantially lower to compensate for the weaker inhibition at `id_x589_mean = 0.2`.
The hope is that `id_r7xb` production remains strong enough even when `id_x589` is between 0.2 and 0.35, allowing `id_x589` to accumulate to higher levels. Once `id_x589` approaches 0.35, the `h_X=6` term should cause a sharp turndown of `id_r7xb` production.

If this model produces oscillations with a higher `id_x589` amplitude, we can then fine-tune the means and amplitudes of `id_r7xb`.
The `id_r7xb` mean in v11 was 0.619 (target 0.548). The `Vmax_R` in v11 was 0.1096.
The new `Vmax_R` (0.05666) is much lower, which should help bring the `id_r7xb` mean down.

## Action

### Code
```python
import libsbml
import pandas as pd
import numpy as np
from scipy.signal import find_peaks

# --- Helper function to add reactions (L2 compatible) ---
def add_reaction_l2(model, r_id, reactants_dict, products_dict, formula_str, reversible=False):
    reaction = model.createReaction()
    reaction.setId(r_id)
    reaction.setReversible(reversible)

    for species_id, stoich in reactants_dict.items():
        reactant = reaction.createReactant()
        reactant.setSpecies(species_id)
        reactant.setStoichiometry(stoich)

    for species_id, stoich in products_dict.items():
        product = reaction.createProduct()
        product.setSpecies(species_id)
        product.setStoichiometry(stoich)

    kinetic_law = reaction.createKineticLaw()
    kinetic_law.setFormula(formula_str)
    return reaction

# --- Load original SBML and experimental data ---
sbml_doc_v17 = libsbml.readSBMLFromString(input_sbml_string)
model_v17 = sbml_doc_v17.getModel()
obs_data = experiment_history['iteration_0']

# --- Define Initial Concentrations ---
IC_P_v17 = 0.89
IC_N_v17 = 0.71

species_plsk = model_v17.getSpecies('id_plsk')
if species_plsk: species_plsk.setInitialConcentration(IC_P_v17)

species_0nyd = model_v17.getSpecies('id_0nyd')
if species_0nyd: species_0nyd.setInitialConcentration(IC_N_v17)
    
model_v17.getSpecies('id_x589').setInitialConcentration(0.0)
model_v17.getSpecies('id_22ta').setInitialConcentration(0.0)
model_v17.getSpecies('id_r7xb').setInitialConcentration(0.36)

# --- Define Parameters for v17 ---
param_definitions_v17 = {
    "Vmax_R": 0.05666032,     # Recalculated: (0.1 * 0.548) * (1 + (0.2/0.35)^6)
    "K_IX": 0.35,             # Changed
    "h_X": 6.0,
    "k_R_deg": 0.1,
    "k_X_prod": 0.0018248175, # (0.200 / 0.548) * 0.005
    "k_X_deg": 0.005,
    "k_PT": 0.08,
    "k_N": 0.06
}

for p_id, p_val in param_definitions_v17.items():
    param = model_v17.getParameter(p_id)
    if not param:
        param = model_v17.createParameter()
        param.setId(p_id)
        param.setConstant(True)
    param.setValue(p_val)

# --- Add Reactions for v17 (same structure) ---
formula_Rprod = "Vmax_R / (1 + (id_x589 / K_IX)^h_X)"
add_reaction_l2(model_v17, "R_Rprod", {}, {"id_r7xb": 1}, formula_Rprod)

formula_Rdeg = "k_R_deg * id_r7xb"
add_reaction_l2(model_v17, "R_Rdeg", {"id_r7xb": 1}, {}, formula_Rdeg)

formula_Xprod = "k_X_prod * id_r7xb"
add_reaction_l2(model_v17, "R_Xprod", {}, {"id_x589": 1}, formula_Xprod)

formula_Xdeg = "k_X_deg * id_x589"
add_reaction_l2(model_v17, "R_Xdeg", {"id_x589": 1}, {}, formula_Xdeg)

formula_PT = "k_PT * id_plsk * id_r7xb"
add_reaction_l2(model_v17, "R_PT", {"id_plsk": 1}, {"id_22ta": 1}, formula_PT)

formula_N = "k_N * id_0nyd * id_r7xb"
add_reaction_l2(model_v17, "R_N", {"id_0nyd": 1}, {}, formula_N)

# --- Generate SBML string for Model v17 ---
writer = libsbml.SBMLWriter()
sbml_v17_string = writer.writeSBMLToString(sbml_doc_v17)
shared_variables.add("hypothetical_sbml_v17", sbml_v17_string)

# --- Simulate Model v17 ---
sim_data_v17 = simulate(sbml_v17_string)
shared_variables.add("sim_data_v17", sim_data_v17)
print("Simulation of Model v17 completed.")

# --- Analyze Simulation Results ---
sim_data_v17_adjusted = sim_data_v17.copy()
sim_data_v17_adjusted['id_plsk'] = sim_data_v17['id_plsk'] - IC_P_v17
sim_data_v17_adjusted['id_0nyd'] = sim_data_v17['id_0nyd'] - IC_N_v17

r7xb_sim_stats_v17 = sim_data_v17['id_r7xb'].describe()
x589_sim_stats_v17 = sim_data_v17['id_x589'].describe()

print(f"\nObserved id_r7xb: mean={obs_data['id_r7xb'].mean():.3f}, std={obs_data['id_r7xb'].std():.3f}, min={obs_data['id_r7xb'].min():.3f}, max={obs_data['id_r7xb'].max():.3f}")
print(f"Simulated id_r7xb (v17): mean={r7xb_sim_stats_v17['mean']:.3f}, std={r7xb_sim_stats_v17['std']:.3f}, min={r7xb_sim_stats_v17['min']:.3f}, max={r7xb_sim_stats_v17['max']:.3f}")

print(f"Observed id_x589: mean={obs_data['id_x589'].mean():.3f}, std={obs_data['id_x589'].std():.3f}, min={obs_data['id_x589'].min():.3f}, max={obs_data['id_x589'].max():.3f}")
print(f"Simulated id_x589 (v17): mean={x589_sim_stats_v17['mean']:.3f}, std={x589_sim_stats_v17['std']:.3f}, min={x589_sim_stats_v17['min']:.3f}, max={x589_sim_stats_v17['max']:.3f}")

if r7xb_sim_stats_v17['std'] > 0.15 and x589_sim_stats_v17['std'] > 0.065:
    print("Simulated id_r7xb and id_x589 (v17) show significant dynamic/oscillatory behavior.")
    time_points = sim_data_v17['Time'].to_numpy()
    r7xb_values = sim_data_v17['id_r7xb'].to_numpy()
    peaks_r7xb, _ = find_peaks(r7xb_values, height=r7xb_sim_stats_v17['mean'], distance=800) 
    
    if len(peaks_r7xb) > 1:
        peak_times_r7xb = time_points[peaks_r7xb]
        periods_r7xb = np.diff(peak_times_r7xb)
        print(f"Simulated peak times for id_r7xb: {peak_times_r7xb}")
        print(f"Simulated periods for id_r7xb: {periods_r7xb}, mean period: {np.mean(periods_r7xb):.2f}")
    else:
        print("Not enough peaks found for id_r7xb to estimate period reliably.")
else:
    print("Simulated id_r7xb or id_x589 (v17) is not strongly oscillating with these parameters.")

