Physical limits to biochemical signaling

Theory

Binding to a Single Receptor. Consider a binding site for signaling molecules, and let the fractional occupancy of the site be n. If we do not worry about the discreteness of this one site, or about the fluctuations in concentration c of the signaling molecule, we can write a kinetic equation

This equation describes the kinetics whereby the system comes to equilibrium. The free energy F associated with binding, which is given by the difference in the free energies of the unbound and bound states of the receptor, is related to the rate constant through detailed balance,

If we imagine that thermal fluctuations can lead to small changes δk ₊ and δk _- in the rate constants, we can linearize Eq. 2 to obtain

But from Eq. 3 we have

Applying this constraint to Eq. 4, we find that the individual rate constant fluctuations cancel, and all that remains is the fluctuation in the thermodynamic binding energy δF; the resulting equation can be written in the form

It is useful to note the analogy between this chemical kinetic problem and the Langevin equation (15) for the position, X(t), of an overdamped Brownian particle bound by a Hookean spring. The spring generates a restoring force proportional to position, -κX, and as the particle moves through the fluid it experiences a viscous drag with drag coefficient γ, so that the (Newtonian) equation of motion becomes

where f(t) is a fluctuating force. The dissipative and fluctuating parts of the force on the Brownian particle are related through the fluctuation-dissipation theorem

where k _B is the Boltzmann constant and T is the temperature; angle brackets denote ensemble averages. Intuitively, this relation is a consequence of the fact that the fluctuating and the dissipative forces both arise because of collisions of the Brownian particle with the molecules of the fluid.

More generally, the linear response, X(t), of a system from equilibrium due to the thermodynamically conjugate “force,” F(t), defines the generalized susceptibility, α(t),

where we have taken 〈X(t)〉 = 0 (Fig. 2). The generalized susceptibility depends on the properties of the system and completely characterizes its response to small external perturbations. Fourier transforming

the response to an external force near equilibrium becomes . In its general form, the fluctuation-dissipation theorem relates the imaginary part of the generalized susceptibility, , which determines how much energy is dissipated by a system as heat due to an external force, to the power spectrum of the spontaneous fluctuations of the corresponding coordinate, X, for the closed system in thermal equilibrium

where Im [...] refers to the imaginary part. In the present chemical system, the “coordinates” are the concentrations of the interacting species, or equivalently the fractional occupancy of receptors, the phenomenological “equations of motion” are the chemical kinetic equations, and the thermodynamically conjugate “forces” are the free-energy differences among the species (17, 18) (Table 1).

Fourier transforming, from Eq. 6 we find the generalized susceptibility, , describing the response of the coordinate n to its conjugate force F,

The fluctuation-dissipation theorem relates this response function to the power spectrum of fluctuations in the occupancy, n,

where the total variance is

and the correlation time is given by τ_c = (k ₊ c + k _-)^-1. These are the usual results for switching in a Markovian way between two states; here it follows from the “macroscopic” kinetic equations, as in Eq. 2, plus the fact that binding is an equilibrium process. Note that in using the fluctuation-dissipation theorem to arrive at this result, no assumptions are required about the underlying statistics of transitions between the bound and unbound states of the receptor. The Markovian nature of these transitions is reflected in the macroscopic chemical kinetic equations.

Coupled Binding and Diffusion. The same methods can be used in the more general case where the concentration itself has dynamics due to diffusion. Now we write

where the receptor is located at , and

where the last term expresses the “injection” of one molecule at the point as it unbinds from the receptor. Linearizing the equations as before, and solving Eq. 18 by transforming to spatial Fourier variables, we find the linear response function

where c̃ is the mean concentration, and

We note that by obtaining the spatial Fourier transform of Eq. 18 over infinite volume, we are assuming the number of ligand molecules to be infinite at constant concentration. Hence, we are considering only the regime where the number of ligand molecules exceeds the number of receptors. The “self-energy” ∑ (ω) is ultraviolet divergent, which can be traced to the delta function in Eq. 18; we have assumed that the receptor is infinitely small. A more realistic treatment would give the receptor a finite size, which is equivalent to cutting off the k integrals at some (large) Λ ∼ π/a, with a the linear dimension of the receptor.

