Trapping reactions with subdiffusive traps and particles characterized by different anomalous diffusion exponents

A number of results for reactions involving subdiffusive species all with the same anomalous exponent gamma have recently appeared in the literature and can often be understood in terms of a subordination principle whereby time t in ordinary diffusion is replaced by t^gamma. However, very few results are known for reactions involving different species characterized by different anomalous diffusion exponents. Here we study the reaction dynamics of a (sub)diffusive particle surrounded by a sea of (sub)diffusive traps in one dimension. We find rigorous results for the asymptotic survival probability of the particle in most cases, with the exception of the case of a particle that diffuses normally while the anomalous diffusion exponent of the traps is smaller than 2/3.Comment: To appear in Phys. Rev.

Driven diffusion in a periodically compartmentalized tube: homogeneity versus intermittency of particle motion

We study the effect of a driving force F on drift and diffusion of a point Brownian particle in a tube formed by identical ylindrical compartments, which create periodic entropy barriers for the particle motion along the tube axis. The particle transport exhibits striking features: the effective mobility monotonically decreases with increasing F, and the effective diffusivity diverges as F → ∞, which indicates that the entropic effects in diffusive transport are enhanced by the driving force. Our consideration is based on two different scenarios of the particle motion at small and large F, homogeneous and intermittent, respectively. The scenarios are deduced from the careful analysis of statistics of the particle transition times between neighboring openings. From this qualitative picture, the limiting small-F and large-F behaviors of the effective mobility and diffusivity are derived analytically. Brownian dynamics simulations are used to find these quantities at intermediate values of the driving force for various compartment lengths and opening radii. This work shows that the driving force may lead to qualitatively different anomalous transport features, depending on the geometry design

Single Stranded DNA Translocation Through A Nanopore: A Master Equation Approach

We study voltage driven translocation of a single stranded (ss) DNA through a membrane channel. Our model, based on a master equation (ME) approach, investigates the probability density function (pdf) of the translocation times, and shows that it can be either double or mono-peaked, depending on the system parameters. We show that the most probable translocation time is proportional to the polymer length, and inversely proportional to the first or second power of the voltage, depending on the initial conditions. The model recovers experimental observations on hetro-polymers when using their properties inside the pore, such as stiffness and polymer-pore interaction.Comment: 7 pages submitted to PR

Thermodynamics of Competitive Molecular Channel Transport: Application to Artificial Nuclear Pores

In an analytical model channel transport is analyzed as a function of key parameters, determining efficiency and selectivity of particle transport in a competitive molecular environment. These key parameters are the concentration of particles, solvent-channel exchange dynamics, as well as particle-in-channel- and interparticle interaction. These parameters are explicitly related to translocation dynamics and channel occupation probability. Slowing down the exchange dynamics at the channel ends, or elevating the particle concentration reduces the in-channel binding strength necessary to maintain maximum transport. Optimized in-channel interaction may even shift from binding to repulsion. A simple equation gives the interrelation of access dynamics and concentration at this transition point. The model is readily transferred to competitive transport of different species, each of them having their individual in-channel affinity. Combinations of channel affinities are determined which differentially favor selectivity of certain species on the cost of others. Selectivity for a species increases if its in-channel binding enhances the species' translocation probablity when compared to that of the other species. Selectivity increases particularly for a wide binding site, long channels, and fast access dynamics. Recent experiments on competitive transport of in-channel binding and inert molecules through artificial nuclear pores serve as a paradigm for our model. It explains qualitatively and quantitatively how binding molecules are favored for transport at the cost of the transport of inert molecules

Trapping of a random walk by diffusing traps

We present a systematic analytical approach to the trapping of a random walk by a finite density rho of diffusing traps in arbitrary dimension d. We confirm the phenomenologically predicted e^{-c_d rho t^{d/2}} time decay of the survival probability, and compute the dimension dependent constant c_d to leading order within an eps=2-d expansion.Comment: 16 pages, to appear in J. Phys.

