Kinetic timing: a novel mechanism that improves the accuracy of GTPase timers in endosome fusion and other biological processes

Molecular mechanisms of precise timing in cell lysis

Many biological systems exhibit precise timing of events, and one of the most known examples is cell lysis, which is a process of breaking bacterial host cells in the virus infection cycle. However, the underlying microscopic picture of precise timing remains not well understood. We present a novel theoretical approach to explain the molecular mechanisms of effectively deterministic dynamics in biological systems. Our hypothesis is based on the idea of stochastic coupling between relevant underlying biophysical and biochemical processes that lead to noise cancellation. To test this hypothesis, we introduced a minimal discrete-state stochastic model to investigate how holin proteins produced by bacteriophages break the inner membranes of gram-negative bacteria. By explicitly solving this model, the dynamic properties of cell lysis are fully evaluated, and theoretical predictions quantitatively agree with available experimental data for both wild-type and holin mutants. It is found that the observed threshold-like behavior is a result of the balance between holin proteins entering the membrane and leaving the membrane during the lysis. Theoretical analysis suggests that the cell lysis achieves precise timing for wild-type species by maximizing the number of holins in the membrane and narrowing their spatial distribution. In contrast, for mutated species, these conditions are not satisfied. Our theoretical approach presents a possible molecular picture of precise dynamic regulation in intrinsically random biological processes.

Universal energy-accuracy tradeoffs in nonequilibrium cellular sensing

We combine stochastic thermodynamics, large deviation theory, and information theory to derive fundamental limits on the accuracy with which single cell receptors can estimate external concentrations. As expected, if the estimation is performed by an ideal observer of the entire trajectory of receptor states, then no energy consuming nonequilibrium receptor that can be divided into bound and unbound states can outperform an equilibrium two-state receptor. However, when the estimation is performed by a simple observer that measures the fraction of time the receptor is bound, we derive a fundamental limit on the accuracy of general nonequilibrium receptors as a function of energy consumption. We further derive and exploit explicit formulas to numerically estimate a Pareto-optimal tradeoff between accuracy and energy. We find this tradeoff can be achieved by nonuniform ring receptors with a number of states that necessarily increases with energy. Our results yield a thermodynamic uncertainty relation for the time a physical system spends in a pool of states and generalize the classic Berg-Purcell limit [H. C. Berg and E. M. Purcell, Biophys. J. 20, 193 (1977)0006-349510.1016/S0006-3495(77)85544-6] on cellular sensing along multiple dimensions.

The Rho GTPase signalling pathway in urothelial carcinoma

Urothelial carcinoma remains a clinical challenge: non-muscle-invasive disease has a high rate of recurrence and risk of progression, and outcomes for patients with advanced disease are poor, owing to a lack of effective systemic therapies. The Rho GTPase family of enzymes was first identified >30 years ago and contains >20 members, which are divided into eight subfamilies: Cdc42, Rac, Rho, RhoUV, RhoBTB, RhoDF, RhoH, and Rnd. Rho GTPases are molecular on-off switches, which are increasingly being understood to have a critical role in a number of cellular processes, including cell migration, cell polarity, cell adhesion, cell cycle progression, and regulation of the cytoskeleton. This switch is an evolutionarily conserved system in which GTPases alternate between GDP-bound (inactive) and GTP-bound (active) forms. The activities of these Rho GTPases are many, context-dependent, and regulated by a number of proteins that are being progressively elucidated. Aberrations of the Rho GTPase signalling pathways have been implicated in various malignancies, including urothelial carcinoma, and understanding of the role of Rho GTPases in these diseases is increasing. This signalling pathway has the potential for therapeutic targeting in urothelial carcinoma. Research in this area is nascent, and much work is necessary before current laboratory-based research can be translated into the clinic.

