Nonmonotonic Dependence of Random Migration upon Density of Cells on Biomaterials

Cell migration is a basic biological process essential for physiological homeostasis and disease pathogenesis. It is interesting that random migration of a cell on an extracellular matrix or a biomaterial obeys the diffusion equation of Brownian particles proposed by Einstein in 1905, from which diffusivity can be used to quantify the migration rate. While the complexity of density dependence of diffusivity has been pointed out for cells, a function simply relating migration rate to cell density has never been reported. Herein we show that, unlike the diffusion of an abiotic particle, the migration rate of a living cell changes with cell density nonmonotonically, and a quantitative relation between migration rate and cell density is established by us, resulting in a product equation. The term d_max, namely, cell density for the fastest migration, is further defined and justified based on both real-time observations of cells and Monte Carlo simulations of model "living particles". The maximum migration rate is interpreted by the combination of volume-exclusion and autocrine effects, representing physical and biological effects, respectively. The regulation works universally across different cell types, culture media, and biomaterial surfaces examined by us, while the concrete values of d_max depend on these conditions. This fundamental study is helpful for understanding dynamic cell behaviors under material microenvironments and for the design of advanced biomaterials for tissue regeneration or drug carriers.