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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.
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.
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.
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.
The biased random-walk chemotaxis of the bacterium Escherichia coli is a remarkably effective method of navigation based on random trial-and-error responding rather than steering. Humans restricted to the same mode of responding are able to navigate to target locations, just like the bacterium. This mode of navigation can be modeled as an input control process that selectively retains favorable and rejects unfavorable consequences of the random responses. The selection process is determined by the internal organization of the system rather than the external influence of the environment (as in natural selection or reinforcement).
Bacterial migration is important in understanding many practical problems ranging from disease pathogenesis to the bioremediation of hazardous waste in the environment. Our laboratory has been successful in quantifying bacterial migration in fluid media through experiment and the use of population balance equations and cellular level simulations that incorporate parameters based on a fundamental description of the microscopic motion of bacteria. The present work is part of an effort to extend these results to bacterial migration in porous media. Random walk algorithms have been used successfully to date in nonbiological contexts to obtain the diffusion coefficient for disordered continuum problems. This approach has been used here to describe bacterial motility. We have generated model porous media using molecular dynamics simulations applied to a fluid with equal sized spheres. The porosity is varied by allowing different degrees of sphere overlap. A random walk algorithm is applied to simulate bacterial migration, and the Einstein relation is used to calculate the effective bacterial diffusion coefficient. The tortuosity as a function of particle size is calculated and compared with available experimental results of migration of Pseudomonas putida in sand columns. Tortuosity increases with decreasing obstacle diameter, which is in agreement with the experimental results.
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 distribution 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.
An equation describing a two-dimensional random walk with non-uniform step length is derived and applied to a cinephotomicrographic analysis of human granulocyte movement. The method is also applied to a generally recognized chemotactic movement.
We determined the folding of chromosomes in interphase nuclei by measuring the distance between points on the same chromosome. Over 25,000 measurements were made in G0/G1 nuclei between DNA sequences separated by 0.15-190 megabase pairs (Mbp) on three human chromosomes. The DNA sequences were specifically labeled by fluorescence in situ hybridization. The relationship between mean-square interphase distance and genomic separation has two linear phases, with a transition at approximately 2 Mbp. This biphasic relationship indicates the existence of two organizational levels at scales > 100 kbp. On one level, chromatin appears to be arranged in large loops several Mbp in size. Within each loop, chromatin is randomly folded. On the second level, specific loop-attachment sites are arranged to form a supple, backbonelike structure, which also shows characteristic random walk behavior. This random walk/giant loop model is the simplest model that fully describes the observed large-scale spatial relationships. Additional evidence for large loops comes from measurements among probes in Xq28, where interphase distance increases and then locally decreases with increasing genomic separation.
The random walk function is a mathematical function derived from studies of the mass transport and flow of diffusible materials through tubes. Approximations to the function were first used some time ago in the field of cardiac tracer dilution curves, but in the absence of rapid and reproducible curve fitting the method never became commonplace. The current study uses the latest curve-fitting techniques and shows how the method may be used with precision in the analysis of time-activity curves from dynamic oesophageal and blood flow studies. The physiological basis of the method is given and parameters obtained which relate to both the rate of flow and the local dispersion of the bolus.
A theorem for Poisson convergence on realizations of two-dimensional Branching Random Walks with an underlying continuous time Markov Branching Process is proved. This result can be used to gain an approximation for the number of cells having sustained a certain deficiency after a long time in multistage carcinogenesis.
Accurate survival prediction in cancer remains challenging due to the sparsity of somatic mutation profiles and the failure of existing models to capture higher-order gene-gene dependencies. Network diffusion methods such as Random Walk with Restart (RWR) can propagate mutation signals across protein-protein interaction (PPI) networks to address sparsity, yet their integration within a deep learning Cox survival framework has not been comprehensively benchmarked across multiple cancer cohorts. We present RCoxNet, a deep learning framework that maps somatic mutation profiles onto a ConsensusPathDB-derived PPI network via RWR, selects prognostic genes by log-rank filtering, and processes network-informed mutation scores through three fully connected hidden layers feeding into a Cox proportional hazards output. RCoxNet was evaluated on The Cancer Genome Atlas (TCGA) cohorts for four cancer types (breast invasive carcinoma [BRCA], lung adenocarcinoma [LUNG], glioblastoma multiforme [GBM], and ovarian serous cystadenocarcinoma [OV]) using 20 independent random splits. The model achieved mean C-index values of 0.807 ± 0.044 (BRCA), 0.750 ± 0.039 (LUNG), 0.704 ± 0.041 (GBM), and 0.668 ± 0.036 (OV), consistently outperforming DeepSurv, Cox-nnet, SurvivalNet, Cox Elastic-Net (Cox-EN), and DeepHit, with statistically significant gains over Cox-EN, Cox-nnet, SurvivalNet, and DeepHit across the majority of cohorts. RCoxNet demonstrates that embedding sparse mutation profiles into a PPI network context substantially improves cancer survival prediction and yields biologically interpretable prognostic features relevant to precision oncology.
A new technique for the excitation of arbitrary shapes is proposed. It is based on a parallel sequence of small tip angle RF pulses and gradient pulses. The small tip angle rotations co-add yielding a 90 degrees excitation pulse within the selected excitation profile while outside the profile, the rotations cancel each other. A full theory of the completely arbitrary regional volume excitation (CARVE) method is presented and experimentally verified. In CARVE, k-space is discrete because the RF is applied in pulses. The discrete character of k-space permits an arbitrary trajectory for the k-space walk. The optimal random trajectory is found by minimizing the gradient load using simulated annealing. It is shown, both theoretically and experimentally, that such a trajectory is much better than any other systematic or random trajectory in k-space.
