## Abstract

The reliable detection of environmental molecules in the presence of noise is an important cellular function, yet the underlying computational mechanisms are not well understood. We introduce a model of two interacting sensors which allows for the principled exploration of signal statistics, cooperation strategies and the role of energy consumption in optimal sensing, quantified through the mutual information between the signal and the sensors. Here we report that in general the optimal sensing strategy depends both on the noise level and the statistics of the signals. For joint, correlated signals, energy consuming (nonequilibrium), asymmetric couplings result in maximum information gain in the low-noise, high-signal-correlation limit. Surprisingly we also find that energy consumption is not always required for optimal sensing. We generalise our model to incorporate time integration of the sensor state by a population of readout molecules, and demonstrate that sensor interaction and energy consumption remain important for optimal sensing.

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## Introduction

Cells are surrounded by a cocktail of chemicals, which carry important information, such as the number of nearby cells, the presence of foreign material, and the location of food sources and toxin. The ability to reliably measure chemical concentrations is thus essential to cellular function. In fact, cells can exhibit extremely high sensitivity in chemical sensing, for example, our immune response can be triggered by only one foreign ligand^{1} and *Escherichia coli* chemotaxis responds to nanomolar changes in chemical concentration^{2}. But how does cellular machinery achieve such sensitivity?

One strategy is to consume energy: molecular motors metabolise ATPs to drive cell movement and cell division, and kinetic proofreading employs nonequilibrium biochemical networks to increase enzyme–substrate specificity^{3}. Indeed, the role of energy consumption in enhancing the sensitivity of chemosensing is the subject of several studies^{4,5,6,7,8}. However, whether nonequilibrium sensing can supersede equilibrium limits to performance is unknown^{9,10}.

Interactions also directly influence sensitivity, and receptor cooperativity is a biologically plausible strategy for suppressing noise^{11,12,13}. These results, however, apply in steady state^{11} and it is independent receptor that maximise the signal-to-noise ratio under a finite integration time^{14,15} even when receptor interactions are coupled to energy consumption^{16}. More generally, a trade-off exists between noise-reduction and available resources, such as integration time and the number of readout molecules^{6,7}. It is therefore important to examine how sensor circuit sensitivity depends on the level of noise and the structure of the signals without a priori fixing the interactions or the energy consumption.

We introduce a general model for nonequilibrium coupled binary sensors. Specialising to the case of two sensors, we obtain the steady state distribution of the two-sensor states for a specified signal. We then determine the sensing strategy that maximises the mutual information for a given noise level and signal prior. We find that the optimal sensing strategy depends on both the noise level and signal statistics. In particular, energy consumption can improve sensing performance in the low-noise, high-signal-correlation limit but is not always required for optimal sensing. Finally, we generalise our model to include time averaging of the sensor state by a population of readout molecules, and show that optimal sensing remains reliant on sensor interaction and energy consumption.

## Results

### Model overview

We consider a simple system of two information processing units (sensors), an abstraction of a pair of coupled chemoreceptors or two transcriptional regulations with cross-feedback (Fig. 1a). The sensor states depend on noises, signals (e.g., chemical changes) and sensor interactions, which can couple to energy consumption. Instead of the signal-to-noise ratio, we use the mutual information between the signals and the states of the system as the measure of sensing performance. Physically, the mutual information corresponds to the reduction in the uncertainty (entropy) of the signal (input) once the system state (output) is known. In the absence of signal integration, the mutual information between the signals and sensors is also the maximum information the system can learn about the signals as noisy downstream networks can only further degrade the signals. However, computing mutual information requires the knowledge of the prior distribution of the signals. Importantly, the prior encodes some of the information about the signal, e.g., signals could be more likely to take certain values or drawn from a set of discrete values. Although the signal prior in cellular sensing is generally unknown, one simple, physically plausible choice is the Gaussian distribution, which is the least informative distribution for a given mean and variance.

