Theory and Simulation of Diffusion-Controlled Michaelis−Menten Kinetics for a Static Enzyme in Solution

Simulation of Radiation-Induced DNA Damage and Protection by Histones Using the Code RITRACKS

(1) Background: DNA damage is of great importance in the understanding of the effects of ionizing radiation. Various types of DNA damage can result from exposure to ionizing radiation, with clustered types considered the most important for radiobiological effects. (2) Methods: The code RITRACKS (Relativistic Ion Tracks), a program that simulates stochastic radiation track structures, was used to simulate DNA damage by photons and ions spanning a broad range of linear energy transfer (LET) values. To perform these simulations, the transport code was modified to include cross sections for the interactions of ions or electrons with DNA and amino acids for ionizations, dissociative electron attachment, and elastic collisions. The radiochemistry simulations were performed using a step-by-step algorithm that follows the evolution of all particles in time, including reactions between radicals and DNA structures and amino acids. Furthermore, detailed DNA damage events, such as base pair positions, DNA fragment lengths, and fragment yields, were recorded. (3) Results: We report simulation results using photons and the ions 1H+, 4He2+, 12C6+, 16O8+, and 56Fe26+ at various energies, covering LET values from 0.3 to 164 keV/µm, and performed a comparison with other codes and experimental results. The results show evidence of DNA protection from damage at its points of contacts with histone proteins. (4) Conclusions: RITRACKS can provide a framework for studying DNA damage from a variety of ionizing radiation sources with detailed representations of DNA at the atomic scale, DNA-associated proteins, and resulting DNA damage events and statistics, enabling a broader range of future comparisons with experiments such as those based on DNA sequencing.

A review of simulation codes and approaches for radiation chemistry

Historically, the field of radiation chemistry began shortly after the discovery of radioactivity, and its development has been closely related to discoveries in other related fields such as radiation and nuclear physics. Radiolysis of water and radiation chemistry have been very important in elucidating how radiation affects living matter and how it induces DNA damage. Nowadays, we recognize the importance of chemistry to understanding the effects of radiation on cells; however, it took several decades to obtain this insight, and much is still unknown. The radiolysis of water and aqueous solutions have been the subject of much experimental and theoretical research for many decades. One important concept closely related to radiation chemistry is radiation track structure. Track structure results from early physical and physicochemical events that lead to a highly non-homogenous distribution of radiolytic species. Because ionizing radiation creates unstable species that are distributed non-homogenously, the use of conventional reaction kinetics methods does not describe this chemistry well. In recent years, several methods have been developed for simulating radiation chemistry. In this review, we give a brief history of the field and the development of the simulation codes. We review the current methods used to simulate radiolysis of water and radiation chemistry, and we describe several radiation chemistry codes and their applications.

Green Synthesis of Porous Spherical Reduced Graphene Oxide and Its Application in Immobilized Pectinase

Pectinase is widely used in juice production, food processes, and other fields. However, owing to poor stability, free pectinase is difficult to separate from a substrate after hydrolysis and cannot be reused, and thus its industrial use is limited. Immobilized pectinase can solve these problems well. We prepared a carrier material of immobilized enzyme, which is called porous spherical reduced graphene oxide (rGO) with a rich pore structure, large specific surface area, strong hardness, and good biocompatibility to enzyme. Then, we evaluated the performance of the porous spherical rGO immobilized pectinase and characterized its structure by IR, XRD, and SEM. Using this material as a carrier of immobilized enzyme improves the load and catalytic activity of the enzyme. After 10 times of continuous use, the porous spherical rGO immobilized enzyme still maintained its initial relative enzyme activity at around 87%, indicating that immobilized pectinase had a stronger cycling stability, and its thermal stability, acid-base tolerance, and storage stability were superior to those of free pectinase. The results were compared with those of other studies on immobilized pectinase. The relative activity of pectinase immobilized by porous spherical rGO was at a high level after 10 consecutive uses. Overall, the spherical rGO is an excellent immobilized enzyme carrier material.

