Resolving the binding-kinase discrepancy in bacterial chemotaxis: A nonequilibrium allosteric model and the role of energy dissipation

TL;DR

This paper introduces a nonequilibrium allosteric model driven by energy dissipation to explain bacterial chemotaxis signaling, resolving previous inconsistencies and broadening understanding of sensor-kinase systems.

Contribution

The authors develop a nonequilibrium allosteric model that incorporates energy dissipation, successfully explaining experimental data and extending to other sensor-kinase systems.

Findings

Equilibrium models cannot explain the asymmetric shift in kinase response.

Energy dissipation is essential for sensitivity and response amplitude.

The model applies to other bacterial sensor-kinase systems like DosP.

Abstract

The Escherichia coli chemotaxis signaling pathway has served as a model system for studying the adaptive sensing of environmental signals by large protein complexes. The chemoreceptors control the kinase activity of CheA in response to the extracellular ligand concentration and adapt across a wide concentration range by undergoing methylation and demethylation. Methylation shifts the kinase response curve by orders of magnitude in ligand concentration while incurring a much smaller change in the ligand binding curve. Here, we show that this asymmetric shift in binding and kinase response is inconsistent with equilibrium allosteric models regardless of parameter choices. To resolve this inconsistency, we present a nonequilibrium allosteric model that explicitly includes the dissipative reaction cycles driven by ATP hydrolysis. The model successfully explains all existing measurements for both aspartate and serine receptors. Our results suggest that while ligand binding controls the equilibrium balance between the ON and OFF states of the kinase, receptor methylation modulates the kinetic properties (e.g., the phosphorylation rate) of the ON state. Furthermore, sufficient energy dissipation is necessary for maintaining and enhancing the sensitivity range and amplitude of the kinase response. We demonstrate that the nonequilibrium allosteric model is broadly applicable to other sensor-kinase systems by successfully fitting previously unexplained data from the DosP bacterial oxygen-sensing system. Overall, this work provides a new perspective on cooperative sensing by large protein complexes and opens up new research directions for understanding their microscopic mechanisms through simultaneous measurements and modeling of ligand binding and downstream responses.

Equations

$$H=(-\mu +E_{0,1}{s}_1)\sum _{i=1}^N{\sigma }_i+\sum _{i=1}^{M-1}(E_i{s}_i+E_{i,i+1}{s}_i{s}_{i+1})+E_M{s}_M.$$

$$\begin{array}{rl}⟨s_{j+1}⟩& =⟨s_{j+1}|s_j=0⟩P(s_j=0)+⟨s_{j+1}|s_j=1⟩P(s_j=1)\\ & =⟨s_{j+1}|s_j=0⟩(1-⟨s_j⟩)+⟨s_{j+1}|s_j=1⟩⟨s_j⟩\\ & =C_0+C_1⟨s_j⟩,\end{array}$$

$$\frac{\text{expectationvalue}{s_{j+1}}_{\max }-\text{expectationvalue}{s}_{j+1}}{\text{expectationvalue}{s_{j+1}}_{\max }-\text{expectationvalue}{s_{j+1}}_{\min }}=\frac{\text{expectationvalue}{s_{j+1}}_{\max }-\text{expectationvalue}{s}_{j+1}}{\text{expectationvalue}{s_{j+1}}_{\max }-\text{expectationvalue}{s_{j+1}}_{\min }}.$$

$$H=\sum _{i=1}^N(E_s{s}_i+E_0{s}_i{\sigma }_i-\mu {\sigma }_i)-\sum _{i\ne j}{J}_{ij}{s}_i{s}_j.$$

$$H_{\mathrm{eff}}=-\sum _{i\ne j}{J}_{ij}{s}_i{s}_j+h_{\mathrm{eff}}\sum _i{s}_i+N\log (1+e^{\mu }),h_{\mathrm{eff}}=E_s+\log \frac{1+e^{\mu }}{1+e^{\mu -E_0}}.$$

$$\text{expectationvalue}\sigma =\tilde{\sigma }\text{quantity}(\text{expectationvalue}s,t,[L{]}_{1/3},[L{]}_{2/3}),$$

$$\text{expectationvalue}a=\frac{1}{1+\mathcal{P}},\text{where }\mathcal{P}=\frac{{\alpha }_0+1}{{\alpha }_0{e}^{E_{p1}}+e^{E_{p0}}},{\alpha }_0=e^{-E_s}\text{quantity}(\frac{1+[L]/K_i}{1+e^{-E_0}[L]/K_i}{)}^{-N}.$$

$$\text{expectationvalue}\sigma =\frac{[L]}{K_i+[L]}(1-⟨s⟩)+\frac{[L]}{e^{E_0}{K}_i+[L]}⟨s⟩.$$