### Nonequilibrium coupled sensors

We provide an overview of our model in Fig. 1b. Here, a sensor complex is a network of interacting sensors, each endowed with binary states *s* = ±1, e.g., whether a receptor or gene regulation is active. The state of each sensor depends on that of the others through interactions, and on the local bias fields generated by a signal; for example, an increase in ligand concentration favours the occupied state of a chemoreceptor. Owing to noise, the sensor states are not deterministic so that the probability of every state is finite. We encode the effects of signals, interactions and intrinsic noise in the inversion rate—the rate at which a sensor switches its state. We define the inversion rate for the *i*th sensor

where *S* = {*s*_{i}} denotes the present state of the sensor system, *H* = {*h*_{i}} the signal, *J*_{ij} the interactions, and *β* the sensor reliability (i.e., the inverse intrinsic noise level). The transition rate determines the lifetime, and thus the likelihood, of each state *S*. In the above form, the coupling to the signal, *h*_{i}*s*_{i}, favours alignment between the sensor *s*_{i} and the signal *h*_{i}, whereas the interaction *J*_{ij} > 0 (*J*_{ij} < 0) encourages correlation (anticorrelation) between the sensors *s*_{i} and *s*_{j}. The constant \({{\mathcal{N}}}_{H}\) sets the overall timescale but drops out in steady state, which is characterised by the ratios of the transition rates. In the context of chemosensing, the signal {*h*_{i}} parametrises the concentration change of one type of ligand when all sensors in the sensing complex respond to the same chemical, and of multiple ligands when the sensors exhibit different ligand specificity.

Given an input signal *H*, the conditional probability of the states of the sensor complex in steady state *P*_{S∣H} is obtained by balancing the probability flows into and out of each state while conserving the total probability ∑_{S}*P*_{S∣H} = 1,

Here and in the following, the state vector *S*^{i} is related to *S* by the inversion of the sensor *i*, *s*_{i} → −*s*_{i}, whereas all other sensors remain in the same configuration.

In equilibrium, detailed balance imposes an additional constraint forbidding net probability flow between any two states,

and this condition can only be satisfied by symmetric interactions *J*_{ij} = *J*_{ji} (see, Coupling symmetry and detailed balance in Methods). We define the equilibrium free energy

such that the inversion rate depends on the initial and final states of the system only through the change in free energy

Together with the detailed balance condition (Eq. (3)), this equation leads directly to the Boltzmann distribution \({P}_{S| H}^{\,\text{eq}\,}={e}^{-\beta {F}_{S| H}}/{{\mathcal{Z}}}_{H}^{\,\text{eq}\,}\) with the partition function \({{\mathcal{Z}}}_{H}^{\,\text{eq}\,}\). When constrained to equilibrium couplings, this model has been previously investigated in the context of optimal coding by a network of spiking neurons^{17}. Asymmetric interactions *J*_{ij} ≠ *J*_{ji} break detailed balance, resulting in a nonequilibrium steady state (see, Coupling symmetry and detailed balance in Methods).

We specialise to the case of two coupled sensors *S* = (*s*_{1}, *s*_{2}), belonging to one of the four states: −−, −+, ++ and +− (Fig. 1b). For convenience, we introduce two new variables, the coupling *J* and nonequilibrium drive *t*, and parametrise *J*_{12} and *J*_{21} such that *J*_{21} = *J* − *t* ∕ 2 and *J*_{12} = *J* + *t* ∕ 2 (Fig. 1b). The effects of the bias fields (*h*_{1}, *h*_{2}), coupling *J* and nonequilibrium drive *t* are summarised in Fig. 1c. Compared with the equilibrium inversion rate [Eq. (5)], a finite nonequilibrium drive leads to a modification of the form

where *S* → *S*^{i} is cyclic if it corresponds to one of the transitions in the cycle −− → −+ → ++ → +− → −−, and anticyclic otherwise. Recalling that this probability flow vanishes in equilibrium, it is easy to see that, depending on whether *t* is positive or negative, the nonequilibrium inversion rates result in either cyclic or anticyclic steady state probability flow.

A net probability flow in steady state leads to power dissipation. By analogy with Eq. (5), we write down the effective change in free energy of a transition *S* → *S*^{i},

That is, the system loses energy of 4*t* per complete cycle. To conserve total energy, the sensor complex must consume the same amount of energy it dissipates to the environment. The nonequilibrium drive also modifies the steady state probability distribution. Solving Eq. (2), we have (see also, Steady state master equation in Methods)

where *F*_{S∣H} denotes the free energy in equilibrium [Eq. (4)]. The nonequilibrium effects are encoded in the noise-dependent term

and note that *δ**F*_{S∣H} → 0 as *t* → 0.