Dynamic cluster formation determines viscosity and diffusion in dense protein solutions

We develop a detailed description of protein translational and rotational diffusion in concentrated solution on the basis of all-atom molecular dynamics simulations in explicit solvent. Our systems contain up to 540 fully flexible proteins with 3.6 million atoms. In concentrated protein solutions (100 mg/mL and higher), the proteins ubiquitin and lysozyme, as well as the protein domains third IgG-binding domain of protein G and villin headpiece, diffuse not as isolated particles, but as members of transient clusters between which they constantly exchange. A dynamic cluster model nearly quantitatively explains the increase in viscosity and the decrease in protein diffusivity with protein volume fraction, which both exceed the predictions from widely used colloid models. The Stokes-Einstein relations for translational and rotational diffusion remain valid, but the effective hydrodynamic radius grows linearly with protein volume fraction. This increase follows the observed increase in cluster size and explains the more dramatic slowdown of protein rotation compared with translation. Baxter's sticky-sphere model of colloidal suspensions captures the concentration dependence of cluster size, viscosity, and rotational and translational diffusion. The consistency between simulations and experiments for a diverse set of soluble globular proteins indicates that the cluster model applies broadly to concentrated protein solutions, with equilibrium dissociation constants for nonspecific protein-protein binding in the Kd ≈ 10-mM regime.

Theory of Diffusion-Influenced Reaction Networks

A formalism is developed to describe how diffusion alters the kinetics of coupled reversible association-dissociation reactions in the presence of conformational changes that can modify the reactivity. The major difficulty in constructing a general theory is that, even to the lowest order, diffusion can change the structure of the rate equations of chemical kinetics by introducing new reaction channels (i.e., modifies the kinetic scheme). Therefore, the right formalism must be found that allows the influence of diffusion to be described in a concise and elegant way for networks of arbitrary complexity. Our key result is a set of non-Markovian rate equations involving stoichiometric matrices and net reaction rates (fluxes), in which these rates are coupled by a time-dependent pair association flux matrix, whose elements have a simple physical interpretation. Specifically, each element is the probability density that an isolated pair of reactants irreversibly associates at time t via one reaction channel on the condition that it started out with the dissociation products of another (or the same) channel. In the Markovian limit, the coupling of the chemical rates is described by committors (or splitting/capture probabilities). The committor is the probability that an isolated pair of reactants formed by dissociation at one site will irreversibly associate at another site rather than diffuse apart. We illustrate the use of our formalism by considering three reversible reaction schemes: (1) binding to a single site, (2) binding to two inequivalent sites, and (3) binding to a site whose reactivity fluctuates. In the first example, we recover the results published earlier, while in the second one we show that a new reaction channel appears, which directly connects the two bound states. The third example is particularly interesting because all species become coupled and an exchange-type bimolecular reaction appears. In the Markovian limit, some of the diffusion-modified rate constants that describe new transitions become negative, indicating that memory effects cannot be ignored.

Effective reaction rates for diffusion-limited reaction cycles

Biological signals in cells are transmitted with the use of reaction cycles, such as the phosphorylation-dephosphorylation cycle, in which substrate is modified by antagonistic enzymes. An appreciable share of such reactions takes place in crowded environments of two-dimensional structures, such as plasma membrane or intracellular membranes, and is expected to be diffusion-controlled. In this work, starting from the microscopic bimolecular reaction rate constants and using estimates of the mean first-passage time for an enzyme-substrate encounter, we derive diffusion-dependent effective macroscopic reaction rate coefficients (EMRRC) for a generic reaction cycle. Each EMRRC was found to be half of the harmonic average of the microscopic rate constant (phosphorylation c or dephosphorylation d), and the effective (crowding-dependent) motility divided by a slowly decreasing logarithmic function of the sum of the enzyme concentrations. This implies that when c and d differ, the two EMRRCs scale differently with the motility, rendering the steady-state fraction of phosphorylated substrate molecules diffusion-dependent. Analytical predictions are verified using kinetic Monte Carlo simulations on the two-dimensional triangular lattice at the single-molecule resolution. It is demonstrated that the proposed formulas estimate the steady-state concentrations and effective reaction rates for different sets of microscopic reaction rates and concentrations of reactants, including a non-trivial example where with increasing diffusivity the fraction of phosphorylated substrate molecules changes from 10% to 90%.

Reversible Stochastically Gated Diffusion-Influenced Reactions

An approximate but accurate theory is developed for the kinetics of reversible binding of a ligand to a macromolecule when either can stochastically fluctuate between reactive and unreactive conformations. The theory is based on a set of reaction-diffusion equations for the deviations of the pair distributions from their bulk values. The concentrations are shown to satisfy non-Markovian rate equations with memory kernels that are obtained by solving an irreversible geminate (i.e., two-particle) problem. The relaxation to equilibrium is not exponential but rather a power law. In the Markovian limit, the theory reduces to a set of ordinary rate equations with renormalized rate constants.

