[论文解读] Open reaction-diffusion systems: bridging probabilistic theory and simulations across scales

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

Gene Regulatory Network Analysis被引用 5

一句话总结

论文提出化学扩散主方程（CDME）来描述粒子层面的开放反应扩散系统，然后展示如何将该概率框架与宏观以浓度为基础的模型以及多尺度模拟中的储库耦合联系起来。

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.

研究动机与目标

• 在生物学和社会现象中存在的开放反应扩散系统的动机，以及对一致开放系统建模的需求。
• 形成一个处理因反应而导致系统维度变化的概率性、粒子层面的描述。
• 开发一个框架，将粒子层面的动力学与宏观储库耦合并推导宏观方程。
• 建立一个工作流程，从CDME在线性和非线性反应中恢复基于浓度的RD等式。

提出的方法

• 将CDME呈现为rho_n(x^(n))的密度等级序列，表示在位置x^(n)处有n个粒子的概率。
• 通过标准布朗运动项描述扩散，并加上包含损失与获得分量的反应算子。
• 给出简单降解/产生和双分子反应，以及多种物种的一般化的明确CDME形式。
• 通过边界层边界条件使用离散或连续速率，模拟储库交换来包含开边界。
• 显示在CDME的平均场极限下，线性和非线性情况下的宏观RD-PDE如何出现。
• 通过极限过程，将粒子层面的参数与扩散影响/开放反应中的宏观速率联系起来。

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

实验结果

研究问题

• RQ1 如何在粒子层面以概率方式表述具有变化粒子数的反应扩散动力学？
• RQ2 如何将与宏观储库的相互作用一致地纳入粒子层面的CDME？
• RQ3 如何从CDME在线性和非线性反应中恢复基于浓度的开放RD方程？
• RQ4 在扩散影响/开放系统中，微观参数与宏观速率之间的关系是什么？

主要发现

• CDME在一个无限维相空间中提供线性演化框架，耦合扩散与反应以及变化的粒子数。
• 开放边界通过与储库相关的边界层建模，得到与Dirac-δ表示一致的进出速率的显式形式。
• CDME的平均场极限恢复经典运算的降解与产生的RD-PDE，并在开放设置中产生扩散影响的反应项。
• 对于非线性类双分子反应，在大拷贝数假设下建立微观到宏观映射，将速率常数与宏观动力学联系起来。

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