Chemotaxis, or directed motion in chemical gradients, is critical for various biological processes. Many eukaryotic cells perform spatial sensing, i.e. they detect gradients by comparing spatial differences in binding occupancy of chemosensory receptors across their membrane. In many theoretical models of spatial sensing, it is assumed, for the sake of simplicity, that the receptors concerned do not move. However, in reality, receptors undergo diverse modes of diffusion, and can traverse considerable distances in the time it takes such cells to turn in an external gradient. This sets a physical limit on the accuracy of spatial sensing, which we explore using a model in which receptors diffuse freely over the membrane. We find that the Fisher information carried in binding and unbinding events decreases monotonically with the diffusion constant of the receptors.
Chemotaxis, involving movements that are oriented relative to chemical gradients, is an important and widespread phenomenon among eukaryotic cells. This mechanism is critical to neuronal growth cones navigating to their targets in the developing nervous system [1,2], neutrophils migrating to the site of inflammation  and sperm cells swimming towards an egg . In the first steps of chemotaxis, ligand molecules in the environment bind to membrane chemoreceptors, activating them and triggering downstream signalling events . In spatial sensing, cells estimate the external concentration difference across their spatial extent, a comparison whose corruption by noise from the stochastic nature of ligand binding and downstream signalling has been well studied [6–11]. In order to interpret information about binding, cells need to have information about the positions of the receptors concerned. These are generally in constant motion in the membrane, which is itself a highly complex structure that is compartmentalized on multiple scales [12,13]. Receptors undergo diverse modes of diffusion, including fast and slow diffusion, confinement and drift [14–17]; further, differential fluidity of membrane lipids can affect the lateral movement of proteins embedded in the membrane , and the diffusive properties of receptors can also be influenced by oligomerization or association with the membrane skeleton [14,18]. Because movement of receptors owing to diffusion represents a smearing out of the spatial information, they can provide about local concentration, diffusion will reduce the accuracy of gradient sensing. However, the importance of this depends on exactly how much information is lost; a quantity that has not previously been calculated. Receptor diffusion poses a challenge that cannot be addressed by previous models of gradient sensing which assume immobile receptors and thus a fixed binding probability for each receptor. ‘Snapshot’ measurements no longer suffice for the cell to infer the gradient, as information is lost about how long each receptor has been unbound and where it has been. Instead, cells must integrate over time to estimate the gradient efficiently.
Here, we model the effects of receptor diffusion on a one-dimensional ‘cell’, under the assumption that it is only bound receptors that signal and thus reveal their locations, leaving the cell uncertain about the locations of unbound receptors. In particular, we extend a previous model  which assumed that the cell has knowledge of the amount of time the receptors are bound and where the bindings happen, and we compare the accuracy of gradient sensing for receptors that are mobile versus immobile. We show analytically how receptor diffusion reduces the accuracy of gradient sensing, and demonstrate that the reduction increases monotonically with the diffusion constant.
2.1. Immobile receptors
In common with many earlier models, here we take a normative or ‘ideal observer’ approach to gradient sensing. In particular, we consider the limits to the information available to a cell for decision-making, without addressing the mechanisms by which a cell could access that information via downstream signalling mechanisms. Many such previous models of gradient sensing have assumed, for the sake of simplicity, that receptors are immobile [6,9,10,20,21]. In these models, each receptor has a fixed probability of binding with ligand molecules which depends on the local ligand concentration at the receptor's position. This gives a probability distribution over possible binding patterns, and allows the cell to estimate the gradient from a ‘snapshot’ measurement of the binding pattern of all the receptors at a single moment. However, if the assumption of immobility is not imposed, each receptor will have a probability of binding that depends on its motion across the cell and thus through the concentration field.
An important alternative to the snapshot approach is to assume that the cell infers the gradient by integrating sequences of binding and unbinding events over time [19,22]. This leads to an upper limit on the information the cell can obtain assuming it knows both the unbound time and the number of bound/unbound transitions. However, these studies also assumed receptors were immobile. Here, we therefore extend the model of Mortimer et al.  to examine the effects of receptor mobility.
