Single-molecule enzymology: stochastic Michaelis-Menten kinetics

Spatial models of pattern formation during phagocytosis

Phagocytosis, the biological process in which cells ingest large particles such as bacteria, is a key component of the innate immune response. Fcγ receptor (FcγR)-mediated phagocytosis is initiated when these receptors are activated after binding immunoglobulin G (IgG). Receptor activation initiates a signaling cascade that leads to the formation of the phagocytic cup and culminates with ingestion of the foreign particle. In the experimental system termed "frustrated phagocytosis", cells attempt to internalize micropatterned disks of IgG. Cells that engage in frustrated phagocytosis form "rosettes" of actin-enriched structures called podosomes around the IgG disk. The mechanism that generates the rosette pattern is unknown. We present data that supports the involvement of Cdc42, a member of the Rho family of GTPases, in pattern formation. Cdc42 acts downstream of receptor activation, upstream of actin polymerization, and is known to play a role in polarity establishment. Reaction-diffusion models for GTPase spatiotemporal dynamics exist. We demonstrate how the addition of negative feedback and minor changes to these models can generate the experimentally observed rosette pattern of podosomes. We show that this pattern formation can occur through two general mechanisms. In the first mechanism, an intermediate species forms a ring of high activity around the IgG disk, which then promotes rosette organization. The second mechanism does not require initial ring formation but relies on spatial gradients of intermediate chemical species that are selectively activated over the IgG patch. Finally, we analyze the models to suggest experiments to test their validity.

Computational Aspects of Single-Molecule Kinetics for Coupled Catalytic Cycles: A Spectral Analysis

Catalysis from single active sites is analyzed using methods developed from single-molecule kinetics. Using a stochastic Markov-state description, the observable properties of general catalytic networks of reactions are expressed using an eigenvalue decomposition of the transition matrix for the Markov process. By the use of a sensitivity analysis, the necessary eigenvalues and eigenvectors are related to the energies of controlling barriers and wells located along the reaction routes. A generalization of the energetic span theory allows the eigenvalues to be computed from several activation energies corresponding to distinct barrier-well pairings. The formalism is demonstrated for model problems and for a physically realistic mechanism for an alkene hydrogenation reaction on a single-atom catalyst. The spectral analysis permits a hierarchy of timescales to be identified from the single-molecule signal, which correspond to specific relaxation modes in the network.

Reaction-path statistical mechanics of enzymatic kinetics

We introduce a reaction-path statistical mechanics formalism based on the principle of large deviations to quantify the kinetics of single-molecule enzymatic reaction processes under the Michaelis-Menten mechanism, which exemplifies an out-of-equilibrium process in the living system. Our theoretical approach begins with the principle of equal a priori probabilities and defines the reaction path entropy to construct a new nonequilibrium ensemble as a collection of possible chemical reaction paths. As a result, we evaluate a variety of path-based partition functions and free energies by using the formalism of statistical mechanics. They allow us to calculate the timescales of a given enzymatic reaction, even in the absence of an explicit boundary condition that is necessary for the equilibrium ensemble. We also consider the large deviation theory under a closed-boundary condition of the fixed observation time to quantify the enzyme-substrate unbinding rates. The result demonstrates the presence of a phase-separation-like, bimodal behavior in unbinding events at a finite timescale, and the behavior vanishes as its rate function converges to a single phase in the long-time limit.

Biosensors based on peptide exposure show single molecule conformations in live cells

We describe an approach to study the conformation of individual proteins during single particle tracking (SPT) in living cells. "Binder/tag" is based on incorporation of a 7-mer peptide (the tag) into a protein where its solvent exposure is controlled by protein conformation. Only upon exposure can the peptide specifically interact with a reporter protein (the binder). Thus, simple fluorescence localization reflects protein conformation. Through direct excitation of bright dyes, the trajectory and conformation of individual proteins can be followed. Simple protein engineering provides highly specific biosensors suitable for SPT and FRET. We describe tagSrc, tagFyn, tagSyk, tagFAK, and an orthogonal binder/tag pair. SPT showed slowly diffusing islands of activated Src within Src clusters and dynamics of activation in adhesions. Quantitative analysis and stochastic modeling revealed in vivo Src kinetics. The simplicity of binder/tag can provide access to diverse proteins.

