Michaelis-Menten Mechanism Research Articles

Reserving Electrons in Cofactor Decorated Coordination Capsules for Biomimetic Electrosynthesis of α-Hydroxy/amino Esters.

Sustainable electricity-to-chemical conversion via the utilization of artificial catalysts inspired by redox biological systems holds great significance for catalyzing synthesis. Herein, we develop a biomimetic electrosynthesis strategy mediated by a nicotinamide adenine dinucleotide (NADH) mimic-containing coordination capsule for efficiently producing α-hydroxy/amino esters. The coordination saturated metal centers worked as an electron relay to consecutively accept single electrons while donating two electrons to the NAD+ mimics simultaneously. The protonation of the intermediate generated active NADH mimics for biomimetic hydrogenation of the substrates via the conventional enzymatic manifold with or without the presence of natural enzymes. The pocket of the capsule encapsulated the substrate and enforced the close proximity between the substrate and the NADH mimics, forming a preorganized intermediate to shift the redox potential by 0.4 V anodically. The cobalt capsule gave methyl mandelate over a range of applied potentials, with an improved yield of 92% when operated at -1.2 V compared to that of Hantzsch ester or natural NADH. Kinetic experiments revealed a Michaelis-Menten mechanism with a Km of 7.5 mM and a Kcat of 1.1 × 10-2 s-1. This extended strategy in tandem with an enzyme exhibited a TON of 650 molE-1 with an initial TOF of 185 molE-1·h-1, outperforming relevant Rh-mediated enzymatic electrosynthesis systems and providing an attractive avenue toward advanced artificial electrosynthesis.

Enzymatic Activation and Continuous Electrochemical Production of Methane from Dilute CO2 Sources with a Self-Healing Capsule.

Converting dilute CO2 source into value-added chemicals and fuels is a promising route to reduce fossil fuel consumption and greenhouse gas emission, but integrating electrocatalysis with CO2 capture still faced marked challenges. Herein, we show that a self-healing metal-organic macrocycle functionalized as an electrochemical catalyst to selectively produce methane from flue gas and air with the lowest applied potential so far (0.06 V vs reversible hydrogen electrode, RHE) through an enzymatic activation fashion. The capsule emulates the enzyme' pocket to abstract one in situ-formed CO2-adduct molecule with the commercial amino alcohols, forming an easy-to-reduce substrate-involving clathrate to combine the CO2 capture with electroreduction for a thorough CO2 reduction. We find that the self-healing system exhibited enzymatic kinetics for the first time with the Michaelis-Menten mechanism in the electrochemical reduction of CO2 and maintained a methane Faraday efficiency (FE) of 74.24% with a selectivity of over 99% for continuous operation over 200 h. A consecutive working lab at 50 mA·cm-2, in an eleven-for-one (10 h working and 1 h healing) electrolysis manner, gives a methane turnover number (TON) of more than 10,000 within 100 h. The integrated electrolysis with CO2 capture facilitates the thorough reduction of flue gas (ca. 13.0% of CO2) and first time of air (ca. 400 ppm of CO2 to 42.7 mL CH4 from 1.0 m3 air). The new self-healing strategy of molecular electrocatalyst with an enzymatic activation manner and anodic shifting of the applied potentials provided a departure from the existing electrochemical catalytic techniques.

Abstract A003: Unwinding the complexities of helicases as compelling drug targets in oncology

