A nonequilibrium allosteric model for receptor-kinase complexes: The role of energy dissipation in chemotaxis signaling.

The Escherichia coli chemotaxis signaling pathway has served as a model system for 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 the disproportionate shift in binding and kinase response is inconsistent with equilibrium allosteric models. To resolve this inconsistency, we present a nonequilibrium allosteric model that explicitly includes the dissipative reaction cycles driven by adenosine triphosphate (ATP) hydrolysis. The model successfully explains all existing joint measurements of ligand binding, receptor conformation, and kinase activity for both aspartate and serine receptors. Our results suggest that the receptor complex acts as an enzyme: Receptor methylation modulates the ON-state kinetics of the kinase (e.g., phosphorylation rate), while ligand binding controls the equilibrium balance between kinase ON/OFF states. Furthermore, sufficient energy dissipation is responsible 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 nonequilibrium physics perspective on cooperative sensing by large protein complexes and opens up research directions for understanding their microscopic mechanisms through simultaneous measurements and modeling of ligand binding and downstream responses.

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

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.

Sequential modification of bacterial chemoreceptors is key for achieving both accurate adaptation and high gain.

Many regulatory and signaling proteins have multiple modification sites. In bacterial chemotaxis, each chemoreceptor has multiple methylation sites that are responsible for adaptation. However, whether the ordering of the multisite methylation process affects adaptation remains unclear. Furthermore, the benefit of having multiple modification sites is also unclear. Here, we show that sequentially ordered methylation/demethylation is critical for perfect adaptation; adaptation accuracy decreases as randomness in the multisite methylation process increases. A tradeoff between adaptation accuracy and response gain is discovered. We find that this accuracy-gain tradeoff is lifted significantly by having more methylation sites, but only when the multisite modification process is sequential. Our study suggests that having multiple modification sites and a sequential modification process constitute a general strategy to achieve both accurate adaptation and high response gain simultaneously. Our theory agrees with existing data and predictions are made to help identify the molecular mechanism underlying ordered covalent modifications.

A dual regulation mechanism of histidine kinase CheA identified by combining network-dynamics modeling and system-level input-output data.

It is challenging to decipher molecular mechanisms in biological systems from system-level input-output data, especially for complex processes that involve interactions among multiple components. We addressed this general problem for the bacterial histidine kinase CheA, the activity of which is regulated in chemotaxis signaling complexes by bacterial chemoreceptors. We developed a general network model to describe the dynamics of the system, treating the receptor complex with coupling protein CheW and the P3P4P5 domains of kinase CheA as a regulated enzyme with two substrates, ATP and P1, the phosphoryl-accepting domain of CheA. Our simple network model allowed us to search hypothesis space systematically. For different and progressively more complex regulation schemes, we fit our models to a large set of input-output data with the aim of identifying the simplest possible regulation mechanisms consistent with the data. Our modeling and analysis revealed novel dual regulation mechanisms in which receptor activity regulated ATP binding plus one other process, either P1 binding or phosphoryl transfer between P1 and ATP. Strikingly, in our models receptor control affected the kinetic rate constants of substrate association and dissociation equally and thus did not alter the respective equilibrium constants. We suggest experiments that could distinguish between the two dual-regulation mechanisms. This systems-biology approach of combining modeling and a large input-output dataset should be applicable for studying other complex biological processes.

Pendular behavior of public transport networks.

In this paper, we propose a methodology that bears close resemblance to the Fourier analysis of the first harmonic to study networks subjected to pendular behavior. In this context, pendular behavior is characterized by the phenomenon of people's dislocation from their homes to work in the morning and people's dislocation in the opposite direction in the afternoon. Pendular behavior is a relevant phenomenon that takes place in public transport networks because it may reduce the overall efficiency of the system as a result of the asymmetric utilization of the system in different directions. We apply this methodology to the bus transport system of Brasília, which is a city that has commercial and residential activities in distinct boroughs. We show that this methodology can be used to characterize the pendular behavior of this system, identifying the most critical nodes and times of the day when this system is in more severe demanded.

Analysis of etching at a solid-solid interface.

We present a method to derive an analytical expression for the roughness of an eroded surface whose dynamics are ruled by cellular automaton. Starting from the automaton, we obtain the time evolution of the height average and height variance (roughness). We apply this method to the etching model in 1+1 dimensions, and then we obtain the roughness exponent. Using this in conjunction with the Galilean invariance we obtain the other exponents, which perfectly match the numerical results obtained from simulations. These exponents are exact, and they are the same as those exhibited by the Kardar-Parisi-Zhang (KPZ) model for this dimension. Therefore, our results provide proof for the conjecture that the etching and KPZ models belong to the same universality class. Moreover, the method is general, and it can be applied to other cellular automata models.

Physiological aging as an infinitesimally ratcheted random walk.

The distribution of a population throughout the physiological age of the individuals is very relevant information in population studies. It has been modeled by the Langevin and the Fokker-Planck equations. A major problem with these equations is that they allow the physiological age to move back in time. This paper proposes an Infinitesimally ratcheted random walk as a way to solve that problem. Two mathematical representations are proposed. One of them uses a nonlocal scalar field. The other one is local, but involves a multicomponent field of speed states. These two formulations are compared to each other and to the Fokker-Planck equation. The relevant properties are discussed. The dynamics of the mean and variance of the population age resulting from the two proposed formulations are obtained.

Effects of adaptation in maintaining high sensitivity over a wide range of backgrounds for Escherichia coli chemotaxis.

