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# Limits to the precision of gradient sensing with spatial communication and temporal integration

Edited by Herbert Levine, Rice University, Houston, TX, and approved October 8, 2015 (received for review May 18, 2015)

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## Significance

Knowing which way to move is crucial for many biological processes, from organismal development to migration of cancer cells and from motion of microbes to wound healing. To find their preferred directions, biological systems compare concentrations of a chemical cue at their different edges. The comparison requires information from these different locations to be communicated to the same place. However, all communication is noisy (just think of the childhood game “telephone”). This communication noise, as well as noise in the individual measurements themselves, sets the accuracy of the direction sensing. Here we quantify the importance of the communication noise and propose a mechanism that can improve the accuracy of direction sensing.

## Abstract

Gradient sensing requires at least two measurements at different points in space. These measurements must then be communicated to a common location to be compared, which is unavoidably noisy. Although much is known about the limits of measurement precision by cells, the limits placed by the communication are not understood. Motivated by recent experiments, we derive the fundamental limits to the precision of gradient sensing in a multicellular system, accounting for communication and temporal integration. The gradient is estimated by comparing a “local” and a “global” molecular reporter of the external concentration, where the global reporter is exchanged between neighboring cells. Using the fluctuation–dissipation framework, we find, in contrast to the case when communication is ignored, that precision saturates with the number of cells independently of the measurement time duration, because communication establishes a maximum length scale over which sensory information can be reliably conveyed. Surprisingly, we also find that precision is improved if the local reporter is exchanged between cells as well, albeit more slowly than the global reporter. The reason is that whereas exchange of the local reporter weakens the comparison, it decreases the measurement noise. We term such a model “regional excitation–global inhibition.” Our results demonstrate that fundamental sensing limits are necessarily sharpened when the need to communicate information is taken into account.

Cells sense spatial gradients in environmental chemicals with remarkable precision. A single amoeba, for example, can respond to a difference of roughly 10 attractant molecules between the front and the back of the cell (1). Cells are even more sensitive when they are in a group: Cultures of many neurons respond to chemical gradients equivalent to a difference of only one molecule across an individual neuron’s axonal growth cone (2), clusters of malignant lymphocytes have a wider chemotactic sensitivity than single cells (3), and groups of communicating epithelial cells detect gradients that are too weak for a single cell to detect (4). More generally, collective chemosensing properties are often very distinct from those in individual cells (3, 5⇓–7). These observations have generated a renewed interest in the question of what sets the fundamental limit to the precision of gradient sensing in large, spatially extended, often collective sensory systems.

Fundamentally, sensing a stationary gradient requires at least two measurements to be made at different points in space. The precision of these two or more individual measurements bounds the gradient sensing precision (8, 9). In its turn, each individual measurement is limited by the finite number of molecules within the detector volume and the ability of the detector to integrate over time, a point first made by Berg and Purcell (BP) (8). More detailed calculations of gradient sensing by specific geometries of receptors have since confirmed that the precision of gradient sensing remains limited by an expression of the BP type (10⇓–12).

However, absent in this description is the fundamental recognition that for the gradient to be measured, information about multiple spatially separated measurements must be communicated to a common location. This point is particularly evident in the case of multicellular sensing: If two cells at either edge of a population measure concentrations that are different, neither cell “knows” this fact until the information is shared. This is also important for a single cell: Information from receptors on either side of a cell must be transported, e.g., via diffusive messenger molecules, to the location of the molecular machinery that initiates the phenotypic response. How is the precision of gradient sensing affected by this fundamental communication requirement?

As discussed recently in the context of instantaneous measurements (4), the communication imposes important limitations. First, detection of an internal diffusive messenger by cellular machinery introduces its own BP-type limit on gradient sensing. Because the volume of an internal detector must be smaller than that of the whole system, and diffusion in the cytoplasm is often slow, such an intrinsic BP limitation could be dramatic. Second, in addition to the detection noise, the strength of the communication itself may be hampered over long distances by messenger turnover. This imposes a finite length scale over which communication is reliable, with respect to the molecular noise. However, the communicating cells can integrate the signals over time (8), improving detection of even very weak messages. To what extent such integration and the ensuing temporal correlations between the ligand and the communication molecules change the communication constraints has not yet been addressed.