Abstract In recent years, helicases have moved to the forefront of research as novel drug targets for a variety of human diseases, including cancer. As critical components of the cell cycle and DNA repair, these enzymes offer an attractive point of therapeutic intervention. Recent bioinformatic endeavours such as the Cancer DepMap have further highlighted the essentiality of these proteins in the progression of numerous cancers. Inhibitors of Werner helicase (WRN) entered Phase I evaluation in 2023, after studies highlighted its role in tumours with microsatellite instability (MSI) and impaired mismatch repair (MMR). Building on these advances, helicases now represent a significant drug target class and are the subject of intense research. However, these targets present significant complexities in their prosecution. Hit finding and robust validation of these putative active compounds remains challenging, with high attrition rates and significant sensitivity to false positives. Resultant hits require thorough biological characterisation via both biochemical and biophysical methods, alongside determination of selectivity against other closely related family members, to increase confidence in their provenance before commencing a detailed drug discovery project. To exemplify these approaches, we have profiled three commercially available WRN inhibitors from two chemical series through a selection of biochemical and biophysical assays. The three compounds exhibit excellent selectivity against 6 distinct helicase/translocase targets. In Michaelis-Menten mechanism of Inhibition studies, we defined contrasting mechanisms of inhibition for the two series and were also able to identify time dependent inhibition with one compound series. Analysing binding interactions of the compounds against protein and protein:substrate complexes, via a fluorescent thermal shift assay (FTSA) and microscale thermophoresis (MST), enabled us to further understand and interrogate the mechanism of inhibition. Herein, we describe some of our experiences, learnings and best practices when prosecuting these compelling, but challenging, targets. Citation Format: Yael Mamane, Zoe Caple, Katie Chapman, Ian Henderson, Allan Jordan, Vanessa Lyne, Cynthia Okoye, Laurent Rigoreau, Ganesh Kadamur, Stuart Thomson, Chris Tomlinson, Jana Wolf. Unwinding the complexities of helicases as compelling drug targets in oncology [abstract]. In: Proceedings of the AACR Special Conference in Cancer Research: Expanding and Translating Cancer Synthetic Vulnerabilities; 2024 Jun 10-13; Montreal, Quebec, Canada. Philadelphia (PA): AACR; Mol Cancer Ther 2024;23(6 Suppl):Abstract nr A003.

A novel copper ion enhanced electrochemical DNA biosensor for the determination of epinephrine

A novel electrochemical biosensor was developed for the detection of epinephrine (EP) by immobilizing double-strand DNA (dsDNA) bound with copper ions on a gold electrode (Cu2+/dsDNA/MCH/AuE). The electrochemical behavior of EP at Cu2+/dsDNA/MCH/AuE was examined, and the results demonstrated a significant enhancement in the electrocatalytic oxidation peak current of EP due to the formation of a stable G-Cu(II)-G sandwich structure between Cu2+ and guanine at the modified electrode. The modification process of the electrode was characterized by scanning electron microscopy, infrared spectroscopy, electrochemical impedance spectroscopy, and differential pulse voltammetry. A study on the effect of pH in phosphate buffer solution on the electrochemical oxidation of EP indicated that the catalytic oxidation process was pH-dependent. A plot of catalytic current versus EP concentration exhibited a dual-linear relationship within two ranges: 1.0–12.5 μM and 12.5–1000.0 μM, with correlation coefficients of 0.995 and 0.997, respectively. The limit of detection was determined to be 47 nM (S/N = 3). According to the calculated Hill coefficient (0.99), it can be concluded that the electrocatalytic process followed the Michaelis-Menten kinetic mechanism. The maximum catalytic current Im was 25 μA, while the apparent Michaelis-Menten constant Km was 1.425 mM. These findings indicated excellent electrocatalytic activity of the modified electrode towards oxidation of EP. The developed biosensor successfully detected EP in spiked mouse serum as well as epinephrine hydrochloride injection with high selectivity, sensitivity, stability, and accuracy.