Steady-state joint distribution for first-order stochastic reaction kinetics

While the analytical solution for the marginal distribution of a stochastic chemical reaction network has been extensively studied, its joint distribution, i.e., the solution of a high-dimensional chemical master equation, has received much less attention. Here we develop an alternative method of computing the exact joint distributions of a wide class of first-order stochastic reaction systems in steady-state conditions. The effectiveness of our method is validated by applying it to four gene expression models of biological significance, including models with 2A peptides, nascent mRNA, gene regulation, translational bursting, and alternative splicing.

Modelling single-molecule kinetics of helicase translocation using high-resolution nanopore tweezers (SPRNT)

Single-molecule picometer resolution nanopore tweezers (SPRNT) is a technique for monitoring the motion of individual enzymes along a nucleic acid template at unprecedented spatiotemporal resolution. We review the development of SPRNT and the application of single-molecule kinetics theory to SPRNT data to develop a detailed model of helicase motion along a single-stranded DNA substrate. In this review, we present three examples of questions SPRNT can answer in the context of the Superfamily 2 helicase Hel308. With Hel308, SPRNT's spatiotemporal resolution enables resolution of two distinct enzymatic substates, one which is dependent upon ATP concentration and one which is ATP independent. By analyzing dwell-time distributions and helicase back-stepping, we show, in detail, how SPRNT can be used to determine the nature of these observed steps. We use dwell-time distributions to discern between three different possible models of helicase backstepping. We conclude by using SPRNT's ability to discern an enzyme's nucleotide-specific location along a DNA strand to understand the nature of sequence-specific enzyme kinetics and show that the sequence within the helicase itself affects both step dwell-time and backstepping probability while translocating on single-stranded DNA.

Motion magnification analysis of microscopy videos of biological cells

It is well recognized that isolated cardiac muscle cells beat in a periodic manner. Recently, evidence indicates that other, non-muscle cells, also perform periodic motions that are either imperceptible under conventional lab microscope lens or practically not easily amenable for analysis of oscillation amplitude, frequency, phase of movement and its direction. Here, we create a real-time video analysis tool to visually magnify and explore sub-micron rhythmic movements performed by biological cells and the induced movements in their surroundings. Using this tool, we suggest that fibroblast cells perform small fluctuating movements with a dominant frequency that is dependent on their surrounding substrate and its stiffness.

Words of advice: teaching enzyme kinetics

Enzymology is concerned with the study of enzyme structure, function, regulation and kinetics. It is an interdisciplinary subject that can be treated as an exclusive sphere of exhaustive inquiry within mathematical, physico-chemical and biological sciences. Hence, teaching of enzymology, in general, and enzyme kinetics, in particular, should be undertaken in an interdisciplinary manner for a holistic appreciation of this subject. Further, analogous examples from everyday life should form an integral component of the teaching for an intuitive grasp of the subject matter. Furthermore, simulation-based appreciation of enzyme kinetics should be preferred over simplifying assumptions and approximations of traditional enzyme kinetics teaching. In this Words of Advice, I outline the domain depth of enzymology across the various disciplines and provide initial ideas on how appropriate analogies can provide firm insights into the subject. Further, I demonstrate how an intuitive feel for the subject can help not only in grasping abstract concepts but also extending it in experimental design and subsequent interpretation. Use of simulations in grasping complex concepts is also advocated given the advantages this medium offers over traditional approaches involving images and molecular models. Furthermore, I discuss the merits of incorporating the historical backdrop of major discoveries in enzymological teaching. We, at AstraZeneca, have experimented with this approach with the desired outcome of generating interest in the subject from people practising diverse disciplines.

