  Motor proteins are active enzymatic molecules that drive a variety of
biological processes, including transfer of genetic information, cellular
transport, cell motility and muscles contraction. It is known that these
biological molecular motors usually perform their cellular tasks by acting
collectively, and there are interactions between individual motors that specify
the overall collective behavior. One of the fundamental issues related to the
collective dynamics of motor proteins is the question if they function at
stationary-state conditions. To investigate this problem, we analyze a
relaxation to the stationary state for the system of interacting molecular
motors. Our approach utilizes a recently developed theoretical framework, which
views the collective dynamics of motor proteins as a totally asymmetric simple
exclusion process of interacting particles, where interactions are taken into
account via a thermodynamically consistent approach. The dynamics of relaxation
to the stationary state is analyzed using a domain-wall method that relies on a
mean-field description, which takes into account some correlations. It is found
that the system quickly relaxes for repulsive interactions, while attractive
interactions always slow down reaching the stationary state. It is also
predicted that for some range of parameters the fastest relaxation might be
achieved for a weak repulsive interaction. Our theoretical predictions are
tested with Monte Carlo computer simulations. The implications of our findings
for biological systems are briefly discussed.
