Generalized run-and-turn motions: From bacteria to Lévy walks

Swimming bacteria exhibit a repertoire of motility patterns, in which persistent motion is interrupted by turning events. What are the statistical properties of such random walks? If some particular instances have long been studied, the general case where turning times do not follow a Poisson process has remained unsolved. We present a generic extension of the continuous time random walks formalism relying on operators and noncommutative calculus. The approach is first applied to a unimodal model of bacterial motion. We examine the existence of a minimum in velocity correlation function and discuss the maximum of diffusivity at an optimal value of rotational diffusion. The model is then extended to bimodal patterns and includes as particular cases all swimming strategies: run-and-tumble, run-stop, run-reverse and run-reverse-flick. We characterize their velocity correlation functions and investigate how bimodality affects diffusivity. Finally, the wider applicability of the method is illustrated by considering curved trajectories and Lévy walks. Our results are relevant for intermittent motion of living beings, be they swimming micro-organisms or crawling cells.

Figure 1

Top: Schematics for the run time distribution $\psi$ and the derived quantities ${\psi }_{\alpha }, {\psi }_{\beta }$, and ${\psi }_{\chi }$ defined in the text and given by Eq. (1). $t_{\mathrm{in}}$ is the randomly chosen initial time, each dot indicates a jump event. Bottom: Graphical interpretation of the two terms appearing in $P(\mathbf{q},s)$ as given by Eq. (7).

Figure 2

Unimodal run-and-turn motion in dimension $d=2$, and its parameters. Runs are characterized by a velocity $v$, a rotational diffusion coefficient $D_{\mathrm{r}}$, and a run time distribution $\psi \left(t\right)$. Turning events, represented by dots, are specified by the distribution of turning angle $h\left(\vartheta \right)$.

Figure 3

Velocity correlation function for unimodal run-reverse motion with vanishing $D_{\mathrm{r}}$ and a gamma distribution of run time. The latter, with mean $\tau$ and parameter $\gamma$, is shown in the inset. The curve with $\gamma ={10}^3$ is rescaled for clarity.

Figure 4

Minimum value in velocity correlation function $C\left(t\right)/v^2$, as a function of rotational diffusion ${\tilde{D}}_{\mathrm{r}}=(d-1)D_{\mathrm{r}}$. The run time is gamma distributed with mean $\tau$ and parameter $\gamma$.

Figure 5

Dimensionless diffusivity as a function of rotational diffusion coefficient for a run-reverse pattern ($\alpha =-1$), and gamma distributed run time. There is a maximum for $\gamma \ge {\gamma }_c=2$. Inset: critical value ${\gamma }_c\left(\alpha \right)$ above which a maximum exists.

Figure 6

Optimal rotational diffusion coefficient ${\tilde{D}}_{\mathrm{r}}^{*}$ (left) and maximal diffusivity ${\overline{\mathcal{D}}}^{*}$ (right), as a function of $\alpha$ for gamma distributed run time.

Figure 7

Model of bimodal run-and-turn motion. Two modes alternate, whose parameter sets are different.

Figure 8

Normalized velocity correlation function $C_{\mathrm{nor}}\left(t\right)=C\left(t\right)/\langle v^2\rangle$ for four types of swimming patterns. The run time is gamma distributed with $\gamma =1$ (Poissonian case) or $\gamma =2$ and has the same mean $\tau$ in all modes. For the run-and-tumble, $\alpha =0.33$ as in $\mathit{\text{E.}}\mathit{\text{coli}}$. The rotational diffusion coefficient is $\tau D_{\mathrm{r}}=0.1$.

Figure 9

Diffusivity ratio $\mathcal{R}$ between bimodal and unimodal motions, for bimodality in angle (left) and velocity (right). In the latter, the pattern is run-reverse ($\alpha =-1$). In both cases, $d=3, \tau D_{\mathrm{r}}=0.1$, and $\gamma =2$.

Figure 10

Diffusivity ratio $\mathcal{R}$ as a function of rotational diffusion, for the run-reverse-flick pattern (left) and the two-velocity run-reverse with $v_1/v_2=2$ (right).

Figure 11

Two extensions of the run-and-turn model. Left: Curved runs with constant mean curvature $\mathrm{\Omega }$. In the absence of rotational diffusion, trajectories are circular. Right: Lévy walks, in which motion is ballistic with a heavy tail in the distribution of run time.

Figure 12

Diffusivity as a function of angular velocity for a two-dimensional unimodal run-reverse with curved runs. Run time is gamma distributed with parameter $\gamma$ and $D_{\mathrm{r}}=0$. Inset: diffusivity is maximized by a finite angular velocity ${\mathrm{\Omega }}^{*}$ when $\gamma >{\gamma }_c\left(\alpha \right)$.

Figure 13

Comparison between simulation (points) and analytical result (line) for the velocity correlation function in cases $A$ to $D$ described in Table 2. The maximum absolute difference is less than ${10}^{-5}$.