Nonequilibrium thermodynamics and nonlinear kinetics in a cellular signaling switch

Information Transmission in G Protein-Coupled Receptors

G protein-coupled receptors (GPCRs) are the largest class of receptors in the human genome and constitute about 30% of all drug targets. In this article, intended for a non-mathematical audience, both experimental observations and new theoretical results are compared in the context of information transmission across the cell membrane. The amount of information actually currently used or projected to be used in clinical settings is a small fraction of the information transmission capacity of the GPCR. This indicates that the number of yet undiscovered drug targets within GPCRs is much larger than what is currently known. Theoretical studies with some experimental validation indicate that localized heat deposition and dissipation are key to the identification of sites and mechanisms for drug action.

Temporal Self-Regulation of Mechanical Properties via Catalytic Amyloid Polymers of a Short Peptide

We report a short peptide that accessed dynamic catalytic polymers to demonstrate four-stage (sol-gel-weak gel-strong gel) temporal self-regulation of its mechanical properties. The peptide exploited its intrinsic catalytic capabilities of manipulating C-C bonds (retro-aldolase-like) that resulted in a nonlinear variation in the catalytic rate. The seven-residue sequence exploited two lysines for binding and cleaving the thermodynamically activated substrate that subsequently led to the self-regulation of the mechanical strengths of the polymerized states as a function of time and reaction progress. Interestingly, the polymerization events were modulated by the different catalytic potentials of the two terminal lysines to cleave the substrate, covalently trap the electrophilic products, and subsequently control the mechanical properties of the system.

Model of ligand-triggered information transmission in G-protein coupled receptor complexes

We present a model for the effects of ligands on information transmission in G-Protein Coupled Receptor (GPCR) complexes. The model is built ab initio entirely on principles of statistical mechanics and tenets of information transmission theory and was validated in part using agonist-induced effector activity and signaling bias for the angiotensin- and adrenergic-mediated signaling pathways, with in vitro observations of phosphorylation sites on the C tail of the GPCR complex, and single-cell information-transmission experiments. The model extends traditional kinetic models that form the basis for many existing models of GPCR signaling. It is based on maximizing the rates of entropy production and information transmission through the GPCR complex. The model predicts that (1) phosphatase-catalyzed reactions, as opposed to kinase-catalyzed reactions, on the C-tail and internal loops of the GPCR are responsible for controlling the signaling activity, (2) signaling favors the statistical balance of the number of switches in the ON state and the number in the OFF state, and (3) biased-signaling response depends discontinuously on ligand concentration.

Information thermodynamics of encoding and encoders

Non-isolated systems have diverse coupling relations with the external environment. These relations generate complex thermodynamics and information transmission between the system and its environment. The framework depicted in the current research attempts to glance at the critical role of the internal orders inside the non-isolated system in shaping the information thermodynamics coupling. We characterize the coupling as a generalized encoding process, where the system acts as an information thermodynamics encoder to encode the external information based on thermodynamics. We formalize the encoding process in the context of the nonequilibrium second law of thermodynamics, revealing an intrinsic difference in information thermodynamics characteristics between information thermodynamics encoders with and without internal correlations. During the information encoding process of an external source Y, specific sub-systems in an encoder X with internal correlations can exceed the information thermodynamics bound on ( X , Y ) and encode more information than system X works as a whole. We computationally verify this theoretical finding in an Ising model with a random external field and a neural data set of the human brain during visual perception and recognition. Our analysis demonstrates that the stronger internal correlation inside these systems implies a higher possibility for specific sub-systems to encode more information than the global one. These findings may suggest a new perspective in studying information thermodynamics in diverse physical and biological systems.

Dynamic modeling of the activated sludge microbial growth and activity under exposure to heavy metals

The presence of heavy metals in the environment can lead to ecological and health problems. The evolution of biological systems, such as activated sludge, exposed to heavy metals is still underexplored. Therefore, this study sought to develop a model of microorganism activity and growth in activated sludge and used it to investigate the toxicity of five metals: Cu, Cd, Ni, Zn, and Ag. Patterns in the evolution of the toxic effects caused by these metals were similar at the beginning of exposure. Differences in toxicity between metal ions were noted for longer exposure times. Changes in model parameters indicate the influence of metal ions on the mass and energy balance of living cells. Decreases in new enzyme units and biomass production yields in contaminated activated sludge indicate a shift from anabolic reactions to metal homeostasis and resistance.

Dynamic bistable switches enhance robustness and accuracy of cell cycle transitions

Bistability is a common mechanism to ensure robust and irreversible cell cycle transitions. Whenever biological parameters or external conditions change such that a threshold is crossed, the system abruptly switches between different cell cycle states. Experimental studies have uncovered mechanisms that can make the shape of the bistable response curve change dynamically in time. Here, we show how such a dynamically changing bistable switch can provide a cell with better control over the timing of cell cycle transitions. Moreover, cell cycle oscillations built on bistable switches are more robust when the bistability is modulated in time. Our results are not specific to cell cycle models and may apply to other bistable systems in which the bistable response curve is time-dependent.

Many systems in nature show bistability, which means they can evolve to one of two stable steady states under exactly the same conditions. Which state they evolve to depends on where the system comes from. Such bistability underlies the switching behavior that is essential for cells to progress in the cell division cycle. A quick switch happens when the cell jumps from one steady state to another steady state. Typical of this switching behavior is its robustness and irreversibility. In this paper, we expand this viewpoint of the dynamics of the cell cycle by considering bistable switches which themselves are changing in time. This gives the cell an extra layer of control over transitions both in time and in space, and can make those transitions more robust. Such dynamically changing bistability can appear very naturally. We show this in a model of mitotic entry, in which we include a nuclear and cytoplasmic compartment. The activity of a crucial cell cycle protein follows a bistable switch in each compartment, but the shape of its response is changing in time as proteins are imported into and exported from the nucleus.

