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Proc Natl Acad Sci U S A. Oct 14, 2008; 105(41): 15749–15754.
Published online Oct 8, 2008. doi:  10.1073/pnas.0804688105
PMCID: PMC2572938
Biophysics

Accuracy of direct gradient sensing by single cells

Abstract

Many types of cells are able to accurately sense shallow gradients of chemicals across their diameters, allowing the cells to move toward or away from chemical sources. This chemotactic ability relies on the remarkable capacity of cells to infer gradients from particles randomly arriving at cell-surface receptors by diffusion. Whereas the physical limits of concentration sensing by cells have been explored, there is no theory for the physical limits of gradient sensing. Here, we derive such a theory, using as models a perfectly absorbing sphere and a perfectly monitoring sphere, which, respectively, infer gradients from the absorbed surface particle density or the positions of freely diffusing particles inside a spherical volume. We find that the perfectly absorbing sphere is superior to the perfectly monitoring sphere, both for concentration and gradient sensing, because previously observed particles are never remeasured. The superiority of the absorbing sphere helps explain the presence at the surfaces of cells of signal-degrading enzymes, such as PDE for cAMP in Dictyostelium discoideum (Dicty) and BAR1 for mating factor α in Saccharomyces cerevisiae (budding yeast). Quantitatively, our theory compares favorably with recent measurements of Dicty moving up a cAMP gradient, suggesting these cells operate near the physical limits of gradient detection.

Keywords: chemotaxis, receptors, noise

Cells are able to sense gradients of chemical concentration with extremely high sensitivity. This is done either directly, by measuring spatial gradients across the cell diameter, or indirectly, by temporally sensing gradients while moving. In temporal sensing, a cell modifies its swimming behavior according to whether a chemical concentration is rising or falling in time (1). This mode of sensing is typical of small, fast moving bacteria such as Escherichia coli, which can respond to changes in concentration as low as 3.2 nM attractant aspartate (2). In contrast, direct spatial sensing is prevalent among larger, single-celled eukaryotic organisms such as the slime mold Dictyostelium discoideum (Dicty) and the yeast Saccharomyces cerevisiae (3,4). Dicty cells are able to sense a concentration difference of only 1–5% across the cell (5), corresponding to a difference in receptor occupancy between front and back of only five receptors (6). Spatial sensing is also performed with high accuracy by cells of the immune system including neutrophils and lymphocytes (7), as well as by growing synaptic cells and tumor cells. While there has been great progress in understanding the limits of concentration sensing and signaling in bacteria such as E. coli (813), very little is know about the theoretical 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 gradient (Fig. 1, symbols). Chemotactic Index is defined as the distance moved in the direction of the cAMP gradient divided by the total distance moved. To obtain the data in Fig. 1, van Haastert and Postma used a pipette containing different concentrations of cAMP. Diffusion of cAMP out of the pipette established a steady-state cAMP gradient, with magnitude a function of distance from the pipette. Chemotaxis was observed for cells as far as 700 μm from a pipette filled with 10−4 M cAMP, corresponding to a mean concentration of 7 nM and a gradient of only 0.01 nM/μm. This remarkable chemotactic ability raises the question – how closely does gradient sensing by Dicty cells compare with the fundamental limits on gradient sensing set by diffusion?

Fig. 1.
Comparison between the Chemotactic Index determined by experiment (symbols and dashed lines) and our theory (solid curves). Chemotactic Index is defined as the distance moved by a cell in the direction of a gradient divided by the total distance moved. ...

Here, we derive the fundamental limits of gradient sensing by using two models for cells: a perfectly absorbing sphere and a perfectly monitoring sphere. Within the theory, gradients are estimated by comparing the discrete positions of particles, either absorbed on the surface of the sphere or measured inside the spherical volume, with the expected continuous distribution originating from a particular gradient. We find that a perfectly absorbing sphere is superior to a perfectly monitoring sphere for sensing both concentrations and gradients (Table 1), because previously observed particles are never remeasured. Quantitatively, our theory (Fig. 1, solid curves) compares favorably with recent measurements of Dicty cells migrating to a cAMP-filled pipette (6), suggesting that the chemotactic ability of Dicty approaches the fundamental limits set by diffusion.

Table 1.
Uncertainty of concentration and gradient sensing

Limits of Concentration Sensing

In this section, we consider the limits of concentration sensing set by particle diffusion. Consider as a measurement device a spherical cell of radius a that can measure the local concentration of a certain dissolved chemical. Such an idealized device may make measurements by using two different strategies: (i) The device can either act as a perfectly absorbing sphere and record the number of absorbed particles on its surface or (ii) act as a perfectly monitoring sphere and count the number of particles inside its volume. In either case, from the number of particles, an estimate of the chemical concentration can be obtained. However, these estimates have an intrinsic uncertainty due to the randomness of particle diffusion.