Quantifying the flux as the driving force for nonequilibrium dynamics and thermodynamics in non-Michaelis-Menten enzyme kinetics

The driving force for active physical and biological systems is determined by both the underlying landscape and nonequilibrium curl flux. While landscape can be experimentally quantified from the histograms of the collected real-time trajectories of the observables, quantifying the experimental flux remains challenging. In this work, we studied the single-molecule enzyme dynamics of horseradish peroxidase with dihydrorhodamine 123 and hydrogen peroxide (H2O2) as substrates. Surprisingly, significant deviations in the kinetics from the conventional Michaelis-Menten reaction rate were observed. Instead of a linear relationship between the inverse of the enzyme kinetic rate and the inverse of substrate concentration, a nonlinear relationship between the two emerged. We identified nonequilibrium flux as the origin of such non-Michaelis-Menten enzyme rate behavior. Furthermore, we quantified the nonequilibrium flux from experimentally obtained fluorescence correlation spectroscopy data and showed this flux to led to the deviations from the Michaelis-Menten kinetics. We also identified and quantified the nonequilibrium thermodynamic driving forces as the chemical potential and entropy production for such non-Michaelis-Menten kinetics. Moreover, through isothermal titration calorimetry measurements, we identified and quantified the origin of both nonequilibrium dynamic and thermodynamic driving forces as the heat absorbed (energy input) into the enzyme reaction system. Furthermore, we showed that the nonequilibrium driving forces led to time irreversibility through the difference between the forward and backward directions in time and high-order correlations were associated with the deviations from Michaelis-Menten kinetics. This study provided a general framework for experimentally quantifying the dynamic and thermodynamic driving forces for nonequilibrium systems.

A new Michaelis-Menten equation valid everywhere multi-scale dynamics prevails

The Michaelis-Menten reaction scheme is among the most influential models in the field of biochemistry, since it led to a very popular expression for the rate of an enzyme-catalysed reaction. After the realisation that this expression is valid in a limited region of the parameter space, two additional expressions were later introduced. The range of validity of these three expressions has been studied thoroughly, since the significance of a reliable rate is not based only on the accuracy of its predictive abilities but also on the physical insight that is acquired in the process of its construction. Here a new expression for the rate is introduced that is valid in practically the full parameter space and reduces to the expressions in the literature, when considering appropriate limits. The new expression is produced by employing algorithmic tools for asymptotic analysis, so that its construction is not hindered by the dimensional form and the complexity of the full model or by a wide parameter range of interest. These tools can be employed for the derivation of enzyme rate expressions of much more complex kinetics mechanisms.

High-temperature ratchets driven by deterministic and stochastic fluctuations

We consider the overdamped dynamics of a Brownian particle in an arbitrary spatial periodic and time-dependent potential on the basis of an exact solution for the probability density in the form of a power series in the inverse friction coefficient. The expression for the average velocity of a Brownian ratchet is simplified in the high-temperature consideration when only the first terms of the series can be used. For the potential of an additive-multiplicative form (a sum of a time-independent contribution and a time-dependent multiplicative perturbation), general explicit expressions are obtained which allow comparative analysis of frequency dependencies of the average velocity, implying deterministic and stochastic potential energy fluctuations. For qualitative and quantitative analysis of these dependences, we choose illustrative examples for spatial harmonic fluctuations: with deterministic time dependences of a relaxation type and stochastic time dependences describing Markovian dichotomous and harmonic noise processes. We explore the influence of fluctuation types on the ratchet effect and demonstrate its enhancement in the case of harmonic noise.

A much better replacement of the Michaelis–Menten equation and its application

Michaelis–Menten equation is a basic equation of enzyme kinetics and gives acceptable approximations of real chemical reaction processes. Analyzing the derivation of this equation yields the fact that its good performance of approximating real reaction processes is due to Michaelis–Menten curve (8). This curve is derived from Quasi-Steady-State Assumption (QSSA), which has been proved always true and called Quasi-Steady-State Law by Banghe Li et al. [Quasi-steady state laws in enzyme kinetics, J. Phys. Chem. A 112(11) (2008) 2311–2321].

Here, we found a polynomial equation with total degree of four [Formula: see text] (14), which gives more accurate approximation of the reaction process in two aspects: during the quasi-steady-state of the reaction, Michaelis–Menten curve approximates the reaction well, while our equation [Formula: see text] gives better approximation; near the end of the reaction, our equation approaches the end of the reaction with a tangent line the same to that of the reaction process trajectory simulated by mass action, while Michaelis–Menten curve does not. In addition, our equation [Formula: see text] differs to Michaelis–Menten curve less than the order of [Formula: see text] as [Formula: see text] approaches [Formula: see text].

