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Accuracy of direct gradient sensing by single cellsRobert G. Endres∗†‡ and Ned S. Wingreen∗§

∗Department of Molecular Biology, Princeton University, Princeton, NJ 08544-1014; †Division of Molecular Biosciences, Imperial College London, London SW72AZ, United Kingdom; and ‡Centre for Integrated Systems Biology at Imperial College, Imperial College London, London SW7 2AZ, United Kingdomb;

Edited by José N. Onuchic, University of California at San Diego, La Jolla, CA, and approved August 27, 2008 (received for review May 14, 2008)

Many types of cells are able to accurately sense shallow gradientsof chemicals across their diameters, allowing the cells to movetoward or away from chemical sources. This chemotactic abilityrelies on the remarkable capacity of cells to infer gradients fromparticles randomly arriving at cell-surface receptors by diffusion.Whereas the physical limits of concentration sensing by cells havebeen explored, there is no theory for the physical limits of gradientsensing. Here, we derive such a theory, using as models a perfectlyabsorbing sphere and a perfectly monitoring sphere, which, respec-tively, infer gradients from the absorbed surface particle densityor the positions of freely diffusing particles inside a spherical vol-ume. We find that the perfectly absorbing sphere is superior to theperfectly monitoring sphere, both for concentration and gradientsensing, because previously observed particles are never remea-sured. The superiority of the absorbing sphere helps explain thepresence at the surfaces of cells of signal-degrading enzymes, suchas PDE for cAMP in Dictyostelium discoideum (Dicty) and BAR1for mating factor α in Saccharomyces cerevisiae (budding yeast).Quantitatively, our theory compares favorably with recent mea-surements of Dicty moving up a cAMP gradient, suggesting thesecells operate near the physical limits of gradient detection.

chemotaxis | receptors | noise

C ells are able to sense gradients of chemical concentration withextremely high sensitivity. This is done either directly, by mea-

suring spatial gradients across the cell diameter, or indirectly, bytemporally sensing gradients while moving. In temporal sensing, acell modifies its swimming behavior according to whether a chem-ical concentration is rising or falling in time (1). This mode ofsensing is typical of small, fast moving bacteria such as Escherichiacoli, which can respond to changes in concentration as low as 3.2nM attractant aspartate (2). In contrast, direct spatial sensing isprevalent among larger, single-celled eukaryotic organisms suchas the slime mold Dictyostelium discoideum (Dicty) and the yeastSaccharomyces cerevisiae (3, 4). Dicty cells are able to sense aconcentration difference of only 1–5% across the cell (5), cor-responding to a difference in receptor occupancy between frontand back of only five receptors (6). Spatial sensing is also per-formed with high accuracy by cells of the immune system includingneutrophils and lymphocytes (7), as well as by growing synap-tic cells and tumor cells. While there has been great progress inunderstanding the limits of concentration sensing and signalingin bacteria such as E. coli (8–13), very little is know about thetheoretical limits of direct gradient sensing by eukaryotic cells.

In a recent set of experiments, van Haastert and Postma (6)measured the Chemotactic Index of Dicty cells in a cAMP gradi-ent (Fig. 1, symbols). Chemotactic Index is defined as the distancemoved in the direction of the cAMP gradient divided by thetotal distance moved. To obtain the data in Fig. 1, van Haastertand Postma used a pipette containing different concentrations ofcAMP. Diffusion of cAMP out of the pipette established a steady-state cAMP gradient, with magnitude a function of distance fromthe pipette. Chemotaxis was observed for cells as far as 700 µmfrom a pipette filled with 10−4 M cAMP, corresponding to a meanconcentration of 7 nM and a gradient of only 0.01 nM/µm. Thisremarkable chemotactic ability raises the question – how closelydoes gradient sensing by Dicty cells compare with the fundamentallimits on gradient sensing set by diffusion?