Inference of complex reaction mechanisms applying model reduction techniques

Both structural (number of species and reactions) and temporal (extremely diverse reaction rates) aspects of complexity are considered when describing large chemical reaction networks. A consistent way to make model reduction is to construct the invariant manifold, which describes the asymptotic system behavior. Preliminary approximations to SIM (Slow Invariant Manifold) are constructed using the model reduction techniques (MRTs): the Quasi-Equilibrium Manifold (QEM), the Spectral Quasi-Equilibrium Manifold (SQEM), and the Intrinsic Low-Dimension Manifold (ILDM). In this paper, the activities of the concerned species and the overall dynamics of the system are examined. Two examples are used to demonstrate the techniques: the Michaelis–Menten mechanism, which is a single reaction mechanism, and a multi-route route reaction mechanism. The behavior of each species on the available route is covered separately. As a result, the reduced invariant solution curve of several approaches is illustrated, along with a comparison of these methods in various graphs. Sensitivity analysis is applied using the SimBiology toolbox in MATLAB to monitor the role of each parameter involved. All the results of model reduction techniques are simulated through MATLAB.

Kinetically promoted hydrogen generation by Ru nanoparticles decorated CoB2O4 on mesoporous carbon spheres with rich oxygen vacancies for NaBH4 hydrolysis

Developing heterostructured catalysts with highly active and reusable is an urgent task to achieve green hydrogen production via NaBH4. Herein, we proposed a novel NaCl template method, which involves solid-state physical grinding of metal salts, reducing agents and prepared carbon spheres with the participation of sodium chloride to synthesize Ru-particles decorated CoB2O4 supported on hollow mesoporous carbon nanospheres (Ru/CoB2O4@C) catalysts for efficient and durable H2 generation through alkaline hydrolysis of NaBH4. The optimized catalyst with a Ru content of 3.0 wt% exhibits a high hydrogen generation rate of 8139 mL min−1gcat-1 and a low activation energy of 33.2 kJ mol−1 for NaBH4 hydrolysis, which is one of the most efficient catalysts reported recently. The extraordinary performance is mainly attributed to the synergistic effect between Ru and CoB2O4 species and abundant oxygen vacancies, which facilitate charge redistribution and provide more active sites, thereby enhancing the catalytic activity. Density functional theory calculations support the proposed Michaelis-Menten mechanism, where BH4− and H2O are adsorbed on electron-rich Ru and electron-deficient CoB2O4 surfaces, respectively, and the reaction energy barrier for the rupture of the B-H bond as the rate-determining step is only 1.37 eV, synergistically promoting the hydrolysis of NaBH4. Our study provides an innovative method for designing efficient and stable catalysts for green hydrogen generation.

Facile green synthesis of wasted hop-based zinc oxide nanozymes as peroxidase-like catalysts for colorimetric analysis.

Hops are a common ingredient in beer production, and a considerable quantity of hops is usually discarded as a waste material once the brewing process is completed. Transforming this waste material into valuable nanomaterials offers a sustainable approach that has the potential to significantly mitigate environmental impact. Herein, a facile and green protocol for the production of zinc oxide nanozymes (ZnO NZs) using wasted hop extract (WHE) as a natural precursor was demonstrated. The process involved a hydrothermal synthesis method followed by a calcination step to form the final ZnO NZs. The results revealed that lupulon, the main β-acid in hops, particularly the phenolic hydroxy group, is primarily responsible for the biosynthesis of ZnO NZs. The WHE-ZnO NZs exhibited exceptional peroxidase-like (POD-like) activity and served as effective catalysts for the oxidation of 3,3,5,5-tetramethylbenzidine (TMB) in the presence of hydrogen peroxide (H2O2). Analysis of the catalytic mechanism revealed that the POD-like activity of these WHE-ZnO NZs originated from their ability to expedite the transfer of electrons between TMB and H2O2, resulting in the enzymatic kinetics following the standard Michaelis-Menten mechanism. Furthermore, we developed a straightforward and user-friendly colorimetric technique for detecting both H2O2 and glucose. By utilizing the WHE-ZnO NZs as POD-like catalysts, we achieved a linear detection range of 1-1000 μM and a limit of detection of 0.24 μM (S/N = 3) for H2O2 detection and a linear range of 0-100 mM and a detection limit of 16.73 μM (S/N = 3) for glucose detection. These results highlighted the potential applications of our waste-to-resource approach for nanozyme synthesis in the field of analytical chemistry.