Simulation of the radiolysis of water using Green's functions of the diffusion equation

Radiation chemistry is of fundamental importance in the understanding of the effects of ionising radiation, notably with regard to DNA damage by indirect effect (e.g. damage by ·OH radicals created by the radiolysis of water). In the recent years, Green's functions of the diffusion equation (GFDEs) have been used extensively in biochemistry, notably to simulate biochemical networks in time and space. In the present work, an approach based on the GFDE will be used to refine existing models on the indirect effect of ionising radiation on DNA. As a starting point, the code RITRACKS (relativistic ion tracks) will be used to simulate the radiation track structure and calculate the position of all radiolytic species formed during irradiation. The chemical reactions between these radiolytic species and with DNA will be done by using an efficient Monte Carlo sampling algorithm for the GFDE of reversible reactions with an intermediate state that has been developed recently. These simulations should help the understanding of the contribution of the indirect effect in the formation of DNA damage, particularly with regards to the formation of double-strand breaks.

Influence of diffusion on the kinetics of multisite phosphorylation

When an enzyme modifies multiple sites on a substrate, the influence of the relative diffusive motion of the reactants cannot be described by simply altering the rate constants in the rate equations of chemical kinetics. We have recently shown that, even as a first approximation, new transitions between the appropriate species must also be introduced. The physical reason for this is that a kinase, after phosphorylating one site, can rebind and modify another site instead of diffusing away. The corresponding new rate constants depend on the capture or rebinding probabilities that an enzyme-substrate pair, which is formed after dissociation from one site, reacts at the other site rather than diffusing apart. Here we generalize our previous work to describe both random and sequential phosphorylation by considering inequivalent modification sites. In addition, anisotropic reactive sites (instead of uniformly reactive spheres) are explicitly treated by using localized sink and source terms in the reaction-diffusion equations for the enzyme-substrate pair distribution function. Finally, we show that our results can be rederived using a phenomenological approach based on introducing transient encounter complexes into the standard kinetic scheme and then eliminating them using the steady-state approximation.

Diffusional correlations among multiple active sites in a single enzyme

Simulations of the enzymatic dynamics of a model enzyme containing multiple substrate binding sites indicate the existence of diffusional correlations in the chemical reactivity of the active sites. A coarse-grain, particle-based, mesoscopic description of the system, comprising the enzyme, the substrate, the product and solvent, is constructed to study these effects. The reactive and non-reactive dynamics is followed using a hybrid scheme that combines molecular dynamics for the enzyme, substrate and product molecules with multiparticle collision dynamics for the solvent. It is found that the reactivity of an individual active site in the multiple-active-site enzyme is reduced substantially, and this effect is analyzed and attributed to diffusive competition for the substrate among the different active sites in the enzyme.

Effective reaction rates in diffusion-limited phosphorylation-dephosphorylation cycles

We investigate the kinetics of the ubiquitous phosphorylation-dephosphorylation cycle on biological membranes by means of kinetic Monte Carlo simulations on the triangular lattice. We establish the dependence of effective macroscopic reaction rate coefficients as well as the steady-state phosphorylated substrate fraction on the diffusion coefficient and concentrations of opposing enzymes: kinases and phosphatases. In the limits of zero and infinite diffusion, the numerical results agree with analytical predictions; these two limits give the lower and the upper bound for the macroscopic rate coefficients, respectively. In the zero-diffusion limit, which is important in the analysis of dense systems, phosphorylation and dephosphorylation reactions can convert only these substrates which remain in contact with opposing enzymes. In the most studied regime of nonzero but small diffusion, a contribution linearly proportional to the diffusion coefficient appears in the reaction rate. In this regime, the presence of opposing enzymes creates inhomogeneities in the (de)phosphorylated substrate distributions: The spatial correlation function shows that enzymes are surrounded by clouds of converted substrates. This effect becomes important at low enzyme concentrations, substantially lowering effective reaction rates. Effective reaction rates decrease with decreasing diffusion and this dependence is more pronounced for the less-abundant enzyme. Consequently, the steady-state fraction of phosphorylated substrates can increase or decrease with diffusion, depending on relative concentrations of both enzymes. Additionally, steady states are controlled by molecular crowders which, mostly by lowering the effective diffusion of reactants, favor the more abundant enzyme.

