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PubMed2004

Persistence exponents and the statistics of crossings and occupation times for Gaussian stationary processes.

We consider the persistence probability, the occupation-time distribution, and the distribution of the number of zero crossings for discrete or (equivalently) discretely sampled Gaussian stationary processes (GSPs) of zero mean. We first consider the Ornstein-Uhlenbeck process, finding expressions for the mean and variance of the number of crossings and the "partial survival" probability. We then elaborate on the correlator expansion developed in an earlier paper [G. C. M. A. Ehrhardt and A. J. Bray, Phys. Rev. Lett. 88, 070602 (2002)] to calculate discretely sampled persistence exponents of GSPs of known correlator by means of a series expansion in the correlator. We apply this method to the processes d(n)x/dt(n)=eta(t) with n>/=3, incorporating an extrapolation of the series to the limit of continuous sampling. We then extend the correlator method to calculate the occupation-time and crossing-number distributions, as well as their partial-survival distributions and the means and variances of the occupation time and number of crossings. We apply these general methods to the d(n)x/dt(n)=eta(t) processes for n=1 (random walk), n=2 (random acceleration), and larger n, and to simple diffusion from random initial conditions in one to three dimensions. The results for discrete sampling are extrapolated to the continuum limit where possible.

Journal Article↗

PubMed2000

How efficiently Do three pointlike particles sample phase space?

We show that the continuous phase space of a hard particle system can be mapped onto a discrete but infinite phase space. For three pointlike particles confined to a ring, the evolution of the system maps onto a chaotic walk on a hexagonal lattice. This facilitates direct measurement of the departure of the system from its original configuration. In special cases of mass ratios the phase space becomes closed and finite (nonergodic). There are qualitative differences between this chaotic walk and a random walk, in particular a more rapid sampling of phase space.

Journal Article↗

PubMed1993

On the existence of scaling in DNA sequences.

A re-evaluation of the properties of the random walks generated from DNA sequences reveals that previously reported scaling (appropriate to either pure random walks or long-range correlations) cannot be justified. Thus our analysis cannot support the claim that different evolutionary categories exhibit different scaling.

Base Sequence↗

PubMed2000

A generalized transport model for biased cell migration in an anisotropic environment.

A generalized transport model is derived for cell migration in an anisotropic environment and is applied to the specific cases of biased cell migration in a gradient of a stimulus (taxis; e.g., chemotaxis or haptotaxis) or along an axis of anisotropy (e.g., contact guidance). The model accounts for spatial or directional dependence of cell speed and cell turning behavior to predict a constitutive cell flux equation with drift velocity and diffusivity tensor (termed random motility tensor) that are explicit functions of the parameters of the underlying random walk model. This model provides the connection between cell locomotion and the resulting persistent random walk behavior to the observed cell migration on longer time scales, thus it provides a framework for interpreting cell migration data in terms of underlying motility mechanisms.

Anisotropy↗

PubMed1996

A modified gambler's ruin model of polyethylene chains in the amorphous region.

Polyethylene chains in the amorphous region between two crystalline lamellae M unit apart are modeled as random walks with one-step memory on a cubic lattice between two absorbing boundaries. These walks avoid the two preceding steps, though they are not true self-avoiding walks. Systems of difference equations are introduced to calculate the statistics of the restricted random walks. They yield that the fraction of loops is (2M - 2)/(2M + 1), the fraction of ties 3/(2M + 1), the average length of loops 2M - 0.5, the average length of ties 2/3M2 + 2/3M - 4/3, the average length of walks equals 3M - 3, the variance of the loop length 16/15M3 + O(M2), the variance of the tie length 28/45M4 + O(M3), and the variance of the walk length 2M3 + O(M2).

