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PubMed1980

A random walk model for the analysis of the isoprenaline-stimulated proliferative response of the rat submaxillary gland.

In a series of papers, Hodgson, Radley and Koschel have studied the proliferative response of the rat submaxillary gland to the drug isoprenaline. They found that whereas the mitotic activity of the gland is normally very low, a single injection of isoprenaline will stimulate a wave of DNA synthesis and repeated administration at daily intervals, or longer, results in further waves of DNA synthesis. An explanation of these observations was proposed by Radley, Hodgson & Koschel (1976) who introduced the concept that administration of Isoprenaline resulted in a period of enhanced 'biochemical activity' of the cells. In this paper we present a stochastic model, based on a random walk model introduced by Hopper & Brockwell (1978), to represent quantitatively the concept of 'biochemical activity'. It is shown that the parameters of this model can be adjusted to fit the observed data. It is noted that the simpler Smith & Martin model (1973), which is a special case of our random walk model, can also be fitted to the data. However, there is experimental evidence that contradicts an assumption of the Smith & Martin model, but agrees with the predictions of our model. The random walk model also reveals that the different responses of the cell population to different injection schedules cannot be fully explained by changed in the cell population's biochemical activity distribution.

Animals↗
PubMed1988

Interrelations between random walks on diagrams (graphs) with and without cycles.

Three topics are discussed. A discrete-state, continuous-time random walk with one or more absorption states can be studied by a presumably new method: some mean properties, including the mean time to absorption, can be found from a modified diagram (graph) in which each absorption state is replaced by a one-way cycle back to the starting state. The second problem is a random walk on a diagram (graph) with cycles. The walk terminates on completion of the first cycle. This walk can be replaced by an equivalent walk on a modified diagram with absorption. This absorption diagram can in turn be replaced by another modified diagram with one-way cycles back to the starting state, just as in the first problem. The third problem, important in biophysics, relates to a long-time continuous walk on a diagram with cycles. This diagram can be transformed (in two steps) to a modified, more-detailed, diagram with one-way cycles only. Thus, the one-way cycle fluxes of the original diagram can be found from the state probabilities of the modified diagram. These probabilities can themselves be obtained by simple matrix inversion (the probabilities are determined by linear algebraic steady-state equations). Thus, a simple method is now available to find one-way cycle fluxes exactly (previously Monte Carlo simulation was required to find these fluxes, with attendant fluctuations, for diagrams of any complexity). An incidental benefit of the above procedure is that it provides a simple proof of the one-way cycle flux relation Jn +/- = IIn +/- sigma n/sigma, where n is any cycle of the original diagram.

Algorithms↗
PubMed1999

Spatial instabilities in reaction random walks with direction-independent kinetics.

We study spatial instabilities in reacting and diffusing systems, where diffusion is modeled by a persistent random walk instead of the usual Brownian motion. Perturbations in these reaction walk systems propagate with finite speed, whereas in reaction-diffusion systems localized disturbances affect every part instantly, albeit with heavy damping. We present evolution equations for reaction random walks whose kinetics do not depend on the particles' direction of motion. The homogeneous steady state of such systems can undergo two types of transport-driven instabilities. One type of bifurcation gives rise to stationary spatial patterns and corresponds to the Turing instability in reaction-diffusion systems. The other type occurs in the ballistic regime and leads to oscillatory spatial patterns; it has no analog in reaction-diffusion systems. The conditions for these bifurcations are derived and applied to two model systems. We also analyze the stability properties of one-variable systems and find that small wavelength perturbations decay in an oscillatory manner.

Journal Article↗
PubMed1988

Further properties of random walks on diagrams (graphs) with and without cycles.

Three problems are considered. The first is the relation between ensemble-averaged state probabilities in a random walk with absorption and time-averaged state probabilities in the corresponding closed diagram. The second problem is concerned with random walks on diagrams with cycles in which the cycle completion rates and probabilities may depend on the "remainder" after the previously completed cycle. The final topic is a study of cycle completions prior to absorption for diagrams that involve both cycles and absorption (e.g., a cycling enzyme that binds a dead-end inhibitor or poison in one of its states).

Biometry↗
PubMed1977

On the mapping of chains of first order chemical reactions on random walks.

We present the transient solution to a random walk problem which characterizes chemical kinetic processes in which a particular state may have a finite number of internal states. While there are many chemical and physical examples, we introduce the solution through models of biophysical interest.

Biological Transport↗
PubMed2002

Optimization of goal-directed movements in the cerebellum: a random walk hypothesis.

