Alignment interaction and band formation in assemblies of autochemorepulsive walkers

Chemotaxis refers to the motion of an organism induced by chemical stimuli and is a motility mode shared by many living species that has been developed by evolution to optimize certain biological processes such as foraging or immune response. In particular, autochemotaxis refers to chemotaxis mediated by a cue produced by the chemotactic particle itself. Here, we investigate the collective behavior of autochemotactic particles that are repelled by the cue and therefore migrate preferentially towards low-concentration regions. To this end, we introduce a lattice model inspired by the true self-avoiding walk which reduces to the Keller-Segel model in the continuous limit, for which we describe the rich phase behavior. We first rationalize the chemically mediated alignment interaction between walkers in the limit of stationary concentration fields, and then describe the various large-scale structures that can spontaneously form and the conditions for them to emerge, among which we find stable bands traveling at constant speed in the direction transverse to the band.

Figure 1

Concentration gradient $\nabla c_{\infty }$, where the color indicates its norm and the arrow its direction. The lower-half plan ($y<0$) corresponds to $D_c=0.1$ while the upper half corresponds to $D_c=0.5$.

Figure 2

Persistence length $l_p$ as a function $D_c$ and various values of $\beta$, where we have subtracted the value ${l_p}_0=1/3$ reached for $\beta =0$. The inset shows the value of $D_c$ that maximizes the persistence length as a function of $\beta$ (solid line), together with the corresponding value of the persistence length $\mathrm{\Delta }{l}_p^{*}=l_p^{*}-{l_p}_0$ (dashed line).

Figure 3

Alignment probability ${\mathcal{A}}_{\perp }$ for $\beta =100$. Because of the symmetry of the field by reflection with the line ${\mathrm{\Delta }}_y=-{\mathrm{\Delta }}_x$, we show it for $D_c=0.1$ in the lower left plane and $D_c=0.5$ in the upper right plane. The position $\mathrm{\Delta }=0$ is indicated by the white circle. An animation of an alignment event for $\theta =\pi /2$ is provided in the Supplemental Material [40].

Figure 4

Alignment probability ${\mathcal{A}}_{\leftrightharpoons }$ for $\beta =100$ and $D_c=0.1$ (upper panel) and $D_c=0.5$ (lower panel).

Figure 5

Persistence length $l_p^{\left(2\right)}$ for $\beta =100$ and $D_c=0.1$ (upper panel) and $D_c=0.5$ (lower panel).

Figure 6

${max}_{\mathrm{\Delta }}{l}_p^{\left(2\right)}-l_p$ (solid lines) and ${\delta }^{*}$ (dotted lines with squares) as a function of $D_c$ for various values of $\beta$. The inset shows the value of $D_c$ that maximizes this function (solid line) and the corresponding value of the persistence length from which is subtracted the persistence length due to self-interaction (dashed line).

Figure 7

Mean-square displacement of a single walker for $D_c=1$ and various values of $\beta$. The limit cases $\langle \mathbf{r}{\left(t\right)}^2\rangle =t$ and $\langle \mathbf{r}{\left(t\right)}^2\rangle =t^2$ are shown in dashed and dotted lines, respectively.

Figure 8

Mean-square displacement for $D_c=1, \beta =100$, and various values of $\varrho$. The limit cases $\langle \mathbf{r}{\left(t\right)}^2\rangle =t$ and $\langle \mathbf{r}{\left(t\right)}^2\rangle =t^2$ are shown in dashed and dotted lines, respectively.

Figure 9

Diagrams in the $(\varrho ,\beta )$ plane showing various phases found in the system for $D_c=0.1$ (left panel), $D_c=0.5$ (center panel), and $D_c=1$ (right panel). Note that both axes are in logarithmic scale.

Figure 10

Typical configuration of the cluster phase ($\varrho =0.001, \beta =4640, D_c=0.5, L_x=400, L_y=100$). The upper panel shows the positions of walkers and their traveling directions along the $x$ axis while the lower panel shows the corresponding concentration field. A full animation is provided in the Supplemental Material [40].

Figure 11

Different formations of bands for $D_c=0.5, L_x=400, L_y=100$ and various couples $(\varrho ,\beta )$ (upper block, $\varrho =0.11,\beta =1000$; center block, $\varrho =0.1,\beta =215$; lower block, $\varrho =0.18,\beta =1000$). In each example we show snapshots of the total orientation along the $x$ axis (upper left panel) and concentration field (lower left panel), with the corresponding time series of $N_{\to }, N_{\uparrow }, N_{\leftarrow }$, and $N_{\downarrow }$ as a function of time (upper right panel) and the profiles ${\psi }_{\mathbf{v}}$ for all four directions (lower right panel). Animations of all cases are provided in the Supplemental Material [40].

Figure 12

Series of snapshots in a bouncing event between two bands. The background color codes the value of the concentration field (note that the scale is not the same in all snapshots to increase the contrast), while the dots indicate the position of walkers, the color of which refers to their position in the first snapshot. A full animation is provided in the Supplemental Material [40].

Figure 13

Snapshot of a system in the oscillating band phase ($\varrho =0.56, \beta =1000, D_c=1$). The upper panel shows the direction of particles along the $x$ axis while the lower panel shows the corresponding concentration field. An animation of the formation of the phase is provided in the Supplemental Material [40]. We also show ${\psi }_{\mathbf{v}}$ for all four directions.

Figure 14

Setup for the stability analysis for three different values of $D_c$. The color codes for the concentration field. Positions of all walkers are indicated by the white circles except for $W_{0,1,2,3}$ which are specified by special markers. The preferred directions for the next time step are indicated by arrows. The ones of $W_{0,1,2,3}$ are reported as a function of $D_c$ in the lower panel. Three different regimes can clearly be identified.