Collective gradient sensing with limited positional information

Emiliano Perez Ipina∗

Department of Physics & Astronomy, Johns Hopkins University, Baltimore, MD.

Brian A. Camley†

Department of Physics & Astronomy and Biophysics, Johns Hopkins University, Baltimore, MD.(Dated: October 22, 2021)

Eukaryotic cells sense chemical gradients to decide where and when to move. Clusters of cellscan sense gradients more accurately than individual cells by integrating measurements of the con-centration made across the cluster. Is this gradient sensing accuracy impeded when cells havelimited knowledge of their position within the cluster, i.e. limited positional information? We applymaximum likelihood estimation to study gradient sensing accuracy of a cluster of cells with finitepositional information. If cells must estimate their location within the cluster, this lowers the ac-curacy of collective gradient sensing. We compare our results with a tug-of-war model where cellsrespond to the gradient by a mechanism of collective guidance without relying on their positionalinformation. As the cell positional uncertainty increases, there is a trade-off where the tug-of-warmodel responds more accurately to the chemical gradient. However, for sufficiently large cell clustersor shallow chemical gradients, the tug-of-war model will always be suboptimal to one that integratesinformation from all cells, even if positional uncertainty is high.

I. INTRODUCTION

Eukaryotic cells may directionally migrate by sensingexternal cues of many types – mechanical, electrical, to-pographical, or chemical [1], and in these processes theiraccuracy is often limited by basic physical principles [2].The best-known of these processes is chemotaxis, wherecells sense and follow gradients in concentrations of chem-ical cues – an example of gradient sensing. Gradientsensing and directed migration are essential to many fun-damental biological processes, such as immune response[3], embryonic development [4, 5], and cancer metastasis[6, 7]. In these processes, cells may migrate individu-ally or collectively – in sheets, streams, or small clusters[8–10]. Cell cluster migration may be particularly im-portant in cancer, where small clusters of tumor cells areassociated with more harmful metastasis [11]. There areseveral advantages for cells to migrates as groups [12],one of these being gradient sensing. In several biologicalsystems [5, 13, 14], groups of cells can chemotax wheresingle cells fail to do so – cells work collectively to im-prove their ability to sense a gradient [15].

How this enhanced sensing arises is not fully under-stood, and may not have a universal mechanism [15], buttwo types of cell clusters seem to primarily sense usingmeasurements of the concentration at the cluster edge.In clusters of neural crest cells responding to gradientsof Sdf1, edge cells are polarized out from the cluster cen-ter by contact inhibition of locomotion [5], suggestinga model of chemotaxis as a tug-of-war by the perime-ter cells [16]. A similar edge-driven mechanism is ob-served in gradient-sensing lymphocyte clusters [13, 17].

∗ emperipi@jhu.edu† bcamley@jhu.edu

These observations suggest that in these cell types onlyedge cells are involved in cluster chemotaxis. Why useonly the edge cells to sense the gradient instead of all ofthem? Our initial intuition is that using only the edgecells wastes information [18]. In earlier work, we found afundamental limit on gradient sensing accuracy [19], ob-serving that collective chemotaxis was likely limited bycell-to-cell variability. In that model, measurements fromall cells and their relative positions are used to computethe best estimate of the gradient direction.

Why would clusters use sensing mechanisms driven byedge cells, like the tug-of-war mechanism, instead of theoptimal use of all cells? One hypothesis is that the bene-fit of the extra information is small. Another hypothesisis that there are relevant sources of noise that are not in-cluded in Ref. [19]. A major assumption implicit in ourearlier work [19] is that cells “know” their position withina cluster – the best estimator of the chemical gradient in-volves a sum weighted by a cell’s position relative to thecluster center. Cells can measure their position withinan embryo or aggregate by measuring diffusible [20] ormechanical [21] signals but this positional information islimited [20, 22–24]. Cells in Drosophila embryos canmeasure their position to the order of a cell length [20],using temporal and possibly spatial averaging of gradedsignals [20, 25, 26]. In this paper, we explore the op-timal strategies for collective sensing of gradients whencells have limited information on their position in thecluster, showing that positional uncertainty reduces gra-dient sensing accuracy. We also compare our model toan extension of the tug-of-war model of [16], where onlyedge cells respond to chemoattractant and no positionalinformation is required. We find that these tug-of-warstrategies are optimal for small clusters or strong gradi-ents. However, for a sufficiently large cluster the benefitsof using all cells’ data will eventually outweigh the errorfrom finite positional information.

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II. MODELS AND RESULTS

A. Maximum likelihood estimates of gradient withlimited positional information

We consider N circular cells of radius Rcell arrangedin 2D clusters under a linear chemoattractant gradientg = g (cos(φ), sin(φ)), see Fig. 1a. g and φ are thegradient magnitude or steepness and direction respec-tively. The chemoattractant concentration at cell i isc(ri) = c0(1+g(ri−rcm)), where ri is the cell’s position,and rcm is the position of the cluster center of mass,computed as rcm = 1

N

∑ri. c0 is the chemoattractant

concentration at the cluster center of mass. Cells per-form a collective measurement of the gradient by inte-grating individual cells’ measurements of local chemoat-tractant concentration, which we denote by Mi. Cellssense the concentration by binding the chemoattractantmolecules to receptors at their surface, which is subjectto fluctuations due to the molecules’ diffusion in reachingreceptors, and intrinsic ligand-receptor kinetics [26–29].The reading concentration error of a cell at position riwith nr receptors that bind ligand molecules with dis-sociation constant KD, can be expressed as δc2(ri) =c(ri)nrKD

(c(ri) + KD)2 [19, 26]. In addition, even geneti-cally identical cells do not respond in the same way tothe chemoattractant due to cell-cell variability (CCV) re-sulting from fluctuations in their internal molecular ma-chinery [19]. Then, the concentration measurement ofcell i is,

Mi =c(ri) + δc(ri)ηi

c+ σ∆ξi, (1)

where ηi and ξi are independent Gaussian noises withmean value 0 and variance 1, c is the mean concentra-tion over the cluster, and σ∆ is the CCV standard devia-

tion. Note that c = 1N

∑Ni c(ri) = c0. For typical values

of σ∆ ≥ 0.05 (i.e. 5% cell-to-cell variability), ligand-receptor noise is less critical than CCV noise [19].

The gradient sensing error can be obtained by combin-ing the measurements in Eq. (1) through the maximumlikelihood estimation (MLE) method, as in [19]. How-ever, this implicitly assumes that the positions of thecells are known, as each measurement Mi correspond toa given position ri within the gradient.

Here we extend [19] by assuming that cells have lim-ited positional information. As a result, cells need toestimate their positions, r∗i , in addition to the local con-centration. We assume that cells get their positional in-formation from an independent process different from thesensing concentration such that r∗i and Mi are indepen-dent. Thus, the measured position for the cell i is,

r∗i = ri +Aiξri , (2)

where Ai is a matrix such that Σi = AiATi is the covari-ance matrix of the positional errors, and ξri is a two

dimensional vector of uncorrelated normal distributednumbers. XT denotes the transpose of X.

Following the same procedure as in [19], we applythe maximum likelihood estimator (MLE) method, tocompute the gradient sensing error limits. We start bywriting the probability distribution for a cell i to mea-sure a signal Mi and a position r∗i given the gradient gand its true position ri, p(Mi, r

∗i |g, ri). As Mi and r∗i

are independent, this probability function can be bro-ken into the product of probabilities p(Mi, r

∗i |g, ri) =

p(Mi|g, ri)p(r∗i |ri). Given that Eq. (1) takes the sumof two uncorrelated Gaussian random variables, which isalso Gaussian, then, the probability distribution for Mi

is

p(Mi|g, ri) =1√

2πhiexp

[− (Mi − µi)2

2hi

], (3)

where µi = 1+g ·δri, and hi = (δci/c0)2 +σ2∆. Next, the

probability distribution that a cell i measures a positionr∗i given the true position ri is,

p(r∗i |ri) =1√

(2π)2|Σi|exp

[− (r∗i − ri)Σ

−1i (r∗i − ri)

T

2

].