Perfectly Absorbing Sphere.

For the perfectly absorbing sphere the uncertainty in measuring a background chemical concentration c is straightforward to derive. At steady state, the average particle current impinging on the sphere is J = 4πDac, where D is the chemical diffusion constant. The average number of particles absorbed in time T is N = 4πDacT. Because the particles are independent, N is Poisson distributed, i.e., left angle bracketN)2right angle bracket = left angle bracketNright angle bracket. Therefore, the perfectly absorbing sphere has a concentration-measurement uncertainty of

equation image

Perfectly Monitoring Sphere.

The perfectly monitoring sphere was introduced by Berg and Purcell (8) as a parameter-free model for a cell that “perfectly” binds and releases all ligands that contact its surface. To quantify the time a diffusing particle spends in the cell's vicinity and is therefore capable of being measured, Berg and Purcell treated the cell as a permeable sphere that infers the particle concentration by counting the number of particles N inside its volume, and improves accuracy by averaging over several statistically independent measurements. A simple estimate for the resulting uncertainty in concentration can be obtained as follows: the number N is Poisson distributed and the cell counts appoximately N = a3c particles in its volume at any time. During a time T, the cell can make Nmeas = T/(a2/D) independent measurements, where a2/D is the typical turnover time for particles inside the sphere, leading to

equation image

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

equation image

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

However, notice that the concentration-measurement uncertainty of the perfectly absorbing sphere is actually smaller than that of a perfectly monitoring sphere of the same size, because the perfectly absorbing sphere removes particles from the environment, and hence does not measure the same particle more than once.

Limits of Gradient Sensing

Now consider the perfectly absorbing sphere and the perfectly monitoring sphere as devices for measuring the local gradient of a certain dissolved chemical. In both cases, measurements of discrete particles can be compared with the expected continuous distribution of particles originating from a particular gradient (Fig. 2A), and hence, the gradient can be estimated. Here, we present, in brief, a theoretical derivation of the intrinsic uncertainty of gradient sensing [for details see supporting information (SI) Appendix (PDF)]. We find that the intrinsic uncertainty is independent of the actual gradient present, and is always much smaller (by a factor of 7/60 [similar, equals]12%) for the perfectly absorbing sphere.

Fig. 2.
Idealized models for gradient sensing by a cell. The gradient points along the z axis, which is shown horizontally. (A) Continuum model for a perfectly absorbing sphere. The mean particle current density j(θ) impinging on the sphere has axial ...

Perfectly Absorbing Sphere.

The average particle current density j=Dc impinging on the surface of a perfectly absorbing sphere of radius a at steady state follows from the diffusion equation, [nabla]2c = 0, and is given in polar coordinates by

equation image

where c0 is a constant background concentration, c.gif" border="0" alt="c" title=""/>r is the background gradient, and e(θ,φ)=(cosφsinθ,sinφsinθ,cosθ) is a unit vector normal to the surface of the sphere (see Fig. 2A). To best estimate the chemical gradient from an observed discrete density of particles absorbed at the surface of the sphere during time T (Fig. 2B), fit the observed density σTobs=i=1Nδ(r-ri), where N is the total number of absorbed particles, to the expected density j(θ,ϕ)T from Eq. 4. Because the estimates of the components of the gradient in the x,y, and z directions are independent, without loss of generality, we consider only the gradient estimate in the z direction, i.e., cz = [partial differential]c/[partial differential]z, and later generalize to an arbitrary gradient. From the best fit, the estimate for the gradient in the z direction after absorption of particles for a time T is given by

equation image

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

equation image

The derivation of Eq. 6 made use of the independence of the particles to factorize the expectation value as i=1NijNcosθicosθj=N(N-1)cosθ2=N2cosθ2, because the number of absorbed particles N is Poisson distributed. We also used left angle bracketNright angle bracket = 4πDac0T, as well as left angle bracketcos2θright angle bracket = 1/3. (The relation left angle bracketcos2θiright angle bracket = left angle bracketcos2θright angle bracket for absorbed particles holds even in the presence of a true gradient in the z direction because the gradient-weighted contribution left angle bracketcos3θright angle bracket is zero.)

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

equation image

with the factor of 3 arising because each component of the gradient contributes independently to the total uncertainty. This result for the uncertainty in gradient sensing is independent of the magnitude of the actual gradient present, including the case when no actual gradient is present. Curiously, the result is numerically identical to the concentration-measurement uncertainty (Eq. 1).

