Migration of individual microvessel endothelial cells: stochastic model and parameter measurement
Authors: Stokes Cynthia L. , Lauffenburger Douglas A. , Williams Stuart K. 
Published: April 2021

ABSTRACT
Analysis of cell motility effects in physiological processes can be facilitated by a mathematical model capable of simulating individual cell movement paths. A quantitative description of motility of individual cells would be useful, for example, in the study of the formation of new blood vessel networks in angiogenesis by microvessel endothelial cell (MEC) migration. In this paper we propose a stochastic mathematical model for the random motility and chemotaxis of single cells, and evaluate migration paths of MEC in terms of this model. In our model, cell velocity under random motility conditions is described as a persistent random walk using the Omstein-Uhlenbeck (O.U) process. Two parameters quantify this process: the magnitude of random movement accelerations, α, and a decay rate constant for movement velocity, β. Two other quantities often used in measurements of individual cell random motility properties - cell speed, S, and persistence time in velocity, Pv- can be defined in terms of the fundamental stochastic parameters α and β by: S=√ (α/β) and Pv=l/p. We account for chemotactic cell movement in chemoattractant gradients by adding a directional bias term to the O-U process. The magnitude of the directional bias is characterized by the chemotactic responsiveness, K. A critical advantage of the proposed model is that it can generate, using experimentally measured values of α, β and K, computer simulations of theoretical individual cell paths for use in evaluating the role of cell migration in specific physiological processes.
We have used the model to assess MEC migration in the presence or absence of the angiogenic stimulus acidic fibroblast growth factor (aFGF). Time-lapse video was used to observe and track the paths of cells moving in various media, and the mean square displacement was measured from these paths. To test the validity of the model, we compared the mean square displacement measurements of each cell with model predictions of that displacement The comparison indicates that the O-U process provides a satisfactory description of the random migration at this level of comparison. Using nonlinear regression in these comparisons, we measured the magnitude of random accelerations, a, and the velocity decay rate constant β, for each cell path. We consequently obtained values for the derived quantities, speed and persistence time. In control medium, we find that α=250±100 μm2h−3 and β=0.22±0.03h-1, while in stimulus medium (control plus unpurified aFGF) α=1900±720μm2h-3 and β=0.99±0.37h-1. These results indicate that both random acceleration and velocity decay rate are enhanced by aFGF. From the perspective of the derived quantities, cell speed is increased (from 25 to 42 μmh-1) but persistence time is decreased (from 5.4 to 2.9 h) by this chemical stimulus. These results suggest that the intracellular mechanisms that control rate of movement of MEC may be different from those that control movement direction. We also estimated a value for the chemotactic responsiveness K by relating computer-simulated cell paths to previous measurements of population chemotactic migration in aFGF gradients. A value kao=2400 μm2h-2, where ao, is the source attractant concentration, was obtained for the chemotactic responsiveness. The ratio of chemotactic to random migration, represented by kao/ (α/β)=Ka/S2, is approximately 1.5, demonstrating that MEC display a numerically significant degree of directional sensitivity to aFGF.