(4)The likelihood of parameters g, {ri} given thatthe clusters measures values {Mi} and {r∗i } isL(g, {ri}|{Mi, r

∗i }) ≡ p({Mi, r

∗i }|g, {ri}). Assuming the

measurement each cell performs is independent, thislikelihood then factorizes into a product over cells i,L(g, ri|{Mi, r

∗i }) =

∏i p(Mi|g, ri)p(r∗i |ri). Estimators of

the gradient magnitude g and orientation φ can be ob-tained by maximizing this likelihood. However, we aremore interested in the best possible accuracy for an un-biased estimator g, which is given by the Cramer-Raobound [30],

σ2g ≡ 〈(g − g)2〉 =

(I−1

)g,g, (5)

where I−1 is the inverse of the Fisher information matrix,

given by Iα,β = −⟨∂2 lnL∂α∂β

⟩, with α and β are parameters

of the likelihood function, i.e. g and {ri}.It is possible to analytically compute the Fisher infor-

mation I and thus the best possible accuracy for mea-suring the chemical gradient in the limit of positionalinformation in terms of a sum over cell positions and thepositional information at each location. These resultsare presented in full detail in Appendix A. However, theresults are much more intuitively understandable with afew key assumptions.

B. Simplest case: shallow gradients and constantpositional error

Cells will likely have different positional uncertainties,depending on the underlying process through which cellsget their positional information, such as an external or

3

w/o pos errorsw/ pos errorsshallow gradient

(a) (b)

(c) (d)

Positional error

Sensing concentration error

w/ pos errorsw/o pos errors

shallow gradient

w/ pos errorsw/o pos errors

shallow gradient

FIG. 1. (a) Cluster under a linear gradient and a representa-tion of the two sources of uncertainties: sensing concentrationand positional errors. (b) Gradient sensing error increaseswith positional uncertainty. (c)–(d) Positional uncertain-ties becomes more dominant in steeper gradients. (c)Gradient sensing error normalized by its value in the absenceof positional errors, σ2

g(∆r = 0), as a function of the gradientsteepness (blue solid line). The black dashed-line indicatesthe limit where the gradient sensing error is dominated bypositional errors and it follows ∝ g2∆r2. (d) Gradient sens-ing relative error σg/g as a function of the gradient steep-ness. Parameters: number of cells N = 37 (correspondingto an hexagonal cluster of Q = 3 layers), Rcell = 10µm, (b)g = 0.005µm−1, (c)–(d) ∆r = 10µm.

a self-generated signal. To gain understanding, we willfirst look at the simplest case by assuming all the cellshave the same positional covariance matrix, Σi = Σ,which we also assume is isotropic Σls = ∆r2δls, wherel, s are matrix element indexes. (We study clusters withvarying positional uncertainty in Section II C). Withinthis section, the limited positional information is onlycharacterized by a single number, ∆r – the error inmeasuring a cell’s position. In addition to assuminga constant isotropic error, we simplify our results byassuming that the gradient is relatively shallow – i.e.that each cell has the same reading concentration errors,

δc2i ≈ δc2

= c0(c0+KD)2

nrKD. Finally, we consider that our

cluster of cells is roughly circular: the isotropy of theproblem then makes

∑i δr

2xi ≈

∑i δr

2yi ≈

12

∑i |δri|2,

and that∑i δrxiδryi ≈ 0. (In practice, for the calcula-

tions we present here, we show results for hexagonallypacked clusters of layers of cells, as in [16, 19]). With allthis, we can simplify the results of Appendix A to find thebest possible error in measuring the gradient magnitudeg as:

σ2g ≈

1

χ

(h+ (g∆r)

2), , (6)

where h = δc2/c20 + σ2

∆ and χ = 12

∑i |δri|2. Moreover,

we find that the gradient direction error is equal to thegradient magnitude relative error,

σφ =σgg. (7)

See Appendix A for detailed derivation of Eqs. (6)–(7).How does the presence of positional uncertainties affect

gradient sensing? In Fig. 1(b) we show that the gradientsensing error, σ2

g , as a function of the positional uncer-tainty ∆r, using the generic solution of Eq. (5) (see Ap-pendix A) and the shallow gradient approximation fromEq. (6). Unsurprisingly, positional uncertainties increasegradient sensing error – the more uninformed cells areabout their positions, the worse an estimate the clustermakes. This added error from positional uncertainty in-creases as ∆r2 in the shallow gradient limit (Eq. 6). Wesee in Fig. 1 that the additional uncertainty from finitepositional information is significant when ∆r is of orderof the cell size.

Eq. (6) also tells us there are two terms controllingthe estimate error. The first, h, is the contribution fromthe cells reading and reporting their local concentration.The second term, is the positional uncertainty contri-bution to the gradient sensing error, controlled by g∆r.The relative weight of these two terms allows to iden-tify two regimes in which the gradient sensing is limitedin one case by the reading concentration fluctuations,(g∆r)2 � h, and on the other case by the positional un-certainties, (g∆r)2 � h. When positional uncertaintiesare smaller than reading concentration errors, we recoverthe results from [19], σ2

g ≈ 1χ h. In the other extreme,

(g∆r)2 � h, gradient sensing is primarily limited by po-sitional information, σ2

g ≈ 1χg

2∆r2.

The gradient steepness g, controls the relative impor-tance of positional information: the steeper the gradientis, the larger the contribution from the positional uncer-tainty. In Fig. 1(c) we show the gradient sensing error σ2

g

normalized by the bound in absence of positional uncer-tainty σ2

g(∆r = 0) increases with the gradient steepnessg. In steeper gradients, small errors in measuring a cell’sposition lead to larger differences between the concen-tration at the true position, c(ri), and at the estimatedposition, c(r∗i ). On the other hand, this also implies, thatin the limit of g → 0, where error bounds are most im-portant (where single cells fail to follow a gradient whileclusters do), that positional information does not play asignificant role.

In analogy to single-cell gradient sensing, it would benatural to assume that the relative error, σg/g, whichcontrols the angular uncertainty σφ, always gets betterwith the increasing g – i.e. that cells can better mea-sure the angle to the gradient direction if the gradient issteeper. This is true for clusters without positional error[19]: σg/g is plotted as the red line in Fig. 1(d). However,in the presence of positional uncertainties, the relative er-ror is limited at large g, converging to a constant value∆r/√χ in the case of a shallow gradient (Fig. 1(d)). This

4

surprise occurs because the increasing error due to posi-tional uncertainty σ2

g ∼ g2 in Eq. (6) exactly balances

the g2 in the denominator of σ2g/g

2.

C. Non-uniform positional information

In the above section we assumed that all cells have thesame positional errors. However, positional uncertaintyis not necessarily uniform within the cluster. If cells gettheir positional information from a secreted factor or me-chanical signal, the distance from the source may influ-ence the accuracy ∆r (see, e.g. [31]). In addition, obtain-ing positional information may have an associated costand therefore, it may be optimal for the cell cluster toonly have some cells measure their location. We study aprototypical example of both cell specialization and po-sitional errors varying from cell to cell by introducing asecond type of cells which have more positional informa-tion. These “informed” cells have a positional error ∆rinf

that is smaller than “normal” cells, ∆rinf ≤ ∆r. Then,we distribute a fraction f of informed cells at differentpositions over the cluster to see the effect of positionaluncertainty localization. We follow three different typesof distributions: informed cells are distributed (i) at theedge, (ii) at the center, or (iii) randomly over the cluster,see Fig. 2(a). We use the general solution for the gradi-ent sensing error given by Eqs. (A4)–(A6). We note thatthere are often many cells that are equidistant from thecluster center or edge, so unless a layer of the hexagonalpacking is completely filled, there will be multiple pos-sible configurations at a particular fraction of informedcells f (Fig. 2a, bottom row). To address this, we sam-ple 100 realizations for each distribution. Fig 2 shows anaverage over these realizations. Gradient sensing has thelowest error when informed cells are closer to the edgesand the largest when they are placed closer to the cen-ter of mass. This is illustrated in Fig. 2(b), that showsthe gradient sensing error normalized by the case f = 0,(σ2g0

= σ2g(f = 0)), as a function of the fraction of in-

formed cells. For all three types of distributions, increas-ing the fraction of informed cells reduces the error of g.Gradient sensing error takes the same value for the threedistributions, in the extreme cases of no informed cells(f = 0) and all cells informed (f = 1), but following dif-ferent paths in between. Informed cells distributed closeto the edge lead to the lowest sensing errors of the threedistributions. Similarly, reducing the positional uncer-tainty of the informed cells results in higher accuracy ingradient sensing, which is again greater when informedcells are distributed closer to the edge (Fig. 2(c)).

