A cellular solution to an information-processing problem Garud Iyengara and Madan Raob,c,1 a
Industrial Engineering and Operations Research, Columbia University, New York, NY 10027; bRaman Research Institute, Bangalore 560080, India; and National Centre for Biological Sciences (TIFR), Bangalore 560065, India
c
Edited† by Albert Libchaber, The Rockefeller University, New York, NY, and approved July 21, 2014 (received for review April 14, 2014)
Signaling receptors on the cell surface are mobile and have evolved to efficiently sense and process mechanical or chemical information. We pose the problem of identifying the optimal strategy for placing a collection of distributed and mobile sensors to faithfully estimate a signal that varies in space and time. The optimal strategy has to balance two opposing objectives: the need to locally assemble sensors to reduce estimation noise and the need to spread them to reduce spatial error. This results in a phase transition in the space of strategies as a function of sensor density and efficiency. We show that these optimal strategies have been arrived at multiple times in diverse cell biology contexts, including the stationary lattice architecture of receptors on the bacterial cell surface and the active clustering of cell-surface signaling receptors in metazoan cells. protein sensors
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he molecular characteristics of signaling receptors and their spatiotemporal organization have evolved to optimize different facets of information processing at the cell surface. A canonical information-processing problem involves designing strategies for a collection of distributed, noisy, mobile sensors to faithfully estimate a signal or function that varies in space and time (1). This problem appears naturally in many contexts, biological and nonbiological: (i) chemoattractant protein sensors on the bacteria cell surface (2, 3); (ii) galectin-glycoprotein assemblies designed for effective immune response on the surface of metazoan cells (4, 5); (iii) ligand-activated signaling protein receptors on the surface of eukaryotic cells (6–10); (iv) coclustering of integrin receptors to faithfully read and discriminate the rigidity and chemistry of a substrate (11); (v) clustering of e-cadherin receptors for effective adherence at cell– cell junctions (12); and even (vi) radio frequency (RF) sensor networks monitoring the environment or mobile targets (13). In the signal-processing community, this problem is known as data fusion or more generally information fusion (14, 15); however typical applications do not consider mobile sensors. In this paper we show how biology has, on multiple occasions, arrived at a solution to this optimization problem. The optimal solution needs to balance two opposing objectives, the need to locally assemble sensors to reduce estimation noise and the need to spread them out for broader spatial coverage. We show that in the space of strategies, this leads to a phase transition as a function of sensor density, sensor characteristics, and function properties. At very low sensor density, the optimal design corresponds to freely diffusing sensors. For sensor density above a threshold, there are two different optimal solutions as a function of a dimensionless parameter constructed from the sensor advection velocity and the correlation length and time of the incident signal. One optimal solution is that the sensors are static and located on a regular lattice grid. This is the strategy used in bacteria, such as Escherichia coli, to organize their chemoattractant receptors in a regular lattice array (3, 16), and in metazoan cells, where galectin-glycoproteins are organized in a lattice on the cell surface to effect an optimal immune response (4, 5). To realize this strategy, the cell needs to provide a rigid cortical scaffold that holds the receptors in place. Another optimal
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solution is to make the receptors mobile in such a way that a fraction of them form multiparticle nanoclusters, which then break up and reform randomly, the rest being uniformly distributed. Recent studies on the steady-state distribution of several cell-surface proteins reveal a stereotypical distribution of a fixed fraction of monomers and dynamic nanoclusters (6–9), and our information theoretic perspective could provide a general explanation for this. To realize this dynamic strategy, the cell surface needed to be relieved of the constraints imposed by the rigid scaffold and to be more regulatable. This strategy change needed the innovation of motor proteins and dynamic actin filaments, a regulated actomyosin machinery fueled by ATP, and a coupling of components of the cell surface to this cortical dynamic actin (17). Coordinated Signal Estimation Problem Consider a collection of Np mobile sensors in a finite 2D space of size L × L, such as protein receptors on a cell membrane (Fig. 1A), with sensor density ρ = Np/L2. The external signal monitored by the sensors (ligand field in Fig. 1A) is a continuous function f(x, t) of space and time, with ξ(τ) denoting the length (time) scale of signal variation. We assume that the sensors are equipped with an internal clock that allows them to sample the signal values at precise instances in time tm. This can arise, for instance, from a minimum informationprocessing cycle time, which involves binding of the receptor to a ligand and resulting conformational change, signal transmission downstream and resetting time. New signals cannot be processed during this cycle time. We will also implicitly assume that the location of each sensor is known. The sensors are inherently noisy; the output of the sensor is the function f(x, t) corrupted by additive noise, i.e., the output from a Significance Cell-surface signaling receptors are organized into different architectures that have been arrived at multiple times in diverse contexts. To understand the trade-offs that lead to these architectures, we pose the generic information-processing problem of identifying the optimal strategy for distributed mobile noisy sensors to faithfully “read” an incoming signal that varies in space–time. This involves balancing two opposing requirements: clustering noisy sensors to reduce statistical error and spreading sensors to enhance spatial coverage, resulting in a phase transition that explains the frequent reemergence of a set of architectures. Our results extend to a variety of engineering and communication applications that involve mobile and distributed sensing, and suggest that biology might offer solutions to hard optimization problems that arise in these applications. Author contributions: G.I. and M.R. designed research, performed research, analyzed data, and wrote the paper. The authors declare no conflict of interest. †
This Direct Submission article had a prearranged editor.