Perfectly Monitoring Sphere.

Here, we extend Berg and Purcell's analysis of the perfectly monitoring sphere (“perfect instrument”) to include gradient sensing. Specifically, we assume that the monitoring sphere measures not only the number, but also the positions of all particles in its volume (Fig. 2C). The best estimate of the gradient is obtained by fitting a concentration gradient with a c=c0+c.gif" border="0" alt="c" title=""/>r·r.gif" border="0" alt="[r with right arrow above]" title=""/> to the observed time-averaged number density 1Tdtρobs(t)=1Tdti=1Nδ(r-ri(t)), obtained by measuring the exact positions of all the particles inside the volume of the sphere for a time T. As above we focus on one component of the gradient, namely the gradient in the z direction cz, and obtain as a best estimate

equation image

where the integral ∫dV is over the volume of the sphere. We are interested in the variance of this estimated gradient

equation image

where we have used ∫dVz2 = 4πa5/15 and have defined

equation image

namely, mz,T is the time-averaged total z coordinate of particles inside the sphere. The expectation values in Eq. 9 are therefore given by

equation image

where the quantity mz(t) is the total of the z coordinates of all the particles inside the sphere at time t. To calculate mz(t), we consider the sphere embedded inside a much larger volume containing a total of M particles. Then, mz(t)=i=1Mzi(t), where zi is the z coordinate of particle i if this particle is inside the sphere and is zero otherwise. On average there will be N=43πa3c0 particles inside the sphere at any time t. The auto correlation function left angle bracketmz(t)mz(t′)right angle bracket of particles inside the sphere at time t and time t′ can consequently be calculated as

equation image

where we have defined u(t - t′) = left angle bracketz(t)z(t′)right angle bracket for a single particle. Substituting Eqs. 11 and 12 into Eq. 9 results in

equation image

By defining a correlation time τz=(1/a2)0dτu(τ) for the coordinate z(t), the double time integral in Eq. 13 can be simplified, provided the time T is much larger than τz. By using time-reversal symmetry u(τ) = u(−τ) for equilibrium diffusion (assuming small gradients), the variance simplifies to

equation image

The remaining task is to calculate τz, the probability that a particle with coordinate z inside the sphere at time t = 0 is still (or again) inside the sphere at a later time τ. We first consider the case in which the background chemical concentration is uniform, and later consider the presence of an actual gradient. Based on the solution of the diffusion equation in three dimensions, if a unit amount of chemical is released at point r, the concentration at point r at a later time τ is given by

equation image

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

equation image

the correlation time τz can be expressed as a volume integral over the sphere (the initial coordinate r is uniform in the sphere because we assume a uniform background chemical concentration)

equation image

where r=|r|. The function

equation image

is analogous to the potential of a charge density in electrostatics, specifically the charge density ρ(z′) = z′ = ρ(r′,θ′) = r′cosθ′ for r′a and ρ(r′) = 0 for r′ > a. To solve the final integral in Eq. 17, we perform a multipole expansion of ψ(r,θ) in terms of Legendre polynomials Pl(cosθ), exploiting the rotational symmetry of ψ(r,θ) about the z axis (14), leading to

equation image

Now, consider the contribution from an additional gradient with zero mean over the volume of the sphere. We need to perform the integrals over a non uniform distribution in Eq. 17. Because a gradient along the z axis contributes a factor cosθ, which leads to the vanishing integral 0πsinθcos3θdθ=0, we conclude that only the constant background contributes to the uncertainty in the gradient measurement. Therefore, Eq. 19 for τz remains true even when an actual gradient is present.

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

equation image

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

Comparison with Experiment

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

To obtain the optimal Chemotactic Index CI, we assume that after averaging for a time T, a cell moves at a constant velocity in the direction of the estimated gradient. If we take the actual gradient to point in the z direction, then the chemotactic index for one run i is simply cosθi, where θi is the angle between the true gradient and the estimated gradient. If the velocity and run time are the same for each run, leading to a constant run length l, then the average Chemotactic Index is given by

equation image

To evaluate left angle bracketcosθiright angle bracket for a perfectly absorbing sphere, we use our result (Eq. 7) for the variance of the estimated gradient in each direction, e.g., (δcx,y,z)2=c0/(12πDa3T). Assuming a Gaussian distribution with these variances, as well as an actual gradient with mean value cz0 in the z direction, the two-dimensional distribution of estimated gradients cr=(cx,cz) is given by

equation image

where σx,z=(δcx,z)2. From this distribution, we can obtain the optimal Chemotactic Index