Steady State Kinetics for Enzymes with Multiple Binding Sites Upstream of the Catalytic Site

The Michaelis–Menten mechanism, which describes the binding of a substrate to an enzyme, is a simplification of the process on a molecular scale. A more detailed model should include the binding of the substrate to precatalytic binding sites (PCBSs) prior to the transition to the catalytic site. Our work shows that the incorporation of PCBSs, in steady-state conditions, generates a Michaelis–Menten-type expression, in which the kinetic parameters KM and Vmax adopt more complex expressions than in the model without PCBSs. The equations governing reaction kinetics can be seen as generalized symmetries, relative to time translation actions over the state space of the underlying chemical system. The study of their structure and defining parameters can be interpreted as looking for invariants associated with these time evolution actions. The expression of KM decreases as the number of PCBSs increases, while Vmax reaches a minimum when the first PCBSs are incorporated into the model. To evaluate the trend of the dynamic behavior of the system, numerical simulations were performed based on schemes with different numbers of PCBSs and six conditions of kinetic constants. From these simulations, with equal kinetic constants for the formation of the Substrate/PCBS complex, it is observed that KM and Vmax are lower than those obtained with the Michaelis–Menten model. For the model with PCBSs, the Vmax reaches a minimum at one PCBS and that value is maintained for all of the systems evaluated. Since KM decreases with the number of PCBSs, the catalytic efficiency increases for enzymes fitting this model. All of these observations are consistent with the general equation obtained. This study allows us to explain, on the basis of the PCBS to KM and Vmax ratios, the effect on enzyme parameters due to mutations far from the catalytic site, at sites involved in the first enzyme/substrate interaction. In addition, it incorporates a new mechanism of enzyme activity regulation that could be fundamental to search for new activity-modulating sites or for the design of mutants with modified enzyme parameters.

Importance of diffusional constraints for the quantitative evaluation of calibration curves of enzymatic micro- and nanoelectrochemical sensors

Electrochemical calibration curves recorded at enzyme-modified micro- or nanoelectrodes are often quantitatively analysed using graphical approaches. In this case the information is directly derived from the calibration curve using a frequent practice although it is rooted on extrapolating the procedures valid when the enzymes as well as their substrate and cofactors are homogeneously distributed in a solution. Here, using a simple but highly representative model we demonstrate that this practice yields incorrect interpretation of the experimental results even for simple Michaelis-Menten mechanisms. This model makes it possible to establish correct relationships linking calibration currents and bulk substrate concentrations by a simple method allowing to account for the biases due to diffusional constraints.

Novel carbon nanozymes with enhanced phosphatase-like catalytic activity for antimicrobial applications

In this work, Sulfur and Nitrogen co-doped carbon nanoparticles (SN-CNPs) were synthesized by hydrothermal method using dried beet powder as the carbon source. TEM and AFM images indicated that these SN-CNPs form a round-shape ball with an approximate diameter of 50 nm. The presence of Sulfur and Nitrogen in these carbon-based nanoparticles was confirmed by FTIR and XPS analyses. These SN-CNPs were found to have strong phosphatase-like enzymatic activity. The enzymatic behavior of SN-CNPs follows the Michaelis–Menten mechanism with greater vmax and much lower Km values compared to alkaline phosphatase. Their antimicrobial properties were tested on E. coli and L. lactis, with MIC values of 63 μg mL−1 and 250 μg mL−1, respectively. SEM and AFM images of fixed and live E. coli cells revealed that SN-CNPs strongly interacted with the outer membranes of bacterial cells, significantly increasing the cell surface roughness. The chemical interaction between SN-CNPs and phospholipid modeled using quantum mechanical calculations further support our hypothesis that the phosphatase and antimicrobial properties of SN-CNPs are due to the thiol group on the SN-CNPs, which is a mimic of the cysteine-based protein phosphatase. The present work is the first to report carbon-based nanoparticles with strong phosphatase activity and propose a phosphatase natured antimicrobial mechanism. This novel class of carbon nanozymes has the potential to be used for effective catalytic and antibacterial applications.Graphical abstract

Modifying Enzymatic Substrate Binding within a Metal-Organic Capsule for Supramolecular Catalysis.