Here, we derive the fundamental limits of gradient sensingby using two models for cells: a perfectly absorbing sphere anda perfectly monitoring sphere. Within the theory, gradients areestimated by comparing the discrete positions of particles, eitherabsorbed on the surface of the sphere or measured inside thespherical volume, with the expected continuous distribution origi-nating from a particular gradient. We find that a perfectly absorb-ing sphere is superior to a perfectly monitoring sphere for sensingboth concentrations and gradients (Table 1), because previouslyobserved particles are never remeasured. Quantitatively, our the-ory (Fig. 1, solid curves) compares favorably with recent mea-surements of Dicty cells migrating to a cAMP-filled pipette (6),suggesting that the chemotactic ability of Dicty approaches thefundamental limits set by diffusion.

Limits of Concentration SensingIn this section, we consider the limits of concentration sensing setby particle diffusion. Consider as a measurement device a spher-ical cell of radius a that can measure the local concentration ofa certain dissolved chemical. Such an idealized device may makemeasurements by using two different strategies: (i) The devicecan either act as a perfectly absorbing sphere and record the num-ber of absorbed particles on its surface or (ii) act as a perfectlymonitoring sphere and count the number of particles inside its vol-ume. In either case, from the number of particles, an estimateof the chemical concentration can be obtained. However, theseestimates have an intrinsic uncertainty due to the randomness ofparticle diffusion.

Perfectly Absorbing Sphere. For the perfectly absorbing sphere theuncertainty in measuring a background chemical concentrationc is straightforward to derive. At steady state, the average parti-cle current impinging on the sphere is J = 4πDac, where D isthe chemical diffusion constant. The average number of particlesabsorbed in time T is N = 4πDacT . Because the particles are inde-pendent, N is Poisson distributed, i.e., 〈(δN)2〉 = 〈N〉. Therefore,the perfectly absorbing sphere has a concentration-measurementuncertainty of

〈(δc)2〉c2 = 〈(δN)2〉

〈N〉2 = 14πDacT

. [1]

Perfectly Monitoring Sphere. The perfectly monitoring sphere wasintroduced by Berg and Purcell (8) as a parameter-free modelfor a cell that “perfectly” binds and releases all ligands that con-tact its surface. To quantify the time a diffusing particle spendsin the cell’s vicinity and is therefore capable of being measured,Berg and Purcell treated the cell as a permeable sphere that infersthe particle concentration by counting the number of particles N

Author contributions: R.G.E. and N.S.W. designed research, performed research, con-tributed new analytic tools, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.§To whom correspondence should be addressed. E-mail: wingreen@princeton.edu.

This article contains supporting information online at www.pnas.org/cgi/content/full/0804688105/DCSupplemental.

© 2008 by The National Academy of Sciences of the USA

www.pnas.org / cgi / doi / 10.1073 / pnas.0804688105 PNAS October 14 , 2008 vol. 105 no. 41 15749–15754

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Fig. 1. Comparison between the Chemotactic Index determined by experi-ment (symbols and dashed lines) and our theory (solid curves). ChemotacticIndex is defined as the distance moved by a cell in the direction of a gradi-ent divided by the total distance moved. Experimental data were obtainedby van Haastert and Postma (6) from Dictyostelium discoideum cells migrat-ing toward pipettes containing four different cAMP concentrations, 0.1 µM(squares), 1 µM (triangles), 10 µM (diamonds), 100 µM (circles). The theoret-ical curves were obtained for a perfectly absorbing sphere by using a singlefitting parameter Da3T = 1.2 ·105 µm5, corresponding to, e.g., a cAMP diffu-sion constant of D = 300 µm2/s, a cell radius of a = 5 µm, and an averagingtime T = 3.2 s, using the gradient profiles from ref. 6 and the Chemotac-tic Index from Eq. 21. Experimentally, the Chemotactic Index only reachesapproximately 0.9 at zero distance, so we rescale our theory curves by 0.9.(Inset) Chemotactic Index as a function of (c0

z )2/c0 in units of nM/(µm)2,

where c0z and c0 are the gradient and concentration, respectively.

inside its volume, and improves accuracy by averaging over sev-eral statistically independent measurements. A simple estimatefor the resulting uncertainty in concentration can be obtained asfollows: the number N is Poisson distributed and the cell countsappoximately N = a3c particles in its volume at any time. Duringa time T , the cell can make Nmeas = T/(a2/D) independent mea-surements, where a2/D is the typical turnover time for particlesinside the sphere, leading to