By considering the above merits of [Formula: see text], we suggest it as a replacement of Michaelis–Menten curve. Intuitively, this new equation is more complex and harder to understand. But, just because of its complexity, it provides more information about the rate constants than Michaelis–Menten curve does.

Finally, we get a better replacement of the Michaelis–Menten equation by combing [Formula: see text] and the equation [Formula: see text].

Structural conditions on complex networks for the Michaelis-Menten input-output response

The Michaelis-Menten (MM) fundamental formula describes how the rate of enzyme catalysis depends on substrate concentration. The familiar hyperbolic relationship was derived by timescale separation for a network of three reactions. The same formula has subsequently been found to describe steady-state input-output responses in many biological contexts, including single-molecule enzyme kinetics, gene regulation, transcription, translation, and force generation. Previous attempts to explain its ubiquity have been limited to networks with regular structure or simplifying parametric assumptions. Here, we exploit the graph-based linear framework for timescale separation to derive general structural conditions under which the MM formula arises. The conditions require a partition of the graph into two parts, akin to a "coarse graining" into the original MM graph, and constraints on where and how the input variable occurs. Other features of the graph, including the numerical values of parameters, can remain arbitrary, thereby explaining the formula's ubiquity. For systems at thermodynamic equilibrium, we derive a necessary and sufficient condition. For systems away from thermodynamic equilibrium, especially those with irreversible reactions, distinct structural conditions arise and a general characterization remains open. Nevertheless, our results accommodate, in much greater generality, all examples known to us in the literature.

Bottom-up single-molecule strategy for understanding subunit function of tetrameric β-galactosidase

In this paper, we report an example of the engineered expression of tetrameric β-galactosidase (β-gal) containing varying numbers of active monomers. Specifically, by combining wild-type and single-nucleotide polymorphism plasmids at varying ratios, tetrameric β-gal was expressed in vitro with one to four active monomers. The kinetics of individual enzyme molecules revealed four distinct populations, corresponding to the number of active monomers in the enzyme. Using single-molecule-level enzyme kinetics, we were able to measure an accurate in vitro mistranslation frequency (5.8 × 10^-4 per base). In addition, we studied the kinetics of the mistranslated β-gal at the single-molecule level.

In Silico Prediction of Large-Scale Microbial Production Performance: Constraints for Getting Proper Data-Driven Models

Industrial bioreactors range from 10.000 to 700.000 L and characteristically show different zones of substrate availabilities, dissolved gas concentrations and pH values reflecting physical, technical and economic constraints of scale-up. Microbial producers are fluctuating inside the bioreactors thereby experiencing frequently changing micro-environmental conditions. The external stimuli induce responses on microbial metabolism and on transcriptional regulation programs. Both may deteriorate the expected microbial production performance in large scale compared to expectations deduced from ideal, well-mixed lab-scale conditions. Accordingly, predictive tools are needed to quantify large-scale impacts considering bioreactor heterogeneities. The review shows that the time is right to combine simulations of microbial kinetics with calculations of large-scale environmental conditions to predict the bioreactor performance. Accordingly, basic experimental procedures and computational tools are presented to derive proper microbial models and hydrodynamic conditions, and to link both for bioreactor modeling. Particular emphasis is laid on the identification of gene regulatory networks as the implementation of such models will surely gain momentum in future studies.

Simplification of Markov chains with infinite state space and the mathematical theory of random gene expression bursts

Here we develop an effective approach to simplify two-time-scale Markov chains with infinite state spaces by removal of states with fast leaving rates, which improves the simplification method of finite Markov chains. We introduce the concept of fast transition paths and show that the effective transitions of the reduced chain can be represented as the superposition of the direct transitions and the indirect transitions via all the fast transition paths. Furthermore, we apply our simplification approach to the standard Markov model of single-cell stochastic gene expression and provide a mathematical theory of random gene expression bursts. We give the precise mathematical conditions for the bursting kinetics of both mRNAs and proteins. It turns out that random bursts exactly correspond to the fast transition paths of the Markov model. This helps us gain a better understanding of the physics behind the bursting kinetics as an emergent behavior from the fundamental multiscale biochemical reaction kinetics of stochastic gene expression.