Entropy production in mesoscopic stochastic thermodynamics: nonequilibrium kinetic cycles driven by chemical potentials, temperatures, and mechanical forces

Nonequilibrium thermodynamics (NET) investigates processes in systems out of global equilibrium. On a mesoscopic level, it provides a statistical dynamic description of various complex phenomena such as chemical reactions, ion transport, diffusion, thermochemical, thermomechanical and mechanochemical fluxes. In the present review, we introduce a mesoscopic stochastic formulation of NET by analyzing entropy production in several simple examples. The fundamental role of nonequilibrium steady-state cycle kinetics is emphasized. The statistical mechanics of Onsager's reciprocal relations in this context is elucidated. Chemomechanical, thermomechanical, and enzyme-catalyzed thermochemical energy transduction processes are discussed. It is argued that mesoscopic stochastic NET in phase space provides a rigorous mathematical basis of fundamental concepts needed for understanding complex processes in chemistry, physics and biology. This theory is also relevant for nanoscale technological advances.

Dispersion for two classes of random variables: general theory and application to inference of an external ligand concentration by a cell

We derive expressions for the dispersion for two classes of random variables in Markov processes. Random variables such as current and activity pertain to the first class, which is composed of random variables that change whenever a jump in the stochastic trajectory occurs. The second class corresponds to the time the trajectory spends in a state (or cluster of states). While the expression for the first class follows straightforwardly from known results in the literature, we show that a similar formalism can be used to derive an expression for the second class. As an application, we use this formalism to analyze a cellular two-component network estimating an external ligand concentration. The uncertainty related to this external concentration is calculated by monitoring different random variables related to an internal protein. We show that, inter alia, monitoring the time spent in the phosphorylated state of the protein leads to a finite uncertainty only if there is dissipation, whereas the uncertainty obtained from the activity of the transitions of the internal protein can reach the Berg-Purcell limit even in equilibrium.

Reinitiation enhances reliable transcriptional responses in eukaryotes

Gene transcription is a noisy process carried out by the transcription machinery recruited to the promoter. Noise reduction is a fundamental requirement for reliable transcriptional responses which in turn are crucial for signal transduction. Compared with the relatively simple transcription initiation in prokaryotes, eukaryotic transcription is more complex partially owing to its additional reinitiation mechanism. By theoretical analysis, we showed that reinitiation reduces noise in eukaryotic transcription independent of the transcription level. Besides, a higher reinitiation rate enables a stable scaffold complex an advantage in noise reduction. Finally, we showed that the coupling between scaffold formation and transcription can further reduce transcription noise independent of the transcription level. Furthermore, compared with the reinitiation mechanism, the noise reduction effect of the coupling can be of more significance in the case that the transcription level is low and the intrinsic noise dominates. Our results uncover a mechanistic route which eukaryotes may use to facilitate a more reliable response in the noisy transcription process.

Promoter-mediated transcriptional dynamics

Genes in eukaryotic cells are typically regulated by complex promoters containing multiple binding sites for a variety of transcription factors, but how promoter dynamics affect transcriptional dynamics has remained poorly understood. In this study, we analyze gene models at the transcriptional regulation level, which incorporate the complexity of promoter structure (PS) defined as transcriptional exits (i.e., ON states of the promoter) and the transition pattern (described by a matrix consisting of transition rates among promoter activity states). We show that multiple exits of transcription are the essential origin of generating multimodal distributions of mRNA, but promoters with the same transition pattern can lead to multimodality of different modes, depending on the regulation of transcriptional factors. In turn, for similar mRNA distributions in the models, the mean ON or OFF time distributions may exhibit different characteristics, thus providing the supplemental information on PS. In addition, we demonstrate that the transcriptional noise can be characterized by a nonlinear function of mean ON and OFF times. These results not only reveal essential characteristics of promoter-mediated transcriptional dynamics but also provide signatures useful for inferring PS based on characteristics of transcriptional outputs.

SUMO as a nuclear hormone receptor effector: New insights into combinatorial transcriptional regulation

Animal development is driven by robust, cell-specific gene expression programs. Understanding mechanistically how a single transcription factor (TF) can govern distinct programs with exquisite precision is a major challenge. We view TFs as signal integrators, taking information from co-regulator interactions, post-translational modifications, other transcription factors, chromatin state, DNA sequence and in some cases, specific noncovalent ligands, to determine the collection of genes regulated by a TF at any given time. Here, we describe a reductionist approach to combinatorial transcriptional regulation, focusing on a single C. elegans TF, the nuclear hormone receptor NHR-25, and a single post-translational modification, SUMO. We suggest that the ratio of sumoylated to unsumoylated NHR-25 could specify a switch-like cell-fate decision during vulval development. Direct examination of this "SUMO ratio" in vivo is challenging and we discuss possible solutions going forward. We also consider how sumoylation of multiple substrates might be coordinated during vulval development. Finally, we note that iteration of this approach could leverage our sumoylation findings to define the roles of other effectors of NHR-25 in the developing vulva and in other tissues.

