  Molecular motors are specialized proteins which perform active, directed
transport of cellular cargoes on cytoskeletal filaments. In many cases, cargo
motion powered by motor proteins is found to be bidirectional, and may be
viewed as a biased random walk with fast unidirectional runs interspersed with
slow `tug-of-war' states. The statistical properties of this walk are not known
in detail, and here, we study memory and bias, as well as directional
correlations between successive runs in bidirectional transport. We show, based
on a study of the direction reversal probabilities of the cargo using a purely
stochastic (tug-of-war) model, that bidirectional motion of cellular cargoes
is, in general, a correlated random walk. In particular, while the motion of a
cargo driven by two oppositely pulling motors is a Markovian random walk,
memory of direction appears when multiple motors haul the cargo in one or both
directions. In the latter case, the Markovian nature of the underlying single
motor processes is hidden by internal transitions between degenerate run and
pause states of the cargo. Interestingly, memory is found to be a non-monotonic
function of the number of motors. Stochastic numerical simulations of the
tug-of-war model support our mathematical results and extend them to
biologically relevant situations.