We first describe the original model in Mortimer et al.  (figure 1). The cell is represented as a one-dimensional array of uniformly distributed receptors whose binding and unbinding provide information about the concentration field that the cell experiences. The receptor state is modelled as a continuous time, two-state Markov process. The transition rate from bound to unbound is r– and the transition rate from unbound to bound is , where C is the local concentration of ligand at the location of the receptor. The dissociation constant is Writing γ = C/Kd for the dimensionless concentration, we have the relationship between the on- and off-rates: The cell has N receptors at positions normalized by the cell diameter and the linear concentration field is with μ being the gradient, or the fractional change of the concentration, that the cell has to estimate.
It was assumed in Mortimer et al.  that all receptors start out unbound. The times of binding and unbinding events were written as and , where j is the jth receptor and i is the ith transition. Similarly, the positions of the receptors at those times were denoted and The cell was assumed to know about the timings and positions of the binding/unbinding events of the receptors via downstream signalling mechanisms that were not specified.
This implies that binding and unbinding are Poisson processes with rates r– and r– γ(x). For a Poisson process with rate r, the probability density of an event happening at a particular time t after another event at time 0 is r exp(−rt). Therefore, the probability density for the jth receptor to remain bound from to isand denoting γj = γ(xj), the probability density of it being unbound from to is
Therefore, the likelihood of observing a time series given concentration γ(xj) at the location of receptor j is 2.1where Mj is the number of unbound–bound transitions, and is the number of bound–unbound transitions (this equals Mj or Mj − 1, because each binding event, with the exception of the last, must be followed by an unbinding event). are the total time unbound and bound, respectively, for receptor j. A maximum-likelihood approach suggests finding the value of the parameter of interest that makes the likelihood of observing the data as high as possible; this typically leads to an optimal unbiased estimate. Recalling that where μ is the gradient, and making the approximation , we set the derivative of the log-likelihood to zero,
Thus, the maximum-likelihood estimate of the gradient is given by 2.2
Equation (2.2) implies that the association between the receptor position and its unbound time (but not its bound time) carries gradient information, similar to a previous result that found unbound intervals carry concentration information . A useful metric to measure the amount of information that an observation carries about an unknown parameter is the Fisher information, defined as the expected value of the second derivative of the log-likelihood functionAs time increases, the variance of the maximum-likelihood estimate approaches the limit set by the inverse of the Fisher information2.3As the average time, it takes the receptor to become bound is 1/γr– and the average time the receptor remains bound is 1/r–, during the integration time T, the average number of unbound–bound transitions will be approximatelyHence, the Fisher information can be approximated as2.4
2.2. Mobile receptors
We now consider the more realistic case in which receptors are free to diffuse on the cell surface, starting from uniform randomly distributed initial positions. For the sake of simplicity, we consider the case that receptors continuously diffuse and ignore periods of confinement or drift. As unbinding events are independent of the receptor's position, their probability remains unchanged from above.
