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PubMed2002

Multiparticle random walks on a deformable medium.

Multiparticle random walks on a deformable medium have been investigated in (2+1) dimensions. The time evolution of the particle distribution is studied. The results show that the randomly distributed particles in the beginning will be self-organized into a cluster pattern in the intermediate stage, and then return to the random distribution pattern in the late stage. The dependence of the clustering degree on the stiffness parameter of medium alpha, stability parameter of systems beta, and average particle density rho(0) is also investigated. There exists an optimal clustering stability beta(p), at which the system has the strongest clustering ability and corresponds to a maximum clustering coefficient Gamma(*)(p). The dependence of the optimal clustering coefficient Gamma(*)(p) on the stiffness alpha and particle density rho(0) is obtained, and the landscape of the medium generated by particles is also investigated.

Journal Article↗
PubMed2002

Random walks on a ( 2+1)-dimensional deformable medium.

A model of random walks on a deformable medium is proposed in 2+1 dimensions. The behavior of the walk is characterized by the stability parameter beta and the stiffness exponent alpha. The average square end-to-end distance l approximately equals (2nu) and the average number of visited sites approximately equals (k) are calculated. As beta increases, for each alpha there exists a critical transition point beta(c) from purely random walks ( nu = 1/2 and k approximate to 1) to compact growth ( nu = 1/3 and k = 2/3). The relationship between beta(c) and alpha can be expressed as beta(c) = e(alpha). The landscape generated by a walk is also investigated by means of the visit-number distribution N(n)(beta). There exists a scaling relationship of the form N(n)(beta)approximately n(-2)f(n/beta(z)).

Journal Article↗
PubMed2003

Dispersal of spores following a persistent random walk.

A model of a persistent random walk is used to describe the transport and deposition of the spore dispersal process. In this model, the spore particle flies along straight line trajectories, with constant speed v, which are interrupted by scattering, originating from interaction of spores with the field and wind variations, which randomly change its direction. To characterize the spore dispersal gradients, we have derived analytical expressions of the deposition probability epsilon (r|v) of airborne spores as a function of the distance r from the spore source in an infinite free space and in a disk of radius R with an absorbing edge that mimics an agricultural field surrounded with fields of nonhost plants and bare land. It is found in the free space that epsilon (r|v) approximately e(-alphar/l), with alpha a function of l(d)/l, where l and l(d) are the scattering and deposition mean free paths, respectively. In the disk, however, epsilon (r|v) is an infinite series of Bessel functions and, exhibits three regimes: absorbing (R l(d)).

Air↗
PubMed1986

A mathematical model applying the random-walk method to the environment of a neuron.

A mathematical model describing the supply and demand relationships existing in the environment of a brain cell (neuron of the cerebral cortex) was developed. The stochastic random-walk technique was applied to the representation and solution of the system which consisted of a neuron being supplied with nutrients by an adjoining capillary. The random-walk method incorporated a uniformly generated random number which was weighted by the normal distribution curve to determine the random walk of a molecule. The resultant weighted value was designated as defining the motion of any particular species in space. The distribution curve was a function of diffusivity and time. The method allowed the tracking of individual molecules as they proceeded through the metabolic reactions in the cell. Oxygen, glucose, carbon dioxide and lactate were selected as the primary components of study, since they represent the major input and output parameters of metabolism inside the cell. The consumption and/or production of these components were dependent on probability values assigned to each metabolic reaction into which they entered. The solution of the model was based on the number of molecules existing in the tissue as a function of PO2, (partial pressure of oxygen), glucose level, etc. The model was very sensitive to perturbations of metabolic scheme parameters and to PO2 levels in the capillary. The model predicted an excess of O2, (oxygen) in the tissue. The effects of edema on intercapillary distances as well as changes in the size and number of mitochondria within the neuron were examined using the model.(ABSTRACT TRUNCATED AT 250 WORDS)

Animals↗
PubMed2001

Simple model of a random walk with arbitrarily long memory.

We present a generalization of the persistent random-walk model in which the step at time n depends on the state of the step at time n-T, for arbitrary T. This gives rise to arbitrarily long memory effects, yet by an appropriate transformation the model is tractable by essentially the same techniques applicable to the usual persistent random-walk problem. We apply our results to the specific case of delayed "step" persistence, and analyze its asymptotic statistical properties.

Journal Article↗
PubMed2003

Asymmetric random walk in a reaction intermediate of homologous recombination.

At an intermediate step of the homologous recombination between two double-stranded DNA molecules, a point (often called Holliday structure) connecting two strands coming from two recombining partners migrates along the homologous region. Assuming random walk of a connecting point, we previously explained the dependence of recombination frequency on the homology length observed in vivo. In this model, the random walk was assumed to be symmetric in that the forward transition rate equals the backward one. According to observations in vitro, however, catalysed migration appears unidirectional. Taking into account possible asymmetry, we thus reformulate our random walk model to reexamine the observations in vivo. We also derive some theoretical results to analyse dynamic processes observed in vitro.

Animals↗
PubMed2001

Anomalous roughness, localization, and globally constrained random walks.