Reconciling the concurrent fast and slow cycling of proteins on gene promoters

During gene transcription, proteins appear to cycle on and off some gene promoters with both long (tens of minutes) and short periods (no more than several minutes). The essence of these phenomena still remains unclear. Here, we propose a stochastic model for the state evolution of promoters in terms of DNA-protein interactions. The model associates the characteristics of microscopic molecular interactions with macroscopic measurable quantities. Through theoretical derivation, we reconcile the contradictory viewpoints on the concurrent fast and slow cycling; both the cycling phenomena are further reproduced by fitting simulation results to the experimental data on the pS2 gene. Our results suggest that the fast cycling dictates how the proteins behave on the promoter and that stable binding hardly occurs. Different kinds of proteins rapidly bind/unbind the promoter at distinct transcriptional stages fulfilling specific functions; this feature is essentially manifested as the slow cycling of proteins when detected by chromatin immunoprecipitation assays. Thus, the slow cycling represents neither stable binding of proteins nor external modulation of the fast cycling. This work also reveals the relationship between the essence and measurement of transcriptional dynamics.

Polyubiquitin chain assembly and organization determine the dynamics of protein activation and degradation

Protein degradation via ubiquitination is a major proteolytic mechanism in cells. Once a protein is destined for degradation, it is tagged by multiple ubiquitin (Ub) molecules. The synthesized polyubiquitin chains can be recognized by the 26S proteosome where proteins are degraded. These chains form through multiple ubiquitination cycles that are similar to multi-site phosphorylation cycles. As kinases and phosphatases, two opposing enzymes (E3 ligases and deubiquitinases DUBs) catalyze (de)ubiquitination cycles. Although multi-ubiquitination cycles are fundamental mechanisms of controlling protein concentrations within a cell, their dynamics have never been explored. Here, we fill this knowledge gap. We show that under permissive physiological conditions, the formation of polyubiquitin chain of length greater than two and subsequent degradation of the ubiquitinated protein, which is balanced by protein synthesis, can display bistable, switch-like responses. Interestingly, the occurrence of bistability becomes pronounced, as the chain grows, giving rise to “all-or-none” regulation at the protein levels. We give predictions of protein distributions under bistable regime awaiting experimental verification. Importantly, we show for the first time that sustained oscillations can robustly arise in the process of formation of ubiquitin chain, largely due to the degradation of the target protein. This new feature is opposite to the properties of multi-site phosphorylation cycles, which are incapable of generating oscillation if the total abundance of interconverted protein forms is conserved. We derive structural and kinetic constraints for the emergence of oscillations, indicating that a competition between different substrate forms and the E3 and DUB is critical for oscillation. Our work provides the first detailed elucidation of the dynamical features brought about by different molecular setups of the polyubiquitin chain assembly process responsible for protein degradation.

Statistics and Related Topics in Single-Molecule Biophysics

Since the universal acceptance of atoms and molecules as the fundamental constituents of matter in the early twentieth century, molecular physics, chemistry and molecular biology have all experienced major theoretical breakthroughs. To be able to actually "see" biological macromolecules, one at a time in action, one has to wait until the 1970s. Since then the field of single-molecule biophysics has witnessed extensive growth both in experiments and theory. A distinct feature of single-molecule biophysics is that the motions and interactions of molecules and the transformation of molecular species are necessarily described in the language of stochastic processes, whether one investigates equilibrium or nonequilibrium living behavior. For laboratory measurements following a biological process, if it is sampled over time on individual participating molecules, then the analysis of experimental data naturally calls for the inference of stochastic processes. The theoretical and experimental developments of single-molecule biophysics thus present interesting questions and unique opportunity for applied statisticians and probabilists. In this article, we review some important statistical developments in connection to single-molecule biophysics, emphasizing the application of stochastic-process theory and the statistical questions arising from modeling and analyzing experimental data.

