Concentration effects on the rates of irreversible diffusion-influenced reactions

The generalized method of separation of variables for diffusion-influenced reactions: Irreducible Cartesian tensor technique

Motivated by the various applications of the trapping diffusion-influenced reaction theory in physics, chemistry, and biology, this paper deals with irreducible Cartesian tensor (ICT) technique within the scope of the generalized method of separation of variables (GMSV). We provide a survey from the basic concepts of the theory and highlight the distinctive features of our approach in contrast to similar techniques documented in the literature. The solution to the stationary diffusion equation under appropriate boundary conditions is represented as a series in terms of ICT. By means of proved translational addition theorem, we straightforwardly reduce the general boundary value diffusion problem for N spherical sinks to the corresponding resolving infinite set of linear algebraic equations with respect to the unknown tensor coefficients. These coefficients exhibit an explicit dependence on the arbitrary three-dimensional configurations of N sinks with different radii and surface reactivities. Our research contains all relevant mathematical details such as terminology, definitions, and geometrical structure, along with a step by step description of the GMSV algorithm with the ICT technique to solve the general diffusion boundary value problem within the scope of Smoluchowski's trapping model.

Accurate analytical calculation of the rate coefficient for the diffusion-controlled reactions due to hyperbolic diffusion

Using an approach based on the diffusion analog of the Cattaneo-Vernotte differential model, we find the exact analytical solution to the corresponding time-dependent linear hyperbolic initial boundary value problem, describing irreversible diffusion-controlled reactions under Smoluchowski's boundary condition on a spherical sink. By means of this solution, we extend exact analytical calculations for the time-dependent classical Smoluchowski rate coefficient to the case that includes the so-called inertial effects, occurring in the host media with finite relaxation times. We also present a brief survey of Smoluchowski's theory and its various subsequent refinements, including works devoted to the description of the short-time behavior of Brownian particles. In this paper, we managed to show that a known Rice's formula, commonly recognized earlier as an exact reaction rate coefficient for the case of hyperbolic diffusion, turned out to be only its approximation being a uniform upper bound of the exact value. Here, the obtained formula seems to be of great significance for bridging a known gap between an analytically estimated rate coefficient on the one hand and molecular dynamics simulations together with experimentally observed results for the short times regime on the other hand. A particular emphasis has been placed on the rigorous mathematical treatment and important properties of the relevant initial boundary value problems in parabolic and hyperbolic diffusion theories.

Interplay of reactive interference and crowding effects in the diffusion-influenced reaction kinetics

We investigate the interplay of reactive interference and crowding effects in the irreversible diffusion-influenced bimolecular reactions of the type A+B→P+B by using the Brownian dynamics simulation method. It is known that the presence of nonreactive crowding agents retards the reaction rate when the volume fraction of the crowding agents is large enough. On the other hand, a high concentration of B is known to increase the reaction rate more than expected from the mass action law, although the B's may also act as crowders. Therefore, it would be interesting to see which effect dominates when the number density of B as well as the number density of the crowders increases. We will present an approximate theory that provides a reasonable account for the Brownian dynamics simulation results.

Particle-Based Simulation Reveals Macromolecular Crowding Effects on the Michaelis-Menten Mechanism

Many computational models for analyzing and predicting cell physiology rely on in vitro data collected in dilute and controlled buffer solutions. However, this can mislead models because up to 40% of the intracellular volume—depending on the organism, the physiology, and the cellular compartment—is occupied by a dense mixture of proteins, lipids, polysaccharides, RNA, and DNA. These intracellular macromolecules interfere with the interactions of enzymes and their reactants and thus affect the kinetics of biochemical reactions, making in vivo reactions considerably more complex than the in vitro data indicates. In this work, we present a new, to our knowledge, type of kinetics that captures and quantifies the effect of volume exclusion and other spatial phenomena on the kinetics of elementary reactions. We further developed a framework that allows for the efficient parameterization of these kinetics using particle simulations. Our formulation, entitled generalized elementary kinetics, can be used to analyze and predict the effect of intracellular crowding on enzymatic reactions and was herein applied to investigate the influence of crowding on phosphoglycerate mutase in Escherichia coli, which exhibits prototypical reversible Michaelis-Menten kinetics. Current research indicates that many enzymes are reaction limited and not diffusion limited, and our results suggest that the influence of fractal diffusion is minimal for these reaction-limited enzymes. Instead, increased association rates and decreased dissociation rates lead to a strong decrease in the effective maximal velocities V_max and the effective Michaelis-Menten constants K_M under physiologically relevant volume occupancies. Finally, the effects of crowding were explored in the context of a linear pathway, with the finding that crowding can have a redistributing effect on the effective flux responses in the case of twofold enzyme overexpression. We suggest that this framework, in combination with detailed kinetics models, will improve our understanding of enzyme reaction networks under nonideal conditions.