Nonequilibrium thermodynamics and nonlinear kinetics in a cellular signaling switch

Intrinsic noise, dissipation cost, and robustness of cellular networks: the underlying energy landscape of MAPK signal transduction

We develop a probabilistic method for analyzing global features of a cellular network under intrinsic statistical fluctuations, which is important when there are finite numbers of molecules. By making a self-consistent mean field approximation of splitting the variables in order to reduce the large number of degrees of freedom, which is reasonable for a not very strongly interacting network, we discovered that the underlying energy landscape of the mitogen-activated protein kinases (MAPKs) signal transduction network (with experimentally measured or inferred parameters such as chemical reaction rate coefficients in the network) is funneled toward a global minimum characterized by the nonequilibrium steady-state fixed point of the system at the end of the signal transduction process. For this system, we also show that the energy landscape is robust against intrinsic fluctuations and random perturbation to the inherent chemical reaction rates. The ratio of the slope versus the roughness of the energy landscape becomes a quantitative measure of robustness and stability of the network. Furthermore, we quantify the dissipation cost of this nonequilibrium system through entropy production, caused by the nonequilibrium flux in the system. We found that a lower dissipation cost corresponds to a more robust network. This least dissipation property might provide a design principle for robust and functional networks. Finally, we find the possibility of bistable and oscillatory-like solutions, which are important for cell fate decisions, upon perturbations. The method described here can be used in a variety of biological networks.

Least dissipation cost as a design principle for robustness and function of cellular networks

From a study of the budding yeast cell cycle, we found that the cellular network evolves to have the least cost for realizing its biological function. We quantify the cost in terms of the dissipation or heat loss characterized through the steady-state properties: the underlying landscape and the associated flux. We found that the dissipation cost is intimately related to the stability and robustness of the network. With the least dissipation cost, the network becomes most stable and robust under mutations and perturbations on the sharpness of the response from input to output as well as self-degradations. The least dissipation cost may provide a general design principle for the cellular network to survive from the evolution and realize the biological function.

Waiting cycle times and generalized Haldane equality in the steady-state cycle kinetics of single enzymes

Enzyme kinetics are cyclic. A more realistic reversible three-step mechanism of the Michaelis-Menten kinetics is investigated in detail, and three kinds of waiting cycle times T, T+, T- are defined. It is shown that the mean waiting cycle times T, T+, and T- are the reciprocal of the steady-state cycle flux Jss, the forward steady-state cycle flux Jss+ and the backward steady-state cycle flux Jss, respectively. We also show that the distribution of T+ conditioned on T+<T- is identical to the distribution of T- conditioned on T-< T+, which is referred to as the generalized Haldane equality. Consequently, the mean waiting cycle time of T+ conditioned on T+<T- (T+|T+<T-) and the one of T- conditioned on T-<T+ (T-|T-<T+) are both just the same as T. In addition, the forward and backward stepping probabilities p+ and p- are also defined and discussed, especially their relationship with the cycle fluxes and waiting cycle times. Furthermore, we extend the same results to the n-step cycle, and finally, experimental and theoretically based evidence are also included.

Synchronized dynamics and non-equilibrium steady states in a stochastic yeast cell-cycle network

Applying the mathematical circulation theory of Markov chains, we investigate the synchronized stochastic dynamics of a discrete network model of yeast cell-cycle regulation where stochasticity has been kept rather than being averaged out. By comparing the network dynamics of the stochastic model with its corresponding deterministic network counterpart, we show that the synchronized dynamics can be soundly characterized by a dominant circulation in the stochastic model, which is the natural generalization of the deterministic limit cycle in the deterministic system. Moreover, the period of the main peak in the power spectrum, which is in common use to characterize the synchronized dynamics, perfectly corresponds to the number of states in the main cycle with dominant circulation. Such a large separation in the magnitude of the circulations, between a dominant, main cycle and the rest, gives rise to the stochastic synchronization phenomenon.