Voluntary goal-directed movements, such as arm reaching, are nearly optimized in terms of smoothness over the entire movement. Such smoothness is lost with cerebellar dysfunction, suggesting the essential role of the cerebellum in optimizing movement. However, it is still not clear how the cerebellum contributes to achieving smoothness over an entire movement. A recent study has shown that such smoothness of movement can be achieved by reducing the variance of errors at the end of the movement. Here, I hypothesize that the terminal errors conveyed by climbing fibers in the cerebellum serve to reduce not only the mean error, but also the variance of the error, through a process analogous to the random walk through movement control candidates. In the random walk, the direction of each step is randomly determined, but the size of each step is determined by the error at the end of each trial.

Cerebellum↗
PubMed2000

Random walks in the space of conformations of toy proteins.

Monte Carlo dynamics of the lattice toy protein of 48 monomers is interpreted as a random walk in an abstract (discrete) space of conformations. To test the geometry of this space, we examine the return probability P(T), which is the probability to find the polymer in the native state after T Monte Carlo steps, provided that it starts from the native state at the initial moment. Comparing computational data with the theoretical expressions for P(T) for random walks in a variety of different spaces, we show that conformation spaces of polymer loops may have nontrivial dimensions and exhibit negative curvature characteristics of Lobachevskii (hyperbolic) geometry.

Computer Simulation↗
PubMed2002

Directed random walks in continuous space.

The investigation on diffusion with directed motion in a two-dimensional continuous space is completed by using the model of the continuous directed random walks. The average square end-to-end distance approximately t(2nu) is calculated. The results show that this type of walks belongs asymptotically to the same class (nu=1.0) as the ballistic motions. For short time, we observe a crossover from purely random walks (nu=0.5) to ballistic motions (nu=1.0). The dependence of the crossover on the direction parameter theta is studied. There exists a scaling relation of the form approximately tf(t/theta(-2)). The return probability P00(t) is also investigated and the scaling form similar to is obtained.

Journal Article↗
PubMed2003

Directed random walks on directed percolation clusters.

The characteristics of directed random walks on directed percolation clusters are numerically studied. For two-dimensional clusters grown at the critical probability p(c), it is shown that the distance d of the directed random walkers from their most probable end point is determined by a probability distribution p(d) approximately d(-(1+w/nu)), where the values of w and nu are close to the values of the known exponents: w approximately 0.50 and nu approximately 0.63. This probability distribution is independent of the cluster's length t up to d values comparable to the cluster's width approximately t(nu). The results are shown to be consistent with a tree description of the directed percolation clusters.

Journal Article↗
PubMed1999

Effect of DNA sequence divergence on homologous recombination as analyzed by a random-walk model.

A point connecting a pair of homologous regions of DNA duplexes moves along the homology in a reaction intermediate of the homologous recombination. Formulating this movement as a random walk, we were previously successful at explaining the dependence of the recombination frequency on the homology length. Recently, the dependence of the recombination frequency on the DNA sequence divergence in the homologous region was investigated experimentally; if the methyl-directed mismatch repair (MMR) system is active, the logarithm of the recombination frequency decreases very rapidly with an increase of the divergence in a low-divergence regime. Beyond this regime, the logarithm decreases slowly and linearly with the divergence. This "very rapid drop-off" is not observed when the MMR system is defective. In this article, we show that our random-walk model can explain these data in a straightforward way. When a connecting point encounters a diverged base pair, it is assumed to be destroyed with a probability that depends on the level of MMR activity.

DNA↗
PubMed2000

Kinetics of stochastically gated diffusion-limited reactions and geometry of random walk trajectories

In this paper we study the kinetics of diffusion-limited, pseudo-first-order A+B-->B reactions in situations in which the particles' intrinsic reactivities are not constant but vary randomly in time. That is, we suppose that the particles are bearing "gates" which fluctuate in time, randomly and independently of each other, between two states-an active state, when the reaction may take place between A and B particles appearing in close contact; and a blocked state, when the reaction is completely inhibited. We focus here on two customary limiting cases of pseudo-first-order reactions-the so-called target annihilation and the Rosenstock trapping model-and consider four different particular models, such that the A particle can be either mobile or immobile or gated or ungated, and ungated or gated B particles can be fixed at random positions or move randomly. All models are formulated on a d-dimensional regular lattice, and we suppose that the mobile species perform independent, homogeneous, discrete-time lattice random walks. The model involving a single, immobile, ungated target A and a concentration of mobile, gated B particles is solved exactly. For the remaining three models we determine exactly, in the form of rigorous lower and upper bounds showing the same N dependence, the large-N asymptotical behavior of the probability that the A particle survives until the Nth step. We also realize that for all four models studied here the A particle survival probability can be interpreted as the moment generating function of some functionals of random walk trajectories, such as, e. g., the number of self-intersections, the number of sites visited exactly a given number of times, the "residence time" on a random array of lattice sites, etc. Our results thus apply to the asymptotic behavior of corresponding generating functions which are not known as yet.

