Abstract
In many biological systems, the movement of individual agents is characterized having multiple qualitatively distinct behaviors that arise from a variety of biophysical states. For example, in cells the movement of vesicles, organelles, and other intracellular cargo is affected by their binding to and unbinding from cytoskeletal filaments such as microtubules through molecular motor proteins. A typical goal of theoretical or numerical analysis of models of such systems is to investigate effective transport properties and their dependence on model parameters. While the effective velocity of particles undergoing switching diffusion dynamics is often easily characterized in terms of the long-time fraction of time that particles spend in each state, the calculation of the effective diffusivity is more complicated because it cannot be expressed simply in terms of a statistical average of the particle transport state at one moment of time. However, it is common that these systems are regenerative, in the sense that they can be decomposed into independent cycles marked by returns to a base state. Using decompositions of this kind, we calculate effective transport properties by computing the moments of the dynamics within each cycle and then applying renewal reward theory. This method provides a useful alternative large-time analysis to direct homogenization for linear advection–reaction–diffusion partial differential equation models. Moreover, it applies to a general class of semi-Markov processes and certain stochastic differential equations that arise in models of intracellular transport. Applications of the proposed renewal reward framework are illustrated for several case studies such as mRNA transport in developing oocytes and processive cargo movement by teams of molecular motor proteins.
Highlights
Microscale biological agents frequently change biophysical state, which results in significant changes in their movement behavior
1.2 Summary of Method Based on Regeneration Cycles. These foregoing methods (Pavliotis and Stuart 2008; Ciocanel et al 2017; Bressloff and Newby 2013) rely on the fully Markovian structure of the dynamics, with the state-switching process in particular taking the form of a continuous-time Markov chain with exponentially distributed state durations
We allow for more general random spatial dynamics within a state that need not be fully Markovian
Summary
Microscale biological agents frequently change biophysical state, which results in significant changes in their movement behavior. Intracellular cargo, for example, switches among active transport, diffusive transport, and paused states, each resulting from different mechanochemical configurations of the cargo, cytoskeletal filaments, and the molecular motors that bind them (Hancock 2014; Bressloff and Newby 2013) Models for this behavior can be either deterministic (typically partial differential equations, PDEs) or stochastic (often continuous-time Markov chains, CTMCs, or stochastic differential equations, SDEs) depending on whether the investigation focuses on population properties (deterministic methods) or individual paths (stochastic methods). The study of deterministic models often relies on numerical simulation using PDE integration methods (Wang et al 2003; Cox and Matthews 2002; Trong et al 2015), while stochastic models are simulated with Monte Carlo/Gillespie algorithms (Müller et al 2008; Kunwar and Mogilner 2010; Müller et al 2010; Allard et al 2019) to generate individual trajectories that are analyzed statistically These computations can be quite costly, especially when one wants to understand how bulk transport properties (like effective velocity or diffusivity) depend on individual model parameters. Asymptotic analysis allows for explicit approximation of transport properties, which can validate, complement, or even replace numerical simulations (Reed et al 1990; Brooks 1999; Pavliotis 2005; Pavliotis and Stuart 2008; Popovic et al 2011; McKinley et al 2012; Bressloff and Xu 2015; Ciocanel et al 2017)
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