equation image

where y = (cz0)2/(4σz2 and I0(1) is the first (second)-order modified Bessel function of the first kind. Fig. 1 shows a comparison of the optimal CI (solid curves) with the data of ref. 6. Importantly, the comparison relies on only a single global fitting parameter representing gradient-sensing ability, namely, the product Da3T, where D is the diffusion constant, a is the cell diameter, and T is the averaging time. Based on the estimates D = 300μm2/s and a = 5μm (6), the averaging time is predicted to be about T = 3.2 s. (The perfect-monitor model yields an identical curve, but with a longer inferred averaging time T = 27.5 s.) The theory for the optimal Chemotactic Index matches the experiment rather well. Eq. 23 further predicts that the Chemotactic Index depends on the gradient cz0 and the concentration c0 only through the combination (cz0)2/c0 (Fig. 1 Inset). Intuitively, (cz0)2/c0 measures the signal-to-noise ratio—the signal is proportional to the true gradient |cz0|, whereas the noise from particle diffusion scales as c0. More generally, the optimal Chemotactic Index depends on all variables only through the combination (cz0)2Da3T/c0. Moreover, the theory predicts the full distribution of run angles (see SI Appendix, Fig. S1 Inset), which can be obtained by integrating Eq. 22 in the radial direction for each angle θ.

Discussion

Many types of cells are known to measure spatial chemical gradients directly with high accuracy. In particular, D. discoideum (Dicty) is well known to measure extremely shallow cAMP gradients 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 the immune system including neutrophils and lymphocytes (7). The question arises: What are the fundamental limits of gradient sensing set by chemical diffusion? Here, we derived such limits by using as model cells a perfectly absorbing sphere and a perfectly monitoring sphere (8). Within the theory, gradients are estimated by comparing the discrete distribution of observed locations of particles, either absorbed on the surface of the sphere (Fig. 2B) or measured inside the sphere (Fig. 2C), with the expected continuous distribution originating from a gradient (Fig. 2A). We find that a perfectly absorbing sphere is superior to a perfectly monitoring sphere for concentration and gradient sensing by respective factors of 12/5 (= 2.4) and 60/7 (≈ 8.6) (Table 1), because the perfectly absorbing sphere prevents rebinding of already measured particles. Indeed, the results presented here for the perfectly absorbing sphere represent the true fundamental limits of both concentration and gradient sensing by cells. Our theory for the limits of gradient sensing compares favorably with recent measurements by van Haastert and Postma (6) of Dicty cells migrating to a cAMP-filled pipette (Fig. 1), suggesting that Dicty chemotaxis approaches the fundamental limits set by cAMP diffusion.

The marked superiority of the perfect absorber for concentration and gradient sensing leads us to conjecture that cells may have developed mechanisms to absorb ligands so as to prevent their rebinding. Such absorption could be implemented by ligand or ligand–receptor internalization or by degradation of bound ligands. Even degradation of ligands without measurement could be advantageous. For example, a perfect absorber that measures only a fraction f of incident particles has the same uncertainties given in Table 1 but with an effective measurement time Teff = fT. Such an absorbing cell that measures only 12% of absorbed particles can still measure gradients as accurately as a perfectly monitoring sphere. Similarly, an absorbing sphere of radius a with two small measurement patches at its poles of radius s (s [double less-than sign] a), i.e., with a measuring surface-area fraction s2/(2a2), effectively reduces the averaging time for pole-to-pole gradients to 3s2T/(2a2) (SI Appendix). Consequently, a measuring surface fraction as small as 4% yields the same uncertainty as the monitoring sphere.

In fact, there are numerous examples in biology of ligand–receptor internalization (16) and ligand degradation on cell surfaces, which we speculate might be related to gradient sensing. (i) Although many G protein-coupled receptors are internalized by 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 (1921). One form is membrane-bound (mPDE) and effectively turns Dicty into an absorber, whereas the other form is soluble (ePDE). The membrane-bound form mPDE only accumulates during cell aggregration, supporting the idea that degradation of cAMP at the membrane helps accurate gradient sensing and navigation. Indeed, cells lacking mPDE display cell-autonomous chemotaxis defects even in mixed aggregates with isogenic wild-type cells (21). Interestingly, there is good evidence that Dicty cells do carry out G 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 mating response including directed growth (“shmooing”) toward a potential mating partner. Ligand-bound Ste2 undergoes internalization by endocytosis (23). Furthermore, the protease Bar1 degrades α-pheromone externally (24, 25), and may be largely membrane associated (26). (iii) There are many examples of ligand–receptor internalization in developmental biology. For example, primordial germ cells in zebrafish migrate toward the chemokine SDF-1a that activates the receptor CXCR4b. Ligand-induced CXCR4b internalization is required for precise arrival of germ cells at their target destination (27). These examples suggest a correlation between ligand internalization/degradation and the accuracy of cell polarization and movement. In presenting these admittedly speculative examples, our hope is to raise interest, across fields, in how the constraints of gradient sensing accuracy may have shaped cellular sensing systems.