Characteristics of the neural coding of causality

While causality processing is an essential cognitive capacity of the neural system, a systematic understanding of the neural coding of causality is still elusive. We propose a physically fundamental analysis of this issue and demonstrate that the neural dynamics encodes the original causality between external events near homomorphically. The causality coding is memory robust for the amount of historical information and features high precision but low recall. This coding process creates a sparser representation for the external causality. Finally, we propose a statistic characterization for the neural coding mapping from the original causality to the coded causality in neural dynamics.

History-dependent nonequilibrium conformations of a highly confined polymer globule in a sphere

Chromatin undergoes condensation-decondensation processes repeatedly during its cell lifetime. The spatial organization of chromatin in nucleus resembles the fractal globule, of which structure significantly differs from an equilibrium polymer globule. There have been efforts to develop a polymer globule model to describe the fractal globulelike structure of tightly packed chromatin in nucleus. However, the transition pathway of a polymer toward a globular state has been often ignored. Because biological systems are intrinsically in nonequilibrium states, the transition pathway that the chromatin would take before reaching the densely packaged globule should be of importance. In this study, by employing a simple polymer model and Langevin dynamics simulations, we investigate the conformational transition of a single polymer from a swollen coil to a compact globule. We aim to elucidate the effect of transition pathways on the final globular structure. We show that a fast collapse induces a nonequilibrium structure even without a specific intramolecular interaction and that its relaxation toward an equilibrium globule is extremely slow. Due to a strong confinement, the fractal globule never relaxes into an equilibrium state during our simulations such that the globular structure becomes dependent on the transition pathway.

ATP-powered molecular recognition to engineer transient multivalency and self-sorting 4D hierarchical systems

Biological systems organize multiple hierarchical structures in parallel, and create dynamic assemblies and functions by energy dissipation. In contrast, emerging artificial non-equilibrium self-assembling systems have remained relatively simplistic concerning hierarchical design, and non-equilibrium multi-component systems are uncharted territory. Here we report a modular DNA toolbox allowing to program transient non-equilibrium multicomponent systems across hierarchical length scales by introducing chemically fueled molecular recognition orchestrated by reaction networks of concurrent ATP-powered ligation and cleavage of freely programmable DNA building blocks. Going across hierarchical levels, we demonstrate transient side-chain functionalized nucleic acid polymers, and further introduce the concept of transient cooperative multivalency as a key to bridge length scales to pioneer fuel-driven encapsulation, self-assembly of colloids, and non-equilibrium transient narcissistic colloidal self-sorting on a systems level. The fully programmable and functionalizable DNA components pave the way to design chemically fueled 4D (3 space, 1 time) molecular multicomponent systems and autonomous materials.

Interlinked GTPase cascades provide a motif for both robust switches and oscillators

GTPases regulate a wide range of cellular processes, such as intracellular vesicular transport, signal transduction and protein translation. These hydrolase enzymes operate as biochemical switches by toggling between an active guanosine triphosphate (GTP)-bound state and an inactive guanosine diphosphate (GDP)-bound state. We compare two network motifs, a single-species switch and an interlinked cascade that consists of two species coupled through positive and negative feedback loops. We find that interlinked cascades are closer to the ideal all-or-none switch and are more robust against fluctuating signals. While the single-species switch can only achieve bistability, interlinked cascades can be converted into oscillators by tuning the cofactor concentrations, which catalyse the activity of the cascade. These regimes can only be achieved with sufficient chemical driving provided by GTP hydrolysis. In this study, we present a thermodynamically consistent model that can achieve bistability and oscillations with the same feedback motif.

Interrogating metabolism as an electron flow system

Metabolism is generally considered as a neatly organised system of modular pathways, shaped by evolution under selection for optimal cellular growth. This view falls short of explaining and predicting a number of key observations about the structure and dynamics of metabolism. We highlight these limitations of a pathway-centric view on metabolism and summarise studies suggesting how these could be overcome by viewing metabolism as a thermodynamically and kinetically constrained, dynamical flow system. Such a systems-level, first-principles based view of metabolism can open up new avenues of metabolic engineering and cures for metabolic diseases and allow better insights to a myriad of physiological processes that are ultimately linked to metabolism. Towards further developing this view, we call for a closer interaction among physical and biological disciplines and an increased use of electrochemical and biophysical approaches to interrogate cellular metabolism together with the microenvironment in which it exists.

Relatively slow stochastic gene-state switching in the presence of positive feedback significantly broadens the region of bimodality through stabilizing the uninduced phenotypic state

Within an isogenic population, even in the same extracellular environment, individual cells can exhibit various phenotypic states. The exact role of stochastic gene-state switching regulating the transition among these phenotypic states in a single cell is not fully understood, especially in the presence of positive feedback. Recent high-precision single-cell measurements showed that, at least in bacteria, switching in gene states is slow relative to the typical rates of active transcription and translation. Hence using the lac operon as an archetype, in such a region of operon-state switching, we present a fluctuating-rate model for this classical gene regulation module, incorporating the more realistic operon-state switching mechanism that was recently elucidated. We found that the positive feedback mechanism induces bistability (referred to as deterministic bistability), and that the parameter range for its occurrence is significantly broadened by stochastic operon-state switching. We further show that in the absence of positive feedback, operon-state switching must be extremely slow to trigger bistability by itself. However, in the presence of positive feedback, which stabilizes the induced state, the relatively slow operon-state switching kinetics within the physiological region are sufficient to stabilize the uninduced state, together generating a broadened parameter region of bistability (referred to as stochastic bistability). We illustrate the opposite phenotype-transition rate dependence upon the operon-state switching rates in the two types of bistability, with the aid of a recently proposed rate formula for fluctuating-rate models. The rate formula also predicts a maximal transition rate in the intermediate region of operon-state switching, which is validated by numerical simulations in our model. Overall, our findings suggest a biological function of transcriptional “variations” among genetically identical cells, for the emergence of bistability and transition between phenotypic states.