### Mutual information

We quantify sensing performance through the mutual information between the signal and sensor complex *I*(*S*; *H*), which measures the reduction in the uncertainty (entropy) in the signal *H* once the system state *S* is known and vice versa. For convenience, we introduce the “output” and “noise” entropies where output entropy is the entropy of the two-sensor state distribution \({\mathcal{S}}[{P}_{S}]={\mathcal{S}}[{\sum }_{H}{P}_{H}{P}_{S| H}]\), whereas the noise entropy is defined as the average entropy of the conditional probability of sensor states \({\sum }_{H}{P}_{H}{\mathcal{S}}[{P}_{S| H}]\). Here, *P*_{H} is the prior distribution from which a signal is drawn and the entropy of a distribution is defined by \({\mathcal{S}}[{P}_{X}]=-{\sum }_{X}{P}_{X}{\mathrm{log}\,}_{2}{P}_{X}\). In terms of the output and noise entropies, the mutual information is given by

and we seek the sensing strategy (the coupling *J* and nonequilibrium drive *t*) that maximises the mutual information for given reliability *β* and signal priors *P*_{H}. In practice, we solve this optimisation problem by a numerical search in the *J*–*t* parameter space using standard numerical-analysis software (see, Code availability for an example code for numerical optimisation of mutual information).

### Correlated signals

The bias fields at two sensors are generally different, for example, chemoreceptors with distinct ligand specificity or exposure, and we consider signals *H* = (*h*_{1}, *h*_{2}), drawn from a correlated bivariate Gaussian distribution (Fig. 2a),

where *α* ∈ [−1, 1] is the correlation between *h*_{1} and *h*_{2}. When we maximise the mutual information in the *J*–*t* parameter space, we find that the mutual information is maximised by an equilibrium system (*t*^{*} = 0) for small *β*. In Fig. 2b, we show that the optimal strategy is cooperative (*J* > 0) at small *β* and switches to anticooperative (*J* < 0) around *β* ~ 1. Below a certain value of *β*, the optimal coupling diverges *J*^{*} → *∞* (region I in Fig. 2b). In addition, sensor cooperativity is less effective for less-correlated signals because a cooperative strategy relies on output suppression (which reduces both noise and output entropies). This strategy works well for more correlated signals as they carry less information (low signal entropy), which can be efficiently encoded by fewer output states. Thus, a reduction in noise entropy increases mutual information despite the decrease in output entropy. This is not the case for less-correlated signals, which carry more information (higher entropy) and which require more output states to encode effectively. As sensors become less noisy, the optimal strategy is nonequilibrium (*t*^{*} ≠ 0; region III in Fig. 2b) only when the signal redundancy, i.e., the mutual information between the input signals *I*(*h*_{1}; *h*_{2}), is relatively high. The sensing strategies *t* = ±∣*t*^{*}∣ are time-reversed partners of one another both of which yield the same mutual information. This symmetry results from the fact that the signal prior *P*_{H} (Eq. (10)) is invariant under *h*_{1} ↔ *h*_{2}, hence the freedom in the choice of dominant sensor (i.e., we can either make *J*_{12} > *J*_{21} or *J*_{12} < *J*_{21}).

Although Fig. 2b shows the results for positively correlated signals (*α* > 0), the optimal sensing strategies for anticorrelated signals (*α* < 0) exhibit the same dependence on sensor reliability and the signal redundancy but with the same nonequilibrium drive *t* = ±∣*t*^{*}∣ and the optimal coupling *J*^{*} that is opposite to that in the case *α* > 0.

### Perfectly correlated signals

To understand the mechanisms behind the optimal sensing strategy for correlated signals, we consider the limiting case of completely redundant Gaussian signals (*h*_{1} = *h*_{2}), which captures most of the phenomenology depicted in Fig. 2b. We find that nonequilibrium drive allows further improvement on equilibrium sensors only for *β* > 1.7 and that the nonequilibrium gain remains finite as *β* → *∞* (Fig. 3a). In Fig. 3b, we show the optimal parameters for both equilibrium and nonequilibrium sensing. The optimal coupling diverges for sensors with *β* < 1, decreases with increasing *β* and exhibits a sign change at *β* = 1.4. For *β* > 1.7, the nonequilibrium drive is finite and the couplings are distinct \({J}^{* }\; \ne\; {J}_{\,\text{eq}\,}^{* }\! \).

