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PubMed2004

Polymers confined between two parallel plane walls.

Single three-dimensional polymers confined to a slab, i.e., to the region between two parallel plane walls, are studied by Monte Carlo simulations. They are described by N-step walks on a simple cubic lattice confined to the region 1< or = z < or = D. The simulations cover both regions D< >RF (where RF approximately Nnu is the Flory radius, with nu approximately 0.587), as well as the cross-over region in between. Chain lengths are up to N=80 000, slab widths up to D=120. In order to test the analysis program and to check for finite size corrections, we actually studied three different models: (a) ordinary random walks (mimicking Theta polymers); (b) self-avoiding walks; and (c) Domb-Joyce walks with the self-repulsion tuned to the point where finite size corrections for free (unrestricted) chains are minimal. For the simulations we employ the pruned-enriched-Rosenbluth method with Markovian anticipation. In addition to the partition sum (which gives us a direct estimate of the forces exerted onto the walls), we measure the density profiles of monomers and of end points transverse to the slab, and the radial extent of the chain parallel to the walls. All scaling laws and some of the universal amplitude ratios are compared to theoretical predictions.

Journal Article↗
PubMed1999

Adaptive walks on time-dependent fitness landscapes.

The idea of adaptive walks on fitness landscapes as a means of studying evolutionary processes on large time scales is extended to fitness landscapes that are slowly changing over time. The influence of ruggedness and of the amount of static fitness contributions is investigated for model landscapes derived from Kauffman's NK landscapes. Depending on the amount of static fitness contributions in the landscape, the evolutionary dynamics can be divided into a percolating and a nonpercolating phase. In the percolating phase, the walker performs a random walk over the regions of the landscape with high fitness.

Biological Evolution↗
PubMed2006

Microsaccades are triggered by low retinal image slip.

Even during visual fixation of a stationary target, our eyes perform rather erratic miniature movements, which represent a random walk. These "fixational" eye movements counteract perceptual fading, a consequence of fast adaptation of the retinal receptor systems to constant input. The most important contribution to fixational eye movements is produced by microsaccades; however, a specific function of microsaccades only recently has been found. Here we show that the occurrence of microsaccades is correlated with low retinal image slip approximately 200 ms before microsaccade onset. This result suggests that microsaccades are triggered dynamically, in contrast to the current view that microsaccades are randomly distributed in time characterized by their rate-of-occurrence of 1 to 2 per second. As a result of the dynamic triggering mechanism, individual microsaccade rate can be predicted by the fractal dimension of trajectories. Finally, we propose a minimal computational model for the dynamic triggering of microsaccades.

Algorithms↗
PubMed2000

Diffusion-limited predator-prey dynamics in Euclidean environments: an allometric individual-based model.

We claim that diffusion-limited rates of reaction can be an explanation for the altered population dynamics predicted by models incorporating local interactions and limited individual mobility. We show that the predictions of a spatially explicit, individual-based model result from reduced rates of predation and reproduction caused by limited individual mobility and patchiness. When these reduced rates are used in a mean-field model, there is better agreement with the predictions of the simulation model incorporating local interactions. We also explain previous findings regarding the effects of dimensionality on population dynamics in light of diffusion-limited reactions and Pólya random walks. In particular, we demonstrate that 3D systems are better "stirred" than 2D systems and consequently have a reduced tendency for diffusion-limited interaction rates.

Animals↗
PubMed2005

"Burnt-bridge" mechanism of molecular motor motion.

Motivated by a biased diffusion of molecular motors with the bias dependent on the state of the substrate, we investigate a random walk on a one-dimensional lattice that contains weak links (called "bridges") which are affected by the walker. Namely, a bridge is destroyed with probability when p the walker crosses it; the walker is not allowed to cross it again and this leads to a directed motion. The velocity of the walker is determined analytically for equidistant bridges. The special case of p = 1 is more tractable--both the velocity and the diffusion constant are calculated for uncorrelated locations of bridges, including periodic and random distributions.

Journal Article↗
PubMed1992

Lateral diffusion and aggregation. A Monte Carlo study.

