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PubMed2004

Exploratory behavior, trap models, and glass transitions.

A random walk is performed on a disordered landscape composed of N sites randomly and uniformly distributed inside a d-dimensional hypercube. The walker hops from one site to another with probability proportional to exp[-betaE(D)], where beta=1/T is the inverse of a formal temperature and E(D) is an arbitrary cost function which depends on the hop distance D. Analytic results indicate that, if E(D)=D(d) and N--> infinity, there exists a glass transition at beta(d)=pi(d/2)/[(d/2)Gamma(d/2)]. Below T(d), the average trapping time diverges and the system falls into an out-of-equilibrium regime with aging phenomena. A Lévy flight scenario and applications of exploratory behavior are considered.

Journal Article↗

PubMed2000

Quantitative analysis of adhesion-mediated cell migration in three-dimensional gels of RGD-grafted collagen.

Adhesion-mediated migration is required in a number of physiological and pathological processes. A further quantitative understanding of the relationship between cell migration and cell-substratum adhesiveness may aid in therapeutic or tissue engineering applications. The aim of this work was to quantify three-dimensional cell migration as a function of increasing cell-substratum adhesiveness within reconstituted collagen gels. Cell-substratum adhesiveness was controlled by grafting additional adhesive peptides containing the well-characterized arginine-glycine-aspartic acid sequence to collagen. The three-dimensional migration of multiple individual cells was tracked in real time in an automated fashion for extended periods. Cell displacements were statistically analyzed and fit to a correlated persistent random walk model to estimate root-mean-square speed, directional persistence time, and random motility coefficient. Based on model parameter estimates, cell speed was found to be a monotonically decreasing function of increasing substratum adhesiveness, while the directional persistence time and random motility coefficient exhibited a biphasic dependence, with maximum values at approximately intermediate concentrations of grafted adhesive peptide and hence intermediate cell-substratum adhesiveness. In conclusion, these studies suggest an optimal adhesiveness for three-dimensional random migration, consistent with previous studies on two-dimensional surfaces. However, the maximum in random motility corresponded to a maximum in directional persistence, not in cell speed.

Animals↗

PubMed2005

Slow transport by continuous time quantum walks.

Continuous time quantum walks (CTQWs) do not necessarily perform better than their classical counterparts, the continuous time random walks (CTRWs). For one special graph, where a recent analysis showed that in a particular direction of propagation the penetration of the graph is faster by CTQWs than by CTRWs, we demonstrate that in another direction of propagation the opposite is true. In this case a CTQW initially localized at one site displays a slow transport. We furthermore show that when the CTQW's initial condition is a totally symmetric superposition of states of equivalent sites, the transport gets to be much more rapid.

Journal Article↗

PubMed2000

Strong anomaly in diffusion generated by iterated maps.

We investigate the diffusion generated deterministically by periodic iterated maps that are defined by x(t+1) = x(t)+ax(z)(t)exp[-(b/x(t))(z-1)], z>1. It is shown that the obtained mean squared displacement grows asymptotically as sigma(2)(t) approximately ln (1/(z-1))(t) and that the corresponding propagator decays exponentially with the scaling variable |x|/square root of (sigma(2)(t))]. This strong diffusional anomaly stems from the anomalously broad distribution of waiting times in the corresponding random walk process and leads to a behavior obtained for diffusion in the presence of random local fields. A scaling approach is introduced which connects the explicit form of the maps to the mean squared displacement.

Journal Article↗

PubMed1992

Determination of the sequence coding for the beta subunit of the human high-affinity IgE receptor.

The cDNA encoding the beta subunit of the human high-affinity IgE receptor was cloned by a combination of various polymerase chain reactions (PCR). A major portion of the beta cDNA was amplified using primers homologous within the sequences of rat and mouse. The 3' unknown sequence was preferentially amplified using the RNA template-specific PCR and the improved two-step PCR. The 5' unknown sequence was specifically amplified by our newly developed PCR walking. Random heptanucleotides tagged with a unique sequence at the 5' end were used as the walking primer. Finally, the entire coding region was amplified and sequenced. The two extracellular loops of the human beta subunit were the least homologous to those of rat and mouse.

