A random walk method for computing genetic location scores.

Calculation of location scores is one of the most computationally intensive tasks in modern genetics. Since these scores are crucial in placing disease loci on marker maps, there is ample incentive to pursue such calculations with large numbers of markers. However, in contrast to the simple, standardized pedigrees used in making marker maps, disease pedigrees are often graphically complex and sparsely phenotyped. These complications can present insuperable barriers to exact likelihood calculations with more than a few markers simultaneously. To overcome these barriers we introduce in the present paper a random walk method for computing approximate location scores with large numbers of biallelic markers. Sufficient mathematical theory is developed to explain the method. Feasibility is checked by small-scale simulations for two applications permitting exact calculation of location scores.     

A Turing model with correlated random walk

 If in the classical Turing model the diffusion process (Brownian motion) is replaced by a more general correlated random walk, then the parameters describing spatial spread are the particle speeds and the rates of change in direction. As in the Turing model, a spatially constant equilibrium can become unstable if the different species have different turning rates and different speeds. Furthermore, a Hopf bifurcation can be found if the reproduction rate of the activator is greater than its rate of change of direction, and oscillating patterns are possible. Received 24 February 1995; received in revised form 6 September 1995     

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.     

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.     

Identifying diseases-related metabolites using random walk

Background

Metabolites disrupted by abnormal state of human body are deemed as the effect of diseases. In comparison with the cause of diseases like genes, these markers are easier to be captured for the prevention and diagnosis of metabolic diseases. Currently, a large number of metabolic markers of diseases need to be explored, which drive us to do this work.

Methods

The existing metabolite-disease associations were extracted from Human Metabolome Database (HMDB) using a text mining tool NCBO annotator as priori knowledge. Next we calculated the similarity of a pair-wise metabolites based on the similarity of disease sets of them. Then, all the similarities of metabolite pairs were utilized for constructing a weighted metabolite association network (WMAN). Subsequently, the network was utilized for predicting novel metabolic markers of diseases using random walk.

Results

Totally, 604 metabolites and 228 diseases were extracted from HMDB. From 604 metabolites, 453 metabolites are selected to construct the WMAN, where each metabolite is deemed as a node, and the similarity of two metabolites as the weight of the edge linking them. The performance of the network is validated using the leave one out method. As a result, the high area under the receiver operating characteristic curve (AUC) (0.7048) is achieved. The further case studies for identifying novel metabolites of diabetes mellitus were validated in the recent studies.

Conclusion

In this paper, we presented a novel method for prioritizing metabolite-disease pairs. The superior performance validates its reliability for exploring novel metabolic markers of diseases.

     

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.     

Molecular motors: thermodynamics and the random walk

The biochemical cycle of a molecular motor provides the essential link between its thermodynamics and kinetics. The thermodynamics of the cycle determine the motor's ability to perform mechanical work, whilst the kinetics of the cycle govern its stochastic behaviour. We concentrate here on tightly coupled, processive molecular motors, such as kinesin and myosin V, which hydrolyse one molecule of ATP per forward step. Thermodynamics require that, when such a motor pulls against a constant load f, the ratio of the forward and backward products of the rate constants for its cycle is exp [-(DeltaG + u(0)f)/kT], where -DeltaG is the free energy available from ATP hydrolysis and u(0) is the motor's step size. A hypothetical one-state motor can therefore act as a chemically driven ratchet executing a biased random walk. Treating this random walk as a diffusion problem, we calculate the forward velocity v and the diffusion coefficient D and we find that its randomness parameter r is determined solely by thermodynamics. However, real molecular motors pass through several states at each attachment site. They satisfy a modified diffusion equation that follows directly from the rate equations for the biochemical cycle and their effective diffusion coefficient is reduced to D-v(2)tau, where tau is the time-constant for the motor to reach the steady state. Hence, the randomness of multistate motors is reduced compared with the one-state case and can be used for determining tau. Our analysis therefore demonstrates the intimate relationship between the biochemical cycle, the force-velocity relation and the random motion of molecular motors.     

