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.

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Cellular cargo transported bidirectionally along microtubules by dynein and kinesin can be organized by spatially nonuniform upstream regulation or can self-organize. A mathematical model of early endosome transport in fungal hyphae demonstrates that spatiotemporally uniform regulation results in cargo dynamics consistent with experiment.

Spatio-temporal dynamics associated with a class of coupled membrane-bulk PDE-ODE models in one spatial dimension is analyzed using a combination of linear stability theory, numerical bifurcation software, and full time-dependent simulations. In our simplified 1-D setting, the mathematical model consists of two dynamically active membranes, separated spatially by a distance 2L, that are coupled together through a linear bulk diffusion field, with a fixed bulk decay rate. The coupling of the bulk and active membranes arises through both nonlinear flux boundary conditions for the bulk diffusion field together with feedback terms, depending on the local bulk concentration, to the dynamics on each membrane. For this class of models, it is shown both analytically and numerically that bulk diffusion can trigger a synchronous oscillatory instability in the temporal dynamics associated with the two active membranes. For the case of a single active component on each membrane, and in the limit L → ∞, rigorous spectral results for the linearization around a steady-state solution, characterizing the possibility of Hopf bifurcations and temporal oscillations in the membranes, are obtained. For finite L, a weakly nonlinear theory, accounting for eigenvalue-dependent boundary conditions appearing in the linearization, is developed to predict the local branching behavior near the Hopf bifurcation point. The analytical theory, together with numerical bifurcation results and full numerical simulations of the PDE-ODE system, are undertaken for various coupled membrane-bulk systems, including two specific biologically relevant applications. Our results show the existence of a wide parameter range where stable synchronous oscillatory dynamics in the two membranes can occur.

Oscillatory dynamics associated with the coupled membrane-bulk PDE-ODE model of Gomez et al. [Phys. Rev. Lett., 98(16), (2007), 168303] in one spatial dimension is analyzed using a combination of asymptotic analysis, linear stability theory, and numerical bifurcation software. The mathematical model consists of two dynamically active membranes with Fitzhugh-Nagumo kinetics, separated spatially by a distance L, that are coupled together through a diffusion field that occupies the bulk region 0 < x < L. The flux of the diffusion field on the membranes at x = 0 and x = L provides feedback to the local dynamics on the membranes. In the absence of membrane-bulk coupling the membrane kinetics have a stable fixed point. The effect of bulk diffusion is to trigger either synchronous and asynchronous oscillations in the two membranes. In the singular limit of slow-fast membrane dynamics, and with only one diffusing species in the bulk, phase diagrams in parameter space showing where either synchronous or asynchronous oscillations occur, together with the corresponding Hopf frequencies at onset, are provided analytically. When the membrane kinetics is not of slow-fast type, a numerical study of the stability problem together with the numerical bifurcation software XPPAUT [7] is used to to construct global bifurcation diagrams of steady-states and the bifurcating periodic solution branches for the case of either one or two diffusing species in the bulk. Overall, our results show the existence of a wide parameter range where stable synchronous oscillatory dynamics in the two membranes can occur. Predictions from the analytical and bifurcation theory are confirmed with full numerical simulations of the PDE-ODE system.