Revealing dynamics of helicase translocation on single-stranded DNA using high-resolution nanopore tweezers

Enzymes that operate on DNA or RNA perform the core functions of replication and expression in all of biology. To gain high-resolution access to the detailed mechanistic behavior of these enzymes, we developed single-molecule picometer-resolution nanopore tweezers (SPRNT), a single-molecule technique in which the motion of polynucleotides through an enzyme is measured by a nanopore. SPRNT reveals two mechanical substates of the ATP hydrolysis cycle of the superfamily 2 helicase Hel308 during translocation on single-stranded DNA (ssDNA). By analyzing these substates at millisecond resolution, we derive a detailed kinetic model for Hel308 translocation along ssDNA that sheds light on how superfamily 1 and 2 helicases turn ATP hydrolysis into motion along DNA. Surprisingly, we find that the DNA sequence within Hel308 affects the kinetics of helicase translocation.

Noise slows the rate of Michaelis-Menten reactions

Microscopic randomness and the small volumes of living cells combine to generate random fluctuations in molecule concentrations called "noise". Here I investigate the effect of noise on biochemical reactions obeying Michaelis-Menten kinetics, concluding that substrate noise causes these reactions to slow. I derive a general expression for the time evolution of the joint probability density of chemical species in arbitrarily connected networks of non-linear chemical reactions in small volumes. This equation is a generalization of the chemical master equation (CME), a common tool for investigating stochastic chemical kinetics, extended to reaction networks occurring in small volumes, such as living cells. I apply this equation to a generalized Michaelis-Menten reaction in an open system, deriving the following general result: 〈p〉≤p¯ and 〈s〉≥s¯, where s¯ and p¯ denote the deterministic steady-state concentration of reactant and product species, respectively, and 〈s〉 and 〈p〉 denote the steady-state ensemble average over independent realizations of a stochastic reaction. Under biologically realistic conditions, namely when substrate is degraded or diluted by cell division, 〈p〉≤p¯. Consequently, noise slows the rate of in vivo Michaelis-Menten reactions. These predictions are validated by extensive stochastic simulations using Gillespie's exact stochastic simulation algorithm. I specify the conditions under which these effects occur and when they vanish, therefore reconciling discrepancies among previous theoretical investigations of stochastic biochemical reactions. Stochastic slowdown of reaction flux caused by molecular noise in living cells may have functional consequences, which the present theory may be used to quantify.

Simplification of irreversible Markov chains by removal of states with fast leaving rates

In the recent work of Ullah et al. (2012a), the authors developed an effective method to simplify reversible Markov chains by removal of states with low equilibrium occupancies. In this paper, we extend this result to irreversible Markov chains. We show that an irreversible chain can be simplified by removal of states with fast leaving rates. Moreover, we reveal that the irreversibility of the chain will always decrease after model simplification. This suggests that although model simplification can retain almost all the dynamic information of the chain, it will lose some thermodynamic information as a trade-off. Examples from biology are also given to illustrate the main results of this paper.

Nonequilibrium Enhances Adaptation Efficiency of Stochastic Biochemical Systems

Adaptation is a crucial biological function possessed by many sensory systems. Early work has shown that some influential equilibrium models can achieve accurate adaptation. However, recent studies indicate that there are close relationships between adaptation and nonequilibrium. In this paper, we provide an explanation of these two seemingly contradictory results based on Markov models with relatively simple networks. We show that as the nonequilibrium driving becomes stronger, the system under consideration will undergo a phase transition along a fixed direction: from non-adaptation to simple adaptation then to oscillatory adaptation, while the transition in the opposite direction is forbidden. This indicates that although adaptation may be observed in equilibrium systems, it tends to occur in systems far away from equilibrium. In addition, we find that nonequilibrium will improve the performance of adaptation by enhancing the adaptation efficiency. All these results provide a deeper insight into the connection between adaptation and nonequilibrium. Finally, we use a more complicated network model of bacterial chemotaxis to validate the main results of this paper.

