Feedback control architecture and the bacterial chemotaxis network

Abstract

Bacteria move towards favourable and away from toxic environments by changing their swimming pattern. This response is regulated by the chemotaxis signalling pathway, which has an important feature: it uses feedback to 'reset' (adapt) the bacterial sensing ability, which allows the bacteria to sense a range of background environmental changes. The role of this feedback has been studied extensively in the simple chemotaxis pathway of Escherichia coli. However it has been recently found that the majority of bacteria have multiple chemotaxis homologues of the E. coli proteins, resulting in more complex pathways. In this paper we investigate the configuration and role of feedback in Rhodobacter sphaeroides, a bacterium containing multiple homologues of the chemotaxis proteins found in E. coli. Multiple proteins could produce different possible feedback configurations, each having different chemotactic performance qualities and levels of robustness to variations and uncertainties in biological parameters and to intracellular noise. We develop four models corresponding to different feedback configurations. Using a series of carefully designed experiments we discriminate between these models and invalidate three of them. When these models are examined in terms of robustness to noise and parametric uncertainties, we find that the non-invalidated model is superior to the others. Moreover, it has a 'cascade control' feedback architecture which is used extensively in engineering to improve system performance, including robustness. Given that the majority of bacteria are known to have multiple chemotaxis pathways, in this paper we show that some feedback architectures allow them to have better performance than others. In particular, cascade control may be an important feature in achieving robust functionality in more complex signalling pathways and in improving their performance.

Figure 1. A cascade control system.

The subsystem is slow relative to . Cascade control involves placing a negative feedback loop (dashed line) around the fast secondary module. This scheme helps reduce the sensitivity of the system's output to uncertainties in the subsystems and .

Figure 2. Chemotaxis in R. sphaeroides.

(A) The chemotaxis pathway in R. sphaeroides as currently understood, including the forward chemotaxis pathway previously proposed . MCP: transmembrane methyl accepting chemotaxis protein, Tlp: cytoplasmic methyl accepting chemotaxis protein, A: CheA histidine protein kinase, W: CheW a linker protein between receptors and CheA, Y: the response regulator CheY, B: the response regulator CheB, R: the methyltransferase CheR. P indicates a phosphoryl group. The number in subscript denotes one of the multiple homologues in R. sphaeroides. The flagella motor is shown at the right of the figure. (B) The possible de-methylation feedback structures for the phosphorylated proteins CheB₁-P and CheB₂-P in R. sphaeroides. Each possible connection is denoted by a (red) thick solid, dashed or dotted line. Possible models involve combinations of these four lines. Interactions from the phosphotransfer network are shown in (black) thin dashed arrows, receptor activation/de-activation is denoted by (black) thin solid lines.

Figure 3. The speed of response of each cluster to input signals.

The response of the normalized CheY₆-P concentration to a step decrease, at time 10 seconds, in the number of active receptors at the polar cluster (from  = 1 µM to  = 0 µM, dashed) and at the cytoplasmic cluster (from  = 1 µM to  = 0 µM, solid). Such a decrease in active receptors can be due to a step increase in sensed ligand. A step decrease in active polar cluster receptors results in a slower fall in the normalized CheY₆-P concentration (90%-10% fall time: 50.57 sec) than would an identical change in the number of active cytoplasmic cluster receptors (90%-10% fall time: 21.98 sec).

Figure 4. Model invalidation.

Top left: Simulations of the wild type Models I–IV and with CheR₃ deleted in response to 100 µM of ligand added at 100 seconds and removed at 220 seconds. Top right: Average responses of wild type cells and CheR₃ deletion cells in a tethered cell assay with 100 µM of propionate added at 100 seconds and removed at 220 seconds. Bottom left: Simulations of the wild type Models I–IV (dashed line) and with CheB₁ deleted in response to 100 µM of ligand added at 100 seconds and removed at 220 seconds. Bottom right: Average responses of wild type cells and CheB₁ deletion cells in a tethered cell assay with 100 µM of propionate added at 100 seconds and removed at 220 seconds. Cells rotate counter clockwise hence negative Hz values are observed. Ligand addition is marked by grey shading.

