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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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Optimal Filtering Cutoff Frequency Is Determined by the Signal and Noise(A) Insight into the relevant features of the optimal estimator can be obtained by considering a simplified model that assumes the cell measures L + v, where L is the true ligand concentration. The system dL/dt = w acts as an integrator and has signal PSD dependent on the rotational diffusion coefficient (blue solid line). The observed signal L + v includes the effect of the binding noise v which is assumed to be a white noise process (blue dashed line). For this model, the optimal filter for the estimation of L, given L + v, is a first-order, low-pass filter with a cutoff frequency (ωcf) related to the covariances of w and v:(Protocol S1). Graphically, this is determined by the intersection of the signal and noise PSDs (point A). Increasing the noise variance (red dashed line) decreases the optimal cutoff frequency because the filter must become more restrictive to eliminate the additional noise (point B). If the signal PSD is then increased (red solid line), for example, by increasing the effect of rotational diffusion, the cutoff frequency increases (point C). Parameters used for point A: u = 20 μm/s, Dr = 0.16 rad2/s, τ = 1 s, RT = 2.5 μM, k−/k+ = 100 μM, g = 0.03 μM/μm, and binding is assumed at steady state with a ligand value of L0 = 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.
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pcbi-0020154-g007: Optimal Filtering Cutoff Frequency Is Determined by the Signal and Noise(A) Insight into the relevant features of the optimal estimator can be obtained by considering a simplified model that assumes the cell measures L + v, where L is the true ligand concentration. The system dL/dt = w acts as an integrator and has signal PSD dependent on the rotational diffusion coefficient (blue solid line). The observed signal L + v includes the effect of the binding noise v which is assumed to be a white noise process (blue dashed line). For this model, the optimal filter for the estimation of L, given L + v, is a first-order, low-pass filter with a cutoff frequency (ωcf) related to the covariances of w and v:(Protocol S1). Graphically, this is determined by the intersection of the signal and noise PSDs (point A). Increasing the noise variance (red dashed line) decreases the optimal cutoff frequency because the filter must become more restrictive to eliminate the additional noise (point B). If the signal PSD is then increased (red solid line), for example, by increasing the effect of rotational diffusion, the cutoff frequency increases (point C). Parameters used for point A: u = 20 μm/s, Dr = 0.16 rad2/s, τ = 1 s, RT = 2.5 μM, k−/k+ = 100 μM, g = 0.03 μM/μm, and binding is assumed at steady state with a ligand value of L0 = 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.

Mentions: 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 Figure 7A, which highlights the contributions of the noise and signal power spectral densities (PSDs) to this tradeoff. As binding noise increases, the noise PSD shifts upward, and the optimal cutoff frequency (ωcf) decreases, thus yielding a longer time over which to average ligand measurements so that the additional noise may be attenuated. This trend was seen in the chemotaxis simulations (Figure 4D). As fluctuations of the signal increase (e.g., by increasing the effect of rotational diffusion), the signal PSD shifts upward, and ωcf increases to yield shorter averaging times that are necessary for observing the more rapidly changing signal; see Figure 4E.


Optimal noise filtering in the chemotactic response of Escherichia coli.

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

Optimal Filtering Cutoff Frequency Is Determined by the Signal and Noise(A) Insight into the relevant features of the optimal estimator can be obtained by considering a simplified model that assumes the cell measures L + v, where L is the true ligand concentration. The system dL/dt = w acts as an integrator and has signal PSD dependent on the rotational diffusion coefficient (blue solid line). The observed signal L + v includes the effect of the binding noise v which is assumed to be a white noise process (blue dashed line). For this model, the optimal filter for the estimation of L, given L + v, is a first-order, low-pass filter with a cutoff frequency (ωcf) related to the covariances of w and v:(Protocol S1). Graphically, this is determined by the intersection of the signal and noise PSDs (point A). Increasing the noise variance (red dashed line) decreases the optimal cutoff frequency because the filter must become more restrictive to eliminate the additional noise (point B). If the signal PSD is then increased (red solid line), for example, by increasing the effect of rotational diffusion, the cutoff frequency increases (point C). Parameters used for point A: u = 20 μm/s, Dr = 0.16 rad2/s, τ = 1 s, RT = 2.5 μM, k−/k+ = 100 μM, g = 0.03 μM/μm, and binding is assumed at steady state with a ligand value of L0 = 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.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC1636674&req=5

pcbi-0020154-g007: Optimal Filtering Cutoff Frequency Is Determined by the Signal and Noise(A) Insight into the relevant features of the optimal estimator can be obtained by considering a simplified model that assumes the cell measures L + v, where L is the true ligand concentration. The system dL/dt = w acts as an integrator and has signal PSD dependent on the rotational diffusion coefficient (blue solid line). The observed signal L + v includes the effect of the binding noise v which is assumed to be a white noise process (blue dashed line). For this model, the optimal filter for the estimation of L, given L + v, is a first-order, low-pass filter with a cutoff frequency (ωcf) related to the covariances of w and v:(Protocol S1). Graphically, this is determined by the intersection of the signal and noise PSDs (point A). Increasing the noise variance (red dashed line) decreases the optimal cutoff frequency because the filter must become more restrictive to eliminate the additional noise (point B). If the signal PSD is then increased (red solid line), for example, by increasing the effect of rotational diffusion, the cutoff frequency increases (point C). Parameters used for point A: u = 20 μm/s, Dr = 0.16 rad2/s, τ = 1 s, RT = 2.5 μM, k−/k+ = 100 μM, g = 0.03 μM/μm, and binding is assumed at steady state with a ligand value of L0 = 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.
Mentions: 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 Figure 7A, which highlights the contributions of the noise and signal power spectral densities (PSDs) to this tradeoff. As binding noise increases, the noise PSD shifts upward, and the optimal cutoff frequency (ωcf) decreases, thus yielding a longer time over which to average ligand measurements so that the additional noise may be attenuated. This trend was seen in the chemotaxis simulations (Figure 4D). As fluctuations of the signal increase (e.g., by increasing the effect of rotational diffusion), the signal PSD shifts upward, and ωcf increases to yield shorter averaging times that are necessary for observing the more rapidly changing signal; see Figure 4E.

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