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To whom correspondence should be addressed. Email:
[email protected].
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 1073/pnas.1406608111/-/DCSupplemental.
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sensor at space–time location (y, u) is f(y, u) + σ pζ, where ζ ∼ N (0, 1), is a standard normal random variable. The source of this noise is context dependent; in mobile RF sensors the source of noise is typically thermal, and in protein receptors on the cell surface the source of the noise is conformation fluctuations—the sensors randomly flip between a “reading” (active) and a “nonreading” (inactive) conformation. Suppose there are s(y, u) sensors at location y at time u, then the signal read at location (y, u) is
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function “imaging” Fig. 1. Schematic of the optimization problem. (A) Mobile protein sensors on the cell surface read the concentration of an extracellular ligand field, which varies in space and time. (B) Dynamic integrin receptors form dynamic clusters to read the mechanical and chemical properties of the substrate as the cell spreads on a substrate coated with components of the extracellular matrix. (C) Imaging a function f(x, t) by noisy sensors can be done more optimally by sensor movement, shown here as different sensor locations at different time slices t1, t2, t3, . . ..
Iyengar and Rao
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where the exponent γ characterizes sensor correlation. When the sensors are independent, γ = 0, and γ < 0 corresponds to the case where the sensors have redundant information and γ > 0 corresponds to sensor cooperativity where cluster formation can reduce the sampling noise. In the RF sensor setting, although the signal sensed by each sensor is the same, thermal noise is independent across sensors; consequently, averaging the output across colocalized sensors reduces the impact of noise. For coclustered protein sensors, the impact of noise reduces via allosteric feedback (18). Clustering sensors reduces noise at a sampling location; however, it also reduces the total number of sampling locations. An illustrative analogy to keep in mind is that the sensors are “imaging” the function (Fig. 1C), with the feature that the sensors that are taking the image are possibly noisy. To overcome the noise, the sensors would need to cluster, however at the expense of less coverage of the entire image plane. We would like to investigate how mobile sensors can mitigate some of the loss due to clustering. Our goal here is to characterize the placement strategy of the mobile sensors that minimizes the expected distortion E½dðf ; ^f Þ between the true signal and its approximation constructed from the signal values sampled by the sensors. Clearly, the optimal sensor placement strategy is constrained by the physical mechanisms available for sensor movement. We first study an idealized “mean-field” model where the movement of sensors is unconstrained, and then generalize to a physically “realistic” stochastic model for the movement of sensors. In this paper we focus on the problem of minimizing expected distortion E½dðf ; ^f Þ without considering the cost of discriminating or “decoding” the information in the observed values ^f ðy; uÞ (19, 20). Idealized Model Here we consider optimal sensor organization when there are no physical constraints on sensor transport. This idealized model allows us to make accurate analytical predictions (SI Appendix) that serve as a useful guide to the more realistic stochastic model. The input signal f is taken from the class of functions spatial correlation ξ and piecewise constant with period τ (SI Appendix). We compare the performance of two different signal acquisition architectures, stationary and mobile. In the mobile architecture sensors move with velocity v, and can move in a coordinated fashion. (In the SI Appendix we consider another form of mobile architecture where the sensor movement approximates diffusion– advection transport.) We show that the optimal architecture has a phase transition from stationary to mobile architecture as a function of the sensor density ρ, sensor velocity v, sensor sampling time tm, and the correlation length and time ξ and τ. The noisy sensors sample the function every tm s. Because there are τm = τ/tm sampling instants in each signal period, the statistical error in the function estimate ^f ðyÞ is given by σ 2 ð yÞ = σ 2p tm =τ · sð yÞ−ð1+γÞ=2 , where s(y) denotes the number of sensors at location y and γ ≥ −1 is a parameter that controls sensor interaction. At locations x where there are no sensors, we PNAS | August 26, 2014 | vol. 111 | no. 34 | 12403
