One of the characteristics that distinguishes living things from non-living things is motility. On the cellular level, the motility or non-motility of different types of cells can be life building, life-saving or life-threatening. A thorough study of cell motion is needed to help understand the underlying mechanisms that enhance or prohibit cell motion. We introduce a discrete centroid model of cell motion in the context of a generalized random walk. We find an approximation for the theoretical mean square displacement (MSD) that uses a subset of the state space to estimate the MSD for the entire space. We give some intuition as to why this is an unexpectedly good estimate. A lower and upper bound for the MSD is also given. We extend the centroid model to an ODE model and use it to analyze the distribution of focal adhesion (FA) lifetimes gathered from experimental data. We found that in all but one case a unimodal, non-symmetric gamma distribution is a good match for the experimental data. We use a detach-rate function in the ODE model to determine how long a FA will persist before it detaches. A detach-rate function that is dependent on both force and time produces distributions with a best fit gamma curve that closely matches the data. Using the data gathered from the matching simulations, we calculate both the cell speed and mean FA lifetime and compare them. Where available, we also compare this relationship to that of the experimental data and find that the simulation reasonably matches it in most cases. In both the simulations and experimental data, the cell speed and mean FA lifetime are related, with longer mean lifetimes being indicative of slower speeds. We suspect that one of the main predictors of cell speed for migrating cells is the distribution of the FA lifetimes.



College and Department

Physical and Mathematical Sciences; Mathematics



Date Submitted


Document Type





Cell motion, biology, dynamical systems, stochastic process, focal adhesions