Journal Article↗

PubMed2000

Delayed stochastic systems

Noise and time delay are two elements that are associated with many natural systems, and often they are sources of complex behaviors. Understanding of this complexity is yet to be explored, particularly when both elements are present. As a step to gain insight into such complexity for a system with both noise and delay, we investigate such delayed stochastic systems both in dynamical and probabilistic perspectives. A Langevin equation with delay and a random-walk model whose transition probability depends on a fixed time-interval past (delayed random walk model) are the subjects of in depth focus. As well as considering relations between these two types of models, we derive an approximate Fokker-Planck equation for delayed stochastic systems and compare its solution with numerical results.

Journal Article↗

PubMed2007

Automatic detection of single fluorophores in live cells.

Recent developments in light microscopy enable individual fluorophores to be observed in aqueous conditions. Biological molecules, labeled with a single fluorophore, can be localized as isolated spots of light when viewed by optical microscopy. Total internal reflection fluorescence microscopy greatly reduces background fluorescence and allows single fluorophores to be observed inside living cells. This advance in live-cell imaging means that the spatial and temporal dynamics of individual molecules can be measured directly. Because of the stochastic nature of single molecule behavior a statistically meaningful number of individual molecules must be detected and their separate trajectories in space and time stored and analyzed. Here, we describe digital image processing methods that we have devised for automatic detection and tracking of hundreds of molecules, observed simultaneously, in vitro and within living cells. Using this technique we have measured the diffusive behavior of pleckstrin homology domains bound to phosphoinositide phospholipids at the plasma membrane of live cultured mammalian cells. We found that mobility of these membrane-bound protein domains is dominated by mobility of the lipid molecule to which they are attached and is highly temperature dependent. Movement of PH domains isolated from the tail region of myosin-10 is consistent with a simple random walk, whereas, diffusion of intact PLC-delta1 shows behavior inconsistent with a simple random walk. Movement is rapid over short timescales but much slower at longer timescales. This anomalous behavior can be explained by movement being restricted to membrane regions of 0.7 microm diameter.

Artificial Intelligence↗

PubMed2006

Low-frequency vibratory exercise reduces the risk of bone fracture more than walking: a randomized controlled trial.

BACKGROUND: Whole-body vibration (WBV) is a new type of exercise that has been increasingly tested for the ability to prevent bone fractures and osteoporosis in frail people. There are two currently marketed vibrating plates: a) the whole plate oscillates up and down; b) reciprocating vertical displacements on the left and right side of a fulcrum, increasing the lateral accelerations. A few studies have shown recently the effectiveness of the up-and-down plate for increasing Bone Mineral Density (BMD) and balance; but the effectiveness of the reciprocating plate technique remains mainly unknown. The aim was to compare the effects of WBV using a reciprocating platform at frequencies lower than 20 Hz and a walking-based exercise programme on BMD and balance in post-menopausal women. METHODS: Twenty-eight physically untrained post-menopausal women were assigned at random to a WBV group or a Walking group. Both experimental programmes consisted of 3 sessions per week for 8 months. Each vibratory session included 6 bouts of 1 min (12.6 Hz in frequency and 3 cm in amplitude with 60 degrees of knee flexion) with 1 min rest between bouts. Each walking session was 55 minutes of walking and 5 minutes of stretching. Hip and lumbar BMD (g.cm-2) were measured using dual-energy X-ray absorptiometry and balance was assessed by the blind flamingo test. ANOVA for repeated measurements was adjusted by baseline data, weight and age. RESULTS: After 8 months, BMD at the femoral neck in the WBV group was increased by 4.3% (P = 0.011) compared to the Walking group. In contrast, the BMD at the lumbar spine was unaltered in both groups. Balance was improved in the WBV group (29%) but not in the Walking group. CONCLUSION: The 8-month course of vibratory exercise using a reciprocating plate is feasible and is more effective than walking to improve two major determinants of bone fractures: hip BMD and balance.

Age Factors↗

PubMed2005

Exploring complex networks by walking on them.