Identifying the mechanism underlying the coexistence of multiple stable phenotypic states has been a challenging scientific problem for more than half a century, and an appropriate mathematical model at the single-cell level is also in high demand. Single-cell measurements conducted in the past ten years have shown that gene-state switching is slow relative to the typical rates of active transcription and translation; hence the recently proposed fluctuating-rate model is a good candidate for describing the single-cell dynamics. We use the classic gene regulation module of the lac operon as an archetype and build a specific fluctuating-rate model based on the recently identified operon-state switching mechanism. This model is analyzed to dissect the interplay between positive feedback and the stochastic switching of gene states in the emergence of bistability/multistablity and the transition between phenotypic states. We show that relatively slow operon-state switching stabilizes the uninduced state and that the positive feedback stabilizes the induced state. Thus, the parameter range for bistability is significantly broadened. In addition, recently proposed landscape theory and rate formula predict opposite phenotype-transition rate dependence on operon-state switching rates for the two types of bistability.

Derivation of Markov processes that violate detailed balance

Time-reversal symmetry of the microscopic laws dictates that the equilibrium distribution of a stochastic process must obey the condition of detailed balance. However, cyclic Markov processes that do not admit equilibrium distributions with detailed balance are often used to model systems driven out of equilibrium by external agents. I show that for a Markov model without detailed balance, an extended Markov model can be constructed, which explicitly includes the degrees of freedom for the driving agent and satisfies the detailed balance condition. The original cyclic Markov model for the driven system is then recovered as an approximation at early times by summing over the degrees of freedom for the driving agent. I also show that the widely accepted expression for the entropy production in a cyclic Markov model is actually a time derivative of an entropy component in the extended model. Further, I present an analytic expression for the entropy component that is hidden in the cyclic Markov model.

Processes on the emergent landscapes of biochemical reaction networks and heterogeneous cell population dynamics: differentiation in living matters

The notion of an attractor has been widely employed in thinking about the nonlinear dynamics of organisms and biological phenomena as systems and as processes. The notion of a landscape with valleys and mountains encoding multiple attractors, however, has a rigorous foundation only for closed, thermodynamically non-driven, chemical systems, such as a protein. Recent advances in the theory of nonlinear stochastic dynamical systems and its applications to mesoscopic reaction networks, one reaction at a time, have provided a new basis for a landscape of open, driven biochemical reaction systems under sustained chemostat. The theory is equally applicable not only to intracellular dynamics of biochemical regulatory networks within an individual cell but also to tissue dynamics of heterogeneous interacting cell populations. The landscape for an individual cell, applicable to a population of isogenic non-interacting cells under the same environmental conditions, is defined on the counting space of intracellular chemical compositions x = (x₁,x₂, … ,x_N ) in a cell, where x_ℓ is the concentration of the ℓth biochemical species. Equivalently, for heterogeneous cell population dynamics x_ℓ is the number density of cells of the ℓth cell type. One of the insights derived from the landscape perspective is that the life history of an individual organism, which occurs on the hillsides of a landscape, is nearly deterministic and 'programmed', while population-wise an asynchronous non-equilibrium steady state resides mostly in the lowlands of the landscape. We argue that a dynamic 'blue-sky' bifurcation, as a representation of Waddington's landscape, is a more robust mechanism for a cell fate decision and subsequent differentiation than the widely pictured pitch-fork bifurcation. We revisit, in terms of the chemostatic driving forces upon active, living matter, the notions of near-equilibrium thermodynamic branches versus far-from-equilibrium states. The emergent landscape perspective permits a quantitative discussion of a wide range of biological phenomena as nonlinear, stochastic dynamics.

Mapping quorum sensing onto neural networks to understand collective decision making in heterogeneous microbial communities

Microbial communities frequently communicate via quorum sensing (QS), where cells produce, secrete, and respond to a threshold level of an autoinducer (AI) molecule, thereby modulating gene expression. However, the biology of QS remains incompletely understood in heterogeneous communities, where variant bacterial strains possess distinct QS systems that produce chemically unique AIs. AI molecules bind to 'cognate' receptors, but also to 'non-cognate' receptors found in other strains, resulting in inter-strain crosstalk. Understanding these interactions is a prerequisite for deciphering the consequences of crosstalk in real ecosystems, where multiple AIs are regularly present in the same environment. As a step towards this goal, we map crosstalk in a heterogeneous community of variant QS strains onto an artificial neural network model. This formulation allows us to systematically analyze how crosstalk regulates the community's capacity for flexible decision making, as quantified by the Boltzmann entropy of all QS gene expression states of the system. In a mean-field limit of complete cross-inhibition between variant strains, the model is exactly solvable, allowing for an analytical formula for the number of variants that maximize capacity as a function of signal kinetics and activation parameters. An analysis of previous experimental results on the Staphylococcus aureus two-component Agr system indicates that the observed combination of variant numbers, gene expression rates and threshold concentrations lies near this critical regime of parameter space where capacity peaks. The results are suggestive of a potential evolutionary driving force for diversification in certain QS systems.