Stochastic dynamic study of optical transition properties of single GFP-like molecules

Due to high fluctuations and quantum uncertainty, the processes of single-molecules should be treated by stochastic methods. To study fluorescence time series and their statistical properties, we have applied two stochastic methods, one of which is an analytic method to study the off-time distributions of certain fluorescence transitions and the other is Gillespie's method of stochastic simulations. These methods have been applied to study the optical transition properties of two single-molecule systems, GFPmut2 and a Dronpa-like molecule, to yield results in approximate agreement with experimental observations on these systems. Rigorous oscillatory time series of GFPmut2 before it unfolds in the presence of denaturants have not been obtained based on the stochastic method used, but, on the other hand, the stochastic treatment puts constraints on the conditions under which such oscillatory behavior is possible. Furthermore, a sensitivity analysis is carried out on GFPmut2 to assess the effects of transition rates on the observables, such as fluorescence intensities.

Minimal moment equations for stochastic models of biochemical reaction networks with partially finite state space

Many stochastic models of biochemical reaction networks contain some chemical species for which the number of molecules that are present in the system can only be finite (for instance due to conservation laws), but also other species that can be present in arbitrarily large amounts. The prime example of such networks are models of gene expression, which typically contain a small and finite number of possible states for the promoter but an infinite number of possible states for the amount of mRNA and protein. One of the main approaches to analyze such models is through the use of equations for the time evolution of moments of the chemical species. Recently, a new approach based on conditional moments of the species with infinite state space given all the different possible states of the finite species has been proposed. It was argued that this approach allows one to capture more details about the full underlying probability distribution with a smaller number of equations. Here, I show that the result that less moments provide more information can only stem from an unnecessarily complicated description of the system in the classical formulation. The foundation of this argument will be the derivation of moment equations that describe the complete probability distribution over the finite state space but only low-order moments over the infinite state space. I will show that the number of equations that is needed is always less than what was previously claimed and always less than the number of conditional moment equations up to the same order. To support these arguments, a symbolic algorithm is provided that can be used to derive minimal systems of unconditional moment equations for models with partially finite state space.

Variance-corrected Michaelis-Menten equation predicts transient rates of single-enzyme reactions and response times in bacterial gene-regulation

Many chemical reactions in biological cells occur at very low concentrations of constituent molecules. Thus, transcriptional gene-regulation is often controlled by poorly expressed transcription-factors, such as E.coli lac repressor with few tens of copies. Here we study the effects of inherent concentration fluctuations of substrate-molecules on the seminal Michaelis-Menten scheme of biochemical reactions. We present a universal correction to the Michaelis-Menten equation for the reaction-rates. The relevance and validity of this correction for enzymatic reactions and intracellular gene-regulation is demonstrated. Our analytical theory and simulation results confirm that the proposed variance-corrected Michaelis-Menten equation predicts the rate of reactions with remarkable accuracy even in the presence of large non-equilibrium concentration fluctuations. The major advantage of our approach is that it involves only the mean and variance of the substrate-molecule concentration. Our theory is therefore accessible to experiments and not specific to the exact source of the concentration fluctuations.

Active Degradation Explains the Distribution of Nuclear Proteins during Cellular Senescence

The amount of cellular proteins is a crucial parameter that is known to vary between cells as a function of the replicative passages, and can be important during physiological aging. The process of protein degradation is known to be performed by a series of enzymatic reactions, ranging from an initial step of protein ubiquitination to their final fragmentation by the proteasome. In this paper we propose a stochastic dynamical model of nuclear proteins concentration resulting from a balance between a constant production of proteins and their degradation by a cooperative enzymatic reaction. The predictions of this model are compared with experimental data obtained by fluorescence measurements of the amount of nuclear proteins in murine tail fibroblast (MTF) undergoing cellular senescence. Our model provides a three-parameter stationary distribution that is in good agreement with the experimental data even during the transition to the senescent state, where the nuclear protein concentration changes abruptly. The estimation of three parameters (cooperativity, saturation threshold, and maximal velocity of the reaction), and their evolution during replicative passages shows that only the maximal velocity varies significantly. Based on our modeling we speculate the reduction of functionality of the protein degradation mechanism as a possible competitive inhibition of the proteasome.