Supramolecular catalysis is established to modify reaction kinetics by substrate encapsulation, but manipulating the thermodynamics of electron-transfer reactions remains unexplored. Herein, we reported a new microenvironment-shielding approach to induce an anodic shift in the redox potentials of hydrazine substrates, reminiscent of the enzymatic activation for N-N bond cleavage within a metal-organic capsule H1. Equipped with the catalytic active cobalt sites and substrate-binding amide groups, H1 encapsulated the hydrazines to form the substrate-involving clathration intermediate, triggering the catalytic reduction N-N bond cleavage when electrons were acquired from the electron donors. Compared with the reduction of free hydrazines, the conceptual molecular confined microenvironment decreases the Gibbs free energy (up to -70 kJ mol-1), which is relevant to the initial electron-transfer reaction. Kinetic experiments demonstrate a Michaelis-Menten mechanism, which involves the formation of the pre-equilibrium of substrate-binding, followed by bond cleavage. Then, the distal N is released as NH3 and the product is squeezed. Integrating fluorescein into H1 enabled the photoreduction of N2H4 with an initial rate of ca. 1530 nmol min-1 into ammonia, comparable to that of natural MoFe proteins; thus, the approach provides an attractive manifold toward mimicking enzymatic activation.

Natural Parameter Conditions for Singular Perturbations of Chemical and Biochemical Reaction Networks

We consider reaction networks that admit a singular perturbation reduction in a certain parameter range. The focus of this paper is on deriving "small parameters" (briefly for small perturbation parameters), to gauge the accuracy of the reduction, in a manner that is consistent, amenable to computation and permits an interpretation in chemical or biochemical terms. Our work is based on local timescale estimates via ratios of the real parts of eigenvalues of the Jacobian near critical manifolds. This approach modifies the one introduced by Segel and Slemrod and is familiar from computational singular perturbation theory. While parameters derived by this method cannot provide universal quantitative estimates for the accuracy of a reduction, they represent a critical first step toward this end. Working directly with eigenvalues is generally unfeasible, and at best cumbersome. Therefore we focus on the coefficients of the characteristic polynomial to derive parameters, and relate them to timescales. Thus, we obtain distinguished parameters for systems of arbitrary dimension, with particular emphasis on reduction to dimension one. As a first application, we discuss the Michaelis-Menten reaction mechanism system in various settings, with new and perhaps surprising results. We proceed to investigate more complex enzyme catalyzed reaction mechanisms (uncompetitive, competitive inhibition and cooperativity) of dimension three, with reductions to dimension one and two. The distinguished parameters we derive for these three-dimensional systems are new. In fact, no rigorous derivation of small parameters seems to exist in the literature so far. Numerical simulations are included to illustrate the efficacy of the parameters obtained, but also to show that certain limitations must be observed.

The Determination of All Rate Constants in the Enzyme Catalyzed Reactions Based on Michaelis-Menten Mechanism

The Determination of All Rate Constants in the Enzyme Catalyzed Reactions Based on Michaelis-Menten Mechanism

Optimization and Determination of Kinetic Parameters of the Synthesis of 5-Lauryl-hydroxymethylfurfural Catalyzed by Lipases

