[Paper Review] Open reaction-diffusion systems: bridging probabilistic theory and simulations across scales

Mauricio J. del Razo, Kostré, Margarita|arXiv (Cornell University)|Apr 10, 2024

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TL;DR

The paper develops the chemical diffusion master equation (CDME) to describe open reaction-diffusion systems at the particle level, then shows how to connect this probabilistic framework to macroscopic concentration-based models and reservoir interactions for multiscale simulations.

ABSTRACT

Reaction-diffusion processes are the foundational model for a diverse range of complex systems, ranging from biochemical reactions to social agent-based phenomena. The underlying dynamics of these systems occur at the individual particle/agent level, and in realistic applications, they often display interaction with their environment through energy or material exchange with a reservoir. This requires intricate mathematical considerations, especially in the case of material exchange since the varying number of particles/agents results in ``on-the-fly'' modification of the system dimension. In this work, we first overview the probabilistic description of reaction-diffusion processes at the particle level, which readily handles varying number of particles. We then extend this model to consistently incorporate interactions with macroscopic material reservoirs. Based on the resulting expressions, we bridge the probabilistic description with macroscopic concentration-based descriptions for linear and nonlinear reaction-diffusion systems, as well as for an archetypal open reaction-diffusion system. Using these mathematical bridges across scales, we finally develop numerical schemes for open reaction-diffusion systems, which we implement in two illustrative examples. This work establishes a methodological workflow to bridge particle-based probabilistic descriptions with macroscopic concentration-based descriptions of reaction-diffusion in open settings, laying the foundations for a multiscale theoretical framework upon which to construct theory and simulation schemes that are consistent across scales.

Motivation & Objective

• Motivate open reaction-diffusion systems present in biology and social phenomena and the need for consistent open-system modeling.
• Formulate a probabilistic, particle-level description that handles changing system dimension due to reactions.
• Develop a framework to couple particle-level dynamics with macroscopic reservoirs and derive macroscopic equations.
• Establish a workflow to recover concentration-based RD equations from the CDME for linear and nonlinear reactions.

Proposed method

• Present the CDME as a hierarchy of densities rho_n(x^(n)) representing probability of n particles at positions x^(n).
• Describe diffusion via standard Brownian terms plus reaction operators that include loss and gain components.
• Provide explicit CDME forms for simple degradation/creation and for bimolecular reactions, including multi-species generalization.
• Incorporate open boundaries by modeling reservoir exchange through boundary-layer boundary conditions using discrete or continuous rates.
• Show how macroscopic RD-PDEs arise as mean-field limits from the CDME for linear and nonlinear cases.
• Connect particle-level parameters to macroscopic rates in diffusion-influenced/open reactions via limit procedures.

Figure 1: Structure of the phase space of the CDME. a. Phase space for a system with one chemical species $A$ . The structure of the phase space consists of discrete sets of continuous diffusion domains $\mathbb{X}$ depending on the number of particles. The transitions between sets depend on the spe

Experimental results

Research questions

• RQ1 How can reaction-diffusion dynamics with changing particle number be formulated probabilistically at the particle level?
• RQ2 How can interactions with macroscopic reservoirs be incorporated consistently into the particle-level CDME?
• RQ3 How can one recover macroscopic concentration-based open RD equations from the CDME for linear and nonlinear reactions?
• RQ4 What is the relation between microscopic parameters and macroscopic rates in diffusion-influenced/open systems?

Key findings

• The CDME provides a linear evolution framework in an infinite-dimensional phase space that couples diffusion and reactions with varying particle numbers.
• Open boundaries are modeled via reservoir-linked boundary layers, yielding explicit forms for in/out rates consistent with Dirac-delta representations.
• Mean-field limits of the CDME recover classical RD-PDEs for degradation and creation, and yield diffusion-influenced reaction terms in open settings.
• For nonlinear bimolecular-like reactions, a microscopic-to-macroscopic mapping is established under large-copy-number assumptions, linking rate constants to macroscopic kinetics.

Figure 2: Diagram representing loss from and gain into the $n$ particle state due to the reaction $kA\rightarrow lA$ assuming $k>l$ . The loss $\mathcal{L}_{n}\rho_{n}$ must depend on the reactant’s positions, so it is a function of $x^{(n)}$ , while the gain $\mathcal{G}_{n}\rho_{n+k-l}$ depends on