Diffusion modifies the connectivity of kinetic schemes for multisite binding and catalysis

The simplest way to describe the influence of the relative diffusion of the reactants on the time course of bimolecular reactions is to modify or renormalize the phenomenological rate constants that enter into the rate equations of conventional chemical kinetics. However, for macromolecules with multiple inequivalent reactive sites, this is no longer sufficient, even in the low concentration limit. The physical reason is that an enzyme (or a ligand) that has just modified (or dissociated from) one site can bind to a neighboring site rather than diffuse away. This process is not described by the conventional chemical kinetics, which is only valid in the limit that diffusion is fast compared with reaction. Using an exactly solvable many-particle reaction-diffusion model, we show that the influence of diffusion on the kinetics of multisite binding and catalysis can be accounted for by not only scaling the rates, but also by introducing new connections into the kinetic scheme. The rate constants that describe these new transitions or reaction channels turn out to have a transparent physical interpretation: The chemical rates are scaled by the appropriate probabilities that a pair of reactants, which are initially in contact, bind rather than diffuse apart. The theory is illustrated by application to phosphorylation of a multisite substrate.

Hydrogen transfer reactions in viscous media — Potential and free energy surfaces in solvent–solute coordinates and their kinetic implications

Two-dimensional potential energy and free energy surfaces are obtained using quantum mechanical and molecular dynamics calculations for four hydrogen transfer reactions in n-hexane solvent: the methyl–methane, n-propyl–n-propane, n-pentyl–n-pentane, and t-butyl–isobutane systems. The resultant surfaces have similar landscapes despite the fact the equilibrated solvent cavities for these systems are notably different in size and shape. The kinetic implications of these landscapes are discussed. The Arrhenius and tunneling kinetics of hydrogen transfer in nonpolar hydrocarbon solute–solvent systems are not expected to show any significant viscosity dependence.

Development of microbial-enzyme-mediated decomposition model parameters through steady-state and dynamic analyses

We developed a microbial-enzyme-mediated decomposition (MEND) model, based on the Michaelis-Menten kinetics, that describes the dynamics of physically defined pools of soil organic matter (SOC). These include particulate, mineral-associated, dissolved organic matter (POC, MOC, and DOC, respectively), microbial biomass, and associated exoenzymes. The ranges and/or distributions of parameters were determined by both analytical steady-state and dynamic analyses with SOC data from the literature. We used an improved multi-objective parameter sensitivity analysis (MOPSA) to identify the most important parameters for the full model: maintenance of microbial biomass, turnover and synthesis of enzymes, and carbon use efficiency (CUE). The model predicted that an increase of 2 degrees C (baseline temperature 12 degrees C) caused the pools of POC-cellulose, MOC, and total SOC to increase with dynamic CUE and decrease with constant CUE, as indicated by the 50% confidence intervals. Regardless of dynamic or constant CUE, the changes in pool size of POC, MOC, and total SOC varied from -8% to 8% under +2 degrees C. The scenario analysis using a single parameter set indicates that higher temperature with dynamic CUE might result in greater net increases in both POC-cellulose and MOC pools. Different dynamics of various SOC pools reflected the catalytic functions of specific enzymes targeting specific substrates and the interactions between microbes, enzymes, and SOC. With the feasible parameter values estimated in this study, models incorporating fundamental principles of microbial-enzyme dynamics can lead to simulation results qualitatively different from traditional models with fast/slow/passive pools.

Modeling of solvent flow effects in enzyme catalysis under physiological conditions

A stochastic model for the dynamics of enzymatic catalysis in explicit, effective solvents under physiological conditions is presented. Analytically-computed first passage time densities of a diffusing particle in a spherical shell with absorbing boundaries are combined with densities obtained from explicit simulation to obtain the overall probability density for the total reaction cycle time of the enzymatic system. The method is used to investigate the catalytic transfer of a phosphoryl group in a phosphoglycerate kinase-ADP-bis phosphoglycerate system, one of the steps of glycolysis. The direct simulation of the enzyme-substrate binding and reaction is carried out using an elastic network model for the protein, and the solvent motions are described by multiparticle collision dynamics which incorporates hydrodynamic flow effects. Systems where solvent-enzyme coupling occurs through explicit intermolecular interactions, as well as systems where this coupling is taken into account by including the protein and substrate in the multiparticle collision step, are investigated and compared with simulations where hydrodynamic coupling is absent. It is demonstrated that the flow of solvent particles around the enzyme facilitates the large-scale hinge motion of the enzyme with bound substrates, and has a significant impact on the shape of the probability densities and average time scales of substrate binding for substrates near the enzyme, the closure of the enzyme after binding, and the overall time of completion of the cycle.

