  Cell polarity and movement are fundamental to many biological functions.
Experimental and theoretically studies have indicated that interactions of
certain proteins lead to the cell polarization which plays a key role in
controlling the cell movement. We study the cell polarity and movement based on
a class of biophysical models that consist of reaction-diffusion equations for
different proteins and the dynamics of moving cell boundary. Such a moving
boundary is often simulated by a phase-filed model. We first apply the matched
asymptotic analysis to give a rigorous derivation of the sharp-interface model
of the cell boundary from a phase-field model. We then develop a robust
numerical approach that combines the level-set method to track the sharp
boundary of a moving cell and accurate discretization techniques for solving
the reaction-diffusion equations on the moving cell region. Our extensive
numerical simulations predict the cell polarization under various kinds of
stimulus, and capture both the linear and circular trajectories of a moving
cell for a long period of time. In particular, we have identified some key
parameters controlling different cell trajectories that are less accurately
predicted by reduced models. Our work has linked different models and also
developed tools that can be adapted for the challenging three-dimensional
simulations.