We carry out a comparative study of the problem of a walker searching several typical complex networks. The search efficiency is evaluated for various strategies. Having no knowledge of the global properties of the underlying networks and the optimal path between any two given nodes, it is found that the best search strategy is the self-avoiding random walk. The preferentially self-avoiding random walk does not help in improving the search efficiency further. In return, topological information of the underlying networks may be drawn by comparing the results of the different search strategies.

Journal Article↗

PubMed2006

Model averaging in linkage analysis.

Methods for genetic linkage analysis are traditionally divided into "model-dependent" and "model-independent," but there may be a useful place for an intermediate class, in which a broad range of possible models is considered as a parametric family. It is possible to average over model space with an empirical Bayes prior that weights models according to their goodness of fit to epidemiologic data, such as the frequency of the disease in the population and in first-degree relatives (and correlations with other traits in the pleiotropic case). For averaging over high-dimensional spaces, Markov chain Monte Carlo (MCMC) has great appeal, but it has a near-fatal flaw: it is not possible, in most cases, to provide rigorous sufficient conditions to permit the user safely to conclude that the chain has converged. A way of overcoming the convergence problem, if not of solving it, rests on a simple application of the principle of detailed balance. If the starting point of the chain has the equilibrium distribution, so will every subsequent point. The first point is chosen according to the target distribution by rejection sampling, and subsequent points by an MCMC process that has the target distribution as its equilibrium distribution. Model averaging with an empirical Bayes prior requires rapid estimation of likelihoods at many points in parameter space. Symbolic polynomials are constructed before the random walk over parameter space begins, to make the actual likelihood computations at each step of the random walk very fast. Power analysis in an illustrative case is described. (c) 2006 Wiley-Liss, Inc.

Algorithms↗

PubMed1987

Transport of platelets in flowing blood.

Distribution and transport of platelets in flowing blood were studied experimentally using suspensions of washed red cells and fluorescent latex beads as platelet analogues. Distributions of the platelet analogues were obtained from stroboscopic epifluorescence photomicrographs of flow in 50-micron channels and from images of the cut cross sections of cryogenically frozen thin-walled 200-micron tubes. Concentration profiles of platelet analogues had a substantial near-wall excess for situations with a substantial hematocrit (greater than 10%) and a substantial wall shear rate (greater than 400 s-1). The viscosity of the suspending fluid was found to affect the size of the near-wall excess and its shear-dependent onset. Additionally, the shear-rate dependence of the near-wall excess did not occur with suspensions of hardened red cells. The excess extended a substantial distance from the wall in the 200-micron tubes and a portion of the profile could be fitted to an exponential curve. The random walk model that is used to describe enhanced platelet diffusion is envisioned as a walk (lateral platelet motion) caused by shear-induced collisions with red cells. A more comprehensive random walk model that includes biased collisions produces an effective lateral motion of convective nature in addition to a diffusional motion; it is used to explain the observed nonuniform distributions of platelet analogues.

Animals↗

PubMed2000

Phototactic orientation mechanism in the ciliate Fabrea salina, as inferred from numerical simulations.

The marine ciliate Fabrea salina shows a clear positive phototaxis, but the mechanism by which a single cell is able to detect the direction of light and orient its swimming accordingly is still unknown. A simple model of phototaxis is that of a biased random walk, where the bias due to light can affect one or more of the parameters that characterize a random walk, i.e., the mean speed, the frequency distribution of the angles of directional changes and the frequency of directional changes. Since experimental evidence has shown no effect of light on the mean speed of Fabrea salina, we have excluded models depending on this parameter. We have, therefore, investigated the phototactic orientation of Fabrea salina by computer simulation of two simple models, the first where light affects the frequency distribution of the angles of directional changes (model M1) and the second where the light bias modifies the frequency of directional changes (model M2). Simulated M1 cells directly orient their swimming towards the direction of light, regardless of their current swimming orientation; simulated M2 cells, on the contrary, are unable to actively orient their motion, but remain locked along the light direction once they find it by chance. The simulations show that these two orientation models lead to different macroscopic behaviours of the simulated cell populations. By comparing the results of the simulations with the experimental ones, we have found that the phototactic behaviour of real cells is more similar to that of the M2 model.