Stochastic Thermodynamics of Learning

Virtually every organism gathers information about its noisy environment and builds models from those data, mostly using neural networks. Here, we use stochastic thermodynamics to analyze the learning of a classification rule by a neural network. We show that the information acquired by the network is bounded by the thermodynamic cost of learning and introduce a learning efficiency η≤1. We discuss the conditions for optimal learning and analyze Hebbian learning in the thermodynamic limit.

Fundamental principles of energy consumption for gene expression

How energy is consumed in gene expression is largely unknown mainly due to complexity of non-equilibrium mechanisms affecting expression levels. Here, by analyzing a representative gene model that considers complexity of gene expression, we show that negative feedback increases energy consumption but positive feedback has an opposite effect; promoter leakage always reduces energy consumption; generating more bursts needs to consume more energy; and the speed of promoter switching is at the cost of energy consumption. We also find that the relationship between energy consumption and expression noise is multi-mode, depending on both the type of feedback and the speed of promoter switching. Altogether, these results constitute fundamental principles of energy consumption for gene expression, which lay a foundation for designing biologically reasonable gene modules. In addition, we discuss possible biological implications of these principles by combining experimental facts.

Distinguishing the rates of gene activation from phenotypic variations

Background

Stochastic genetic switching driven by intrinsic noise is an important process in gene expression. When the rates of gene activation/inactivation are relatively slow, fast, or medium compared with the synthesis/degradation rates of mRNAs and proteins, the variability of protein and mRNA levels may exhibit very different dynamical patterns. It is desirable to provide a systematic approach to identify their key dynamical features in different regimes, aiming at distinguishing which regime a considered gene regulatory network is in from their phenotypic variations.

Results

We studied a gene expression model with positive feedbacks when genetic switching rates vary over a wide range. With the goal of providing a method to distinguish the regime of the switching rates, we first focus on understanding the essential dynamics of gene expression system in different cases. In the regime of slow switching rates, we found that the effective dynamics can be reduced to independent evolutions on two separate layers corresponding to gene activation and inactivation states, and the transitions between two layers are rare events, after which the system goes mainly along deterministic ODE trajectories on a particular layer to reach new steady states. The energy landscape in this regime can be well approximated by using Gaussian mixture model. In the regime of intermediate switching rates, we analyzed the mean switching time to investigate the stability of the system in different parameter ranges. We also discussed the case of fast switching rates from the viewpoint of transition state theory. Based on the obtained results, we made a proposal to distinguish these three regimes in a simulation experiment. We identified the intermediate regime from the fact that the strength of cellular memory is lower than the other two cases, and the fast and slow regimes can be distinguished by their different perturbation-response behavior with respect to the switching rates perturbations.

Conclusions

We proposed a simulation experiment to distinguish the slow, intermediate and fast regimes, which is the main point of our paper. In order to achieve this goal, we systematically studied the essential dynamics of gene expression system when the switching rates are in different regimes. Our theoretical understanding provides new insights on the gene expression experiments.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-015-0172-0) contains supplementary material, which is available to authorized users.

Thermodynamic uncertainty relation for biomolecular processes

Biomolecular systems like molecular motors or pumps, transcription and translation machinery, and other enzymatic reactions, can be described as Markov processes on a suitable network. We show quite generally that, in a steady state, the dispersion of observables, like the number of consumed or produced molecules or the number of steps of a motor, is constrained by the thermodynamic cost of generating it. An uncertainty ε requires at least a cost of 2k(B)T/ε2 independent of the time required to generate the output.

Bistability: requirements on cell-volume, protein diffusion, and thermodynamics

Bistability is considered wide-spread among bacteria and eukaryotic cells, useful e.g. for enzyme induction, bet hedging, and epigenetic switching. However, this phenomenon has mostly been described with deterministic dynamic or well-mixed stochastic models. Here, we map known biological bistable systems onto the well-characterized biochemical Schlögl model, using analytical calculations and stochastic spatiotemporal simulations. In addition to network architecture and strong thermodynamic driving away from equilibrium, we show that bistability requires fine-tuning towards small cell volumes (or compartments) and fast protein diffusion (well mixing). Bistability is thus fragile and hence may be restricted to small bacteria and eukaryotic nuclei, with switching triggered by volume changes during the cell cycle. For large volumes, single cells generally loose their ability for bistable switching and instead undergo a first-order phase transition.

Hsp70 chaperones are non-equilibrium machines that achieve ultra-affinity by energy consumption

70-kDa Heat shock proteins are ATP-driven molecular chaperones that perform a myriad of essential cellular tasks. Although structural and biochemical studies have shed some light on their functional mechanism, the fundamental issue of the role of energy consumption, due to ATP-hydrolysis, has remained unaddressed. Here we establish a clear connection between the non-equilibrium nature of Hsp70, due to ATP hydrolysis, and the determining feature of its function, namely its high affinity for its substrates. Energy consumption can indeed decrease the dissociation constant of the chaperone-substrate complex by several orders of magnitude with respect to an equilibrium scenario. We find that the biochemical requirements for observing such ultra-affinity coincide with the physiological conditions in the cell. Our results rationalize several experimental observations and pave the way for further analysis of non-equilibrium effects underlying chaperone functions.

DOI:http://dx.doi.org/10.7554/eLife.02218.001

Proteins perform numerous essential tasks in cells. Most of these tasks require the protein to have a very specific structure, which is maintained by a balance of chemical and physical interactions. However, this delicate balance is vulnerable to excessive heat, changes in the pH of the cell, and certain chemicals. As a consequence, proteins could lose their specific structure and stop working.

Cells employ a group of specialized proteins—called chaperones—to check that other proteins have the correct structure, and to ‘refold’ those that do not. The Hsp70 chaperone family needs energy to do its job, and it gets this energy from a molecule called ATP. However, the exact way that Hsp70s work and use this energy is not fully understood.

