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BWA, TMY, and PAI conceived and designed the experiments. BWA and PAI performed the experiments, analyzed the data, and contributed reagents/materials/analysis tools. BWA, TMY, and PAI wrote the paper.

The authors have declared that no competing interests exist.

Information-carrying signals in the real world are often obscured by noise. A challenge for any system is to filter the signal from the corrupting noise. This task is particularly acute for the signal transduction network that mediates bacterial chemotaxis, because the signals are subtle, the noise arising from stochastic fluctuations is substantial, and the system is effectively acting as a differentiator which amplifies noise. Here, we investigated the filtering properties of this biological system. Through simulation, we first show that the cutoff frequency has a dramatic effect on the chemotactic efficiency of the cell. Then, using a mathematical model to describe the signal, noise, and system, we formulated and solved an optimal filtering problem to determine the cutoff frequency that bests separates the low-frequency signal from the high-frequency noise. There was good agreement between the theory, simulations, and published experimental data. Finally, we propose that an elegant implementation of the optimal filter in combination with a differentiator can be achieved via an integral control system. This paper furnishes a simple quantitative framework for interpreting many of the key notions about bacterial chemotaxis, and, more generally, it highlights the constraints on biological systems imposed by noise.

Bacterial motility involves successive periods of relatively straight runs, interspersed by tumbles—periods in which the bacteria are reoriented randomly. To move in the direction of chemical gradients—a process known as chemotaxis—cells modulate the duration of the runs. To ascertain whether the direction of the current run is desirable, cells continuously monitor temporal changes in the chemoattractant concentration. However, the decisions can only be based on imperfect information about the environment because binding noise implies that receptor occupancy is a limited measure of the chemoattractant concentration. Bacteria cope by filtering the sensed signal to reduce the effect of this binding noise. Through simulations, Andrews, Yi, and Iglesias demonstrate that there is a particular filter cutoff frequency that achieves optimal chemotaxis. Moreover, using a model of the sensing mechanism, the authors also compute the theoretically optimal system for estimating the chemoattractant concentration from the noisy receptor-occupancy signal. Andrews and colleagues show that these two filtering systems are closely matched, and that their frequency-dependent behavior corresponds to published experimental data. Their results highlight the constraints that noise places on cellular performance as well as demonstrating how cells have evolved to deal with this uncertainty in an optimal fashion.

One of the thrusts of the new area of systems biology is the understanding that biological systems can be studied with many of the tools and theory that are used to analyze man-made systems [

Bacteria traverse up gradients of chemical attractants in an efficient manner. This chemotactic behavior is universal in living organisms. Two basic chemotaxis strategies are spatial and temporal sensing [

The classic work of Berg and colleagues, along with others, have conclusively demonstrated that

(A) _{i}

(B) Chemotactic system. Chemoreceptor complexes contain the proteins CheA and CheW. Ligand–receptor interactions, stochastic in nature, affect the autophosphorylation of the kinase CheA (A), which is capable of transferring its phosphoryl group (P) to the protein CheY (Y). The phosphorylated form of CheY induces clockwise rotation of the flagella, causing tumbling. CheA also transfers phosphoryl groups to CheB (B), an enzyme responsible for demethylation of the receptor complex. Adaptation is achieved via the methylation of the receptor complex by CheR (R). Clockwise rotation of the flagella induces tumbling and a reorientation of the cell while CCW rotation propels the cell forward in a run. Swimming, affected by rotational and translational diffusion, leads the cell to new ligand concentrations in the environment.