Amino Acid Sequence↗

PubMed1998

Homologous chromosome pairing in Drosophila melanogaster proceeds through multiple independent initiations.

The dynamics by which homologous chromosomes pair is currently unknown. Here, we use fluorescence in situ hybridization in combination with three-dimensional optical microscopy to show that homologous pairing of the somatic chromosome arm 2L in Drosophila occurs by independent initiation of pairing at discrete loci rather than by a processive zippering of sites along the length of chromosome. By evaluating the pairing frequencies of 11 loci on chromosome arm 2L over several timepoints during Drosophila embryonic development, we show that all 11 loci are paired very early in Drosophila development, within 13 h after egg deposition. To elucidate whether such pairing occurs by directed or undirected motion, we analyzed the pairing kinetics of histone loci during nuclear cycle 14. By measuring changes of nuclear length and correlating these changes with progression of time during cycle 14, we were able to express the pairing frequency and distance between homologous loci as a function of time. Comparing the experimentally determined dynamics of pairing to simulations based on previously proposed models of pairing motion, we show that the observed pairing kinetics are most consistent with a constrained random walk model and not consistent with a directed motion model. Thus, we conclude that simple random contacts through diffusion could suffice to allow pairing of homologous sites.

Animals↗

PubMed2005

Statistical mechanical load balancer for the web.

The maximum entropy principle from statistical mechanics states that a closed system attains an equilibrium distribution that maximizes its entropy. We first show that for graphs with fixed number of edges one can define a stochastic edge dynamic that can serve as an effective thermalization scheme, and hence, the underlying graphs are expected to attain their maximum-entropy states, which turn out to be Erdös-Rényi (ER) random graphs. We next show that (i) a rate-equation-based analysis of node degree distribution does indeed confirm the maximum-entropy principle, and (ii) the edge dynamic can be effectively implemented using short random walks on the underlying graphs, leading to a local algorithm for the generation of ER random graphs. The resulting statistical mechanical system can be adapted to provide a distributed and local (i.e., without any centralized monitoring) mechanism for load balancing, which can have a significant impact in increasing the efficiency and utilization of both the Internet (e.g., efficient web mirroring), and large-scale computing infrastructure (e.g., cluster and grid computing).

Journal Article↗

PubMed2002

Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations.

We propose diffusionlike equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which, correspondingly, cannot be viewed as self-affine random processes possessing a unique Hurst exponent. We prove the positivity of the solutions of the proposed equations and establish their relation to the continuous-time random walk theory. We show that the distributed-order time fractional diffusion equation describes the subdiffusion random process that is subordinated to the Wiener process and whose diffusion exponent decreases in time (retarding subdiffusion). This process may lead to superslow diffusion, with the mean square displacement growing logarithmically in time. We also demonstrate that the distributed-order space fractional diffusion equation describes superdiffusion phenomena with the diffusion exponent increasing in time (accelerating superdiffusion).

Journal Article↗

PubMed2006

Analysis of spike statistics in neuronal systems with continuous attractors or multiple, discrete attractor States.

Attractor networks are likely to underlie working memory and integrator circuits in the brain. It is unknown whether continuous quantities are stored in an analog manner or discretized and stored in a set of discrete attractors. In order to investigate the important issue of how to differentiate the two systems, here we compare the neuronal spiking activity that arises from a continuous (line) attractor with that from a series of discrete attractors. Stochastic fluctuations cause the position of the system along its continuous attractor to vary as a random walk, whereas in a discrete attractor, noise causes spontaneous transitions to occur between discrete states at random intervals. We calculate the statistics of spike trains of neurons firing as a Poisson process with rates that vary according to the underlying attractor network. Since individual neurons fire spikes probabilistically and since the state of the network as a whole drifts randomly, the spike trains of individual neurons follow a doubly stochastic (Poisson) point process. We compare the series of spike trains from the two systems using the autocorrelation function, Fano factor, and interspike interval (ISI) distribution. Although the variation in rate can be dramatically different, especially for short time intervals, surprisingly both the autocorrelation functions and Fano factors are identical, given appropriate scaling of the noise terms. Since the range of firing rates is limited in neurons, we also investigate systems for which the variation in rate is bounded by either rigid limits or because of leak to a single attractor state, such as the Ornstein-Uhlenbeck process. In these cases, the time dependence of the variance in rate can be different between discrete and continuous systems, so that in principle, these processes can be distinguished using second-order spike statistics.