One major puzzle is how Hsp70 binds to a protein to fold it up. Previous experiments suggested that this binding is particularly effective if Hsp70 can adopt different structures as part of a complex cycle governed by ATP. Now, De Los Rios and Barducci reveal that the energy released from breaking down ATP molecules enables this extra-efficient binding to occur. According to the proposed model, this is possible under some conditions that are normally found in cells. These include having many more Hsp70 proteins than target proteins, and producing energy at extremely high rates from ATP. The specific kinetic properties of the different structures Hsp70 can form are also crucial.

More generally, the principle that energy consumption enhances binding could be extended beyond chaperone proteins and represent a general mechanism for other biomolecular systems.

DOI:http://dx.doi.org/10.7554/eLife.02218.002

Chemical sensing by nonequilibrium cooperative receptors

Cooperativity arising from local interactions in equilibrium receptor systems provides gain, but does not increase sensory performance, as measured by the signal-to-noise ratio (SNR) due to a fundamental tradeoff between gain and intrinsic noise. Here we allow sensing to be a nonequilibrium process and show that energy dissipation cannot circumvent the fundamental tradeoff, so that the SNR is still optimal for independent receptors. For systems requiring high gain, nonequilibrium 2D-coupled receptors maximize the SNR, revealing a new design principle for biological sensors.

The energy costs of insulators in biochemical networks

Complex networks of biochemical reactions, such as intracellular protein signaling pathways and genetic networks, are often conceptualized in terms of modules--semiindependent collections of components that perform a well-defined function and which may be incorporated in multiple pathways. However, due to sequestration of molecular messengers during interactions and other effects, collectively referred to as retroactivity, real biochemical systems do not exhibit perfect modularity. Biochemical signaling pathways can be insulated from impedance and competition effects, which inhibit modularity, through enzymatic futile cycles that consume energy, typically in the form of ATP. We hypothesize that better insulation necessarily requires higher energy consumption. We test this hypothesis through a combined theoretical and computational analysis of a simplified physical model of covalent cycles, using two innovative measures of insulation, as well as a possible new way to characterize optimal insulation through the balancing of these two measures in a Pareto sense. Our results indicate that indeed better insulation requires more energy. While insulation may facilitate evolution by enabling a modular plug-and-play interconnection architecture, allowing for the creation of new behaviors by adding targets to existing pathways, our work suggests that this potential benefit must be balanced against the metabolic costs of insulation necessarily incurred in not affecting the behavior of existing processes.

Energetic costs of cellular computation

Cells often perform computations in order to respond to environmental cues. A simple example is the classic problem, first considered by Berg and Purcell, of determining the concentration of a chemical ligand in the surrounding media. On general theoretical grounds, it is expected that such computations require cells to consume energy. In particular, Landauer's principle states that energy must be consumed in order to erase the memory of past observations. Here, we explicitly calculate the energetic cost of steady-state computation of ligand concentration for a simple two-component cellular network that implements a noisy version of the Berg-Purcell strategy. We show that learning about external concentrations necessitates the breaking of detailed balance and consumption of energy, with greater learning requiring more energy. Our calculations suggest that the energetic costs of cellular computation may be an important constraint on networks designed to function in resource poor environments, such as the spore germination networks of bacteria.

Potential flux landscapes determine the global stability of a Lorenz chaotic attractor under intrinsic fluctuations

We developed a potential flux landscape theory to investigate the dynamics and the global stability of a chemical Lorenz chaotic strange attractor under intrinsic fluctuations. Landscape was uncovered to have a butterfly shape. For chaotic systems, both landscape and probabilistic flux are crucial to the dynamics of chaotic oscillations. Landscape attracts the system down to the chaotic attractor, while flux drives the coherent motions along the chaotic attractors. Barrier heights from the landscape topography provide a quantitative measure for the robustness of chaotic attractor. We also found that the entropy production rate and phase coherence increase as the molecular numbers increase. Power spectrum analysis of autocorrelation function provides another way to quantify the global stability of chaotic attractor. We further found that limit cycle requires more flux and energy to sustain than the chaotic strange attractor. Finally, by detailed analysis we found that the curl probabilistic flux may provide the origin of the chaotic attractor.

Landscape topography determines global stability and robustness of a metabolic network

Metabolic networks have gained broad attention in recent years as a result of their important roles in biological systems. However, how to quantify the global stability of the metabolic networks is still challenging. We develop a probabilistic landscape approach to investigate the global natures of the metabolic system under external fluctuations. As an example, we choose a model of the carbohydrate metabolism and the anaplerotic synthesis of oxalacetate in Aspergillus niger under conditions of citric acid accumulation to explore landscape topography. The landscape has a funnel shape, which guarantees the robustness of system under fluctuations and perturbations. Robustness ratio (RR), defined as the ratio of gap between lowest potential and average potential versus roughness measured by the dispersion or square root of variations of potentials, can be used to quantitatively evaluate the global stability of metabolic networks, and the larger the RR value, the more stable the system. Results of the entropy production rate imply that nature might evolve such that the network is robust against perturbations from environment or network wirings and performs specific biological functions with less dissipation cost. We also carried out a sensitivity analysis of parameters and uncovered some key network structure factors such as kinetic rates or wirings connecting the protein species nodes, which influence the global natures of the system. We found there is a strong correlation between the landscape topography and the input-output response. The more stable and robust the metabolic network is, the sharper the response is.