To compare between clusters with informed cells andcluster with a single cell type population, we intro-duce the “equivalent” constant-positional-error cluster,in which all the cells have ∆r = (1 − f)∆r + f∆rinf

(dashed black line in Fig. 2(b)–(d)). Results show thatplacing informed cells closer to the edge enhances gra-dient sensing when compared to the equivalent cluster,

(a) (b)

(c) (d)

Cell with small positional uncertainty

Cell with large positional uncertainty

Center Random Edge

Lower error

Higher error

FIG. 2. Gradient sensing errors for a cluster with a fractionf of informed cells. (a) Informed cells are distributed overthe cluster in three distinct manners: closer to the cluster’scenter, randomly, or closer to the cluster’s edge. The rowwith the small clusters illustrates the existence of multipleequivalent ways of distributing the informed cells. (b)–(c)Gradient sensing variance, σ2

g , normalized by the case of noninformed cells, σ2

g0 = σ2g(f = 0) as a function of the fraction

of informed cells (b) and the ratio of the two types of cellpositional uncertainties (c). (d) Relative difference betweeninformed cluster and the equivalent constant-positional-error

cluster, δσg =σg(∆r)−σg(∆rinf ,∆r)

σg(∆rinf ,∆r), as a function of the gradi-

ent steepness. δσg > 0 (δσg < 0) indicates a lower (higher)gradient sensing error for the cluster with the informed cellscompared to the uniform positional error cluster. The dashedblack line indicates δσg = 0. For (b)–(c)–(d) color codes arethe same. Solid lines with symbols represent the three dif-ferent distribution patterns: Center (red squares), Random(light blue crosses), and Edge (dark blue circles). The dashedblack line represents the equivalent one type of cells clusterwith positional uncertainty equal to ∆r = f∆rinf +(1−f)∆r.Parameters: cluster size N = 91 (Q = 5 layers in the hexag-onal cluster), ∆r = 2Rcell, (b) ∆rinf = 0.5∆r, (c) f = 0.25,(d) f = 0.25, ∆rinf = 0.2∆r, and ∆r = 20µm. Curves showan average of 100 realizations.

while doing it closer to the cluster’s center or randomlyresults in a worse or similar performance, respectively.To further explore the role of informed cells and theirdistribution inside the cluster, in Fig. 2(d) we showsthe relative difference between the gradient sensing er-rors for the cluster with informed cells and its constant-

positional-error equivalent, δσg =σg(∆r)−σg(∆rinf ,∆r)

σg(∆rinf ,∆r), as

a function of the gradient steepness. Results show thatincreasing the gradient steepness enhances the trends de-scribed before, in which the benefit of reduced positionalerror is larger for cells at the cluster edge. This obser-vation agrees with our previous results that showed thatpositional errors have a higher effect over gradient sensingin steeper gradients. Interestingly, in steeper gradients,even randomly distributed informed cells perform better

5

Small cluster of informed cells

Cluster center of mass

(a) (b)

Lower error

Higher error

Lower error

Higher error

FIG. 3. (a) Schematic cell cluster with a group of fourinformed cells located at a distance r from the cluster’scenter and direction θ with respect to the chemoattractantgradient. On the Right: A non exhaustive representationof different configurations for the small cluster. (b) Rel-ative gradient sensing steepness (left panel) and direction,

δσφ =σφ(∆r)−σφ(∆rinf ,∆r)

σφ(∆rinf ,∆r), (right panel) errors when the

informed group of cells is displaced in different directions:θ = 0 (red line and squares), θ = π/4 (light blue line andcrosses) and θ = π/2 (dark blue line and circles). Same asin Fig. 2, δσg,φ > 0 (δσg,φ < 0) represent a lower (higher)error relative to the constant-positional-error equivalent clus-ter. The dashed black line indicates δσg,φ = 0 and also rep-resent the equivalent cluster without informed cells and posi-tional uncertainty ∆r. Parameters: cluster size N = 91 and∆rinf = 0.5∆r. Curves show an average of 100 realizations.

than the equivalent cluster (light blue line in Fig. 2(d)).The reason is that by randomly distributing the informedcells, a few will lie close to the edge of the cluster, andthose are the ones that dominate and end up having amajor weight in sensing the gradient.

Together, these results show that positional informa-tion becomes more relevant at the edge of the clusterand is even more significant in steeper gradients. Theimportance of the edge is large enough that evenhaving a few informed cells randomly localized to theedge can significantly decrease error. If determin-ing cell position is difficult, our results suggest thatinstead of maximizing positional information evenlyacross the entire cluster, cells may improve in gradi-ent detection by prioritizing the localization of edge cells.

We argued that informed cells are generally most ben-eficially placed at the cluster edge. How does gradientsensing accuracy change if we place these cells anisotrop-ically? We study the change in the errors for gradientmagnitude g and direction φ when a small group of four“informed cells” are displaced from the center to the edgeof the cluster in different directions, see Fig. 3(a). Wenote that for this case, σφ 6= σg/g, due to anisotropy; weuse the full analytical solution of Appendix A to computethe bounds σg and σφ. As in the previous case, there aremultiple equivalent ways to place a cluster of informedcells at a position (r, θ), see Fig. 3(a). To account forsuch variability, we averaged over 100 realization for eachcondition pair (r, θ). We find that the errors for the es-timators of the gradient magnitude and direction evolvedifferently depending on the orientation of this cluster ofcells, Fig. 3(b). Moving the small informed cluster along

the axis parallel to the gradient direction, θ = 0, leadsto an improvement in sensing the gradient steepness g.By contrast, moving along the axis perpendicular to thegradient, θ = π/2, sensing the orientation of the gradi-ent is improved. Finally, there is a compromise situationwhen moving along θ = π/4, where both gradient steep-ness and direction sensing are improved, but to a lesserextent. These results support again that positional in-formation is more significant at the edge of the cluster,but also that the orientation with respect to the gradientmatters. In chemotaxis, for instance, cells need to sensethe direction rather than the steepness of the gradient.Therefore, a cluster might strive to locate cells at theedge of the axis perpendicular to the gradient directionto enhance its ability to chemotax. However, this couldonly work in the case that the cluster has previous knowl-edge of the gradient direction. Our findings are similarin concept to the ideas reported in [32] for single cells, inwhich elongated cells sense the gradient steepness and di-rection with different accuracies depending on the orien-tation of the cell. Note that in comparing Fig. 3(b) panelsthat δσg(0) 6= δσφ(π/2) and δσg(π/4) 6= δσφ(π/4). Thisis due to hexagonal clusters not having perfect circularsymmetry, being more elongated, in our case, in the di-rection parallel to the gradient.