Open-system nonequilibrium steady state: statistical thermodynamics, fluctuations, and chemical oscillations

Gibbsian equilibrium statistical thermodynamics is the theoretical foundation for isothermal, closed chemical, and biochemical reaction systems. This theory, however, is not applicable to most biochemical reactions in living cells, which exhibit a range of interesting phenomena such as free energy transduction, temporal and spatial complexity, and kinetic proofreading. In this article, a nonequilibrium statistical thermodynamic theory based on stochastic kinetics is introduced, mainly through a series of examples: single-molecule enzyme kinetics, nonlinear chemical oscillation, molecular motor, biochemical switch, and specificity amplification. The case studies illustrate an emerging theory for the isothermal nonequilibrium steady state of open systems.

Phosphorylation energy hypothesis: open chemical systems and their biological functions

Biochemical systems and processes in living cells generally operate far from equilibrium. This review presents an overview of a statistical thermodynamic treatment for such systems, with examples from several key components in cellular signal transduction. Open-system nonequilibrium steady-state (NESS) models are introduced. The models account quantitatively for the energetics and thermodynamics in phosphorylation-dephosphorylation switches, GTPase timers, and specificity amplification through kinetic proofreading. The chemical energy derived from ATP and GTP hydrolysis establishes the NESS of a cell and makes the cell--a mesoscopic-biochemical reaction system that consists of a collection of thermally driven fluctuating macromolecules--a genetically programmed chemical machine.

Nonequilibrium thermodynamics and energy efficiency in weight loss diets

Carbohydrate restriction as a strategy for control of obesity is based on two effects: a behavioral effect, spontaneous reduction in caloric intake and a metabolic effect, an apparent reduction in energy efficiency, greater weight loss per calorie consumed. Variable energy efficiency is established in many contexts (hormonal imbalance, weight regain and knock-out experiments in animal models), but in the area of the effect of macronutrient composition on weight loss, controversy remains. Resistance to the idea comes from a perception that variable weight loss on isocaloric diets would somehow violate the laws of thermodynamics, that is, only caloric intake is important ("a calorie is a calorie"). Previous explanations of how the phenomenon occurs, based on equilibrium thermodynamics, emphasized the inefficiencies introduced by substrate cycling and requirements for increased gluconeogenesis. Living systems, however, are maintained far from equilibrium, and metabolism is controlled by the regulation of the rates of enzymatic reactions. The principles of nonequilibrium thermodynamics which emphasize kinetic fluxes as well as thermodynamic forces should therefore also be considered. Here we review the principles of nonequilibrium thermodynamics and provide an approach to the problem of maintenance and change in body mass by recasting the problem of TAG accumulation and breakdown in the adipocyte in the language of nonequilibrium thermodynamics. We describe adipocyte physiology in terms of cycling between an efficient storage mode and a dissipative mode. Experimentally, this is measured in the rate of fatty acid flux and fatty acid oxidation. Hormonal levels controlled by changes in dietary carbohydrate regulate the relative contributions of the efficient and dissipative parts of the cycle. While no experiment exists that measures all relevant variables, the model is supported by evidence in the literature that 1) dietary carbohydrate, via its effect on hormone levels controls fatty acid flux and oxidation, 2) the rate of lipolysis is a primary target of insulin, postprandial, and 3) chronic carbohydrate-restricted diets reduce the levels of plasma TAG in response to a single meal. In summary, we propose that, in isocaloric diets of different macronutrient composition, there is variable flux of stored TAG controlled by the kinetic effects of insulin and other hormones. Because the fatty acid-TAG cycle never comes to equilibrium, net gain or loss is possible. The greater weight loss on carbohydrate restricted diets, popularly referred to as metabolic advantage can thus be understood in terms of the principles of nonequilibrium thermodynamics and is a consequence of the dynamic nature of bioenergetics where it is important to consider kinetic as well as thermodynamic variables.

