  Actin filaments are critical components of the eukaryotic cytoskeleton,
playing important roles in a number of cellular functions, such as cell
migration, organelle transport, and mechanosensation. They are helical polymers
with a well-defined polarity, composed of globular monomers that bind
nucleotides in one of three hydrolysis states (ATP, ADP-Pi, or ADP). Mean-field
models of the dynamics of actin polymerization have succeeded in, among other
things, determining the nucleotide profile of an average filament and resolving
the mechanisms of accessory proteins, however these models require numerical
solution of a high-dimensional system of nonlinear ODE's. By truncating a set
of recursion equations, the Brooks-Carlsson model reduces dimensionality to 11,
but it remains nonlinear and does not admit an analytical solution, hence,
significantly hindering understanding of its resulting dynamics. In this work,
by taking advantage of the fast timescales of the hydrolysis states of the
filament tips, we propose two model reduction schemes that achieve low
dimensionality and linearity. We provide an exact solution of the resulting
linear equations and use it to shed light on the dynamical behaviors of the
full BC model, highlighting the relative ordering of the timescales of various
collective processes, and explaining some unusual dependence of the
steady-state behavior on initial conditions.