Kinetics of stochastically-gated diffusion-limited reactions and geometry of random walk trajectories

In this paper we study the kinetics of diffusion-limited, pseudo-first-order A + B -> B reactions in situations in which the particles' intrinsic reactivities vary randomly in time. That is, we suppose that the particles are bearing "gates" which interchange randomly and independently of each other between two states - an active state, when the reaction may take place, and a blocked state, when the reaction is completly inhibited. We consider four different models, such that the A particle can be either mobile or immobile, gated or ungated, as well as ungated or gated B particles can be fixed at random positions or move randomly. All models are formulated on a $d$-dimensional regular lattice and we suppose that the mobile species perform independent, homogeneous, discrete-time lattice random walks. The model involving a single, immobile, ungated target A and a concentration of mobile, gated B particles is solved exactly. For the remaining three models we determine exactly, in form of rigorous lower and upper bounds, the large-N asymptotical behavior of the A particle survival probability. We also realize that for all four models studied here such a probalibity can be interpreted as the moment generating function of some functionals of random walk trajectories, such as, e.g., the number of self-intersections, the number of sites visited exactly a given number of times, "residence time" on a random array of lattice sites and etc. Our results thus apply to the asymptotical behavior of the corresponding generating functions which has not been known as yet.Comment: Latex, 45 pages, 5 ps-figures, submitted to PR

Universal fluctuations in the support of the random walk

A random walk starts from the origin of a d-dimensional lattice. The occupation number n(x,t) equals unity if after t steps site x has been visited by the walk, and zero otherwise. We study translationally invariant sums M(t) of observables defined locally on the field of occupation numbers. Examples are the number S(t) of visited sites; the area E(t) of the (appropriately defined) surface of the set of visited sites; and, in dimension d=3, the Euler index of this surface. In d > 3, the averages (t) all increase linearly with t as t-->infinity. We show that in d=3, to leading order in an asymptotic expansion in t, the deviations from average Delta M(t)= M(t)-(t) are, up to a normalization, all identical to a single "universal" random variable. This result resembles an earlier one in dimension d=2; we show that this universality breaks down for d>3.Comment: 17 pages, LaTeX, 2 figures include

Kinetics of Anchoring of Polymer Chains on Substrates with Chemically Active Sites

We consider dynamics of an isolated polymer chain with a chemically active end-bead on a 2D solid substrate containing immobile, randomly placed chemically active sites (traps). For a particular situation when the end-bead can be irreversibly trapped by any of these sites, which results in a complete anchoring of the whole chain, we calculate the time evolution of the probability $P_{ch}(t)$ that the initially non-anchored chain remains mobile until time $t$. We find that for relatively short chains $P_{ch}(t)$ follows at intermediate times a standard-form 2D Smoluchowski-type decay law $ln P_{ch}(t) \sim - t/ln(t)$, which crosses over at very large times to the fluctuation-induced dependence $ln P_{ch}(t) \sim - t^{1/2}$, associated with fluctuations in the spatial distribution of traps. We show next that for long chains the kinetic behavior is quite different; here the intermediate-time decay is of the form $ln P_{ch}(t) \sim - t^{1/2}$, which is the Smoluchowski-type law associated with subdiffusive motion of the end-bead, while the long-time fluctuation-induced decay is described by the dependence $ln P_{ch}(t) \sim - t^{1/4}$, stemming out of the interplay between fluctuations in traps distribution and internal relaxations of the chain.Comment: Latex file, 19 pages, one ps figure, to appear in PR

Pascal Principle for Diffusion-Controlled Trapping Reactions

"All misfortune of man comes from the fact that he does not stay peacefully in his room", has once asserted Blaise Pascal. In the present paper we evoke this statement as the "Pascal principle" in regard to the problem of survival of an "A" particle, which performs a lattice random walk in presence of a concentration of randomly moving traps "B", and gets annihilated upon encounters with any of them. We prove here that at sufficiently large times for both perfect and imperfect trapping reactions, for arbitrary spatial dimension "d" and for a rather general class of random walks, the "A" particle survival probability is less than or equal to the survival probability of an immobile target in the presence of randomly moving traps.Comment: 4 pages, RevTex, appearing in PR