Hydroxymethylfurfural esters (HMF-esters) have great potential for additive development; for this reason, the goal of this work was to study the optimization of the esterification conversion of HFM and lauric acid using two lipases: the Novozym 435® biocatalyst and immobilized lipase from Thermomyces lanuginosus (TL). For the optimization of conversion, a three-level three-factorial Box–Behnken experimental design was used. The models achieved a good fit (R2 over 90%) for reactions catalyzed with Novozym 435® and immobilized TL lipase. The best conversion, 78.4%, was achieved with immobilized TL lipase using 30 mM HMF, 16 U of biocatalytic activity, and 50 °C. The kinetic parameters without inhibition by the substrate were determined using the Michaelis–Menten mechanism, whereby VMax for both biocatalysts reached the highest values at 50 °C, and the highest enzyme–substrate affinities (low Km) were reached at temperatures of 30 °C and 40 °C. It can be concluded that immobilized TL lipase has the potential to catalyze this reaction since, under optimal reaction conditions, an 80.6% conversion (value predicted) could be achieved.

Form and Function in Biological Macromolecules: Kinetic Stability is Key, with Oblique Roles for Intramolecularity and Hydrophobicity in Enzyme Catalysis

Certain structure-reactivity aspects of biological macromolecules, with particular emphasis on protein folding and enzyme catalysis, are discussed herein. Furthermore, the role played by the hydrophobic effect and intramolecularity in enzymic reactivity are evaluated afresh, with new insights of much importance in chemical biology.
 Thus, the sum of the energies of the hydrogen bonds constituting the tertiary structures of proteins, determines the overall Gibbs energy of activation for loss of conformational integrity. As protein molecules of even modest size consist of a relatively large number of intramolecular hydrogen bonding interactions, the activation barrier to even partial unfolding of the α-helices and β-sheets forming the tertiary structure would be prohibitively high under normal conditions.
 The resulting kinetic stability conserves the natural conformation of a protein molecule established at the ribosomal site of synthesis, carrying the molecule through the thick-and-thin of a range of metabolic pathways during its ‘journey of life’. However, protein molecules also acquire flexibility via ‘strain delocalization’ (Ramachandran plots being relevant), thus enabling stabilization of multiple transition states along a pathway (particularly in case of covalent enzyme-substrate complexes).
 Two mechanistic features of enzyme catalysis that have been exhaustively studied are intramolecularity and the hydrophobic effect. Although intramolecularity has for long been touted as the origin of enzymic reactivity, this can be challenged on fundamental physical-organic grounds. Intriguingly, however, the collapse of the classical Michaelis-Menten mechanism for enzyme catalysis leads to a reconsideration of the role of intramolecularity, although not as hitherto envisaged. Thus, a majority of enzymes apparently form covalent enzyme-substrate complexes—possibly also exergonically—so the subsequent reactions at the active site may well benefit from the traditional propinquity effect: The critical caveat would be the highly exergonic formation of final products.
 It is argued that the hydrophobic effect—although intuitively reasonable—is difficult to pin down quantitatively, model systems (including micelles) leading to inconsistent and debatable results. However, the hydrophobic effect likely contributes to enzymic reactivity along with charge-relay via the proteinic backbone.

Chemistry students’ conceptual difficulties and problem solving behavior in chemical kinetics, as a component of an introductory physical chemistry course

Abstract The identification of undergraduate chemistry students’ conceptual difficulties and common mistakes with basic concepts and problems in chemical kinetics provided the aim for this study, which involved 2nd-year/4th semester students who had passed the chemical kinetics component of a physical chemistry course. The study involved the analysis, evaluation and interpretation of students’ answers to the final examination in chemical kinetics. Three achievement groups, for the various topics, were identified: Group A, high achievement (mean ≈ 85%): (a) the steps in a chain-reaction mechanism, (b) integrated 1st- and 2nd-order rate laws; and (c) the Lindemann–Hinshelwood mechanism. Group B, intermediate achievement (mean ≈ 74%): (a) half-life, (b) instantaneous rate and the extent of reaction variable (ξ), (c) the Michaelis–Menten mechanism, and (d) theoretical rate law not asking for a final formula. Group C, low achievement (mean ≈ 54%): (a) experimental rate law and the reaction rate constant on the basis of an experimental-data table, (b) extracting the theoretical rate law, and (c) the Arrhenius equation. Students’ errors and misconceptions have also been identified. Successful students tended to respond well to straightforward questions on the theory of the subject, but had difficulties when solving problems. It is essential that teachers understand the potential of their students, especially possible misconceptions they may hold, and the teaching approaches that may contribute to overcoming the student difficulties. Problems in chemical kinetics can be very demanding both in terms of algebraic manipulations and conceptually. Teaching should focus on problem solving, with the emphasis on students themselves trying to solve the problems.