D. Collective chemotaxis without positionalinformation: tug–of–war model

In our above approach, a group of cells senses a chem-ical gradient collectively by making local measurementsof the chemoattractant concentration and and their po-sitions. Then, by applying the maximum likelihood es-timator method, we find the best possible measurementof the gradient the group of cells can make. However, weshowed that this measurement is constrained by the lim-ited positional information available to the cells. Here,we introduce a different mechanism by which cells senseand follow the gradient without needing cells to mea-sure their locations, based on the model in [16]. An in-teresting feature of this model, which makes it differentfrom the previous one, is that it does not depend on thepositions of the cells, but only on the direction to thenearest cells. Cells interact by contact inhibition loco-motion (CIL) [33, 34], in which contacting cells polarizeaway from each other. This interaction dynamic leadsto a cluster where only cells at the edge are polarized[5, 13, 16], see Fig. 4(a). Cells on the inside of the clus-ter, completely surrounded by neighboring cells, do notpolarize, whereas cells on the edges, which only see neigh-boring cells on the interior side, polarize away from thecluster. Another important property of the model is thatcell CIL is modulated by the local chemoattractant con-centration. In this way, those cells at the front (up thegradient) of the cluster polarize and pull stronger thanthe ones on the back (down the gradient), resulting ina net movement toward up the gradient in a tug–of–war

6

Cell polarity by CIL in the absence of a gradient

Cells sense different concentrations

+

Polarity proportional to cell concentration

MLE preferred

Tug-of-war preferred

trade-off point

FIG. 4. Tug–of–war model (a) Scheme representation of the tug–of–war model. (b) Gradient sensing variance as a functionof the positional uncertainty for the tug–of–war model (red solid line) and the MLE model (blue solid line). The black dashedline marks the crossover and the trade-off positional uncertainty between the two models. (c)–(d) Trade–off value as a functionof: (c) the cluster number of cells, and (d) the gradient steepness. (e) Phase diagram of the trade–off values for differentcluster’s sizes and gradient steepness. The dashed red line indicates a boundary given by ∆rt = 2Rcell. Positional errors, ∆r,and trade–off values, ∆rt, are re-scaled in of cell’s radius. Parameters: (b) same as Fig. 1(b), (c) g = 0.005µm−1, and (d)N = 37 (Q = 3).

like dynamic. From now on we will refer to this model asthe “tug–of–war” model.

Our goal in studying the tug-of-war model will be toextend the results of [16] so that they can be directly com-pared with our maximum likelihood results above, whichrequires us to include fluctuations in concentration dueto receptor-ligand binding as well as cell-to-cell variabil-ity. Following [16], we start from the assumption thatcells behave as stochastic particles. The motility of cell iresults from the balance between the intracellular forces,Fij , due to cell-cell adhesion and volume exclusion, andthe cell’s polarity pi,

∂tri = pi +∑i6=j

Fij . (8)

Computing the cluster velocity by summing over all cellsin Eq. (8) and noting that Fij = −Fji, we find

vc =1

N

∑i

pi. (9)

Cell polarity obeys the differential equation,

∂tpi = − 1

τppi + σpεi(t) + βiqi, (10)

where τp is the characteristic time it takes for the po-larity to relax to its steady state value, and σp is themagnitude of a fluctuating noise that can drive the po-larity away from its steady-state value. εi(t) is a vectorGaussian Langevin noise for cell i that fulfills 〈εi(t)〉 = 0and 〈εiµ(t)εjν(t′)〉 = δµνδijδ(t− t′), with i, j cell indexes,while µ, ν are indexes for the Cartesian coordinates x, y.

The last term in Eq. (10) comes from the CIL interac-tions, where βi is the cell’s susceptibility to CIL, and thevector qi is the resulting direction of contact interactionof cell i and its neighbors, qi =

∑i∼j(ri − rj)/|ri − rj |,

where i ∼ j represent the sum over the cell’s neigh-bors, defined as those cells within a distance 2.1Rcell.Note that qi ≈ 0 for interior cells and qi 6= 0 foredge cells. The tug-of-war model of [16] assumed thatsusceptibility of the cell i is proportional to chemoat-tractant concentration, βi = β c(ri, t). Here, we musthandle the variability in the measured concentration dueto both ligand-receptor binding and cell-to-cell variabil-ity. We assume that cells polarize in response to whatthey believe the chemoattractant concentration to be,Mic, instead of the true concentration value at the cellpositions, c(ri, t). This will add an additional sourceof noise to Eq. 10, which we will now write explicitly.If the susceptibility of cell i is βi(t) = βc0Mi(t), we

7

can write Mi(t) = c(ri, t)/c0 + Ξi(t). Here, Ξi(t) =δci/c0ζ

ci (t) + σ∆ζ

∆i (t) is the noise in Mi. Note that, un-

like Eq. 1, we must specify the time-dependence of theerrors due to ligand-receptor binding and cell-to-cell vari-ation. We explicitly introduce ζci (t) to measure the fluc-tuations due to concentration sensing and ζ∆

i (t) for thefluctuations in cell-to-cell variability. We can character-ize the time scale over which the errors due to receptor-ligand concentration and cell-to-cell variability are per-sistent by the correlation functions 〈ζc(t)ζc(0)〉 = Cc(t)and 〈ζ∆(t)ζ∆(0)〉 = C∆(t). Then we have 〈Ξi(t)〉 = 0and 〈Ξi(t)Ξj(0)〉 =

(δc2i /c

20Cc(t) + σ2

∆C∆(t))δij . Next,

we insert the expression for the polarity susceptibilityβi(t) = βc(ri) + βc0Ξi(t), into Eq. (10) to arrive at

∂tpi = − 1

τp(pi − γc(ri, t)qi) + σpεi(t) + βc0qiΞi(t),

(11)where we have defined γ = τpβ.

We can solve Eq. 11 by directly integrating it. If weassume that τp is small enough for the polarity to relaxto its steady-state solution before the cluster reorganizes,so the contact vectors qi are fixed, then pi is given by

pi(t) =µi + σp

∫ t

0

dt′e−(t−t′)/τpεi(t′)

+ βc0qi

∫ t

0

dt′e−(t−t′)/τpΞi(t′) (12)

where µi = γc(ri)qi and we chose initial conditionspi(0) = µi. From Eq. (12), it is possible to computethe mean and variance of pi(t). We assume that corre-lations decay exponentially, so Cs(t) = e−t/τs , with τsthe correlation time and the index s = {c,∆}. Our finalresults will not be crucially dependent on τc, τ∆, so thisis primarily a convenience to make the analytical calcu-lations simple. Then, recalling that εi(t) is a Gaussianuncorrelated noise with mean zero, we compute the meanand the covariance of pi(t) (see Appendix B),

〈pi(t)〉 =µi,

〈(pµi(t)− 〈pµi〉)(pνi(0)− 〈pνi〉)〉 =σ2pτp

2δµν

(1− e−2t/τp

)+β2qµiqνi

(δc2iT(t, τc) + c20σ

2∆T(t, τ∆)

),

where

T(t, τs) =τ2p

1 + υs

(1 +

1

1− υs×((1 + υs)e

−2t/τp − 2e−(1+υs)t/τp)),

and υs =τpτs

. Note that typical values for the polariza-

tion relaxation time τp are ∼ 20 minutes [16]. This ismuch longer than the typical correlation times for thechemoattractant–cell receptors binding dynamics whichis on the order of the τc ∼ 1s [19]. However, recall thatcell concentration sensing is limited by CCV which has

a much slower correlation time than can reach up to 48h in human cells [19, 35]. To calculate the variance ofpi at a reasonable steady-state, then, we should think ofthe CCV noise as constant in time, and wait for times tmuch longer than the other relaxation times. Explic-itly, this means τ∆ � t � τp � τc. In this limit,

T(t, τ∆) ≈ τ2p

1+τp/τ∆≈ τ2

p and T(t, τc) ≈τ2p

1+τp/τc. As

a result, the polarity covariance is 〈(pµi − 〈pµi〉)(pνi −〈pνi〉)〉 = V iµν , where V iµν =

(γ2c20hTiqµiqνi +

σ2pτp2 δµν

)and hTi =

δc2i /c20

1+τp/τc+

σ2∆

1+τp/τ∆≈ σ2

∆. Essentially, the vari-

ance of the polarity has two terms: the time-averagednoise from measuring concentration, which is propor-tional to hTi – which we have named to make clear it iseffectively a time average of hi. The second arises fromthe added noise σp in Eq. 10, which is a source of noisenot included in the maximum likelihood model.