〈(δc)2〉c2 = 〈(δN)2〉

N2 = 1NmeasN

≈ 1DacT

. [2]

Berg and Purcell (8) derived the exact concentration-measurement uncertainty for a perfectly monitoring sphere (“per-fect instrument”) from the time correlations of particles inside thesphere, and obtained

〈(δc)2〉c2 = 3

5πDacT, [3]

which is identical to the estimate in Eq. 2 up to a numericalprefactor.

However, notice that the concentration-measurement uncer-tainty of the perfectly absorbing sphere is actually smaller thanthat of a perfectly monitoring sphere of the same size, because theperfectly absorbing sphere removes particles from the environ-ment, and hence does not measure the same particle more thanonce.

Limits of Gradient SensingNow consider the perfectly absorbing sphere and the perfectly moni-toring sphere as devices for measuring the local gradient of a certaindissolved chemical. In both cases, measurements of discrete par-ticles can be compared with the expected continuous distributionof particles originating from a particular gradient (Fig. 2A), andhence, the gradient can be estimated. Here, we present, in brief, atheoretical derivation of the intrinsic uncertainty of gradient sens-ing [for details see supporting information (SI) Appendix (PDF)].We find that the intrinsic uncertainty is independent of the actualgradient present, and is always much smaller (by a factor of 7/60� 12%) for the perfectly absorbing sphere.

Perfectly Absorbing Sphere. The average particle current density �j =−D �∇c impinging on the surface of a perfectly absorbing sphereof radius a at steady state follows from the diffusion equation,∇2c = 0, and is given in polar coordinates by

j(θ , φ) = Dc0

a+ 3D�cr · �e(θ , φ), [4]

where c0 is a constant background concentration, �cr is the back-ground gradient, and �e(θ , φ) = (cos φ sin θ , sin φ sin θ , cos θ) is aunit vector normal to the surface of the sphere (see Fig. 2A). Tobest estimate the chemical gradient from an observed discrete den-sity of particles absorbed at the surface of the sphere during timeT (Fig. 2B), fit the observed density σ obs

T = ∑Ni=1 δ(�r−�ri), where N

is the total number of absorbed particles, to the expected densityj(θ , φ)T from Eq. 4. Because the estimates of the components ofthe gradient in the x, y, and z directions are independent, withoutloss of generality, we consider only the gradient estimate in thez direction, i.e., cz = ∂c/∂z, and later generalize to an arbitrarygradient. From the best fit, the estimate for the gradient in thez direction after absorption of particles for a time T is given by

cz =∫

σ obsT cos θ dA4πDa2T

=∑N

i=1 cos θi

4πDa2T, [5]

where θi is the polar angle of the ith absorbed particle. We are inter-ested in the uncertainty (accuracy) of the gradient measurement,which is given by the variance

〈(δcz)2〉 = ⟨c2

z

⟩ − ⟨cz

⟩2

=⟨∑N

i=1 cos2 θi

⟩+

⟨∑Ni=1

∑Ni�=j cos θi cos θj

⟩(4πDa2T)2 −

⟨∑Ni=1 cos θi

⟩2

(4πDa2T)2

Table 1. Uncertainty of concentration and gradient sensing

Measurement uncertainty Perfect absorber Perfect monitor Ratio absorber/monitor

Concentration:〈(δc)2〉

c20

14πDac0T

(8)3

5πDac0T125

= 2.4

Gradient:〈(δc�r )2〉(c0/a)2

14πDac0T

157πDac0T

607

≈ 8.6

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=⟨∑N

i=1 cos2 θi

⟩(4πDa2T)2 = 〈N〉〈cos2 θ〉

(4πDa2T)2 = c0

12πDa3T. [6]

The derivation of Eq. 6 made use of the independence of the parti-cles to factorize the expectation value as 〈∑N

i=1∑N

i�=j cos θi cos θj〉 =〈N(N − 1)〉〈cos θ〉2 = 〈N〉2〈cos θ〉2, because the number ofabsorbed particles N is Poisson distributed. We also used 〈N〉 =4πDac0T , as well as 〈cos2 θ〉 = 1/3. (The relation 〈cos2 θi〉 =〈cos2 θ〉 for absorbed particles holds even in the presence of atrue gradient in the z direction because the gradient-weightedcontribution 〈cos3 θ〉 is zero.)