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.

Fast reactions with non-interacting species in stochastic reaction networks.

We consider stochastic reaction networks modeled by continuous-time Markov chains. Such reaction networks often contain many reactions, potentially occurring at different time scales, and have unknown parameters (kinetic rates, total amounts). This makes their analysis complex. We examine stochastic reaction networks with non-interacting species that often appear in examples of interest (e.g. in the two-substrate Michaelis Menten mechanism). Non-interacting species typically appear as intermediate (or transient) chemical complexes that are depleted at a fast rate. We embed the Markov process of the reaction network into a one-parameter family under a two time-scale approach, such that molecules of non-interacting species are degraded fast. We derive simplified reaction networks where the non-interacting species are eliminated and that approximate the scaled Markov process in the limit as the parameter becomes small. Then, we derive sufficient conditions for such reductions based on the reaction network structure for both homogeneous and time-varying stochastic settings, and study examples and properties of the reduction.

Chemical Reaction Rate from a (Semiempirical) Dynamical Point of View

The aim of the present paper is to check - or better to confirm - the mathematical validity of chemical reaction rate, faced as a set of differential equations. Firstly one - way elementary reactions are considered, in the most general case. Secondly the same thing is done with two-way (opposing) elementary reactions. At this stage, we show that the two – way reaction, as we mean it, is compatible with the reduction of the total Gibbs energy as expected in every natural process. As an example of a two way elementary reaction of a completely solvable problem we give the hydrolysis of sucrose to glucose and fructose, where the “inversion” of sucrose is examined not only with the initial linear reaction of “Wilhelmy” (1850), but also with the two way nonlinear reaction introduced. Finally the validity of the mathematical model is checked for more complex cases such as the Michaelis-Menten mechanism or reactions in solution, where it is found that the two cases, apparently are four – dimensional while in reality are two – dimensional (after the “subtraction” of the constraints of “motion”) and naturally cannot exhibit chaotic behavior. In all cases the treatment is not one-hundred-percent mathematically austere but it has also arbitrary although reasonable hypotheses.

Stochastic time-dependent enzyme kinetics: Closed-form solution and transient bimodality.

We derive an approximate closed-form solution to the chemical master equation describing the Michaelis-Menten reaction mechanism of enzyme action. In particular, assuming that the probability of a complex dissociating into an enzyme and substrate is significantly larger than the probability of a product formation event, we obtain expressions for the time-dependent marginal probability distributions of the number of substrate and enzyme molecules. For delta function initial conditions, we show that the substrate distribution is either unimodal at all times or else becomes bimodal at intermediate times. This transient bimodality, which has no deterministic counterpart, manifests when the initial number of substrate molecules is much larger than the total number of enzyme molecules and if the frequency of enzyme-substrate binding events is large enough. Furthermore, we show that our closed-form solution is different from the solution of the chemical master equation reduced by means of the widely used discrete stochastic Michaelis-Menten approximation, where the propensity for substrate decay has a hyperbolic dependence on the number of substrate molecules. The differences arise because the latter does not take into account enzyme number fluctuations, while our approach includes them. We confirm by means of a stochastic simulation of all the elementary reaction steps in the Michaelis-Menten mechanism that our closed-form solution is accurate over a larger region of parameter space than that obtained using the discrete stochastic Michaelis-Menten approximation.