Biased random walk models for chemotaxis and related diffusion approximations

Summary Stochastic models of biased random walk are discussed, which describe the behavior of chemosensitive cells like bacteria or leukocytes in the gradient of a chemotactic factor. In particular the turning frequency and turn angle distributions are derived from certain biological hypotheses on the background of related experimental observations. Under suitable assumptions it is shown that solutions of the underlying differential-integral equation approximately satisfy the well-known Patlak-Keller-Segel diffusion equation, whose coefficients can be expressed in terms of the microscopic parameters. By an appropriate energy functional a precise error estimation of the diffusion approximation is given within the framework of singular perturbation theory.     

Chemosensory responses of microorganisms: Asymmetric diffusion as the limit of a biased random walk

A macroscopic asymmetric diffusion equation to model the responses of microbial populations to chemical attractants and repellents is derived from a biased random walk model for the motion of individual cells. The equation is different from the well-known Keller-Segel equation which contains a Fickian diffusion term. The implications of this difference for selected problems of biological interest are considered. In particular the aggregation of a population of microorganisms in a region of high concentration of attractant is discussed. Some similarities and limitations of both models are noted.     

Drug-target interaction prediction by random walk on the heterogeneous network

Predicting potential drug-target interactions from heterogeneous biological data is critical not only for better understanding of the various interactions and biological processes, but also for the development of novel drugs and the improvement of human medicines. In this paper, the method of Network-based Random Walk with Restart on the Heterogeneous network (NRWRH) is developed to predict potential drug-target interactions on a large scale under the hypothesis that similar drugs often target similar target proteins and the framework of Random Walk. Compared with traditional supervised or semi-supervised methods, NRWRH makes full use of the tool of the network for data integration to predict drug-target associations. It integrates three different networks (protein-protein similarity network, drug-drug similarity network, and known drug-target interaction networks) into a heterogeneous network by known drug-target interactions and implements the random walk on this heterogeneous network. When applied to four classes of important drug-target interactions including enzymes, ion channels, GPCRs and nuclear receptors, NRWRH significantly improves previous methods in terms of cross-validation and potential drug-target interaction prediction. Excellent performance enables us to suggest a number of new potential drug-target interactions for drug development.     

The random walk of Azospirillum brasilense

The bacterium Azospirillum brasilense has been frequently studied in laboratory experiments. It performs movements in space where long forward and backward runs on a straight line occur simultaneously with slow changes of direction of the line. A model is presented in which a correlated random walk on a line is joined to diffusion on a sphere of directions. For this transport system, a hierarchy of moment approximations is derived, ranging from a hyperbolic system with four dependent variables to a scalar damped wave equation (telegraph equation) and then to a single diffusion equation for particle density. The original parameters are compounded in the diffusion quotient. The effects of these parameters, such as particle speed or turning rate, on the diffusion coefficient are discussed in detail.     

Youden Index and optimal cut-point estimated from observations affected by a lower limit of detection

The receiver operating characteristic (ROC) curve is used to evaluate a biomarker's ability for classifying disease status. The Youden Index (J), the maximum potential effectiveness of a biomarker, is a common summary measure of the ROC curve. In biomarker development, levels may be unquantifiable below a limit of detection (LOD) and missing from the overall dataset. Disregarding these observations may negatively bias the ROC curve and thus J. Several correction methods have been suggested for mean estimation and testing; however, little has been written about the ROC curve or its summary measures. We adapt non-parametric (empirical) and semi-parametric (ROC-GLM [generalized linear model]) methods and propose parametric methods (maximum likelihood (ML)) to estimate J and the optimal cut-point (c *) for a biomarker affected by a LOD. We develop unbiased estimators of J and c * via ML for normally and gamma distributed biomarkers. Alpha level confidence intervals are proposed using delta and bootstrap methods for the ML, semi-parametric, and non-parametric approaches respectively. Simulation studies are conducted over a range of distributional scenarios and sample sizes evaluating estimators' bias, root-mean square error, and coverage probability; the average bias was less than one percent for ML and GLM methods across scenarios and decreases with increased sample size. An example using polychlorinated biphenyl levels to classify women with and without endometriosis illustrates the potential benefits of these methods. We address the limitations and usefulness of each method in order to give researchers guidance in constructing appropriate estimates of biomarkers' true discriminating capabilities.     

