Kinetics of diffusion-assisted reactions in microheterogeneous systems

Multiscale Approach for Computing Gated Ligand Binding from Molecular Dynamics and Brownian Dynamics Simulations

We develop an approach to characterize the effects of gating by a multiconformation protein consisting of macrostate conformations that are either accessible or inaccessible to ligand binding. We first construct a Markov state model of the apo-protein from atomistic molecular dynamics simulations from which we identify macrostates and their conformations, compute their relative macrostate populations and interchange kinetics, and structurally characterize them in terms of ligand accessibility. We insert the calculated first-order rate constants for conformational transitions into a multistate gating theory from which we derive a gating factor γ that quantifies the degree of conformational gating. Applied to HIV-1 protease, our approach yields a kinetic network of three accessible (semi-open, open, and wide-open) and two inaccessible (closed and a newly identified, "parted") macrostate conformations. The parted conformation sterically partitions the active site, suggesting a possible role in product release. We find that the binding kinetics of drugs and drug-like inhibitors to HIV-1 protease falls in the slow gating regime. However, because γ = 0.75, conformational gating only modestly slows ligand binding. Brownian dynamics simulations of the diffusional association of eight inhibitors to the protease─having a wide range of experimental association constants (∼10⁴-10¹⁰ M^-1 s^-1)─yields gated rate constants in the range of ∼0.5-5.7 × 10⁸ M^-1 s^-1. This indicates that, whereas the association rate of some inhibitors could be described by the model, for many inhibitors either subsequent conformational transitions or alternate binding mechanisms may be rate-limiting. For systems known to be modulated by conformational gating, the approach could be scaled computationally efficiently to screen association kinetics for a large number of ligands.

Bimolecular photo-induced electron transfer enlightened by diffusion

Photochemical electron transfer between freely diffusing molecules has been studied extensively. Here, we try to elucidate how much these works have contributed to the understanding of electron transfer. To this end, we have revisited the work performed in the experimental and theoretical areas of concern from the beginning of the 20th century up to the present day. We present a critical look at the major contributions and compile the current picture of a variety of phenomena around electron transfer in solution. This is based on two main developments, besides the theory of Marcus: encounter theories of diffusion and laser techniques in time-resolved spectroscopy.

Diffusion toward non-overlapping partially reactive spherical traps: Fresh insights onto classic problems

Several classic problems for particles diffusing outside an arbitrary configuration of non-overlapping partially reactive spherical traps in three dimensions are revisited. For this purpose, we describe the generalized method of separation of variables for solving boundary value problems of the associated modified Helmholtz equation. In particular, we derive a semi-analytical solution for the Green function that is the key ingredient to determine various diffusion-reaction characteristics such as the survival probability, the first-passage time distribution, and the reaction rate. We also present modifications of the method to determine numerically or asymptotically the eigenvalues and eigenfunctions of the Laplace operator and the Dirichlet-to-Neumann operator in such perforated domains. Some potential applications in chemical physics and biophysics are discussed, including diffusion-controlled reactions for mortal particles.

Bimolecular multistage diffusion-influenced chemical reactions proceeding from different sites in solutions. I. Rate constants

General matrix algebraic equations for calculating rate constants of multistage diffusion-influenced reactions (involving bimolecular exchange reactions as elementary stages) in liquid solutions that proceed from different active sites in the immediate vicinity of the contact of reactants have been obtained on the basis of the kinematic approximation developed by the authors earlier. The equations make it possible to express rate constants of any multistage multisite bimolecular reaction between non-identical reactants in terms of the defined reaction constants and stationary Green functions averaged over reaction sites and completely determined by molecular motion of reactants or their molecular groups. The asymptotic behavior of these rate constants as they attain their steady-state values on completion of the transient stage is established. It is shown that it coincides with the corresponding exact time asymptote. Calculations are made with some specific two-stage (three-channel) bimolecular reactions as an example.