To analyze constraints on gradient sensing in spatially extended systems with temporal integration, we use a minimal model of collective sensing based on the local excitation–global inhibition (LEGI) approach (13). This sensory mechanism uses a local and a global internal reporter of the external concentration, where only the global reporter is exchanged and averaged among neighboring cells. Comparison of the two reporters then measures whether the local concentration is above or below the average and hence whether the cell is on the high-concentration edge of the population. We analyze the model, using a fluctuation–dissipation framework (14), to derive the precision with which a chemical gradient can be estimated over long observation times. In the case where the need to communicate is ignored, the precision would grow indefinitely with the number of cells. In contrast, we find that communication imposes limits on sensing even for long measurement times, although slightly different from those on instantaneous sensing (4). Furthermore, the analysis reveals a counterintuitive strategy for optimizing the precision. We find that if the local reporter is also exchanged, at a fraction of the rate of the global reporter, the precision can be significantly enhanced. Even though such exchange makes the two compared concentrations more similar, which weakens the comparison, it reduces the measurement noise of the local reporter. This tradeoff leads to an optimal ratio of exchange rates that maximizes sensory precision. We discuss how our model is realized in classic gradient sensing systems and how its additional predictions could be tested experimentally.

## Results

Spatially extended gradient sensors come in different forms, from large spatially extended cells (15, 16) to groups of neighboring cells or nuclei (4, 17⇓–19). We intend to develop a theory that accounts for both structures simultaneously. For this, we view the spatially extended gradient sensor as consisting of compartments. These can be whole cells or their parts, but we refer to them as cells from now on. The limit of a single-cell, homogeneous, spatially extended sensor can be obtained by taking the compartment size to zero, while keeping the overall sensor length constant.

There is a diffusible chemical whose concentration varies linearly in space. The chemical gradient defines a direction within the sensor, and we focus on a chain of cells along this direction (Fig. 1*A*). Numbering the cells from *N*, each cell experiences a local concentration *g* is the concentration gradient, and *a* is the linear size of each cell. We choose without loss of generality to have *N*, which is then at the higher edge of the gradient. We focus on this cell because we imagine it will be the first to initiate a phenotypic response, such as proliferating or directed motility. Finally, we focus on the limit

### An Idealized Detector.

First, we consider the case when the two edge cells form an idealized detector, in the sense that each cell counts every external molecule in its vicinity and one cell knows instantly and perfectly the count of the other (Fig. 1*B*). The gradient could then be estimated by the difference in the concentration measurements made by the two cells (8, 9). The mean of this difference is

Berg and Purcell (8) showed that the fractional error in each measurement is not smaller than *D* is the diffusion constant of the ligand, and *T* is the time over which the measurement is integrated. This expression has an intuitive interpretation: The fractional error is at least as large as the Poisson counting noise, which scales inversely with the number of molecular counts. The number of counts that can be made in a time *T* is given by the number of molecules in the vicinity of a cell at a given time, roughly

The error in the gradient estimate is then given by *a* in Eq. **1**. This result has been derived more rigorously (10), and apart from a constant prefactor, Eq. **1** indeed provides the estimation error in the limit of large detector separation and fast detection kinetics. More complex geometries, such as rings of detectors (10), or detectors distributed over the surface of a circle (11) or a sphere (12), have also been considered, and Eq. **1** again emerges as the corresponding bound, with the length scale *a* replaced by that dictated by the specific geometry.