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construct an approximation ^f ðxÞ by “mapping” it to the location y that minimizes the error " # 2 ^ eðjxjsÞ = min max E f ðxÞ − f ð yÞ y:sðyÞ>0 f ∈F
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and the scaled sensor speed θ ≡ vτξ (Fig. 2A). In SI Appendix we compute the optimal error eps ðδ; λÞ of the stationary mechanism as a function of (δ, λ) at finite θ. The error of the perfect mobile architecture is a decreasing function of θ, and approaches a bound when θ → ∞. In the limitpof θ = ffi∞, the mobile architecture is ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi always optimal when λ > 3 ð1 + γÞδ, or equivalently, ρξ2 ≤ 9ð1 +1 γÞδ, i.e., the density of sensors is sufficiently small. In the limit θ → 0, the mobile architecture is never optimal. Because the derivative of the error jumps discontinuously across this boundary, this corresponds to the first-order phase transition. An immediate prediction is that the optimal strategy for sensors with small sampling time δ (high on–off rates of ligand-activated receptors) is the mobile active cluster architecture (Fig. 2A). Realistic Stochastic Model The Voronoi centers, i.e., the foci in the idealized model, are a purely geometrical construct. These foci can be physically realized by having focusing regions or signaling platforms (SP) that colocate sensors at their cores. We imbue the dynamics of both sensors and SPs with physical realism, while allowing for stochasticity. Although this generalization is not amenable to simple analytic treatment, we make considerable progress using Monte Carlo simulations. 12404 | www.pnas.org/cgi/doi/10.1073/pnas.1406608111
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The error e(xjy) has two components: the first is the spatial error associated with function decorrelation over the correlation length ξ, and the second is the statistical error associated with sensor Pnoise. We minimize maxx{e(xjy)} subject to the constraint that y sðyÞ = ρL2 , and s(y) being a nonnegative integer. In some cases, in addition to the distribution s(y), the density ρ itself could be a decision variable, although, over different time scales. In this case, the optimization problem takes the form minρ{e*(ρ) + κρ}, where e*(ρ) is the optimal error as a function of the density ρ, and κ is the cost of producing more sensors. In the limit of large L, an optimal solution is to tessellate space with identical Voronoi cells V, with sensors located at the centroids of V (SI Appendix). Because the signal decorrelation error then depends on the distance kx − yk from a measurement location y; it follows that the ideal Voronoi tessellation corresponds to the hexagonal packing with regular hexagons H(r) with a radius r that is function of the correlation length ξ, the density ρ, and the sensor correlation γ. The stationary mechanism clearly minimizes the statistical error, because it maximizes the number of independent measurements at each sampling location y. However, it is possible that mobile sensors, which are able to sample the function at more locations, can significantly reduce the spatial error, although at the cost of increasing the statistical error. In the perfect mobile architecture, the sensors move with a maximum velocity v and can be organized at specified locations in a coordinated fashion. Note that in this idealized model we assume that the sensor measurements are synchronous. A more refined model would allow for sensors to be asynchronous and establish that clustering in time is optimal in some parameter regimes. The optimal trade-off between spatial sampling vs. temporal sampling is governed by the sensor correlation γ, the scaled 2σ 2 sampling time δ ≡ ppffiffi · tτm , the scaled intersensor distance λ ≡ p1ffiffiffiffiffi2 , 3 3
