Persistent random walk of cells involving anomalous effects and random death

The purpose of this paper is to implement a random death process into a persistent random walk model which produces sub-ballistic superdiffusion (Lévy walk). We develop a stochastic two-velocity jump model of cell motility for which the switching rate depends upon the time which the cell has spent moving in one direction. It is assumed that the switching rate is a decreasing function of residence (running) time. This assumption leads to the power law for the velocity switching time distribution. This describes the anomalous persistence of cell motility: the longer the cell moves in one direction, the smaller the switching probability to another direction becomes. We derive master equations for the cell densities with the generalized switching terms involving the tempered fractional material derivatives. We show that the random death of cells has an important implication for the transport process through tempering of the superdiffusive process. In the long-time limit we write stationary master equations in terms of exponentially truncated fractional derivatives in which the rate of death plays the role of tempering of a Lévy jump distribution. We find the upper and lower bounds for the stationary profiles corresponding to the ballistic transport and diffusion with the death-rate-dependent diffusion coefficient. Monte Carlo simulations confirm these bounds.

Figure 1

Monte Carlo profile involving $N={10}^4$ cells at time $t={10}^3$ (black line), stationary upper bound (48) (blue top dot-dashed line), and lower bound (54) (red lower dot-dashed line) profiles with parameters: the anomalous exponent $\mu =0.2,$ the death rate $\theta =0.01,$ the time unit ${\tau }_0=1$, and the cell velocity $v=1.$