Diffusion-influenced reaction rates for active "sphere-prolate spheroid" pairs and Janus dimers

The purpose of this paper is twofold. First, we provide a concise introduction to the generalized method of separation of variables for solving diffusion problems in canonical domains beyond conventional arrays of spheres. Second, as an important example of its application in the theory of diffusion-influenced reactions, we present an exact solution of the axially symmetric problem on diffusive competition in an array of two active particles (including Janus dumbbells) constructed of a prolate spheroid and a sphere. In particular, we investigate how the reaction rate depends on sizes of active particles, spheroid aspect ratio, particles' surface reactivity, and distance between their centers.

Growth kinetics of circular liquid domains on vesicles by diffusion-controlled coalescence

Motivated by recent experiments on multicomponent membranes, the growth kinetics of domains on vesicles are theoretically studied. It is known that the steady-state rate of coalescence cannot be obtained by taking the long-time limit of the coalescence rate when the membrane is regarded as an infinite two-dimensional (2D) system. The steady-state rate of coalescence is obtained by explicitly taking into account the spherical vesicle shape. Using the expression for the 2D diffusion coefficient obtained in the limit of small domain size, an analytical expression for domain growth kinetics is obtained when a circular shape is always maintained. For large domains, the growth kinetics are discussed by investigating the size dependence of the coalescence rate, using the expression for the diffusion coefficient of arbitrary domain size.

Synthesis and photophysical properties of a covalently bonded palladiummeso-sulfophenylporphyrin-poly(vinyl alcohol)polymer with potential applications as an oxygen sensor

The synthesis and photochemical characterization of a covalently bonded palladium meso-sulfophenylporphyrin-poly(vinyl alcohol)polymer ( PVA - PdTPPS ), made via reaction of the chlorosulfonyl derivative of Pd meso-tetrakis-phenylporphyrin with the alcohol group of the polymer to form the sulfonate ester, is described. Absorption, emission and phosphorescence lifetime data are reported. In both the solid film and in aqueous solution phosphorescence decays in the absence of oxygen are non-monoexponential and suggest heterogeneity in the decay and oxygen quenching kinetics. As a solid film there is very little phosphorescence quenching by oxygen at moderate pressures of O2(0-30 Torr), but in aqueous solution oxygen quenches at a rate of 7.5 (±0.5) × 108M . s−1. The synthetic route described is versatile enough to be used with a variety of metalloporphyrins and polymers and this allows the possibility of extensive tuning in terms of lumophore lifetime, polarity, solubility, location in heterogeneous media, and compatibility with other polymers.

Quantitative understanding of cell signaling: the importance of membrane organization

Systems biology modeling of signal transduction pathways traditionally employs ordinary differential equations, deterministic models based on the assumptions of spatial homogeneity. However, this can be a poor approximation for certain aspects of signal transduction, especially its initial steps: the cell membrane exhibits significant spatial organization, with diffusion rates approximately two orders of magnitude slower than those in the cytosol. Thus, to unravel the complexities of signaling pathways, quantitative models must consider spatial organization as an important feature of cell signaling. Furthermore, spatial separation limits the number of molecules that can physically interact, requiring stochastic simulation methods that account for individual molecules. Herein, we discuss the need for mathematical models and experiments that appreciate the importance of spatial organization in the membrane.

Tracer diffusion in silica inverse opals

We employed fluorescence correlation spectroscopy (FCS) to study the diffusion of small fluorescence tracers in liquid filled silica inverse opals. The inverse opals consisted of a nanoporous silica scaffold spanning a hexagonal crystal of spherical voids of 360 nm diameter connected by circular pores of 70 nm diameter. The diffusion of Alexa Fluor 488 in water and of perylene-3,4,9,10-tetracarboxylic diimide (PDI) in toluene was studied. Three diffusion modes could be distinguished: (1) Free diffusion limited by the geometric constraints given by the inverse opal, where, as compared to the free solution, this diffusion is slowed down by a factor of 3-4, (2) slow diffusion inside the nanoporous matrix of the silica scaffold, and (3) diffusion limited by adsorption. On the length scale of the focus of a confocal microscope of roughly 400 nm diffusion was non-Fickian in all cases.

Single molecule kinetics under stochastic gating

In this note, we generally calculate the statistical blinking properties of single molecule based on the generating function approach developed recently. Specifically, we investigate the statistical variables of single molecule blinking kinetics under the stochastic gating.