Binding events are treated as a Cox process, i.e. an inhomogeneous Poisson process whose intensity is also stochastic. Consider a single receptor (thus temporarily ignoring subscript j) that starts from initial position x(0) = x0 and diffuses freely by Brownian motion independently of the binding (although the binding is not independent of the motion). In this case, the binding rate λ(t) is a function of the random position x(t) of the receptor at time t: . We use the concept of the Brownian bridge, i.e. Brownian motion fixed at two ends and . According to the Cox process, the conditional density of a binding event at given the unbinding time and the path of the receptor is2.5where the first term accounts for the binding at and the second for the absence of binding over the time period . Marginalizing over the unknown trajectory ωi, the conditional likelihood given only the positions at the unbinding and binding times is2.6where is the binding rate, and the expectation is taken with respect to the diffusion bridge, but subjected to the boundary condition −0.5 < x < 0.5, as the receptors can diffuse only on the cell. We now proceed to find the expectation terms in the expression above2.7The receptors undergo Brownian motion: where W(t) is a standard Brownian motion and D is the diffusion constant. Only paths that satisfy the condition −0.5 < x(t) < 0.5 contribute to the expectation. We assume that the concentration is sufficiently high such that the probability of paths beyond this boundary is very small (i.e. and R is the radius of the cell). This assumption implies that including such paths does not affect the calculation. We discretize t into n intervals of :
For brevity, we write , , , . The distribution of x(t) at time is normal with meanand the covariance between x(s) and x(t) with s < t is
Therefore, xk's are the components of a multivariate normal vector with mean
and n × n covariance matrix
We know that the sum of the components of a multivariate normal vector has a univariate normal distribution with meanand covarianceTherefore, is also a normal random variable with mean and variance . Thus, follows a lognormal distribution with mean m and variance νThe likelihood of a full observation The maximum-likelihood estimate of the gradient is
A caveat of this calculation is that it assumes the cell knows the time series and locations of each receptor individually. That is, a pair of unbinding and binding events can be associated with each other, even though the receptor may have mingled with other receptors in the meantime. This is a reasonable assumption when diffusion is not too rapid or receptor density is low, such that the receptors are far enough apart and move sufficiently small distances for the cell to be able to distinguish between them after their unbinding period.
The Fisher information for this case is2.8
As expected, we can see that as D increases, the Fisher information decreases and when D = 0, this expression reduces to equation (2.3). This expression cannot be evaluated analytically, but needs to be approximated. In order to approximate the first term, we recall that and t+–t− is an exponentially distributed variable with mean , thus . Thus,2.9
We adopt the approach of Mora & Wingreen  to approximate the second term as follows2.10with
This approximation leads to2.11
We now need to calculate subjected to the boundary condition −0.5 < x(t) < 0.5. We employ the reflection principle to calculate this term. For a freely diffusing receptor without any boundary condition, the probability density function of x(t) given the initial position x0 is
If the receptor is reflected back when it reaches the boundary ±0.5, then the probability density function of x(t) is equal to that of a Brownian motion starting at x0 plus those of its reflections across the boundary.
We assume that receptors cannot diffuse too much while being unbound, and thus include only two reflection terms. The mean square displacement is thenThese approximations are good as long as . Diffusion constants of relevant receptors such as epidermal growth factor receptor (EGFR) and platelet-derived factor receptors are of order 10−2 µm2, whereas the dissociation constants are of order 10−1 s, suggesting that this approximation is valid at Kd and higher concentrations [24–28].
We plot the quality of the approximations as a function of what is a convenient quantity, the normalized diffusion constant in figure 2. The unit of this normalized diffusion constant is R2/t, where R is the radius of the cell (which we assume for numerical calculations to be 5 µm) and t is the duration of each time step of our simulation (set in our case to 0.02 s). The concentration is 10 Kd (i.e. the normalized concentration is γ0 = 10) and the gradient is 10% across the cell surface. Here, we used a discrete approximation to diffusion by initializing the positions of the receptors as uniform random numbers between −0.5 and 0.5 and, at each time step, making each move left or right a fixed amount , being ‘reflected’ if it hits −0.5 or 0.5. The probability of being bound at each time step is , if currently unbound, and the probability of becoming unbound is r–, if currently bound. We assume off-rates of 0.001–0.01 per time step, equivalent to 0.05–0.5 s−1. In our simulation, a normalized diffusion constant of 10−4 corresponds to an actual diffusion constant of 0.125 µm2 s−1. In 20 000 time steps (400 s of real time) of each simulation, with this diffusion constant, the mean square displacement of each receptor is 2R2 = 50 µm2. Although this is only a one-dimensional model, the parameters are biologically plausible.
