Berg and Purcell [Biophys. J. 20, 193 (1977)] calculated how the accuracy of concentration sensing by single-celled organisms is limited by noise from the small number of counted molecules. Here we generalize their results to the sensing of concentration ramps, which is often the biologically relevant situation (e.g. during bacterial chemotaxis). We calculate lower bounds on the uncertainty of ramp sensing by three measurement devices: a single receptor, an absorbing sphere, and a monitoring sphere. We contrast two strategies, simple linear regression of the input signal versus maximum likelihood estimation, and show that the latter can be twice as accurate as the former. Finally, we consider biological implementations of these two strategies, and identify possible signatures that maximum likelihood estimation is implemented by real biological systems.
Deep Dive into Limits of sensing temporal concentration changes by single cells.
Berg and Purcell [Biophys. J. 20, 193 (1977)] calculated how the accuracy of concentration sensing by single-celled organisms is limited by noise from the small number of counted molecules. Here we generalize their results to the sensing of concentration ramps, which is often the biologically relevant situation (e.g. during bacterial chemotaxis). We calculate lower bounds on the uncertainty of ramp sensing by three measurement devices: a single receptor, an absorbing sphere, and a monitoring sphere. We contrast two strategies, simple linear regression of the input signal versus maximum likelihood estimation, and show that the latter can be twice as accurate as the former. Finally, we consider biological implementations of these two strategies, and identify possible signatures that maximum likelihood estimation is implemented by real biological systems.
Cells are able to sense concentration gradients with high accuracy. Large eukaryotic cells such as the amoeba Dictyostelium discoideum and the budding yeast Saccharomyces cerevisiae can sense very shallow spatial gradients by comparing concentrations across their lengths [1]. By contrast, small motile bacteria such as Escherichia coli detect spatial gradients indirectly by measuring concentration ramps (temporal concentration changes) as they swim [2], and can respond to concentrations as low as 3.2 nM-about three molecules per cell volume [3]. The noise arising from the small number of detected molecules sets a fundamental physical limit on the accuracy of concentration sensing, as originally shown in the seminal work of Berg and Purcell [4,5]. This approach was recently extended to derive a fundamental bound on the accuracy of direct spatial gradient sensing [6]. However, no theory exists for the physical limit of ramp sensing, which is what bacteria actually do when they chemotact. In this Letter, we present such a theory for different measurement devices, from a single receptor to an entire cell. We contrast two strategies: linear regression (LR) of the input signal (in line with Berg and Purcell) and maximum likelihood estimation (MLE) [7,8], a method from statistics to optimally fit a model to data, revealing an up to twofold advantage for the latter. Finally, we introduce a biochemical signaling network, similar to the E. coli chemotaxis system, that outputs an estimate of the ramp rate. Consistent with the derived theoretical bounds, we find that a mechanism emulating MLE yields twofold higher accuracy that one emulating LR. However, this improved performance has a cost: either storage of signaling proteins near the receptors, or irreversibility of the receptor cycle with concomitant energy consumption.
Sensing small numbers of molecules implies relative noise ∼n -1/2 , where n is the number of detected molecules. Berg and Purcell (BP) calculated how this noise affects the accuracy of concentration sensing [4]. They considered three types of measurement devices: a single receptor, a perfectly absorbing sphere, and a per- fectly monitoring sphere. Following their approach, we investigate ramp sensing by these three devices when presented with a concentration c(t) = c 0 + c 1 t, as schematized in Fig. 1.
A single receptor [Fig. 1(a)] binds particles at rate k + c(t) and unbinds them at rate k -. Following BP, we assume that diffusion is fast enough that the receptor never rebinds the same particle. An ideal observer has access to the binary time series s(t) of receptor occupancy between -T /2 and T /2. The lengths of bound and unbound invervals have exponential distributions with means 1/k -and 1/k + c, respectively. Throughout, we assume that the ramp is shallow, c 1 T c 0 , and that the observation time is long compared to receptor kinetics,
In BP, the true concentration c is estimated from the fraction of time the receptor is bound, s = 1 T T /2 -T /2 dt s(t), which is equal to the equilibrium occupancy in the limit of large times:
where • represents an ensemble average. Following a similar strategy, we can estimate the ramp rate by performing the linear regression of s(t) to s 0 + s 1 t:
from which the concentration and the ramp rate are estimated using (A18) as:
The uncertainties of these estimates can be calculated from the time correlations of receptor occupancy (see Appendix A 1 a), yielding:
where n is the total number of binding events in the time T . Note that the result for c 0 is precisely that of BP [4,8].
In [8], it was shown that the accuracy of concentration sensing could be improved using maximum likelihood estimation. In this scheme, the parameters of the model are chosen to maximize the probability (“likelihood”) that the observed data was generated by the model. Can we also improve the accuracy of ramp sensing over LR by using this method? The time trace s(t) can be characterized by the series of binding (t + i ) and subsequent unbinding (t - i ) times, i = 1, . . . , n. The probability of the data within our model is [8]:
where T b is the total bound time. The concentration and the ramp rate, c 0 and c 1 , are the model parameters.
Given the times of the events, the likelihood is maximized with respect to c 0 and c 1 by solving ∂P /∂c 0 = 0 and ∂P /∂c 1 = 0, from which the maximum likelihood estimate
) is obtained. In general these equations have no simple solution, but we can obtain the average behavior by exploiting the fact that binding and unbinding are fast with respect to concentration changes, i.e. that the receptor remains adiabatically in equilibrium with the concentration c(t). We can thus simplify the sum and product in (5):
where s(t) is the equilibrium occupancy at time t, given by (A18) with c = c0 + c1 t, where c0 and c1 are the true parameters that generated the data. Applying this approximation to ∂P /∂c 0 , ∂P /∂c 1 , we confirm that c MLE 0 = c0 and
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