SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 438-499, January 2021.
We analyze a class of cell-bulk coupled PDE-ODE models, motivated by quorum-sensing and diffusion-mediated behavior in microbial systems, that characterize communication between localized spatially segregated dynamically active signaling compartments or “cells” that have a permeable boundary. In this model, the cells are disks of a common radius $\varepsilon \ll 1$ and they are spatially coupled through a passive extracellular bulk diffusion field with diffusivity $D$ in a bounded 2-D domain. Each cell secretes a signaling chemical into the bulk region at a constant rate and receives a feedback of the bulk chemical from the entire collection of cells. This global feedback, which activates signaling pathways within the cells, modifies the intracellular dynamics according to the external environment. The cell secretion and global feedback are regulated by permeability parameters across the cell membrane. For arbitrary reaction-kinetics within each cell, the method of matched asymptotic expansions is used in the limit $\varepsilon\ll 1$ of small cell radius to construct steady-state solutions of the PDE-ODE model and to derive a nonlinear globally coupled eigenvalue problem (GCEP) that characterizes the linear stability properties of the steady-states. The analysis and computation of the nullspace of the GCEP as parameters are varied is central to the linear stability analysis. In the limit of large bulk diffusivity $D={D_0/\nu}\gg 1$, where $\nu\equiv {-1/\log\varepsilon}$, an asymptotic analysis of the PDE-ODE model leads to a limiting ODE system for the spatial average of the concentration in the bulk region that is coupled to the intracellular dynamics within the cells. Results from the linear stability theory and ODE dynamics are illustrated for Sel'kov reaction-kinetics, where the kinetic parameters are chosen so that each cell is quiescent when uncoupled from the bulk medium. For various specific spatial configurations of cells, the linear stability theory is used to construct phase diagrams in parameter space characterizing where a switch-like emergence of intracellular oscillations can occur through a Hopf bifurcation. The effect of the membrane permeability parameters, the reaction-kinetic parameters, the bulk diffusivity, and the spatial configuration of cells on both the emergence and synchronization of the oscillatory intracellular dynamics, as mediated by the bulk diffusion field, is analyzed in detail. The linear stability theory is validated from full numerical simulations of the PDE-ODE system, and from the reduced ODE model when $D$ is large.

中文翻译：

---

小型信号隔室之间扩散介导的二维PDE-ODE模型的同步和振荡动力学

SIAM应用动力系统杂志，第20卷，第1期，第438-499页，2021年1月。
我们分析了由群体感应和微生物系统中扩散介导的行为所激发的一类细胞-体-体耦合的PDE-ODE模型，该模型表征了具有可渗透边界的局部空间分隔的动态活动信号传递隔室或“细胞”之间的通信。在该模型中，细胞是具有共同半径$ \ varepsilon \ ll 1 $的圆盘，并且它们通过有界2-D域中具有扩散率$ D $的无源细胞外体扩散场在空间上耦合。每个细胞以恒定的速率将信号化学物质分泌到主体区域中，并从整个细胞集合中接收主体化学物质的反馈。该全局反馈激活细胞内的信号传导途径，根据外部环境改变细胞内动力学。细胞分泌和整体反馈受跨细胞膜通透性参数的调节。对于每个单元内的任意反应动力学，在小单元半径的极限$ \ varepsilon \ ll $中使用匹配渐近展开的方法构造PDE-ODE模型的稳态解并得出非线性全局耦合表征稳态线性稳定性的特征值问题（GCEP）。随着参数的变化，GCEP零空间的分析和计算对于线性稳定性分析至关重要。在大体积扩散系数$ D = {D_0 / \ nu} \ gg 1 $的极限下，其中$ \ nu \ equiv {-1 / \ log \ varepsilon} $，PDE-ODE模型的渐近分析导致了一个有限的ODE系统，该系统的体积平均值与细胞内的细胞内动力学相关，是整个区域内浓度的空间平均值。线性稳定性理论和ODE动力学的结果说明了Sel'kov反应动力学，其中选择了动力学参数，以使每个单元格从大容量介质中解耦时都处于静止状态。对于细胞的各种特定空间配置，线性稳定性理论用于在参数空间中构造相图，以表征其中通过Hopf分支可以发生细胞内振荡的开关状出现的特征。膜渗透性参数，反应动力学参数，体积扩散率的影响，并详细分析了由体扩散场介导的振荡细胞内动力学的出现和同步过程中细胞的空间构型。线性稳定性理论通过PDE-ODE系统的完整数值模拟以及$ D $大时的简化ODE模型进行了验证。