Figure 4a compares the output and noise entropies of equilibrium and nonequilibrium sensing at optimal with that of noninteracting sensors for a representative *β* = 4. Here, anticooperativity (\({J}_{\,\text{eq}\,}^{* }\,<\, 0\)) enhances mutual information in equilibrium sensing by maximising the output entropy, whereas nonequilibrium drive produces further improvement by lowering noise entropy. Compared with the noninteracting case Fig. 4b, optimal equilibrium sensing distributes the probability of the output states, *P*_{S}, more evenly (Fig. 4c), resulting in higher output entropy. This is because a negative coupling *J* < 0 favours the states +− and −+, which are much less probable than ++ and −− in a noninteracting system subject to perfectly correlated signals (Fig. 4b). However, this also leads to higher noise entropy, as the states +− and −+ are equally likely for a given signal (Fig. 4c). By lifting the degeneracy between the states +− and −+, nonequilibrium sensing suppresses noise entropy while maintaining a relatively even distribution of output states, Fig. 4d. As a result, for signals with low redundancy (*I*(*h*_{1}; *h*_{2}) ≲ 1), a nonequilibrium strategy allows no further improvement (Fig. 2b) because the states of a sensor complex are less likely to be degenerate (since the probability that *h*_{1} ≈ *h*_{2} is smaller).

Although anticooperativity increases output entropy more than noise entropy at *β* = 4, it is not the optimal strategy for *β* < 1.4. For noisy sensors, a positive coupling *J* > 0 yields higher mutual information, Fig. 3b. This is because when the noise level is high, the output entropy is nearly saturated and an increase in mutual information must result primarily from the reduction of noise entropy by suppressing some output states–in this case, the states +− and −+ are suppressed by *J* > 0.

We emphasise that the nonequilibrium gain is not merely a result of an additional sensor parameter. Instead of nonequilibrium drive we can introduce signal-independent biases on each sensor and keep the entire sensory complex in equilibrium. Such intrinsic biases can lower noise entropy in optimal sensing by breaking the degeneracy between the states +− and −+ (Fig. 4b, c). However, favouring one sensor state over the other results in a greater decrease in output entropy and hence lower mutual information (see, Supplementary Fig. 1).

Our analysis does not rely on the specific Gaussian form of the prior distribution. Indeed, for correlated signals the nonequilibrium improvement in the low-noise, high-correlation limit is generic for most continuous priors (see, Supplementary Fig. 2).

### Time integration

Cells do not generally have direct access to the receptor state. Instead, chemosensing relies on downstream readout molecules whose activation and decay couple to sensor states. Repeated interactions between receptors and a readout population provide a potential noise-reduction strategy through time averaging, which can compete with sensor cooperativity and energy consumption^{14,16}. We generalise our model to incorporate time integration of the sensor state by a population of readout molecules and demonstrate that sensor coupling and nonequilibrium drive remains essential to optimal sensing.

We consider a system of binary sensors *S*, coupled to signals *H* and a readout population *r*. We expand our original model (Nonequilibrium coupled sensors) to include readout activation (*r* → *r* + 1) and decay (*r* → *r* − 1), resulting in modified transition rates:

where *S* → *S*^{i} denotes sensor inversion *s*_{i} → − *s*_{i}, *r* the readout population, *b*_{i} the sensor-specific bias and \({{\mathcal{N}}}_{H}\) the overall timescale constant. We also introduce the sensor-dependent differential readout potential *δ**μ*_{i}; the sensor state \({s}_{i}={\rm{sgn}}(\delta {\mu }_{i})\) favours a larger readout population (by increasing activation rate and suppressing decays), whereas \({s}_{i}=-{\rm{sgn}}(\delta {\mu }_{i})\) biases the readout towards a smaller population. This allows the readout population to store samplings of sensor states over time, thus providing a physical mechanism for time integration of sensor states. The readout population in turn affects sensor inversions: the larger the readout population, the more favourable \({s}_{i}={\rm{sgn}}(\delta {\mu }_{i})\) over \({s}_{i}=-{\rm{sgn}}(\delta {\mu }_{i})\). This two-way interplay between sensors and readouts is essential for a consistent equilibrium description, for one-way effects (e.g., sampling sensor states without altering the sensor complex) require Maxwell’s demon—a nonequilibrium process not described by the model.

