Breakdown of Ovchinnikov-Zeldovich Segregation in the A+B-->0 Reaction under Lévy Mixing

Transient dynamics in a nonequilibrium superdiffusive reaction-diffusion process: Nonequilibrium random search as a case study

Transient regimes, often difficult to characterize, can be fundamental in establishing final steady states features of reaction-diffusion phenomena. This is particularly true in ecological problems. Here, through both numerical simulations and an analytic approximation, we analyze the transient of a nonequilibrium superdiffusive random search when the targets are created at a certain rate and annihilated upon encounters (a key dynamics, e.g., in biological foraging). The steady state is achieved when the number of targets stabilizes to a constant value. Our results unveil how key features of the steady state are closely associated to the particularities of the initial evolution. The searching efficiency variation in time is also obtained. It presents a rather surprising universal behavior at the asymptotic limit. These analyses shed some light into the general relevance of transients in reaction-diffusion systems.

Spontaneous repulsion in the A+B→0 reaction on coupled networks

We study the transient dynamics of an A+B→0 process on a pair of randomly coupled networks, where reactants are initially separated. We find that, for sufficiently small fractions q of cross couplings, the concentration of A (or B) particles decays linearly in a first stage and crosses over to a second linear decrease at a mixing time t_{x}. By numerical and analytical arguments, we show that for symmetric and homogeneous structures t_{x}∝(〈k〉/q)log(〈k〉/q) where 〈k〉 is the mean degree of both networks. Being this behavior is in marked contrast with a purely diffusive process, where the mixing time would go simply like 〈k〉/q, we identify the logarithmic slowing down in t_{x} to be the result of a spontaneous mechanism of repulsion between the reactants A and B due to the interactions taking place at the networks' interface. We show numerically how this spontaneous repulsion effect depends on the topology of the underlying networks.

Reaction-diffusion processes on interconnected scale-free networks

We study the two-particle annihilation reaction A+B→∅ on interconnected scale-free networks, using different interconnecting strategies. We explore how the mixing of particles and the process evolution are influenced by the number of interconnecting links, by their functional properties, and by the interconnectivity strategies in use. We show that the reaction rates on this system are faster than what was observed in other topologies, due to the better particle mixing that suppresses the segregation effect, in line with previous studies performed on single scale-free networks.

Incomplete mixing and reactions in laminar shear flow

Incomplete mixing of reactive solutes is well known to slow down reaction rates relative to what would be expected from assuming perfect mixing. In purely diffusive systems, for example, it is known that small initial fluctuations in reactant concentrations can lead to reactant segregation, which in the long run can reduce global reaction rates due to poor mixing. In contrast, nonuniform flows can enhance mixing between interacting solutes. Thus, a natural question arises: Can nonuniform flows sufficiently enhance mixing to restrain incomplete mixing effects and, if so, under what conditions? We address this question by considering a specific and simple case, namely, a laminar pure shear reactive flow. Two solution approaches are developed: a Lagrangian random walk method and a semianalytical solution. The results consistently highlight that if shear effects in the system are not sufficiently strong, incomplete mixing effects initially similar to purely diffusive systems will occur, slowing down the overall reaction rate. Then, at some later time, dependent on the strength of the shear, the system will return to behaving as if it were well mixed, but represented by a reduced effective reaction rate.

Mixing-Driven Equilibrium Reactions in Multidimensional Fractional Advection Dispersion Systems

We study instantaneous, mixing-driven, bimolecular equilibrium reactions in a system where transport is governed by a multidimensional space fractional dispersion equation. The superdiffusive, nonlocal nature of the system causes the location and magnitude of reactions that take place to change significantly from a classical Fickian diffusion model. In particular, regions where reaction rates would be zero for the Fickian case become regions where the maximum reaction rate occurs when anomalous dispersion operates. We also study a global metric of mixing in the system, the scalar dissipation rate and compute its asymptotic scaling rates analytically. The scalar dissipation rate scales asymptotically as t^-(d+α)/α , where d is the number of spatial dimensions and α is the fractional derivative exponent.

Nonuniversality and the role of tails in reaction-subdiffusion fronts

Recently there has been a certain controversy about the scaling properties of reaction-subdiffusion fronts. Some works seem to suggest that these fronts should move with constant speed, as do classical reaction-diffusion fronts, while other authors have predicted propagation failure, i.e., that the front speed tends asymptotically to zero. In the present work we confirm by Monte Carlo experiments that the two situations can actually occur depending on the way the reaction process is implemented. Also, we present a general analytical model that includes these two different behaviors as particular cases. From our analysis, we reach two main conclusions. First, the differences found in the scaling properties show the lack of universality of reaction-subdiffusion fronts. Second, we prove that, contrary to the widespread belief, the tail of the waiting time distributions is not always decisive to determine the speed of these fronts, but sometimes it plays just a marginal role in the front dynamics.

Wave-front dynamics in systems with directional anomalous diffusion

In this paper we study the solutions of a generalized reaction-diffusion system with a bistable reaction term, and considering directional anomalous diffusion. We use the well-known properties of fractional derivatives to model asymmetric anomalous diffusion, and obtain traveling wave solutions that propagate in a direction that depends on the metastability of the front, the fractional exponent and the asymmetry of the diffusion.

