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Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation data.

Seungjoon Lee1, Yorgos M Psarellis2, Constantinos I Siettos3

  • 1Department of Applied Data Science, San José State University, San Jose, USA.

Journal of Mathematical Biology
|June 21, 2023
PubMed
Summary

We developed a machine learning framework to discover macroscopic chemotactic Partial Differential Equations (PDEs) from bacterial motility simulations. This approach enables data-driven discovery of complex biological behaviors.

Keywords:
ChemotaxisInverse problemsMachine learningMultiscale methodsNumerical analysisPartial differential equationsStochastic simulations

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Area of Science:

  • Computational Biology
  • Biophysics
  • Applied Mathematics

Background:

  • Bacterial motility, like that of Escherichia coli, is a complex phenomenon driven by individual cell behaviors.
  • Macroscopic models, such as chemotactic Partial Differential Equations (PDEs), are crucial for understanding collective bacterial movement.
  • Existing models often rely on simplified assumptions or approximations for closure relations.

Purpose of the Study:

  • To develop a machine learning framework for discovering macroscopic chemotactic PDEs and their closure relations from individual-based simulations.
  • To enable data-driven derivation of effective, coarse-grained models from fine-scale biophysical simulations.
  • To investigate both black-box and gray-box machine learning approaches for PDE discovery.

Main Methods:

  • Utilized high-fidelity, individual-based stochastic simulations of Escherichia coli motility, incorporating underlying biophysics.
  • Employed a hybrid continuum-Monte Carlo simulation model with parameters informed by experimental data.
  • Applied machine learning regressors, including feedforward neural networks and Gaussian Processes, to learn PDEs from collective observables.

Main Results:

  • Successfully discovered effective, coarse-grained chemotactic PDEs belonging to the Keller-Segel class.
  • Demonstrated the ability to learn both black-box (fully data-driven) and gray-box (partially known structure) PDE laws.
  • Showcased data-driven corrections to analytically known, approximate closure relations.

Conclusions:

  • The proposed machine learning framework provides a powerful tool for data-driven discovery of macroscopic PDEs from complex biological systems.
  • This approach bridges the gap between individual-level biophysics and collective emergent behavior.
  • The framework facilitates the refinement of existing models and the discovery of novel mathematical descriptions for biological phenomena.