Stochastic quasi-steady state approximations for asymptotic solutions of the chemical master equation

In this paper, we propose two methods to carry out the quasi-steady state approximation in stochastic models of enzyme catalytic regulation, based on WKB asymptotics of the chemical master equation or of the corresponding partial differential equation for the generating function. The first of the methods we propose involves the development of multiscale generalisation of a WKB approximation of the solution of the master equation, where the separation of time scales is made explicit which allows us to apply the quasi-steady state approximation in a straightforward manner. To the lowest order, the multi-scale WKB method provides a quasi-steady state, Gaussian approximation of the probability distribution. The second method is based on the Hamilton-Jacobi representation of the stochastic process where, as predicted by large deviation theory, the solution of the partial differential equation for the corresponding characteristic function is given in terms of an effective action functional. The optimal transition paths between two states are then given by those paths that maximise the effective action. Such paths are the solutions of the Hamilton equations for the Hamiltonian associated to the effective action functional. The quasi-steady state approximation is applied to the Hamilton equations thus providing an approximation to the optimal transition paths and the transition time between two states. Using this approximation we predict that, unlike the mean-field quasi-steady approximation result, the rate of enzyme catalysis depends explicitly on the initial number of enzyme molecules. The accuracy and validity of our approximated results as well as that of our predictions regarding the behaviour of the stochastic enzyme catalytic models are verified by direct simulation of the stochastic model using Gillespie stochastic simulation algorithm.

Tracking electrons in biological macromolecules: from ensemble to single molecule

Nature utilizes oxido-reductases to cater to the energy demands of most biochemical processes in respiratory species. Oxido-reductases are capable of meeting this challenge by utilizing redox active sites, often containing transition metal ions, which facilitate movement and relocation of electrons/protons to create a potential gradient that is used to energize redox reactions. There has been a consistent struggle by researchers to estimate the electron transfer rate constants in physiologically relevant processes. This review provides a brief background on the measurements of electron transfer rates in biological molecules, in particular Cu-containing enzymes, and highlights the recent advances in monitoring these electron transfer events at the single molecule level or better to say, at the individual event level.

A queueing approach to multi-site enzyme kinetics

Multi-site enzymes, defined as where multiple substrate molecules can bind simultaneously to the same enzyme molecule, play a key role in a number of biological networks, with the Escherichia coli protease ClpXP a well-studied example. These enzymes can form a low latency 'waiting line' of substrate to the enzyme's catalytic core, such that the enzyme molecule can continue to collect substrate even when the catalytic core is occupied. To understand multi-site enzyme kinetics, we study a discrete stochastic model that includes a single catalytic core fed by a fixed number of substrate binding sites. A natural queueing systems analogy is found to provide substantial insight into the dynamics of the model. From this, we derive exact results for the probability distribution of the enzyme configuration and for the distribution of substrate departure times in the case of identical but distinguishable classes of substrate molecules. Comments are also provided for the case when different classes of substrate molecules are not processed identically.

Enzyme nanoarchitectonics: organization and device application

Fabrication of ultrasmall functional machines and their integration within ultrasmall areas or volumes can be useful for creation of novel technologies. The ultimate goal of the development of ultrasmall machines and device systems is to construct functional structures where independent molecules operate as independent device components. To realize exotic functions, use of enzymes in device structures is an attractive solution because enzymes can be regarded as efficient machines possessing high reaction efficiencies and specificities and can operate even under ambient conditions. In this review, recent developments in enzyme immobilization for advanced functions including device applications are summarized from the viewpoint of micro/nano-level structural control, or nanoarchitectonics. Examples are roughly classified as organic soft matter, inorganic soft materials or integrated/organized media. Soft matter such as polymers and their hybrids provide a medium appropriate for entrapment and encapsulation of enzymes. In addition, self-immobilization based on self-assembly and array formation results in enzyme nanoarchitectures with soft functions. For the confinement of enzymes in nanospaces, hard inorganic mesoporous materials containing well-defined channels play an important role. Enzymes that are confined exhibit improved stability and controllable arrangement, which are useful for formation of functional relays and for their integration into artificial devices. Layer-by-layer assemblies as well as organized lipid assemblies such as Langmuir-Blodgett films are some of the best media for architecting controllable enzyme arrangements. The ultrathin forms of these films facilitate their connection with external devices such as electrodes and transistors. Artificial enzymes and enzyme-mimicking catalysts are finally briefly described as examples of enzyme functions involving non-biological materials. These systems may compensate for the drawbacks of natural enzymes, such as their instabilities under harsh conditions. We believe that enzymes and their mimics will be freely coupled, organized and integrated upon demand in near future technologies.