Exact asymptotics for nonradiative migration-accelerated energy transfer in one-dimensional systems

We study direct energy transfer by multipolar or exchange interactions between diffusive excited donor and diffusive unexcited acceptors. Extending over the case of long-range transfer of an excitation energy a nonperturbative approach by Bray and Blythe [Phys. Rev. Lett. 89, 150601 (2002)], originally developed for contact diffusion-controlled reactions, we determine exactly long-time asymptotics of the donor decay function in one-dimensional systems.

Attractive membrane domains control lateral diffusion

Lipid membranes play a fundamental role in vital cellular functions such as signal transduction. Many of these processes rely on lateral diffusion within the membrane, generally a complex fluid containing ordered microdomains. However, little attention has been paid to the alterations in transport dynamics of a diffusing species caused by long-range interactions with membrane domains. In this paper, we address the effect of such interactions on diffusive transport by studying lateral diffusion in a phase-separated Langmuir phospholipid monolayer via single-particle tracking. We find that attractive dipole-dipole interactions between condensed phase domains and diffusing probe beads lead to transient confinement at the phase boundaries, causing a transition from two- to one-dimensional diffusion. Using Brownian dynamics simulations, the long-term diffusion constant for such a system is found to have a sensitive, Boltzmann-like, dependence on the interaction strength. In addition, this interaction strength is shown to be a strong function of the ratio of domain to particle size. As similar interactions are expected in biological membranes, the modulation of diffusive transport dynamics by varying interaction strength and/or domain size may offer cells selective spatial and temporal control over signaling processes.

A biological interpretation of transient anomalous subdiffusion. II. Reaction kinetics

Reaction kinetics in a cell or cell membrane is modeled in terms of the first passage time for a random walker at a random initial position to reach an immobile target site in the presence of a hierarchy of nonreactive binding sites. Monte Carlo calculations are carried out for the triangular, square, and cubic lattices. The mean capture time is expressed as the product of three factors: the analytical expression of Montroll for the capture time in a system with a single target and no binding sites; an exact expression for the mean escape time from the set of lattice points; and a correction factor for the number of targets present. The correction factor, obtained from Monte Carlo calculations, is between one and two. Trapping may contribute significantly to noise in reaction rates. The statistical distribution of capture times is obtained from Monte Carlo calculations and shows a crossover from power-law to exponential behavior. The distribution is analyzed using probability generating functions; this analysis resolves the contributions of the different sources of randomness to the distribution of capture times. This analysis predicts the distribution function for a lattice with perfect mixing; deviations reflect imperfect mixing in an ordinary random walk.

Concentration-dependent self-diffusion of liquids in nanopores: a nuclear magnetic resonance study

Nuclear magnetic resonance has been applied to study the details of molecular motion of low-molecular-weight polar and nonpolar organic liquids in nanoporous silicon crystals of straight cylindrical pore morphology at different pore loadings. Effective self-diffusion coefficients as obtained using the pulsed field gradient nuclear magnetic resonance method were found to pass through a maximum with increasing concentration for all liquids under study. Taking account of a concentration-dependent coexistence of capillary condensed, adsorbed and gaseous phases a generalized model for the effective self-diffusion coefficient was developed and shown to satisfactorily explain the experimental results. An explicit use of the adsorption isotherm properties within the model extends its applicability to the mesoporous range and highlights the role of surface interaction for the transport of molecules in small pores. The problem of surface diffusion and diffusion of multilayered molecules is also addressed.

Gated binding of ligands to HIV-1 protease: Brownian dynamics simulations in a coarse-grained model

The internal motions of proteins may serve as a "gate" in some systems, which controls ligand-protein association. This study applies Brownian dynamics simulations in a coarse-grained model to study the gated association rate constants of HIV-1 proteases and drugs. The computed gated association rate constants of three protease mutants, G48V/V82A/I84V/L90M, G48V, and L90M with three drugs, amprenavir, indinavir, and saquinavir, yield good agreements with experiments. The work shows that the flap dynamics leads to "slow gating". The simulations suggest that the flap flexibility and the opening frequency of the wild-type, the G48V and L90M mutants are similar, but the flaps of the variant G48V/V82A/I84V/L90M open less frequently, resulting in a lower gated rate constant. The developed methodology is fast and provides an efficient way to predict the gated association rate constants for various protease mutants and ligands.

