Random Walks: Theory and Selected Applications

Membrane Catalyzed Formation of Nucleotide Clusters and Their Role in the Origins of Life: Insights from Molecular Simulations and Lattice Modeling

One of the mysteries in studying the molecular "Origin of Life" is the emergence of RNA and RNA-based life forms, where nonenzymatic polymerization of nucleotides is a crucial hypothesis in formation of large RNA chains. The nonenzymatic polymerization can be mediated by various environmental settings, such as cycles of hydration and dehydration, temperature variations, and proximity to a variety of organizing matrices, such as clay, salt, fatty acids, lipid membrane, and mineral surface. In this work, we explore the influence of different phases of the lipid membrane toward nucleotide organization and polymerization in a simulated prebiotic setting. Our molecular simulations quantify the localization propensity of a mononucleotide, uridine monophosphate (UMP), in distinct membrane settings. We perform all-atom molecular dynamics (MD) simulations to estimate the role of the monophasic and biphasic membranes in modifying the behavior of UMPs localization and their clustering mechanism. Based on the interaction energy of mononucleotides with the membrane and their diffusion profile from our MD calculations, we developed a lattice-based model to explore the thermodynamic limits of the observations made from the MD simulations. The mathematical model substantiates our hypothesis that the lipid layers can act as unique substrates for "catalyzing" polymerization of mononucleotides due to the inherent spatiotemporal heterogeneity and phase change behavior.

Universal to nonuniversal transition of the statistics of rare events during the spread of random walks

Through numerous experiments that analyzed rare event statistics in heterogeneous media, it was discovered that in many cases the probability density function for particle position, P(X,t), exhibits a slower decay rate than the Gaussian function. Typically, the decay behavior is exponential, referred to as Laplace tails. However, many systems exhibit an even slower decay rate, such as power-law, log-normal, or stretched exponential. In this study, we utilize the continuous-time random walk method to investigate the rare events in particle hopping dynamics and find that the properties of the hop size distribution induce a critical transition between the Laplace universality of rare events and a more specific, slower decay of P(X,t). Specifically, when the hop size distribution decays slower than exponential, such as e^{-|x|^{β}} (β>1), the Laplace universality no longer applies, and the decay is specific, influenced by a few large events, rather than by the accumulation of many smaller events that give rise to Laplace tails.

Complete visitation statistics of one-dimensional random walks

We develop a framework to determine the complete statistical behavior of a fundamental quantity in the theory of random walks, namely, the probability that n_{1},n_{2},n_{3},... distinct sites are visited at times t_{1},t_{2},t_{3},.... From this multiple-time distribution, we show that the visitation statistics of one-dimensional random walks are temporally correlated, and we quantify the non-Markovian nature of the process. We exploit these ideas to derive unexpected results for the two-time trapping problem and to determine the visitation statistics of two important stochastic processes, the run-and-tumble particle and the biased random walk.

Structural, dynamic, and hydration properties of quercetin and its aggregates in solution

Quercetin is a flavonoid present in the human diet with multiple health benefits. Quercetin solutions are inhomogeneous even at very low concentrations due to quercetin's tendency to aggregate. We simulate, using molecular dynamics, three systems of quercetin solutions: infinite dilution, 0.22 M, and 0.46 M. The systems at the two highest concentrations represent regions of the quercetin aggregates, in which the concentration of this molecule is unusually high. We study the behavior of this molecule, its aggregates, and the modifications in the surrounding water. In the first three successive layers of quercetin hydration, the density of water and the hydrogen bonds formations between water molecules are smaller than that of bulk. Quercetin has a hydrophilic surface region that preferentially establishes donor hydrogen bonds with water molecules with relative frequencies from 0.12 to 0.46 at infinite dilution. Also, it has two hydrophobic regions above and below the planes of its rings, whose first hydration layers are further out from quercetin (≈0.3 Å) and their water molecules do not establish hydrogen bonds with it. Water density around the hydrophobic regions is smaller than that of the hydrophilic. Quercetin molecules aggregate inπ-stacking configurations, with a distance of ≈0.37 nm between the planes of their rings, and form bonds between their hydroxyl groups. The formation of quercetin aggregates decreases the hydrogen bonds between quercetin and the surrounding water and produces a subdiffusive behavior in water molecules. Quercetin has a subdiffusive behavior even at infinite dilution, which increases with the number of molecules within the aggregates and the time they remain within them.