Animals↗

PubMed1996

Random root movements in weightlessness.

The dynamics of root growth was studied in weightlessness. In the absence of the gravitropic reference direction during weightlessness, root movements could be controlled by spontaneous growth processes, without any corrective growth induced by the gravitropic system. If truly random of nature, the bending behavior should follow so-called 'random walk' mathematics during weightlessness. Predictions from this hypothesis were critically tested. In a Spacelab ESA-experiment, denoted RANDOM and carried out during the IML-2 Shuttle flight in July 1994, the growth of garden cress (Lepidium sativum) roots was followed by time lapse photography at 1-h intervals. The growth pattern was recorded for about 20 h. Root growth was significantly smaller in weightlessness as compared to gravity (control) conditions. It was found that the roots performed spontaneous movements in weightlessness. The average direction of deviation of the plants consistently stayed equal to zero, despite these spontaneous movements. The average squared deviation increased linearly with time as predicted theoretically (but only for 8-10 h). Autocorrelation calculations showed that bendings of the roots, as determined from the 1-h photographs, were uncorrelated after about a 2-h interval. It is concluded that random processes play an important role in root growth. Predictions from a random walk hypothesis as to the growth dynamics could explain parts of the growth patterns recorded. This test of the hypothesis required microgravity conditions as provided for in a space experiment.

Brassicaceae↗

PubMed1996

Multifractals, encoded walks and the ergodicity of protein sequences.

A variety of statistical methods have been developed to explore correlations in protein and nucleic acid sequences. Such correlations have important implications for the evolution and stability of these macromolecules. Recently, a number of fractal analyses of sequence data have been developed. These analyses have considerable appeal as they are extremely sensitive to long range correlations and to hierarchical structures. One such analysis decodes sequence information into a random walk and the statistics of the resulting random walk is investigated. Anomalous scaling of such walks has been interpreted as indicative of a fractal structure. Alternatively, a generalized box counting analysis of decoded sequences can be used to establish multifractal properties. In this work, the connection between these two seemingly disparate approaches is established. This connection is exploited to investigate correlations in protein sequences. An ensemble consisting of a comprehensive data set of representative protein sequences is analyzed to establish the ergodicity of protein sequences. The implications of this ergodicity for information theoretical approaches to protein structure prediction is explored.

Amino Acid Sequence↗

PubMed1988

A stochastic model for leukocyte random motility and chemotaxis based on receptor binding fluctuations.

Two central features of polymorphonuclear leukocyte chemosensory movement behavior demand fundamental theoretical understanding. In uniform concentrations of chemoattractant, these cells exhibit a persistent random walk, with a characteristic "persistence time" between significant changes in direction. In chemoattractant concentration gradients, they demonstrate a biased random walk, with an "orientation bias" characterizing the fraction of cells moving up the gradient. A coherent picture of cell movement responses to chemoattractant requires that both the persistence time and the orientation bias be explained within a unifying framework. In this paper, we offer the possibility that "noise" in the cellular signal perception/response mechanism can simultaneously account for these two key phenomena. In particular, we develop a stochastic mathematical model for cell locomotion based on kinetic fluctuations in chemoattractant/receptor binding. This model can simulate cell paths similar to those observed experimentally, under conditions of uniform chemoattractant concentrations as well as chemoattractant concentration gradients. Furthermore, this model can quantitatively predict both cell persistence time and dependence of orientation bias on gradient size. Thus, the concept of signal "noise" can quantitatively unify the major characteristics of leukocyte random motility and chemotaxis. The same level of noise large enough to account for the observed frequency of turning in uniform environments is simultaneously small enough to allow for the observed degree of directional bias in gradients.