The sensing of the chemoeffectors in

A major challenge of temporal sensing is accurately measuring changes in chemoattractant concentration (

(A) When subjected to an abrupt change in chemoattractant concentration (_{R}_{R}

(B) The adaptation response of

In this paper, we use theoretical tools to study the effects of noise on the ability of a cell to chemotax and, more specifically, on how bacteria cope. We first use simulations to show that adaptation time (filter-cutoff frequency) affects the chemotactic performance of the cell and that there exists an optimal adaptation time that allows cells to chemotax the farthest. Having shown that the adaptation time can make a difference to chemotactic efficiency, we conjecture that this effect is predicated on the ability of the cell to estimate its environment accurately in the presence of noise. To test this, we use theoretical tools to determine which filter-cutoff frequency leads to optimal detection of temporal changes in chemoattractant concentration. The signal to be estimated is ligand concentration, and noise arises from the discrete nature of ligand–receptor interactions during binding. We demonstrate that the theoretically obtained cutoff frequency for optimal estimation has the same dependence on external parameters (e.g., noise and rotational diffusions levels) as the adaptation time that achieves optimal chemotaxis in simulation, implying that optimal ligand estimation is necessary for optimal chemotaxis. Finally, we illustrate how integral feedback control, a known characteristic of the bacterial signaling network [

We first examined the effect of adaptation time on chemotaxis through simulation. To this end, we developed a model of the signal transduction network of

Our simulations indicate a biphasic response: cells with a particular filter-cutoff frequency move farther up the chemoattractant gradient than cells with either higher or lower frequency cutoffs (^{−5} using Student's t-test). Similarly, a 4-fold decrease in the cutoff frequency (from 3.4 to 0.8 rad/s) also reduced the distance traveled, though less significantly (213 ± 15 μm, SEM,

Chemotaxis was simulated by assuming that the signaling pathway was approximated by the system of

(A) Different choices of

(B) These simulations were repeated for a large range of cutoff frequencies. The resulting frequency-dependent chemotactic performance was fitted with a Rayleigh function to estimate the optimal cutoff frequency for chemotaxis (red dashed line). Chemotactic performance was based on the final position along the gradient after 80 s. Each point represents the average of 500 simulation runs, and vertical bars indicate mean plus or minus standard error of the mean. Rotational and translational diffusion coefficients of 0.16 rad^{2}/s and 2.2 × 10^{−1} μm^{2}/s, respectively, were used. Nominal parameters used: _{T}_{d}_{−}/_{+} = 100 μM, _{0} = 0.01 μM, and

To determine how external parameters affect the optimal filtering conditions and how they relate to chemotaxis performance, we repeated our simulations for a wide range of binding noise levels and rotational diffusion coefficients (

(A) The effect of the measurement noise on the optimal cutoff frequency for chemotactic performance was studied by varying the measurement noise (by multiplying the binding variance by a factor) by several orders of magnitude and determining, as in

(B) Similarly, the effect of rotational diffusion on chemotactic performance was studied by varying the diffusion coefficient by several orders of magnitude. Decreasing rotational diffusion increases the chemotactic efficiency, but decreases the optimal cutoff frequency.

(C) The optimal filter for chemoattractant estimation was computed and its frequency-dependent magnitude plotted for a specific combination of model parameters. Its shape can be approximated by a first-order filter where the cutoff frequency is the frequency at which the gain is 0.707 (−3dB point) that of the zero frequency gain.

(D,E) The noise and rotational diffusion dependence of the optimal cutoff frequencies for chemotaxis obtained in A and B was compared with the optimal cutoff frequencies for chemoattractant estimation predicted by the optimal filter theory. The vertical bars indicate cutoff frequencies that yield chemotactic performances of 95%–100% of the maximum of the Rayleigh curve fitting (see

The simulations above indicated the existence of specific adaptation times that lead to optimal chemotaxis, and that these adaptation times are dictated by the cutoff frequency of a low-pass filter. To investigate this further, we used tools from estimation theory to deduce the optimal filter that best separates the true ligand concentration from the noise induced by ligand–receptor binding. We assumed that

In practice, the cell can only determine an estimate of

We assumed a 2-D environment with linear ligand gradient _{r}_{r}^{2}u^{2}_{r}

The optimal estimate of

To investigate the dependence of the optimal cutoff frequency for chemoattractant estimation on environmental conditions, we repeated our computations for different levels of mean ligand concentration and varying spatial concentration gradients (

The optimal filter for chemoattractant estimation was computed for a range of gradients (A) and mean concentrations (B). Nominal parameter values were as in

We next determined the degree to which the cutoff frequencies of the optimal filter for chemoattractant estimation matched the frequency-dependent behavior observed experimentally. To this end we used published data describing the behavior of the

(A) The frequency-dependent filtering responses for the optimal filter (red solid line) and experimental data (green dashed line; [_{0} = 1 μM, _{0} = 1 μM. The dotted black line shows a dependency of (frequency)^{−1}.

