  It was previously believed that diffusion of a tracer particle in a
viscoelastic material should be of the fractional Brownian motion (fBm) type.
This is due to the long-term memory in the response of such materials to
mechanical perturbations. Surprisingly, the diffusion of a tracer particle in a
network of a purified protein, actin, was found to conform to the continuous
time random walk (CTRW) type in one study that focused on the ensemble average
characteristics. Here, we analyze the dynamics of two differently sized tracer
particles in actin networks of different mesh sizes. We find that the ratio of
tracer particle size to the characteristic length scale of a bio-polymer
network plays a crucial role in determining the type of diffusion the particle
performs. We find that the tracer particle diffusion has features of fBm when
the particle is large compared to the mesh size, of normal diffusion when the
particle is much smaller than the mesh size, and of CTRW in between these two
limits. Based on our findings, we propose and verify numerically a new model
for the tracer particle's motion in all regimes. Our model suggests that
diffusion in actin networks consists of fBm of the tracer particle coupled with
occasional caging events with power-law-distributed escape times.