We further assume a finite readout population *r* ≤ *r*_{0}, and that the readout activation and decay are intrinsically stochastic. Consequently, a readout population has a limited memory for past sensor states. Indeed, readout stochasticity sets a timescale beyond, which an increase in measurement time cannot improve sensing performance^{6}. To investigate this fundamental limit, we let the measurement time be much longer than any stochastic timescale. In this case, the sensor-readout joint distribution *P*_{r,S∣H} satisfies the steady state master equation with the transition rates in Eqs. (11), (12) (see, Steady state master equation in Methods).

When *J*_{ij} = *J*_{ji}, the steady state distribution obeys the detailed balance condition (see, Coupling symmetry and detailed balance in Methods) and is given by

with the free energy

This distribution results in \({P}_{r| S,H}^{\,\text{eq}}={P}_{r| S}^{\text{eq}\,}\), implying a Markov chain *H* → *S* → *r* (*H* affects *r* only via *S*), hence the data processing inequality *I*(*S*; *H*) ≥ *I*(*r*; *H*). That is, in equilibrium, time integration of sensor states cannot produce a readout population that contain more information about signals than the sensor states^{7} (see, also refs. ^{18,19}). This result applies to any equilibrium sensing complexes (see, Equilibrium time integration in Methods).

We now specialise to the case of two coupled sensors *S* = (*s*_{1}, *s*_{2}) and introduce two new variables, Δ and *δ*, defined via

For this parametrisation, the effective chemical potentials for readout molecules are given by

where *μ*_{S} = ∑_{i}*δ**μ*_{i}*s*_{i}. We see that *Δ* (*δ*) parametrises how different the sensor states ++ and −− ( +− and −+ ) appear to the readout population. To maximise utilisation of readout states, we set the sensors biases to *b*_{i} = *δ**μ*_{i}*r*_{0} ∕ 2 where *r*_{0} is the maximum readout population. For optimal equilibrium sensing, we maximise mutual information *I*(*r*; *H*) by varying *J* and *δ* under the constraint *t* = 0, and the noninteracting case corresponds to *J* = *t* = *δ* = 0.

Figure 5a depicts the readout-signal mutual information for perfectly correlated Gaussian signals for the cases of optimal equilibrium sensing (solid) and noninteracting sensors (dashed). Independent sensors are suboptimal for all *β*, i.e., we can always increase mutual information by tuning *δ* and *J* away from zero. At maximum mutual information we find *δ* = 0 and *J* ≠ 0. In Fig. 5b, we show the mutual information as a function of *J* (at *δ* = 0). We see that \({J}_{\,\text{eq}\,}^{* }\to \infty\) in the noisy limit (low *β*) and \({J}_{\,\text{eq}\,}^{* }\,<\, 0\) in the low-noise limit (high *β*). This crossover from cooperativity to anticooperativity is consistent with our results in Correlated signals and Perfectly correlated signals (see also, Figs. 2 and 3b).

To reveal the mechanism behind optimal equilibrium sensing in the low-noise limit, we examine the joint probability distribution *P*(*r*, *h*) at *β* = 4 for noninteracting (Fig. 5c) and optimal equilibrium sensors (Fig. 5d). We see that anticooperativity increases mutual information by distributing the output (readout) states more efficiently (cf. Fig. 4b, c). Noninteracting sensors partition outputs into large and small readout populations, which corresponds to positive and negative signals, respectively. This is because correlated signals favour the chemical potentials *μ*_{++} = − *μ*_{−−} = Δ, which bias the readout population towards *r* = 0 and *r* = *r*_{0}, and suppress *μ*_{+−} = −*μ*_{−+} = *δ* = 0, which encourage evenly distributed readout states. By adopting an anticooperative strategy (*J* < 0) to counter signal correlation, optimal equilibrium sensors can use more output states (on average) to encode the signal. The increase in accessible readout states also raises noise entropy, but the increase in output entropy dominates in the low-noise limit, resulting in higher mutual information.

Finally, we demonstrate that energy consumption can further enhance sensing performance. Figure 5e shows *P*(*r*, *h*) for a nonequilibrium sensor complex. We see that nonequilibrium drive lifts the degeneracy in intermediate readout states (0 < *r* < *r*_{0}), leading to a much more effective use of output states. For the nonequilibrium sensor complex in Fig. 5e, we obtain *I*_{neq}(*r*; *h*) = 1.75 bits, compared with \({I}_{\,\text{eq}\,}^{* }(r;h)=0.96\) bits for optimal equilibrium sensors (Fig. 5a, b) at the same sensor reliability (*β* = 4). We note that this nonequilibrium gain relies also on *δ* ≠ 0 to distinguish the sensor states +− and −+. The staircase of readout states in Fig. 5e corresponds to the anticorrelated sensor states +− and −+ which do not always favour higher readouts at positive signals (see also, Supplementary Figs. 3, 4).