Then, with the distribution of polarities in hand, fromEq. (9) we proceed to compute the expected value andvariance of the cluster velocity. In the case of the ex-pected value, pointing out that

∑i qi = 0, we arrive at

the following simple expression,

〈vc〉 = γc0Mg, (13)

where Mµν = 1N

∑i δrµiqνi . For the variance we have,

∆v2cµν =

γ2c20N2

∑i

hTi qµiqνi +τpσ

2p

2Nδµν . (14)

Now, we ask the question, what is the gradient sensingerror for this tug–of–war model? We use Eq. (13) andEq. (14) to propagate errors and compute the gradientsensing error for the tug–of–war model (Appendix B),

σ2g ≈

2

(∑i qi · δri)

2

(∑i

|qi|2hTi +σ2pτp

γ2c20N

), (15)

under the assumption of isotropic clusters, as in Eq. 6.How can we make a consistent comparison between

the tug-of-war model and the maximum likelihood esti-mation? There are two key differences. First, the tug-of-war model includes an additional source of noise – thepolarity noise σp representing fluctuations in cell polar-ity arising from factors outside of concentration sensing.Second, the tug-of-war model, as a dynamical model, ex-plicitly includes an average over a characteristic time τp,while the maximum likelihood estimate is based on a sin-gle snapshot of the measured concentration. The firstissue is simple to deal with: since the MLE method re-turns a lower bound for the gradient sensing error, it ismost comparable to deal with the best possible situa-tion for the tug–of–war model, i.e., no polarity fluctua-tions, σp = 0. The issue of time-averaging is less clear.

Eq. (15) depends on hTi =δc2i /c

20

1+τp/τc+

σ2∆

1+τp/τ∆. To com-

pare with the instantaneous snapshot, we would have tochoose τp � τc, τ∆, in which case hTi ≈ hi. However, this

8

is inconsistent with our estimates above where τp ∼ 20minutes and τc ∼ 1 second. In practice, however, this dis-tinction is not very important, because both hTi and hiare dominated by the CCV noise, so hTi ≈ hi ≈ h ≈ σ2

∆.We will then directly assume that hTi = hi = h, so thatthe tug-of-war and MLE results are directly comparable[36]. With these results, we can compute the uncertaintyin the estimation of g arising from the tug-of-war modelas:

σ2g ≈

1

χtowh, (16)

where χtow =(∑i δri·qi)

2

2∑i |qi|2

. Note how similar Eq. (16)

is to Eq. (6) when ∆r = 0. The only difference is inthe geometrical factors χ and χtow. The prefactor χtow

involves sums over the CIL directional vectors qi, whichare only non zero on the edges cells, supporting againthat only those cells contribute to sensing the gradientin the tug–of–war model.

We now want to compare the tug–of–war model andthe MLE method to see which of them, and under whatconditions, senses the gradient more accurately. Fig. 4(b)shows the gradient sensing error as a function of posi-tional uncertainties for both models. In this case, sincethe tug–of–war model does not depend on positional un-certainties its gradient sensing error is constant. In ab-sence of positional errors, the MLE method makes abetter estimation of the gradient than the tug–of–warmodel, since this last one only uses information from theedge cells in contrast with the MLE model that uses allthe cells. However, as cells become less informed abouttheir positions, a trade-off occurs where, beyond a certainpositional uncertainty value, ∆rt, the tug-of-war modelturns out to be a better estimator of the gradient. Thiscrossover exists since the tug-of-war model does not needto estimate cell positions and is therefore independentof positional uncertainties. Under the shallow gradientapproximation, from Eq. (6) and Eq. (16), we can findanalytically the crossover between both models,

∆rt ≈

√h(

χχtow− 1)

g. (17)

Using maximum likelihood estimation, increasing thecluster’s size can compensate for limitations in cells po-sitional information, see Fig. 4(c). This result can beeasily predicted from Eq. (17), by considering the scal-ing of the geometrical factors χ and χtow with the clus-ter radius, Rcluster. We expect χ = 1

2

∑i |δri|2 to scale

as χ ∼ R4cluster [19]. We find this by approximating

the sum over cells as an integral over a roughly circu-lar cluster,

∑iR

2cell ≈

∫Rcluster

d2r, so χ = 12

∑i |δri|2 ∼

R−2cell

∫Rcluster

d2rr2 ∼ R4cluster. For the tug-of-war model,

χtow =(∑i δri·qi)

2

2∑i |qi|2

. For a roughly circular cluster, qi will

be along the direction δri for cells at the edge, and zero

for cells in the interior. Edge cells will have roughly thesame magnitude of qi, |q|. Thus, δri · qi ≈ |q|Rcluster foredge cells, and χtow ≈ (Nedge|q|Rcluster)

2/2Nedge|q|2 ∼NedgeR

2cluster. As, for a roughly circular cluster, we ex-

pect the number of cells on the edge Nedge to be pro-portional to 2πRcluster, we see χtow ∼ R3

cluster. Then,χ/χtow ∝ Rcluster and we get from Eq. (17) that the∆rt shifts to larger values with the cluster size, (∆rt ∝R

1/2cluster).On the other hand, the gradient steepness plays an

opposite role in shifting the trade-off values. In a steepergradient, the MLE method becomes more sensitive touncertainties in cell positions. In this scenario, thetug-of-war model benefits from this limitation in theMLE method and senses the gradient more accurately,even in the presence of small positional errors. As aconsequence, the trade-off values shift towards smallerpositional uncertainties with the gradient steepness, seeFig. 4(d). We can summarize everything we describedabove with the phase diagram for the trade–off valuesshown in Fig. 4(e). From the figure, we see that areas inblue represent conditions in which the trade–off valuesare small. In these areas, a group of cells would be moreprompt to uses the tug–of–war method rather than theMLE to sense the gradient. On the other hand, red areasshow large trade–off values where it might be optimalfor cells to pay the cost of estimating their positions andfollow the MLE method for gradient sensing. Lastly, itis interesting to note the case of N = 7. This clustersize corresponds to the 1–layer hexagonal cluster, wherealmost all cells are edge cells, and there is only oneinterior cell. The interior cell is located at the cluster’scenter, so that δr = 0, and therefore has no contributionto the gradient sensing in the MLE method (recall thatχ =

∑i |δri|2/2 in the shallow gradient, see also for

the general solution Eqs. (A4)–(A6)). Consequently,both models use the same cells to sense the gradient,except that the MLE approach is affected by positionaluncertainty, whereas the tug-of-war is not. For thisreason, even small uncertainties in cell positions aresufficient for the tug–of–war model to outperformthe MLE and become the best gradient estimator.In the case of the shallow gradient approximation, wehave that χ = χtow and from Eq. (17) we obtain ∆rt ≈ 0.

III. DISCUSSION

We have studied the role of limited positional informa-tion in collective gradient sensing. Within the contextof gradient sensing, how cells get their positional infor-mation – or if they estimate the gradient without thisinformation – is still an open question. Positional in-formation within a group of cells has been studied inthe context of developmental biology, where an exter-nal morphogen signal determine cell’s fate [20, 22, 24];cells may also sense their position within a colony by

9

sensing mechanical stresses [21]. Moving from a sensedchemical or mechanical signal to positional knowledgewithin a cluster is a complex multistage process [24].Consequently, attaining precise positional information isa time costly process, and there are essential tradeoffsbetween the rate at which a pattern can be establishedand its precision[37]. Establishing a reproducible gra-dient is likely even more challenging for migratory cellsin clusters [5, 13], as cell position and number of cellsin the cluster all evolve over time. However, there is apotential candidate for a morphogen-like source of posi-tional information within neural crest cell cluster chemo-taxis: the complement fragment C3a, which acts as a“co-attractant” [38]. Earlier simulations show that theco-attractant C3a can be established in a graded fashion[39]. Here, we have suppressed the details of how po-sitional information is established, neglecting potentialissues with establishing a reliable gradient, and focusedon the unavoidable impact of limited positional informa-tion in collective gradient sensing. Consequently, theseresults should in a sense be considered the best case forexploiting positional information: if the tradeoff argu-ment of Fig. 4 predicts that maximum likelihood sens-ing using limited positional information is optimal, weshould be skeptical about this. Contrarily, if the trade-off suggests MLE is inferior to the tug-of-war, we shouldbe highly confident that this is true. For this reason, itseems unlikely that collective gradient sensing by smallclusters, such as border cell migration [4, 40, 41], reliessignificantly on positional information.