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“Realistic” model Fig. 2. Phase diagram of the idealized and realistic models. (A) Phase diagram of the idealized model as ap function of the (scaled) intersensor distance λ and sensor ffiffiffiffiffiffiffi sampling time δ, ðλ ≡ 1= ρξ2 ,δ ≡ σ 2p tm =τÞ, shows a first-order transition between the stationary lattice (blue) and mobile (active) cluster (red) architectures, for a given maximal scaled speed θ and sensor correlation γ. The threshold value of λ is a decreasing function of θ, the dashed curve shows the limiting phase boundary when θ → ∞. (B) Phase diagram of the realistic of (scaled) model as a function ð1+γÞ intersensor distance η and Péclet number θ, η ≡ ðσ 2f =σ 2p Þ=ðρξ2 Þ 2 ,θ ≡ ðPe τÞ=ξ , for independent uncorrelated sensors γ = 0 using Monte Carlo simulations. Phase diagram for γ ≠ 0 shown in SI Appendix, Fig. S2. The phase boundaries are first order as seen from the intersections of the error branches as one moves along a cut across the phase diagram (SI Appendix, Fig. S3). The measurement frequency τm = 10 for all of the simulations. Values for the remaining four parameters, Pe, τ, ξ, and ρ, were varied within a range and combined to construct the pair (η, θ). In the simulations (and idealized analysis), the sensors are treated as point particles. However, at high enough sensor density (low η), one needs to account for steric constraints arising from the finite size of sensors (SI Appendix). This leads to an extension of the phase diagram, shown beyond the hashed transition.
In addition to having a fixed density ρ of sensors that diffuse with a diffusion coefficient D, there is a fixed uniform density nsp = Nsp/L2 of SPs of size R. The SPs capture sensors with an activation rate pa and advect them to their cores with a speed v, we define an active Péclet number Pe = vR/D (21), which measures the relative contribution of advection to diffusion. The SPs are Iyengar and Rao
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Phase Diagram. We define two dimensionless parameters: η ≡ ð1+γÞ ðσ 2f =σ 2p Þ=ðρξ2 Þ 2 that measures the trade-off between sensor noise, density, and signal correlation length ξ; and θ ≡ (Pe τ)/ξ that explores the trade-off between the advection velocity and the correlation length ξ (made dimensionless by setting D = 1/8 (SI Appendix). Based on an earlier study (21), we find that the performance is maximized when the mean SP lifetime τa = τ, the signal correlation time, reflecting a kind of active resonance condition. In the cellular context, this suggests the possibility that the SP lifetime has evolved to match the signal correlation time. All other parameters not included in η and θ were taken to be fixed (see SI Appendix for default values; varying these merely shifts the phase boundaries and does not affect the qualitative features of the phase diagram). As in the idealized model, the optimal strategy in the stochastic model also displays a phase transition with three distinct phases, see Fig. 2B. On decreasing sensor density (i.e., increasing η) at fixed θ, the optimal architecture changes from the stationary lattice phase to an active clustering phase, just as in the idealized model. A further decrease in sensor density results in a reentrant stationary lattice phase, because the remodeling dynamics of SPs does not give an advantage unless the Péclet number (or θ) is high. Finally at still lower density, it is more optimal to move the sensors—at low θ the sensor movement is diffusive and goes over to an active advection– diffusion when θ is high. Robustness of the Optimal Solution. Fig. 3A plots the fraction of sensors in clusters in the active clustering phase as a function of
Variation with Fig. 3. Robustness of optimal solution in the active clustering phase. (A) Fraction of clustered sensors shows a maximum at an activation probability pa ∼ 0.78 for two different values of sensor density ρ. This maximum corresponds to 38% focused in active clusters and 62% freely diffusing on the surface. The error bars on each curve denote the 95% confidence interval. The signal estimation error is a function of the fraction clustered. (B and C) Estimation error in the active clustering phase as a function of the activation
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rate pa for different values of sensor parameters. Note that for convenience, we have normalized the errors so that minimum error in each curve is 1. We define sensitivity to pa as the curvature at the minimum. All other parameters are set equal to their default values (SI Appendix). (B) Variation with sensor density ρ shows that the minimum of the error achieved at p*a ≈ 0:78 is robust. Low density results in high sensitivity. (C) Variation with sensor sampling frequency τm = τ/tm also shows a robust minimum at p*a ≈ 0:78, with high τm resulting in lower sensitivity.