## Discussion

We introduce a minimal model of a sensor complex that encapsulates both sensor interactions and energy consumption. For correlated signals, we find that sensor interactions can increase sensing performance of two binary sensors, as measured by the steady state mutual information between the signal and the states of the sensor complex.

This result highlights sensor cooperativity as a biologically plausible sensing strategy^{11,12,13}. However, the nature of the optimal sensor coupling does not always reflect the correlation in the signal; for positively correlated signals, the optimal sensing strategy changes from cooperativity to anticooperativity as the noise level decreases, see also ref. ^{17}. Anticooperativity emerges as the optimal strategy through countering the redundancy in correlated signals by suppressing correlated outputs, and thus redistributing the output states more evenly. The same principle also applies to population coding in neural networks^{17}, positional information coding by the gap genes in the *Drosophila* embryo^{20,21,22} and time-telling from multiple readout genes^{23}. Surprisingly, we find that energy consumption leads to further improvement only when the noise level is low and the signal redundancy high.

We find that sensor coupling and energy consumption remain important for optimal sensing under time integration of the sensor state—a result contrary to earlier findings that a cooperative strategy is suboptimal even when sensor interaction can couple to nonequilibrium drive^{14,16}. This discrepancy results from an assumption of continuous, deterministic time integration that requires an infinite supply of readout molecules and external nonequilibrium processes, and which also leads to an underestimation of noise in the output; we make no such assumption in our model. In addition, we use the data processing inequality to show for any sensing system that time integration cannot improve sensing performance unless energy consumption is allowed either in sensor coupling or downstream networks (see also refs. ^{5,7}).

Our work highlights the role of signal statistics in the context of optimal sensing. We show that a signal prior distribution is an important factor in determining the optimal sensing strategy as it sets the amount of information carried by a signal. With a signal prior, we quantify sensing performance by mutual information, which is a generalisation of linear approximations used in previous works^{6,7,8,11,14,16}.

To focus on the effects of nonequilibrium sensor cooperation, we neglect the possibility of signal crosstalk and the presence of false signals. Limited sensor-signal specificity places an additional constraint on sensing performance^{24} (but see, ref. ^{25}). Previous works have shown that kinetic proofreading schemes^{3} can mitigate this problem for isolated chemoreceptors that bind to correct and incorrect ligands^{26,27}. Our model can be easily generalised to include crosstalk, and it would be interesting to investigate whether nonequilibrium sensor coupling can provide a way to alleviate the problem of limited specificity.

Although we considered a simple model, our approach provides a general framework for understanding collective sensing strategies across different biological systems from chemoreceptors to transcriptional regulation to a group of animals in search of mates or food. In particular, possible future investigations include the mechanisms behind collective sensing strategies in more complex, realistic models, non-binary sensors, adaptation, and generalisation to a larger number of sensors. It would also be interesting to study the channel capacity in the parameter spaces of both the sensor couplings and the signal prior, an approach that has already led to major advances in the understanding of gene regulatory networks^{28}. Finally, the existence of optimal collective sensing strategies necessitates a characterisation of the learning rules that gives rise to such strategies.

## Methods

### Steady state master equation

The steady state probability distribution satisfies a linear matrix equation

where *p*_{j} denotes the probability of the state *j*. The matrix *W* is defined such that

where Γ_{j→i} denotes the transition rate from the states *j* to *i* and Γ_{j→j} = 0. The solution of Eq. (16) corresponds to a direction in the null space of the linear operator *W*. For two coupled sensors considered in Nonequilibrium coupled sensors, Eq. (16) (Eq. (2)) becomes a set of four simultaneous equations, which we solve analytically for a solution (Eq. (7)) that also satisfies the constraints of a probability distribution (∑_{j}*p*_{j} = 1, \({p}_{i}\in {\mathbb{R}}\) and *p*_{i} ≥ 0 for all *i*). In a larger system an analytical solution is not practical. For example, in ‘Time integration’ we consider a 44-state system of two sensors with at most 10 readout molecules. In this case we obtain the null space of the matrix *W* from its singular value decomposition, which can be computed with a standard numerical software.

