  In many biological systems, the movement of individual agents is commonly
characterized as having multiple qualitatively distinct behaviors that arise
from various biophysical states. This is true for vesicles in intracellular
transport, micro-organisms like bacteria, or animals moving within and
responding to their environment. For example, in cells the movement of
vesicles, organelles and other cargo are 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 the effective transport properties and their
dependence on model parameters. While the effective velocity of particles
undergoing switching diffusion 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 framework are illustrated for case studies such as
mRNA transport in developing oocytes and processive cargo movement by teams of
motor proteins.