Cell Movement↗

PubMed2000

Critical properties of a branched polymer growth model.

We study the branched polymer growth model (BPGM) introduced by Lucena et al. [Phys. Rev. Lett. 72, 230 (1994)] in two dimensions. First the BPGM was simulated in very large lattices with concentrations of impurities q=0 and q=0.2. The scaling of the mass in chemical space gives accurate estimates of the critical branching probabilities b(c) and of the chemical dimensions Dc at criticality, improving previous results. Estimates of the fractal dimension D(F) at criticality are consistent with a universal value along the critical line. Our results for q=0 suggest small deviations of Dc and D(F) from the percolation values. We also simulated the BPGM in finite lattices of lengths between L=32 and L=512 for the same concentrations q. Using finite-size scaling techniques, we confirm the previous estimates of D(F) and the universality along the critical line, and obtain the correlation exponent nu=1.43+/-0.06. It proves that the BPGM is not in the same universality class of percolation in two dimensions. Finally, we simulate random walks on the critical polymers grown in very large lattices with q=0 and q=0.2, and obtain the random walk dimension Dw and the spectral dimension Ds. Dw is larger and Ds is smaller than the corresponding values in critical percolation clusters, due to the lower connectivity of the polymers. The scaling relation Ds=2D(F)/Dw is not satisfied, as observed in other tree-like structures.

Journal Article↗

PubMed2001

Random spread on the family of small-world networks.

We present analytical and numerical results of a random walk on the family of small-world graphs. The average access time shows a crossover from regular to random behavior with increasing distance from the starting point of the random walk. We introduce an independent step approximation, which enables us to obtain analytic results for the average access time. We observe a scaling relation for the average access time in the degree of the nodes. The behavior of the average access time as a function of p shows striking similarity with that of the characteristic length of the graph. This observation may have important applications in routing and switching in networks with a large number of nodes.

Journal Article↗

PubMed2002

Immediate and long-term effects of ankle-foot orthosis on muscle activity during walking: a randomized study of patients with unilateral foot drop.

OBJECTIVES: To determine (1) whether use of an ankle-foot orthosis (AFO) by patients with ankle dorsiflexor paresis leads to decreased muscle activity, immediately or 6 weeks after AFO use, and (2) whether this decrease (if present) differs between healthy and paretic subjects. DESIGN: Cross-sectional and longitudinal randomized case-control study. SETTING: Rehabilitation research center in the Netherlands. PARTICIPANTS: Fourteen healthy persons and 29 patients with foot drop. INTERVENTIONS: Muscle activity was measured by surface electromyography. Electromyographic reproducibility was tested in 14 healthy volunteers walking with and without AFO. Acute changes in muscle activity from AFO use were compared between the 14 healthy persons and the 29 patients with foot drop. Adaptation effects of AFO use after 6 weeks were studied in 29 patients, randomly chosen 16 of whom had started using an AFO at the first measurement. MAIN OUTCOME MEASURES: Amount of change in mean rectified electromyographic activity (delta value) between walking with and without AFO. Follow-up measurements were conducted after 3 and 6 weeks. RESULTS: Correlation coefficients, reflecting within-subject reproducibility, varied between.68 and.96 (mean,.86). In patients and healthy subjects, tibialis anterior muscle activity decreased by 7% and 20% (P = .01, P = .04), respectively, when using an AFO. In patients, this decrease was measured in the overall activity during the gait cycle; in healthy subjects, it was measured in the first 15% of the gait cycle. Overall electromyographic activity did not change during 6 weeks; delta values per muscle did not change during follow-up in the AFO group. CONCLUSION: AFO use immediately reduced muscle activity of the ankle dorsiflexors. However, using an AFO for 6 weeks did not lead to a generally lower electromyographic activity level nor did the amount of activity reduction accumulate in comparison with patients who did not use an AFO. It is, therefore, safe to use an AFO, even with recently paretic patients.

Adult↗