Cell type determines the light-induced endosomal escape kinetics of multifunctional mesoporous silica nanoparticles

We investigated uptake and individual endosome lysis events in fibroblast, normal, and carcinoma cell lines using a colloidal mesoporous silica (CMS) nanoparticle (NP)-based reporter system with a covalently attached photosensitizer. Endosome lysis was induced through the activation of protoporphyrin IX (PpIX). Surprisingly, this release-on-demand system resulted in more broadly distributed lysis times than expected, particularly for Renca, a renal carcinoma cell line. An analysis of the NP load per endosome, endosome size, and uptake characteristics indicate that Renca cells not only take up a lower amount of NPs in comparison with the fibroblast cells but also have larger endosomes and a lower NP load per endosome. We then created a stochastic model detailing steps downstream of uptake to understand how much factors that cannot be directly measured, such as variations in the PpIX load per NP, affect the lysis time distributions. Model results indicate that the distributions are primarily determined by the endosome properties, rather than variations across NPs.

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.

GTP: gatekeeper for autophagy

Chaperone-mediated autophagy (CMA) is involved in cellular quality control, and in the response to stress, deterioration in CMA contributes to the aging phenotype and to various disorders. The paper by Bandyopadhyay et al. in this issue of Molecular Cell demonstrates the role of GTP binding by elongation factor 1 alpha (EF1alpha) in regulating the lysosomal uptake of CMA substrates.

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.

Fast changes in the functional status of release sites during short-term plasticity: involvement of a frequency-dependent bypass of Rac at Aplysia synapses

Synaptic transmission can be described as a stochastic quantal process defined by three main parameters: N, the number of functional release sites; P, the release probability; and Q, the quantum of response. Many changes in synaptic strength that are observed during expression of short term plasticity rely on modifications in P. Regulation of N has been also suggested. We have investigated at identified cholinergic inhibitory Aplysia synapses the cellular mechanism of post-tetanic potentiation (PTP) expressed under control conditions or after N has been depressed by applying lethal toxin (LT) from Clostridium sordellii or tetanus toxin (TeNT). The analysis of the Ca(2+) dependency, paired-pulse ratio and variance to mean amplitude relationship of the postsynaptic responses elicited at distinct extracellular [Ca(2+)]/[Mg(2+)] elicited during control post-tetanic potentiation (PTP(cont)) indicated that PTP(cont) is mainly driven by an increase in release probability, P. The PTP expressed at TeNT-treated synapses (PTP(TeNT)) was found to be similar to PTP(cont), but scaled to the extent of reduction in N produced by TeNT. Despite LT inducing a decrease in N as TeNT does, the PTP expressed at LT-treated synapses (PTP(LT)) was characterized by exceptionally large amplitude and bi-exponential time course, as compared to PTP(cont) or the PTP(TeNT). Analysis of the Ca(2+) dependency of PTP(LT), paired-pulse ratio and fluctuations in amplitude of the postsynaptic responses elicited during PTP(LT) or the variance to mean amplitude relationship of time-locked postsynaptic responses in a series of subsequent PTP(LT) indicated that an N-driven change is involved in the early phase (1 s time scale) of PTP(LT), while at a later stage PTP(LT) is composed of both N and P increases. Our results suggest that fast switching on of the functional status of the release sites occurs also during the early events of PTP(cont). The early N-driven phase of PTP(LT) is likely to be a functional recovery of the release sites silenced by Rac inactivation. This effect did not appear to result from reversion of LT inhibitory action but from bypassing the step regulated by Rac. Altogether the data suggest that Rac and the intracellular pathway which allows the bypassing of Rac are key players in new forms of short-term plasticity that rely on fast, activity-dependent changes in the functional status of the release sites.

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.

