Survival probabilities in three-dimensional random walks

We evaluate the survival probability for random walkers on lattices doped with traps. We consider nearest-neighborstep walks and walks mediated bi multipolar interactions, for which the probability of steps of length r is proportional to r^?s. The decay law due to trapping is calculated for the diamond, simple cubic and body- and face-centered cubic lattices. We establish validity domains of approximate expressions.     

Shape asymmetry of star-branched random walks and nonreversal random walks

Star-branched random walks with 3, 4, 6, 8 and 12 arms (the total chain-length ranging from N = 49 to 1925) have been produced and analysed with respect to their instantaneous shape. The short-chain behaviour of nonreversal random walk stars (NRRWs) embedded in various lattices is compared to that of star-branched freely jointed (off-lattice) chains (RWs). While for all types of NRRW-stars examined as well as for RW-stars with bonds of constant length shape-asymmetry increases with increasing chain-lengths, the opposite behaviour is found for RW-stars with Gaussian-distributed bond-lengths. The amount of short-chain effects is strongly dependent on the number of arms and on the lattice type used. In the limit of infinitely large molecules, however, quantities characteristic of the shape converge to common values for all types of RWs and NRRWs examined.     

Resistance distance in graphs and random walks

We study the resistance distance on connected undirected graphs, linking this concept to the fruitful area of random walks on graphs. We provide two short proofs of a general lower bound for the resistance, or Kirchhoff index, of graphs on N vertices, as well as an upper bound and a general formula to compute it exactly, whose complexity is that of inverting an N×N matrix. We argue that the formulas for the resistance in the case of the Platonic solids can be generalized to all distance‐transitive graphs. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 29–33, 2001     

Loop-erased self-avoiding random walks in four and five dimensions

Monte Carlo simulations of loop-erased self-avoiding random walks in four and five dimensions were performed, using two distinct algorithms. We find consistency between these methods in their estimates of critical exponents. The upper critical dimension for this phenomenon is four, and it has been shown that the mean square end-to-end distance grows as n(log n)^α. It has recently been established that the mean square end-to-end distance is asymptotically bounded by n(log n)^1/3 (see Ref. 21). Our results show that asymptotic convergence to n(log n)^1/3 in fact obtains and does so rather quickly. In five dimensions we examine the rate of asymptotic convergence to the mean-field model. © 1996 by John Wiley & Sons, Inc.     

Two-particle continuous-time random walks and binary reactions in disordered media

The relation between unary-(trappihg) and binary (mutual annihilation) reactions in disordered systems is studied in the framework of the continuous-time random walk. It is found that if the waiting-time distribution of the walk has infinite moments, a time-independent binary rate constant may exist even though a unary one does not.     

Propagators and related descriptors for non-Markovian asymmetric random walks with and without boundaries

There are many current applications of the continuous-time random walk (CTRW), particularly in describing kinetic and transport processes in different chemical and biophysical phenomena. We derive exact solutions for the Laplace transforms of the propagators for non-Markovian asymmetric one-dimensional CTRW's in an infinite space and in the presence of an absorbing boundary. The former is used to produce exact results for the Laplace transforms of the first two moments of the displacement of the random walker, the asymptotic behavior of the moments as t-->infinity, and the effective diffusion constant. We show that in the infinite space, the propagator satisfies a relation that can be interpreted as a generalized fluctuation theorem since it reduces to the conventional fluctuation theorem at large times. Based on the Laplace transform of the propagator in the presence of an absorbing boundary, we derive the Laplace transform of the survival probability of the random walker, which is then used to find the mean lifetime for terminated trajectories of the random walk.     

Short-chain behaviour of star-branched nonreversal random walks embedded in a tetrahedral lattice

Analytical results are presented for the mean-square dimensions of star-branched nonreversal random walk polymers on a tetrahedral lattice considering short-chain effects and the influence of the core (centre) of the star. They are checked against numerical results obtained from highly precise Monte Carlo calculations. Short-chain effects are studied in detail. In addition, the deviation from proportionality between mean-square dimensions and the total chain-length is evaluated and compared to that for self-avoiding chains and stars near theta-conditions, the total chain-lengths ranging from 125 to 3845 segments.     