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     

A random walk model of oligodendrocyte generation in vitro and associated estimation problems.

A branching stochastic process proposed earlier to model oligodendrocyte generation by O-2A progenitor cells under in vitro conditions does not allow invoking the maximum likelihood techniques for estimation purposes. To overcome this difficulty, we propose a partial likelihood function based on an embedded random walk model of clonal growth and differentiation of O-2A progenitor cells. Under certain conditions, the partial likelihood function yields consistent estimates of model parameters. The usefulness of this approach is illustrated with computer simulations and data analyses.     

Centrosome-dependent anisotropic random walk of cytoplasmic vesicles

We approach the problem of an apparently random movement of small cytoplasmic vesicles and its relationship to centrosome functioning. Motion of small vesicles in the cytoplasm of BSC-1 cells was quantified using computer-assisted microscopy. The vesicles move across the cytoplasm frequently changing their directions with negligible net displacement. The autocorrelation function for consecutive velocities of individual vesicles becomes indistinguishable from zero in 10s. Variance in the displacement is proportional to time. The motion of vesicles is anisotropic: It has diffusivity along the radii drawn from the centrosome several times higher than the tangential diffusivity. This anisotropy is abolished by ultraviolet microbeam irradiation of the centrosome when the microtubule array loses radial structure. We conclude that the motion of the vesicles in the cytoplasm can be described as diffusion-like random walk with centrosome-dependent anisotropy. The present analysis quantitatively corroborates the 'trial and error' model of vesicular transport.     

Caribou movement as a correlated random walk

Movement is a primary mechanism coupling animals to their environment, yet there exists little empirical analysis to test our theoretical knowledge of this basic process. We used correlated random walk (CRW) models and satellite telemetry to investigate long-distance movements of caribou, the most vagile, non-volant terrestrial vertebrate in the world. Individual paths of migratory and sedentary female caribou were quantified using measures of mean move length and angle, and net squared displacements at each successive move were compared to predictions from the models. Movements were modelled at two temporal scales. For paths recorded through one annual cycle, the CRW model overpredicted net displacement of caribou through time. For paths recorded over shorter intervals delineated by seasonal behavioural changes of caribou, there was excellent correspondence between model predictions and observations for most periods for both migratory and sedentary caribou. On the smallest temporal scale, a CRW model significantly overpredicted displacements of migratory caribou during 3 months following calving; this was also the case for sedentary caribou in late summer, and in late winter. In all cases of overprediction there was significant positive autocorrelation in turn direction, indicating that movements were more tortuous than expected. In one case of underprediction, significant negative autocorrelation of sequential turn direction was evident, indicating that migratory caribou moved in straightened paths during spring migration to calving grounds. Results are discussed in light of known migration patterns and possible limiting factors for caribou, and indicate the applicability of CRW models to animal movement at vast spatial and temporal scales, thus assisting in future development of more sophisticated models of population spread and redistribution for vertebrates. Received: 14 July 1999 / Accepted: 15 November 1999     

Temporal dynamics of recurrent airway symptoms and cellular random walk.

Asthma is a complex chronic inflammatory disease of the small airways that has dramatically increased in prevalence in industrialized countries during the last decades. Risk factors for adult asthma have been related to the complex array of gene-environment interactions and exposure of the immune system to allergens in early childhood. In genetically predisposed subjects, continuous exposure to environmental agents such as allergens or infections can lead to recurrent airway symptoms characterized by recurrent episodes of airway inflammation and bronchoconstriction with clinical symptoms of cough, dyspnea, or wheezing. In this study, we report that the longterm temporal dynamics of recurrent airway symptoms in a population of unselected infants display a complex intermittent pattern and that the distribution of interepisode intervals follows a power law. We interpret the data by using a model of the dynamics of attack episodes in which an attack is triggered by an avalanche of airway constrictions. We map the dynamics of this model to the known problem of a random walk in the presence of an absorbing boundary in which the walker corresponds to the fluctuations in contractile state of airway smooth muscle cells. These findings may provide new insight into the mechanisms of otherwise unexplained symptom episodes.     

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.