Improving and Testing Software for Hybrid Mathematical Models and Machine-Learning to Predict Cell Network Dynamics

[cannin]

Background

Uncovering the equations that govern the network dynamics of biological systems is both a major challenge and incredibly important because it reveals the mechanisms and patterns that drive complex underlying processes. Various strategies have been developed to discover these equations. For example, our team has developed CellBox, a hybrid approach that combines explicit mathematical models of cell dynamics with a machine-learning framework. Other related approaches that fall under the umbrella of symbolic regression to search for mathematical expressions that best fits a given dataset.

Goal

The goal of the project existing quantitative biological models described using systems biology standards in combination with existing "symbolic regression" methods to both test and improve these methods.

Difficulty Level: Hard

Difficulty is based on the having and understanding of symbolic regression and an understanding biological mathematical modeling to make sense of resulting models.

Size and Length of Project

• Large: 350 hours

Skills

• Essential skills: Python
• Nice to have skills: Symbolic regression frameworks, biological mathematical modeling

Public Repository

• https://github.com/sanderlab/CellBox (https://pubmed.ncbi.nlm.nih.gov/33373583/)

Potential Mentors

• Augustin Luna (augustin AT nih.gov)

[RitikaSen27]

Hi,

I’m very interested in working on this project. I’ve started reading the CellBox paper and exploring the repository to understand how explicit ODE-based models are combined with machine-learning optimization.

I’m particularly interested in evaluating symbolic regression methods on existing systems biology models (e.g., SBML/ODE models) to test whether known governing equations can be rediscovered from simulated or experimental data.

As a first step, I’m planning to:

Review existing symbolic regression frameworks in Python (e.g., PySR, gplearn)

Reproduce simple benchmark models (e.g., small signaling networks)

Compare recovered equations with ground-truth models

I’d love feedback on whether this approach aligns with the project goals, and if there are specific models or datasets you’d recommend starting with.

[PrasannaPal21]

Hello @cannin,

I'm Prasanna, interested in contributing to CellBox.

I noticed the project doesn't have unit tests for individual modules. I've been working on adding a testing infrastructure with pytest - covering config parsing, ODE solvers, loss functions, and data validation (41 tests total).

Branch here: https://github.com/PrasannaPal21/CellBox/tree/testing-infra
Readme : https://github.com/PrasannaPal21/CellBox/blob/testing-infra/tests/README.md

Would you be open to a PR for this? Happy to discuss if you'd like more details.

[Ramrajnagar]

Hi @cannin and mentors,

I’m very interested in contributing to this project and have started exploring the CellBox repository and paper to understand the hybrid ODE + ML framework. I have strong experience in Python and numerical computing, and I’m currently learning symbolic regression techniques such as PySR and gplearn to better align with this project’s goals.

I would like to contribute in the following ways initially:

Setting up reproducible pipelines for testing symbolic regression on known ODE-based biological models

Evaluating recovery of ground-truth equations using simulated datasets

Improving experiment tracking, benchmarking scripts, and result validation

Helping with documentation and code quality improvements if needed

As a starting step, I plan to reproduce simple signaling network models and compare recovered equations with original system dynamics. Please let me know if there are preferred datasets, benchmarks, or beginner-friendly tasks you would recommend.

Looking forward to contributing and learning from the team. Thank you!

[cannin]

One starting point for anyone interested would be the repressilator SBML model: https://github.com/combine-org/combine-notebooks/tree/main/resources (BIOMD0000000012_urn.xml)

[PrasannaPal21]

Thank you @cannin for the pointer on the repressilator model
I managed to reproduce the system dynamics using the provided XML. I'm getting stable oscillations with a period of about 120 mins, it was around 130's in the paper.

I also plotted the phase space just to see the limit cycles.

planning to feed this data into PySR now and see if I can reverse engineer the Hill equations. And then I'll try adding some noice to the data

please let me know if you have anything for me before I proceed.

[Vinayaktoor]

Reproducing the limit cycle is a good sanity check. One thing I ran into is that symbolic regression can recover qualitatively correct oscillations while still learning dynamically unstable or non-identifiable equations. Evaluating local vs global validity and noise sensitivity made a big difference in separating superficially correct models from genuinely meaningful ones.

For completeness, here’s a short-horizon comparison using SINDy.

Polynomial sparse regression recovers the local dynamics reasonably well, but becomes unstable over longer horizons
which is expected given the mismatch between polynomial libraries and bounded Hill-type repression. This contrast is exactly why we moved to rational/PySR models for long-term behavior.

[PrasannaPal21]

hello @cannin, @khanspers

I ran a set of experiments to stress-test symbolic regression (PySR) on the Repressilator model and understand where it starts to break down.

1. Rediscovering the Equation (using PySR)
   I first fed the clean simulated data (species concentrations vs. time) into PySR. It successfully recovered the underlying biological law, the Hill function structure, directly from the data.

2. Verifying the Learned Model
   I then simulated the system using the recovered equation. The learned dynamics (blue dashed) closely match the original simulation (black), this ensured that the equation which i got from PySR was matching to the original one.

3. Added some Noice to the data
   After introducing 10% Gaussian noise to mimic realistic conditions, computing derivatives (dX/dt) with finite differences (np.gradient) became highly unstable. The noise was amplified during differentiation, effectively turning the target signal into waste for the regression model.

what I understood -
Naive symbolic regression ends up fitting noise-amplified derivatives instead of the true dynamics, which makes it unreliable.

This points me to 'Grey-Box Modeling' as the next step: inject known biological priors (like Hill functions) and let the model focus on fitting parameters. This should be more robust in realistic experimental noise.