Aggregation in a lipid bilayer is modeled as cluster-cluster aggregation on a square lattice. In the model, clusters carry out a random walk on the lattice, with a diffusion coefficient inversely proportional to mass. On contact, they adhere with a prescribed probability, rigidly and irreversibly. Monte Carlo calculations show that, as expected, rotational diffusion of the aggregating species is highly sensitive to the initial stages of aggregation. Lateral diffusion of an inert tracer obstructed by the aggregate is a sensitive probe of the later stages of aggregation. Cluster-cluster aggregates are much more effective barriers to lateral diffusion of an inert tracer than the same area fraction of random point obstacles is, but random point obstacles are more effective barriers than the same area fraction of compact obstacles. The effectiveness of aggregates as obstacles is discussed in terms of particle-particle correlation functions and fractal dimensions. Results are applicable to aggregation of membrane proteins, and at least qualitatively to aggregation of gel-phase lipid during lateral phase separation.

Diffusion↗
PubMed1998

Persistence of orientation toward a temporarily invisible landmark in Drosophila melanogaster.

In arena experiments with the walking fruit fly, we found a remarkable persistence of orientation toward a landmark that disappeared during the fly's approach. The directional stability achieved by 'after-fixation' allows a fly to continue pursuit under natural conditions, where a selected target is frequently concealed by surrounding structures. The persistence of after-fixation was investigated in Buridan's paradigm, where a fly walks persistently back and forth between two inaccessible landmarks. Upon disappearance of a selected target, the flies maintained their intended course for more than 15 body lengths of approximately 2.5 mm in about 50% of the trials. About 13% even exceeded 75 body lengths. About 88% of the approaches clustered in equal portions around peaks at 2.4 s and 8.6 s. About 12% of the approaches persisted even longer. In contrast, a single peak at about 2.2 s is sufficient to describe the persistence of orientation in a random walk. The ability to pursue an invisible landmark is disturbed neither by a transient angular deviation from the course toward this landmark, when this target disappeared, nor by a distracting second landmark. Accordingly, after-fixation seems to require an internal representation of the direction toward the concealed target, and idiothetical course control to maintain this direction.

Animals↗
PubMed2003

A mathematical model of single target site location by Brownian movement in subcellular compartments.

The location of distinct sites is mandatory for many cellular processes. In the subcompartments of the cell nucleus, only very small numbers of diffusing macromolecules and specific target sites of some types may be present. In this case, we are faced with the Brownian movement of individual macromolecules and their "random search" for single/few specific target sites, rather than bulk-averaged diffusion and multiple sites. In this article, I consider the location of a distant central target site, e.g. a globular protein, by individual macromolecules executing unbiased (i.e. drift-free) random walks in a spherical compartment. For this walk-and-capture model, the closed-form analytic solution of the first passage time probability density function (p.d.f.) has been obtained as well as the first and second moment. In the limit of a large ratio of the radii of the spherical diffusion space and central target, well-known relations for the variance and the first two moments for the exponential p.d.f. were found to hold with high accuracy. These calculations reinforce earlier numerical results and Monte Carlo simulations. A major implication derivable from the model is that non-directed random movement is an effective means for locating single sites in submicron-sized compartments, even when the diffusion coefficients are comparatively small and the diffusing species are present in one copy only. These theoretical conclusions are underscored numerically for effective diffusion constants ranging from 0.5 to 10.0 microm(2) s(-1), which have been reported for a couple of nuclear proteins in their physiological environment. Spherical compartments of submicron size are, for example, the Cajal bodies (size: 0.1-1.0 microm), which are present in 1-5 copies in the cell nucleus. Within a small Cajal body of radius 0.1 microm a single diffusing protein molecule (with D=0.5 microm(2) s(-1)) would encounter a medium-sized protein of radius 2.5 nm within 1 s with a probability near certainty (p=0.98).

Animals↗
PubMed2007

A new approach to study fibroblast migration.

This paper presents a new approach to study cell migration. Human tendon fibroblasts were plated on silicone membranes coated with 10 microg/ml ProNectin-F. The silicone surfaces were micro-fabricated with parallel microgrooves, with 10 microm ridge and groove width, and 3 microm groove depth. Fibroblasts grown in the microgrooves had an elongated shape and oriented along the microgroove direction. They also moved along the same direction instead of "random walk" when cells migrate on smooth culture surfaces. In response to TGF-beta1 (5 ng/ml) treatment, these fibroblasts on the microgrooved surfaces were differentiated into myofibroblasts, as judged by an elevated expression of alpha-smooth muscle actin (alpha-SMA), a specific marker for myofibroblasts. Moreover, these myofibroblasts were found to be approximately 30% less motile compared to that of untreated fibroblasts. Thus, use of microgrooved surface may be an effective approach to detect difference in cell motility because cell migration on the microgrooved surface is one dimensional and hence easier to be quantified than two-dimensional random movement on conventional smooth culture surfaces.