```
# Observation

## Code Stdout
Simulation of Model v17 completed.

Observed id_r7xb: mean=0.548, std=0.180, min=0.297, max=0.939
Simulated id_r7xb (v17): mean=0.554, std=0.015, min=0.360, max=0.567
Observed id_x589: mean=0.200, std=0.071, min=0.000, max=0.438
Simulated id_x589 (v17): mean=0.161, std=0.050, min=0.000, max=0.199
Simulated id_r7xb or id_x589 (v17) is not strongly oscillating with these parameters.

## 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. (hypothetical_sbml_v4, sim_data_v4, hypothetical_sbml_v5, sim_data_v5, hypothetical_sbml_v6, sim_data_v6, hypothetical_sbml_v7, sim_data_v7, hypothetical_sbml_v8, sim_data_v8, hypothetical_sbml_v9, sim_data_v9, hypothetical_sbml_v10, sim_data_v10, hypothetical_sbml_v11, sim_data_v11, hypothetical_sbml_v12, sim_data_v12, hypothetical_sbml_v13, sim_data_v13, hypothetical_sbml_v14, sim_data_v14, hypothetical_sbml_v15, sim_data_v15, hypothetical_sbml_v16, sim_data_v16, hypothetical_sbml_v17, sim_data_v17)

### 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

Model v17 (with `K_IX = 0.35`, `h_X = 6.0`) also failed to produce oscillations and `id_x589_max` was still stuck at ~0.2. This is perplexing. The strategy of raising `K_IX` to allow `id_x589` to rise higher before inhibition becomes strong is not working as expected with `h_X=6`.

It seems the high Hill coefficient (`h_X=6`) creates an extremely sharp switch. If `K_IX` is too low (e.g., 0.2), `id_x589` gets clamped at `K_IX`. If `K_IX` is too high (e.g., 0.35 or 0.4), the inhibition threshold is so high that `id_x589` (whose mean is targeted at 0.2) never reaches it strongly enough to cause a downturn in `id_r7xb`, leading to a stable state.

Let's revisit Model v11, which had the best `id_r7xb` std (0.157), and reconsider its parameters:
*   `Vmax_R = 0.1096`, `K_IX = 0.2`, `h_X = 6.0`
*   `k_R_deg = 0.1` (timescale 10)
*   `k_X_prod = 0.0018248`, `k_X_deg = 0.005` (timescale 200)
*   `k_PT = 0.08`, `k_N = 0.06`

Simulated results for v11:
*   `id_r7xb`: mean=0.619 (obs: 0.548), std=0.157 (obs: 0.180), min=0.360 (obs: 0.297), max=1.090 (obs: 0.939)
*   `id_x589`: mean=0.185 (obs: 0.200), std=0.040 (obs: 0.071), min=0.000 (obs: 0.000), max=0.200 (obs: 0.438)

The main issues are:
1.  `id_r7xb` mean, min, and max are all too high.
2.  `id_x589` std and max are too low. `id_x589_max` is stuck at `K_IX`.

To address `id_r7xb` levels being too high:
-   Decrease `Vmax_R` (production of `id_r7xb`).
-   Increase `k_R_deg` (degradation of `id_r7xb`).

To address `id_x589` amplitude being too low:
-   This is the core problem. If `id_x589_max` is stuck at `K_IX=0.2` because `h_X=6` is too steep, we need to allow `id_x589` to rise above 0.2 without completely shutting down `id_r7xb`.

What if we try a combination:
-   A slightly higher `K_IX` than 0.2 (e.g., `K_IX = 0.22` as in v16, or `K_IX = 0.24`).
-   A slightly lower `h_X` than 6 (e.g., `h_X = 4` or `h_X = 5`). Model v16 had `K_IX=0.22, h_X=3` and didn't work well. Model v13 had `K_IX=0.2, h_X=4` and also didn't work well.

Let's try to make the inhibition less "binary" by reducing `h_X` significantly, but keep `K_IX` at 0.2.
If `h_X` is lower, say `h_X=2` (Michaelis-Menten like for dimer binding):
-   Inhibition term at `id_x589_mean=0.2, K_IX=0.2`: `(0.2/0.2)^2 = 1`.
-   `Vmax_R` would be `0.1096` (same as v11, as inhibition at mean is 1).
-   This "softer" inhibition might allow `id_r7xb` to stay higher for longer even as `id_x589` rises past 0.2, potentially allowing `id_x589` to reach a higher peak.

Parameters for Model v18 (based on v11, but with `h_X=2`):
*   `IC_P_v18 = 0.89`, `IC_N_v18 = 0.71`
*   `Vmax_R = 0.1096` (from v11, as `(K_IX/K_IX)^h_X` is still 1)
*   `K_IX = 0.2`
*   `h_X = 2.0` (significantly reduced from 6.0)
*   `k_R_deg = 0.1`
*   `k_X_prod = 0.0018248175`
*   `k_X_deg = 0.005`
*   `k_PT = 0.08`, `k_N = 0.06`

This is a direct test of the Hill coefficient's role in clamping `id_x589_max`.
If `id_x589_max` increases with `h_X=2`, then we are on the right track.
The `id_r7xb` mean might still be high with this `Vmax_R`, but the priority is to get `id_x589` to oscillate with a larger amplitude.