Our results show that whether cells should use posi-tional information in estimating a chemical gradient ori-entation depends not only on the amount of positionalinformation but also on parameters like the cluster sizeand the gradient steepness. Large clusters overcome po-sitional uncertainties by having numerous cells, i.e. moreindependent measurements, combining their estimates oftheir positions and the local concentration of chemoat-tractant. This large benefit from gaining informationfrom all cells and weighting them according to positioncan outweigh the costs of having limited positional infor-mation. This is especially apparent in comparison withthe tug-of-war model, which only uses information fromthe small fraction of the cells at the cluster’s edge.

Gradient steepness is also a key parameter. Finite po-sitional information limits the accuracy of collective gra-dient sensing by a group of cells – but this limitationbecomes more significant for steeper gradients, since dif-ference in positions lead to larger changes in the chemoat-tractant concentration. Remarkably, for shallow gradi-ents, (g → 0), the limit where collective gradient sensingis most important, as single cells can become ineffectivesensors, positional information turns out not to be rele-vant.

While our simplest analytical results are derived for acluster with a constant positional information ∆r, we cangeneralize our results beyond this point, finding that theeffect of limited positional information differs depending

on a cell’s position within the cluster. Positional uncer-tainties in cells farther away from the cluster center ofmass have a greater effect on the accuracy of gradientdetection – again, suggesting that edge cells must playa large role in sensing. Our observations suggest that agroup of cells may benefit from having specialized cellscapable of sensing their position more accurately thanthe rest. The idea that cells may specialize depending onposition in the cluster or chemoattractant concentrationhas also been suggested in other contexts [17, 42–44].

In addition to the maximum likelihood estimation of[19], which requires positional information, and the tug-of-war models [13, 16, 17], which do not, there are manyother models of collective gradient sensing [15]. Onebroad category of a collective sensing mechanism thatis not affected by positional information is collective lo-cal excitation–global inhibition (LEGI), where a localreporter reads out local chemoattractant concentrationand a global inhibitor measures the global concentra-tion [39, 45–48]. In LEGI, cell–cell communication im-poses a maximum length scale at which gradient sensinginformation is reliably shared, leading to a saturation ofgradient sensing accuracy with cell number [46], compat-ible with experiments on mammary organoids [14] butnot lymphocyte clusters. While we have not treated theLEGI model in detail here, we argue that, similarly withthe tug-of-war model, it does not require the cluster toknow cell positions – but it does not gain the benefitsof measuring signals over the entire cluster. We wouldsimilarly expect LEGI sensing to be suboptimal to themaximum likelihood estimator for large clusters, shallowgradients, or low positional uncertainty.

Our results show that a sensing mechanism that doesnot rely on positional information, such as tug-of-war,is optimal for sufficiently small clusters or sufficientlylarge gradients. In Figure 4, we have shown a phase di-agram assuming that positional information is accurateto roughly a cell diameter, finding that tug-of-war is al-ways optimal for clusters of seven cells, and will becomeoptimal at large gradients for larger clusters. The experi-ments that originally suggested tug-of-war models whereonly edged cells participate in sensing and responding tothe gradient [5, 13] involved tens to hundreds of cells. Ifthe tug-of-war behavior represents an evolutionary opti-mum, it may reflect either a low ability to gain positionalinformation or a typical need to follow relatively steepgradients.

ACKNOWLEDGEMENTS

BAC acknowledges support from NSF grant PHY1915491. We thank Kurmanbek Kaiyrbekov for a closereading of the manuscript.

10

Appendix A: Derivation of the Fisher Informationmatrix and computing of the lower bound for the

gradient sensing error through the maximumlikelihood estimation method

We apply MLE to obtain the gradient sensing error ofa group of cells that sense their local chemoattractantconcentration and have limited positional information.Each cell i performs two independent measurements, ameasurement of the local concentration Mi and a mea-surement of its position r∗i given by Eqs. (1) and (2)respectively. From the Results section, we know we can

write the likelihood function as, L(g, {ri}|{Mi, r∗i }) =∏

i p(Mi|g, ri)p(r∗i |ri). In general, it is easier to workwith the log of the likelihood instead, since the productbetween the probabilities becomes a sum. Then, consid-ering a gradient g = gxx + gy y and a covariance matrixfor the positional error,

Σi =

[σ2xi ρiσxiσyi

ρiσxiσyi σ2yi

], (A1)

we find the log of the likelihood to be

logL(g, {ri}|{Mi, r∗i }) =− 3

2

∑i

log(2π)− 1

2

∑i

log hi −∑i

(Mi − µi)2

2hi− 1

2

∑i

log(σ2xiσ

2yi(1− ρ

2i ))

−∑i

1

2(1− ρ2i )

[(r∗xi − rxi)

2

σ2xi

− 2ρi(r∗xi − rxi)(r

∗yi − ryi)

σxiσyi+

(r∗yi − ryi)2

σ2yi

]. (A2)

where µi = 1 + g · δri and hi = (δci/c0)2 + σ2∆ are the

mean and variance of the local concentration measure-ment performed by cell i.

From Eq. (A2), we could find the global maximum toobtain the maximum likelihood estimator. Here, we arenot interested in the estimator itself but in its fluctua-tions. We want to compute the estimator error, whichwe know must converge to the Cramer-Rao bound, givenby the inverse of the Fisher information matrix, (I)

−1.

This bound limits the accuracy of any unbiased measure-

ment of the gradient g [30]. This bound puts a limit onminimal errors of any unbiased estimator of a parameter(e.g. an estimator α) away from the true value of thatparameter (α). This bound is, for parameters α and β,

〈(α− α)(β − β)〉 =(I−1

)α,β

, (A3)

Recalling the Fisher information definition, Iα,β =

−⟨∂2 lnL∂α∂β

⟩, we next take partial derivatives of Eq. (A2),

and compute the expectation values, getting

I =

∑i fiδr

2xi

∑i fiδrxiδryi gxf1δrx1

gyf1δrx1. . . gxfNδrxN gyfNδrxN∑

i fiδrxiδryi∑i fiδr

2yi gxf1δry1

gyf1δry1. . . gxfNδryN gyfNδryN

gxf1δrx1gxf1δry1

g2xf1 + m1

σ2x1

gxgyf1 − m1ρ1

σx1σy1

. . . 0 0

gyf1δrx1gyf1δry1

gxgyf1 − m1ρ1

σx1σy1

g2yf1 + m1

σ2y1

. . . 0 0

......

......

. . ....

...gxfNδrxN gxfNδryN 0 0 . . . g2

xfN + mNσ2xN

gxgyfN − mNρNσxN σyN

gyfNδrxN gyfNδryN 0 0 . . . gxgyfN − mNρNσxN σyN

g2yfN + mN

σ2yN

where fi =(a+µi)

2((a+3µi)2+2anrµi)+2a2n2

rσ2∆

2(µi(µi+a)2+anrσ2∆)

2 , mi =

1(1−ρ2

i ), a = KD/c0, and δri = ri − rcm. The order

of the Fisher information matrix elements correspondto: {gx, gy, rx1 , ry1 , . . . , rxN , ryN }, such that, as for exam-

ple, I1,1 = −⟨∂2 lnL∂g2x

⟩, I1,2 = −

⟨∂2 lnL∂gx∂gy

⟩, I1,2(N+1) =

−⟨

∂2 lnL∂gx∂ryN

⟩. The derivation of the Fisher information

matrix from the likelihood function is not hard, but takessome calculation effort. To gain some intuition about theshape it takes, we point out some aspects. Note that gand ri are only related by the likelihood function throughthe term g·δri, which appears inside the expressions of hiand µi. This explains that the fi parameters are in everyelement of the Fisher information matrix, since takingtwo times the derivatives with respect to g or ri leads to

11

the same factor. Also, from the same term we can under-stand the full first two rows/columns in the Fisher infor-mation matrix, since these correspond to the derivativeswith respect to g and ri. Lastly, note that the elementsrelated to the double derivatives with respect to positionshave a contribution coming from the gradient terms, plusthe contribution from the positional uncertainty terms ofthe likelihood function.