Deciphering pore-level precipitation mechanisms

Mineral precipitation and dissolution in aqueous solutions has a significant effect on solute transport and structural properties of porous media. The understanding of the involved physical mechanisms, which cover a large range of spatial and temporal scales, plays a key role in several geochemical and industrial processes. Here, by coupling pore scale reactive transport simulations with classical nucleation theory, we demonstrate how the interplay between homogeneous and heterogeneous precipitation kinetics along with the non-linear dependence on solute concentration affects the evolution of the system. Such phenomena are usually neglected in pure macroscopic modelling. Comprehensive parametric analysis and comparison with laboratory experiments confirm that incorporation of detailed microscale physical processes in the models is compulsory. This sheds light on the inherent coupling mechanisms and bridges the gap between atomistic processes and macroscopic observations.

Anomalous transport in the crowded world of biological cells

A ubiquitous observation in cell biology is that the diffusive motion of macromolecules and organelles is anomalous, and a description simply based on the conventional diffusion equation with diffusion constants measured in dilute solution fails. This is commonly attributed to macromolecular crowding in the interior of cells and in cellular membranes, summarizing their densely packed and heterogeneous structures. The most familiar phenomenon is a sublinear, power-law increase of the mean-square displacement (MSD) as a function of the lag time, but there are other manifestations like strongly reduced and time-dependent diffusion coefficients, persistent correlations in time, non-Gaussian distributions of spatial displacements, heterogeneous diffusion and a fraction of immobile particles. After a general introduction to the statistical description of slow, anomalous transport, we summarize some widely used theoretical models: Gaussian models like fractional Brownian motion and Langevin equations for visco-elastic media, the continuous-time random walk model, and the Lorentz model describing obstructed transport in a heterogeneous environment. Particular emphasis is put on the spatio-temporal properties of the transport in terms of two-point correlation functions, dynamic scaling behaviour, and how the models are distinguished by their propagators even if the MSDs are identical. Then, we review the theory underlying commonly applied experimental techniques in the presence of anomalous transport like single-particle tracking, fluorescence correlation spectroscopy (FCS) and fluorescence recovery after photobleaching (FRAP). We report on the large body of recent experimental evidence for anomalous transport in crowded biological media: in cyto- and nucleoplasm as well as in cellular membranes, complemented by in vitro experiments where a variety of model systems mimic physiological crowding conditions. Finally, computer simulations are discussed which play an important role in testing the theoretical models and corroborating the experimental findings. The review is completed by a synthesis of the theoretical and experimental progress identifying open questions for future investigation.

Attracted diffusion-limited aggregation

In this paper we present results of extensive Monte Carlo simulations of diffusion-limited aggregation (DLA) with a seed placed on an attractive plane as a simple model in connection with the electrical double layers. We compute the fractal dimension of the aggregated patterns as a function of the attraction strength α. For the patterns grown in both two and three dimensions, the fractal dimension shows a significant dependence on the attraction strength for small values of α and approaches that of the ordinary two-dimensional (2D) DLA in the limit of large α. For the nonattracting case with α = 1, our results in three dimensions reproduce the patterns of 3D ordinary DLA, while in two dimensions our model leads to the formation of a compact cluster with dimension 2. For intermediate α, the 3D clusters have a quasi-2D structure with a fractal dimension very close to that of the ordinary 2D DLA. This allows one to control the morphology of a growing cluster by tuning a single external parameter α.

Random walks and random numbers from supercontinuum generation

We report a numerical study showing how the random intensity and phase fluctuations across the bandwidth of a broadband optical super-continuum can be interpreted in terms of the random processes of random walks and Lévy flights. We also describe how the intensity fluctuations can be applied to physical random number generation. We conclude that the optical supercontinuum provides a highly versatile means of studying and generating a wide class of random processes at optical wavelengths.

Fractal structure of a three-dimensional brownian motion on an attractive plane

Consider a brownian particle in three dimensions which is attracted by a plane with a strength proportional to some dimensionless parameter α. We investigate the fractal spatial structure of the visited lattice sites in a cubic lattice by the particle around and on the attractive plane. We compute the fractal dimensions of the set of visited sites both in three dimensions and on the attractive plane, as a function of the strength of attraction α. We also investigate the scaling properties of the size distribution of the clusters of nearest-neighbor visited sites on the attractive plane and compute the corresponding scaling exponent τ as a function of α. The fractal dimension of the curves surrounding the clusters is also computed for different values of α, which, in the limit α→∞, tends to that of the outer perimeter of planar brownian motion, i.e., the self-avoiding random walk (SAW). We find that all measured exponents depend significantly on the strength of attraction.

