Optimal resource allocation in cellular sensing systems

Abstract

Living cells deploy many resources to sense their environments, including receptors, downstream signaling molecules, time, and fuel. However, it is not known which resources fundamentally limit the precision of sensing, like weak links in a chain, and which can compensate each other, leading to trade-offs between them. We present a theory for the optimal design of the large class of sensing systems in which a receptor drives a push-pull network. The theory identifies three classes of resources that are required for sensing: receptors and their integration time, readout molecules, and energy (fuel turnover). Each resource class sets a fundamental sensing limit, which means that the sensing precision is bounded by the limiting resource class and cannot be enhanced by increasing another class--the different classes cannot compensate each other. This result yields a previously unidentified design principle, namely that of optimal resource allocation in cellular sensing. It states that, in an optimally designed sensing system, each class of resources is equally limiting so that no resource is wasted. We apply our theory to what is arguably the best-characterized sensing system in biology, the chemotaxis network of Escherichia coli. Our analysis reveals that this system obeys the principle of optimal resource allocation, indicating a selective pressure for the efficient design of cellular sensing systems.

Keywords: cell signaling; chemotaxis; design principles; information transmission; thermodynamics.

Fig. 1.

The chemotaxis network of E. coli obeys the principle of optimal resource allocation, which states that in an optimally designed system each cellular resource is equally limiting. (A) Cartoon of the sensing system. The receptor is via the adaptor protein CheW associated with the kinase CheA. This complex, coarse-grained as R in our model, can bind extracellular ligand L and activate the intracellular messenger protein CheY (x in our model) by phosphorylating it; phosphorylated CheY controls the rotation direction of the motor. Deactivation, i.e., dephosphorylation, of CheY is catalyzed by the phosphatase CheZ; the effect of CheZ is coarse-grained into the deactivation rate. The proteins CheR and CheB, which implement adaptation, have been omitted, because we are interested in the lower bound on the accuracy of sensing in static environments. (B) The principle of optimal resource allocation, Eq. 5, predicts that the number of CheY proteins, $X_T$, scales linearly with the number of receptor–CheA complexes, $R_T$, with a slope given by the relaxation time of the signaling network, ${\tau }_r$, over the correlation time of the receptor ligand-binding state, ${\tau }_c$. Plotted are data from ref. for two E. coli strains under two different growth conditions; the number of CheA dimers is a proxy for the number of receptor–CheA complexes. The line is a best fit to the data, having a slope of $\approx 3$. The resource allocation principle, Eq. 5, thus predicts that ${\tau }_r/{\tau }_c\approx 3$. This is on the same order of magnitude as that given by the relaxation time, ${\tau }_r\approx 100\hspace{0.17em}\text{ms}$ (24), and correlation time ${\tau }_c\approx 10\hspace{0.17em}\text{ms}$, estimated from the measured receptor–ligand dissociation constant (25) and association rate (26).

Fig. 2.

Sensing at the molecular level. The sensing precision in terms of the rate constants $\left\{k_i\right\}$ (A) does not reveal the resource requirements (Eq. S8). To reveal these, the signaling network is viewed as a device that discretely samples the ligand-binding state of the receptor. The accuracy of sensing depends on how the samples are taken (B and C), erased (D), and on how reliable they are (E). (B) The ligand-bound receptor drives the modification of a downstream readout (i.e., the push–pull network $RL+x\to RL+x^{\ast }$). (C) The signaling network in B discretely samples the receptor state, illustrated for one receptor. The states of the receptor over time are encoded in the states of the N molecules that collided with it: the readout is modified if the receptor is bound; otherwise, it is unmodified. Molecules that collide with the unbound receptor are indistinguishable from those that have never collided, leading to an additional error. (D) Active molecules can be degraded, erasing samples. (E) All reactions are in principle reversible, compromising the encoding of the receptor state into the readout. The sensing error is determined by collective variables that reveal the resource requirements for sensing: the probability p that the receptor is bound to ligand, the receptor–ligand correlation time ${\tau }_c$, the flux $\dot{n}$, the relaxation time ${\tau }_r$, and the free-energy drops $\mathrm{\Delta }{\mu }_1$ and $\mathrm{\Delta }{\mu }_2$ across the activation and deactivation reactions of the readout, respectively.

Fig. 3.

Trade-offs in nonequilibrium sensing. (A) When two resources A and B compensate each other, one resource can always be decreased without affecting the sensing error, by increasing the other resource; concomitantly, increasing a resource will always reduce the sensing error. When both resources are instead fundamental, the sensing error is bounded by the limiting resource and cannot be reduced by increasing the other. (B and C) The three classes time/receptor copies, copies of downstream molecules, and energy are all required for sensing, with no trade-offs among them (Fig. 4). The minimum sensing error obtained by minimizing Eq. 2 is plotted for different combinations of (B) $X_T$ and w, and (C) $R_T\left(1+{\tau }_r/{\tau }_c\right)$ and w (SI Text). The curves track the bound for the limiting resource indicated by the gray lines, showing that the resources do not compensate each other. The plot for the minimum sensing error as a function of $R_T\left(1+{\tau }_r/{\tau }_c\right)$ and $X_T$ is identical to that of (C) with w replaced by $X_T$. (D) The energy requirements for sensing. In the irreversible regime $\left(\mathrm{\Delta }\mu \to \mathrm{\infty }\right)$, the work to take one sample of a ligand-bound receptor, $w/\left(p{\overline{N}}_{\text{eff}}\right)$, equals $\mathrm{\Delta }\mu$, because each sample requires the turnover of one fuel molecule, consuming $\mathrm{\Delta }\mu$ of energy. In the quasiequilibrium regime $\left(\mathrm{\Delta }\mu \to 0\right)$, each effective sample of the bound receptor requires $4{\mathit{k}}_{\mathit{B}}\mathit{T}$, which defines the fundamental lower bound on the energy requirement for taking a sample. When $\mathrm{\Delta }\mu =0$, the network is in equilibrium and both w and ${\overline{N}}_{\text{eff}}$ are 0. ATP hydrolysis provides $20{\mathit{k}}_{\mathit{B}}\mathit{T}$, showing that phosphorylation of readout molecules makes it possible to store the receptor state reliably. The results are obtained from Eq. 3 with $\mathrm{\Delta }{\mu }_1=\mathrm{\Delta }{\mu }_2=\mathrm{\Delta }\mu /2$. (E) Sampling more than once per correlation time requires more resources, although the benefit is marginal. As the sampling rate is increased by increasing the readout copy number $X_T$, the number of independent measurements ${\overline{N}}_I$ saturates at the Berg–Purcell limit $R_T{\tau }_r/{\tau }_c$, but the energy consumption and protein cost $\left(\propto X_T\right)$ continue to rise.

Fig. 4.

The relationship between resources and the precision of biochemical sensing. The sensing precision is fundamentally limited by time and receptor copies, energy, and copies of downstream readouts. These three classes of resources cannot compensate each other, and it is the limiting resource that sets the fundamental limit to the precision of sensing. Within each class, however, trade-offs are possible: Power can be traded against speed to meet the energy requirement for reaching a desired sensing accuracy, whereas time can be traded against the number of receptors.