[Paper Review] Taking the reaction-diffusion master equation to the microscopic limit

Paul Sjöberg, Otto G. Berg+1 more|ArXiv.org|May 28, 2009

Gene Regulatory Network Analysis14 references3 citations

TL;DR

This paper resolves inconsistencies between the reaction-diffusion master equation (RDME) and microscopic diffusion-limited kinetics by deriving spatially dependent association and dissociation rate constants from a microscopic model. It shows that RDME rates must depend on spatial discretization, approaching microscopic constants at fine resolution and macroscopic 3D limits at coarse resolution, with no limit in 2D—enabling physically consistent stochastic spatial modeling.

ABSTRACT

The reaction-diffusion master equation (RDME) is commonly used to model processes where both the spatial and stochastic nature of chemical reactions need to be considered. We show that the RDME in many cases is inconsistent with a microscopic description of diffusion limited chemical reactions and that this will result in unphysical results. We describe how the inconsistency can be reconciled if the association and dissociation rates used in the RDME are derived from the underlying microscopic description. These rate constants will however necessarily depend on the spatial discretization. At fine spatial resolution the rates approach the microscopic rate constants defined at the reaction radius. At low resolution the rates converge to the macroscopic diffusion limited rate constants in 3D, whereas there is no limiting value in 2D. Our results make it possible to develop spatially discretized reaction-diffusion models that correspond to a well-defined microscopic description. We show that this is critical for a correct description of 2D systems and systems that require high spatial resolution in 3D.

Motivation & Objective

• To identify inconsistencies between the RDME and microscopic descriptions of diffusion-limited reactions.
• To resolve the mismatch between RDME rate constants and microscopic reaction kinetics.
• To derive rate constants for the RDME that are consistent with a well-defined microscopic model.
• To establish how these derived rates depend on spatial discretization in 2D and 3D systems.
• To enable accurate, physically meaningful spatially discretized reaction-diffusion simulations in biological contexts.

Proposed method

• Derives microscopic association and dissociation rate constants from a hard-sphere collision model with a reaction radius.
• Calculates the effective RDME rate constants by averaging over the spatial discretization grid.
• Analyzes the dependence of these rates on grid size in both 2D and 3D.
• Shows that at fine resolution, RDME rates converge to the microscopic rate constants defined at the reaction radius.
• Demonstrates that at coarse resolution, rates in 3D converge to macroscopic diffusion-limited rate constants.
• Identifies that no limiting value exists for RDME rates in 2D as resolution decreases.

Experimental results

Research questions

• RQ1 How do RDME rate constants compare to those derived from a microscopic description of diffusion-limited reactions?
• RQ2 What is the dependence of RDME rate constants on spatial discretization scale?
• RQ3 Do RDME rates converge to macroscopic or microscopic limits at coarse or fine resolution?
• RQ4 Why does the RDME fail to describe 2D systems consistently at low resolution?
• RQ5 Can a physically consistent RDME formulation be derived by linking it to microscopic kinetics?

Key findings

• RDME rates derived from microscopic kinetics are inherently dependent on spatial discretization.
• At fine spatial resolution, RDME rates converge to the microscopic rate constants defined at the reaction radius.
• At coarse resolution in 3D, RDME rates converge to the macroscopic diffusion-limited rate constants.
• In 2D, no limiting value exists for RDME rates as spatial resolution decreases.
• The inconsistency between standard RDME and microscopic models leads to unphysical results in spatially discretized simulations.
• Correct modeling of 2D and high-resolution 3D systems requires rate constants derived from the microscopic description.