Action Potentials↗

PubMed2002

Self-avoiding walks on Sierpinski lattices in two and three dimensions.

The scaling properties of linear polymers on deterministic fractal structures, modeled by self-avoiding random walks (SAW) on Sierpinski lattices in two and three dimensions, are studied. To this end, all possible SAW configurations of N steps are enumerated exactly and averages over suitable sets of starting lattice points for the walks are performed to extract the mean quantities of interest reliably. We determine the critical exponent describing the mean end-to-end chemical distance (-)l(N) after N steps and the corresponding distribution function, P(S)(l,N). A des Cloizeaux-type relation between the exponent characterizing the asymptotic shape of the distribution, for l-->0 and N--> infinity, and the one describing the total number of SAW of N steps is suggested and supported by numerical results. These results are confronted with those obtained recently on the backbone of the incipient percolation cluster, where the corresponding exponents are very well described by a generalized des Cloizeaux relation valid for statistically self-similar structures.

Journal Article↗

PubMed1985

Motion of mitochondria in cultured cells quantified by analysis of digitized images.

Translational movements of mitochondria in cultured rat liver cells were characterized quantitatively by using a video camera to detect and a video digitizer-computer system to analyze fluorescent images of mitochondria stained with rhodamine 123. The centroids of the images of individual mitochondria were determined at selected time intervals and the paths followed by the mitochondria were defined by the paths of the centroids. The predominant translation motion of the mitochondria satisfied the formal conditions of a Brownian random walk for a free particle, although in several cases there was a slow drift superimposed on the random motion. The apparent diffusion coefficients were approximately 5 X 10(-12) cm2 s-1, and the drift speeds approximately 2 X 10(-3) micron s-1.

Animals↗

PubMed2002

Topographical control of human neutrophil motility on micropatterned materials with various surface chemistry.

Controlling cell responses to an implantable material is essential to tissue engineering. Because the surface is in direct contact with cells, both chemical and topographical properties of a material surface can play a crucial role. In this study, parallel ridges/grooves were micropatterned on glass surfaces using photosensitive polyimide to create transparent substrates. The migratory behavior of live human neutrophils on the patterned surfaces was observed using a light microscope with transmitted light source. The width (2 microm) and length (400 microm) of the ridges were kept constant. The height (5 or 3 microm) and the repeat spacing (6-14 microm) of the ridges were systematically changed to investigate the effect of microgeometry on neutrophil migration. In addition, the effect of surface chemistry on neutrophil migration was studied by deposition of a thin layer of "inert", biocompatible metal such as Au-Pd alloy and titanium on patterned substrates. More than 95% of neutrophils moved in the direction of the long axis of ridges/grooves regardless of the topographical geometry and chemistry, consistent with a phenomenon termed "contact guidance". Therefore, cell migration was characterized using a one-dimensional persistent random walk. The rate of cell movement was strongly dependent on the topographical microgeometry of the ridges. The random motility coefficient mu, 9.8 x 10(-9) cm2/s, was the greatest at a ridge height of 5 microm and spacing of 10 microm, about 10 times faster than on smooth glass surface. The Au-Pd coating did not change neutrophil migratory behavior on patterned surfaces, whereas titanium decreased cell motility substantially. The results of this study suggest that optimization of both surface chemistry and topography may be important when designing biomaterials for tissue engineering. In addition, parallel ridges/grooves can be used to control the direction and rate of cell migration on the surface.