Full text

Supplemental Information: Resolving the binding-kinase discrepancy in bacterial chemotaxis: A nonequilibrium allosteric model and the role of energy dissipation

David Hathcock

These two authors contributed equally

IBM T. J. Watson Research Center, Yorktown Heights, NY 10598

  

Qiwei Yu

These two authors contributed equally

IBM T. J. Watson Research Center, Yorktown Heights, NY 10598

Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ 08544

  

Bernardo A. Mello

International Center of Physics, Physics Institute, University of Brasilia, Brasilia, Brazil

  

Divya N. Amin

Department of Biochemistry, University of Missouri, Columbia, MO 65211

  

Gerald L. Hazelbauer

Department of Biochemistry, University of Missouri, Columbia, MO 65211

  

Yuhai Tu

IBM T. J. Watson Research Center, Yorktown Heights, NY 10598

I Testing equilibrium allosteric models

I.1 MWC model: fitting ligand binding leads to inconsistent kinase activity prediction

In similar spirit to Fig. 1B of the main text, we use the MWC model to fit the binding curves and plot the resulting kinase activity curves. The results are shown in Fig. S1, which is another demonstration that the MWC model is inconsistent with the data.

I.2 Linear dependence between additional degrees of freedom

In the main text [Eq. (7)], we argued that coupling an additional equilibrium degree of freedom (e.g. kinase activity) to the MWC model produces a response that is linearly related to the MWC activity. Here we show this result is quite general and holds for any chain of binary variables coupled via the following Hamiltonian,

[TABLE]

As in the main text, $\sigma_{i}=0,1$ denotes the receptor occupancy, which depends on ligand concentration $[L]$ and dissociation constant $K_{i}$ through the chemical potential $\mu=\log([L]/K_{i})$. The variables $s_{i}=0,1$ are activities for various components of the system. For example, these might represent different conformational changes along the receptor body or kinase-controlling tip as well as conformational changes of the kinase itself, each coupled in a chain via equilibrium mechanisms. For each $s_{i}$, $E_{i}$ is the energy difference between the active and inactive states, and $E_{i,i+1}$ is the coupling energy between neighboring conformations.

We can relate the average states of neighboring conformational degrees of freedom using conditional probability:

[TABLE]

where $C_{0}$, $C_{1}$ are constants independent of $\mu$. The final step relies on the interactions between activities being equilibrium, so that the conditional expectation $\expectationvalue{s_{j+1}|s_{j}=0,1}$ is independent of $\mu$. For nonequilibrium models, including the model in the main text, this is not generally true: the conditional expectation can be a function of $\expectationvalue{s_{j}}$ and therefore maintain $\mu$-dependence. Given the linear relationship between $\expectationvalue{s_{j+1}}$ and $\expectationvalue{s_{j}}$ in equilibrium models, the response curves must be identical after normalization,

[TABLE]

Since linear proportionality applies to any pair of neighboring activities, it extends to the entire chain. Therefore, if a system is governed by equilibrium interactions between neighboring conformational states, any measurement of activity can be captured by an effective MWC model.

I.3 Ising models fail to explain both ligand binding and kinase response

Ising models are also frequently used to study the behavior of chemoreceptor clusters [2, 3]. In contrast to the MWC model, which assumes all-or-none activity within a cluster, Ising models allow each receptor to be independently active or inactive, based on its occupancy. The receptors interact cooperatively via equilibrium mechanisms (perhaps reflective of the lattice spatial structure present in real chemoreceptor complexes). A general Ising Hamiltonian for a chemoreceptor cluster is given by,

[TABLE]

The energy parameters have the same meanings as their MWC counterparts: $\mu=\log([L]/K_{i})$ is the chemical potential (which depends on ligand concentration $[L]$ and dissociation constant $K_{i}$) and $E_{s}$ is the energy difference between active and inactive states, which is increased by $E_{0}$ when the receptor is occupied. Each receptor $s_{i}$ reacts individually to its occupancy $\sigma_{i}$, but the receptors interact cooperatively via the coupling $J_{ij}>0$.