The approximate analytical values of the Fisher information come within 20% of their numerically simulated values (quantified via the inverse of the variance of the maximum-likelihood estimates over 2000 trials lasting 20 000 time steps). The quality of the approximation worsens as increases. One reason for this is that the estimate of only included two reflection terms, whereas higher-order terms could become more significant in that regime. A famous theorem in statistics called the Cramer–Rao lower bound implies that the reciprocal of the Fisher information is a lower bound on the variance of the estimates (equation (2.3)). For example, if the Fisher information for a gradient of 10%, this means the lower limit to the variance is 0.0025. Gradient estimates will be approximately normally distributed, with an unbiased mean of 10% and a standard deviation of 5%. Figure 1 implies that to achieve the same accuracy as in the case that the receptors are static, diffusing receptors force the cell to integrate over a longer time. Figures 3 and 4 show that as the concentration or the off-rate increases, more binding and unbinding events will happen, so that the measurement contains more information.
3. Delayed signalling
So far, we have assumed that as soon as the receptors become bound, they start signalling immediately. However, in reality, it takes time for the receptor to change its conformational state or for the downstream cytosolic molecules to be recruited to the activated receptor. Therefore, there might be a delay between ligand binding and the initiation of the signalling cascade. It has been suggested that bound receptors slow down significantly, so that the cell can pinpoint exactly where the extracellular signal is coming from . Whether this has any benefit for gradient sensing is not clear. By calculating the Fisher information in the case that the receptors can still move freely after being bound, we find that the cell gains no extra information about the gradient if the bound receptors are immobilized, because it is only unbound times which carry information.
We now consider the consequence of a time delay between the binding event and the report of the location, implying that the latter does not reflect exactly where the receptor was at the time of binding. We denote the positions at which the transduction pathways for receptor i are activated and inactivated as Now, instead of knowing exactly where the binding and unbinding events occur on the surface, the cell only has access to partial information about these locations, represented by Gaussian distributions If we assume that the time delay td is constant (a stochastic delay is also possible), then the uncertainty in the location of the binding/unbinding events will be We can now examine how this extra noise adds to the uncertainty of the gradient estimate. The conditional probability of binding now has an expectation taken over all possible paths with all possible endpoints where . The Fisher information is nowwhere is the Fisher information given in equation (2.8), because and . This means the delay in signalling does not add any noise to gradient sensing. Thus, surprisingly, in this framework, confining the cytoplasmic signal to the precise place where the extracellular signal was received (i.e. immobilizing the receptor following binding) is not substantially beneficial to the cell.
The lipid bilayer that forms cell membranes has been extensively investigated. Among other key characteristics, it has been found that membrane-bound receptors display complex behaviours, including hop diffusion, drift and confinement . Here, we investigated the role of this receptor diffusion in spatial gradient sensing, exploring for simplicity the scenario in which receptors diffuse freely. Diffusion reduces the positional information that receptors carry about the signal they receive from the environment, and so potentially degrades the quality of gradient measurement. The speed of diffusion of receptors can vary depending on receptor and cell type. However, well-known receptors in chemotaxis such as EGFR in human mammary epithelial cells, TrkA in growth cones and PDGF in fibroblasts have diffusion constants of the order of 10−1 to 10−2 µm2 s−1 [25,29,16]. In the typical time it takes cells to respond to chemical gradients, i.e. around a few minutes, the mean square displacement of receptors can be comparable to cell size (several micrometres). These parameter ranges suggest that diffusion of receptors can contribute to a significant loss of information during that integration time.
We modelled the cell as a one-dimensional array of independent chemoreceptors whose bindings with ligand molecules depend on their stochastic positions on the cell surface. We considered the regime of fast diffusion of the ligand molecules, and thus neglected rebinding of particles. We assumed that only bound receptors signal their positions to the cell, with the timings of the bindings also being known, either precisely or after a time delay. We derived an approximation to the Fisher information about the concentration gradient, and showed the approximation error to be small given a long integration time. One caveat with the model is that the cell has to associate times and positions with particular receptors. If the receptors are sparse and far apart, then this would be straightforward. However, if in a typical binding time the receptors can intermingle, then this is harder. It may be that the cell carries out a simpler computation that will underperform the maximum-likelihood estimate. Alternatively, receptors may signal constitutively, and so provide a means of being tracked. The gradient measurement is based on a quantity that depends on both the unbound intervals and the receptor positions (equation (2.2)). If the receptors can move and yet only snapshot measurements are used, the cell will have lost the information as to how long each receptor has been unbound or where it has been. Thus, unlike the immobile case, receptor diffusion therefore necessitates integration of information over a time window.
