  During cell migration, cells become polarized, change their shape, and move
in response to various internal and external cues. Cell polarization is defined
through the spatio-temporal organization of molecules such as PI3K or small
GTPases, and is determined by intracellular signaling networks. It results in
directional forces through actin polymerization and myosin contractions. Many
existing mathematical models of cell polarization are formulated in terms of
reaction-diffusion systems of interacting molecules, and are often defined in
one or two spatial dimensions. In this paper, we introduce a 3D
reaction-diffusion model of interacting molecules in a single cell, and find
that cell geometry has an important role affecting the capability of a cell to
polarize, or change polarization when an external signal changes direction. Our
results suggest a geometrical argument why more roundish cells can repolarize
more effectively than cells which are elongated along the direction of the
original stimulus, and thus enable roundish cells to turn faster, as has been
observed in experiments. On the other hand, elongated cells preferentially
polarize along their main axis even when a gradient stimulus appears from
another direction. Furthermore, our 3D model can accurately capture the effect
of binding and unbinding of important regulators of cell polarization to and
from the cell membrane. This spatial separation of membrane and cytosol, not
possible to capture in 1D or 2D models, leads to marked differences of our
model from comparable lower-dimensional models.