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allowed to breakup and reform randomly at a new location with a lifetime taken from an exponential distribution with mean τa. Each active sensor becomes inactive with probability 1 − pa. We will call this the active clustering strategy. In eukaryotic cells, such SPs may be formed from the active restructuring of the actin cortex adjoining the cell membrane, which drives nanoclustering of passive molecules (17, 21). Alternatively, a fraction of sensors could switch to being SPs and start drawing in other sensors in their vicinity, either by inducing local restructuring of the actin cortex (active molecules; refs. 17, 21) or by elaborate [possibly multivalent (22)] protein–protein interactions. This is the most general physically realizable dynamical setting; the stationary and mobile strategies explored in the idealized model can be obtained as limiting cases of this—the stationary lattice corresponds to permanently fixing the SPs at regular spatial positions, and the mobile architecture corresponds to coordinated movement of SPs. Passive diffusion corresponds to setting nsp = 0. The signal f(x, t) to be estimated is drawn from a Gaussian random field with variance σ 2f and correlation length (time) ξ (τ). This function class is a stochastic version of the function class considered in the idealized model. The sensors make measurements at intervals the mean of which is tm; thus on an average, the sensor makes τ/tm measurements within the correlation time period τ. The signal f(x, t) is estimated from the measurements ^f ðy; uÞ assuming that the correlation structure of the signal is known. Let the estimate of f ðx; tÞ denote ðx; tÞ P denote the error the function value and let eðx; tÞ = f ðx; tÞ − f P at location (x, t). The mean error e = T ·1L2 x∈L Tt=1 eðx; tÞ is an average over all space and measurement epochs.
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Fig. 4. Estimation error in the active clustering phase as a function of the activation rate pa for different values of SP parameters. Note that for convenience, we have normalized the errors so that minimum error in each curve is 1. We define sensitivity to pa as the curvature at the minimum. All other parameters are set equal to their default values (SI Appendix). (A) Variation with Péclet number Pe shows a more sensitive response with the optimal probability dropping from p* a ≈ 0:75 to 0.68 as Pe increases. This is because at high Pe, the sensors get tightly focused very quickly; consequently, one needs a larger fraction of freely diffusing sensors to refocus after SP remodeling. The error is more sensitive for larger Pe. (B) Variation with SP break time τa shows that the optimal probability drops from p*a ≈ 0:75 to 0.65 as τa increases. High τa tightly focuses the sensors, however, it is also associated with lower sensitivity. SI Appendix, Figs. S4 and S5 contain more details of the robustness of the optimal solutions.
the activation rate pa. The maximum in this plot occurs at pa = 0.78, corresponding to 38% focused in active clusters and 62% freely diffusing on the surface; this maximum fraction is very weakly sensitive to the overall sensor density ρ. Plotting the estimation error versus pa for different values of sensor characteristics, we find, quite remarkably, that the minimum error is obtained when pa ∼ 0.78 (Fig. 3 B and C, and SI Appendix, Figs. S4 and S5), i.e., this optimal cluster fraction is robust and fairly independent of sensor parameters, such as the sampling frequency τm, sensor density ρ, density of SPs nsp , and the sensor correlation γ. We also ran simulations where the probability of error was minimized with respect to both pa and density of SPs, nsp, and found that in this case both the fraction in clusters (∼ 38%) and the number in a cluster (= 6 ± 2) are robust to variations in the other sensor parameters. This result is in contrast to the idealized model, where the optimal pa = 1. It does, however, show a dependence on the active Péclet number, Pe, and SP remodeling time τa (Fig. 4 A and B). Biology Solves an Optimization Problem In this paper we show that the optimal solution to the coordinated signal estimation problem encountered by a collection of mobile sensors is determined by the trade-off between the spatial decorrelation and statistical noise in the sensors. This generic estimation problem appears naturally in a variety of engineering situations such as in communication networks and signal processing, but as discussed in the Introduction, it is the biological context that we wish to highlight here. It is quite remarkable that this generic signal estimation problem exhibits sharp phase transitions, and that every phase has a realization in a specific cell biology context. For instance, the stationary lattice architecture is optimal when ρξ2 1, i.e., either the sensor density or the spatial correlation length of the signal is large, a condition that is met by the chemotactic receptors on the bacterial cell surface, such as E. coli (3); reassembly does not give an advantage. Indeed, following the initial proposal (3), there have been spectacular demonstrations of the hexagonal lattice arrangement of chemotactic receptors in a variety of bacterial species (16). Eukaryotic cells have also independently, and at multiple times, arrived at this strategy in contexts where both the space and time correlations ξ and τ are large. Aggregates of galectin-glycan make use of multivalent interactions to organize themselves in a large lattice array on the surface of metazoan cells. This architecture allows them to optimize the dual requirements of high 12406 | www.pnas.org/cgi/doi/10.1073/pnas.1406608111
