Nonclassical kinetics in three dimensions: Simulations of elementary A+B and A+A reactions

Sources of anomalous diffusion on cell membranes: a Monte Carlo study

A stochastic random walk model of protein molecule diffusion on a cell membrane was used to investigate the fundamental causes of anomalous diffusion in two-dimensional biological media. Three different interactions were considered: collisions with fixed obstacles, picket fence posts, and capture by, or exclusion from, lipid rafts. If motion is impeded by randomly placed, fixed obstacles, we find that diffusion can be highly anomalous, in agreement with previous studies. In contrast, collision with picket fence posts has a negligible effect on the anomalous exponent at realistic picket fence parameters. The effects of lipid rafts are more complex. If proteins partition into lipid rafts there is a small to moderate effect on the anomalous exponent, whereas if proteins are excluded from rafts there is a large effect on the anomalous exponent. In combination, these mechanisms can explain the level of anomaly in experimentally observed membrane diffusion, suggesting that anomalous diffusion is caused by multiple mechanisms whose effects are approximately additive. Finally, we show that the long-range diffusion rate, D(macro), estimated from fluorescence recovery after photobleaching studies, can be much smaller than D(micro), the small-scale diffusion rate, and is highly sensitive to obstacle densities and other impeding structures.

Effect of a slit-shaped trap on depletion kinetics within a microchannel

The diffusion-limited trapping reaction kinetics of the growth of the depletion zone within and around a "slit-shaped" trap in a flat microchannel was studied experimentally and numerically. In the experiment, an ellipse-shaped laser beam acted as a slit trap in a long, flat capillary, and the trapping reaction is photobleaching of fluorescein dye. The parameter studied was the theta distance, i.e., the distance from the trap to the point where the reactant concentration has been locally depleted to the specific survival fraction [theta] of its initial bulk value. When the trap is perfect, then, due to the geometry of the trap and the reactor, as many as three time regimes can be found, with up to two crossover transitions. The number of crossovers is determined by the relative sizes of the trap and the microreactor. In the case of two crossovers, we show that the first crossover relates to the length of the trap, while the second crossover relates to the width of the reactor. When the slit trap is imperfect and its width cannot be neglected, as is the case in the experiments, a nontrivial early behavior is observed, followed by two regions in time, separated by a single crossover only.

Molecular-dynamics simulations for nonclassical kinetics of diffusion-controlled bimolecular reactions

Molecular-dynamics simulations are presented for the diffusion-controlled bimolecular reaction A+B<==>C in two and three dimensions. The reactants and solvent molecules are modeled as spheres interacting via continuous potential-energy functions. The interaction potential between two reactants contains a deep well that results in a reaction. When the solvent concentration is low and the reactant dynamics is essentially ballistic, the system reaches equilibrium rapidly, and the reaction follows classical kinetics with exponential decay to the equilibrium. When the solvent concentration is high the particles enter the normal diffusion regime quickly and nonclassical behavior is observed, i.e., the reactant concentrations approach equilibrium as t(-d/2) where d is the dimensionality of space. When the reaction well depth is large, however, the reaction becomes irreversible within the simulation time. In this case the reactant concentrations decay as t(-d/4). Interestingly this behavior is also observed at intermediate times for reversible reactions.

Depletion kinetics in the photobleaching trapping reaction inside a flat microchannel

The diffusion-limited kinetics of the growth of a depletion zone around a static point trap in a thin, long channel geometry was studied using a laser photobleaching experiment of fluorescein dye inside a flat rectangular capillary. The dynamics of the depletion zone was monitored by the theta distance, defined as the distance from the trap to the point where the reactant concentration has been locally depleted to the specified survival fraction (theta) of its initial bulk value. A dimensional crossover from two dimensions to one dimension, due to the finite width of the reaction zone, was observed. We define a "parallel" and a "perpendicular" theta distance, along the slab long and short dimensions, respectively, and study their time development as a means to study the asymmetrical nature of the slab geometry. For all theta values, the crossover occurs concurrently for both theta distances when the depletion zone touches the boundary for the first time. We derive theoretical expressions for this geometry and compare them with the experimental data. We also obtain important insight from the ratio of the reactant concentration profiles in the parallel and perpendicular directions. Exact enumeration and Monte Carlo simulations support the anomalous depletion scaling results. Nevertheless, the crossover time (tau(c)) is still found to scale with the width (W) of the rectangular reaction zone as tau(c) approximately W2 , as expected from the basic Einstein diffusion law.

