Stochasticity in Biophysical Systems: Searching, Aging, and Spreading

The role of stochasticity throughout many biophysical systems is of great importance. From
evolution to foraging, from the spreading of viruses to cell fate, all hinge in one way or
another on inherent stochasticity. The aim of this thesis is to explore the tools often used
to quantify randomness in nature, and employ these tools on a selected range of biophysical
systems (Papers I-IV). In these papers we cover many-bodied systems with keen focus on two
areas: single-file diffusion (SFD), and epidemiology.
In a SFD system particles in a 1D channel are allowed to diffuse but cannot occupy the same
space, thus the particles maintain their order for all time. SFD occurs often in biology, for
example, it is often used as an abstract representation of protein motion on crowded DNA,
the motivation for the first two papers. In Paper I we analyze the first passage time density
(FPTD) of a tracer particle in homogeneous and heterogeneous systems and how they link
to fractional Brownian motion particles (non-Markovian diffusive particles). In Paper II we
extend the model to allow flanking particles of the tracer to enter/leave the 1D channel with
a given rate, and investigate how this affects the FPTD. Paper III is similar to the first,
but the particles are all functionalized: their waiting time between movement is taken from
a power-law density, not an exponential (as in Papers I and II). Through a simple scaling
argument we analyze the tracer dynamics, and seek to provide a mechanism for "aging",
logarithmically slow dynamics seen in certain physical systems.
In the second area, Paper IV, we explore the stochastic spreading of viruses on metapop-
ulations. We provide an analytical method (in an area saturated by numerical techniques)
to model the spread of a susceptible-infected-susceptible virus on a general network of large
populations, connected through a travel rate matrix.