Quantifying robustness and dissipation cost of yeast cell cycle network: the funneled energy landscape perspectives

We study the origin of robustness of yeast cell cycle cellular network through uncovering its underlying energy landscape. This is realized from the information of the steady-state probabilities by solving a discrete set of kinetic master equations for the network. We discovered that the potential landscape of yeast cell cycle network is funneled toward the global minimum, G1 state. The ratio of the energy gap between G1 and average versus roughness of the landscape termed as robustness ratio (RR) becomes a quantitative measure of the robustness and stability for the network. The funneled landscape is quite robust against random perturbations from the inherent wiring or connections of the network. There exists a global phase transition between the more sensitive response or less self-degradation phase leading to underlying funneled global landscape with large RR, and insensitive response or more self-degradation phase leading to shallower underlying landscape of the network with small RR. Furthermore, we show that the more robust landscape also leads to less dissipation cost of the network. Least dissipation and robust landscape might be a realization of Darwinian principle of natural selection at cellular network level. It may provide an optimal criterion for network wiring connections and design.

Potential energy landscape and robustness of a gene regulatory network: toggle switch

Finding a multidimensional potential landscape is the key for addressing important global issues, such as the robustness of cellular networks. We have uncovered the underlying potential energy landscape of a simple gene regulatory network: a toggle switch. This was realized by explicitly constructing the steady state probability of the gene switch in the protein concentration space in the presence of the intrinsic statistical fluctuations due to the small number of proteins in the cell. We explored the global phase space for the system. We found that the protein synthesis rate and the unbinding rate of proteins to the gene were small relative to the protein degradation rate; the gene switch is monostable with only one stable basin of attraction. When both the protein synthesis rate and the unbinding rate of proteins to the gene are large compared with the protein degradation rate, two global basins of attraction emerge for a toggle switch. These basins correspond to the biologically stable functional states. The potential energy barrier between the two basins determines the time scale of conversion from one to the other. We found as the protein synthesis rate and protein unbinding rate to the gene relative to the protein degradation rate became larger, the potential energy barrier became larger. This also corresponded to systems with less noise or the fluctuations on the protein numbers. It leads to the robustness of the biological basins of the gene switches. The technique used here is general and can be applied to explore the potential energy landscape of the gene networks.

A Quasistationary Analysis of a Stochastic Chemical Reaction: Keizer’s Paradox

For a system of biochemical reactions, it is known from the work of T.G. Kurtz [J. Appl. Prob. 8, 344 (1971)] that the chemical master equation model based on a stochastic formulation approaches the deterministic model based on the Law of Mass Action in the infinite system-size limit in finite time. The two models, however, often show distinctly different steady-state behavior. To further investigate this "paradox," a comparative study of the deterministic and stochastic models of a simple autocatalytic biochemical reaction, taken from a text by the late J. Keizer, is carried out. We compute the expected time to extinction, the true stochastic steady state, and a quasistationary probability distribution in the stochastic model. We show that the stochastic model predicts the deterministic behavior on a reasonable time scale, which can be consistently obtained from both models. The transition time to the extinction, however, grows exponentially with the system size. Mathematically, we identify that exchanging the limits of infinite system size and infinite time is problematic. The appropriate system size that can be considered sufficiently large, an important parameter in numerical computation, is also discussed.

Metabolic futile cycles and their functions: a systems analysis of energy and control

It has long been hypothesised that futile cycles in cellular metabolism are involved in the regulation of biochemical pathways. Following the work of Newsholme and Crabtree, a quantitative theory was developed for this idea based on open-system thermodynamics and metabolic control analysis. It is shown that the stoichiometric sensitivity of an intermediary metabolite concentration with respect to changes in steady-state flux is governed by the effective equilibrium constant of the intermediate formation, and the equilibrium can be regulated by a futile cycle. The direction of the shift in the effective equilibrium constant depends on the direction of operation of the futile cycle. High stoichiometric sensitivity corresponds to ultrasensitivity of an intermediate concentration to net flow through a pathway; low stoichiometric sensitivity corresponds to super-robustness of concentration with respect to changes in flux. Both cases potentially play important roles in metabolic regulation. Futile cycles actively shift the effective equilibrium by expending energy; the magnitude of changes in effective equilibria and sensitivities is a function of the amount of energy used by a futile cycle. This proposed mechanism for control by futile cycles works remarkably similar to kinetic proofreading in biosynthesis. The sensitivity of the system is also intimately related to the rate of concentration fluctuations of intermediate metabolites. The possibility of different roles for the two major mechanisms within cellular biochemical regulation, namely reversible chemical modifications via futile cycles and shifting equilibrium by macromolecular binding, are discussed.

