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Optimal noise filtering in the chemotactic response of Escherichia coli.

Andrews BW, Yi TM, Iglesias PA - PLoS Comput. Biol. (2006)

Bottom Line: Through simulation, we first show that the cutoff frequency has a dramatic effect on the chemotactic efficiency of the cell.There was good agreement between the theory, simulations, and published experimental data.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.

View Article: PubMed Central - PubMed

Affiliation: Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, Maryland, United States of America.

ABSTRACT
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.

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Effect of Adaptation Time on Filtering Capabilities(A) When subjected to an abrupt change in chemoattractant concentration (x(t)), the output (A(t)) of the model of the E. coli signaling pathway (Materials and Methods) responds with an initial transient burst which decays exponentially. The adaptation time depends on the component levels of the signaling pathway such as the methylation rate γR [13,14]. Faster methylation rates (γR) yield shorter adaptation times but result in noisier activity levels. Thus, the filtering capabilities of E. coli are determined by the time it takes to adapt to step inputs of ligand.(B) The adaptation response of E. coli is representative of a system consisting of a low-pass filter (k/(s + k)) coupled with a differentiator (s) (inset). A negative gain is used here to mimic the activity response of E. coli to positive changes in ligand [38]. As in the case of the E. coli model, the output of the low-pass filter plus differentiator (y(t)) is a filtered version of the derivative of the input signal. Smaller filter cutoff frequencies (smaller k), which correspond to longer averaging times, yield less noisy outputs. Red dashed lines indicate the time it takes the mean output of the filter to reach 95% of its steady state level. Although longer averaging times help reduce noise, they result in a slower response: the output takes longer to approach zero after the step.
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pcbi-0020154-g002: Effect of Adaptation Time on Filtering Capabilities(A) When subjected to an abrupt change in chemoattractant concentration (x(t)), the output (A(t)) of the model of the E. coli signaling pathway (Materials and Methods) responds with an initial transient burst which decays exponentially. The adaptation time depends on the component levels of the signaling pathway such as the methylation rate γR [13,14]. Faster methylation rates (γR) yield shorter adaptation times but result in noisier activity levels. Thus, the filtering capabilities of E. coli are determined by the time it takes to adapt to step inputs of ligand.(B) The adaptation response of E. coli is representative of a system consisting of a low-pass filter (k/(s + k)) coupled with a differentiator (s) (inset). A negative gain is used here to mimic the activity response of E. coli to positive changes in ligand [38]. As in the case of the E. coli model, the output of the low-pass filter plus differentiator (y(t)) is a filtered version of the derivative of the input signal. Smaller filter cutoff frequencies (smaller k), which correspond to longer averaging times, yield less noisy outputs. Red dashed lines indicate the time it takes the mean output of the filter to reach 95% of its steady state level. Although longer averaging times help reduce noise, they result in a slower response: the output takes longer to approach zero after the step.

Mentions: A major challenge of temporal sensing is accurately measuring changes in chemoattractant concentration (dL/dt) in the presence of noise. Fluctuations, such as those intrinsic to the ligand–receptor binding process, are amplified by this temporal differentiation. A traditional fix in engineering is to use a low-pass filter to remove the high-frequency noise from the lower-frequency signal; the E. coli signaling network implements such a mechanism (Figure 2). The key design criterion is to determine the appropriate cutoff frequency for the filter. In the time domain, this roughly corresponds to selecting the optimal time over which to measure, and average, chemoattractant. The length of time over which the signal is averaged is inversely proportional to the filter-cutoff frequency and determines the adaptation time of a differentiating system to step inputs. This averaging time may also be thought of as a memory length because the cell must “remember” previous values of the input to compute the average. If the averaging or adaptation time is too short, then the noise is not filtered out; alternatively, if the time is too long, then the bacteria cannot detect real changes in the gradient [15,16].


Optimal noise filtering in the chemotactic response of Escherichia coli.

Andrews BW, Yi TM, Iglesias PA - PLoS Comput. Biol. (2006)

Effect of Adaptation Time on Filtering Capabilities(A) When subjected to an abrupt change in chemoattractant concentration (x(t)), the output (A(t)) of the model of the E. coli signaling pathway (Materials and Methods) responds with an initial transient burst which decays exponentially. The adaptation time depends on the component levels of the signaling pathway such as the methylation rate γR [13,14]. Faster methylation rates (γR) yield shorter adaptation times but result in noisier activity levels. Thus, the filtering capabilities of E. coli are determined by the time it takes to adapt to step inputs of ligand.(B) The adaptation response of E. coli is representative of a system consisting of a low-pass filter (k/(s + k)) coupled with a differentiator (s) (inset). A negative gain is used here to mimic the activity response of E. coli to positive changes in ligand [38]. As in the case of the E. coli model, the output of the low-pass filter plus differentiator (y(t)) is a filtered version of the derivative of the input signal. Smaller filter cutoff frequencies (smaller k), which correspond to longer averaging times, yield less noisy outputs. Red dashed lines indicate the time it takes the mean output of the filter to reach 95% of its steady state level. Although longer averaging times help reduce noise, they result in a slower response: the output takes longer to approach zero after the step.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC1636674&req=5

pcbi-0020154-g002: Effect of Adaptation Time on Filtering Capabilities(A) When subjected to an abrupt change in chemoattractant concentration (x(t)), the output (A(t)) of the model of the E. coli signaling pathway (Materials and Methods) responds with an initial transient burst which decays exponentially. The adaptation time depends on the component levels of the signaling pathway such as the methylation rate γR [13,14]. Faster methylation rates (γR) yield shorter adaptation times but result in noisier activity levels. Thus, the filtering capabilities of E. coli are determined by the time it takes to adapt to step inputs of ligand.(B) The adaptation response of E. coli is representative of a system consisting of a low-pass filter (k/(s + k)) coupled with a differentiator (s) (inset). A negative gain is used here to mimic the activity response of E. coli to positive changes in ligand [38]. As in the case of the E. coli model, the output of the low-pass filter plus differentiator (y(t)) is a filtered version of the derivative of the input signal. Smaller filter cutoff frequencies (smaller k), which correspond to longer averaging times, yield less noisy outputs. Red dashed lines indicate the time it takes the mean output of the filter to reach 95% of its steady state level. Although longer averaging times help reduce noise, they result in a slower response: the output takes longer to approach zero after the step.
Mentions: A major challenge of temporal sensing is accurately measuring changes in chemoattractant concentration (dL/dt) in the presence of noise. Fluctuations, such as those intrinsic to the ligand–receptor binding process, are amplified by this temporal differentiation. A traditional fix in engineering is to use a low-pass filter to remove the high-frequency noise from the lower-frequency signal; the E. coli signaling network implements such a mechanism (Figure 2). The key design criterion is to determine the appropriate cutoff frequency for the filter. In the time domain, this roughly corresponds to selecting the optimal time over which to measure, and average, chemoattractant. The length of time over which the signal is averaged is inversely proportional to the filter-cutoff frequency and determines the adaptation time of a differentiating system to step inputs. This averaging time may also be thought of as a memory length because the cell must “remember” previous values of the input to compute the average. If the averaging or adaptation time is too short, then the noise is not filtered out; alternatively, if the time is too long, then the bacteria cannot detect real changes in the gradient [15,16].

Bottom Line: Through simulation, we first show that the cutoff frequency has a dramatic effect on the chemotactic efficiency of the cell.There was good agreement between the theory, simulations, and published experimental data.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.

View Article: PubMed Central - PubMed

Affiliation: Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, Maryland, United States of America.

ABSTRACT
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.

Show MeSH
Related in: MedlinePlus