Integrating out the binding degrees of freedom leads to a standard Ising model with an effective field,

[TABLE]

We will consider the 1D (periodic) $J_{ij}=J(\delta_{i,j-1}+\delta_{i,j+1})/2$ and mean field $J_{ij}=J/(2N)$ Ising models. Given Eq. (S5), the partition function $Z=\sum_{\{s_{i}\}}\exp(-H_{\mathrm{eff}})$ is well known for each of these cases, from which the binding and activity can be readily computed $\expectationvalue{s}=-\differential{\log Z}/\differential{E_{s}}$ and $\expectationvalue{\sigma}=\differential{\log Z}/\differential{\mu}$. We omit the expressions here due to their complexity.

Given expressions for $\expectationvalue{s}$ and $\expectationvalue{\sigma}$, we can derive a parametric relation between the two $\expectationvalue{\sigma}=\tilde{\sigma}(\expectationvalue{s},s_{\min},s_{\max}|J,N)$, similar to that obtained in the main text for the MWC model. For the Ising models, $\expectationvalue{s}$ is generally not invertible, so we determine the inferred occupancy $\tilde{\sigma}$ numerically for a given receptor-receptor coupling $J$ and system size $N$. Fig. S2 shows the parametric test for 1D and mean field Ising models applied to the Amin and Hazelbauer measurements [1]. For any choice of system size and receptor coupling, the data lie well off the diagonal, indicating these equilibrium models cannot simultaneously explain the binding and activity measurements.

II The nonequilibrium allosteric model captures binding and activity of other signaling proteins

II.1 E. coli oxygen-sensing protein DosP

E. coli DosP is a c-di-GMP phosphodiesterase, whose enzymatic activity can be significantly enhanced by the binding of oxygen to DosP. Previous experiments measured the ligand binding and phosphodiesterase activity at different oxygen concentrations and demonstrated that the results are incompatible with an equilibrium model [4]. The same study proposed that the inconsistency with the equilibrium model could be attributed to “memory effects” [4], which suggests that the system operates out of equilibrium. Here, we show that these measurements can be consistently explained under the framework of the nonequilibrium allosteric model presented in this work.

First, we carry out a parametric test to examine whether the measurements are consistent with an equilibrium MWC model with $N=4$ (DosP are known to form tetramers [5]). Due to the type of data available, the parametric test used here is slightly different from the one for the chemoreceptor measurements. Specifically, we infer the mean ligand occupancy $\tilde{\sigma}$ from the mean activity $\expectationvalue{s}$ using the MWC model:

[TABLE]

where $[L]_{1/3}$ and $[L]_{2/3}$ are the ligand concentration required to achieve $1/3$ or $2/3$ of normalized activity and $t=\frac{s_{\mathrm{max}}}{s_{\mathrm{min}}}$ is the ratio of maximum and minimum activities, which was measured to be 17 [4]. The inferred occupancy $\tilde{\sigma}$ can be obtained by solving Eqs. (2) and (3) of the main text. As shown by Fig. 6A of the main text, the parametric test found all data points deviating from the diagonal $\expectationvalue{\sigma}_{\mathrm{exp}}=\tilde{\sigma}$, revealing that the measurements are inconsistent with the equilibrium MWC model.

The parametric test implies that the binding and activity data cannot be simultaneously explained under the framework of the MWC model, no matter how the fitting is done. To demonstrate this, we show representative fits of the MWC model to the data in Fig. S3. When fit to ligand binding (Fig. S3A, blue line), the model incorrectly predicts the half-maximal concentration for activity (orange line). Conversely, the model does not capture the half-maximal concentration for binding when fit to the activity curve (Fig. S3B). When fit to both curves, the model predicts the correct half-maximal concentrations but completely misses the sharpness of the curves, which describes cooperativity (Fig. S3C). There exist other ways to fit the data, for example, by assigning different weights to activity and binding, respectively. Nonetheless, none of them will be consistent with the measurements as demonstrated by the parametric test in Fig. 6A of the main text.

Next, we show that the nonequilibrium allosteric model is capable of simultaneously capturing binding and activity (Fig. 6B in the main text). Here, we use a slightly generalized version of the model shown in the main text by allowing hydrolysis for both receptor conformation states $s=0$ and $s=1$, whose rates are given by $k_{0}=k_{z}e^{E_{p0}}$ and $k_{1}=k_{z}e^{E_{p1}}$, respectively. The model is shown in Fig. S4. In the large dissipation limit, the mean activity is

[TABLE]

The mean ligand occupancy is given by the solution to the MWC model,

[TABLE]

where $\expectationvalue{s}=\frac{\alpha_{0}}{1+\alpha_{0}}$ is the mean receptor activity. The model successfully fits the data as shown in Fig. 6B in the main text.