Exact quantum,quasiclassical forward and reverse reaction probabilities for a colldmear asymmetric model reaction

Detailed quasiclassical and time-independent quantum reaction probabilities are given for a surface on which large discrepancies between quasiclassical and wavepacket results have previously been found. The quasiclassical results are shown to agree relatively reasonably with the oscillation-averaged time-independent quantum ones if the quasiclassical reverse probabilities are chosen in the threshold region.     

The shape of linear and star-branched nonreversal random walks embedded in a tetrahedral lattice

The shape-asymmetry of linear and star-branched nonreversal random walk polymers on a tetrahedral lattice is studied by means of a Monte Carlo simulation. Properties characteristic of the instantaneous shape based on the mean-square radius of gyration and its principal components as well as based on the end-to-end distance are discussed. Short-chain effects are characterized by polynomial expressions (power series of N⁻¹), simultaneously by extrapolation yielding a result in the limit of infinitely large configurations.     

Bounds for the Kirchhoff index of regular graphs via the spectra of their random walks

Using probabilistic tools, we give tight upper and lower bounds for the Kirchhoff index of any d‐regular N‐vertex graph in terms of d, N, and the spectral gap of the transition probability matrix associated to the random walk on the graph. We then use bounds of the spectral gap of more specialized graphs, available in the literature, in order to obtain upper bounds for the Kirchhoff index of these specialized graphs. As a byproduct, we obtain a closed‐form formula for the Kirchhoff index of the d‐dimensional cube in terms of the first inverse moment of a positive binomial variable. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Computerized evaluation of mean residence times in multicompartmental linear system and pharmacokinetics

Deriving mean residence times (MRTs) is an important task both in pharmacokinetics and in multicompartmental linear systems. Taking as starting point the analysis of MRTs in open or closed (Garcia-Meseguer et al., Bull Math Biol 2003, 65, 279) multicompartmental linear systems, we implement a versatile software, using the Visual Basic 6.0 language for MS-Windows, that is easy to use and with a user-friendly format for the input of data and the output of results. For any multicompartmental linear system of up to 512 compartments, whether closed or open, with traps or without traps and with zero input in one or more of the compartments, this software allows the user to obtain the symbolic expressions, in the most simplified form, and/or the numerical values of the MRTs in any of its compartments, in the entire system or in a part of the system. As far as we known from the literature, such a software has not been implemented before. The advantage of the present software is that it reduces on the work time needed and minimizes the human errors that are frequent in compartmental systems even those that are relatively staightforward. The software bioCelTer, along with instructions, can be downloaded from http://oretano.iele-ab.uclm.es/~fgarcia/bioCelTer/.     

Exact analytical expressions for the potential of electrical double layer interactions for a sphere-plate system

Exact, closed-form analytical expressions are presented for evaluating the potential energy of electrical double layer (EDL) interactions between a sphere and an infinite flat plate for three different types of interactions: constant potential, constant charge, and an intermediate case as given by the linear superposition approximation (LSA). By taking advantage of the simpler sphere-plate geometry, simplifying assumptions used in the original Derjaguin approximation (DA) for sphere-sphere interaction are avoided, yielding expressions that are more accurate and applicable over the full range of κa. These analytical expressions are significant improvements over the existing equations in the literature that are valid only for large κa because the new equations facilitate the modeling of EDL interactions between nanoscale particles and surfaces over a wide range of ionic strength.     

Discussion of several analytical approximate expressions for the eigenvalues of the bounded harmonic oscillator and hydrogen atom

Several approximate analytical formulas for the multidimensional isotropic bounded oscillators and the bounded hydrogen atom are compared. Numerical results show that the coth z method is in both cases better than the Padé approximants method. Perturbational polynomials, necessary in order to build the approximate eigenvalues, are obtained through the hypervirial perturbative method.     