Eq. **1** can be combined with the mean *D*) or are more sharply graded (*g*) or if the detectors are larger (*a*), are better separated (*N*), or integrate longer (*T*). However, the SNR decreases for a larger background concentration (**1** and **2** are conceptually equivalent only in the case of low background concentration, when the difference **1**, here we are concerned with the opposite case: the fundamental limits to the detection of small gradients on a large background. Therefore, from here on we focus on the SNR, Eq. **2**.

### Accounting for the Need to Communicate.

Eq. **2** cannot be a fundamental limit because it neglects a critical aspect of gradient sensing: the need to communicate information from multiple detectors to a common location. Indeed, the idealized detector implies the existence of a “spooky action at a distance” (20), i.e., an unknown, instantaneous, and error-free communication mechanism. What are the limits to gradient sensing when communication is properly accounted for?

To answer this, a model of gradient sensing must be assumed. A naive model would allow each cell access to information about the input measured and broadcast by every other cell. This would require a number of private communication channels that grow with the number of cells, which is not plausible. A realistic alternative that would involve just one message being communicated is for each cell to have access to some aggregate, average information, to which all comparisons are made. There are a few such models (21⇓–23), and our choice among them is guided by the fact that collective detection of weak gradients is observed in steady state and over a wide range of background concentrations in both neurons (2) and epithelial cells (4). This supports an adaptive spatial (rather than temporal) sensing, such as can be implemented by the local excitation–global inhibition (LEGI) mechanism (13).

The LEGI model is illustrated in Fig. 1*C*. Each cell contains receptors that bind and unbind external molecules with rates *α* and *μ*, respectively. Bound receptors (*R*) activate both a local (*X*) and a global (*Y*) intracellular species with rate *β*. Deactivation of *X* and *Y* occurs spontaneously with rate *ν*. Whereas *X* is confined to each cell, *Y* is exchanged between neighboring cells with rate *γ*, which provides the cell–cell communication. *X* then excites a downstream species whereas *Y* inhibits it (LEGI). Conceptually, *X* measures the local concentration of external molecules, whereas *Y* represents their spatially averaged concentration. If the local concentration is higher than the average (i.e., the excitation exceeds the inhibition), then the cell is at the higher edge of the gradient. Although such comparison of the excitation and the inhibition can be done by many different molecular mechanisms (13), here we are interested in the limit of shallow gradients. In this limit, biochemical reactions doing the comparison can be linearized around the small difference of *X* and *Y*, and the comparison is equivalent to subtracting *Y* from *X* (4). Therefore, we take this difference,

Because we are interested in the limits to sensory precision, we focus on the most sensitive regime, the linear response regime, where the effects of saturation are neglected. Introducing *R*, *X*, and *Y* in the *n*th cell, the stochastic model dynamics are*n*th cell. The matrix *n*th cell (14), **2** for the idealized detector.

**3** in steady state:

To find the noise, we calculate the power spectra of *r*, *x*, and *y*. As explained below and argued for in Discussion, we assume that the measurement integration time *T* is longer than the receptor equilibration time (*SI Text*, we solve for **3** around its means and Fourier transforming in time and space, which yields a relationship between **8** requires inverting a Toeplitz marix (a matrix with constant diagonals), which has a known inversion algorithm (26). The result is*a* appears because we cut off the wavevector integrals at the maximal value *δ*-correlated noises in the Langevin approximation in Eq. **3**. In deriving Eq. **9**, we have made the first of our timescale assumptions, namely

The second step is to calculate power spectra for *SI Text*, we calculate these directly from the Fourier transform of Eq. **3** and the noise correlations in Eq. **4**, which propagate via the same matrix **5** and **10**, together with Eqs. **6** and **9**, give the **10**, we have made our second timescale assumption, namely

The SNR is compared with the result for the idealized detector (Eq. **2**) in Fig. 2. We see that whereas the SNR for the idealized detector increases indefinitely with the number of cells *N*, the SNR for the model with communication and temporal integration saturates, as in the no-integration case (4). This is our first main finding: Communication leads to a maximum precision of gradient sensing, which a multicellular system cannot surpass no matter how large it grows. The reason is that communication is not infinitely precise over large length scales. In the next section, we make this point clear by deriving a simple fundamental expression for the maximum value of the SNR.