Actins↗
PubMed1984

Polydispersity effects on head-to-tail polymerization of F-actin. A theoretical treatment.

A theoretical description of the release of labeled subunits of F-actin that undergoes head-to-tail polymerization utilizing random-walk theory is extended to the case of a finite-length polymer. An exact expression is given for the number of released subunits in a suitable form for numerical calculation. By comparing the results for a finite-length polymer with that for an infinite-length case, the limitation of the infinite-length approximation is readily known. The results for finite-length polymers are applied to systems with polydispersity. It is shown that the number of labeled subunits released from F-actin depends strongly on the distribution functions of the polymer lengths and the values of the transition probabilities of the random walkers.

Actins↗
PubMed2006

Quantifying polypeptide conformational space: sensitivity to conformation and ensemble definition.

Quantifying the density of conformations over phase space (the conformational distribution) is needed to model important macromolecular processes such as protein folding. In this work, we quantify the conformational distribution for a simple polypeptide (N-mer polyalanine) using the cumulative distribution function (CDF), which gives the probability that two randomly selected conformations are separated by less than a "conformational" distance and whose inverse gives conformation counts as a function of conformational radius. An important finding is that the conformation counts obtained by the CDF inverse depend critically on the assignment of a conformation's distance span and the ensemble (e.g., unfolded state model): varying ensemble and conformation definition (1 --> 2 A) varies the CDF-based conformation counts for Ala(50) from 10(11) to 10(69). In particular, relatively short molecular dynamics (MD) relaxation of Ala(50)'s random-walk ensemble reduces the number of conformers from 10(55) to 10(14) (using a 1 A root-mean-square-deviation radius conformation definition) pointing to potential disconnections in comparing the results from simplified models of unfolded proteins with those from all-atom MD simulations. Explicit waters are found to roughen the landscape considerably. Under some common conformation definitions, the results herein provide (i) an upper limit to the number of accessible conformations that compose unfolded states of proteins, (ii) the optimal clustering radius/conformation radius for counting conformations for a given energy and solvent model, (iii) a means of comparing various studies, and (iv) an assessment of the applicability of random search in protein folding.

Computer Simulation↗
PubMed1995

Finger-like lysing patterns of blood clots.

One-dimensional modeling of fibrinolysis (Senf, 1979; Zidansek and Blinc, 1991; Diamond and Anand, 1993) has accounted for the dissolution velocity, but the shape of the lysing patterns can be explained only by two- or three- drug-induced blood clot dissolution patterns obtained by proton nuclear magnetic resonance imaging, which can be described by the enzyme transport-limited system of fibrinolytic chemical equations with diffusion and perfusion terms (Zidansek and Blinc, 1991) in the reaction time approximation if the random character of gel porosity is taken into account. A two-dimensional calculation based on the Hele-Shaw random walk models (Kadanoff, 1985; Liang, 1986) leads to fractal lysing patterns as, indeed, is observed. The fractal dimension of the experimental lysing patterns changes from 1.2 at the beginning of the experiments to a maximum of approximately 1.3 in the middle and then decreases toward one when the clot is recanalized.

Fibrinogen↗
PubMed1994

The structure of human S-phase chromosome fibres.

Recent in situ hybridization studies suggested that within the range of 0.1-1.0 Mb, human interphase chromosomes follow a random walk model (i.e. they behave as flexible polymers without major constraints). However, chromosome structure may differ in the G1, S, and G2 phases, and phase-specific constraints may be masked if the chromosome analysis does not discriminate between the phases. Therefore, using confocal microscopy, we examined the structure of S-phase chromosomes labelled with 5-iododeoxyuridine after prolonged treatment with 5-fluorodeoxyuridine. In the S-phase, labelled 0.32 mu chromosome fibres mostly appear as semi-circles with an average diameter of 0.83 +/- 0.03 mu. These semi-circles are joined together to form different 3D structures, and two semicircles frequently adopt s- or omega-like conformations involving about 2.5 mu of the chromosome contour length (L). Morphometric analysis of the S-phase fibres suggests that our data fit both the random flexible polymer model and also a model in which two constrained semi-circles are attached to each other by a flexible joint, thus eliminating constraints at long distances (L more than 2 mu).