If we imagine mechanisms that read out the receptor occupancy and average over a time τ long compared with the correlation time τ_c of the noise, then the relevant quantity is the low frequency limit of the noise spectrum. Hence, we are interested in

and

Applying the fluctuation-dissipation theorem once again, we find

The total variance in occupancy is unchanged because it is an equilibrium property of the system. Coupling to concentration fluctuations does serve to renormalize the correlation time of the noise, τ_c → τ_c [1 + ∑(0)]. The new τ_c can be written as

so the second term is a lower bound on τ_c, independent of the kinetic parameters k _±,

Again, the relevant quantity is the low-frequency limit of the noise spectrum,

If we average for a time τ, then the root-mean-square error in our estimate of n will be

and we see that this noise level has a minimum value independent of the kinetic parameters k _±,

To relate these results back to the discussion by Berg and Purcell (6), we note that an overall change in concentration is equivalent to a change in F by an amount equal to the change in chemical potential, so that Δc/c̄ ≡ ΔF/k _B T. This equivalence means that there is an effective spectral density of noise in measuring c

where the “noise force” spectrum S_F(ω) is given by the fluctuation-dissipation theorem as

In the present case, we find that

As before, the accuracy of a measurement that integrates for a time τ is set by

and we find again a lower bound that is determined only by the physics of diffusion,

Note that this result is (up to a factor of ) exactly the Berg-Purcell formula in Eq. 1.

Binding to Multiple, Noninteracting Receptors. To complete the derivation of Berg and Purcell's original results (6), we consider a collection of m receptor sites at positions , where μ = 1, 2,..., m:

We can solve Eq. 35 by going to a spatial Fourier representation as before, and we find

where Λ is the cut-off wave number; as before, the cut-off arises to regulate the delta function in Eq. 35 and is related to the size of the individual receptor. In the limit (ω/D)^1/2 ≪ 1, for ν ≠ μ with ν, μ = 1,..., m, we have

Combining this equation with the Fourier transform of Eq. 34 and summing to find the total occupancy δÑ(ω) = δn _μ(ω) of the receptor cluster, we obtain

In cluster geometries such that the innermost sum is independent of , we can write the sum as

where

and this simplification allows us to solve Eq. 38 directly to find the response of δÑ to the force . Then, as before, we use the fluctuation-dissipation theorem to find the spectrum of and convert that to an equivalent concentration error as in Eq. 31. The result is

We note that whereas the first term is positive and depends on the details of the chemical kinetics of ligand-receptor binding, the second term defines a lower bound on the measurement accuracy of the ligand concentration by the receptor cluster that depends only on the physics of diffusion. As an example, for receptors of radius b uniformly distributed around a ring of radius a > b (Fig. 3), this lower bound is

where for m ≫ 1, we have used φ(m) ≈ mg ₀/a; g ₀ is a geometric factor of order unity for typical cluster geometries and receptor distributions (W.B. and S.S., unpublished results). With an increasing number of receptors, m, the accuracy in measuring the substrate concentration ultimately is limited by the linear size of the cluster. Extension of this result to cooperatively interacting receptors is treated separately elsewhere (W.B. and S.S., unpublished results); remarkably, we find that this lower bound in measurement accuracy persists, independent of the details of cooperative interactions among the cluster subunits.

Summary. For both a single receptor and an array of receptors, we find that the sensitivity of signaling is limited, and this limit can be described as an effective noise in concentration, δc _rms, determined by

In this summary, F({k_j}, c̄, m) depends on the details of the kinetic interactions between the signaling molecules and its receptor through the kinetic parameters, {k_j}, the substrate concentration, c̄, and the number of receptors, m. The important point is that this term is positive, so that even if we do not know the details of the ligand-receptor chemical kinetics, we know that the noise level can never be smaller than that set by the second term. In the second term ℓ is an effective size for the receptor or the receptor array; then, to within a factor, this term is exactly that written down by Berg and Purcell (Eq. 1) for a perfect concentration measuring device. Thus, although the original Berg and Purcell (6) arguments made use of very specific assumptions, we see that in the general case the details of the chemical signaling only add to the noise level.

Comparison with Experiment

Here, we consider two experimentally well-characterized examples to demonstrate how the theoretical limits on receptor occupancy noise and the resulting precision of concentration measurements compare with the performance of real cellular signaling systems.

Regulation of Gene Expression in Bacteria. Expression of genes is controlled in part by the occupancy of specific sites in the promotor regions adjacent to the sequences of DNA that code for protein (19). Reversing the usual picture of changes in transcription factor concentration as driving changes in gene expression, we can view gene expression as a sensor for the concentration of the transcription factor proteins that bind to the promoter site. In a bacterium like E. coli, transcription factors are present in N _TF ∼ 100 copies in a cell of volume of ∼1 μm³ (20); presumably, the concentration of free transcription factor molecules is smaller than N _TF/V. Diffusion constants for small proteins in the E. coli cytoplasm are D ∼ 3 μm²/s (21); promoter sites have a linear dimension a ∼ 3 nm, and putting these factors together, we find the crucial combination of parameters πDac̄ ≲ 3 s^-1.