Spatial modeling of dimerization reaction dynamics in the plasma membrane: Monte Carlo vs. continuum differential equations

Bimolecular reactions in the plasma membrane, such as receptor dimerization, are a key signaling step for many signaling systems. For receptors to dimerize, they must first diffuse until a collision happens, upon which a dimerization reaction may occur. Therefore, study of the dynamics of cell signaling on the membrane may require the use of a spatial modeling framework. Despite the availability of spatial simulation methods, e.g., stochastic spatial Monte Carlo (MC) simulation and partial differential equation (PDE) based approaches, many biological models invoke well-mixed assumptions without completely evaluating the importance of spatial organization. Whether one is to utilize a spatial or non-spatial simulation framework is therefore an important decision. In order to evaluate the importance of spatial effects a priori, i.e., without performing simulations, we have assessed the applicability of a dimensionless number, known as second Damköhler number (Da), defined here as the ratio of time scales of collision and reaction, for 2-dimensional bimolecular reactions. Our study shows that dimerization reactions in the plasma membrane with Da approximately >0.1 (tested in the receptor density range of 10(2)-10(5)/microm(2)) require spatial modeling. We also evaluated the effective reaction rate constants of MC and simple deterministic PDEs. Our simulations show that the effective reaction rate constant decreases with time due to time dependent changes in the spatial distribution of receptors. As a result, the effective reaction rate constant of simple PDEs can differ from that of MC by up to two orders of magnitude. Furthermore, we show that the fluctuations in the number of copies of signaling proteins (noise) may also depend on the diffusion properties of the system. Finally, we used the spatial MC model to explore the effect of plasma membrane heterogeneities, such as receptor localization and reduced diffusivity, on the dimerization rate. Interestingly, our simulations show that localization of epidermal growth factor receptor (EGFR) can cause the diffusion limited dimerization rate to be up to two orders of magnitude higher at higher average receptor densities reported for cancer cells, as compared to a normal cell.

Ion–electron recombination on silica gel surfaces: experiment and modelling

Kinetics on silica gel and other solid, porous surfaces are often complex. In this paper we have studied the decay kinetics of radical cations produced following multiphoton ionisation on silica gel, and have characterised these using an empirical model. Trends in kinetics have been observed both as a function of concentration and of temperature. Concentration dependent studies suggest heterogeneity of surface adsorption, both in terms of the nature of adsorption sites and aggregation effects. Temperature dependent studies show that the activation energies for surface diffusion correlate with the size of the radical cation, suggesting that its movement rather than that of the electron dominates the observed kinetics. Monte Carlo simulations have been shown to give useful qualitative insights into the interpretation of the extracted parameters, in particular into how apparent distributions of rate constants can arise as a result of low surface dimensionality.

Non-Markovian stochastic Liouville equation and its Markovian representation

The non-Markovian variant of the stochastic Liouville equation (SLE) is studied within the continuous time random walk approach (CTRWA). The CTRWA-based non-Markovian SLE is shown to be equivalently represented by the corresponding conventional Markovian SLE. This Markovian representation provides a rigorous method for deriving the non-Markovian SLE and allows for a physically clear interpretation of the specific features of this SLE. It also enables one to develop convenient non-Markovian models useful for applications, some of which are discussed in detail. Special attention is given to the discussion of anomalous long-tailed CTRW processes and non-Markovian SLE. The obtained results are applied to the analysis of the effect of rate fluctuations on chemical reaction kinetics. It is shown, in particular, that the anomalous fluctuations not only influence the reaction rate but also change the reaction kinetics itself.