Continuous-time random walk models of DNA electrophoresis in a post array: part II. Mobility and sources of band broadening

Using the two-state, continuous-time random walk model, we develop expressions for the mobility and the plate height during DNA electrophoresis in an ordered post array that delineate the contributions due to (i) the random distance between collisions and (ii) the random duration of a collision. These contributions are expressed in terms of the means and variances of the underlying stochastic processes, which we evaluate from a large ensemble of Brownian dynamics simulations performed using different electric fields and molecular weights in a hexagonal array of 1 μm posts with a 3 μm center-to-center distance. If we fix the molecular weight, we find that the collision frequency governs the mobility. On the contrary, the average collision duration is the most important factor for predicting the mobility as a function of DNA size at constant Péclet number. The plate height is reasonably well described by a single post rope-over-pulley model, provided that the extension of the molecule is small. Our results only account for dispersion inside the post array and thus represent a theoretical lower bound on the plate height in an actual device.

Lévy walks and scaling in quenched disordered media

We study Lévy walks in quenched disordered one-dimensional media, with scatterers spaced according to a long-tailed distribution. By analyzing the scaling relations for the random-walk probability and for the resistivity in the equivalent electric problem, we obtain the asymptotic behavior of the mean-square displacement as a function of the exponent characterizing the scatterers distribution. We demonstrate that in quenched media different average procedures can display different asymptotic behavior. In particular, we estimate the moments of the displacement averaged over processes starting from scattering sites. Our results are compared with numerical simulations, with excellent agreement.

Polymer length distributions for catalytic polymerization within mesoporous materials: non-Markovian behavior associated with partial extrusion

We analyze a model for polymerization at catalytic sites distributed within parallel linear pores of a mesoporous material. Polymerization occurs primarily by reaction of monomers diffusing into the pores with the ends of polymers near the pore openings. Monomers and polymers undergo single-file diffusion within the pores. Model behavior, including the polymer length distribution, is determined by kinetic Monte Carlo simulation of a suitable atomistic-level lattice model. While the polymers remain within the pore, their length distribution during growth can be described qualitatively by a Markovian rate equation treatment. However, once they become partially extruded, the distribution is shown to exhibit non-Markovian scaling behavior. This feature is attributed to the long-tail in the "return-time distribution" for the protruding end of the partially extruded polymer to return to the pore, such return being necessary for further reaction and growth. The detailed form of the scaled length distribution is elucidated by application of continuous-time random walk theory.

Persistence effects in deterministic diffusion

In systems that exhibit deterministic diffusion, the gross parameter dependence of the diffusion coefficient can often be understood in terms of random-walk models. Provided the decay of correlations is fast enough, one can ignore memory effects and approximate the diffusion coefficient according to dimensional arguments. By successively including the effects of one and two steps of memory on this approximation, we examine the effects of "persistence" on the diffusion coefficients of extended two-dimensional billiard tables and show how to properly account for these effects using walks in which a particle undergoes jumps in different directions with probabilities that depend on where they came from.

Scaling property of flux fluctuations from random walks

We study dynamical scaling of flux fluctuation sigma(t) from the one-random-walker model on regular lattices and complex networks and compare it to the surface width W(t) of a corresponding growth model. On the regular lattices, we analytically show that sigma(t) undergoes a crossover from the nontrivial scaling regime to the trivial one by increasing time t, and we verify the results by numerical simulations. In contrast to the results on the regular lattices, sigma(t) does not show any crossover behavior on complex networks and satisfies the scaling relation sigma(t) approximately t(1/2) for any t. On the other hand, we show that W(t) of the corresponding model on complex networks has two different scaling regimes, W approximately t(1/2) for t<<N and W(t) approximately t for t>>N .

Scaling and data collapse for the mean exit time of asset prices

We study theoretical and empirical aspects of the mean exit time (MET) of financial time series. The theoretical modeling is done within the framework of continuous time random walk. We empirically verify that the mean exit time follows a quadratic scaling law and it has associated a prefactor which is specific to the analyzed stock. We perform a series of statistical tests to determine which kind of correlation are responsible for this specificity. The main contribution is associated with the autocorrelation property of stock returns. We introduce and solve analytically both two-state and three-state Markov chain models. The analytical results obtained with the two-state Markov chain model allows us to obtain a data collapse of the 20 measured MET profiles in a single master curve.

