Exact solution for the geminate ABCD reaction

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.

Surface reaction-diffusion kinetics on lattice at the microscopic scale

Microscopic models of reaction-diffusion processes on the cell membrane can link local spatiotemporal effects to macroscopic self-organized patterns often observed on the membrane. Simulation schemes based on the microscopic lattice method (MLM) can model these processes at the microscopic scale by tracking individual molecules, represented as hard spheres, on fine lattice voxels. Although MLM is simple to implement and is generally less computationally demanding than off-lattice approaches, its accuracy and consistency in modeling surface reactions have not been fully verified. Using the Spatiocyte scheme, we study the accuracy of MLM in diffusion-influenced surface reactions. We derive the lattice-based bimolecular association rates for two-dimensional (2D) surface-surface reaction and one-dimensional (1D) volume-surface adsorption according to the Smoluchowski-Collins-Kimball model and random walk theory. We match the time-dependent rates on lattice with off-lattice counterparts to obtain the correct expressions for MLM parameters in terms of physical constants. The expressions indicate that the voxel size needs to be at least 0.6% larger than the molecule to accurately simulate surface reactions on triangular lattice. On square lattice, the minimum voxel size should be even larger, at 5%. We also demonstrate the ability of MLM-based schemes such as Spatiocyte to simulate a reaction-diffusion model that involves all dimensions: three-dimensional (3D) diffusion in the cytoplasm, 2D diffusion on the cell membrane, and 1D cytoplasm-membrane adsorption. With the model, we examine the contribution of the 2D reaction pathway to the overall reaction rate at different reactant diffusivity, reactivity, and concentrations.

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%.

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.

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.

Diffusional effects on the reversible excited-state proton transfer. From experiments to Brownian dynamics simulations

We have studied an excited state proton transfer (ESPT) from the cationic "super" photoacid N-methyl 6-hydroxyquinolinium perfluorobutane sulfonate to non-aqueous solvents using picosecond and nanosecond time-resolved fluorescence spectroscopy. Upon the photoinduced adiabatic deprotonation from the hydroxyl moiety, a quinolinium zwitterion with a highly anisotropic charge distribution is formed. Due to the complexity of the resultant photodissociated system, the typical description of the reversible ESPT within the framework of the Spherically Symmetric Diffusion Problem (SSDP) is not possible. Additional complications are caused by the presence of a counteranion particle which affects the proton mobility. To better understand the ESPT process, we have performed extensive Brownian dynamics (BD) simulations of this three-body system as a tool to reveal the nature of the nonstationary interaction potentials and to elucidate the role of a counterion in the diffusion and reactive properties of the proton. Moreover, our results demonstrated that the anisotropy of the potential force can be taken into account after adapting this force for use in the SSDP. The results of both BD simulations and SSDP calculation with the adapted force field were used to fit the experimental kinetics of this three-body problem adequately.

A new multicompartmental reaction-diffusion modeling method links transient membrane attachment of E. coli MinE to E-ring formation

Many important cellular processes are regulated by reaction-diffusion (RD) of molecules that takes place both in the cytoplasm and on the membrane. To model and analyze such multicompartmental processes, we developed a lattice-based Monte Carlo method, Spatiocyte that supports RD in volume and surface compartments at single molecule resolution. Stochasticity in RD and the excluded volume effect brought by intracellular molecular crowding, both of which can significantly affect RD and thus, cellular processes, are also supported. We verified the method by comparing simulation results of diffusion, irreversible and reversible reactions with the predicted analytical and best available numerical solutions. Moreover, to directly compare the localization patterns of molecules in fluorescence microscopy images with simulation, we devised a visualization method that mimics the microphotography process by showing the trajectory of simulated molecules averaged according to the camera exposure time. In the rod-shaped bacterium Escherichia coli, the division site is suppressed at the cell poles by periodic pole-to-pole oscillations of the Min proteins (MinC, MinD and MinE) arising from carefully orchestrated RD in both cytoplasm and membrane compartments. Using Spatiocyte we could model and reproduce the in vivo MinDE localization dynamics by accounting for the previously reported properties of MinE. Our results suggest that the MinE ring, which is essential in preventing polar septation, is largely composed of MinE that is transiently attached to the membrane independently after recruited by MinD. Overall, Spatiocyte allows simulation and visualization of complex spatial and reaction-diffusion mediated cellular processes in volumes and surfaces. As we showed, it can potentially provide mechanistic insights otherwise difficult to obtain experimentally.

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.

Diffusion-influenced excited-state reversible transfer reactions, A*+B⇌C*+D, with two different lifetimes: Theories and simulations

We report accurate Brownian simulation results for the kinetics of the pseudo-first-order diffusion-influenced excited-state reversible transfer reaction A(*) + Bright harpoon over left harpoonC(*) + D with two different lifetimes using two different propagation algorithms. The results are used to test approximate solutions for this many-particle problem. Available theories fail when one of the two reactions or (decay) rate constants is large. To remedy this situation, we develop two uniform approximations, which are based on introducing a generalized Smoluchowski term into the relaxation-time approximation. The best of these is the extended unified theory of reversible target reactions, which reduces correctly in all limits and exhibits superior agreement with simulations.

Exact solution of the excited-state geminate A*+B⇄C*+D reaction with two different lifetimes and quenching

The authors obtain, in the Laplace transform space, the exact analytic solution for the Green function and survival probabilities for the excited-state diffusion-influenced reversible geminate reaction, A*+B <==> C*+D, with two different lifetimes and in the presence of an added quenching process. This extends a previous investigation by Popov and Agmon [J. Chem. Phys. 117, 5770 (2002)] of the ground-state reaction without quenching. The long-time asymptotic behavior of the survival probabilities is obtained in the time domain. It is found to be different from the equal-lifetime case. This paper also provides a useful short-time approximation for the kinetics.