In three dimensions, the total uncertainty of the gradient,normalized by c0/a, is given by

〈(δc�r)2〉(c0/a)2 = 3〈(δcz)2〉

(c0/a)2 = 14πDac0T

, [7]

with the factor of 3 arising because each component of the gradi-ent contributes independently to the total uncertainty. This resultfor the uncertainty in gradient sensing is independent of the mag-nitude of the actual gradient present, including the case whenno actual gradient is present. Curiously, the result is numericallyidentical to the concentration-measurement uncertainty (Eq. 1).

Perfectly Monitoring Sphere. Here, we extend Berg and Purcell’sanalysis of the perfectly monitoring sphere (“perfect instrument”)to include gradient sensing. Specifically, we assume that the moni-toring sphere measures not only the number, but also the positionsof all particles in its volume (Fig. 2C). The best estimate of thegradient is obtained by fitting a concentration gradient with ac = c0 + �cr · �r to the observed time-averaged number density1T

∫dtρobs(t) = 1

T

∫dt

∑Ni=1 δ(�r − �ri(t)), obtained by measuring the

exact positions of all the particles inside the volume of the spherefor a time T . As above we focus on one component of the gradi-ent, namely the gradient in the z direction cz, and obtain as a bestestimate

cz =1T

∫dt

∫dV z ρobs(t)∫dV z2

, [8]

where the integral∫

dV is over the volume of the sphere. We areinterested in the variance of this estimated gradient

〈(δcz)2〉 = ⟨c2

z

⟩ − 〈cz〉2 [9]

=(

154πa5

)2 [⟨m2

z,T

⟩ − 〈mz,T 〉2],where we have used

∫dVz2 = 4πa5/15 and have defined

mz,T = 1T

∫dt

∫dV z ρobs(t), [10]

namely, mz,T is the time-averaged total z coordinate of particlesinside the sphere. The expectation values in Eq. 9 are thereforegiven by

⟨m2

z,T

⟩ = 1T2

⟨(∫dt

∫dV z ρobs(t)

)2⟩

[11]

= 1T2

∫ T

0dt

∫ T

0dt′〈mz(t)mz(t′)〉,

〈mz,T 〉 = 1T

⟨(∫dt

∫dV z ρobs(t)

)⟩

= 1T

∫ T

0dt〈mz(t)〉,

where the quantity mz(t) is the total of the z coordinates of allthe particles inside the sphere at time t. To calculate mz(t), we

Fig. 2. Idealized models for gradient sensing by a cell. The gradient pointsalong the z axis, which is shown horizontally. (A) Continuum model for aperfectly absorbing sphere. The mean particle current density j(θ) imping-ing on the sphere has axial symmetry; θ measures the angle with respect tothe z axis. At steady state, the particle concentration c is zero immediatelyoutside the perfectly absorbing sphere, as shown schematically by the redcurve superposed on the dotted background gradient. (B) Discrete particlemodel for the perfectly absorbing sphere. From the number and positionsof particles absorbed during time T , the background particle concentrationand gradient can be estimated. (C) Perfectly monitoring sphere. Particles dif-fuse in and out of the sphere without resistance. By monitoring, for a timeT , the number and positions of particles inside the sphere, the backgroundconcentration and gradient can be estimated.

consider the sphere embedded inside a much larger volume con-taining a total of M particles. Then, mz(t) = ∑M

i=1 zi(t), where ziis the z coordinate of particle i if this particle is inside the sphereand is zero otherwise. On average there will be 〈N〉 = 4

3 πa3c0

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particles inside the sphere at any time t. The auto correlation func-tion 〈mz(t)mz(t′)〉 of particles inside the sphere at time t and timet′ can consequently be calculated as