Landscape, flux, correlation, resonance, coherence, stability, and key network wirings of stochastic circadian oscillation

Circadian rhythms with a period of ~24 h, are natural timing machines. They are broadly distributed in living organisms, such as Neurospora, Drosophila, and mammals. The underlying natures of the rhythmic behavior have been explored by experimental and theoretical approaches. However, the global and physical natures of the oscillation under fluctuations are still not very clear. We developed a landscape and flux framework to explore the global stability and robustness of a circadian oscillation system. The potential landscape of the network is uncovered and has a global Mexican-hat shape. The height of the Mexican-hat provides a quantitative measure to evaluate the robustness and coherence of the oscillation. We found that in nonequilibrium dynamic systems, not only the potential landscape but also the probability flux are important to the dynamics of the system under intrinsic noise. Landscape attracts the systems down to the oscillation ring while flux drives the coherent oscillation on the ring. We also investigated the phase coherence and the entropy production rate of the system at different fluctuations and found that dissipations are less and the coherence is higher for larger number of molecules. We also found that the power spectrum of autocorrelation functions show resonance peak at the frequency of coherent oscillations. The peak is less prominent for smaller number of molecules and less barrier height and therefore can be used as another measure of stability of oscillations. As a consequence of nonzero probability flux, we show that the three-point correlations from the time traces show irreversibility, providing a possible way to explore the flux from the observations. Furthermore, we explored the escape time from the oscillation ring to outside at different molecular number. We found that when barrier height is higher, escape time is longer and phase coherence of oscillation is higher. Finally, we performed the global sensitivity analysis of the underlying parameters to find the key network wirings responsible for the stability of the oscillation system.

Connection between noise-induced symmetry breaking and an information-decoding function for intracellular networks

The biological function of noise-induced symmetry breaking (NISB) is still unclear even though it may potentially occur in noisy intracellular systems. In this work, I demonstrate that information decoding from a noisy signal is a potential biological function of NISB by revealing that NISB naturally emerges from an optimal information-decoding dynamics and that several intracellular networks can be identified with the information-decoding dynamics. I also propose a mean first passage time profile as a way to experimentally identify NISB.

On the intrinsic inevitability of cancer: from foetal to fatal attraction

The cracks in the paradigm of oncogenic mutations and somatic evolution as driving force of tumorigenesis, lucidly exposed by the dynamic heterogeneity of "cancer stem cells" or the diffuse results of cancer genome sequencing projects, indicate the need for a more encompassing theory of cancer that reaches beyond the current proximate explanations based on individual genetic pathways. One such integrative concept, derived from first principles of the dynamics of gene regulatory networks, is that cancerous cell states are attractor states, just like normal cell types are. Here we extend the concept of cancer attractors to illuminate a more profound property of cancer initiation: its inherent inevitability in the light of metazoan evolution. Using Waddington's Epigenetic Landscape as a conceptual aid, for which we present a mathematical and evolutionary foundation, we propose that cancer is intrinsically linked to ontogenesis and phylogenesis. This explanatory rather than enumerating review uses a formal argumentation structure that is atypical in modern experimental biology but may hopefully offer a new coherent perspective to reconcile many conflicts between new findings and the old thinking in the categories of linear oncogenic pathways.

Potential landscape and probabilistic flux of a predator prey network

Predator-prey system, as an essential element of ecological dynamics, has been recently studied experimentally with synthetic biology. We developed a global probabilistic landscape and flux framework to explore a synthetic predator-prey network constructed with two Escherichia coli populations. We developed a self consistent mean field method to solve multidimensional problem and uncovered the potential landscape with Mexican hat ring valley shape for predator-prey oscillations. The landscape attracts the system down to the closed oscillation ring. The probability flux drives the coherent oscillations on the ring. Both the landscape and flux are essential for the stable and coherent oscillations. The landscape topography characterized by the barrier height from the top of Mexican hat to the closed ring valley provides a quantitative measure of global stability of system. The entropy production rate for the energy dissipation is less for smaller environmental fluctuations or perturbations. The global sensitivity analysis based on the landscape topography gives specific predictions for the effects of parameters on the stability and function of the system. This may provide some clues for the global stability, robustness, function and synthetic network design.

Non-equilibrium phase transition in mesoscopic biochemical systems: from stochastic to nonlinear dynamics and beyond

A theory for an non-equilibrium phase transition in a driven biochemical network is presented. The theory is based on the chemical master equation (CME) formulation of mesoscopic biochemical reactions and the mathematical method of large deviations. The large deviations theory provides an analytical tool connecting the macroscopic multi-stability of an open chemical system with the multi-scale dynamics of its mesoscopic counterpart. It shows a corresponding non-equilibrium phase transition among multiple stochastic attractors. As an example, in the canonical phosphorylation-dephosphorylation system with feedback that exhibits bistability, we show that the non-equilibrium steady-state (NESS) phase transition has all the characteristics of classic equilibrium phase transition: Maxwell construction, a discontinuous first-derivative of the 'free energy function', Lee-Yang's zero for a generating function and a critical point that matches the cusp in nonlinear bifurcation theory. To the biochemical system, the mathematical analysis suggests three distinct timescales and needed levels of description. They are (i) molecular signalling, (ii) biochemical network nonlinear dynamics, and (iii) cellular evolution. For finite mesoscopic systems such as a cell, motions associated with (i) and (iii) are stochastic while that with (ii) is deterministic. Both (ii) and (iii) are emergent properties of a dynamic biochemical network.