Biocompatible Materials↗

PubMed1974

A Markov process of gene frequency change in a geographically structured population.

A Markov process (chain) of gene frequency change is derived for a geographically-structured model of a population. The population consists of colonies which are connected by migration. Selection operates in each colony independently. It is shown that there exists a stochastic clock that transforms the originally complicated process of gene frequency change to a random walk which is independent of the geographical structure of the population. The time parameter is a local random time that is dependent on the sample path. In fact, if the alleles are selectively neutral, the time parameter is exactly equal to the sum of the average local genetic variation appearing in the population, and otherwise they are approximately equal. The Kolmogorov forward and backward equations of the process are obtained. As a limit of large population size, a diffusion process is derived. The transition probabilities of the Markov chain and of the diffusion process are obtained explicitly. Certain quantities of biological interest are shown to be independent of the population structure. The quantities are the fixation probability of a mutant, the sum of the average local genetic variation and the variation summed over the generations in which the gene frequency in the whole population assumes a specified value.

Alleles↗

PubMed2002

Non-Markovian random processes and traveling fronts in a reaction-transport system with memory and long-range interactions.

The problem of finding the propagation rate for traveling waves in reaction-transport systems with memory and long-range interactions has been considered. Our approach makes use of the generalized master equation with logistic growth, hyperbolic scaling, and Hamilton-Jacobi theory. We consider the case when the waiting-time distribution for the underlying microscopic random walk is modeled by the family of gamma distributions, which in turn leads to non-Markovian random processes and corresponding memory effects on mesoscopic scales. We derive formulas that enable us to determine the front propagation rate and understand how the memory and long-range interactions influence the propagation rate for traveling fronts. Several examples involving the Gaussian and discrete distributions for jump densities are presented.

Journal Article↗

PubMed2000

Leukocyte chemosensory migration on vascular prosthetic biomaterial is mediated by an integrin beta2 receptor chain.

The ability of adherent activated leukocytes to migrate on implanted prosthetic biomaterial surfaces may be an early rate-limiting step in eliminating periprosthetic infection. The goal of this study was to explore the molecular mechanism governing leukocyte migration on the implantable cardiovascular prosthetic biomaterial, expanded polytetrafluoroethylene (ePTFE), in response to stimulation by the soluble chemokine, N-formyl-methionyl-leucyl-phenylalanine (fMLP). We used a population level migration assay to study the migration of polymorphonuclear leukocytes (PMN) on ePTFE, overlaid by a gelatin/agar composite. A theoretical random walk model was applied to describe fMLP-induced PMN migration on ePTFE in terms of an objective random cell migration coefficient, mu. Our results show that following stimulation with 0-10(-7) M fMLP, the value of mu ranged from 5.43 x 10(-9) to 1.08 x 10(-7) cm2/s, with a maximum value obtained at 10(-8) M fMLP. We probed the expression levels of various beta2 integrin receptor subunits and their contribution to the migratory function of ePTFE-adherent PMN over a wide range of fMLP concentration. We found that the expression of the integrin beta-chain, CD18, was also maximized at 10(-8) M fMLP, along with only slight changes in the expression of integrin alpha-chains (CD11a,b,c). We report that treatment with antibodies against either beta or combined alpha chains, but not individual alpha chains, inhibited PMN attachment to ePTFE at 10(-8) M fMLP, suggesting the likely role of combined beta2 receptor subunits in early adhesion events following stimulation. However, treatment with only anti-CD18 significantly lowered PMN migration on ePTFE (mu = 5.98 x 10(-9) cm2/s), and this degree of inhibition was much greater than that elicited by the combined treatment with antibodies recognizing all possible alpha-chains. Overall, we conclude that migratory behavior of chemokinetically stimulated PMN on ePTFE is mediated by the integrin beta chain pool, and is only weakly regulated by the integrin alpha chain.

Antigens, CD↗

PubMed2005

Condensation phenomena in nanopores: a Monte Carlo study.