〈mz(t)mz(t′)〉 =⟨

M∑i=1

M∑j=1

zi(t)zj(t′)

⟩

= 43πa3c0 u(t − t′) + 〈mz(t)〉2, [12]

where we have defined u(t − t′) = 〈z(t)z(t′)〉 for a single particle.Substituting Eqs. 11 and 12 into Eq. 9 results in

〈(δcz)2〉 = 75c0

4πa7T2

∫ T

0dt

∫ T

0dt′u0(t − t′). [13]

By defining a correlation time τz = (1/a2)∫ ∞

0 dτu(τ ) for thecoordinate z(t), the double time integral in Eq. 13 can be sim-plified, provided the time T is much larger than τz. By usingtime-reversal symmetry u(τ ) = u(−τ ) for equilibrium diffusion(assuming small gradients), the variance simplifies to

〈(δcz)2〉 = 75c0τz

2πa5T. [14]

The remaining task is to calculate τz, the probability that a par-ticle with coordinate z inside the sphere at time t = 0 is still (oragain) inside the sphere at a later time τ . We first consider the casein which the background chemical concentration is uniform, andlater consider the presence of an actual gradient. Based on thesolution of the diffusion equation in three dimensions, if a unitamount of chemical is released at point �r′, the concentration atpoint �r at a later time τ is given by

f (|�r − �r′|, τ ) =exp

(−|�r−�r′|2

4Dτ

)(4πDτ )

32

. [15]

By using the result for the time integral from ref. 8,∫ ∞

0dτ f (|�r − �r′|, τ ) = 1

4πD|�r − �r′| , [16]

the correlation time τz can be expressed as a volume inte-gral over the sphere (the initial coordinate �r′ is uniform inthe sphere because we assume a uniform background chemicalconcentration)

τz = 143 πa5

∫ ∞

0dτ

∫dV z

∫dV ′ z′ f (|�r − �r′|, τ )

= 316π2Da5

∫dV z

∫dV ′ z′

|�r − �r′|= 3

16π2Da5

∫dV zψ(r, θ), [17]

where r = |�r|. The function

ψ(r, θ) =∫

dV ′ z′

|�r − �r′| =∫

dV ′ r′ cos θ ′

|�r − �r′| [18]

is analogous to the potential of a charge density in electrostatics,specifically the charge density ρ(z′) = z′ = ρ(r′, θ ′) = r′ cos θ ′for r′ ≤ a and ρ(r′) = 0 for r′ > a. To solve the final inte-gral in Eq. 17, we perform a multipole expansion of ψ(r, θ) interms of Legendre polynomials Pl(cos θ), exploiting the rotationalsymmetry of ψ(r, θ) about the z axis (14), leading to

τz = 2a2

105D. [19]

Now, consider the contribution from an additional gradient withzero mean over the volume of the sphere. We need to perform theintegrals over a non uniform distribution in Eq. 17. Because a gra-dient along the z axis contributes a factor cos θ , which leads to thevanishing integral

∫ π

0 sin θ cos3 θdθ = 0, we conclude that only theconstant background contributes to the uncertainty in the gradientmeasurement. Therefore, Eq. 19 for τz remains true even when anactual gradient is present.

The result for τz in Eq. 19 can be used in Eq. 14 to obtain thenormalized uncertainty of gradient measurement by the perfectlymonitoring sphere

〈(δc�r)2〉(c0/a)2 = 3〈(δcz)2〉

(c0/a)2 = 157πDac0T

, [20]

where all three components of the gradient contribute indepen-dently. Hence, the perfectly monitoring sphere is not only inferiorto the perfectly absorbing sphere for concentration sensing by afactor of 12/5 in variance (cf. Eqs. 1 and 3), but is also inferior byan even larger factor of 60/7 for gradient sensing (cf. Eq. 7).

