### Statistical characterization of cell motility

Both in their natural tissue environment and in the artifical
environment of the tissue culture dish, animal tissue cells can exhibit
a highly migratory behavior. Responding to proteins and other signals
in their surrounding, cells continously remodel their cytoskeleton and
energize a host of molecular motors. The resulting forces alter
the cell shape as well as "pull" the cell body forward. Thus, cell
displacements are a high-level manifestation of a vast array of
molecular processes, much like automobile displacements are resulted by
a (probably less complex) underlying machinery.

To characterize the complexity of cell behavior, quantiative measures
are to be extracted from the time-lapse images. Most of these
quantities characterize the cellular trajectories, and thus completely
ignore the complexity associated with changing cell shape. The
trajectories are provided by a tracking procedure, which identifies the
same cells on a number of consecutive images. Related software is
downloadable from here.

Once the spatio-temporal coordinates of cell positions are known,
various statistical quantities can be derived.

Cell velocity distribution gives the empirical distribution
of cell displacements during a suitably short period of time. Most
tissue cells exhibit saltatory motion, that is irregularly alternating
active and passive periods of movements. This wide spectrum of
locomotory activity is reflected in a non-gaussian distribution of
velocities, which is often found to be well approximated by an
exponential distribution. (Related papers: Czirok et al 1998, Hegedus
et al 2000)

Time-averaged cell displacements systematically characterize the
long-term
behavior of cellular motion. While velocities are calculated from the
distance between cell positions at the beginning and at the end of
one-hour-long time periods, this comparision can be made for an
arbitrary long time window. Thus, average (over time and population)
cell displacements are calculated for a range of time periods,
thereby eliminating the choice of a particular time interval length for
velocity calculations. (Related papers:Mehes
et al 2002, Hegedus
et al 2004)

Diffusion index: Beside a simple a natural
way to compare the motility of cell populations, the average
displacement as a function of elapsed time can also be very informative
on
the type of motion: for a highly persistent, linear motion with a
steady velocity the average displacement is proportional with the
elapsed time. In contrast, for a mathematical random walk, where each
step is an independent random variable, the displacement grows with the
square root of time. For a boundend motion (when a cell can not
leave a
certain area), the displacement saturates to a constant value for
long time intervals. Therefore, the functional form of the average
displacement vs time curve yields information on the persistence of
cell motility. (Related papers:Mehes
et al 2002, Mehes
et al 2004)