A “Trimer of Dimers”—Based Model for the Chemotactic Signal Transduction Network in Bacterial Chemotaxis

Abstract

The network that controls chemotaxis in Escherichia coli is one of the most completely characterized signal transduction systems to date. Receptor clustering accounts for characteristics such as high sensitivity, precise adaptation over a wide dynamic range of ligand concentrations, and robustness to variations in the amounts of intracellular proteins. To gain insights into the structure-function relationship of receptor clusters and understand the mechanism behind the high-performance signaling, we develop and analyze a model for a single trimer of dimers. This new model extends an earlier model (Spiro et al. in Proc. Natl. Acad. Sci. 94:7263–7268, 1997) to incorporate the recent experimental findings that the core structure of receptor clusters is the trimer of receptor dimers. We show that the model can reproduce most of the experimentally-observed behaviors, including excitation, adaptation, high sensitivity, and robustness to parameter variations. In addition, the model makes a number of new predictions as to how the adaptation time varies with the expression level of various proteins involved in signal transduction. Our results provide a more mechanistically-based description of the structure-function relationship for the signaling system, and show the key role of the interaction among dimer members of the trimer in the chemotactic response of cells.

Appendix: Sensitivity Analysis

We perform sensitivity analysis for four steps of the signaling pathway—ligand binding, kinase activity regulation, phosphorylation and phosphoryl transfer in the cheRcheB mutant system where expression of CheR and CheB is suppressed and the methylation states of receptors are engineered and do not vary. The mutant cells can respond to but not adapt to chemoattractants. The involved chemical reactions in the system only lie in one slice (whose methylation level is the fixed one) of the network depicted in Fig. 1 excluding CheB-involved phosphoryl transfer. In the analysis, we fix the state at QEQE, and thus the reaction network lies in the m=6 slice in Fig. 1 and the corresponding m=6 parameter sets in Tables 1 and 3 are used in computation. For simplicity, the subindex m of the signaling complex variables is omitted in the section.

In the network, the ligand binding and kinase activity regulation transitions are much faster than autophosphorylation and phosphoryl transfer. Therefore, we apply the QSSA and dissect the network into two relatively independent parts. In the first one, we consider redistribution of signaling complexes in the ligand binding states induced by a chemoattractant stimulus. We use T _n to denote the amount of the signaling complexes with n ligands bound, regardless of the activity state ($T_{n}=T^{i}_{n}+T^{a}_{n}+T^{p}_{n}$). The methylation state is fixed at QEQE and the subindex m is dropped. T _t denotes the conserved total concentration of signaling complexes. The equations that govern the evolution of the amounts in four binding states of signaling complexes are as follows:

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The steady-state solution of ligand occupancy is

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We assume that when redistribution of signaling complexes in the activity states takes place, the ligand binding transitions have reached equilibrium. Then simply, $T^{i}_{n}=(1-p_{n})T_{n}$, and $T^{a}_{n}+T^{p}_{n}=p_{n}T_{n}$, where p _n is the probability of the signaling complex with n ligands bound being active. So, the steady-state solution of activity is

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In the second part, we consider two slow transitions, redistribution of the active signaling complexes in the unphosphorylated and phosphorylated states, and phosphoryl transfer to CheY. We use T ⁱ, T ^a, and T ^p to denote the amounts of the inactive, active-unphosphorylated and active-phosphorylated complexes, respectively, and then $T^{a}=\sum_{n=0}^{3}T^{a}_{n}$, $T^{p}=\sum_{n=0}^{3}T^{p}_{n}$, and $T^{a}_{n}+T^{p}_{n}=p_{n}T_{n}$ hold. The governing equations on T ^a, T ^p. and Y _p are as follows:

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The steady-state solutions of T ^p and Y _p are

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Finally, we apply the definition of dimensionless sensitivity and obtain S(O|L), S(A|O), S(T ^p|A), and S(Y _p |T ^p) as Eqs. (15)–(18) in the text, respectively.

For comparison, we perform a similar analysis in the case of a signaling complex containing a receptor dimer instead of a trimer of receptor dimers. The occupancy and activity are

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The sensitivities of ligand binding and activity regulation are

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The sensitivities of the remaining steps are the same as Eqs. (17) and (18) in the text. The variation in the composition of a signaling complex does not change the formula of the downstream sensitivities, but we need adjust the values of the rate constants in the CheA related reactions due to the change in the stoichiometry of receptors and CheA, and quantitatively it would rescale the downstream sensitivities.

Using a similar technique, we perform a parametric sensitivity analysis for the upstream signaling pathway. Specially, we have interests in the sensitivities of receptor occupancy to ligand dissociation constant $S(O|K_{d_{i}})$ and to cooperativity in ligand affinity S(O|t _i ), where $t_{i}=K_{d_{i}}/K_{d_{i+1}}$ (i=1,2), and in the sensitivities of receptor activity to ligand dissociation constant $S(A|K_{d_{i}})$, to cooperativity in ligand affinity S(A|t _i ), to probability of being active S(A|p _i ), and to cooperativity in activity inhibition S(A|s _i ), where s _i =p _i−1/p _i (i=1,2,3). The formula are Eqs. (66) to (82).

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