### Fundamental Limit to Sensory Precision.

Our results up to this point hold for arbitrary communication strengths and cell numbers and are applicable to multicellular systems or cellular compartments. However, it is instructive to derive the saturating value of the SNR in the limit of large *N* and strong communication, where messenger hopping between cells is equivalent to continuous diffusion. In this limit, and when communication is strong (**6**) reduces to **5** and approximating the sum as an integral in the large *N* limit, the mean becomes **10** results in products of the exponential with the **9**), leading to sums like *SI Text*. The result is**11** is fundamental in the sense that it does not depend on the details of the internal sensory mechanism. Rather, it depends only on the properties of the external signal (*a*), and the fact that information is integrated (*T*) and communicated (*SI Text*). These terms represent intrinsic noise and can in principle be made arbitrarily small by increasing the gain factors **11** is the communicated extrinsic noise, which arises unavoidably from the diffusive fluctuations in the numbers of the ligand molecules being detected. Eq. **11** is shown to bound the exact SNR in Fig. 2.

Comparing Eq. **11** to the expression for the idealized detector (Eq. **2**), we see that the expressions are very similar but contain two important differences. First, whereas Eq. **2** decreases indefinitely with *N*, Eq. **11** remains bounded by *N* (Fig. 2). Evidently, a very large detector is limited in its precision to that of a smaller detector with effective size **11** demonstrates that this noise is present independently of the number of intrinsic signaling molecules in the communication channel. Thus, the fundamental sensory limit is affected not only by the measurement process (as in BP theory), but also unavoidably by the communication process.

The second important difference is that Eq. **2** depends on **11** depends on the effective concentration **12**. **12**. Because intercellular molecular exchange also competes with extracellular molecular diffusion, not all of the measurements made by these **12**. This log arises from the interaction of the *SI Text*). The net result is that, because of the correlations imposed by external diffusion, the number of independent measurements grows sublinearly with the system size. For real biological systems *SI Text*, the asymptotic expansion in Eqs. **11** and **12** is still very accurate in this range. Thus, this logarithmic correction cannot be summarily neglected. Nonetheless, even with the correction, in Eq. **12** the measurement noise in the global species decreases with the communication length scale **11** is dominated by the measurement noise of the local species, i.e., the first term in Eq. **12**.

In deriving the precision of gradient sensing, we have also derived the precision of concentration sensing by communicating cells. Specifically, by focusing only on the global species terms, and following the steps leading to Eq. **11**, we get**13**.

### Optimal Sensing Strategy.

In the previous section, we saw that the limit to the precision of multicellular gradient sensing is dominated by the measurement noise of the local species in the edge cell (Eqs. **11** and **12**). In contrast, the measurement noise of the global species is reduced by the intercellular communication. This finding raises an interesting question: Could the total noise be further reduced if the local species were also exchanged between cells? To explore this possibility, we extend the model in Eqs. **3** and **4** to allow for exchange of the local species at rate **12** remains dominated by the first term, even as local exchange reduces this term according to **11** then becomes

The optimal value **10** (Eq. **S67** in *SI Text*). Fig. 3 shows that

## Discussion

Cellular sensing of spatially inhomogeneous concentrations is a fundamental biological computation, involved in a variety of processes in the development and behavior of living systems. Like binocular vision and stereophonic sound processing, it is a process where the sensing is done by an array of spatially distributed sensors. Thus, the accuracy of sensing is limited in part by the physical properties of the biological machinery that brings together the many spatially distributed measurements. Understanding these limits is a difficult problem.

