  The dynamics of single cell migration on flat surfaces is usually modeled by
a Langevin-like problem consisting of ballistic motion for short periods and
random walk. for long periods. Conversely, recent studies have revealed a
previously neglected random motion at very short intervals, what would rule out
the possibility of defining the cell instantaneous velocity and a robust
measurement procedure. A previous attempt to address this issue considered an
anisotropic migration model, which takes into account a polarization
orientation along which the velocity is well-defined, and a direction
orthogonal to the polarization vector that describes the random walk. Although
the numerically and analytically calculated mean square displacement and
auto-correlation agree with experimental data for that model, the velocity
distribution peaks at zero, which contradicts experimental observations of a
constant drift in the polarization direction. Moreover, Potts model simulations
indicate that instantaneous velocity cannot be measured for any direction.
Here, we consider dynamical equations for cell polarization, which is
measurable and introduce a polarization-dependent displacement, circumventing
the problem of ill defined instantaneous velocity. Polarization is a
well-defined quantity, preserves memory for short intervals, and provides a
robust measurement procedure for characterizing cell migration. We consider
cell polarization dynamics to follow a modified Langevin equation that yields
cell displacement distribution that peaks at positive values, in agreement with
experiments and Potts model simulations. Furthermore, displacement
autocorrelation functions present two different time scales, improving the
agreement between theoretical fits and experiments or simulations.