Comparison with ExperimentVan Haastert and Postma (6) recently measured the Chemotac-tic Index of Dicty cells in a cAMP gradient (6) (Fig. 1, symbols).They used a pipette containing different concentrations of cAMPto establish a distance-dependent steady-state cAMP gradients.The Chemotactic Index was defined as the distance moved by thecell in the direction of the cAMP gradient divided by the total dis-tance moved. How does the observed chemotactic ability of Dictycompare with the fundamental limits on gradient sensing set bydiffusion? To facilitate comparison with the results of van Haastertand Postma (6), we have calculated the optimal Chemotactic Indexfor a cell acting as a perfectly absorbing sphere.

To obtain the optimal Chemotactic Index CI, we assume thatafter averaging for a time T , a cell moves at a constant velocityin the direction of the estimated gradient. If we take the actualgradient to point in the z direction, then the chemotactic index forone run i is simply cos θi, where θi is the angle between the truegradient and the estimated gradient. If the velocity and run timeare the same for each run, leading to a constant run length l, thenthe average Chemotactic Index is given by

CI =∑N

i lz∑Ni l

= l∑N

i cos θi

Nl= 〈cos θi〉. [21]

To evaluate 〈cos θi〉 for a perfectly absorbing sphere, we use ourresult (Eq. 7) for the variance of the estimated gradient in eachdirection, e.g., 〈(δcx,y,z)2〉 = c0/(12πDa3T). Assuming a Gaussiandistribution with these variances, as well as an actual gradient withmean value c0

z in the z direction, the two-dimensional distributionof estimated gradients c�r = (cx, cz) is given by

Pc0z(cx, cz) = 1

2πσxσze−c2

x/(

2σ2x)−(

cz−c0z)2/(

2σ2z), [22]

where σx,z = √〈(δcx,z)2〉. From this distribution, we can obtain theoptimal Chemotactic Index

CI = 〈cos θ〉 =⟨

cz

/√c2

x + c2z

⟩

=√

πy2

e−y[I0(y) + I1(y)], [23]

where y = (c0z )2/(4σ 2

z ) and I0(1) is the first (second)-order modi-fied Bessel function of the first kind. Fig. 1 shows a comparison ofthe optimal CI (solid curves) with the data of ref. 6. Importantly,the comparison relies on only a single global fitting parameterrepresenting gradient-sensing ability, namely, the product Da3T ,where D is the diffusion constant, a is the cell diameter, and T is

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the averaging time. Based on the estimates D = 300µm2/s anda = 5µm (6), the averaging time is predicted to be about T = 3.2s. (The perfect-monitor model yields an identical curve, but witha longer inferred averaging time T = 27.5 s.) The theory for theoptimal Chemotactic Index matches the experiment rather well.Eq. 23 further predicts that the Chemotactic Index depends onthe gradient c0

z and the concentration c0 only through the com-bination (c0

z )2/c0 (Fig. 1 Inset). Intuitively,√

(c0z )2/c0 measures

the signal-to-noise ratio—the signal is proportional to the truegradient |c0

z |, whereas the noise from particle diffusion scales as√c0. More generally, the optimal Chemotactic Index depends on

all variables only through the combination (c0z )2Da3T/c0. More-

over, the theory predicts the full distribution of run angles (seeSI Appendix, Fig. S1 Inset), which can be obtained by integratingEq. 22 in the radial direction for each angle θ .

DiscussionMany types of cells are known to measure spatial chemical gra-dients directly with high accuracy. In particular, D. discoideum(Dicty) is well known to measure extremely shallow cAMP gradi-ents important for fruiting body formation (3, 5, 6) and S. cerevisiae(budding yeast) detects shallow gradients of α mating pheromone(15). Accurate spatial sensing is also performed by cells of theimmune system including neutrophils and lymphocytes (7). Thequestion arises: What are the fundamental limits of gradient sens-ing set by chemical diffusion? Here, we derived such limits by usingas model cells a perfectly absorbing sphere and a perfectly mon-itoring sphere (8). Within the theory, gradients are estimated bycomparing the discrete distribution of observed locations of par-ticles, either absorbed on the surface of the sphere (Fig. 2B) ormeasured inside the sphere (Fig. 2C), with the expected continu-ous distribution originating from a gradient (Fig. 2A). We find thata perfectly absorbing sphere is superior to a perfectly monitoringsphere for concentration and gradient sensing by respective factorsof 12/5 (= 2.4) and 60/7 (≈ 8.6) (Table 1), because the perfectlyabsorbing sphere prevents rebinding of already measured parti-cles. Indeed, the results presented here for the perfectly absorbingsphere represent the true fundamental limits of both concentra-tion and gradient sensing by cells. Our theory for the limits ofgradient sensing compares favorably with recent measurements byvan Haastert and Postma (6) of Dicty cells migrating to a cAMP-filled pipette (Fig. 1), suggesting that Dicty chemotaxis approachesthe fundamental limits set by cAMP diffusion.

