Directional sensing of deformed cells under faint gradients

We consider the physical limit of the directional sensing ability of living cells, as in chemotaxis, under a low concentration and shallow chemoattractant gradient. Elliptic cells sense the direction, which is a stochastic variable of a characteristic distribution with peaks at directions not necessarily to the gradient. The peak positions depend on the information of the gradient that cells use to infer the direction and also the shape and orientation of cells. Cells of different shapes may use different inference strategies to increase their directional sensing performance.

Figure 1

Schematic representation of our model of a eukaryotic chemotactic cell.

Figure 2

Distribution of the estimated direction of chemoattractant gradient. (a)–(p) $\mathcal{P}\left(\theta \right)$ for various concentrations and cell orientations. The chemoattractant gradient direction, $\theta =0$, points to the right in the figure, and $\mathcal{P}\left(\theta \right)$ is represented as the distance from the center (the intersection of the two dotted lines) to the periphery of the red (light gray) region at angle $\theta$. From top to bottom: $\gamma \sim 7.1$, 3.0, 1.0, and 0.31 ($C_0/K_d=1$, 0.05, 0.005, and 0.0005). From left to right: $\varphi =0$, $\pi /6$, $\pi /3$, and $\pi /2$. The open black ellipses indicate cell shape and orientation. (q)–(t) The peak angles ${\theta }_{\mathrm{peak}}$ for which $\mathcal{P}\left(\theta \right)$ is maximal, plotted as functions of $\gamma$. From left to right: $\varphi =0$, $\pi /6$, $\pi /3$, and $\pi /2$. Other parameter values are $s=0.05$, $N=80000$, $r_0=1$ (where 1 distance $\text{unit}=10$ $\mu$m), and $a/b=3$ ($\epsilon \sim 0.94$).

Figure 3

Distribution of the estimated direction when the steepness is known. (a)–(p) ${\mathcal{P}}_s\left(\theta \right)$ for various concentration and cell orientations shown as the blue (dark gray) region. (q)–(t) The peak angles ${\theta }_{\mathrm{peak}}$ plotted as functions of $\gamma$. See Fig. 2 for details.