Adiabatic and non-adiabatic non-equilibrium stochastic dynamics of single regulating genes

We explore the stochastic dynamics of self-regulative genes from fluctuations of molecular numbers and of on and off switching of gene states due to regulatory protein binding/unbinding to the genes. We found when the binding/unbinding is relatively fast (slow) compared with the synthesis/degradation of proteins in adiabatic (nonadiabatic) case the self-regulators can exhibit one or two peak (two peak) distributions in protein concentrations. This phenomena can also be quantified through Fano factors. This shows that even with the same architecture (topology of wiring) networks can have quite different functions (phenotypes), consistent with recent single molecule single gene experiments. We further found the inhibition and activation curves to be consistent with previous results (monomer binding) in adiabatic regime, but, in nonadiabatic regimes, show significantly different behaviors with previous predictions (monomer binding). Such difference is due to the slow (nonadiabatic) dimer binding/unbinding effect, and it has never been reported before. We derived the nonequilibrium phase diagrams of monostability and bistability in adiabatic and nonadiabatic regimes. We studied the dynamical trajectories of the self-regulating genes on the underlying landscapes from nonadiabatic to adiabatic limit, and we provide a global picture of understanding and show an analogy to the electron transfer problem. We studied the stability and robustness of the systems through mean first passage time (MFPT) from one peak (basin of attraction) to another and found both monotonic and nonmonotonic turnover behavior from adiabatic to nonadiabatic regimes. For the first time, we explore global dissipation by entropy production and the relation with binding/unbinding processes. Our theoretical predictions for steady state peaks, fano factos, inhibition/activation curves, and MFPT can be probed and tested from experiments.

Singularity analysis of the AKT signaling pathway reveals connections between cancer and metabolic diseases

Connections between cancer and metabolic diseases may consist in the complex network of interactions among a common set of biomolecules. By applying singularity and bifurcation analysis, the phenotypes constrained by the AKT signaling pathway are identified and mapped onto the parameter space, which include cancer and certain metabolic diseases. By considering physiologic properties (sensitivity, robustness and adaptivity) the AKT pathway must possess in order to efficiently sense growth factors and nutrients, the region of normal responses is located. To optimize these properties, the intracellular concentration of the AKT protein must be sufficiently high to saturate its enzymes; the strength of the positive feedback must be stronger than that of the negative feedback. The analysis illuminates the parameter space and reveals system-level mechanisms in regulating biological functions (cell growth, survival, proliferation and metabolism) and how their deregulation may lead to the development of diseases. The analytical expressions summarize the synergistic interactions among many molecules, which provides valuable insights into therapeutic interventions. In particular, a strategy for overcoming the limitations of mTOR inhibition is proposed for cancer therapy.

The potential landscape of genetic circuits imposes the arrow of time in stem cell differentiation

Differentiation from a multipotent stem or progenitor state to a mature cell is an essentially irreversible process. The associated changes in gene expression patterns exhibit time-directionality. This "arrow of time" in the collective change of gene expression across multiple stable gene expression patterns (attractors) is not explained by the regulated activation, the suppression of individual genes which are bidirectional molecular processes, or by the standard dynamical models of the underlying gene circuit which only account for local stability of attractors. To capture the global dynamics of this nonequilibrium system and gain insight in the time-asymmetry of state transitions, we computed the quasipotential landscape of the stochastic dynamics of a canonical gene circuit that governs branching cell fate commitment. The potential landscape reveals the global dynamics and permits the calculation of potential barriers between cell phenotypes imposed by the circuit architecture. The generic asymmetry of barrier heights indicates that the transition from the uncommitted multipotent state to differentiated states is inherently unidirectional. The model agrees with observations and predicts the extreme conditions for reprogramming cells back to the undifferentiated state.

The chemical master equation approach to nonequilibrium steady-state of open biochemical systems: linear single-molecule enzyme kinetics and nonlinear biochemical reaction networks

We develop the stochastic, chemical master equation as a unifying approach to the dynamics of biochemical reaction systems in a mesoscopic volume under a living environment. A living environment provides a continuous chemical energy input that sustains the reaction system in a nonequilibrium steady state with concentration fluctuations. We discuss the linear, unimolecular single-molecule enzyme kinetics, phosphorylation-dephosphorylation cycle (PdPC) with bistability, and network exhibiting oscillations. Emphasis is paid to the comparison between the stochastic dynamics and the prediction based on the traditional approach based on the Law of Mass Action. We introduce the difference between nonlinear bistability and stochastic bistability, the latter has no deterministic counterpart. For systems with nonlinear bistability, there are three different time scales: (a) individual biochemical reactions, (b) nonlinear network dynamics approaching to attractors, and (c) cellular evolution. For mesoscopic systems with size of a living cell, dynamics in (a) and (c) are stochastic while that with (b) is dominantly deterministic. Both (b) and (c) are emergent properties of a dynamic biochemical network; We suggest that the (c) is most relevant to major cellular biochemical processes such as epi-genetic regulation, apoptosis, and cancer immunoediting. The cellular evolution proceeds with transitions among the attractors of (b) in a "punctuated equilibrium" manner.

First order phase transition and hysteresis in a cell's maintenance of the membrane potential--An essential role for the inward potassium rectifiers

Hysteretic behavior is found experimentally in the transmembrane potential at low extracellular potassium in mouse lumbrical muscle cells. Adding isoprenaline to the external medium eliminates the bistable, hysteretic region. The system can be modeled mathematically and understood analytically with and without isoprenaline. Inward rectifying potassium channels appear to be essential for the bistability. Relations are derived to express the dimensions of the bistable area in terms of system parameters. The selective advantage and evolutionary origin of inward rectifying channels and hysteretic behavior is discussed.