The scaling properties of a random walker subject to the global constraint that it needs to visit each site an even number of times are determined. Such walks are realized in the equilibrium state of one-dimensional surfaces that are subject to dissociative-dimer-type surface dynamics. Moreover, they can be mapped onto unconstrained random walks on a random surface, and the latter corresponds to a non-Hermitian random free fermion model that describes electron localization near a band edge. We show analytically that the dynamic exponent of this random walk is z=d+2 in spatial dimension d. This explains the anomalous roughness, with exponent alpha=1/3, in one-dimensional equilibrium surfaces with dissociative-dimer-type dynamics.

Journal Article↗
PubMed1985

Accumulator and random-walk models of psychophysical discrimination: a counter-evaluation.

In a recent assessment of models of psychophysical discrimination, Heath criticises the accumulator model for its reliance on computer simulation and qualitative evidence, and contrasts it unfavourably with a modified random-walk model, which yields exact predictions, is susceptible to critical test, and is provided with simple parameter-estimation techniques. A counter-evaluation is presented, in which the approximations employed in the modified random-walk analysis are demonstrated to be seriously inaccurate, the resulting parameter estimates to be artefactually determined, and the proposed test not critical. It is pointed out that Heath's specific application of the model is not legitimate, his data treatment inappropriate, and his hypothesis concerning confidence inconsistent with experimental results. Evidence from adaptive performance changes is presented which shows that the necessary assumptions for quantitative analysis in terms of the modified random-walk model are not satisfied, and that the model can be reconciled with data at the qualitative level only by making it virtually indistinguishable from an accumulator process. A procedure for deriving exact predictions for an accumulator process is outlined.

Choice Behavior↗
PubMed2001

Why the null matters: statistical tests, random walks and evolution.

A number of statistical tests have been developed to determine what type of dynamics underlie observed changes in morphology in evolutionary time series, based on the pattern of change within the time series. The theory of the 'scaled maximum', the 'log-rate-interval' (LRI) method, and the Hurst exponent all operate on the same principle of comparing the maximum change, or rate of change, in the observed dataset to the maximum change expected of a random walk. Less change in a dataset than expected of a random walk has been interpreted as indicating stabilizing selection, while more change implies directional selection. The 'runs test' in contrast, operates on the sequencing of steps, rather than on excursion. Applications of these tests to computer generated, simulated time series of known dynamical form and various levels of additive noise indicate that there is a fundamental asymmetry in the rate of type II errors of the tests based on excursion: they are all highly sensitive to noise in models of directional selection that result in a linear trend within a time series, but are largely noise immune in the case of a simple model of stabilizing selection. Additionally, the LRI method has a lower sensitivity than originally claimed, due to the large range of LRI rates produced by random walks. Examination of the published results of these tests show that they have seldom produced a conclusion that an observed evolutionary time series was due to directional selection, a result which needs closer examination in light of the asymmetric response of these tests.

Animals↗
PubMed1995

Random walk and gap plots of DNA sequences.

Genomic sequence analysis is usually performed with the help of specialized software packages written for molecular biologists. The scope of such pre-programmed techniques is quite limited. Because DNA sequences contain a large amount of information, analysis of such sequences without underlying assumptions may provide additional insights. The present article proposes two new graphical representations as examples of such methods. The random walk plot is designed to show the base composition in a compact form, whereas the gap plot visualizes positional correlations. The random walk plot represents the DNA sequence as a curve, a random walk, in a plane. The four possible moves, left/right and up/down, are used to encode the four possible bases. Gap plots provide a tool to exhibit various features in a sequence. They visualize the periodic patterns within a sequence, both with regard to a single type of base or between two types of bases.

Base Composition↗
PubMed1997

An exemplar-based random walk model of speeded classification.

The authors propose and test an exemplar-based random walk model for predicting response times in tasks of speeded, multidimensional perceptual classification. The model combines elements of R. M. Nosofsky's (1986) generalized context model of categorization and G. D. Logan's (1988) instance-based model of automaticity. In the model, exemplars race among one another to be retrieved from memory, with rates determined by their similarity to test items. The retrieved exemplars provide incremental information that enters into a random walk process for making classification decisions. The model predicts correctly effects of within- and between-categories similarity, individual-object familiarity, and extended practice on classification response times. It also builds bridges between the domains of categorization and automaticity.

Concept Formation↗
PubMed2001

Number of times a site is visited in two-dimensional random walks.

In this paper, formulas are derived to compute the mean number of times a site has been visited in a random walk on a two-dimensional lattice. Asymmetric random walks are considered, with or without drift, for different boundary conditions. It is shown that in case of absorbing boundaries the mean number of visits reaches stationary values over the lattice; comparisons with a Monte Carlo simulation are also presented.

Journal Article↗
PubMed2001

Molecular motors: thermodynamics and the random walk.