The nonequilibrium dynamics of condensation phenomena in nanopores is studied via Monte Carlo simulations of a lattice-gas model. Hysteretic behavior of the particle density as a function of the density of a reservoir is obtained for various pore geometries in two and three dimensions. The shape of the hysteresis loops depend on the characteristics of the pore geometry. The evaporation of particles from a pore can be fitted to a stretched exponential decay of the particle density. Phase-separation dynamics inside the pore is effectively described by a random walk of the non-wetting phases. Domain evolution is significantly slowed down in the presence of a random wall-particle potential and gives rise to a temperature-dependent growth exponent. A geometric roughness of the pore wall only delays the onset of a pure domain growth.

Journal Article↗

PubMed2005

Limit theorem for continuous-time quantum walk on the line.

Concerning a discrete-time quantum walk X(t)(d) with a symmetric distribution on the line, whose evolution is described by the Hadamard transformation, it was proved by the author that the following weak limit theorem holds: X(t)(d)/t -->dx/pi(1-x2) square root of (1-2x2) as t --> infinity. The present paper shows that a similar type of weak limit theorem is satisfied for a continuous-time quantum walk X((c) )(t ) on the line as follows: X(t)(c)/t --> dx/pi square root of (1-x2) as t --> infinity. These results for quantum walks form a striking contrast to the central limit theorem for symmetric discrete- and continuous-time classical random walks: Y(t)/square root of (t) --> e(-x2/2)dx/square root of (2pi) as t --> infinity. The work deals also with the issue of the relationship between discrete and continuous-time quantum walks. This topic, subject of a long debate in the previous literature, is treated within the formalism of matrix representation and the limit distributions are exhaustively compared in the two cases.

Journal Article↗

PubMed1991

Migration of individual microvessel endothelial cells: stochastic model and parameter measurement.

Analysis of cell motility effects in physiological processes can be facilitated by a mathematical model capable of simulating individual cell movement paths. A quantitative description of motility of individual cells would be useful, for example, in the study of the formation of new blood vessel networks in angiogenesis by microvessel endothelial cell (MEC) migration. In this paper we propose a stochastic mathematical model for the random motility and chemotaxis of single cells, and evaluate migration paths of MEC in terms of this model. In our model, cell velocity under random motility conditions is described as a persistent random walk using the Ornstein-Uhlenbeck (O-U) process. Two parameters quantify this process: the magnitude of random movement accelerations, alpha, and a decay rate constant for movement velocity, beta. Two other quantities often used in measurements of individual cell random motility properties--cell speed, S, and persistence time in velocity, Pv--can be defined in terms of the fundamental stochastic parameters alpha and beta by: S =square root (alpha/beta) and Pv = 1/beta. We account for chemotactic cell movement in chemoattractant gradients by adding a directional bias term to the O-U process. The magnitude of the directional bias is characterized by the chemotactic responsiveness, kappa. A critical advantage of the proposed model is that it can generate, using experimentally measured values of alpha, beta and kappa, computer simulations of theoretical individual cell paths for use in evaluating the role of cell migration in specific physiological processes. We have used the model to assess MEC migration in the presence of absence of the angiogenic stimulus acidic fibroblast growth factor (aFGF). Time-lapse video was used to observe and track the paths of cells moving in various media, and the mean square displacement was measured from these paths. To test the validity of the model, we compared the mean square displacement measurements of each cell with model predictions of that displacement. The comparison indicates that the O-U process provides a satisfactory description of the random migration at this level of comparison. Using nonlinear regression in these comparisons, we measured the magnitude of random accelerations, alpha, and the velocity decay rate constant, beta, for each cell path. We consequently obtained values for the derived quantities, speed and persistence time. In control medium, we find that alpha = 250 +/- 100 microns 2h-3 and beta = 0.22 +/- 0.03h-1, while in stimulus medium (control plus unpurified aFGF) alpha = 1900 +/- 720 microns 2h-3 and beta = 0.99 +/- 0.37h-1.(ABSTRACT TRUNCATED AT 400 WORDS)

Adipose Tissue↗