First-passage approach for permeable traps

Many reactive processes in complex materials involve absorption of diffusing molecules. Recently, there has been interest in particle interaction with partially absorbing (or permeable) traps. Here, we present a simple and efficient method for accounting for the non-diffusion-limited reaction of particles when the flux of particles to the trap is governed by surface permeability. The trapping probability is determined from a one-dimensional Green's function, which results in a simple algebraic expression. This expression, which applies in the region immediately adjacent to the trap, is then used with a first-passage approach far from the trap. When applied to a suspension of permeable traps, the method is seen to give accurate results over the concentration range. The method is applied to the competition of reactive particles in a suspension of permeable spheres with a reactive continuous phase.

Exchange mechanisms for sodium dodecyl sulfate micelles: high salt concentration

Solute exchange experiments for the pyrene-labeled triglyceride TG-Py solubilized in sodium dodecyl sulfate (SDS) micelles in the presence and absence of salt show that the "observed" rate constant k(obs) for solute exchange varies by over 6 orders of magnitude as the free sodium ion concentration [Na(+)](aq) is varied between 10 and 850 mM. There is a sharp break in the log-log plot of k(obs) versus [Na(+)](aq) in the range of [Na(+)](aq) = 200 mM, with the exchange rate showing a weaker dependence on [Na(+)](aq) above this concentration. Up to 100 mM added NaCl, this exchange takes place essentially exclusively by a micelle fission mechanism in which each submicelle carries off one of the solutes. At higher salt concentrations, a bimolecular process becomes increasingly important. This fusion process, which involves formation of a transient supermicelle followed by fission back to two normal micelles, becomes the dominant process at high salt concentrations. The fission rate appears to level off for salt concentrations above 300-400 mM. These fission and fusion processes are related in an intimate way to the changes in the size and shape of the SDS micelles with increasing salt concentration.

Enhancement of the recollision rate in diffusion-influenced reactions in an inhomogeneous medium

Brownian dynamics simulations were performed to determine the first collision and recollision rates of spherical reagent particles in a reaction medium made heterogeneous by the presence of randomly located inert spherical obstacles in a continuum solvent. The recollision rate vp (and hence the overall reactive collision rate when activation energy is high) was always enhanced by the presence of obstacles, the degree of enhancement increasing with the volume fraction occupied by obstacles (phi) and with decreasing reagent concentration phiR. Enhancement increased with obstacle size at high phiR, and fell with increasing obstacle size at low phiR. The vp-phiR data follow a power law, where the scaling factor betap decreased with decreasing obstacle size and increasing phi, and the prefactor kp initially increased with phi and then fell (except for large obstacles). The behavior of betap appears to be largely due to the obstacles reducing the probability that reagent particles escape from each other after collision, while the dominant factors responsible for the behavior of kp appear to be initially the effect of obstacles in enhancing effective local reagent concentration, and then (for small obstacles), their reduction of the reagent-particle coordination number. As the energy of activation falls, the reactive collision rate becomes less influenced by the reagent recollision rate and more influenced by the rate of first collision. For low-activation-energy reactions, the presence of obstacles depresses the reactive collision rate if reagent concentration is low or if the obstacles are small and their concentration high. The fall in the reactive collision rate with decreasing activation energy is steeper, the lower the reagent concentration and the smaller the obstacles.

Diffusion and trapping in a suspension of spheres with simultaneous reaction in the continuous phase

Much progress has been made in modeling the reaction of Brownian particles with spherical traps. Previously, work has focused on the effective reaction rate of systems of particles that diffuse freely until they are trapped by spheres in the dispersion. A particularly effective and efficient method to describe the reacting system is based on first-passage time distributions, from which an effective reaction rate coefficient of the suspension can be determined. The analysis presented here addresses reaction and diffusion in systems in which particles can undergo reaction in the continuous phase as well as reaction at the sphere surface. The first-passage method is extended to allow reaction or decay of the diffusing species in the continuous phase. The diffusion path is divided into a series of first-passage regions and is considered the probability of the particle being consumed in each of these regions. This allows the determination of the total reaction rate of the suspension (continuous phase reaction plus trapping) and the relative consumption rate in each phase. The extended method is applied to a model system of concentric spheres with a known continuum solution. It is shown to be accurate for consumption of reactant in the continuous phase from approximately 0 to approximately 100%. The method then is applied to a suspension of spheres.