An Asymptotic Analysis of a 2-D Model of Dynamically Active Compartments Coupled by Bulk Diffusion

Abstract

A class of coupled cell-bulk ODE-PDE models is formulated and analyzed in a two-dimensional domain, which is relevant to studying quorum sensing behavior on thin substrates. In this model, spatially segregated dynamically active signaling cells of a common small radius ǫ ≪ 1 are coupled through a passive bulk diffusion field. For this coupled system, the method of matched asymptotic expansions is used to construct steady-state solutions and to formulate a spectral problem that characterizes the linear stability properties of the steady-state solutions, with the aim of predicting whether temporal oscillations can be triggered by the cell-bulk coupling. Phase diagrams in parameter space where such collective oscillations can occur, as obtained from our linear stability analysis, are illustrated for two specific choices of the intracellular kinetics. In the limit of very large bulk diffusion, it is shown that solutions to the ODE-PDE cell-bulk system can be approximated by a finite-dimensional dynamical system. This limiting system is studied both analytically, using a linear stability analysis, and globally, using numerical bifurcation software. For one illustrative example of the theory it is shown that when the number of cells exceeds some critical number, i.e. when a quorum is attained, the passive bulk diffusion field can trigger oscillations that would otherwise not occur without the coupling. Moreover, for two specific models for the intracellular dynamics, we show that there are rather wide regions in parameter space where these triggered oscillations are synchronous in nature. Unless the bulk diffusivity is asymptotically large, it is shown that a clustered spatial configuration of cells inside the domain leads to larger regions in parameter space where synchronous collective oscillations between the small cells can occur. Finally, the linear stability analysis for these cell-bulk models is shown to be qualitatively rather similar to the linear stability analysis of localized spot patterns for activator-inhibitor reaction-diffusion systems in the limit of long-range inhibition and short-range activation.

Spots, traps, and patches: asymptotic analysis of localized solutions to some linear and nonlinear diffusive systems

“…3). For this configuration, the symmetric Green's matrix G λ is also cyclic, and so we can readily calculate its matrix spectrum analytically in terms of the first row of G λ (see §6 of [32]), which we label as…”

“…In contrast to oscillator problems where synchronized oscillations occur by the entrainment of phase and frequencies of individually oscillating elements, the onset of synchronized oscillations for these collections of cells is characterized by a bifurcation, whereby individual cells in a quiescent state become oscillatory and synchronized from the effect of the bulk diffusing species. A coupled cell-bulk PDE model to study the interaction between N dynamically active localized cells and a single bulk diffusible signaling species in a bounded 2-D domain, inspired by the modeling framework of [62] and [63], was formulated in [32]. In non-dimensional form, the model of [32] involves the dimensionless bulk concentration field U , which satisfies…”

“…A coupled cell-bulk PDE model to study the interaction between N dynamically active localized cells and a single bulk diffusible signaling species in a bounded 2-D domain, inspired by the modeling framework of [62] and [63], was formulated in [32]. In non-dimensional form, the model of [32] involves the dimensionless bulk concentration field U , which satisfies…”

Anomalous Scaling of Hopf Bifurcation Thresholds for the Stability of Localized Spot Patterns for Reaction-Diffusion Systems in Two Dimensions

“…The matrix spectrum G λ v j = κ λ,j v j is readily calculated as in section 6 of [8]. The synchronous eigenpair of G λ is Since κ λ,j = κ λ,N +2−j for j = 2, .…”

Ultrasensitivity and noise amplification in a model ofV. harveyiquorum sensing

“…(Examples of spatial models can be found in Refs. [12,[19][20][21][22].) From a dynamical systems perspective, two distinct forms of collective behavior are typically considered: either the population acts as a biochemical switch [12] or as a synchronized biochemical oscillator [14,17,18].…”