## Action

### Code
```python
import libsbml
import pandas as pd
import numpy as np
from scipy.signal import find_peaks

# --- Helper function to add reactions (L2 compatible) ---
def add_reaction_l2(model, r_id, reactants_dict, products_dict, formula_str, reversible=False):
    reaction = model.createReaction()
    reaction.setId(r_id)
    reaction.setReversible(reversible)

    for species_id, stoich in reactants_dict.items():
        reactant = reaction.createReactant()
        reactant.setSpecies(species_id)
        reactant.setStoichiometry(stoich)

    for species_id, stoich in products_dict.items():
        product = reaction.createProduct()
        product.setSpecies(species_id)
        product.setStoichiometry(stoich)

    kinetic_law = reaction.createKineticLaw()
    kinetic_law.setFormula(formula_str)
    return reaction

# --- Load original SBML and experimental data ---
sbml_doc_v18 = libsbml.readSBMLFromString(input_sbml_string)
model_v18 = sbml_doc_v18.getModel()
obs_data = experiment_history['iteration_0']

# --- Define Initial Concentrations ---
IC_P_v18 = 0.89
IC_N_v18 = 0.71

species_plsk = model_v18.getSpecies('id_plsk')
if species_plsk: species_plsk.setInitialConcentration(IC_P_v18)

species_0nyd = model_v18.getSpecies('id_0nyd')
if species_0nyd: species_0nyd.setInitialConcentration(IC_N_v18)
    
model_v18.getSpecies('id_x589').setInitialConcentration(0.0)
model_v18.getSpecies('id_22ta').setInitialConcentration(0.0)
model_v18.getSpecies('id_r7xb').setInitialConcentration(0.36)

# --- Define Parameters for v18 ---
# Based on v11, but h_X changed from 6 to 2.
# Vmax_R and k_X_prod are calculated for target means id_r7xb=0.548, id_x589=0.200
param_definitions_v18 = {
    "Vmax_R": 0.1096,         # 2 * k_R_deg * target_r7xb_mean (since K_IX = target_x589_mean and (1)^h_X = 1)
    "K_IX": 0.2,
    "h_X": 2.0,               # Changed from 6.0
    "k_R_deg": 0.1,
    "k_X_prod": 0.0018248175, # (target_x589_mean / target_r7xb_mean) * k_X_deg
    "k_X_deg": 0.005,
    "k_PT": 0.08,
    "k_N": 0.06
}

for p_id, p_val in param_definitions_v18.items():
    param = model_v18.getParameter(p_id)
    if not param:
        param = model_v18.createParameter()
        param.setId(p_id)
        param.setConstant(True)
    param.setValue(p_val)

# --- Add Reactions for v18 (same structure) ---
formula_Rprod = "Vmax_R / (1 + (id_x589 / K_IX)^h_X)"
add_reaction_l2(model_v18, "R_Rprod", {}, {"id_r7xb": 1}, formula_Rprod)

formula_Rdeg = "k_R_deg * id_r7xb"
add_reaction_l2(model_v18, "R_Rdeg", {"id_r7xb": 1}, {}, formula_Rdeg)

formula_Xprod = "k_X_prod * id_r7xb"
add_reaction_l2(model_v18, "R_Xprod", {}, {"id_x589": 1}, formula_Xprod)

formula_Xdeg = "k_X_deg * id_x589"
add_reaction_l2(model_v18, "R_Xdeg", {"id_x589": 1}, {}, formula_Xdeg)

formula_PT = "k_PT * id_plsk * id_r7xb"
add_reaction_l2(model_v18, "R_PT", {"id_plsk": 1}, {"id_22ta": 1}, formula_PT)

formula_N = "k_N * id_0nyd * id_r7xb"
add_reaction_l2(model_v18, "R_N", {"id_0nyd": 1}, {}, formula_N)

# --- Generate SBML string for Model v18 ---
writer = libsbml.SBMLWriter()
sbml_v18_string = writer.writeSBMLToString(sbml_doc_v18)
shared_variables.add("hypothetical_sbml_v18", sbml_v18_string)

# --- Simulate Model v18 ---
sim_data_v18 = simulate(sbml_v18_string)
shared_variables.add("sim_data_v18", sim_data_v18)
print("Simulation of Model v18 completed.")

# --- Analyze Simulation Results ---
sim_data_v18_adjusted = sim_data_v18.copy()
sim_data_v18_adjusted['id_plsk'] = sim_data_v18['id_plsk'] - IC_P_v18
sim_data_v18_adjusted['id_0nyd'] = sim_data_v18['id_0nyd'] - IC_N_v18

r7xb_sim_stats_v18 = sim_data_v18['id_r7xb'].describe()
x589_sim_stats_v18 = sim_data_v18['id_x589'].describe()

print(f"\nObserved id_r7xb: mean={obs_data['id_r7xb'].mean():.3f}, std={obs_data['id_r7xb'].std():.3f}, min={obs_data['id_r7xb'].min():.3f}, max={obs_data['id_r7xb'].max():.3f}")
print(f"Simulated id_r7xb (v18): mean={r7xb_sim_stats_v18['mean']:.3f}, std={r7xb_sim_stats_v18['std']:.3f}, min={r7xb_sim_stats_v18['min']:.3f}, max={r7xb_sim_stats_v18['max']:.3f}")

print(f"Observed id_x589: mean={obs_data['id_x589'].mean():.3f}, std={obs_data['id_x589'].std():.3f}, min={obs_data['id_x589'].min():.3f}, max={obs_data['id_x589'].max():.3f}")
print(f"Simulated id_x589 (v18): mean={x589_sim_stats_v18['mean']:.3f}, std={x589_sim_stats_v18['std']:.3f}, min={x589_sim_stats_v18['min']:.3f}, max={x589_sim_stats_v18['max']:.3f}")

if r7xb_sim_stats_v18['std'] > 0.15 and x589_sim_stats_v18['std'] > 0.065:
    print("Simulated id_r7xb and id_x589 (v18) show significant dynamic/oscillatory behavior.")
    time_points = sim_data_v18['Time'].to_numpy()
    r7xb_values = sim_data_v18['id_r7xb'].to_numpy()
    peaks_r7xb, _ = find_peaks(r7xb_values, height=r7xb_sim_stats_v18['mean'], distance=800) 
    
    if len(peaks_r7xb) > 1:
        peak_times_r7xb = time_points[peaks_r7xb]
        periods_r7xb = np.diff(peak_times_r7xb)
        print(f"Simulated peak times for id_r7xb: {peak_times_r7xb}")
        print(f"Simulated periods for id_r7xb: {periods_r7xb}, mean period: {np.mean(periods_r7xb):.2f}")
    else:
        print("Not enough peaks found for id_r7xb to estimate period reliably.")
else:
    print("Simulated id_r7xb or id_x589 (v18) is not strongly oscillating with these parameters.")

```
# Observation

## Code Stdout
Simulation of Model v18 completed.

Observed id_r7xb: mean=0.548, std=0.180, min=0.297, max=0.939
Simulated id_r7xb (v18): mean=0.607, std=0.119, min=0.360, max=1.035
Observed id_x589: mean=0.200, std=0.071, min=0.000, max=0.438
Simulated id_x589 (v18): mean=0.180, std=0.041, min=0.000, max=0.200
Simulated id_r7xb or id_x589 (v18) is not strongly oscillating with these parameters.

## 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. (hypothetical_sbml_v4, sim_data_v4, hypothetical_sbml_v5, sim_data_v5, hypothetical_sbml_v6, sim_data_v6, hypothetical_sbml_v7, sim_data_v7, hypothetical_sbml_v8, sim_data_v8, hypothetical_sbml_v9, sim_data_v9, hypothetical_sbml_v10, sim_data_v10, hypothetical_sbml_v11, sim_data_v11, hypothetical_sbml_v12, sim_data_v12, hypothetical_sbml_v13, sim_data_v13, hypothetical_sbml_v14, sim_data_v14, hypothetical_sbml_v15, sim_data_v15, hypothetical_sbml_v16, sim_data_v16, hypothetical_sbml_v17, sim_data_v17, hypothetical_sbml_v18, sim_data_v18)

### 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

Model v18 (reducing `h_X` to 2.0, keeping `K_IX=0.2`) did not improve oscillations and `id_x589_max` was still stuck at 0.2.
This is very challenging. The simple two-gene negative feedback loop is proving hard to tune to match all aspects of the observed data, especially the amplitude of `id_x589`.

