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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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Validation of E. coli Signaling Network Model(A) Response of the E. coli signaling network model is plotted for a 0.1-mM step increase in chemoattractant concentration from 0.2 mM at 10 s. For comparison with experimental data, activity from the model is converted to the probability of CCW flagella rotation by means of a Hill function (Materials and Methods). A transient response after an initial increase in CCW probability adapts to the pre-stimulus level. Adaptation times are similar to experimentally obtained data [20,27] (red dashed line shows data from Figure 2 of [27]).(B) Response to a positive (+0.2 mM) chemoattractant impulse of width 0.5 s is also consistent with experimentally observed behavior [20,27] (red dots show data from Figure 1 of [27]).(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 [17].(D) For step inputs, normalized peak activity of the model (1 – ΔA/Ass, where ΔA is the change in activity from steady-state level Ass) exhibits sensitivity to the size of the step (red line with dots is a guide to the eye). This sensitivity response closely matches data from the “small lattice” model of [36] (blue line with triangles is a fitted sigmoid function). The blue dashed line indicates experimental data from [27] as plotted in Figure 4 of [36] (MeAsp input concentration of [27] is adjusted to that of aspartate that yields an equivalent receptor occupancy). As discussed in [36], the nonzero baseline of our model for large step inputs is due to a fraction of receptors in the cluster always having a nonzero probability of being active.(E,F) Frequency responses of the E. coli signaling network model and the model from [17], linearized about a ligand input of L0 = 1 μM, reveal low-pass characteristics consistent with observations in [27].
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pcbi-0020154-g009: Validation of E. coli Signaling Network Model(A) Response of the E. coli signaling network model is plotted for a 0.1-mM step increase in chemoattractant concentration from 0.2 mM at 10 s. For comparison with experimental data, activity from the model is converted to the probability of CCW flagella rotation by means of a Hill function (Materials and Methods). A transient response after an initial increase in CCW probability adapts to the pre-stimulus level. Adaptation times are similar to experimentally obtained data [20,27] (red dashed line shows data from Figure 2 of [27]).(B) Response to a positive (+0.2 mM) chemoattractant impulse of width 0.5 s is also consistent with experimentally observed behavior [20,27] (red dots show data from Figure 1 of [27]).(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 [17].(D) For step inputs, normalized peak activity of the model (1 – ΔA/Ass, where ΔA is the change in activity from steady-state level Ass) exhibits sensitivity to the size of the step (red line with dots is a guide to the eye). This sensitivity response closely matches data from the “small lattice” model of [36] (blue line with triangles is a fitted sigmoid function). The blue dashed line indicates experimental data from [27] as plotted in Figure 4 of [36] (MeAsp input concentration of [27] is adjusted to that of aspartate that yields an equivalent receptor occupancy). As discussed in [36], the nonzero baseline of our model for large step inputs is due to a fraction of receptors in the cluster always having a nonzero probability of being active.(E,F) Frequency responses of the E. coli signaling network model and the model from [17], linearized about a ligand input of L0 = 1 μM, reveal low-pass characteristics consistent with observations in [27].

Mentions: We compared a variety of different response characteristics of the model to those observed experimentally (Figure 9). Step increases in chemoattractant induce an initial transient decrease in receptor activity, corresponding to an increase in the probability of counterclockwise (CCW) flagella rotation, that decays exponentially to its pre-stimulus steady-state level (Figure 9A). This behavior is consistent with responses observed experimentally [13,20,27] and in previously published models [14,17,21,35–37]. The response to impulses of chemoattractant exhibit an initial spike in CCW probability followed by a decrease below the pre-stimulus level that then increases exponentially back to the pre-stimulus level (Figure 9B). Step and impulse responses agree closely with experimentally observed responses from [27]. Step-decreases in chemoattractant induce a response similar to step-increases with an initial decrease in CCW probability instead of an increase (Figure 9C). Both the adaptation time (the time required for the step response to return to its pre-stimulus level) and response magnitude increase with increasing step size and are asymmetric with respect to positive and negative steps (Figure 9C, [17]). The peak activity response to step inputs exhibits high sensitivity to the size of the step (Figure 9D), as has been observed experimentally [38,39], and is seen in other published models [18,19,35,36,40]. Sensitivity of our model closely matches that of [36].


Optimal noise filtering in the chemotactic response of Escherichia coli.