Using both simulations and the analytical approximations, we found that receptor diffusion can cause a substantial reduction in the quality of gradient sensing. This quality reduces roughly linearly with the diffusion constant of the receptors. This is intuitive, because the cell knows less about where the receptors have been before they became bound. Consistent with a previous paper , the performance increases with ligand concentration and the unbinding rate. This is expected, because the higher these parameters, the more binding/unbinding events occur and the more information the measurement contains. This model also assumes that the cell has infinite precision in measuring the binding delay. A more realistic mechanism for measuring time intervals by the production of second messenger molecules has been discussed in more detail in reference . We have also not considered the issue of noise coming from downstream signalling, for instance from the diffusion of intracellular second messengers, focusing instead on the optimal possible performance only limited to noise owing to ligand binding. Ligands can alter the activation rate of downstream molecules proximal to receptors, as demonstrated in the cAR1 receptor and its G protein in Dictyostelium cells .
Finally, we also considered a more realistic model in which the binding events are not registered immediately by the cell, but only after a time delay. By the time the bound receptors start signalling, they will have moved to another position, implying that the cell will have lost precise knowledge as to where the binding events happened. However, this does not substantially affect the information the measurement carries. This implies that immobilization is unlikely to serve the purpose of preserving the signal, and might rather have a different effect, such as facilitating the interactions between the bound receptors and the cytosolic molecules or other membrane-bound proteins.
Recently, Iyengar & Rao  found that there is a phase transition in sensing strategies as a function of receptor density and efficiency. At low receptor density, the optimal strategy is freely diffusing receptors. At higher density, the optimal solutions are either static receptors on a regular lattice grid or a mix of freely diffusing receptors and clusters. However, this model assumes that each receptor can sample the environment directly at regular time intervals. By contrast, our model is based on the arguably more realistic proposal that the receptors do not sample the gradient itself, but have to infer it from sequences of binding and unbinding events. We also do not assume measurements at regular intervals or that the locations of the receptors are known at all times.
One key simplifying assumption we made is that diffusion is unfettered. In fact, the traditional view of the membrane as a homogeneous fluid has been challenged and replaced by a more sophisticated model incorporating heterogeneous microdomains . Indeed, in a gradient, receptors can have differential binding rates of ligand, changing diffusion constants among different populations of receptors or undergo redistribution [32–34]. The non-uniformity in motion has been hypothesized to be due to the existence of lipid rafts or the compartmentalization actin-based membrane–skeleton ‘fences' and anchored transmembrane protein ‘pickets'. These microdomains might serve as signalling platforms where the activated receptor can interact with cytosolic signalling molecules recruited to the cell membrane. Receptors might change their affinity with lipid rafts upon ligand binding and activate different signalling pathways depending on whether they are in a lipid raft or not. In these circumstances, the diffusion might allow the cell more flexibility to regulate receptor signalling; understanding the consequences of the heterogeneity—for instance, whether receptors can be adaptively slowed in the regions of low signal/noise remains to be investigated.
Funding comes from an Australian Postgraduate Research Scholarship (H.N.), the Gatsby Charitable Foundation (P.D.) and Australian Research Council grant DP110101803 (G.J.G.).
We thank Gergo Bohner for helpful discussions, and five anonymous reviewers for their comments.
- Received October 3, 2014.
- Accepted November 5, 2014.
- © 2014 The Author(s) Published by the Royal Society. All rights reserved.