We are interested in computing the gradient sensingerrors, σ2

gx ≡ 〈(gx − gx)2〉 =

(I−1

)gx,gx

and σ2gy ≡

〈(gy − gy)2〉 =

(I−1

)gy,gy

. Following a guided Gaus-

sian elimination procedure we can obtain the inverse ofthe Fisher information matrix for the elements associ-ated with the gradient (see Appendix D). The maximumlikelihood estimator errors for the gradient are,

σ2gx =

1

S∑i

γiδr2yi , (A4)

σ2gy =

1

S∑i

γiδr2xi , (A5)

σgx,gy =− 1

S∑i

γiδrxiδryi , (A6)

where S =∑i γiδr

2xi

∑i γiδr

2yi −

(∑i γiδrxiδryi

)2

, and

γi = fi1+figTΣig

.

Finally, it is easier to interpret the gradient sensingerrors in terms of the steepness, g, and direction φ, ofthe gradient. We can obtain them by re-parametrizingthe Fisher information matrix as,

Iθ′ = JT Iθ(θ(θ′))J,

where J is the Jacobian matrix, Jij = ∂θi∂θ′j

, and θ =

{gx, gy} and θ′ = {g, φ} are the variables we want totransform. The errors for the gradient steepness and di-rection then are,

σ2g = cos2(φ)σ2

gx + sin2(φ)σ2gy + 2 cos(φ) sin(φ)σgx,gy

(A7)

σ2φ =

sin2(φ)σ2gx + cos2(φ)σ2

gy − 2 cos(φ) sin(φ)σgx,gy

g2.

(A8)

We presented these equations for an isotropic cluster inthe main text, where σgx,gy ≈ 0 and σgx ≈ σgy . In

this case, we find that σ2φ = σ2

g/g2 – but this is only

true for isotropic clusters. Note that Eq. (A8) fails for

large enough σg, since φ is constrained between 0–2π,and consequently, σφ is bounded.

Shallow gradient approximation

In the limit of shallow gradient approximation it is pos-sible to obtain simpler expression for the gradient sens-ing errors. In this limit, gRcluster � 1, meaning that thecell’s ligand-receptor fluctuations can be approximated as

δci ≈ δc =√

1nr

c0(c0+KD)2

KD. Variations of the chemoat-

tractant concentration along the cell’s cluster are small,thus we can assume that all cells have the same reading

concentration errors. Then, hi ≈ h = δc2

+ σ2∆. More-

over, since nr � KD/c0, fi can be approximated as theinverse of the reading concentration error, fi ≈ f = 1/h.Note that the limit of large cell-cell variability, σ∆ �δci/c0, would lead to the same approximation. Assumingconstant isotropic positional errors, such that ρi = 0 andσxi = σyi = ∆r, we find that γi = 1

1/fi+gTΣig≈ 1

h+g2∆r2

is independent of i and thus Eqs. (A4)–(A6) take a sim-plified form,

σ2gx =

∑i δr

2yi

S(h+ g2∆r2

),

σ2gy =

∑i δr

2xi

S(h+ g2∆r2

),

and in polar coordinates,

σ2g = Γg

(h+ g2∆r2

),

σ2φ =

Γφg2

(h+ g2∆r2

),

where Γg =∑i(sin(φ)δrxi−cos(φ)δryi )

2

S , Γφ =∑i(cos(φ)δrxi+sin(φ)δryi )

2

S , and S =∑i δr

2xi

∑i δr

2yi −

(∑i δrxiδryi)

2 are geometrical factors that depend onthe cluster’s shape and the gradient’s orientation. Notethat in the case the cluster have circular symmetry,(∑i δr

2xi ≈

∑i δr

2yi ≈

12

∑i |δri|2,

∑i δrxiδryi ≈ 0),

then Γg ≈ 1χ , and Eq. (6) is recovered.

Appendix B: Auxiliary computations for thetug–of–war model

Details on computation of the mean and covarianceof pi(t)

We want to compute the mean value and variance ofpi. We start from Eq. (12) . The mean value is straightforward,

12

〈pi〉 =⟨µi + σp

∫ t

0

dt′e−(t−t′)/τpεi(t′) + βc0qi

∫ t

0

dt′e−(t−t′)/τpΞi(t′)⟩,

=〈µi〉+ σp

∫ t

0

dt′e−(t−t′)/τp〈εi(t′)〉+ βc0qi

∫ t

0

dt′e−(t−t′)/τp〈Ξi(t′)〉,

=µi.

Recalling that 〈Ξi(t)Ξj(0)〉 =(δc2i /c

20Cc(t) + σ2

∆C∆(t))δij , and that we have assumed

that correlations decay exponentially, so Cs(t) = e−t/τs ,

with τs the correlation time and the index s = {c,∆},we now compute the covariance of pi(t),

〈(pµi(t)− 〈pµi〉)(pνi(t)− 〈pνi〉)〉 = σ2

∫ t

0

dt′∫ t

0

dt′′〈εµi(t′)ενi(t′′)〉e−(t−t′)/τpe−(t−t′′)/τp

+ β2c20qµiqνi

∫ t

0

dt′∫ t

0

dt′′〈Ξi(t′)Ξi(t′′)〉e−(t−t′)/τpe−(t−t′′)/τp

=σ2τp

2+ β2qµiqνiδc

2iT(t, τc) + β2c20qµiqνiσ

2∆T(t, τ∆)

where,

T(t, τs) =τ2p

1 + υs

(1 +

1

1− υs×((1 + υs)e

−2t/τp − 2e−(1+υs)t/τp)),

and υs =τpτs

. T(t, τs)/τ2p represent the time averaging

function for a fluctuating process with correlation timeτs up to a time t. Given that τ∆ � t � τp � τc, then

T(t, τc) ≈τ2p

1+τp/τc≈ τpτc and T(t, τ∆) ≈ τ2

p , so then

T(t, τ∆)� T(t, τc).

Error propagation for the tug-of-war model

To estimate the gradient sensing error for the tug–of–war model we use a simple error propagation method.From Eq. (13), we can write an expression for the gra-dient related to the cluster velocity, g = 1

γc0M−1〈vc〉,

where,

M−1 =N

Sv

[ ∑i δryiqyi −

∑i δryiqxi

−∑i δrxiqyi

∑i δrxiqxi

], (B1)

and Sv =∑i δrxiqxi

∑i δryiqyi −

∑i δrxiqyi

∑i δryiqxi .

Recalling the symmetry of the roughly circularhexagonal clusters, the following relations are fulfilled,∑i δr

2xi ≈

∑i δr

2yi ≈

12

∑i |δri|2 ,

∑i δrxiδryi ≈

0,∑i q

2xi ≈

∑i q

2yi ≈

12

∑i |qi|2,

∑i qxiqyi ≈ 0,

∑i qxiδrxi ≈

∑i qyiδryi ≈

12

∑i qi·δri, and

∑i qxiδryi ≈∑

i qyiδrxi ≈ 0. In addition, note that in such a case, we

can write Sv ≈∑i δrxiqxi

∑i δryiqyi ≈

14 (∑i qi · δri)

2

and,

M−1 ≈ 2N∑i qi · δri

I, (B2)

where I is the 2×2 identity matrix. Then, the expressionfor the gradient takes the following simple form,

g =2N

γc0∑i qi · δri

〈vc〉. (B3)

Eq. (B3) shows that g and 〈vc〉 are proportional to eachother. Then, we can write the variance of g as,

var(g) =

(2N

γc0∑i qi · δri

)2

var(vc). (B4)

Introducing the velocity variance expressions given byEq. (14) in the main text into Eq. (B4), we finally obtainthe gradient sensing variances for the tug–of–war model,

σ2gx ≈ σ

2gy ≈

2

(∑i qi · δri)

2

(∑i

|qi|2hTi +σ2pτp

γ2c20N

),

(B5)

cov(gx, gy) ≈0. (B6)

We note that, though our derivation of this equationdoes involve a sum over cell positions, the cluster cancompute the direction of the gradient through the tug-of-war without explicitly knowing any cell locations.