Here we solved this problem in the case where the communication is diffusive, and one-dimensional distributed measurements are used to calculate external concentration gradients within the LEGI paradigm. We allowed for temporal integration, extending the results of ref. 4. Some of the features of the gradient sensing limit we derived (Eq. **11**), such as the unbounded increase of the SNR with the diffusion coefficient of the ligand or with the integration time, carry over from the Berg–Purcell theory of gradient sensing (12), which does not account for communication. However, our most important finding is that, in contrast to the BP theory, the growth of the sensor array beyond a certain size stops increasing the SNR. The effect is independent of the intrinsic noise in the communication system and thus represents a truly fundamental limitation of diffusive communication for distributed sensing. In particular, it holds for multicellular systems, as well as for large individual cells. Although we derived the limit for a linear signal profile, we anticipate that the limit for a nonlinear profile will be similar, Eq. **11**, but with a different effective concentration

Our results illustrate two important features of temporal averaging by distributed sensors. First, our derivation naturally reveals which timescales are relevant in this process, namely receptor equilibration (*N*) or the communication length (**11** and **12**) and concentration sensing (Eq. **13**), we see that the noise reduction afforded by communication-based averaging is tempered by a factor

Our analysis of the limits on the gradient sensing accuracy has assumed that the integration time is longer than any of the other timescales in the problem, *Dictyostelium*, ^{−1}, and cAMP diffusion constant is ^{2} s^{−1} (30). Thus, ^{−1} (30, 32). The onset of the *Dictyostelium* gradient response takes about 30 s, and the adaptation phase (global inhibition) can be 300 s or longer (16). This validates our assumptions that *T* can be much larger than any of

Similar analysis can be done for multicellular sensors, such as EGF gradient sensing by mammary epithelial organoids in the companion paper (4). There the ligand is EGF, and the messenger is associated with calcium signaling and is able to pass through gap junctions; one possibility is inositol 1,4,5-trisphosphate (IP3). The cell size is ^{−1} (34), and the EGF diffusion constant is ^{2} s^{−1} (35). The turnover rate of IP3 is estimated as ^{−1} (36). Thus,

Another central prediction of our theory is that the gradient sensing is improved by a system with two messengers, exchanged at different rates. We call this mechanism REGI, a generalization of the standard LEGI model. Optimality of REGI follows directly from the interplay between the ligand stochasticity and the communication constraints. Therefore, REGI has not been identified as an optimal strategy in previous studies that neglected either of these two effects. Although the need for spatial averaging of two measurements involved in gradient detection was proposed (38), no model of the communication was suggested, limiting the ability of that analysis to make specific predictions.

Molecular mechanisms supporting REGI are suggested by biophysical mechanisms in many eukaryotic cell types. There activated receptor complexes, which diffuse in the membrane at the rate of *Dictyostelium* cell is a few microns in size, which is a sizable fraction of a cell size of about 10–20 μm. These can be identified with regionally averaging functional units in our terminology. Diffusion of surface receptors and finite membrane rigidity ensure the existence of regional integration in other eukaryotic cells, further supporting REGI over LEGI as a correct model.

REGI emerged from maximizing the SNR in our system, which revealed the optimal rate ratio *g*. In contrast, more physical nonlinear profiles (e.g., exponential, power law, randomly varying, or profiles with extrema) have spatially varying *g*. In these cases, if the size of the group of cells is larger than the correlation length of *g*, then an infinite

REGI can be interpreted as performing a spatial derivative. Specifically, the two-lobed filter *Escherichia coli* chemotaxis. Indeed, the temporal filter of the *E. coli* sensory module is also two-lobed, with the short and long timescales set by ligand statistics and rotational diffusion, respectively (39). Thus, for both spatial and temporal filtering, the choice of the two optimal length scales or timescales is determined by matching the filter to the statistical properties of the signal and the noise (40), which is understood well for *E. coli* (39).