Figure 5. Deletion of CheB₂.

Average responses of CheB₂ deletion cells in a tethered cell assay with 100 µM of propionate added at 200 seconds and removed at 512 seconds. Solid lines: simulations of the Models I–IV with CheB₂ deleted in response to 100 µM of ligand added at 200 seconds and removed at 512 seconds. Cells rotate counter clockwise hence negative Hz values are observed. Ligand addition is marked by grey shading.

Figure 6. Comparison of chemotactic performance.

The four chemotaxis models are simulated in a two-dimensional environment, wherein the chemoattractant concentration L has a ramp profile that varies along the x-direction only, such that L = 100x for x>0 and L = 0 otherwise (left). The simulation output (right) shows the relative average distance travelled up the attractant gradient by ten cells for each of the chemotaxis models.

Figure 7. Response to external ligand variations.

Standard deviations of the flagellar rotation frequencies for each of the four chemotaxis models in response to a noisy ligand input sensed at the polar cluster given by L = max(0,1+η) (where η is a white noise signal with a zero-mean, unity variance Gaussian distribution).

Figure 8. Input-output gains of the two sensing clusters.

Frequency response magnitude plots showing the response of the different models to sinusoidally-varying ligand concentrations modelling noisy ligand input signals. Top: Constant ligand to cytoplasmic cluster and variable ligand to polar cluster (, where ). Bottom: Constant ligand to polar cluster, sinusoidal to cytoplasmic cluster (, where ).

Figure 9. Parametric sensitivity analysis.

Relative sensitivities of the rotation frequency outputs of the different chemotaxis models to changes in the chemotaxis protein copy numbers.

Figure 10. Comparison with engineering systems.

Block diagram representation of a linear system structurally similar to the R. sphaeroides chemotaxis pathway. In this system, gain corresponds to in the chemotaxis model, to , to and to . Levels of CheB₁-P and CheB₂-P exhibit exact adaptation to step changes in ligand concentration . We assume , mirroring the faster dynamics of the cytoplasmic cluster relative to the polar cluster.

Figure 11. Variation of linear system sensitivity under different feedback strengths as a function of frequency.

(A) Bode magnitude plots of the sensitivity function of system (1) with and different values of gain , which corresponds to the feedback strength of CheB₂-P de-methylating active polar cluster receptors. With these gains the system is structurally similar to Model III. (B) Sensitivity function of the block corresponding to the cytoplasmic cluster in the linear model (1), for different values of feedback gain , which corresponds to the feedback strength of CheB₂-P de-methylating active cytoplasmic cluster receptors. The frequency domain sensitivity function is (see Text S1).

Figure 12. Variation of linear system gain magnitude under different feedback strengths as a function of frequency.

Bode magnitude plots of transfer functions from ligand inputs to Y in the linear system (1) corresponding to Models I () and III (). (A) Bode magnitude plots from L to Y. (B) Bode magnitude plots from to Y.

Figure 13. Sensitivity to copy number with varying external feedback.

Sensitivity of the chemotaxis steady state to random changes in copy numbers of chemotaxis proteins under different CheB₂ feedback strengths to the polar cluster. Sensitivity is measured as the ratio of the standard deviation of the steady state to the nominal steady state. Solid line: Sensitivity of the chemotaxis steady state to changes in the copy number of CheA₂ under different strengths of CheB₂ feedback to the polar cluster. Dashed line: Sensitivity of the chemotaxis steady state to changes in the copy number of CheB₁ under different strengths of CheB₂ feedback to the polar cluster.

Figure 14. Sensitivity to copy number with varying internal feedback.

Sensitivity of the chemotaxis steady state to random changes in copy numbers of chemotaxis proteins under different CheB₂ feedback strengths to the cytoplasmic cluster. Sensitivity is measured as the ratio of the standard deviation of the steady state to the nominal steady state. Dashed line: Sensitivity of the chemotaxis steady state to changes in the copy number of CheA₃A₄ under different strengths of CheB₂ feedback to the cytoplasmic cluster. Solid line: Sensitivity of the chemotaxis steady state to changes in the copy number of CheY₆ under different strengths of CheB₂ feedback to the cytoplasmic cluster.