Funneled landscape leads to robustness of cell networks: yeast cell cycle

We uncovered the underlying energy landscape for a cellular network. We discovered that the energy landscape of the yeast cell-cycle network is funneled towards the global minimum (G0/G1 phase) from the experimentally measured or inferred inherent chemical reaction rates. The funneled landscape is quite robust against random perturbations. This naturally explains robustness from a physical point of view. The ratio of slope versus roughness of the landscape becomes a quantitative measure of robustness of the network. The funneled landscape can be seen as a possible realization of the Darwinian principle of natural selection at the cellular network level. It provides an optimal criterion for network connections and design. Our approach is general and can be applied to other cellular networks.

Cellular networks are in general quite robust and perform their biological functions against environmental perturbations. There are so far very few studies of why networks should be robust and perform biological functions from the physical point of view. In this work, Wang, Huang, Xia, and Sun studied the global properties of the network from physical perspectives. The aim of this paper is to provide a conceptual framework and a tool to study the global nature of the cellular network. The main conclusion is that by uncovering the underlying potential landscape of the budding yeast cell cycle the authors show that it is funneled and robust against the perturbation from kinetic rates and environmental disturbances through noise. This provides the physical explanation of the robustness and stability of the network for performing biological functions. They believe the energy landscape is useful in exploring global properties of protein–protein interaction networks. They also believe the funneled landscape may provide a possible quantitative realization of the Darwinian principle of natural selection at the cellular network level. Finally, Wang et al. derived a quantitative criterion for robustness of the network function. This criterion may provide a novel algorithm for optimizing the network connections to improve the design of synthetic networks.

Reducing intrinsic biochemical noise in cells and its thermodynamic limit

In living cells, the specificity of biomolecular recognition can be amplified and the noise from non-specific interactions can be reduced at the expense of cellular free energy. This is the seminal idea in the Hopfield-Ninio theory of kinetic proofreading: The specificity is increased via cyclic network kinetics without altering molecular structures and equilibrium affinites. We show a thermodynamic limit of the specificity amplification with a given amount of available free energy. For a normal cell under physiological condition with sustained phosphorylation potential, this gives a factor of 10(10) as the upper bound in specificity amplification. We also study an optimal kinetic network design that is capable of approaching the thermodynamic limit.

Controlled switching of intrinsic localized modes in a one-dimensional antiferromagnet

Nearly-steady-state locked intrinsic localized modes (ILMs) in the quasi-1D antiferromagnet (C(2)H(5)NH(3))(2)CuCl(4) are detected via four-wave mixing emission or the uniform mode absorption. Exploiting the long-time stability of these locked ILMs, repeatable nonlinear switching is observed by varying the sample temperature, and localized modes with various amplitudes are created by modulation of the microwave driver power. This steady-state ILM locking technique could be used to produce energy localization in other atomic lattices.

Cycle kinetics, steady state thermodynamics and motors-a paradigm for living matter physics

An integration of the stochastic mathematical models for motor proteins with Hill's steady state thermodynamics yields a rather comprehensive theory for molecular motors as open systems in the nonequilibrium steady state. This theory, a natural extension of Gibbs' approach to isothermal molecular systems in equilibrium, is compared with other existing theories with dissipative structures and dynamics. The theory of molecular motors might be considered as an archetype for studying more complex open biological systems such as biochemical reaction networks inside living cells.