Journal Article↗
PubMed1986

A multicomponent, random walk model of transport and metabolism inside a neuron.

A model of multicomponent transport, consumption, and production of metabolites inside a neuron containing discrete mitochondria and glycolytic enzymes is developed using a random walk model of molecular transport. The ratio of anaerobic to aerobic metabolism which maximizes ATP production under normal, ischemic, and anoxic conditions is calculated. The ratio of the number of mitochondria to glycolytic enzymes which maximizes ATP under normal conditions is also calculated. Because the volume of the neuron is fixed, the sum of the number of mitochondria and glycolytic enzymes is fixed. This constraint is incorporated in the optimization process as an interior penalty function. Some of the advantages of employing the random walk technique are simple stoichiometry can be used to model consumption and production of metabolites, the geometry of the enzyme system and their active sites can be easily included in the model, and saturation of enzymes can be more easily modeled.

Animals↗
PubMed2002

Random walk with memory enhancement and decay.

A model of random walk with memory enhancement and decay was presented on the basis of the characteristics of the biological intelligent walks. In this model, the movement of the walker is determined by the difference between the remaining information at the jumping-out site and jumping-in site. The amount of the memory information s(i)(t) at a site i is enhanced with the increment of visiting times to that site, and decays with time t by the rate e(-beta(t)), where beta is the memory decay exponent. When beta=0, there exists a transition from Brownian motion (BM) to the compact growth of walking trajectory with the density of information energy u increasing. But for beta>0, this transition does not appear and the walk with memory enhancement and decay can be considered as the BM of the mass center of the cluster composed of remembered sites in the late stage.

Animals↗
PubMed2001

Random walks of cytoskeletal motors in open and closed compartments.

Random walks of molecular motors, which bind to and unbind from cytoskeletal filaments, are studied theoretically. The bound and unbound motors undergo directed and nondirected motion, respectively. Motors in open compartments exhibit anomalous drift velocities. Motors in closed compartments generate stationary nonequilibrium states with spatially varying densities of the motor concentrations and currents. "Traffic jams" on the filaments lead to a maximum of the motor current at an optimal motor concentration. Quantitative estimates based on experimental data for bound motors indicate that these transport phenomena are accessible to experiments.

Algorithms↗
PubMed2002

Quantification of optical properties of a breast tumor using random walk theory.

For the first time we use a random walk methodology based on time-dependent contrast functions to quantify the optical properties of breast tumors (invasive ductal carcinoma) of two patients. Previously this theoretical approach was successfully applied for analysis of embedded objects in several phantoms. Data analysis was performed on distributions of times of flight for photons transmitted through the breast which were recorded in vivo using a time-domain scanning mammograph at 670 and 785 nm. The size of the tumors, their optical properties, and those of the surrounding tissue were reconstructed at both wavelengths. The tumors showed increased absorption and scattering. From the absorption coefficients at both wavelengths blood oxygen saturation was estimated for the tumors and the surrounding tissue.

Absorption↗
PubMed1995

A random-walk/giant-loop model for interphase chromosomes.

Fluorescence in situ hybridization data on distances between defined genomic sequences are used to construct a quantitative model for the overall geometric structure of a human chromosome. We suggest that the large-scale geometry during the G0/G1 part of the cell cycle may consist of flexible chromatin loops, averaging approximately 3 million bp, with a random-walk backbone. A fully explicit, three-parametric polymer model of this random-walk/giant-loop structure can account well for the data. More general models consistent with the data are briefly discussed.

Base Composition↗
PubMed1988

A random walk model for evaluating clinical trials involving serial observations.

For clinical trials where the variable of interest is ordered and categorical (for example, disease severity, symptom scale), and where measurements are taken at intervals, it might be possible to achieve a greater discrimination between the efficacy of treatments by modelling each patient's progress as a stochastic process. The random walk is a simple, easily interpreted model that can be fitted by maximum likelihood using a maximization routine with inference based on standard likelihood theory. In general the model can allow for randomly censored data, incorporates measured prognostic factors, and inference is conditional on the (possibly non-random) allocation of patients. Tests of fit and of model assumptions are proposed, and application to two therapeutic trials of gastroenterological disorders are presented. The model gave measures of the rate of, and variability in, improvement for patients under different treatments. A small simulation study suggested that the model is more powerful than considering the difference between initial and final scores, even when applied to data generated by a mechanism other than the random walk model assumed in the analysis. It thus provides a useful additional statistical method for evaluating clinical trials.

Algorithms↗