Potential and flux landscapes quantify the stability and robustness of budding yeast cell cycle network

Studying the cell cycle process is crucial for understanding cell growth, proliferation, development, and death. We uncovered some key factors in determining the global robustness and function of the budding yeast cell cycle by exploring the underlying landscape and flux of this nonequilibrium network. The dynamics of the system is determined by both the landscape which attracts the system down to the oscillation orbit and the curl flux which drives the periodic motion on the ring. This global structure of landscape is crucial for the coherent cell cycle dynamics and function. The topography of the underlying landscape, specifically the barrier height separating basins of attractions, characterizes the capability of changing from one part of the system to another. This quantifies the stability and robustness of the system. We studied how barrier height is influenced by environmental fluctuations and perturbations on specific wirings of the cell cycle network. When the fluctuations increase, the barrier height decreases and the period and amplitude of cell cycle oscillation is more dispersed and less coherent. The corresponding dissipation of the system quantitatively measured by the entropy production rate increases. This implies that the system is less stable under fluctuations. We identified some key structural elements for wirings of the cell cycle network responsible for the change of the barrier height and therefore the global stability of the system through the sensitivity analysis. The results are in agreement with recent experiments and also provide new predictions.

Network thermodynamics in the post-genomic era

Network models have been used to study the underlying processes and principles of biological systems for decades, providing many insights into the complexity of life. Biological systems require a constant flow of free energy to drive these processes that operate away from thermodynamic equilibrium. With the advent of high-throughput omics technologies, more and more thermodynamic knowledge about the biological components, processes and their interactions are surfacing that we can integrate using large-scale biological network models. This allows us to ask many fundamental questions about these networks, such as, how far away from equilibrium must the reactions in a network be displaced in order to allow growth, or what are the possible thermodynamic objectives of the cell.

Stochastic bistability and bifurcation in a mesoscopic signaling system with autocatalytic kinase

Bistability is a nonlinear phenomenon widely observed in nature including in biochemical reaction networks. Deterministic chemical kinetics studied in the past has shown that bistability occurs in systems with strong (cubic) nonlinearity. For certain mesoscopic, weakly nonlinear (quadratic) biochemical reaction systems in a small volume, however, stochasticity can induce bistability and bifurcation that have no macroscopic counterpart. We report the simplest yet known reactions involving driven phosphorylation-dephosphorylation cycle kinetics with autocatalytic kinase. We show that the noise-induced phenomenon is correlated with free energy dissipation and thus conforms with the open-chemical system theory. A previous reported noise-induced bistability in futile cycles is found to have originated from the kinase synchronization in a bistable system with slow transitions, as reported here.

Robustness and coherence of a three-protein circadian oscillator: landscape and flux perspectives

Three-protein circadian oscillations in cyanobacteria sustain for weeks. To understand how cellular oscillations function robustly in stochastic fluctuating environments, we used a stochastic model to uncover two natures of circadian oscillation: the potential landscape related to steady-state probability distribution of protein concentrations; and the corresponding flux related to speed of concentration changes which drive the oscillations. The barrier height of escaping from the oscillation attractor on the landscape provides a quantitative measure of the robustness and coherence for oscillations against intrinsic and external fluctuations. The difference between the locations of the zero total driving force and the extremal of the potential provides a possible experimental probe and quantification of the force from curl flux. These results, correlated with experiments, can help in the design of robust oscillatory networks.

Thermodynamic limit of a nonequilibrium steady state: Maxwell-type construction for a bistable biochemical system

We show that the thermodynamic limit of a bistable phosphorylation-dephosphorylation cycle has a selection rule for the "more stable" macroscopic steady state. The analysis is akin to the Maxwell construction. Based on the chemical master equation approach, it is shown that, except at a critical point, bistability disappears in the stochastic model when fluctuation is sufficiently low but unneglectable. Onsager's Gaussian fluctuation theory applies to the unique macroscopic steady state. With an initial state in the basin of attraction of the "less stable" steady state, the deterministic dynamics obtained by the law of mass action is a metastable phenomenon. Stability and robustness in cell biology are emergent stochastic concepts.

Stochastic bifurcation, slow fluctuations, and bistability as an origin of biochemical complexity

We present a simple, unifying theory for stochastic biochemical systems with multiple time-scale dynamics that exhibit noise-induced bistability in an open-chemical environment, while the corresponding macroscopic reaction is unistable. Nonlinear stochastic biochemical systems like these are fundamentally different from classical systems in equilibrium or near-equilibrium steady state whose fluctuations are unimodal following Einstein-Onsager-Lax-Keizer theory. We show that noise-induced bistability in general arises from slow fluctuations, and a pitchfork bifurcation occurs as the rate of fluctuations decreases. Since an equilibrium distribution, due to detailed balance, has to be independent of changes in time-scale, the bifurcation is necessarily a driven phenomenon. As examples, we analyze three biochemical networks of currently interest: self-regulating gene, stochastic binary decision, and phosphorylation-dephosphorylation cycle with fluctuating kinase. The implications of bistability to biochemical complexity are discussed.

Robustness, dissipations and coherence of the oscillation of circadian clock: potential landscape and flux perspectives

Finding the global probabilistic nature of a non-equilibrium circadian clock is essential for addressing important issues of robustness and function. We have uncovered the underlying potential energy landscape of a simple cyanobacteria biochemical network, and the corresponding flux which is the driving force for the oscillation. We found that the underlying potential landscape for the oscillation in the presence of small statistical fluctuations is like an explicit ring valley or doughnut shape in the three dimensional protein concentration space. We found that the barrier height separating the oscillation ring and other area is a quantitative measure of the oscillation robustness and decreases when the fluctuations increase. We also found that the entropy production rate characterizing the dissipation or heat loss decreases as the fluctuations decrease. In addition, we found that, as the fluctuations increase, the period and the amplitude of the oscillations is more dispersed, and the phase coherence decreases. We also found that the properties from exploring the effects of the inherent chemical rate parameters on the robustness. Our approach is quite general and can be applied to other oscillatory cellular network.

PACS Codes: 87.18.-h, 87.18.Vf, 87.18.Yt