Breast Neoplasms↗
PubMedDate not supplied

A generalized formulation of diffusion effects in micron resolution nuclear magnetic resonance imaging.

A generalized formulation of the diffusion related nuclear magnetic resonance (NMR) signal is derived from a random walk model. Previous analyses performed in the NMR spectroscopy were the formulations of the diffusion related signal amplitude at a specific time, such as the spin echo formation time. They are, in general, not applicable to continuous time domain analyses. In this paper, we have extended the theory to the two-dimensional imaging case and derived an analytical formula useful for the computation of the diffusion affected signal as a function of continuous time for a time variant gradient. This formulation will be useful in NMR imaging, especially in NMR microscopy where the diffusion associated signal attenuation is serious due to the strong gradient fields (100-1000 G/cm), and at the same time data are acquired continuously for the acquisition period. In addition to the loss of the resolution and signal-to-noise ratio due to the random phase fluctuation by diffusion, the variation of the intensity during the data acquisition period introduces a line broadening whose full width at half-maximum is found to be much larger than the bandwidth-limited resolution or diffusion related intrinsic resolution. This line spreading effect is integrated in a computer simulation and is evaluated as an integral part of the overall diffusion effects in micron resolution NMR imaging or NMR microscopy.

Computer Simulation↗
PubMed1998

Effects of microgravity on the growth of Lepidium roots.

The normal growth dynamics of plant roots is partly controlled by the gravitational force. In order to study the detailed growth behavior in absence of gravity, the growth of Lepidium sativum roots was recorded by time lapse photography at 1h intervals in a Spacelab ESA-experiment (IML-2). Plants were germinated and kept in microgravity during the experiments, while control roots were at 1 g with normal static gravistimulation. Extended image analyses allowed new information to be achieved about movements of all parts of the roots, extending earlier published results. Root contours were extracted from the images and divided into 0.6mm segments. Deviation angles were calculated for each root segment, both for the first 8-10 h (phase I) and for the last 6-8 h of the experiment (phase II). For phase I, the present analysis confirmed that the average square deviation increased linearly with time for roots in microgravity, while for roots under 1 g conditions it stayed constant. This was consistent with a random walk hypothesis for the bending pattern. In phase II, roots in microgravity stopped their spontaneous curvatures and showed more straight growth or even diminished the root curvatures that had occurred during phase I. Thus, the growth is distinctly different in the two phases and is thought to be controlled by autotropic reactions in phase II. Root hairs developed when the roots passed into phase II. During phase I, the root growth rates were equal in microgravity and on the ground (0.50 mm h-1 with SE 0.04 and 0.51 mm h-1 with SE 0.03, respectively). In phase II the growth rate on the ground was higher than in microgravity (1.44 mm h-1 with SE 0.10 and 1.07 mm h-1 with SE 0.04 in microgravity). Microgravity conditions, therefore, clearly affect Lepidium root growth: In phase I the bending pattern is random in contrast to the normal straight growth under 1g. In phase II the growth rate is reduced, as compared to the growth rate under 1 g.

Brassicaceae↗
PubMed1986

Is ubiquinone diffusion rate-limiting for electron transfer?