Although the absorption of ligands can improve gradient sensing, there is an inherent problem for an absorbing cell to measure a gradient while moving. An absorbing cell moving in a uniform concentration creates an apparent gradient because of an increased flux of incoming particles at its front and a decreased flux of particles at its back (8). By using the model of a spherical cell, the ratio of fluxes between front and back hemispheres is given by

equation image

where v0 is the cell velocity, a is the cell radius, and D is the particle diffusion constant. On the other hand, the flux ratio of a stationary spherical cell in a gradient |c| with uniform background concentration background c0 is given by

equation image

Hence, a moving cell sees an apparent gradient

equation image

As an example, chemotaxis of Dicty to cAMP is observed at a mean concentration of 7 nM in a gradient of only 0.01 nM/μm (6). A Dicty cell moving with a typical speed of 0.2 μm/s at the same mean concentration but without a gradient creates an apparent gradient of about half the real gradient. There are several ways out of this dilemma. (i) Cells could separate measurement from movement at low gradients, e.g., by stopping, measuring the gradient, and then moving. Dicty would only need to stop for approximately a2/D ≈ 0.1 s based on cell radius of a = 5 μm and diffusion constant D = 300 μm2/s. (ii) Cells could sense gradients transverse to their direction of motion. This is particularly advantageous for fast–moving cells (e.g., bacteria) for which the apparent gradient can become >100 times steeper than the actual gradient (8). (Indeed, the oxygen-sensing marine bacterium Thiovulum majus directly senses gradients transverse to its direction of motion (29).) Interestingly, Dicty cells moving on agar in the absence of a gradient appear to combine these two strategies. Qualitatively, the tips of elongated moving cells slow down and flatten, often producing two or more distinct pseudopods. Cells then elongate and move (approximately one cell length) in the direction of one of the pseudopods before the process is repeated (Liang Li and Ted Cox, personal communication). By this strategy, Dicty may avoid locking onto a false, movement-generated gradient. (iii) In principle, cells could compensate for the apparent motion-generated gradient, either by internal signal processing or by external chemical secretion. In fact, Dicty cells do secrete cAMP, primarily from their trailing edge during movement (4), but this cAMP secretion serves dominantly to facilitate cell aggregation, including cells following cells during streaming. Given the complex role of cAMP in Dicty aggregation, studies of Dicty chemotaxis using gradients of folate, which is absorbed (22) but apparently not secreted by Dicty, may ultimately prove simpler to interpret.

Our models of the absorbing and the monitoring spheres neglect all biochemical reactions, such as particle–receptor binding and downstream signaling, which could significantly increase measurement uncertainty beyond the fundamental limits described here. To study the effects of particle–receptor binding, we extended a formalism for the uncertainty of concentration sensing, recently developed by Bialek and Setayeshgar (10), to gradient sensing. We found that the measurement uncertainty allowing ligand rebinding is larger than the measurement uncertainty without rebinding, confirming the superiority of the absorber over the monitor. A number of mechanistic models for gradient sensing and chemotaxis have addressed the important questions of cell polarization, signal amplification, and adaptation (3036), cell movement of individual cells (37,38), cell aggregation with cAMP degradation by PDE (39), as well as sensing of fluctuating concentrations (10,40,41). Our results on the fundamental limits of gradient sensing complement these models, and may ultimately help lead to a comprehensive description of eukaryotic chemotaxis (42).

Finally, we remark that the experiments by van Haastert and Postma used stationary spatial gradients (6). Cells in such gradients 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 the chemotactic index for randomly changing gradients, to find out whether cells indeed use their memory to improve chemotaxis.

Supplementary Material

Supporting Information:

Acknowledgments.

We thank Naama Barkai, John Bonner, Rob Cooper, Ted Cox, Liang Li, Trudi Schupbach, Stanislas Shvartsman, and Monika Skoge for helpful suggestions. This work was supported by the Human Frontier Science Program. R.G.E. was supported by Biotechnological and Biological Sciences Research Council Grant BB/G000131/1 and the Centre for Integrated Systems Biology at Imperial College. N.S.W. was supported by National Science Foundation Grant PHY-065017.

Footnotes

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

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

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