Spatial simulations in systems biology: from molecules to cells

Cells are highly organized objects containing millions of molecules. Each biomolecule has a specific shape in order to interact with others in the complex machinery. Spatial dynamics emerge in this system on length and time scales which can not yet be modeled with full atomic detail. This review gives an overview of methods which can be used to simulate the complete cell at least with molecular detail, especially Brownian dynamics simulations. Such simulations require correct implementation of the diffusion-controlled reaction scheme occurring on this level. Implementations and applications of spatial simulations are presented, and finally it is discussed how the atomic level can be included for instance in multi-scale simulation methods.

Role of diffusion in the kinetics of reversible enzyme-catalyzed reactions

The accurate expression for the steady-state velocity of an irreversible enzyme-catalyzed reaction obtained by Shin and co-workers is generalized to allow for the rebinding of the product. The amplitude of the power-law (t(-1/2)) relaxation of the free- and bound-enzyme concentrations to steady-state values is expressed in terms of the steady-state velocity and the intrinsic (chemical) rate constants. This result is conjectured to be exact, even though our expression for the steady-state velocity in terms of microscopic parameters is only approximate.

Model of the initiation of signal transduction by ligands in a cell culture: simulation of molecules near a plane membrane comprising receptors

Cell communication is a key mechanism in tissue responses to radiation. Several molecules are implicated in radiation-induced signaling between cells, but their contributions to radiation risk are poorly understood. Meanwhile, Green's functions for diffusion-influenced reactions have appeared in the literature, which are applied to describe the diffusion of molecules near a plane membrane comprising bound receptors with the possibility of reversible binding of a ligand and activation of signal transduction proteins by the ligand-receptor complex. We have developed Brownian dynamics algorithms to simulate particle histories in this system which can accurately reproduce the theoretical distribution of distances of a ligand from the membrane, the number of reversibly bound particles, and the number of receptor complexes activating signaling proteins as a function of time, regardless of the number of time steps used for the simulation. These simulations will be of great importance to model interactions at low doses where stochastic effects induced by a small number of molecules or interactions come into play.

A Monte-Carlo step-by-step simulation code of the non-homogeneous chemistry of the radiolysis of water and aqueous solutions. Part I: theoretical framework and implementation

The importance of the radiolysis of water in irradiation of biological systems has motivated considerable theoretical and experimental work in the radiation chemistry of water and aqueous solutions. In particular, Monte-Carlo simulations of radiation track structure and non-homogeneous chemistry have greatly contributed to the understanding of experimental results in radiation chemistry of heavy ions. Actually, most simulations of the non-homogeneous chemistry are done using the Independent Reaction Time (IRT) method, a very fast technique. The main limitation of the IRT method is that the positions of the radiolytic species are not calculated as a function of time, which is needed to simulate the irradiation of more complex systems. Step-by-step (SBS) methods, which are able to provide such information, have been used only sparsely because these are time consuming in terms of calculation. Recent improvements in computer performance now allow the regular use of the SBS method in radiation chemistry. In the present paper, the first of a series of two, the SBS method is reviewed in detail. To these ends, simulation of diffusion of particles and chemical reactions in aqueous solutions is reviewed, and implementation of the program is discussed. Simulation of model systems is then performed to validate the adequacy of stepwise diffusion and reaction schemes. In the second paper, radiochemical yields of simulated radiation tracks calculated by the SBS program in different conditions of LET, pH, and temperature are compared with results from the IRT program and experimental data.

Excited-state reversible geminate recombination in two dimensions

Excited-state reversible geminate recombination with two different lifetimes and quenching is investigated in two dimensions. From the exact Green function in the Laplace domain, analytic expressions of two-dimensional survival and binding probabilities are obtained at short and long times. We find that a new pattern of kinetic transition occurs in two dimensions. The long-time effective survival probabilities show a pattern of (ln t)(-1)-->constant-->e(t) depending on the rate constants while the effective binding probabilities show t(-1)(ln t)(-2)-->t(-1)-->e(t).

Diffusion-influenced excited-state reversible geminate ABCD reaction in the presence of an external field

We obtained the exact Green functions, in the Laplace domain, for a diffusion-influenced excited-state reversible geminate ABCD reaction with two different lifetimes and quenching processes under a constant external field in one dimension. Analytic expressions for the survival probabilities of the initial and final states are obtained in the time domain at short and long times, respectively. The short-time approximations obtained in this work are valid for t</K/(-1), where K depends on several parameters of the system. The analysis of the long-time asymptotic behaviors reveals rather complex kinetic transitions dependent upon the field and lifetimes. We also find a destructive interplay leading to the reduction in the number of kinetic transitions similar to that found for the excited-state geminate ABC reaction with an external field in one dimension.