Let's go back to Model v11, which had the highest `id_r7xb` std (0.157), as the most promising starting point for a final attempt at tuning.
Model v11 parameters:
*   `Vmax_R = 0.1096`, `K_IX = 0.2`, `h_X = 6.0`
*   `k_R_deg = 0.1` (timescale 10)
*   `k_X_prod = 0.0018248175`, `k_X_deg = 0.005` (timescale 200)
*   `k_PT = 0.08`, `k_N = 0.06`

Simulated results for v11:
*   `id_r7xb`: mean=0.619 (obs: 0.548), std=0.157 (obs: 0.180), min=0.360 (obs: 0.297), max=1.090 (obs: 0.939)
*   `id_x589`: mean=0.185 (obs: 0.200), std=0.040 (obs: 0.071), min=0.000 (obs: 0.000), max=0.200 (obs: 0.438)

Primary goals for Model v19 (refining v11):
1.  **Lower `id_r7xb` mean, min, and max** to be closer to observed values.
    -   This primarily involves reducing `Vmax_R`.
2.  **Increase `id_x589` std and max**. This is the hardest part.
    -   If `id_x589_max` is stuck at `K_IX=0.2` due to `h_X=6`, then `id_r7xb` production is nearly zeroed when `id_x589 > 0.2`.
    -   For `id_x589` to rise higher, `id_r7xb` must remain sufficiently high even when `id_x589` is above 0.2.
    -   This implies that the inhibition term `1 / (1 + (id_x589/K_IX)^h_X)` must not go to zero too quickly.
    -   This could be achieved by:
        a.  Increasing `K_IX` (but past attempts showed this can dampen oscillations if `K_IX` is too high).
        b.  Decreasing `h_X` (but `h_X=4` or `h_X=2` didn't help `id_x589_max` break past 0.2).

Let's try a small, careful increase in `K_IX` from 0.2, but keep `h_X=6`.
Perhaps `K_IX = 0.21` or `K_IX = 0.205` - a very slight nudge.
This tiny increase might allow `id_x589` to just barely "escape" being clamped at 0.2, if `id_r7xb` can provide enough "push".

Simultaneously, `Vmax_R` needs to be adjusted.
The `id_r7xb_mean` in v11 was 0.619. Target is 0.548. This is a `(0.619-0.548)/0.619 = 0.1147` or ~11.5% overestimation.
So, `Vmax_R` should be reduced by about 11.5% from v11's `0.1096`.
New `Vmax_R_target = 0.1096 * (1 - 0.115) = 0.1096 * 0.885 = 0.0970`.

Let's try `K_IX = 0.21`.
Recalculate `Vmax_R` for `id_r7xb_mean = 0.548`, `id_x589_mean = 0.200`, `K_IX = 0.21`, `h_X = 6.0`, `k_R_deg = 0.1`:
Inhibition term: `(0.200 / 0.21)^6 = (0.95238)^6 = 0.746`.
`Vmax_R / (1 + 0.746) = 0.1 * 0.548 = 0.0548`.
`Vmax_R = 0.0548 * 1.746 = 0.09568`. This is close to the `0.0970` estimated above.

Recalculate `k_X_prod` for `id_x589_mean = 0.200`, `id_r7xb_mean = 0.548`, `k_X_deg = 0.005`:
`k_X_prod = (0.200 / 0.548) * 0.005 = 0.0018248175`. (Unchanged).

Parameters for Model v19:
*   `IC_P_v19 = 0.89`, `IC_N_v19 = 0.71`
*   `Vmax_R = 0.09568`
*   `K_IX = 0.21` (Slightly increased from 0.2)
*   `h_X = 6.0`
*   `k_R_deg = 0.1`
*   `k_X_prod = 0.0018248175`
*   `k_X_deg = 0.005`
*   `k_PT = 0.08`, `k_N = 0.06`

This is a fine-tuning of v11, primarily adjusting `Vmax_R` to correct the `id_r7xb` mean and slightly increasing `K_IX` in hopes of "unsticking" `id_x589_max`.
The goal is to get `id_r7xb_mean` near 0.548, `id_r7xb_std` near 0.180, `id_r7xb_min` near 0.297, `id_r7xb_max` near 0.939.
And `id_x589_mean` near 0.200, `id_x589_std` near 0.071, `id_x589_max` near 0.438.

This is one of the last few attempts. If this doesn't show significant improvement in `id_x589_max` and overall oscillation quality, I may have to submit the best model found so far (likely v11 or a slight variant if this one is better).

## Action

### Code
```python
import libsbml
import pandas as pd
import numpy as np
from scipy.signal import find_peaks

# --- Helper function to add reactions (L2 compatible) ---
def add_reaction_l2(model, r_id, reactants_dict, products_dict, formula_str, reversible=False):
    reaction = model.createReaction()
    reaction.setId(r_id)
    reaction.setReversible(reversible)

    for species_id, stoich in reactants_dict.items():
        reactant = reaction.createReactant()
        reactant.setSpecies(species_id)
        reactant.setStoichiometry(stoich)

    for species_id, stoich in products_dict.items():
        product = reaction.createProduct()
        product.setSpecies(species_id)
        product.setStoichiometry(stoich)

    kinetic_law = reaction.createKineticLaw()
    kinetic_law.setFormula(formula_str)
    return reaction

# --- Load original SBML and experimental data ---
sbml_doc_v19 = libsbml.readSBMLFromString(input_sbml_string)
model_v19 = sbml_doc_v19.getModel()
obs_data = experiment_history['iteration_0']

# --- Define Initial Concentrations ---
IC_P_v19 = 0.89
IC_N_v19 = 0.71

species_plsk = model_v19.getSpecies('id_plsk')
if species_plsk: species_plsk.setInitialConcentration(IC_P_v19)

species_0nyd = model_v19.getSpecies('id_0nyd')
if species_0nyd: species_0nyd.setInitialConcentration(IC_N_v19)
    
model_v19.getSpecies('id_x589').setInitialConcentration(0.0)
model_v19.getSpecies('id_22ta').setInitialConcentration(0.0)
model_v19.getSpecies('id_r7xb').setInitialConcentration(0.36)

# --- Define Parameters for v19 ---
# Fine-tuning of v11
param_definitions_v19 = {
    "Vmax_R": 0.09568096,     # Recalculated for K_IX=0.21
    "K_IX": 0.21,             # Slightly increased from 0.2
    "h_X": 6.0,
    "k_R_deg": 0.1,
    "k_X_prod": 0.0018248175, # (0.200 / 0.548) * 0.005
    "k_X_deg": 0.005,
    "k_PT": 0.08,
    "k_N": 0.06
}

for p_id, p_val in param_definitions_v19.items():
    param = model_v19.getParameter(p_id)
    if not param:
        param = model_v19.createParameter()
        param.setId(p_id)
        param.setConstant(True)
    param.setValue(p_val)

# --- Add Reactions for v19 (same structure) ---
formula_Rprod = "Vmax_R / (1 + (id_x589 / K_IX)^h_X)"
add_reaction_l2(model_v19, "R_Rprod", {}, {"id_r7xb": 1}, formula_Rprod)

formula_Rdeg = "k_R_deg * id_r7xb"
add_reaction_l2(model_v19, "R_Rdeg", {"id_r7xb": 1}, {}, formula_Rdeg)

formula_Xprod = "k_X_prod * id_r7xb"
add_reaction_l2(model_v19, "R_Xprod", {}, {"id_x589": 1}, formula_Xprod)

formula_Xdeg = "k_X_deg * id_x589"
add_reaction_l2(model_v19, "R_Xdeg", {"id_x589": 1}, {}, formula_Xdeg)

formula_PT = "k_PT * id_plsk * id_r7xb"
add_reaction_l2(model_v19, "R_PT", {"id_plsk": 1}, {"id_22ta": 1}, formula_PT)

formula_N = "k_N * id_0nyd * id_r7xb"
add_reaction_l2(model_v19, "R_N", {"id_0nyd": 1}, {}, formula_N)

# --- Generate SBML string for Model v19 ---
writer = libsbml.SBMLWriter()
sbml_v19_string = writer.writeSBMLToString(sbml_doc_v19)
shared_variables.add("hypothetical_sbml_v19", sbml_v19_string)

# --- Simulate Model v19 ---
sim_data_v19 = simulate(sbml_v19_string)
shared_variables.add("sim_data_v19", sim_data_v19)
print("Simulation of Model v19 completed.")

# --- Analyze Simulation Results ---
sim_data_v19_adjusted = sim_data_v19.copy()
sim_data_v19_adjusted['id_plsk'] = sim_data_v19['id_plsk'] - IC_P_v19
sim_data_v19_adjusted['id_0nyd'] = sim_data_v19['id_0nyd'] - IC_N_v19

r7xb_sim_stats_v19 = sim_data_v19['id_r7xb'].describe()
x589_sim_stats_v19 = sim_data_v19['id_x589'].describe()

print(f"\nObserved id_r7xb: mean={obs_data['id_r7xb'].mean():.3f}, std={obs_data['id_r7xb'].std():.3f}, min={obs_data['id_r7xb'].min():.3f}, max={obs_data['id_r7xb'].max():.3f}")
print(f"Simulated id_r7xb (v19): mean={r7xb_sim_stats_v19['mean']:.3f}, std={r7xb_sim_stats_v19['std']:.3f}, min={r7xb_sim_stats_v19['min']:.3f}, max={r7xb_sim_stats_v19['max']:.3f}")

print(f"Observed id_x589: mean={obs_data['id_x589'].mean():.3f}, std={obs_data['id_x589'].std():.3f}, min={obs_data['id_x589'].min():.3f}, max={obs_data['id_x589'].max():.3f}")
print(f"Simulated id_x589 (v19): mean={x589_sim_stats_v19['mean']:.3f}, std={x589_sim_stats_v19['std']:.3f}, min={x589_sim_stats_v19['min']:.3f}, max={x589_sim_stats_v19['max']:.3f}")

if r7xb_sim_stats_v19['std'] > 0.15 and x589_sim_stats_v19['std'] > 0.065:
    print("Simulated id_r7xb and id_x589 (v19) show significant dynamic/oscillatory behavior.")
    time_points = sim_data_v19['Time'].to_numpy()
    r7xb_values = sim_data_v19['id_r7xb'].to_numpy()
    peaks_r7xb, _ = find_peaks(r7xb_values, height=r7xb_sim_stats_v19['mean'], distance=800) 
    
    if len(peaks_r7xb) > 1:
        peak_times_r7xb = time_points[peaks_r7xb]
        periods_r7xb = np.diff(peak_times_r7xb)
        print(f"Simulated peak times for id_r7xb: {peak_times_r7xb}")
        print(f"Simulated periods for id_r7xb: {periods_r7xb}, mean period: {np.mean(periods_r7xb):.2f}")
    else:
        print("Not enough peaks found for id_r7xb to estimate period reliably.")
else:
    print("Simulated id_r7xb or id_x589 (v19) is not strongly oscillating with these parameters.")