A random walk model of ovum transport

This paper discusses the random nature of ovum transport, and presents a Brownian Motion model of ovum transport in the ampulla and isthmus. A new explanation of the delay in transport at the ampullary-isthmic junction, based on widely differing diffusion coefficients for the ampulla and isthmus, is proposed.     

Simulate the diffusion of hydrated ions by nanofiltration membrane process with random walk

We propose a new diffusion model describing the diffusion behaviours of hydrated ions in the process of nanofiltration (NF) based on the random walk (RW) theory when the NF membrane is uncharged or low charged. In this model, the hydration of ions and their deformation capacity are considered. The structure of the membrane is idealised into a lozenge shape and the diameter of membrane pore is defined as gapsize. A computer program named RW system in chemistry is developed to simulate based on this model. Six familiar ions Li⁺, Na⁺, Mg²⁺, Al³⁺, K⁺ and Ca²⁺ are investigated. Their characteristics are calculated by Gaussian 03, Pople, Inc., Wallingford, CT. The diffusivities of hydrated ions are calculated and discussed. The results show that the hydration of ions cannot be ignored in NF process when the membrane pore size is near the dimensions of the hydrated ions.     

Monitoring receptor-ligand interactions between surfaces by thermal fluctuations

We describe a new method for determining receptor-ligand association/dissociation events across the interface of two surfaces (two-dimensional binding) by monitoring abrupt decrease/resumption in thermal fluctuations of a biomembrane force probe. Our method has been validated by rigorous control experiments and kinetic experiments. We show that cellular on-rate of association can be measured by analysis of intervals from a dissociation event to the next association event (waiting times). Similarly, off-rate of molecular dissociation can be measured by analysis of intervals from an association event to the next dissociation event (bond lifetimes). Different types of molecular bonds could be distinguished by different levels of reduction in thermal fluctuations. This novel method provides a powerful tool to study cell adhesion and signaling mediated by single or multiple receptor-ligand species.     

Cleaning interactions at the southern limit of tropical reef fishes in the Western Atlantic

Cleaning associations are one of the most dynamic and complex mutualistic interactions of reef environments and are often influenced by local conditions. In the Western Atlantic (WE) most studies concentrate in tropical areas, with little attention to subtropical areas. We examined an assemblage of cleaner fish and their clients on the rocky reefs of the coast of Santa Catarina state, South Brazil, the southern limit of tropical reef fishes in the WE. We recorded 150 cleaning interactions, in which four fish species and one shrimp species acted as facultative cleaners. The grunt Anisotremus virginicus and the angelfish Pomacanthus paru serviced most clients. Fifteen fish species acted as clients, among which the most frequent was the planktivorous grunt Haemulon aurolineatum (31%). Cleaning interactions occurred mostly (87%) with non-carnivorous clients and the number of interactions was not related to the abundance of the species involved. The absence of dedicated cleaner fishes at the study sites and the replacement of their roles by facultative cleaners may be related to local conditions, including cold currents and reduction of rock cover. Under these circumstances, clients take advantage of the services offered by facultative cleaners, a characteristic of temperate areas.     

Effects of photoperiod on rat motor activity rhythm at the lower limit of entrainment

The experiment described here studied the rat motor activity pattern as a function of the photoperiod of circadian light-dark cycles in the limits of entrainment (22-and 23-h periods). In most cases, the overt rhythm showed 2 circadian components: 1 that followed the external LD cycle and a 2nd rhythm that was free run. The expression of these components was directly dependent on the photoperiod, and there was a gradual transition in the manifestation of 1 or the other. The component with a period equal to that of the external cycle was more manifested under long photoperiods, while the other 1 was more expressed during short photoperiods. Also, the period of the free-running component was longer under T22 than T23. For each period, the free-running component was longer under a longer photoperiod. At first sight, the presence of these 2 components in most of the rats might appear to be due to the fact that in the limits of entrainment, some rats do not entrain and thus show a free-running rhythm plus masking. However, the gradation observed in the different patterns of the overt motor activity rhythm, especially those patterns related to the different balance between the 2 components and the length of the period of the free-running component under LD as a function of the photoperiod, suggests that the circadian system can be functionally dissociated.     

