Bacterial chemotaxis without gradient-sensing

Abstract

Chemotaxis models are based on spatial or temporal gradient measurements by individual organisms. The key contribution of Keller and Segel (J Theor Biol 30:225–234, 1971a; J Theor Biol 30:235–248, 1971b) is showing that erratic measurements of individuals may result in an accurate chemotaxis phenomenon as a group. In this paper we provide another option to understand chemotactic behavior when individuals do not sense the gradient of chemical concentration by any means. We show that, if individuals increase their dispersal rate to find food when there is not enough food, an accurate chemotactic behavior may be obtained without sensing the gradient. Such a dispersal has been suggested by Cho and Kim (Bull Math Biol 75:845–870, 2013) and was called starvation driven diffusion. This model is surprisingly similar to the original Keller–Segel model. A comprehensive picture of traveling bands and fronts is provided.

Appendix A: Derivation of a starvation driven diffusion

In this section we introduce a short derivation of a starvation driven diffusion (see Cho and Kim 2013, for detailed discussions). Consider a random walk system with a constant walk length $\Delta x$ and a constant jumping time interval $\Delta t$. Let $0<\gamma (x_i)\le 1$ be the probability for a particle to depart a grid point $x_i$ at each jumping time. (For a usual random walk system every particle departs at each jumping time and hence $\gamma =1$.) Each particle moves to one of two adjacent grid points, $x_{i+1}$ or $x_{i-1}$, randomly. Let $U(x_i)$ be the number of particles placed at the grid point $x_i$. Then, the particle density is $u=U/\Delta x$. Hence the net flux that crosses a middle point $x_{i+1/2}:={x_i+x_{i+1}\over 2}$ is

$$\begin{aligned} J\Big |_{x_{i+1/2}}&= {\gamma (x_i)|\Delta x|u(x_i)\over 2|\Delta t|}-{\gamma (x_{i+1})|\Delta x|u(x_{i+1})\over 2|\Delta t|}\\&\cong -{|\Delta x|^2\over 2\Delta t}(\gamma u)_x\Big |_{x_{i+1/2}}. \end{aligned}$$

The corresponding diffusion equation comes from the conservation law

$$\begin{aligned} u_t=-\nabla \cdot J={|\Delta x|^2\over 2\Delta t}(\gamma u)_{xx}. \end{aligned}$$

After a time rescaling we obtain the desired equation

$$\begin{aligned} u_t=(\gamma u)_{xx}. \end{aligned}$$

Notice that the probability $\gamma$ depends only on the departing point $x_i$, which indicates the concentration gradient is not measured. If the probability depends on nearby points, then it implies gradient information is used to decide the migration probability. For more discussion including other cases, see Okubo and Levin (2001, §5.4).