```
# Observation

## Code Stdout
Simulation of Model v19 completed.

Observed id_r7xb: mean=0.548, std=0.180, min=0.297, max=0.939
Simulated id_r7xb (v19): mean=0.612, std=0.127, min=0.360, max=0.954
Observed id_x589: mean=0.200, std=0.071, min=0.000, max=0.438
Simulated id_x589 (v19): mean=0.182, std=0.043, min=0.000, max=0.200
Simulated id_r7xb or id_x589 (v19) is not strongly oscillating with these parameters.

## 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. (hypothetical_sbml_v4, sim_data_v4, hypothetical_sbml_v5, sim_data_v5, hypothetical_sbml_v6, sim_data_v6, hypothetical_sbml_v7, sim_data_v7, hypothetical_sbml_v8, sim_data_v8, hypothetical_sbml_v9, sim_data_v9, hypothetical_sbml_v10, sim_data_v10, hypothetical_sbml_v11, sim_data_v11, hypothetical_sbml_v12, sim_data_v12, hypothetical_sbml_v13, sim_data_v13, hypothetical_sbml_v14, sim_data_v14, hypothetical_sbml_v15, sim_data_v15, hypothetical_sbml_v16, sim_data_v16, hypothetical_sbml_v17, sim_data_v17, hypothetical_sbml_v18, sim_data_v18, hypothetical_sbml_v19, sim_data_v19)

### 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

Model v19 (slight increase of `K_IX` to 0.21, adjustment of `Vmax_R`) did not improve oscillations significantly, and `id_x589_max` is still stuck at 0.2.
-   `id_r7xb` std: 0.127 (v19) vs 0.157 (v11). Obs: 0.180. (Worse than v11)
-   `id_r7xb` mean: 0.612 (v19). Obs: 0.548. (Still high)
-   `id_x589` std: 0.043 (v19). Obs: 0.071. (Still low)
-   `id_x589` max: 0.200 (v19). Obs: 0.438. (Still stuck at mean/`K_IX`)

It appears that with `h_X=6`, the system is extremely sensitive around `K_IX`. If `id_x589` tries to exceed `K_IX`, inhibition is so strong that `id_r7xb` production plummets, preventing `id_x589` from rising further. This effectively "pins" `id_x589_max` at or very near `K_IX`.

Given that `id_x589` needs to reach a peak of ~0.438, this implies that `K_IX` must be set to a value that allows `id_x589` to reach this level before inhibition becomes overwhelming.
If `K_IX` is set to, say, 0.4 (as in v12, which failed), then `id_x589` (with a mean of 0.2) is mostly in the region where inhibition is weak, which also prevents oscillations.

This suggests a fundamental difficulty in achieving both the correct mean for `id_x589` (0.2) and a peak near 0.438 using a simple Hill repression with `h_X=6` if `K_IX` is the main tuning knob for the threshold.

Let's consider the properties of Model v11 again, as it had the best `id_r7xb` std (0.157).
Model v11 parameters:
*   `Vmax_R = 0.1096`, `K_IX = 0.2`, `h_X = 6.0`
*   `k_R_deg = 0.1`
*   `k_X_prod = 0.0018248175`, `k_X_deg = 0.005`
*   `k_PT = 0.08`, `k_N = 0.06`

Simulated v11 `id_r7xb`: mean=0.619 (target 0.548), std=0.157 (target 0.180), min=0.360 (target 0.297), max=1.090 (target 0.939).
Simulated v11 `id_x589`: mean=0.185 (target 0.200), std=0.040 (target 0.071), min=0.000, max=0.200 (target 0.438).

The `id_r7xb` mean, min, and max are all too high. This can be addressed by reducing `Vmax_R`.
The `id_x589` mean is a bit low. This can be addressed by slightly increasing `k_X_prod` or slightly decreasing `k_X_deg` (but `k_X_deg` sets the important timescale).
The main problem is `id_x589_max` and `id_x589_std`.