The folding of chromatin in interphase cell nuclei was studied by fluorescent in situ sequences chromatin according to a random walk model. This model provides the basis for calculating the spacing of sequences along the linear DNA molecule from interphase distance measurements. An interphase mapping strategy based on this model was tested with 13 probes from a 4-megabase pair (Mbp) region of chromosome 4 containing the Huntington disease locus. The results confirmed the locations of the probes and showed that the remaining gap in the published maps of this region is negligible in size. Interphase distance measurements should facilitate construction of chromosome maps with an average marker density of one per 100 kbp, approximately ten times greater than that achieved by hybridization to metaphase chromosome. achieved by hybridization to metaphase chromosomes.
Equations are derived for the probability of n visits to a given state during the course of a random walk on a finite diagram that starts from a specified state and ends with absorption. By deriving the mean number of visits in two different ways, certain conjectures or theorems are encountered that connect properties of different but related diagrams in an interesting way. Other subjects included are (i) number of one-way transitions between two states before absorption; (ii) time dependence of the rate of cycle completions before absorption; and (iii) the relation of this work to the "return process" of Karlin and Taylor.
A Monte-Carlo approach to analysis of dispersion in the tissue of a locally administered drug is presented. The distribution of a drug in the tissue is simulated as a distribution of randomly walking particles. The approach is demonstrated on a simple situation for which both experimental results and an analytical solution are known. The approach can be used in situations, where common numeric methods are difficult to use, especially for analyses of drug transport in an inhomogeneous space, and problems with complex boundary conditions, e.g. in analyses of dispersion of anticancer agents locally applied into tumours.
To study conformational parameters of ligands before and after complex formation with the galactoside-binding agglutinin of Viscum album L. (VAA) in solution, combined computer-assisted random walk molecular mechanics (RAMM) calculations extended by conformational clustering analysis (CCA), molecular dynamics (MD) simulations as well as two-dimensional rotating-frame nuclear Overhauser effect (ROE) and two-dimensional nuclear Overhauser effect (NOE) spectroscopy NMR experiments were employed. Derivatives of the naturally occurring disaccharides Galbeta1-3GlcNAcbeta1-R and Galbeta1-3GalNAcbeta1-R as well as of a synthetic high-affinity binding partner, i.e. the disaccharide Galbeta1-2Galbeta1-R', were chosen as ligands in this study. The disaccharides displayed inherent flexibility in the valley of the global minimum between phi/psi combinations of (40 degrees/60 degrees) and (40 degrees/-60 degrees). Calculations of the de-N-acetylated sugars revealed that presence of this group did not markedly influence the distribution of low-energy conformers in the phi, psi, epsilon plot. Occupation of side minima at phi/psi (180 degrees/0 degrees) or (0 degrees/180 degrees) is either unlikely or low according to the results of MD simulations and RAMM calculations extended by CCA. Notably, these side minima define conformations which are not stable during a MD simulation. Transitions to other minima occur already a few picoseconds after the start of the simulation. NMR experiments of the free-state ligand confirmed the validity of the data sets obtained by the calculations. Following the description of the conformational space in the free-state NMR experiments were performed for these disaccharides complexed with VAA. They yielded two interresidual contacts for Galbeta1-3GlcNAcbeta1-R and Galbeta1-2Galbeta1-R'. The ligand conformations in the complex did not deviate markedly from those of a minimum conformation in the free state. One- and two-dimensional transferred nuclear Overhauser enhancement (TRNOE) experiments at different mixing times excluded the influence of spin-diffusion effects. When the NOE build-up curves in the three studied cases were compared, the residual mobility of the penultimate carbohydrate unit of Galbeta1-3GalNAcbeta1-R was observed to be higher than that of the respective hexopyranose unit of the other two bound ligands. Due to the availability of the conformational parameters of Galbeta1-2Galbeta1-R' in association with a galectin, namely the beta-galactoside-binding protein from chicken liver, it is remarkable to note that this ligand displays different conformations in the binding sites of either the plant or the animal lectin. They correspond to local energy-minimum conformations in the phi,psi, epsilon plot and substantiate differential conformer selection by these two lectins with identical nominal monosaccharide specificity.
PURPOSE: To develop a biophysical model that explains the sizes of radiation-induced hprt deletions. METHODS: Key assumptions: (1) Deletions are produced by two DSB that are closer than an interaction distance at the time of DSB induction; (2) Interphase chromatin is modelled by a biphasic random walk distribution; and (3) Misrejoining of DSB from two separate tracks dominates at low-LET and misrejoining of DSB from a single track dominates at high-LET. RESULTS: The size spectra for radiation-induced total deletions of the hprt gene are calculated. Comparing with the results of Yamada and coworkers for gamma-irradiated human fibroblasts the study finds that an interaction distance of 0.75 microm will fit both the absolute frequency and the size spectrum of the total deletions. It is also shown that high-LET radiations produce, relatively, more total deletions of sizes below 0.5 Mb. The model predicts an essential gene to be located between 2 and 3 Mb from the hprt locus towards the centromere. Using the same assumptions and parameters as for evaluating mutation frequencies, a frequency of intra-arm chromosome deletions is calculated that is in agreement with experimental data. CONCLUSIONS: Radiation-induced total-deletion mutations of the human hprt gene and intrachange chromosome aberrations share a common mechanism for their induction.