The biochemical cycle of a molecular motor provides the essential link between its thermodynamics and kinetics. The thermodynamics of the cycle determine the motor's ability to perform mechanical work, whilst the kinetics of the cycle govern its stochastic behaviour. We concentrate here on tightly coupled, processive molecular motors, such as kinesin and myosin V, which hydrolyse one molecule of ATP per forward step. Thermodynamics require that, when such a motor pulls against a constant load f, the ratio of the forward and backward products of the rate constants for its cycle is exp [-(DeltaG + u(0)f)/kT], where -DeltaG is the free energy available from ATP hydrolysis and u(0) is the motor's step size. A hypothetical one-state motor can therefore act as a chemically driven ratchet executing a biased random walk. Treating this random walk as a diffusion problem, we calculate the forward velocity v and the diffusion coefficient D and we find that its randomness parameter r is determined solely by thermodynamics. However, real molecular motors pass through several states at each attachment site. They satisfy a modified diffusion equation that follows directly from the rate equations for the biochemical cycle and their effective diffusion coefficient is reduced to D-v(2)tau, where tau is the time-constant for the motor to reach the steady state. Hence, the randomness of multistate motors is reduced compared with the one-state case and can be used for determining tau. Our analysis therefore demonstrates the intimate relationship between the biochemical cycle, the force-velocity relation and the random motion of molecular motors.

Adenosine Triphosphate↗
PubMed1991

A random walk method for percutaneous drug absorption pharmacokinetics: application to repeated administration of a therapeutic timolol patch.

A random walk method for predicting percutaneous drug absorption pharmacokinetics was proposed. The profiles predicted by this method were compatible with those predicted by the analytical method. The random walk method is particularly useful for predicting complex processes such as repeated topical application of a drug. The amount of a drug released into skin from four therapeutic timolol patches was measured when the patches were serially applied for 2.5 h each on the same site of six healthy male volunteers. On the average, 33.2, 23.4, 15.1, and 16.5% of the applied dose was released into skin from the first, second, third, and fourth patches, respectively. This pattern was comparable with the predicted profiles (43.9, 30.2, 24.4, and 21.1%) of amounts of drug which were expected to be released from the first to fourth patches into skin, respectively. The estimation method for the normalized skin-capillary boundary clearance is also described and applied in examining the percutaneous absorption of timolol. The estimated value for this parameter was much greater than the diffusion parameter, indicating that the removal process of timolol by the local circulation is much faster than the diffusion process through skin.

Administration, Cutaneous↗
PubMed1998

Time course of reactions controlled and gated by intramolecular dynamics of proteins: predictions of the model of random walk on fractal lattices.

Computer simulations of random walk on the Sierpinski gasket and percolation clusters demonstrate that the short, initial condition-dependent stage of protein involving reactions can dominate the progress of the reaction over the main stage described by the standard kinetics. This phenomenon takes place if the intramolecular conformational transition dynamics modeled by the stochastic process is slow enough and the initial conformational substate of the protein already belongs to the transition state of the reaction. Both conditions are realized in two kinds of experiments: small ligand rebinding to protein after laser flash photolysis and direct recording of single protein channel activity. The model considered suggests simple analytical formulae that can explain the time behavior of the processes observed and its variation with temperature. The initial condition-dependent stage, and not the stage described by the standard kinetics, is expected as responsible for the coupling of component reactions in the complete catalytic cycles and more complex processes of biological free energy transduction.

Journal Article↗
PubMed2000

Random walk with an exponentially varying step

A random walk with exponentially varying step, modeling damped or amplified diffusion, is studied. Each step is equal to the previous one multiplied by a step factor s (0 1/s relating different processes. For s<1/2 and s>2, the process is retrodictive (i.e., every final position can be reached by a unique path) and the set of all possible final points after infinite steps is fractal. For step factors in the interval [1/2,2], some cases result in smooth density distributions, other cases present overlapping self-similarity and there are values of the step factor for which the distribution is singular without a density function.

Journal Article↗
PubMed2002

Exact and efficient discrete random walk method for time-dependent two-dimensional environments.

We present an exact method for speeding up random walk in two-dimensional complicated lattice environments. To this end, we derive the discrete two-dimensional probability distribution function for a diffusing particle starting at the center of a square of linear size s. This is used to propagate random walkers from the center of the square to sites which are nearest neighbors to its perimeter sites, thus saving O(s(2)) steps in numerical simulations. We discuss in detail how this method can be implemented efficiently. We examine its performance in the diffusion limited aggregation model which produces fractal structures, and in a one-sided step-growth model producing compact, fingerlike structures. We show that in both cases, the square propagator method reduces the computational effort by a factor proportional to the linear system size as compared to standard random walk.

Journal Article↗
PubMed2002

Probability distribution of the sizes of the largest erased loops in loop-erased random walks.

We have studied the probability distribution of the perimeter and the area of the kth largest erased loop in loop-erased random walks in two dimensions for k=1 to 3. For a random walk of N steps, for large N, the average value of the kth largest perimeter and area scales as N(5/8) and N, respectively. The behavior of the scaled distribution functions is determined for very large and very small arguments. We have used exact enumeration for N< or =20 to determine the probability that no loop of size greater than l is erased. We show that correlations between loops have to be taken into account to describe the average size of the kth largest erased loops. We propose a one-dimensional Levy walk model that takes care of these correlations. The simulations of this simpler model compare very well with the simulations of the original problem.

Journal Article↗