Extracellular Processing of Molecular Gradients by Eukaryotic Cells Can Improve Gradient Detection Accuracy

Eukaryotic cells sense molecular gradients by measuring spatial concentration variation through the difference in the number of occupied receptors to which molecules can bind. They also secrete enzymes that degrade these molecules, and it is presently not well understood how this affects the local gradient perceived by cells. Numerical and analytical results show that these enzymes can substantially increase the signal-to-noise ratio of the receptor difference and allow cells to respond to a much broader range of molecular concentrations and gradients than they would without these enzymes.

Figure 1

Signal-to-noise ratio (SNR) and model geometry. (a) The SNR is defined as the receptor occupancy difference between the front half and back half of the cell (signal) divided by the noise consisting of receptor shot noise ${\sigma }_{\mathrm{\Delta }R}$ sampled $I$ times and nonreceptor noise ${\sigma }_B$. (b) When the relative gradient $|\overrightarrow{\nabla }c|/c=\text{const}$, the optimal average concentration that maximizes the SNR is $c=K_d$, since the receptor occupancy difference $\mathrm{\Delta }R$ has a maximum when $c=K_d$ and $\mathrm{SNR}\propto \sqrt{\mathrm{\Delta }R}$. (c) Geometry and boundary conditions for 3D numerical simulations; $c=\mathrm{cAMP}$, $p=\mathrm{PDE}$ concentration (not to scale). Constant relative gradient can be set away from the cell (in the bulk), by changing the cAMP concentration on the left boundary $c(-w/2)$, while keeping $c(w/2)=0$. $w=1\mathrm{mm}$.

Figure 2

Fixed PDE secretion flux model with relative gradient on the cell surface of 1%. (a) SNR as a function of PDE secretion flux $P_0$ and cAMP boundary concentration $c(-w/2)$. The SNR can be substantially improved by PDE for a range of parameters $P_0$ and $c(-w/2)$. (b) Horizontal slices from (a). PDE can increase the SNR for $c(-w/2)\gtrsim K_d$. (c) Vertical slices from (a). Increasing $P_0$ shifts the optimal cAMP concentration (maximizing SNR) towards higher values but also broadens the detectable range of cAMP concentrations. The red curve for $P_0={10}^{-14}\mathrm{mol}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}\approx 0$, i.e., matches the SNR in Fig. 1 with $c=c(-w/2)/2$. (d) Broadening of curves from (c) is quantified by calculating the range of $c(-w/2)$ for which $\mathrm{SNR}\ge 1$. (e) Chemotaxis index (CI) calculated as $\mathrm{SNR}/(\mathrm{SNR}+1)$ ([35], §3). (f) Vertical slices from (e).

Figure 3

Signal and noise analysis of the fixed PDE secretion flux model, showing that most of the SNR enhancement results from the increase in signal $\mathrm{\Delta }R$. (a) Mean cAMP concentration $c_0$ and gradient $|\overrightarrow{\nabla }c|$ across the cell surface, as a function of PDE secretion flux $P_0$. When rescaled by their values for $P_0=0$, concentration and gradient curves do not depend on $c(-w/2)$. (b) Signal $\mathrm{\Delta }R(P_0)\propto |\overrightarrow{\nabla }c|/(c_0+K_d{)}^2$. Color legend is the same as in (c). (c) Noise $\sqrt{{\sigma }_B^2+{\sigma }_{\mathrm{\Delta }R}^2/I}$, with lower limit ${\sigma }_B=73$ and upper limit $\sqrt{{\sigma }_B^2+N/(4I)}\approx 134$. (d) Ratio of SNR and SNR with $P_0=0$.

Figure 4

Analytical solutions for the SNR for a fixed PDE concentration model and a cAMP point source model in 3D, showing a similar enhancement. (a),(b) Microfluidic geometry: SNR as a function of PDE concentration $p_0$ and cAMP concentration on the left boundary $c(-w/2)$. (c),(d) cAMP point source $C_0\delta (r)$ located at $\overrightarrow{r}=0$: SNR as a function of cAMP source strength $C_0$ and $p_0$ at a distance $r=225\mu \mathrm{m}$ from the source.