13

Appendix C: Maximum likelihood estimator withtime averaging

In the main text we apply the MLE method to com-pute the gradient sensing error from the individual cellmeasurements of the chemoattractant concentration, Mi,and cell positions, r∗i . These measurements capture a“snapshot” or instant of the cluster state. Later in thetext, we compare the MLE outcomes with the tug–of–war model, which relies on the statistics of the cell po-larities pi obtained for asymptotic times larger than τp.Therefore, we are comparing instantaneous with time av-eraged measurements. We could do this since the sensingconcentration measurements are dominated by the CCVwhich has a correlation times larger than the polarityrelaxation time. Here we extend the MLE method byusing time average measurements in the same way as the

tug–of–war model.We compute the time average for the individual cell

chemoattractant concentration measurements,

MTi (t) =

∫Mi(t

′)KT (t− t′)dt′,

= c(ri)/c0 +

∫Ξi(t

′)KT (t− t′)dt′,

where KT (t) = H(t) 1T e−t/T is a time averaging kernel

and H(t) the Heaviside function. We assume that cellpositions remain fixed during the averaging time T , sothat c(ri, t) = c(ri). Now we compute the mean andvariance for MT

i .

〈MTi 〉 = c(ri)/c0,

⟨(MTi − 〈MT

i 〉)2⟩

=1

T 2

∫ t

0

dt′∫ t

0

dt′′〈Ξi(t′)Ξi(t′′)〉e−(t−t′)/T e−(t−t′′)/T ,

=1

T 2

∫ t

0

dt′∫ t

0

dt′′(δc2ic20Cc(|t′ − t′′|) + σ2

∆C∆(|t′ − t′′|))e−(t−t′)/T e−(t−t′′)/T ,

=1

1 + T/τc

δc2ic20

+1

1 + T/τ∆σ2

∆.

If we substitute the averaging time T = τp, these areprecisely the values hTi which we derived in our tug-of-war model. This shows we can carry over all of our re-sults from the maximum likelihood estimation to matchthe results of the tug-of-war model. However, this is notparticularly important, because T � τc and T � τ∆,the sensing concentration fluctuations can be approxi-

mated by 〈(MTi − 〈MT

i 〉)2〉 ≈ σ2

∆. When we take intoaccount time averaging, the contributions to the sensingconcentration fluctuations are dominated by CCV whileligand-receptor fluctuations are averaged out.

Appendix D: Proof for the analytical solution of theMLE errors

In this appendix we show the derivation of the expres-sions for the MLE errors in the shallow gradient limit.The problem we are solving is very similar to an error–in–variable models of regression, in which the regressorvariable is subject to errors [49]. However, we were notable to find an explicit mapping between these resultsand our question of interest – computing the Fisher in-formation matrix and its inverse specifically for the gxand gy variables. Finding the Fisher information matrixI is relatively straightforward from evaluating derivatives

of the log-likelihood, and it is presented in Appendix A.However, to determine the Cramer-Rao bound, we needto compute the inverse of the Fisher information ma-trix – in principle needing to analytically invert a matrixwhose size scales with the number of cells. This is non-trivial in general. However, for this problem we wereable to use Gaussian elimination to find the elements ofthe inverse of the Fisher information matrix necessary forus, i.e. (I)

−1gxgx

, (I)−1gygy

, and (I)−1gxgy

. Given the almost

tridiagonal shape of the Fisher matrix, we first show theprocedure for how to eliminate the elements of the rowsassociated with a individual cell s and then repeat thesame steps for the rest of the cells.

Recall that the Gaussian elimination method is a lin-ear algebra method to find the solution of a problemA×X = B. This problem can be represented in terms ofan augmented matrix [A|B]. Row operations (i.e. linearcombinations of the rows) are performed, leading even-tually to finding the solution for X when the augmentedmatrix takes the form [I|C], where I is the identity ma-trix, and C =

(A−1

)B = X. In our case, the problem we

want to solve is to find the inverse of the Fisher informa-tion matrix, I ×

(I−1

)= I. To start with the Gaussian

elimination process, we first write the augmented matrix[I|I],

14

Sxx Sxy . . . gxfsδrxs gyfsδrxs . . .Sxy Syy . . . gxfsδrys gyfsδrys . . .

...... . . .

...... . . .

gxfsδrxs gxfsδrys . . . g2xfs + ms

σ2xs

gxgyfs + msρsσxsσys

. . .

gyfsδrxs gyfsδrys . . . gxgyfs + msρsσxsσys

g2yfs + ms

σ2ys

. . .

...... . . .

...... . . .

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 0 . . .0 1 . . ....

... . . .0 0 . . .0 0 . . ....

... . . .

where Sxx =∑i fiδr

2xi , Syy =

∑i fiδr

2yi , and Sxy =∑

i fiδrxiδryi . Since we are only looking for the gradientestimation errors, we solely consider the two first columnsof the right matrix.

Following, we apply the operations to perform an up-per diagonalization of columns associated to the cell s.From now on we use the notation Ri and Im,n to identifythe i–row and the (m,n)–element of the matrix respec-tively. Note that the rows associated with the cell s areR2s+1 and R2s+2.

1. Set element I2s+2,2s+2 = 1: R2s+2 ← R2s+2

I2s+2,2s+2.

2. Set element I2s+1,2s+2 = 0: R2s+1 ← R2s+1 −I2s+1,2s+2R

2s+2.

3. Set element I2,2s+2 = 0: R2 ← R2 − I2,2s+2R2s+2.

4. Set element I1,2s+2 = 0: R1 ← R1 − I1,2s+2R2s+2.

5. Set element I2s+1,2s+1 = 1: R2s+1 ← R2s+1

I2s+1,2s+1.

6. Set element I2,2s+1 = 0: R2 ← R2 − I2,2s+1R2s+1.

7. Set element I1,2s+1 = 0: R1 ← R1 − I1,2s+1R2s+1.

After these steps we arrive to the following matrix,

Sxx − αsδr2xs Sxy − αsδrxsδrys . . . 0 0 . . .

Sxy − αsδrxsδrys Syy − αsδr2ys . . . 0 0 . . .

...... . . .

...... . . .

� � . . . 1 0 . . .� � . . . � 1 . . ....

... . . ....

... . . .

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 0 . . .0 1 . . ....

... . . .0 0 . . .0 0 . . ....

... . . .

αs = f2

sgΣsgT

1+fsgΣsgTand the squares (�) represent ele-

ments with large expressions that are not relevant forthe computation.

Then, we need to repeat steps 1–7 for all s.For each iteration we add a term −αsδrzksδrzls to

the matrix elements {k, l} with k, l = {1, 2} andzk,l = {x, y}. Finally, after repeating the proce-dure for all s, we get that the elements {k, l} resultsin∑i fiδrzxiδrzli −

∑s αsδrzxsδrzls =

∑i γiδrzxiδrzli,

where γi = fi1+figΣigT

, and the matrix reads,∑i γiδr

2xi

∑i γiδrxiδryi . . .∑

i γiδrxiδryi∑i γiδr

2yi . . .

...... . . .

∣∣∣∣∣∣∣1 0 . . .0 1 . . ....

... . . .

Finally, since the first two rows are composed by a

2 × 2 block followed by all zeros to the right, instead ofperforming row operations, we can just invert this 2× 2matrix block to arrive to the final solution for the inverseof the Fisher matrix and the MLE errors.1 0 . . .

0 1 . . ....

... . . .

∣∣∣∣∣∣∣∑i γiδr

2yi

S −∑i γiδrxiδryiS . . .

−∑i γiδrxiδryiS

∑i γiδr

2xi

S . . ....

... . . .

We can be positive that this is the final solution, since fol-lowing with the Gaussian elimination process, no furtheroperations over the first two rows would be necessary,and therefore, they will not be modified.

We have ensured that this result is correct both bychecking with computer algebra packages for small num-bers of cells, as well as explicitly numerically invertingthe Fisher information matrix.

Appendix E: Default parameters

In Table I we show the default values for parameters inthis study with a brief justification of our choices. Valuesdiffering from these are indicated in the text and thefigure captions.

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