Renewal Reward Perspective on Linear Switching Diffusion Systems in Models of Intracellular Transport

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.

Applications of PDEs and Stochastic Modeling to Protein Transport in Cell Biology

“…This approach, as well as the quasi-steady-state methods discussed in the previous section, rely on the Markovian structure of the dynamics, meaning that the stateswitching process is a continuous-time Markov chain with exponentially distributed durations in each state. The stochastic processes framework we extended in [CFKM20] allows for more generalized random dynamics, which in particular need not assume that the particle spends an exponentially-distributed time in each state.…”

“…In [CFKM20], we assume that the cargo switches between dynamic states at the random times {𝑡 𝑘 ∶ 𝑘 ∈ ℕ}, and let {𝐽 𝑘 ∶ 𝑘 ∈ ℕ} denote the state during the 𝑘th time interval [𝑡 𝑘−1 , 𝑡 𝑘 ). We assume that the sequence 𝐽 𝑘 is a timehomogeneous Markov chain and that it has a finite mean time of returning to each particular state (i.e., it is called positive recurrent).…”

“…Node off-track neurofilament on-track neurofilament microtubule The model parameters have the same meanings as in equations (1), and 𝑍 is an independent standard normal random variable. The analysis in [CFKM20] proceeds by interpreting the time duration and spatial displacement in each state as rewards associated to that cargo state. An approach based on setting up moment-generating functions of the reward collected by the cargo until it returns to the base state allows to calculate the moments of the rewards needed to compute the effective transport properties in (12).…”

“…An approach based on setting up moment-generating functions of the reward collected by the cargo until it returns to the base state allows to calculate the moments of the rewards needed to compute the effective transport properties in (12). Interested readers can find additional details in [CFKM20].…”

“…Transport in the context of more complex geometries has been studied using quasi-steady-state analysis and under assumptions of fast switching rates, slow diffusion, and sufficiently small filament density in [BX15], for applications including filaments growing from a microtubule aster in neuron growth cones, or filaments that nucleate from membrane sites in budding yeast. On the other hand, incorporating the dependence on spatial microtubule tracks in the stochastic regenerative cycle framework would require additional assumptions on the spatial scale of transport in a regeneration cycle [CFKM20].…”