Coarse-Grained Limits of Particle-Based Stochastic Reaction-Diffusion Models
Abstract
The dynamics of many biological processes rely on an interplay between spatial transport and chemical reaction. At the scale of a single cell, experiments have demonstrated that many such processes have stochastic dynamics. Particle-based stochastic reaction-diffusion (PBSRD) models are a widely used approach for studying such processes, explicitly modeling the diffusion of, and reactions between, individual molecules. PBSRD models have been successfully applied to a variety of biological problems, including understanding the response of immune cells to foreign antigens, understanding trafficking and signaling across neurological synapses, and understanding how variability of differentiation responses arises within embryonic stem-cell pathways. Due to their mathematical complexity and high dimensionality, PBSRD models are almost entirely studied by Monte Carlo simulation approximating the underlying stochastic process of molecules diffusing and reacting. The computational expense of such methods can greatly limit the size of chemical systems (in each of number of molecules, number of reactions, or physical domain size) that can be studied. One approach to overcoming this challenge is to use more coarse-grained mathematical models that accurately capture the dynamics of the underlying PBSRD model in appropriate physical regimes. Deterministic and stochastic partial differential equation (PDE/SPDE) models are commonly postulated as coarse-grainings of PBSRD models in certain large-population or thermodynamic limits where the population size becomes unbounded but species concentrations are held fixed. However, for the PBSRD models commonly used in biological modeling there is limited rigorous work proving the existence of such deterministic coarse-grained limits (i.e. law of large numbers), developing continuous-noise SPDE corrections to the limits (i.e. central limit theorems), or studying how such coarse-grained models might relate to standard reaction-diffusion PDE models of biological systems. The PIs propose to rigorously establish the large-population limit for PBSRD models commonly used in modeling cellular processes, establish how standard deterministic reaction-diffusion PDE models approximate the derived equations, and investigate continuous-noise corrections to the derived deterministic equations that offer improved accuracy in approximating the underlying PBSRD model. Our approach for developing rigorous coarse-grainings of PBSRD models begins with formulating the dynamics of the diffusing and reacting molecules as measure-valued stochastic processes (MVSPs). Our approach is unique in using a bottom-up hierarchy to rigorously derive from spatial PBSRD models new macroscopic partial integro-differential equations (PIDEs) and stochastic PIDEs that correspond to the true large-population limits, and which correctly account for chemical interactions between particles. Spatial PBSRD models are widely used in studying cellular processes, including investigating interactions between populations of cells, studying how cells respond to extracellular chemical signals, understanding how bacterial cells divide, and studying the spatial control of gene regulation. The methods and coarse-grained models we develop will help facilitate the development of more realistic models of biological processes by allowing for the more accurate representation of systems with large populations. Rigorous results on the large-population limit of non-spatial stochastic chemical kinetics models have been leveraged in developing a variety of new computational and analytical methods. We therefore expect that similar results for PBSRD systems should aid in developing new methods for accurately and efficiently studying spatial models of biological systems.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jul 09, 2020
- Source ID
- W911NF2010244
Entities
People
- Samuel A. Isaacson
Organizations
- Army Contracting Command
- Boston University
- United States Army