The marked superiority of the perfect absorber for concentra-tion and gradient sensing leads us to conjecture that cells mayhave developed mechanisms to absorb ligands so as to preventtheir rebinding. Such absorption could be implemented by ligandor ligand–receptor internalization or by degradation of bound lig-ands. Even degradation of ligands without measurement couldbe advantageous. For example, a perfect absorber that measuresonly a fraction f of incident particles has the same uncertain-ties given in Table 1 but with an effective measurement timeTeff = fT . Such an absorbing cell that measures only 12% ofabsorbed particles can still measure gradients as accurately asa perfectly monitoring sphere. Similarly, an absorbing sphereof radius a with two small measurement patches at its poles ofradius s (s a), i.e., with a measuring surface-area fractions2/(2a2), effectively reduces the averaging time for pole-to-polegradients to 3s2T/(2a2) (SI Appendix). Consequently, a measur-ing surface fraction as small as 4% yields the same uncertainty asthe monitoring sphere.

In fact, there are numerous examples in biology of ligand–receptor internalization (16) and ligand degradation on cell sur-faces, which we speculate might be related to gradient sensing.(i) Although many G protein-coupled receptors are internalizedby endocytosis (17), the cAMP receptor cAR1 in Dicty is not(18). However, Dicty produces two forms of cyclic nucleotide

phosphodiesterase (PDE), which degrade external cAMP (19–21). One form is membrane-bound (mPDE) and effectively turnsDicty into an absorber, whereas the other form is soluble (ePDE).The membrane-bound form mPDE only accumulates during cellaggregration, supporting the idea that degradation of cAMP atthe membrane helps accurate gradient sensing and navigation.Indeed, cells lacking mPDE display cell-autonomous chemotaxisdefects even in mixed aggregates with isogenic wild-type cells (21).Interestingly, there is good evidence that Dicty cells do carry outG protein-coupled receptor-mediated endocytosis of folic acid,another major Dicty chemoattractant (22). (ii) In budding yeast,the receptor Ste2 binds α-factor pheromone, initiating a matingresponse including directed growth (“shmooing”) toward a poten-tial mating partner. Ligand-bound Ste2 undergoes internalizationby endocytosis (23). Furthermore, the protease Bar1 degradesα-pheromone externally (24, 25), and may be largely membraneassociated (26). (iii) There are many examples of ligand–receptorinternalization in developmental biology. For example, primordialgerm cells in zebrafish migrate toward the chemokine SDF-1a thatactivates the receptor CXCR4b. Ligand-induced CXCR4b inter-nalization is required for precise arrival of germ cells at their targetdestination (27). These examples suggest a correlation betweenligand internalization/degradation and the accuracy of cell polar-ization and movement. In presenting these admittedly speculativeexamples, our hope is to raise interest, across fields, in how theconstraints of gradient sensing accuracy may have shaped cellularsensing systems.