The different possible dispositions of the electron transfer components in electron transfer chains are discussed: random distribution of complexes and ubiquinone with diffusion-controlled collisions of ubiquinone with the complexes, random distribution as above, but with ubiquinone diffusion not rate-limiting, diffusion and collision of protein complexes carrying bound ubiquinone, and solid-state assembly. Discrimination among these possibilities requires knowledge of the mobility of the electron transfer chain components. The collisional frequency of ubiquinone-10 with the fluorescent probe 12-(9-anthroyl)stearate, investigated by fluorescence quenching, is 2.3 X 10(9) M-1 sec-1 corresponding to a diffusion coefficient in the range of 10(-6) cm2/sec (Fato, R., Battino, M., Degli Esposti, M., Parenti Castelli, G., and Lenaz, G., Biochemistry, 25, 3378-3390, 1986); the long-range diffusion of a short-chain polar Q derivative measured by fluorescence photobleaching recovery (FRAP) (Gupte, S., Wu, E. S., Höchli, L., Höchli, M., Jacobson, K., Sowers, A. E., and Hackenbrock, C. R., Proc. Natl. Acad. Sci. USA 81, 2606-2610, 1984) is 3 X 10(-9) cm2/sec. The discrepancy between these results is carefully scrutinized, and is mainly ascribed to the differences in diffusion ranges measured by the two techniques; it is proposed that short-range diffusion, measured by fluorescence quenching, is more meaningful for electron transfer than long-range diffusion measured by FRAP, or microcollisions, which are not sensed by either method. Calculation of the distances traveled by random walk of ubiquinone in the membrane allows a large excess of collisions per turnover of the respiratory chain. Moreover, the second-order rate constants of NADH-ubiquinone reductase and ubiquinol-cytochrome c reductase are at least three orders of magnitude lower than the second-order collisional constant calculated from the diffusion of ubiquinone. The activation energies of either the above activities or integrated electron transfer (NADH-cytochrome c reductase) are well above that for diffusion (found to be ca. 1 kcal/mol). Cholesterol incorporation in liposomes, increasing bilayer viscosity, lowers the diffusion coefficients of ubiquinone but not ubiquinol-cytochrome c reductase or succinate-cytochrome c reductase activities. The decrease of activity by ubiquinone dilution in the membrane is explained by its concentration falling below the Km of the partner enzymes. It is calculated that ubiquinone diffusion is not rate-limiting, favoring a random model of the respiratory chain organization.(ABSTRACT TRUNCATED AT 400 WORDS)

Diffusion↗
PubMed2006

Hierarchical structures induce long-range dynamical correlations in written texts.

Thoughts and ideas are multidimensional and often concurrent, yet they can be expressed surprisingly well sequentially by the translation into language. This reduction of dimensions occurs naturally but requires memory and necessitates the existence of correlations, e.g., in written text. However, correlations in word appearance decay quickly, while previous observations of long-range correlations using random walk approaches yield little insight on memory or on semantic context. Instead, we study combinations of words that a reader is exposed to within a "window of attention," spanning about 100 words. We define a vector space of such word combinations by looking at words that co-occur within the window of attention, and analyze its structure. Singular value decomposition of the co-occurrence matrix identifies a basis whose vectors correspond to specific topics, or "concepts" that are relevant to the text. As the reader follows a text, the "vector of attention" traces out a trajectory of directions in this "concept space." We find that memory of the direction is retained over long times, forming power-law correlations. The appearance of power laws hints at the existence of an underlying hierarchical network. Indeed, imposing a hierarchy similar to that defined by volumes, chapters, paragraphs, etc. succeeds in creating correlations in a surrogate random text that are identical to those of the original text. We conclude that hierarchical structures in text serve to create long-range correlations, and use the reader's memory in reenacting some of the multidimensionality of the thoughts being expressed.

Language↗
PubMed2000

Monte carlo simulation of 3-D buffered Ca(2+) diffusion in neuroendocrine cells.

Buffered Ca(2+) diffusion in the cytosol of neuroendocrine cells is a plausible explanation for the slowness and latency in the secretion of hormones. We have developed a Monte Carlo simulation to treat the problem of 3-D diffusion and kinetic reactions of ions and buffers. The 3-D diffusion is modeled as a random walk process that follows the path of each ion and buffer molecule, combined locally with a stochastic treatment of the first-order kinetic reactions involved. Such modeling is able to predict [Ca(2+)] and buffer concentration time courses regardless of how low the calcium influx is, and it is therefore a convenient method for dealing with physiological calcium currents and concentrations. We study the effects of the diffusional and kinetic parameters of the model on the concentration time courses as well as on the local equilibrium of buffers with calcium. An in-mobile and fast endogenous buffer as described by, Biophys. J. 72:674-690) was able to reach local equilibrium with calcium; however, the exogenous buffers considered are displaced drastically from equilibrium at the start of the calcium pulse, particularly below the pores. The versatility of the method also allows the effect of different arrangements of calcium channels on submembrane gradients to be studied, including random distribution of calcium channels and channel clusters. The simulation shows how the particular distribution of channels or clusters can be of relevance for secretion in the case where the distribution of release granules is correlated with the channels or clusters.

Animals↗