If the transcription factor is a repressor then gene expression levels are determined by 1 - n, whereas if it is an activator then expression is related to n. Because δn _rms α n̄ (1 - n̄) (Eq. 28), fractional fluctuations in either A = n or A = 1 - n are determined by

The minimum fluctuations in expression level thus are given by

Recent experiments (22) indicate that E. coli achieves ≈10% precision in control of gene expression at small values of Ā. For this performance to be consistent with the physical limits, the transcription machinery must therefore integrate the promoter site occupancy for times of the order of 1 min, even assuming that the translation from occupancy to expression level itself is noiseless. This integration can be provided by the lifetime of the mRNA transcripts themselves, which is ∼3 min in prokaryotes (23). Recent theoretical work, motivated by ref. 22, has addressed the difference between “intrinsic” sources of noise in the regulation of gene expression in bacteria and “extrinsic” sources of noise that arise from population-level variations among cells (24-26). In this context, our results demonstrate the existence of a previously unappreciated intrinsic noise floor due to diffusion of transcription factors. In principle, this noise floor could be reduced by averaging over longer times, but very recent work on engineered regulatory elements in E. coli (27) shows that the correlation time of intrinsic noise is <10 min, suggesting that the time averaging done by this system is not much more than the minimum required to achieve the observed precision.

Control of the Flagellar Motor by CheY. The output of bacterial chemotaxis is control of the flagellar rotary motor (28). The phosphorylated form of the signaling protein CheY (CheY∼P) binds to the motor and modulates the probability of clockwise vs. counterclockwise rotation (29). Recent measurements (30) show that the probability p of clockwise rotation depends very steeply on the concentration c of CheY∼P,

with h ∼ 10 and c _1/2 ∼ 3 μM. In the phenomenological description of the motor as a simple random telegraph process, switching between clockwise and counterclockwise rotation is governed by Poisson statistics. For c ≈ c _1/2, the switching frequency is measured experimentally to be f ≈ 1.5 s^-1. If we view the motor as a sensor for the internal messenger CheY∼P, then the observed behavior of the motor determines an equivalent noise level of

where τ₀ is the correlation time of the motor state; for the simple telegraph model it can be shown that τ₀ = 2p(1 - p)/f. Using Eq. 46 we find

Thus, for c ≈ c _1/2, a single motor provides a readout of CheY∼P concentration accurate to ≈12% within 2 s.

The motor C ring has a diameter 2a ∼ 45 nm, with m ∼ 34 individual subunits to which the CheY∼P molecules bind (31). From Eq. 42 we find

where we have taken the size of the individual receptor binding site to be b ∼ 1 nm, and D ∼ 3 μm²/s as above. Hence, for the collection of receptors comprising the motor, the physical limit to measurements of the CheY∼P concentration corresponds to ≈4% precision within 2 s.

Taken at face value, our estimates of the actual sensitivity and limiting sensitivity of the motor agree within a factor of three. Recent work shows that at constant CheY∼P concentration the power spectrum of motor bias deviates substantially from the Lorentzian prediction of the Poisson or telegraph model (32); in particular, there is peaking in the spectrum so that the low frequency limiting noise may be lower than estimated from the mean switching frequency. It also might be the case that even the very steep dependence in Eq. 46 is broadened by small errors in the concentration measurements (see note 17 in ref. 30), so that we underestimate the actual sensitivity of the motor. Thus, it is possible that there is an even closer agreement between the cell's performance and the physical limit, which could be tested in experiments pointed more specifically at this issue.

Concluding Remarks

We have derived from statistical mechanics the physical limits to the precision of concentration measurement for biological sensors that rely on the binding of a diffusing ligand to a receptor. Our approach complements and extends the classic work by Berg and Purcell (6), establishing that their intuitive result indeed does set a lower bound on the actual noise level. For a single receptor the accuracy in measurement of concentration is limited by the noise associated with the arrival of discrete substrate molecules at the receptor. Our approach extends in a straight-forward way to multiple receptors without relying on additional considerations; for this case, our result demonstrates more transparently the role of multiple receptors in improving the measurement accuracy, as well as that of correlations in insuring that this improvement saturates at a level set by the receptor cluster size. Relevant internal or external signaling molecules are often present in low copy numbers, and their concentration in turn regulates downstream biochemical networks crucial to the cell's functions. For two experimentally well-studied examples, we show that the cell's performance is close to the physical limits.