Role of Rab5 in insulin receptor-mediated endocytosis and signaling

Activated insulin receptors recruit various intracellular proteins leading to signal generation and endocytic trafficking. Although activated receptors are rapidly internalized into the endocytic compartment and subsequently degraded in lysosomes, the linkage between insulin receptor signaling and endocytosis is not well understood. This study utilizes both overexpression and depletion of Rab5 proteins to show that they play a critical role in both insulin-stimulated fluid phase and receptor-mediated endocytosis. Specifically, Rab5:WT and Rab5:Q79L (a GTP-hydrolysis defective mutant) enhance both types of endocytosis in response to insulin, while Rab5:S34N (a GTP-binding defective mutant) has the opposite effect. Morphological analysis indicates that both Rab5 and insulin receptor are found on early endosomes, but not at the plasma membrane. In addition, expression of Rab5:WT and Rab5:Q79L enhance both Erk1/2 and Akt activation without affecting JN- and p38-kinase activities, while the expression of Rab5:S34N blocks both Erk1/2 and Akt activation. Consistent with these observations, DNA synthesis is also altered by the expression of Rab5:S34N. Taken together, these results demonstrate that Rab5 is required for insulin receptor membrane trafficking and signaling.

Strategies and Tactics in Multiscale Modeling of Cell-to-Organ Systems

Modeling is essential to integrating knowledge of human physiology. Comprehensive self-consistent descriptions expressed in quantitative mathematical form define working hypotheses in testable and reproducible form, and though such models are always "wrong" in the sense of being incomplete or partly incorrect, they provide a means of understanding a system and improving that understanding. Physiological systems, and models of them, encompass different levels of complexity. The lowest levels concern gene signaling and the regulation of transcription and translation, then biophysical and biochemical events at the protein level, and extend through the levels of cells, tissues and organs all the way to descriptions of integrated systems behavior. The highest levels of organization represent the dynamically varying interactions of billions of cells. Models of such systems are necessarily simplified to minimize computation and to emphasize the key factors defining system behavior; different model forms are thus often used to represent a system in different ways. Each simplification of lower level complicated function reduces the range of accurate operability at the higher level model, reducing robustness, the ability to respond correctly to dynamic changes in conditions. When conditions change so that the complexity reduction has resulted in the solution departing from the range of validity, detecting the deviation is critical, and requires special methods to enforce adapting the model formulation to alternative reduced-form modules or decomposing the reduced-form aggregates to the more detailed lower level modules to maintain appropriate behavior. The processes of error recognition, and of mapping between different levels of model complexity and shifting the levels of complexity of models in response to changing conditions, are essential for adaptive modeling and computer simulation of large-scale systems in reasonable time.

Multiscale modeling of cardiac cellular energetics

Multiscale modeling is essential to integrating knowledge of human physiology starting from genomics, molecular biology, and the environment through the levels of cells, tissues, and organs all the way to integrated systems behavior. The lowest levels concern biophysical and biochemical events. The higher levels of organization in tissues, organs, and organism are complex, representing the dynamically varying behavior of billions of cells interacting together. Models integrating cellular events into tissue and organ behavior are forced to resort to simplifications to minimize computational complexity, thus reducing the model's ability to respond correctly to dynamic changes in external conditions. Adjustments at protein and gene regulatory levels shortchange the simplified higher-level representations. Our cell primitive is composed of a set of subcellular modules, each defining an intracellular function (action potential, tricarboxylic acid cycle, oxidative phosphorylation, glycolysis, calcium cycling, contraction, etc.), composing what we call the "eternal cell," which assumes that there is neither proteolysis nor protein synthesis. Within the modules are elements describing each particular component (i.e., enzymatic reactions of assorted types, transporters, ionic channels, binding sites, etc.). Cell subregions are stirred tanks, linked by diffusional or transporter-mediated exchange. The modeling uses ordinary differential equations rather than stochastic or partial differential equations. This basic model is regarded as a primitive upon which to build models encompassing gene regulation, signaling, and long-term adaptations in structure and function. During simulation, simpler forms of the model are used, when possible, to reduce computation. However, when this results in error, the more complex and detailed modules and elements need to be employed to improve model realism. The processes of error recognition and of mapping between different levels of model form complexity are challenging but are essential for successful modeling of large-scale systems in reasonable time. Currently there is to this end no established methodology from computational sciences.