Non-markovian transport of DNA in microfluidic post arrays

We present an analytically solvable model for the transport of long DNA through microfluidic arrays of posts. The motion is a repetitive three-part cycle: (i) collision with the post and extension of the arms; (ii) rope-over-pulley post disengagement; and (iii) a random period of uniform translation before the next collision. This cycle, inspired by geometration, is a nonseparable (Scher-Lax) continuous-time random walk on a lattice defined by the posts. Upon adopting a simple model for the transition probability density on the lattice, we analytically compute the mean DNA velocity and dispersivity in the long-time limit without any adjustable parameters. The results compare favorably with the limited amount of experimental data on separations in self-assembled arrays of magnetic beads. The Scher-Lax formalism provides a template for incorporating more sophisticated microscale models.

Extreme times in financial markets

We apply the theory of continuous time random walks (CTRWs) to study some aspects involving extreme events in financial time series. We focus our attention on the mean exit time (MET). We derive a general equation for this average and compare it with empirical results coming from high-frequency data of the U.S. dollar and Deutsche mark futures market. The empirical MET follows a quadratic law in the return length interval which is consistent with the CTRW formalism.

Fractional diffusion modeling of ion channel gating

An anomalous diffusion model for ion channel gating is put forward. This scheme is able to describe nonexponential, power-law-like distributions of residence time intervals in several types of ion channels. Our method presents a generalization of the discrete diffusion model by Millhauser, Salpeter, and Oswald [Proc. Natl. Acad. Sci. U.S.A. 85, 1503 (1988)] to the case of a continuous, anomalous slow conformational diffusion. The corresponding generalization is derived from a continuous-time random walk composed of nearest-neighbor jumps which in the scaling limit results in a fractional diffusion equation. The studied model contains three parameters only: the mean residence time, a characteristic time of conformational diffusion, and the index of subdiffusion. A tractable analytical expression for the characteristic function of the residence time distribution is obtained. In the limiting case of normal diffusion, our prior findings [Proc. Natl. Acad. Sci. U.S.A. 99, 3552 (2002)] are reproduced. Depending on the chosen parameters, the fractional diffusion model exhibits a very rich behavior of the residence time distribution with different characteristic time regimes. Moreover, the corresponding autocorrelation function of conductance fluctuations displays nontrivial power law features. Our theoretical model is in good agreement with experimental data for large conductance potassium ion channels.

Scaling law in target-hunting processes

We study a hunting process for a target, in which the hunter tracks the goal by smelling odors it emits. The odor intensity is supposed to decrease with the diffusion distance. The Monte Carlo experiment is carried out on a two-dimensional square lattice. Having no idea of the location of the target, the hunter determines its moves only by random attempts in each direction. By sorting the searching time in each simulation and introducing a variable x to reflect the sequence of searching times, we obtain a curve with a wide plateau, indicating the most probable time of successfully finding the target. The simulations reveal a scaling law for the searching time versus the distance to the position of the target. The scaling exponent depends on the sensitivity of the hunter. Our model may be a prototype in studying such searching processes as various food-foraging behaviors of wild animals.

Hopping diffusion of two coupled particles in the random trap model

We show that the hopping dynamics of two strongly connected particles can be mapped exactly to single particle dynamics. In this way we are able to calculate the exact asymptotic diffusion coefficient of two connected particles on a linear chain in the random trap model. In particular we calculate the diffusion coefficient for exponentially distributed site energies and show that there exists a critical temperature below which a subdiffusive behavior appears. It turns out that this critical temperature is twice higher than the critical temperature in the single particle case [S. Havlin, B. L. Trus, and G. H. Weiss, J. Phys. A: Math. Gen. 19, L817 (1986)].

Survival and residence times in disordered chains with bias

We present a unified framework for first-passage time and residence time of random walks in finite one-dimensional disordered biased systems. The derivation is based on the exact expansion of the backward master equation in cumulants. The dependence on the initial condition, system size, and bias strength is explicitly studied for models with weak and strong disorders. Application to thermally activated processes is also developed.