Spatial fluctuations and anomalous reaction order in a reactive scheme involving a cooperative full desorption

Anomalous reaction rates have been found in the hydrogen desorption of H-terminated surfaces in semiconductor epitaxy, with a reaction order shifting from two to one, or even taking fractional values. We analyze the issue in terms of a cooperative full desorption (CFD) reaction A+A--k3-->S+S , coupled to an adsorption reaction S--k1-->A and an alternative desorption route A--k2-->S . Steady state properties of the three-step reactive scheme are analyzed in a one-dimensional lattice in the absence of diffusion. Microscopic Monte Carlo simulations show anomalous spatial distributions of reactants in the stationary state: depending on the reaction rate constants of the overall scheme, either a local "aggregation" or a local "dispersion" of A -particles is observed. The CFD reaction itself is well described by a fractional order kinetics that takes into account these anomalies and that depends on the kinetic rate constants of the overall adsorption-desorption reaction mechanism. The problem is addressed with an analytical approach for the active neighborhood of a reactant, which provides a closed expression of the reaction order as a function of the kinetic parameters. This approach is in excellent agreement with numerical simulations. Spatial correlations, as well as fluctuation correlations, are also formalized in terms of the kinetic constants. We discuss the results in the context of the hydrogen evolution reaction on silicon surfaces.

Reaction kinetics in intracellular environments with macromolecular crowding: simulations and rate laws

We review recent evidence illustrating the fundamental difference between cytoplasmic and test tube biochemical kinetics and thermodynamics, and showing the breakdown of the law of mass action and power-law approximation in in vivo conditions. Simulations of biochemical reactions in non-homogeneous media show that as a result of anomalous diffusion and mixing of the biochemical species, reactions follow a fractal-like kinetics. Consequently, the conventional equations for biochemical pathways fail to describe the reactions in in vivo conditions. We present a modification to fractal-like kinetics following the Zipf-Mandelbrot distribution which will enable the modelling and analysis of biochemical reactions occurring in crowded intracellular environments.

Monte carlo simulations of enzyme reactions in two dimensions: fractal kinetics and spatial segregation

Conventional equations for enzyme kinetics are based on mass-action laws, that may fail in low-dimensional and disordered media such as biological membranes. We present Monte Carlo simulations of an isolated Michaelis-Menten enzyme reaction on two-dimensional lattices with varying obstacle densities, as models of biological membranes. The model predicts that, as a result of anomalous diffusion on these low-dimensional media, the kinetics are of the fractal type. Consequently, the conventional equations for enzyme kinetics fail to describe the reaction. In particular, we show that the quasi-stationary-state assumption can hardly be retained in these conditions. Moreover, the fractal characteristics of the kinetics are increasingly pronounced as obstacle density and initial substrate concentration increase. The simulations indicate that these two influences are mainly additive. Finally, the simulations show pronounced S-P segregation over the lattice at obstacle densities compatible with in vivo conditions. This phenomenon could be a source of spatial self organization in biological membranes.

Single species diffusion-influenced reaction A+A-->alphaA: validity of the smoluchowski approach

We investigate the single species diffusion-influenced reaction, A+A-->alphaA with a finite reactivity in all dimensions. The reaction model includes a pure coagulation (alpha=1) or a pure annihilation (alpha=0) model. We apply the hierarchical Smoluchowski approach to study the dimensional aspects of the fluctuation, reactivity, particle size, and alpha(0</=alpha</=1). The theoretical results are compared with those of the Monte Carlo simulations in one, two, and three regular dimensions. The simulation results reveal that the classical Smoluchowski approach is exact in the short time limit in all dimensions and in the long time limit in three dimensions. The hierarchical Smoluchowski approach is found to be numerically exact at all times in two and three dimensions. A numerical method to obtain the exact result of the annihilation for a finite reactivity in one dimension is presented. We also propose a quite accurate analytic solution for an arbitrary alpha for the infinite reactivity in one dimension.