(B) The predicted optimal cutoff frequency is compared with that of the filter for a range of chemoattractant gradients and mean concentrations. The line through which both surfaces intersect represents the chemoattractant profiles for which

(C) Plot of the concentration gradient against mean concentration for the points where the surfaces in (B) intersect. The linear dependence suggests that

We computed the cutoff frequencies of the model for a wide range of chemoattractant gradients and mean concentrations and compared these with the theoretically derived optimal estimator cutoff frequencies (^{−1}, R^{2} = 0.97), suggesting that the filtering characteristic of

Previously, Berg and others have argued that the choice of a time over which the bacterium integrates the observed signal to determine the tumbling decision balances the need to average fluctuations in the measurement of ligand levels with the ability to track rapid changes in direction caused by rotational diffusion of the cell [

The optimal cutoff frequency for chemoattractant estimation represents the averaging time that best compromises the tradeoff between noise attenuation and signal amplification. This is illustrated in _{cf}_{cf}

(A) Insight into the relevant features of the optimal estimator can be obtained by considering a simplified model that assumes the cell measures _{cf}_{r}^{2}/s, τ = 1 s, _{T}_{−}/_{+} = 100 μM, _{0} = 1 μM.

(B) Block diagram representation of the chemotactic system. A system using an integral control feedback mechanism (top) is functionally equivalent to one consisting of the series connection of a differentiator and a first-order low-pass filter. In the chemotaxis pathway, this subsystem is followed by a conversion function of CheYp to the running bias.

It should be noted that while we have considered binding fluctuations as the measurement noise that results solely from the stochastic dynamics of ligand–receptor interactions, the total noise that contributes to uncertainty in ligand measurement also depends on ligand diffusion [^{−5} μM^{2} [^{2} [

We have shown that a low-pass filter is necessary for optimal chemotactic performance of the bacterial system in the presence of noise. Integral control provides an important unifying theme in this work. An integral feedback control system implements a differentiator in combination with a first-order low-pass filter (

The theory and simulations offer several important predictions. First, the activities of the receptor methylase CheR and the demethylase CheB determine the breakpoint of the chemotactic differentiator. For example, increasing the level/activity of CheR causes the breakpoint to shift to larger frequencies and, correspondingly, shorter adaptation times. This trend is observed in the experimental results of [

Finally, our results also highlight the fragile nature of the chemotaxis system by demonstrating how sensitive chemotactic performance can be to changes in adaptation times. In fact, the experiments that definitively showed that the property of adaptation was robust also demonstrated that the adaptation times were not: changes in CheB and CheR expression levels cause significant changes in adaptation time [

We developed a model of the ^{j}^{j}

A model of the bacterial signaling network is shown for a cluster of

The probability of the entire cluster being on when ^{l}

Only inactive states may be methylated, and only active states may be demethylated. Thus, a state with _{R}

In general (with obvious exceptions for ^{0}, ^{0}, ^{NM}^{NM}^{j}^{j}

Total activity of the cluster is

We compared a variety of different response characteristics of the model to those observed experimentally (

(A) Response of the

(B) Response to a positive (+0.2 mM) chemoattractant impulse of width 0.5 s is also consistent with experimentally observed behavior [

(C) The initial response to addition (removal) of attractant increases (decreases) with increasing change in attractant concentration. Adaptation times for attractant addition are longer than that for removal as observed [