### Coupling symmetry and detailed balance

Here we show that the transition rates in Eq. (1) does not satisfy Kolmogorov’s criterion—a necessary and sufficient condition for detailed balance—unless *J*_{ij} = *J*_{ji}. Consider the sensors *s*_{i} and *s*_{j} in a sensor complex *S* = {*s*_{1}, *s*_{2}, …, *s*_{N}} with *N* > 1. This sensor pair admits four states (*s*_{i}, *s*_{j}) = −−, −+, ++, +−. The transitions between these states form two closed sequences in opposite directions

Kolmogorov’s criterion requires that, for any closed loop, the product of all transition rates in one direction must be equal to the product of all transition rates in the opposite direction—i.e., \({\Gamma }_{a}{\Gamma }_{b}{\Gamma }_{c}{\Gamma }_{d}={\Gamma }_{a^{\prime} }{\Gamma }_{b^{\prime} }{\Gamma }_{c^{\prime} }{\Gamma }_{d^{\prime} }\). For the rates in Eq. (1), we have

therefore, only symmetric interactions satisfy Kolmogorov’s criterion and any asymmetry in sensor coupling necessarily breaks detailed balance. This result holds also for the generalised transition rates in Eqs. (11), (12).

### Equilibrium time integration

Following the analysis in Supplemental Material of ref. ^{7}, we provide a general proof that equilibrium time integration of receptor states cannot generate readout populations that contain more information about the signals than the receptors. We consider a system of receptors *S* = (*s*_{1}, *s*_{2}, …, *s*_{N}), driven by signal *H* = (*h*_{1}, *h*_{2}, …, *h*_{N}) and coupled to readout populations *R* = (*r*_{1}, *r*_{2}, …, *r*_{M}). In equilibrium, this system is described by a free energy

where *f*(*H*, *S*) and *g*(*S*, *R*) describe signal-sensor and sensor-readout couplings, respectively, and include interactions among sensors and readout molecules. The Boltzmann distribution for sensors and readouts reads

with the partition function *Z*_{H}. Therefore, we have

where the summation is over all readout states. This allows the decomposition of the joint distribution, *P*_{R,S∣H} = *P*_{R∣S,H}*P*_{S∣H} = *P*_{R∣S}*P*_{S∣H}, which implies a Markov chain *H* → *S* → *R*—that is, *H* affects *R* only through *S*. (This does not mean *R* does not affect *S*, for *P*_{S∣R,H} still depends on *R*.) From the data processing inequality, it immediately follows that *I*(*H*; *S*) ≥ *I*(*H*; *R*). We emphasise that this constraint applies to any equilibrium sensor complex and downstream networks, which can be described by the free energy in Eq. (20), regardless of the numbers of sensors and readout species, sensor characteristics (e.g., number of states), sensor-signal and sensor-readout couplings (including crosstalk), and interactions among sensors and readout molecules.

## Data availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

## Code availability

A Mathematica code for computing the optimal sensor parameters (*J*^{*}, *t*^{*}) that maximise the mutual information between two coupled sensors and correlated Gaussian signals is available at https://github.com/vn232/NeqCoopSensing.

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## Acknowledgements

We are grateful to Gašper Tkačik and Pieter Rein ten Wolde for useful comments and a critical reading of the manuscript. V.N. acknowledges support from the National Science Foundation under Grants DMR-1508730 and PHY-1734332, and the Northwestern-Fermilab Center for Applied Physics and Superconducting Technologies. G.J.S. acknowledges research funds from Vrije Universiteit Amsterdam and OIST Graduate University. D.J.S. was supported by the National Science Foundation through the Center for the Physics of Biological Function (PHY-1734030) and by a Simons Foundation fellowship for the MMLS. This work was partially supported by the National Institutes of Health under award number R01EB026943 (V.N. and D.J.S.).

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V.N., D.J.S., and G.J.S. conceived the study, interpreted the results and wrote the manuscript. V.N. performed numerical calculations.

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Ngampruetikorn, V., Schwab, D.J. & Stephens, G.J. Energy consumption and cooperation for optimal sensing.
*Nat Commun* **11**, 975 (2020). https://doi.org/10.1038/s41467-020-14806-y

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DOI: https://doi.org/10.1038/s41467-020-14806-y

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