If I cannot get `id_x589_max` to rise significantly above `K_IX=0.2` with `h_X=6`, then the model structure might be missing a detail, or the true parameters are in a very specific, hard-to-find region. Since I cannot add species or change reaction structure fundamentally, parameter tuning is the only option.

Let's try to force `id_x589` to have a higher mean, hoping this pushes its peak up, even if it means `K_IX` is no longer equal to the mean.
Suppose we want `id_x589_mean` to be 0.25 (instead of 0.2), while `K_IX` remains 0.2.
This would mean that, on average, `id_x589` is already in the strong inhibition zone.
This would require a much higher `Vmax_R` to maintain `id_r7xb_mean`.

New target: `id_x589_mean = 0.25`. `id_r7xb_mean = 0.548`.
`K_IX = 0.2`, `h_X = 6.0`.
`k_R_deg = 0.1`, `k_X_deg = 0.005`.

1.  Recalculate `Vmax_R`:
    Inhibition term: `(target_id_x589_mean / K_IX)^h_X = (0.25 / 0.2)^6 = (1.25)^6 = 3.814697`.
    `Vmax_R / (1 + 3.814697) = k_R_deg * id_r7xb_mean`.
    `Vmax_R / 4.814697 = 0.1 * 0.548 = 0.0548`.
    `Vmax_R = 0.0548 * 4.814697 = 0.2638`. This `Vmax_R` is much higher than v11's 0.1096.

2.  Recalculate `k_X_prod`:
    `k_X_prod = (new_target_id_x589_mean / target_id_r7xb_mean) * k_X_deg`
    `k_X_prod = (0.25 / 0.548) * 0.005 = 0.456204 * 0.005 = 0.002281`. (Higher than v11's 0.0018248)

Parameters for Model v20:
*   `IC_P_v20 = 0.89`, `IC_N_v20 = 0.71`
*   `Vmax_R = 0.2638434`
*   `K_IX = 0.2`
*   `h_X = 6.0`
*   `k_R_deg = 0.1`
*   `k_X_prod = 0.0022810219`
*   `k_X_deg = 0.005`
*   `k_PT = 0.08`, `k_N = 0.06`

This strategy is to "force" `id_x589` to operate at a higher average level, hoping its fluctuations will then also be higher and reach the target peak. This is a bit of a brute-force approach to the `id_x589` amplitude problem.
The `id_r7xb` dynamics will be driven by a stronger `Vmax_R` but also face stronger average inhibition.

With few iterations left, this is a high-risk, potentially high-reward change. If it doesn't work, Model v11 remains the best candidate despite its flaws.

## Action

### Code
```python
import libsbml
import pandas as pd
import numpy as np
from scipy.signal import find_peaks

# --- Helper function to add reactions (L2 compatible) ---
def add_reaction_l2(model, r_id, reactants_dict, products_dict, formula_str, reversible=False):
    reaction = model.createReaction()
    reaction.setId(r_id)
    reaction.setReversible(reversible)

    for species_id, stoich in reactants_dict.items():
        reactant = reaction.createReactant()
        reactant.setSpecies(species_id)
        reactant.setStoichiometry(stoich)

    for species_id, stoich in products_dict.items():
        product = reaction.createProduct()
        product.setSpecies(species_id)
        product.setStoichiometry(stoich)

    kinetic_law = reaction.createKineticLaw()
    kinetic_law.setFormula(formula_str)
    return reaction

# --- Load original SBML and experimental data ---
sbml_doc_v20 = libsbml.readSBMLFromString(input_sbml_string)
model_v20 = sbml_doc_v20.getModel()
obs_data = experiment_history['iteration_0']

# --- Define Initial Concentrations ---
IC_P_v20 = 0.89
IC_N_v20 = 0.71

species_plsk = model_v20.getSpecies('id_plsk')
if species_plsk: species_plsk.setInitialConcentration(IC_P_v20)

species_0nyd = model_v20.getSpecies('id_0nyd')
if species_0nyd: species_0nyd.setInitialConcentration(IC_N_v20)
    
model_v20.getSpecies('id_x589').setInitialConcentration(0.0) # Start id_x589 at 0
model_v20.getSpecies('id_22ta').setInitialConcentration(0.0)
model_v20.getSpecies('id_r7xb').setInitialConcentration(0.36)

# --- Define Parameters for v20 ---
# Strategy: Force id_x589 mean higher (to 0.25) while K_IX=0.2
param_definitions_v20 = {
    "Vmax_R": 0.2638434,      # Recalculated for id_x589_mean=0.25, K_IX=0.2
    "K_IX": 0.2,
    "h_X": 6.0,
    "k_R_deg": 0.1,
    "k_X_prod": 0.0022810219, # Recalculated for id_x589_mean=0.25
    "k_X_deg": 0.005,
    "k_PT": 0.08,
    "k_N": 0.06
}

for p_id, p_val in param_definitions_v20.items():
    param = model_v20.getParameter(p_id)
    if not param:
        param = model_v20.createParameter()
        param.setId(p_id)
        param.setConstant(True)
    param.setValue(p_val)

# --- Add Reactions for v20 (same structure) ---
formula_Rprod = "Vmax_R / (1 + (id_x589 / K_IX)^h_X)"
add_reaction_l2(model_v20, "R_Rprod", {}, {"id_r7xb": 1}, formula_Rprod)

formula_Rdeg = "k_R_deg * id_r7xb"
add_reaction_l2(model_v20, "R_Rdeg", {"id_r7xb": 1}, {}, formula_Rdeg)

formula_Xprod = "k_X_prod * id_r7xb"
add_reaction_l2(model_v20, "R_Xprod", {}, {"id_x589": 1}, formula_Xprod)

formula_Xdeg = "k_X_deg * id_x589"
add_reaction_l2(model_v20, "R_Xdeg", {"id_x589": 1}, {}, formula_Xdeg)

formula_PT = "k_PT * id_plsk * id_r7xb"
add_reaction_l2(model_v20, "R_PT", {"id_plsk": 1}, {"id_22ta": 1}, formula_PT)

formula_N = "k_N * id_0nyd * id_r7xb"
add_reaction_l2(model_v20, "R_N", {"id_0nyd": 1}, {}, formula_N)

# --- Generate SBML string for Model v20 ---
writer = libsbml.SBMLWriter()
sbml_v20_string = writer.writeSBMLToString(sbml_doc_v20)
shared_variables.add("hypothetical_sbml_v20", sbml_v20_string)

# --- Simulate Model v20 ---
sim_data_v20 = simulate(sbml_v20_string)
shared_variables.add("sim_data_v20", sim_data_v20)
print("Simulation of Model v20 completed.")

# --- Analyze Simulation Results ---
sim_data_v20_adjusted = sim_data_v20.copy()
sim_data_v20_adjusted['id_plsk'] = sim_data_v20['id_plsk'] - IC_P_v20
sim_data_v20_adjusted['id_0nyd'] = sim_data_v20['id_0nyd'] - IC_N_v20

r7xb_sim_stats_v20 = sim_data_v20['id_r7xb'].describe()
x589_sim_stats_v20 = sim_data_v20['id_x589'].describe()

print(f"\nObserved id_r7xb: mean={obs_data['id_r7xb'].mean():.3f}, std={obs_data['id_r7xb'].std():.3f}, min={obs_data['id_r7xb'].min():.3f}, max={obs_data['id_r7xb'].max():.3f}")
print(f"Simulated id_r7xb (v20): mean={r7xb_sim_stats_v20['mean']:.3f}, std={r7xb_sim_stats_v20['std']:.3f}, min={r7xb_sim_stats_v20['min']:.3f}, max={r7xb_sim_stats_v20['max']:.3f}")

print(f"Observed id_x589: mean={obs_data['id_x589'].mean():.3f}, std={obs_data['id_x589'].std():.3f}, min={obs_data['id_x589'].min():.3f}, max={obs_data['id_x589'].max():.3f}")
print(f"Simulated id_x589 (v20): mean={x589_sim_stats_v20['mean']:.3f}, std={x589_sim_stats_v20['std']:.3f}, min={x589_sim_stats_v20['min']:.3f}, max={x589_sim_stats_v20['max']:.3f}")

if r7xb_sim_stats_v20['std'] > 0.15 and x589_sim_stats_v20['std'] > 0.065:
    print("Simulated id_r7xb and id_x589 (v20) show significant dynamic/oscillatory behavior.")
    time_points = sim_data_v20['Time'].to_numpy()
    r7xb_values = sim_data_v20['id_r7xb'].to_numpy()
    peaks_r7xb, _ = find_peaks(r7xb_values, height=r7xb_sim_stats_v20['mean'], distance=800) 
    
    if len(peaks_r7xb) > 1:
        peak_times_r7xb = time_points[peaks_r7xb]
        periods_r7xb = np.diff(peak_times_r7xb)
        print(f"Simulated peak times for id_r7xb: {peak_times_r7xb}")
        print(f"Simulated periods for id_r7xb: {periods_r7xb}, mean period: {np.mean(periods_r7xb):.2f}")
    else:
        print("Not enough peaks found for id_r7xb to estimate period reliably.")
else:
    print("Simulated id_r7xb or id_x589 (v20) is not strongly oscillating with these parameters.")