Although the absorption of ligands can improve gradient sens-ing, there is an inherent problem for an absorbing cell to measure agradient while moving. An absorbing cell moving in a uniform con-centration creates an apparent gradient because of an increasedflux of incoming particles at its front and a decreased flux of parti-cles at its back (8). By using the model of a spherical cell, the ratioof fluxes between front and back hemispheres is given by

R = 1 + 3av0

D, [24]

where v0 is the cell velocity, a is the cell radius, and D is the particlediffusion constant. On the other hand, the flux ratio of a sta-tionary spherical cell in a gradient | �∇c| with uniform backgroundconcentration background c0 is given by

R = 1 + 3a| �∇c|c0

. [25]

Hence, a moving cell sees an apparent gradient

| �∇c| = v0c0

D. [26]

As an example, chemotaxis of Dicty to cAMP is observed at a meanconcentration of 7 nM in a gradient of only 0.01 nM/µm (6). ADicty cell moving with a typical speed of 0.2 µm/s at the same meanconcentration but without a gradient creates an apparent gradi-ent of about half the real gradient. There are several ways out ofthis dilemma. (i) Cells could separate measurement from move-ment at low gradients, e.g., by stopping, measuring the gradient,and then moving. Dicty would only need to stop for approximatelya2/D ≈ 0.1 s based on cell radius of a = 5 µm and diffusion con-stant D = 300 µm2/s. (ii) Cells could sense gradients transverseto their direction of motion. This is particularly advantageous forfast–moving cells (e.g., bacteria) for which the apparent gradi-ent can become >100 times steeper than the actual gradient (8).(Indeed, the oxygen-sensing marine bacterium Thiovulum majusdirectly senses gradients transverse to its direction of motion (29).)Interestingly, Dicty cells moving on agar in the absence of a gra-dient appear to combine these two strategies. Qualitatively, thetips of elongated moving cells slow down and flatten, often pro-ducing two or more distinct pseudopods. Cells then elongate and

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move (approximately one cell length) in the direction of one ofthe pseudopods before the process is repeated (Liang Li and TedCox, personal communication). By this strategy, Dicty may avoidlocking onto a false, movement-generated gradient. (iii) In prin-ciple, cells could compensate for the apparent motion-generatedgradient, either by internal signal processing or by external chem-ical secretion. In fact, Dicty cells do secrete cAMP, primarily fromtheir trailing edge during movement (4), but this cAMP secretionserves dominantly to facilitate cell aggregation, including cells fol-lowing cells during streaming. Given the complex role of cAMPin Dicty aggregation, studies of Dicty chemotaxis using gradientsof folate, which is absorbed (22) but apparently not secreted byDicty, may ultimately prove simpler to interpret.

Our models of the absorbing and the monitoring spheres neglectall biochemical reactions, such as particle–receptor binding anddownstream signaling, which could significantly increase measure-ment uncertainty beyond the fundamental limits described here.To study the effects of particle–receptor binding, we extended aformalism for the uncertainty of concentration sensing, recentlydeveloped by Bialek and Setayeshgar (10), to gradient sensing. Wefound that the measurement uncertainty allowing ligand rebind-ing is larger than the measurement uncertainty without rebind-ing, confirming the superiority of the absorber over the monitor.

A number of mechanistic models for gradient sensing and chemo-taxis have addressed the important questions of cell polariza-tion, signal amplification, and adaptation (30–36), cell move-ment of individual cells (37, 38), cell aggregation with cAMPdegradation by PDE (39), as well as sensing of fluctuating con-centrations (10, 40, 41). Our results on the fundamental limitsof gradient sensing complement these models, and may ulti-mately help lead to a comprehensive description of eukaryoticchemotaxis (42).

Finally, we remark that the experiments by van Haastert andPostma used stationary spatial gradients (6). Cells in such gra-dients might profit from remembering their direction of motion(43), and evidence for such internal memory was recently obtained(28, 44, 45). It therefore might prove interesting to measure thechemotactic index for randomly changing gradients, to find outwhether cells indeed use their memory to improve chemotaxis.

ACKNOWLEDGMENTS. We thank Naama Barkai, John Bonner, Rob Cooper,Ted Cox, Liang Li, Trudi Schupbach, Stanislas Shvartsman, and Monika Skogefor helpful suggestions. This work was supported by the Human FrontierScience Program. R.G.E. was supported by Biotechnological and BiologicalSciences Research Council Grant BB/G000131/1 and the Centre for IntegratedSystems Biology at Imperial College. N.S.W. was supported by National ScienceFoundation Grant PHY-065017.

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