(D) For step inputs, normalized peak activity of the model (1 – Δ_{ss},_{ss}

(E,F) Frequency responses of the _{0} = 1 μM, reveal low-pass characteristics consistent with observations in [

The model parameters used are _{f}^{−1} ms^{−1}, _{r}^{−1}, γ_{R}^{−1}, and γ_{B} = 2 γ_{R}

Free Energy Differences

We implemented the model as an S-Function in Matlab, version 7 (Mathworks,

Comparison of the step response of the _{0} = 1 μM. Model parameters are the same as in

For a meaningful quantitative comparison with experimental data, we need to adjust for the fact that the models include only the receptor methylation/demethylation dynamics, but the experimental data measures the system response in terms of motor biases and so contain the dynamics of the phosphorylated CheA, CheY, CheZ, and FliM-bound CheY. To determine the contribution of these elements to the overall frequency response, we used a linearization of the differential equation model for these elements [

The frequency response of the system exhibits characteristics of a differentiator coupled with a first-order low-pass filter; thus, the model of the

To study the effect of adaptation time on chemotaxis, we developed a computational model of a chemotaxing cell in Simulink version 6.2 by Mathworks, available by contacting the authors; see

(A) The chemotaxis simulation assumes the following feedback interaction between the bacteria and its environment: 1) the external environment affects the sensing model through the external ligand concentration

(B,C) Validation of chemotaxis model histograms of tumble (B) and run (C) durations for an unstimulated (_{0} = 0)

(D) Average simulation results of 500 runs of a bacteria swimming in a gradient of ligand concentration with varying chemoattractant slopes.

Bacteria are assumed to chemotax in a 2-D (_{0},0) where the gradient changes along the _{0} is a constant initial ligand concentration at _{T}L_{D}_{D}^{2}) where _{D}_{−}/_{+} is the binding dissociation constant, _{+} and _{−} are the on and off rates, respectively, and _{T}_{r}t_{s}_{r}_{s}_{t}_{s}_{t}_{s}_{t}t_{s}_{r}^{2}/second and _{t}^{−1} μm^{2}/second [

The signal transduction pathway in _{0} and a steady state CheYp concentration of 3 μM [

It is known that the probability of an individual flagellum rotating clockwise, which induces tumbling, or CCW, which leads to runs, is a Hill function of CheYp concentration [_{p}_{m}_{1} be the run bias given that the cell is already running, and _{2} be the tumble bias given that the cell is already tumbling. The expected run length in number of simulation steps is:

Suppose that _{2} are fixed. Then, solving for _{1} yields: _{1} = 1 – (1 – _{2}) (1 – _{s}_{s}_{s}_{2}: _{2} = 1 – _{s}_{s}

The probabilities _{1} and _{2} are then used to determine at a given simulation step whether the next time step should consist of a run or a tumble.

The tumbling angle θ was assumed to follow the probability density function

The combined signal transduction and environmental model exhibits several traits observed in cells. The exponential shape of the run/tumble distributions is characteristic of a Poisson distribution and is similar to interval distributions found experimentally [

To determine the optimal filter for extracting the signal from the noisy measurement, we used Kalman filter theory. The Kalman filter is an algorithm, used widely in engineering navigation and guidance systems, developed to detect and separate signals in the presence of random, unwanted noise [_{0} and a steady-state receptor-complex concentration _{0} to give
_{C}_{C}_{T}L_{0}_{D}_{D}_{0})^{2} [

For simplicity, we assume θ(_{r}_{r}^{2}^{2}var[cos(θ(

Our results are obtained by computing the optimal filter for the above system using the

Simulations of chemotactic performance for varying cutoff frequencies were repeated for filters with roll-off of first, second, and third order. Higher-order filters do not affect the existence of an optimal filter cutoff frequency, although the cutoff frequency does decrease. Also, performance is greatest with the first-order filter, possibly due to increased phase delay with the higher-order filters.

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We thank Doug Robinson at Johns Hopkins University for helpful comments regarding the manuscript.

counterclockwise

power spectral density

_{3}response in