```
# Observation

## Code Stdout
Simulation of Model v20 completed.

Observed id_r7xb: mean=0.548, std=0.180, min=0.297, max=0.939
Simulated id_r7xb (v20): mean=0.643, std=0.362, min=0.360, max=2.489
Observed id_x589: mean=0.200, std=0.071, min=0.000, max=0.438
Simulated id_x589 (v20): mean=0.242, std=0.037, min=0.000, max=0.251
Simulated id_r7xb or id_x589 (v20) is not strongly oscillating with these parameters.

## 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. (hypothetical_sbml_v4, sim_data_v4, hypothetical_sbml_v5, sim_data_v5, hypothetical_sbml_v6, sim_data_v6, hypothetical_sbml_v7, sim_data_v7, hypothetical_sbml_v8, sim_data_v8, hypothetical_sbml_v9, sim_data_v9, hypothetical_sbml_v10, sim_data_v10, hypothetical_sbml_v11, sim_data_v11, hypothetical_sbml_v12, sim_data_v12, hypothetical_sbml_v13, sim_data_v13, hypothetical_sbml_v14, sim_data_v14, hypothetical_sbml_v15, sim_data_v15, hypothetical_sbml_v16, sim_data_v16, hypothetical_sbml_v17, sim_data_v17, hypothetical_sbml_v18, sim_data_v18, hypothetical_sbml_v19, sim_data_v19, hypothetical_sbml_v20, sim_data_v20)

### 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

Model v20 (forcing `id_x589_mean` to 0.25) resulted in:
-   `id_r7xb`: mean=0.643 (obs: 0.548), std=0.362 (obs: 0.180), min=0.360 (obs: 0.297), max=2.489 (obs: 0.939).
    -   The `id_r7xb` std is now too high, and max is way too high. The mean is also too high.
-   `id_x589`: mean=0.242 (target 0.25, obs: 0.200), std=0.037 (obs: 0.071), min=0.000, max=0.251 (obs: 0.438).
    -   `id_x589_max` is still pinned at its (new, higher) mean. The std is still too low.

This confirms that the high Hill coefficient (`h_X=6`) with `K_IX` near the mean of `id_x589` effectively clamps `id_x589`'s maximum at its mean. The system does not "overshoot" `K_IX` significantly for `id_x589`.

Given the limited iterations, I must select the best model achieved so far. Model v11 had the highest `id_r7xb_std` (0.157) that was somewhat oscillatory, even though other metrics were off.

Model v11 parameters:
*   `IC_P = 0.89`, `IC_N = 0.71`
*   `Vmax_R = 0.1096`, `K_IX = 0.2`, `h_X = 6.0`
*   `k_R_deg = 0.1`
*   `k_X_prod = 0.0018248175`, `k_X_deg = 0.005`
*   `k_PT = 0.08`, `k_N = 0.06`

Simulated results for v11:
*   `id_r7xb`: mean=0.619 (obs: 0.548), std=0.157 (obs: 0.180), min=0.360 (obs: 0.297), max=1.090 (obs: 0.939)
*   `id_x589`: mean=0.185 (obs: 0.200), std=0.040 (obs: 0.071), min=0.000 (obs: 0.000), max=0.200 (obs: 0.438)

The main deficiencies of v11 are:
1.  `id_r7xb` mean, min, max are too high.
2.  `id_x589` mean is slightly low.
3.  `id_x589` std and max are significantly too low.

Let's try one final adjustment to Model v11 to address the `id_r7xb` levels being too high, and slightly boost `id_x589` mean.
To lower `id_r7xb` mean, min, max: Reduce `Vmax_R`.
The mean was 0.619, target 0.548. Over by `(0.619-0.548)/0.619 = 0.1147` (~11.5%).
So, reduce `Vmax_R` from `0.1096` by ~11.5%: `0.1096 * (1 - 0.115) = 0.0970`. (This was tried in v14, which had lower std).
Let's try a slightly less aggressive reduction, say by 8%.
New `Vmax_R = 0.1096 * 0.92 = 0.100832`.

To slightly increase `id_x589` mean from 0.185 to target 0.200:
`k_X_prod = (target_id_x589_mean / actual_id_r7xb_mean_from_sim) * k_X_deg`.
If we assume the new `Vmax_R` brings `id_r7xb_mean` closer to 0.548 (from 0.619), then the existing `k_X_prod = 0.0018248175` (which was calculated for `id_r7xb_mean=0.548` and `id_x589_mean=0.2`) should be okay.
The previous `id_x589_mean` of 0.185 was likely because `id_r7xb_mean` was actually 0.619, so `(0.0018248175 / 0.005) * 0.619 = 0.36496 * 0.619 = 0.226` - this calculation is wrong.
It should be `id_x589_mean = (k_X_prod / k_X_deg) * id_r7xb_mean`.
So, `0.185 = (0.0018248175 / 0.005) * 0.619 = 0.3649635 * 0.619 = 0.2259`. This is not 0.185.
The `k_X_prod` in v11 was `(0.2/0.548)*0.005 = 0.0018248175`.
If `id_r7xb_mean` (simulated) was 0.619, then `id_x589_mean` (simulated) should have been `(0.0018248175 / 0.005) * 0.619 = 0.2259`.
The reported `id_x589_mean` for v11 was 0.185. This discrepancy is confusing.

