Properties of cooperatively induced phases in sensing models

A large number of eukaryotic cells are able to directly detect external chemical gradients with great accuracy and the ultimate limit to their sensitivity has been a topic of debate for many years. Previous work has been done to understand many aspects of this process but little attention has been paid to the possibility of emergent sensing states. Here we examine how cooperation between sensors existing in a two-dimensional network, as they do on the cell's surface, can both enhance and fundamentally alter the response of the cell to a spatially varying signal. We show that weakly interacting sensors linearly amplify the cell's response to an external gradient while a network of strongly interacting sensors form a collective nonlinear response with two separate domains of active and inactive sensors forming what have called a “1/2-state.” In our analysis we examine the cell's ability to sense the direction of a signal and pay special attention to the substantially different behavior realized in the strongly interacting regime.

Figure 1

A schematic of the cell in a chemical gradient.

Figure 2

The figures were made by placing 5000 sensors in a spatially varying field with strength $p$ at $H_0=0$. Panel (a) shows the activity of the sensors $\langle S\left(\varphi \right)\rangle$ in the plane of an applied external field from $\varphi =0:\pi$ at a fixed gradient steepness of $p=10\%$. Panel (b) shows the average activity asymmetry $M_1$ along the applied gradient direction (normalized by the number of sensors $N$) as a function of the applied gradient strength $p$. In both figures the points are from Monte Carlo simulations while the dotted curves correspond to the linear response (average activity) $\langle S_L\left(\varphi \right)\rangle$ and the solid curves to the collective “1/2-state” response $\langle S_C\left(\varphi \right)\rangle .$

Figure 3

Panel (a) is the uncertainty ${\sigma }_{\widehat{\phi }}^2$ in direction for changing values of asymmetry magnitude $p$ at $H_0=0$. Panel (b) is the uncertainty ${\sigma }_{\widehat{\phi }}^2$ in direction for varying background field levels $H_0=\frac{1}{2}ln\frac{C_0}{K_d}$ at a fixed asymmetry magnitude of $p=10\%$. In both figures the circles, crosses, and triangles are Monte Carlo results for noninteracting, weakly interacting, and strongly interacting (“1/2 state”) systems, respectively, while the dashed and solid lines are the analytical results for the systems with the corresponding interaction strengths (shown in the figure).

Figure 4

Legendre coefficient $c_1\left(J\right)$ for response function expansion as a function of interaction strength $J>H_c$.

Figure 5

Integration constant $f\left(A\right)$ for “1/2 state.”

Figure 6

The formation of a “1/2 state” is shown by watching the receptor asymmetry in time following a shift in field values from $H_0=0.1,p=0$ for $t<0$ to $H_0=0.1,p=0.2$ and held constant for $t>0$. Light lines are individual simulations and dark lines averages each shown to reveal the stochastic nature of the “1/2 state” formation in the strongly coupled regime.

Figure 7

The dynamic rotation of a “1/2 state” in a field of $H_0=0.1,p=0.2$ is shown by watching the receptor asymmetry in time following a shift in field directions from $\widehat{\phi }=0\to \frac{\pi }{4}$ at $t=0$. Light lines are individual simulations and dark lines averages. It is clear from the adaptability and lack of hysteresis that the “1/2-state” dynamics are collective in nature and operate on a time scale distinct from individual receptors.