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. 2008 Jan;4(1):e1.
doi: 10.1371/journal.pcbi.0040001. Epub 2007 Nov 20.

Chemotaxis in Escherichia coli: a molecular model for robust precise adaptation

Clinton H Hansen  1 , Robert G Endres, Ned S Wingreen
Affiliations

Chemotaxis in Escherichia coli: a molecular model for robust precise adaptation

Clinton H Hansen et al. PLoS Comput Biol. 2008 Jan.

Abstract

The chemotaxis system in the bacterium Escherichia coli is remarkably sensitive to small relative changes in the concentrations of multiple chemical signals over a broad range of ambient concentrations. Interactions among receptors are crucial to this sensitivity as is precise adaptation, the return of chemoreceptor activity to prestimulus levels in a constant chemoeffector environment. Precise adaptation relies on methylation and demethylation of chemoreceptors by the enzymes CheR and CheB, respectively. Experiments indicate that when transiently bound to one receptor, these enzymes act on small assistance neighborhoods (AN) of five to seven receptor homodimers. In this paper, we model a strongly coupled complex of receptors including dynamic CheR and CheB acting on ANs. The model yields sensitive response and precise adaptation over several orders of magnitude of attractant concentrations and accounts for different responses to aspartate and serine. Within the model, we explore how the precision of adaptation is limited by small AN size as well as by CheR and CheB kinetics (including dwell times, saturation, and kinetic differences among modification sites) and how these kinetics contribute to noise in complex activity. The robustness of our dynamic model for precise adaptation is demonstrated by randomly varying biochemical parameters.

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Conflict of interest statement

Competing interests. The authors have declared that no competing interests exist.

Figures

Figure 1
Figure 1. Two-State Receptor Complex and Precise-Adaptation Model
(A) Top view of the hexagonal arrangement of the 19-receptor homodimer used in our simulations. Each receptor can bind to either CheR or CheB. Each bound CheR or CheB can then act on an “assistance neighborhood” of adjacent receptors (dashed line) through methylation (CheR) or demethylation (CheB). (B) Side view of receptor complex in the on state (top) and off state (bottom) with, respectively, active and inactive receptor-bound kinases CheA. Active CheA kinases autophosphorylate to CheA-P. In our adaptation model, CheB only demethylates receptors when the complex is in the on state (top), and CheR only methylates receptors when the complex is in the off state (bottom).
Figure 2
Figure 2. Averaged Response Curves for Step Increases in Attractant Concentration (0–100 mM)
A mixed receptor complex of six Tar and 13 Tsr receptors (solid curve) is compared to complexes of 19 Tar receptors with formula image = 0.06 mM (dot-dashed curve) and formula image = 0.02 mM (dashed curve). Other parameters are given in the Model section. Simulations were averaged over 500 runs. (A) Averaged response of complex activity A (Equation 3). Upper curves are each displaced vertically by 0.4. Inset: response curve for one simulation of a single mixed complex. The average activity is superimposed in gray. (B) Averaged methylation m of receptor homodimers. Insets: distribution of methylation levels at 0 mM, 1 mM, and 100 mM.
Figure 3
Figure 3. Adapted Activity as a Function of CheR Binding Rate
Results are shown for ANs of size one, ANs of all adjacent receptors, and the mean field theory. The other enzyme binding rates are formula image and formula image . Inset: 〈CheR〉 (proportion of receptors bound to CheR) as a function of CheR binding rate.
Figure 4
Figure 4. Precision of Adaptation
The adapted activity is shown as a function of MeAsp concentration for variants of the model. (A) Various AN sizes (and mean field theory). For “Half AN,” the assistance neighbors of each receptor were chosen at random (see Model). (B) Different saturation factors M sat, for both CheR and CheB. The rate of methylation/demethylation varies as N/(N + M sat), where N is the number of available modification sites. (C) Different binding/unbinding rates for2 CheR and CheB. The ratio formula image was kept constant at 10. Inset: distribution of receptor methylation levels at 1 mM MeAsp for formula image , and formula image 10−4 s−1.
Figure 5
Figure 5. Distribution of Adapted Complex Activities (Reflecting Distribution of Complex Methylation Levels) at Different MeAsp Concentrations
The MeAsp concentrations are as follows: (A) 0 mM, (B) 1 mM, and (C) 100 mM. Distributions are shown for AN = 1 and full AN at formula image , and for full AN at formula image . The ratio formula image was kept constant at 10.
Figure 6
Figure 6. Variance in Complex Methylation and Activity Levels
Curves are shown for CheR/CheB binding/unbinding rates formula image , and 0.01 s−1, as well as for the theoretical limit from the linear noise approximation (see Model). (A,C) Variance in complex methylation level as a function of the free-energy step δɛ per methyl group (A) and complex size (C) (see Methods). (B, D) Variance in complex activity level as a function of δɛ (C) and complex size (D).
Figure 7
Figure 7. Precision of Adaptation for Receptors with Site-Dependent and Site-Independent Methylation/Demethylation Rates
Shown are site-dependent methylation/demethylation rates with matching ratios (k R = 0.0125,0.025,0.05,0.1 s−1; k B = 0.025,0.05,0.1,0.2 s−1), with inverted ratios (k R = 0.0125,0.25,0.05,0.1 s−1; k B = 0.2,0.1,0.05, 0.025 s−1), for receptors with site-independent methylation/demethylation rates (k R = 0.1 s−1; k B = 0.2 s−1), and for receptors with site-dependent CheR dwell times in the ratio 1:2:4:8 and inverted site-dependent CheB dwell times in the ratio 8:4:2:1. Inset: average site methylation at 1 mM MeAsp for receptors with site-dependent methylation/demethylation rates for both the matching-ratio case (filled bars) and the inverted-ratio case (open bars), and for receptors with inverted site-dependent CheR/CheB dwell times (gray bars).
Figure 8
Figure 8. Adapted Complex Activity versus Concentration of MeAsp (Filled Circles) or Serine (Open Circles)
The saturation factor is M sat = 1, and other parameters are given in the Model section. Setting M sat = 0 would sharpen and shift the drop in activity to higher serine concentration [18]. Inset: experimental measurement of the fractional change in run length versus concentration of aspartate (open circles) or serine (closed circles) [9].
Figure 9
Figure 9. Robustness of Assistance-Neighborhood Model for Adaptation
Ratio of adapted activity at 1 mM MeAsp to adapted activity at 0 mM MeAsp plotted as a function of total parameter variation formula image (see Methods). The scatter plot shows results for 3,000 different parameter sets.
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