A biased random walk model for the trajectories of swimming micro-organisms

The motion of swimming micro-organisms that have a preferred direction of travel, such as single-celled algae moving upwards (gravitaxis) or towards a light source (phototaxis), is modelled as the continuous limit of a correlated and biased random walk as the time step tends to zero. This model leads to a Fokker-Planck equation for the probability distribution function of the orientation of the cells, from which macroscopic parameters such as the mean cell swimming direction and the diffusion coefficient due to cell swimming can be calculated. The model is tested on experimental data for gravitaxis and phototaxis and used to derive values for the macroscopic parameters for future use in theories of bioconvection, for example.     

Isolation by distance in a continuous population under stochastic demographic fluctuations

The local density of individuals is seldom uniform in space and time within natural populations. Yet, formal approaches to the process of isolation by distance in continuous populations have encountered analytical difficulties in describing genetic structuring with demographic heterogeneities, usually disregarding local correlations in the movement and reproduction of genes. We formulate exact recursions for probabilities of identity in continuous populations, from which we deduce definitions of effective dispersal () and effective density (D_e) that generalize results relating spatial genetic structure, dispersal and density in lattice models. The latter claim is checked in simulations where estimates of effective parameters obtained from demographic information are compared with estimates derived from spatial genetic patterns in a plant population evolving in a heterogeneous and dynamic habitat. The simulations further suggest that increasing spatio‐temporal correlations in local density reduce and generally decrease the product , with dispersal kurtosis influencing their sensitivity to density fluctuations. As in the lattice model, the expected relationship between the product and the genetic structure statistic a_r holds under fluctuating density, irrespective of dispersal kurtosis. The product D σ² between observed census density and the observed dispersal rate over one generation will generally be an upwardly biased (up to 400% in simulations) estimator of in populations distributed in spatially aggregated habitats.     

A stochastic limit cycle oscillator model of the EEG

We present an empirical model of the electroencephalogram (EEG) signal based on the construction of a stochastic limit cycle oscillator using Itô calculus. This formulation, where the noise influences actually interact with the dynamics, is substantially different from the usual definition of measurement noise. Analysis of model data is compared with actual EEG data using both traditional methods and modern techniques from nonlinear time series analysis. The model demonstrates visually displayed patterns and statistics that are similar to actual EEG data. In addition, the nonlinear mechanisms underlying the dynamics of the model do not manifest themselves in nonlinear time series analysis, paralleling the situation with real, non-pathological EEG data. This modeling exercise suggests that the EEG is optimally described by stochastic limit cycle behavior.     

Population growth with stochastic fluctuations in the life table

Monte Carlo simulations with the Leslie matrix and similar population models show that as the variance in survivorship or fecundity increases, the expected population growth rate decreases. This is attributed to Jensen's inequality with the observation that the rate of increase is a concave function of age-specific survivorship and fertility rates. Applications of this observation are advised for demographic studies, population simulation, optimal harvest strategies, and natural selection for variance in fertility and survivorship rates.     

A random walk model of fast-phase timing during optokinetic nystagmus

 Most vertebrate animals produce optokinetic nystagmus in response to rotation of their visual surround. Nystagmus consists of an alternation of slow-phase eye rotations, which follow the surround, and fast-phase eye rotations, which quickly reset eye position. The time intervals between fast phases vary stochastically, even during optokinetic nystagmus produced by constant velocity rotation of a uniform surround. The inter-fast-phase interval distribution has a long tail, and intervals that are long relative to the mode become even more likely as constant surround velocity is decreased. This paper provides insight into fast-phase timing by showing that the process of fast-phase generation during constant velocity optokinetic nystagmus is analogous to a random walk with drift toward a threshold. Neurophysiologically, the output of vestibular nucleus neurons, which drive the slow phase, would approximate a random walk with drift because they integrate the noisy, constant surround velocity signal they receive from the visual system. Burst neurons, which fire a burst to drive the fast phase and reset the slow phase, are brought to threshold by the vestibular nucleus neurons. Such a nystagmic process produces stochastically varying inter-fast-phase intervals, and long intervals emerge naturally because, as drift rate (related to surround velocity) decreases, it becomes more likely that any random walk can meander for a long time before it crosses the threshold. The theoretical probability density function of the first threshold crossing times of random walks with drift is known to be that of an inverse Gaussian distribution. This probability density function describes well the distributions of the intervals between fast phases that were either determined experimentally, or simulated using a neurophysiologically plausible neural network model of fast-phase generation, during constant velocity optokinetic nystagmus. Received: 1 June 1995/Accepted in revised form: 15 February 1996