Nonequilibrium thermodynamics and nonlinear kinetics in a cellular signaling switch

We develop a rigorous nonequilibrium thermodynamics for an open system of nonlinear biochemical reactions responsible for cell signal processing. We show that the quality of the biological switch consisting of a phosphorylation-dephosphorylation cycle, such as those in protein kinase cascade, is controlled by the available intracellular free energy from the adenosine triphosphate (ATP) hydrolysis in vivo: DeltaG=k(B)Tln(([ATP]/K(eq)[ADP]), where K(eq) is the equilibrium constant. The model reveals the correlation between the performance of the switch and the level of DeltaG. The result demonstrates the importance of nonequilibrium thermodynamics in analyzing biological information processing, provides its energetic cost, establishes an interplay between signal transduction and energy metabolism in cells, and suggests a biological function for phosphoenergetics in the ubiquitous phosphorylation signaling.

GTP hydrolysis mechanism of Ras-like GTPases

The Ras-like GTPases regulate diverse cellular functions via the chemical cycle of binding and hydrolyzing GTP molecules. They alternate between GTP- and GDP-bound conformations. The GTP-bound conformation is biologically active and promotes a cellular function, such as signal transduction, cytoskeleton organization, protein synthesis/translocation, or a membrane budding/fusion event. GTP hydrolysis turns off the GTPase switch by converting it to the inactive GDP-bound conformation. The fundamental GTP hydrolysis mechanism by these GTPases has generated considerable interest over the last two decades but remained to be firmly established. This review provides an update on the catalytic mechanism with discussions on recent developments from kinetic, structural, and model studies in the context of the various GTP hydrolysis models proposed over the years.

Thermodynamic and kinetic analysis of sensitivity amplification in biological signal transduction

Based on a thermodynamic analysis of the kinetic model for the protein phosphorylation-dephosphorylation cycle, we study the ATP (or GTP) energy utilization of this ubiquitous biological signal transduction process. It is shown that the free energy from hydrolysis inside cells, DeltaG (phosphorylation potential), controls the amplification and sensitivity of the switch-like cellular module; the response coefficient of the sensitivity amplification approaches the optimal 1 and the Hill coefficient increases with increasing DeltaG. We discover that zero-order ultrasensitivity is mathematically equivalent to allosteric cooperativity. Furthermore, we show that the high amplification in ultrasensitivity is mechanistically related to the proofreading kinetics for protein biosynthesis. Both utilize multiple kinetic cycles in time to gain temporal cooperativity, in contrast to allosteric cooperativity that utilizes multiple subunits in a protein.

Fluorescence correlation spectroscopy with high-order and dual-color correlation to probe nonequilibrium steady states

In living cells, biochemical reaction networks often function in nonequilibrium steady states. Under these conditions, the networks necessarily have cyclic reaction kinetics that are maintained by sustained constant input and output, i.e., pumping. To differentiate this state from an equilibrium state without flux, we propose a microscopic method based on concentration fluctuation measurements, via fluorescence correlation spectroscopy, and statistical analyses of high-order correlations and cross correlations beyond the standard fluorescence correlation spectroscopy autocorrelation. We show that, for equilibrium systems with time reversibility, the correlation functions possess certain symmetries, the violation of which is a measure of steady-state fluxes in reaction cycles. This result demonstrates the theoretical basis for experimentally measuring reaction fluxes in a biochemical network in situ and the importance of single-molecule measurements in providing fundamental information on nonequilibrium steady-states in biochemistry.

Single-molecule enzymology: stochastic Michaelis-Menten kinetics

We provide a stochastic analysis of single-molecule enzymatic reactions that follow Michaelis-Menten kinetics. We show that this system can exhibit oscillatory behavior in the non-equilibrium steady-state at appropriate substrate concentrations. The stochastic model includes both enzyme dynamics and substrate turnover kinetics. The relationship between the probability of substrate survival and the time-correlation of enzyme conformation trajectories is discussed. Deterministic kinetics at large substrate concentrations are obtained as a limit of the stochastic model. We suggest that in addition to fluctuating enzyme conformation, the stochastic nature of substrate concentration fluctuations is another possible source of the complex behavior of single-molecule enzyme kinetics.