Non-equilibrium transition state rate theory

Transition state or Kramers' rate theory has been used to quantify the kinetic speed of many chemical, physical and biological equilibrium processes successfully.

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.

Propensity approach to nonequilibrium thermodynamics of a chemical reaction network: controlling single E-coli β-galactosidase enzyme catalysis through the elementary reaction steps

In this work, we develop an approach to nonequilibrium thermodynamics of an open chemical reaction network in terms of the elementary reaction propensities. The method is akin to the microscopic formulation of the dissipation function in terms of the Kullback-Leibler distance of phase space trajectories in Hamiltonian system. The formalism is applied to a single oligomeric enzyme kinetics at chemiostatic condition that leads the reaction system to a nonequilibrium steady state, characterized by a positive total entropy production rate. Analytical expressions are derived, relating the individual reaction contributions towards the total entropy production rate with experimentally measurable reaction velocity. Taking a real case of Escherichia coli β-galactosidase enzyme obeying Michaelis-Menten kinetics, we thoroughly analyze the temporal as well as the steady state behavior of various thermodynamic quantities for each elementary reaction. This gives a useful insight in the relative magnitudes of various energy terms and the dissipated heat to sustain a steady state of the reaction system operating far-from-equilibrium. It is also observed that, the reaction is entropy-driven at low substrate concentration and becomes energy-driven as the substrate concentration rises.

Intracellular viral infection kinetics using a stochastic approach

Stochastic simulation is carried out to investigate intracellular viral reaction kinetics and the time evolution of the average particle number ( N̄) and coefficient variation (CV) for genome, template, and structural protein. The coefficient variation of these components is found to be ordered as: CVtemplate > CVstructural protein > CVgenome. The average particle number is also calculated via a deterministic approach. The magnitude value of the difference between the stochastic and deterministic approaches is found to be N̄template ≈ N̄structural protein > N̄genome. The Poisson algorithm has been used to investigate the number of particles in the dynamics of intracellular viral reaction kinetics. Our results show that the average particle number ( N̄) obtained using the Poisson algorithm for both dynamics and stationary states is le ss than that obtained using the mean field approach relative to the quantity ( N) found via the Gillespie algorithm. Also, the standard deviation of N̄ calculated via the Poisson algorithm is more than that obtained via the Gillespie algorithm. The equilibrium time for the population of particles via the Poisson algorithm is less than that via the Gillespie algorithm and the mean field approach. In the present study, the Gillespie algorithm is more reliable than the Poisson algorithm.

Entropy hysteresis and nonequilibrium thermodynamic efficiency of ion conduction in a voltage-gated potassium ion channel

Here we have studied the nonequilibrium thermodynamic response of a voltage-gated Shaker potassium ion channel using a stochastic master equation. For a constant external voltage, the system reaches equilibrium indicated by the vanishing total entropy production rate, whereas for oscillating voltage the current and entropy production rates show dynamic hysteretic behavior. Here we have shown quantitatively that although the hysteresis loop area vanishes in low and high frequency domains of the external voltage, they are thermodynamically distinguishable. In the very low frequency domain, the system remains close to equilibrium, whereas at high frequencies it goes to a nonequilibrium steady state (NESS) associated with a finite value of dissipation function. At NESS, the efficiency of the ion conduction can also be related with the nonlinear dependence of the dissipation function on the power of the external field. Another intriguing aspect is that, at the high frequency limit, the total entropy production rate oscillates at NESS with half of the time period of the external voltage.