Accurate estimation of the survival probability for trapping in two dimensions

In this work we study the mean survival probability Phi(n,c) of random walks on a two-dimensional lattice in the presence of traps of concentration c, as a function of the number of steps n. The computation of this quantity is performed indirectly by using the distribution of the number of sites visited S(n). In order to achieve an accurate description of this distribution we use a combination of numerical techniques. The method allows an accurate calculation of Phi down to very small values (of the order of 10(-100), for example), which is not possible via direct simulations. The survival probability is analyzed in terms of an asymptotic expansion, following the results of Donsker and Varadhan [Commun. Pure Appl. Math. 28, 525 (1975); 32, 721 (1979)], and by using the outcome of a scaling ansatz, as described in our earlier work.

Trapping and survival probability in two dimensions

We investigate the survival probability Phi(n,c) of particles performing a random walk on a two-dimensional lattice that contains static traps, which are randomly distributed with a concentration c, as a function of the number of steps n. Phi(n,c) is analyzed in terms of a scaling ansatz, which allows us to locate quantitatively the crossover between the Rosenstock approximation (valid only at early times) and the asymptotic Donsker-Varadhan behavior (valid only at long times). While the existence of the crossover has been postulated before, its exact location has not been known. Our scaling hypothesis is based on the mean value of the quantity S(n), the number of sites visited in an n-step walk. We make use of the idea of self-interacting random walks, and a "slithering" snake algorithm, available in the literature, and we are thus able to obtain accurate survival probability data indirectly by Monte Carlo simulation techniques. The crossover can now be determined by our method, and it is found to depend on a combination of c and n. It occurs at small Phi(n,c) values, which is typically the case for large values of n.

Kinetics of diffusion-assisted reactions in microheterogeneous systems

This review is focused on the basic theory of diffusion-assisted reactions in microheterogeneous systems, from porous solids to self-organized colloids and biomolecules. Rich kinetic behaviors observed experimentally are explained in a unified fashion using simple concepts of competing distance and time scales of the reaction and the embedding structure. We mainly consider pseudo-first-order reactions, such as luminescence quenching, described by the Smoluchowski type of equation for the reactant pair distribution function with a sink term defined by the reaction mechanism. Microheterogeneity can affect the microscopic rate constant. It also enters the evolution equation through various spatial constraints leading to complicated boundary conditions and, possibly, to the reduction of dimensionality of the diffusion space. The reaction coordinate and diffusive motion along this coordinate are understood in a general way, depending on the problem at hand. Thus, the evolution operator can describe translational and rotational diffusion of molecules in a usual sense, it can be a discrete random walk operator when dealing with hopping of adsorbates in solids, or it can correspond to conformational fluctuations in proteins. Mathematical formulation is universal but physical consequences can be different. Understanding the principal features of reaction kinetics in microheterogeneous systems enables one to extract important structural and dynamical information about the host environments by analyzing suitably designed experiments, it helps building effective strategies for computer simulations, and ultimately opens possibilities for designing systems with controllable reactivity properties.

Simulation of the wiener sausage

The volume of a region visited by a spherical Brownian particle for a time t, known as the Wiener sausage, is an important random variable characterizing Brownian motion. A Brownian dynamics simulation is used to study statistical properties of the Wiener sausage volume. We show that the probability density is closely approximated by a Gaussian distribution not only at asymptotically long times, but over a wide range of times as well. We also refine the expression for the dispersion by finding a correction term for the long-time asymptotic dependence.

Diffusion-mediated reactions with a time-dependent absorption rate

Diffusion-mediated reactions models are particularly useful for the characterization of physical, chemical, and biological problems. In this paper we present a theoretical study of the absorption probability density, survival probability, and reaction rate for diffusion-mediated reactions models with a time-dependent finite absorption rate (an extension of a model usually referred to as the "imperfect trap model"). The results are obtained by means of the formalism of continuous time random walk on a lattice and considering a general reaction dynamics upon encounter of the reactives. First jump probability densities are included to take initial conditions into account. Previous results presented by Collins and Kimball [J. Colloid. Sci. 4, 425 (1949)] and Noyes [J. Chem. Phys. 22, 1349 (1954)] are reobtained for the particular case of a time-independent absorptivity. Short and long time behaviors are analyzed resulting, in particular, in that the long time behavior of the absorption probability density exhibits the same time dependence as the first passage time density. The results obtained are illustrated by considering a one-dimensional model with consequent discussion.