Let's re-calculate `k_X_prod` strictly based on the *target* means:
`k_X_prod = (target_id_x589_mean=0.2 / target_id_r7xb_mean=0.548) * k_X_deg=0.005 = 0.0018248175`. This is what was used.

The most critical failure is `id_x589_max`. No parameter combination with `h_X=6` and `K_IX` around 0.2 has allowed `id_x589_max` to exceed `K_IX`.
This implies that the model, with these constraints, cannot reproduce this feature.

Given the situation, submitting Model v11 as is, or with a minor tweak to `Vmax_R` to improve `id_r7xb` mean, seems the most pragmatic. The `id_x589` amplitude will remain a known issue.

Let's try Model v21 by taking v11 and reducing `Vmax_R` to target the `id_r7xb` mean more accurately, accepting other flaws.
Target `id_r7xb_mean = 0.548`.
`Vmax_R / (1 + (id_x589_mean_target/K_IX)^h_X) = k_R_deg * id_r7xb_mean_target`
`Vmax_R / (1 + (0.2/0.2)^6) = 0.1 * 0.548`
`Vmax_R / 2 = 0.0548`
`Vmax_R = 0.1096`. This was the value in v11.
The simulation gave `id_r7xb_mean = 0.619`.
This means the effective average inhibition `(1 + (id_x589/K_IX)^h_X)` during oscillation must be less than 2.
If actual mean is 0.619, then `0.1096 / Denom_eff = 0.1 * 0.619 = 0.0619`.
`Denom_eff = 0.1096 / 0.0619 = 1.77`.
This is `(1 + avg_inhibition_term)`. So `avg_inhibition_term = 0.77`.
This is `avg((id_x589/0.2)^6) = 0.77`. This implies `id_x589` spends more time below 0.2 than above it, if it oscillates around 0.2.
The simulated `id_x589_mean` was 0.185, which is below 0.2. This is consistent.

To get `id_r7xb_mean` to be 0.548, we need to scale `Vmax_R` from v11 by `0.548/0.619`.
New `Vmax_R = 0.1096 * (0.548 / 0.619) = 0.1096 * 0.8853 = 0.09703`.

Parameters for Model v21 (final attempt, refinement of v11):
*   `IC_P_v21 = 0.89`, `IC_N_v21 = 0.71`
*   `Vmax_R = 0.09703` (Adjusted from v11 to target `id_r7xb_mean`)
*   `K_IX = 0.2`
*   `h_X = 6.0`
*   `k_R_deg = 0.1`
*   `k_X_prod = 0.0018248175` (Kept, as it targets `id_x589_mean=0.2` if `id_r7xb_mean=0.548`)
*   `k_X_deg = 0.005`
*   `k_PT = 0.08`, `k_N = 0.06`

This model will be submitted.

## Action

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

# --- Helper function to add reactions (L2 compatible) ---
def add_reaction_l2(model, r_id, reactants_dict, products_dict, formula_str, reversible=False):
    reaction = model.createReaction()
    reaction.setId(r_id)
    reaction.setReversible(reversible)

    for species_id, stoich in reactants_dict.items():
        reactant = reaction.createReactant()
        reactant.setSpecies(species_id)
        reactant.setStoichiometry(stoich)

    for species_id, stoich in products_dict.items():
        product = reaction.createProduct()
        product.setSpecies(species_id)
        product.setStoichiometry(stoich)

    kinetic_law = reaction.createKineticLaw()
    kinetic_law.setFormula(formula_str)
    return reaction

# --- Load original SBML ---
sbml_doc_final = libsbml.readSBMLFromString(input_sbml_string)
model_final = sbml_doc_final.getModel()

# --- Define Initial Concentrations ---
IC_P_final = 0.89  # From earlier refinements
IC_N_final = 0.71  # From earlier refinements

species_plsk = model_final.getSpecies('id_plsk')
if species_plsk: species_plsk.setInitialConcentration(IC_P_final)

species_0nyd = model_final.getSpecies('id_0nyd')
if species_0nyd: species_0nyd.setInitialConcentration(IC_N_final)
    
# Ensure other initial concentrations are as per input_sbml_string
model_final.getSpecies('id_x589').setInitialConcentration(0.0)
model_final.getSpecies('id_22ta').setInitialConcentration(0.0)
model_final.getSpecies('id_r7xb').setInitialConcentration(0.36) # From original SBML

# --- Define Parameters for Final Model (based on v11, with Vmax_R adjustment) ---
# These parameters gave id_r7xb_std=0.157 in v11 (obs=0.180)
# Vmax_R adjusted to target id_r7xb_mean=0.548 based on v11's sim mean of 0.619
param_definitions_final = {
    "Vmax_R": 0.097030048,    # 0.1096 * (0.548 / 0.619)
    "K_IX": 0.2,
    "h_X": 6.0,
    "k_R_deg": 0.1,
    "k_X_prod": 0.0018248175, # (0.200 / 0.548) * 0.005
    "k_X_deg": 0.005,
    "k_PT": 0.08,
    "k_N": 0.06
}

for p_id, p_val in param_definitions_final.items():
    param = model_final.getParameter(p_id)
    if not param:
        param = model_final.createParameter()
        param.setId(p_id)
        param.setConstant(True)
    param.setValue(p_val)

# --- Add Reactions for Final Model ---
# R_Rprod: -> id_r7xb (Rate: Vmax_R / (1 + (id_x589/K_IX)^h_X))
formula_Rprod = "Vmax_R / (1 + (id_x589 / K_IX)^h_X)"
add_reaction_l2(model_final, "R_Rprod", {}, {"id_r7xb": 1}, formula_Rprod)

# R_Rdeg: id_r7xb -> (Rate: k_R_deg * id_r7xb)
formula_Rdeg = "k_R_deg * id_r7xb"
add_reaction_l2(model_final, "R_Rdeg", {"id_r7xb": 1}, {}, formula_Rdeg)

# R_Xprod: -> id_x589 (Rate: k_X_prod * id_r7xb)
formula_Xprod = "k_X_prod * id_r7xb"
add_reaction_l2(model_final, "R_Xprod", {}, {"id_x589": 1}, formula_Xprod)

# R_Xdeg: id_x589 -> (Rate: k_X_deg * id_x589)
formula_Xdeg = "k_X_deg * id_x589"
add_reaction_l2(model_final, "R_Xdeg", {"id_x589": 1}, {}, formula_Xdeg)

# R_PT: id_plsk -> id_22ta (Rate: k_PT * id_plsk * id_r7xb)
formula_PT = "k_PT * id_plsk * id_r7xb"
add_reaction_l2(model_final, "R_PT", {"id_plsk": 1}, {"id_22ta": 1}, formula_PT)

# R_N: id_0nyd -> (Rate: k_N * id_0nyd * id_r7xb)
formula_N = "k_N * id_0nyd * id_r7xb"
add_reaction_l2(model_final, "R_N", {"id_0nyd": 1}, {}, formula_N)

# --- Generate SBML string for Final Model ---
writer = libsbml.SBMLWriter()
final_sbml = writer.writeSBMLToString(sbml_doc_final)

# The variable `final_sbml` is now ready for submission.
# print(final_sbml) # For verification if needed, but not for submission output
```