An allosteric model is developed to study the cooperative kinase response of wild-type (wt) Escherichia coli cells to the chemoattractant MeAsp in different ambient MeAsp concentrations. The model, together with wt dose response data, reveals the underlying mechanism for E. coli's ability to maintain high sensitivity over a wide range of backgrounds. We find: 1), Adaptation tunes the system to the steepest part of the dose response curve, where the sensitivity to a given type of stimulus is amplified by the number of corresponding receptors in the (mixed) functional receptor complex. A lower bound on the number of Tar receptor dimers (Na) in the complex Na>approximately 6 is obtained from the measured sensitivity. 2), Accurate adaptation synchronizes the kinase activities from different (uncoupled) receptor complexes in a single cell and is crucial in maintaining the high Hill coefficient in the (population averaged) kinase response curve. 3), The wide dynamic range of the high sensitivity can be explained in our model by either having a very small ratio between ligand dissociation constants of the inactive and the active receptors C=0.006, Na=6, and a (methylation level independent) dissociation constant for the inactive Tar receptor K=18.2 microM or by having K and/or Na increase with receptor methylation level together with a larger value of C>0.01. Specific experiments are suggested to distinguish these two scenarios. 4), The receptor occupancy in a wt cell should also adapt and exhibit a slow (approximately logarithmic) dependence on the ligand concentration in the adapted state; this general prediction can be tested experimentally to verify/falsify our model.

An allosteric model for heterogeneous receptor complexes: understanding bacterial chemotaxis responses to multiple stimuli.

The classical Monod-Wyman-Changeux model for homogeneous allosteric protein complex is generalized in this article to model the responses of heterogeneous receptor complexes to multiple types of ligand stimulus. We show that the recent in vivo experimental data of Escherichia coli chemotaxis responses for mutant strains with different expression levels of the chemo-receptors to different types of stimulus [Sourjik, V. & Berg, H. C. (2004) Nature 428, 437-441] all can be explained consistently within this generalized Monod-Wyman-Changeux model. Based on the model and the existing data, responses of all of the strains (studied in this article) to the presence of any combinations of ligand (Ser and MeAsp) concentrations are predicted quantitatively for future experimental verification. Through modeling the in vivo response data, our study reveals important information about the properties of different types of individual receptors, as well as the composition of the cluster. The energetic contribution of the nonligand binding, cytoplasmic parts of the cluster, such as CheA and CheW, is also discussed. The generalized allosteric model provides a consistent framework in understanding signal integration and differentiation in bacterial chemotaxis. It should also be useful for studying the functions of other heterogeneous receptor complexes.

Effects of receptor interaction in bacterial chemotaxis.

Signaling in bacterial chemotaxis is mediated by several types of transmembrane chemoreceptors. The chemoreceptors form tight polar clusters whose functions are of great biological interest. Here, we study the general properties of a chemotaxis model that includes interaction between neighboring chemoreceptors within a receptor cluster and the appropriate receptor methylation and demethylation dynamics to maintain (near) perfect adaptation. We find that, depending on the receptor coupling strength, there are two steady-state phases in the model: a stationary phase and an oscillatory phase. The mechanism for the existence of the two phases is understood analytically. Two important phenomena in transient response, the overshoot in response to a pulse stimulus and the high gain in response to sustained changes in external ligand concentrations, can be explained in our model, and the mechanisms for these two seemingly different phenomena are found to be closely related. The model also naturally accounts for several key in vitro response experiments and the recent in vivo fluorescence resonance energy transfer experiments for various mutant strains. Quantitatively, our study reveals possible choices of parameters for fitting the existing experiments and suggests future experiments to test the model predictions.

Quantitative modeling of sensitivity in bacterial chemotaxis: the role of coupling among different chemoreceptor species.

We propose a general theoretical framework for modeling receptor sensitivity in bacterial chemotaxis, taking into account receptor interactions, including those among different receptor species. We show that our model can quantitatively explain the recent in vivo measurements of receptor sensitivity at different ligand concentrations for both mutant and wild-type strains. For mutant strains, our model can fit the experimental data exactly. For the wild-type cell, our model is capable of achieving high gain while having modest values of Hill coefficient for the response curves. Furthermore, the high sensitivity of the wild-type cell in our model is maintained for a wide range of ambient ligand concentrations, facilitated by near-perfect adaptation and dependence of ligand binding on receptor activity. Our study reveals the importance of coupling among different chemoreceptor species, in particular strong interactions between the aspartate (Tar) and serine (Tsr) receptors, which is crucial in explaining both the mutant and wild-type data. Predictions for the sensitivity of other mutant strains and possible improvements of our model for the wild-type cell are also discussed.

Perfect and near-perfect adaptation in a model of bacterial chemotaxis.

The signaling apparatus mediating bacterial chemotaxis can adapt to a wide range of persistent external stimuli. In many cases, the bacterial activity returns to its prestimulus level exactly, and this perfect adaptability is robust against variations in various chemotaxis protein concentrations. We model the bacterial chemotaxis signaling pathway, from ligand binding to CheY phosphorylation. By solving the steady-state equations of the model analytically, we derive a full set of conditions for the system to achieve perfect adaptation. The conditions related to the phosphorylation part of the pathway are discovered for the first time, while other conditions are generalizations of the ones found in previous works. Sensitivity of the perfect adaptation is evaluated by perturbing these conditions. We find that, even in the absence of some of the perfect adaptation conditions, adaptation can be achieved with near-perfect precision as a result of the separation of scales in both chemotaxis protein concentrations and reaction rates, or specific properties of the receptor distribution in different methylation states. Since near-perfect adaptation can be found in much larger regions of the parameter space than that defined by the perfect adaptation conditions, their existence is essential to understand robustness in bacterial chemotaxis.