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

Validation of E. coli Signaling Network Model(A) Response of the E. coli signaling network model is plotted for a 0.1-mM step increase in chemoattractant concentration from 0.2 mM at 10 s. For comparison with experimental data, activity from the model is converted to the probability of CCW flagella rotation by means of a Hill function (Materials and Methods). A transient response after an initial increase in CCW probability adapts to the pre-stimulus level. Adaptation times are similar to experimentally obtained data [20,27] (red dashed line shows data from Figure 2 of [27]).(B) Response to a positive (+0.2 mM) chemoattractant impulse of width 0.5 s is also consistent with experimentally observed behavior [20,27] (red dots show data from Figure 1 of [27]).(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 [17].(D) For step inputs, normalized peak activity of the model (1 – ΔA/Ass, where ΔA is the change in activity from steady-state level Ass) exhibits sensitivity to the size of the step (red line with dots is a guide to the eye). This sensitivity response closely matches data from the “small lattice” model of [36] (blue line with triangles is a fitted sigmoid function). The blue dashed line indicates experimental data from [27] as plotted in Figure 4 of [36] (MeAsp input concentration of [27] is adjusted to that of aspartate that yields an equivalent receptor occupancy). As discussed in [36], the nonzero baseline of our model for large step inputs is due to a fraction of receptors in the cluster always having a nonzero probability of being active.(E,F) Frequency responses of the E. coli signaling network model and the model from [17], linearized about a ligand input of L0 = 1 μM, reveal low-pass characteristics consistent with observations in [27].
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Related In: Results  -  Collection

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pcbi-0020154-g009: Validation of E. coli Signaling Network Model(A) Response of the E. coli signaling network model is plotted for a 0.1-mM step increase in chemoattractant concentration from 0.2 mM at 10 s. For comparison with experimental data, activity from the model is converted to the probability of CCW flagella rotation by means of a Hill function (Materials and Methods). A transient response after an initial increase in CCW probability adapts to the pre-stimulus level. Adaptation times are similar to experimentally obtained data [20,27] (red dashed line shows data from Figure 2 of [27]).(B) Response to a positive (+0.2 mM) chemoattractant impulse of width 0.5 s is also consistent with experimentally observed behavior [20,27] (red dots show data from Figure 1 of [27]).(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 [17].(D) For step inputs, normalized peak activity of the model (1 – ΔA/Ass, where ΔA is the change in activity from steady-state level Ass) exhibits sensitivity to the size of the step (red line with dots is a guide to the eye). This sensitivity response closely matches data from the “small lattice” model of [36] (blue line with triangles is a fitted sigmoid function). The blue dashed line indicates experimental data from [27] as plotted in Figure 4 of [36] (MeAsp input concentration of [27] is adjusted to that of aspartate that yields an equivalent receptor occupancy). As discussed in [36], the nonzero baseline of our model for large step inputs is due to a fraction of receptors in the cluster always having a nonzero probability of being active.(E,F) Frequency responses of the E. coli signaling network model and the model from [17], linearized about a ligand input of L0 = 1 μM, reveal low-pass characteristics consistent with observations in [27].
Mentions: We compared a variety of different response characteristics of the model to those observed experimentally (Figure 9). Step increases in chemoattractant induce an initial transient decrease in receptor activity, corresponding to an increase in the probability of counterclockwise (CCW) flagella rotation, that decays exponentially to its pre-stimulus steady-state level (Figure 9A). This behavior is consistent with responses observed experimentally [13,20,27] and in previously published models [14,17,21,35–37]. The response to impulses of chemoattractant exhibit an initial spike in CCW probability followed by a decrease below the pre-stimulus level that then increases exponentially back to the pre-stimulus level (Figure 9B). Step and impulse responses agree closely with experimentally observed responses from [27]. Step-decreases in chemoattractant induce a response similar to step-increases with an initial decrease in CCW probability instead of an increase (Figure 9C). Both the adaptation time (the time required for the step response to return to its pre-stimulus level) and response magnitude increase with increasing step size and are asymmetric with respect to positive and negative steps (Figure 9C, [17]). The peak activity response to step inputs exhibits high sensitivity to the size of the step (Figure 9D), as has been observed experimentally [38,39], and is seen in other published models [18,19,35,36,40]. Sensitivity of our model closely matches that of [36].

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