affinity to ligands and large spatial coverage (4, 5). E-cadherin proteins form a two-dimensional array of microclusters at the junctions of cells, so as to establish reliable cell–cell contact (12). What is even more remarkable is how the information theory perspective brings out the active clustering phase that optimizes the estimation error at low densities. This organizational strategy is realized in a vast variety of signaling systems at the cell surface, such as GPI-anchored proteins (GPI-AP) (7, 8), Ras-signaling proteins (6, 9), glycoproteins, integrin receptors (11, 23), etc., despite the diversity both in their structural forms and in their network of interacting partners. This would argue for a broader conceptual principle underlying the choice of this strategy, such as the one described here. Our study shows that the optimal solution in the active clustering phase has a fixed activation probability pa ∼ 0.78 or, alternatively, a fixed fraction of proteins in active clusters ∼ 38%, the rest being diffusing monomers. Allowing the density of SPs to vary, leads in addition to an optimal number ∼ 6 ± 2 of proteins within a cluster. We find that this optimal fraction is robust over a wide range of parameters such as the density of sensors and sensor characteristics. This not only coincides with the cell-surface distribution of GPI-anchored proteins (7, 8), and Ras-signaling proteins (9), but is consistent with the finding that the fraction of proteins in nanoclusters is maintained over a large variation in cellular expression levels. On the other hand, the optimal solution is relatively sensitive to changes in Pe and SP remodeling time τa; therefore, in order for the cell to provide a reliably stable response over variations in temperature, these parameters should be constant across temperature in the physiological range (21). This is consistent with studies on the remodeling dynamics of GPI-anchored protein receptors in mammalian cells, which showed that the fragmentation–aggregation dynamics of GPI-AP nanoclusters was relatively uniform across 24−40 °C (8, 17). The model predicts that poor (noisier) sensors are more likely to be clustered and low noise sensors are more likely to be diffusing freely on the cell surface. Broad spectrum sensors that bind to a large number of ligands are likely to have lower binding affinity with any ligand; as a consequence they will be noisier and will form clusters. On the other hand, specific sensors are likely to be less noisy, and therefore, diffuse as monomers. The insight that colocating sensors leads to a decrease in the impact of noise, or equivalently, improves information, when the external signal does not change rapidly, may also have implications for organizational decision making. In environments where the information is slowly Iyengar and Rao
ACKNOWLEDGMENTS. We thank Satyajit Mayor and Mukund Thattai for useful discussions. We thank Joseph Mathew for help in preparing the schematic figures. G.I. thanks the National Center for Biological Sciences for hospitality. M.R. acknowledges a grant from the Simons Foundation.
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BIOPHYSICS AND COMPUTATIONAL BIOLOGY
each other and with the active dynamical cytoskeleton juxtaposed to the cell membrane (17). The signal estimation problem posed here can then be viewed as active mechanics of cellular information processing. It is appealing that one might look to biology for insights into solutions of hard optimization problems, arrived at as a result of evolution within an information niche (25).
changing over time and the individuals have access to a very noisy version of the information, the organization should rely on forming relatively large ad hoc teams to counter the lack of information— the teams have to be ad hoc to make sure that one has noise averaging effect. However, when the external environment changes rapidly one has to make do with smaller teams. Similar considerations might operate at a larger scale, in the collective reading of external space–time dependent cues in swarms of organisms (24). Coming back to the cell surface, the space of strategies that we explored are special cases of a general active composite cellsurface model, wherein the constituent molecules interact with
Iyengar and Rao
PNAS | August 26, 2014 | vol. 111 | no. 34 | 12407