Anomalous surfactant diffusion in a living polymer system

Random processes are generally described by Gaussian statistics as formulated by the central limit theorem. However, there exists a large number of exceptions to this rule that can be found in a variety of fields. Diffusion processes are often analyzed by the scaling law <r2> approximately t2beta, where the second moment of the diffusion propagator or molecular mean square displacement, <r2>, in the case of Gaussian diffusion is proportional to t, i.e., beta=1/2. A deviation from Gaussian behavior may be either superdiffusion (beta>1/2) or subdiffusion (beta<1/2). In this paper we demonstrate that all three diffusion regimes may be observed for the surfactant self-diffusion, on the length scale of 10(-6) m and the time scale of 0.02-0.8 s. in a system of wormlike micelles, depending on small variations in the sample composition. The self-diffusion is followed by pulsed gradient NMR where one not only measures the second moment of the diffusion propagator, but actually measures the Fourier transform of the full diffusion propagator itself. A generalized diffusion equation in terms of fractional time derivatives provides a general description of all the different diffusion regimes, and where 1beta can be interpreted as a dynamic fractal dimension. Experimentally, we find beta=1/4 and 3/4, in the regimes of sub- and superdiffusion, respectively. The physical interpretation of the subdiffusion behavior is that the dominating diffusion mechanism corresponds to a lateral diffusion along the contour of the wormlike micelles. Superdiffusion is obtained near the overlap concentration where the average micellar size is smaller so that the center of mass diffusion of the micelles contributes to the transport of surfactant molecules.

Anomalous kinetics of attractive A + B-->0 reactions

We investigate the kinetics of the A + B-->0 reaction with the attractive interaction between opposite species in one spatial dimension. The attractive interaction leads to isotropic diffusions inside segregated single species domains, and accelerates the reactions of opposite species at the domain boundaries. At equal initial densities of and , we analytically and numerically show that the density of particles (rho), the size of domains (l), the distance between the closest neighbor of same species (lAA), and the distance between adjacent opposite species (lAB) scale in time as rho approximately t(-1/3), lAA approximately t(1/3), and l approximately lAB approximately lAB(2/3), respectively. These dynamical exponents define critical behavior distinguished from the class of uniformly driven systems of hard-core particles.

Absence of kinetic effects in reaction-diffusion processes in scale-free networks

We show that the chemical reactions of the model systems of A+A-->0 and A+B-->0 when performed on scale-free networks exhibit drastically different behavior as compared to the same reactions in normal spaces. The exponents characterizing the density evolution as a function of time are considerably higher than 1, implying that both reactions occur at a much faster rate. This is due to the fact that the discerning effects of the generation of a depletion zone (A+A) and the segregation of the reactants (A+B) do not occur at all as in normal spaces. Instead we observe the formation of clusters of A (A+A reaction) and of mixed A and B (A+B reaction) around the hubs of the network. Only at the limit of very sparse networks is the usual behavior recovered.

Catalytic reactions with bulk-mediated excursions: mixing fails to restore chemical equilibrium

In this paper we analyze the effect of the bulk-mediated excursions (BME) of reactive species on the long-time behavior of the catalytic Langmuir-Hinshelwood-like A+B-->0 reactions in systems in which a catalytic plane (CP) is in contact with a liquid phase, containing concentrations of reactive particles. Such BME result from repeated particles desorption from the CP, subsequent diffusion in the liquid phase, and eventual readsorption on the CP away from the initial detachment point. This process leads to an effective superdiffusive transport along the CP. We consider both "batch" reactions, in which all particles of reactive species were initially adsorbed onto the CP, and reactions followed by a steady inflow of particles onto the CP. We show that for batch reactions the BME provide an effective mixing channel and here the mean-field-type behavior emerges. On the contrary, for reaction followed by a steady inflow of particles, we observe essential departures from the mean-field behavior and find that the mixing effect of the BME is insufficient to restore chemical equilibrium. We show that a steady state is established as t--> infinity, in which the limiting value of the mean coverage of the CP depends on the particles' diffusion coefficient in the bulk liquid phase, and that the spatial distributions of adsorbed particles are strongly correlated. Moreover, we show that the relaxation to such a steady state is a power-law function of time, in contrast to the exponential time dependence describing the approach to equilibrium in perfectly stirred systems.

Millisecond kinetics on a microfluidic chip using nanoliters of reagents

This paper describes a microfluidic chip for performing kinetic measurements with better than millisecond resolution. Rapid kinetic measurements in microfluidic systems are complicated by two problems: mixing is slow and dispersion is large. These problems also complicate biochemical assays performed in microfluidic chips. We have recently shown (Song, H.; Tice, J. D.; Ismagilov, R. F. Angew. Chem., Int. Ed. 2003, 42, 768-772) how multiphase fluid flow in microchannels can be used to address both problems by transporting the reagents inside aqueous droplets (plugs) surrounded by an immiscible fluid. Here, this droplet-based microfluidic system was used to extract kinetic parameters of an enzymatic reaction. Rapid single-turnover kinetics of ribonuclease A (RNase A) was measured with better than millisecond resolution using sub-microliter volumes of solutions. To obtain the single-turnover rate constant (k = 1100 +/- 250 s(-1)), four new features for this microfluidics platform were demonstrated: (i) rapid on-chip dilution, (ii) multiple time range access, (iii) biocompatibility with RNase A, and (iv) explicit treatment of mixing for improving time resolution of the system. These features are discussed using kinetics of RNase A. From fluorescent images integrated for 2-4 s, each kinetic profile can be obtained using less than 150 nL of solutions of reagents because this system relies on chaotic advection inside moving droplets rather than on turbulence to achieve rapid mixing. Fabrication of these devices in PDMS is straightforward and no specialized equipment, except for a standard microscope with a CCD camera, is needed to run the experiments. This microfluidic platform could serve as an inexpensive and economical complement to stopped-flow methods for a broad range of time-resolved experiments and assays in chemistry and biochemistry.

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.