Interpreting the REGI model as a spatiotemporal filter suggests experiments that would identify whether a certain biological system employs this mechanism. Such experiments would involve concentration profiles that differ substantially from steady-state linear gradients. For example, subjecting cells to a concentration profile with a spatially localized maximum would allow one to measure both

## SI Text

## Derivation of the Power Spectra

Linearizing Eq. **3** of the main text around its means and Fourier transforming (denoted by ˜) in time and space obtains**S1** for **S5**, insert it into Eq. **S2** to obtain**S7** and **S8** come from writing the volume element

We are interested in the low-frequency limits of *k* to account for the fact that cells have finite extent *a*, making *a*, but the exact shape of the cell will not be important for the limits we take. These expressions allow us to write Eq. **S6** as**S9**, we have assumed that *α* is much smaller than the diffusion-limited association rate *α* is much larger than **S10** and **S12** that *z* as a small parameter.

Solving Eq. **S9** for **S11**, keeping only terms up to first order in the small quantity *z*. From Eq. **S11** we have **S15** and **S16**. Then noting that the second term on the right-hand side of Eq. **S17** is second order in *z*, that equation implies that the diagonals of the inverse are constant. We therefore have the inverse to first order in *z*,**S9** for **9** of the main text.

We now solve for the power spectra for **S3** and **S4**, we obtain**S26** and **S27** follow directly from Fourier transforming Eq. **4** of the main text and using the steady-state means of Eq. **3** of the main text to eliminate **S29** is expected, because the noise arises in reactions in every cell and then propagates to other cells via the same matrix as the means (28). Indeed, this simplifies Eq. **S27** for **10** of the main text.

## SNR in the Many-Cell, Strong-Communication Limit

The variance of the readout is given by Eq. **10** of the main text,**S25** and **S34**, we evaluate Eq. **S33** term by term.

The first, fourth, and fifth terms in Eq. **S33** are straightforward,*N* to approximate the sum in **S33** we split in two,**S35**,*N*. In particular, we have written**S46** is valid for **S33** is then

The second term in Eq. **S33** we also split in two,**S38**,*A*. We split *A* into two equal components,*N*, accounting for the fact that *k* has support on only half of the integers in its range,**S46**. The second integral evaluates to **S57** vanishes exponentially with *N*, and we have**S52** equal to **S33** is then

Finally, collecting the terms in Eqs. **S35**–**S37**, **S48**, and **S61**, the variance in Eq. **S33** becomes*T*, as expected for intrinsic counting noise. The second term in this line is for *y* and is similar, except that because the global species is exchanged over roughly *r*, propagated to *x* and *y*. The third line contains the noise in *c*, propagated to *x* and *y*, which we deem extrinsic, because it originates in the environment and is not under direct control of the cells.

Importantly, the intrinsic noise terms in Eq. **S62** are reducible by increasing the numbers of receptors and local and global species molecules. These molecule numbers are set by the gain factors **S62** vanish as the gain factors grow large. In this limit we are left with only the lower bound set by the extrinsic noise,**11** and **12** in the main text.

## Exact SNR for Regional Excitation–Global Inhibition

In the regional excitation–global inhibition (REGI) strategy, both messengers *X* and *Y* are exchanged between cells, at rates **5** and **6** in the main text)**10** in the main text)**9** in the main text (or equivalently Eq. **S25** here). The SNR is then

The SNR has a maximum as a function of the rate ratio *X* messenger is not exchanged, and we recover the SNR of the local excitation–global inhibition (LEGI) strategy, which is a limiting case of REGI. At *X* and *Y*, and the signal (and therefore the SNR) is 0. The exact location of

We illustrate the dependence of *A*) and *B*). In both cases, we see that the optimal rate ratio *N*th cell. The increase in *X* messenger and thus reduces the noise in the estimate of

## Acknowledgments

We thank Matt Brennan for useful discussions and Thomas Sokolowski for pointing out a more rigorous way to derive the

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: ilya.nemenman{at}emory.edu.

Author contributions: A.M., A.L., and I.N. designed research, performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

See Commentary on page 1471.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1509597112/-/DCSupplemental.

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