Revisiting random walks in fractal media: on the occurrence of time discrete scale invariance

This paper addresses the kinetic behavior of random walks in fractal media. We perform extensive numerical simulations of both single and annihilating random walkers on several Sierpinski carpets, in order to study the time behavior of three observables: the average number of distinct sites visited by a single walker, the mean-square displacement from the origin, and the density of annihilating random walkers. We found that the time behavior of those observables is given by a power law modulated by soft logarithmic-periodic oscillations. We conjecture that logarithmic-periodic oscillations are a manifestation of a time domain discrete scale iNvariance (DSI) that occurs as a consequence of the spatial DSI of the substrate. Our conjecture implies that the logarithmic periods of oscillations in space and time domains are linked by a dynamic exponent z, through z=log(tau)/log(b(1)), where tau and b(1) are the fundamental scaling ratios of the DSI symmetry in the time and space domains, respectively. We use this relationship in order to compute z for different observables and fractals. Furthermore, we check the values obtained with independent measurements provided by the power-law behavior of the mean-square displacement with time [R(2)(t) proportional variant t(2/z)]. The very good agreement obtained between both computations of the z exponent gives strong support to the idea of an intimate interplay between spatial and time symmetry properties that we expect will have a quite general scope. We expect that the application of the outlined concepts in the field of dynamic processes in fractal media will stimulate further research.     

Paramagnetism-based NMR restraints provide maximum allowed probabilities for the different conformations of partially independent protein domains

An innovative analytical/computational approach is presented to provide maximum allowed probabilities (MAPs) of conformations in protein domains not rigidly connected. The approach is applied to calmodulin and to its adduct with alpha-synuclein. Calmodulin is a protein constituted by two rigid domains, each of them composed by two calcium-binding EF-hand motifs, which in solution are largely free to move with respect to one another. We used the N60D mutant of calmodulin, which had been engineered to selectively bind a paramagnetic lanthanide ion to only one of its four calcium binding sites, specifically in the second EF-hand motif of the N-terminal domain. In this way, pseudocontact shifts (pcs's) and self-orientation residual dipolar couplings (rdc's) measured on the C-terminal domain provide information on its relative mobility with respect to the domain hosting the paramagnetic center. Available NMR data for terbium(III) and thulium(III) calmodulin were supplemented with additional data for dysprosium(III), analogous data were generated for the alpha-synuclein adduct, and the conformations with the largest MAPs were obtained for both systems. The MAP analysis for calmodulin provides further information on the variety of conformations experienced by the system. Such variety is somewhat reduced in the calmodulin-alpha-synuclein adduct, which however still retains high flexibility. The flexibility of the calmodulin-alpha-synuclein adduct is an unexpected result of this research.     

Percolation of a collection of finite random walks: a model for gas permeation through thin polymeric membranes

Motivated by recent studies of gas permeation through polymer networks, we consider a collection of ordinary random walks of fixed length ℓ, placed randomly on the bonds of a square lattice. These walks model polymers, each with ℓ segments. Using computer simulations, we find the critical concentration of occupied bonds (i.e., the critical occupation probability) for such a network to percolate the system. Though this threshold decreases monotonically with ℓ, the critical “mass” density, defined as the total number of segments divided by total number of bonds in the system, displays a more complex behavior. In particular, for fixed mass densities, the percolation characteristics of the network can change several times, as shorter polymers are linked to form longer ones.     

Hyperfine splitting in the ESR spectra of random oriented alkali earth biradical chelates

The ESR spectra of rigid glass solutions of biradicals of glyoxal diimine anions and alkali earth cations were recorded. The spectra could be computer simulated only by adding to the spin hamiltonian

= gβH·S + D[S_Z²?$\text{1}\text{2}$S(S+1)] a hyperfine term due to the anisotropic interaction of the nitrogen nuclei. Two different models were tested: the first one with the two radicals in the same plane and the second one with the two radicals perpendicular; agreement with the experiment was found for the latter.     

Contribution to the study of self-avoiding random walks (SARW) confined to strips and capillaries

A flexible chain that can assume all possible positions on a lattice without intersection of segments (self-avoiding random walk, SARW) has been used for a long time as a model of polymers in highly dilute solution, taking into account the excluded volume. In this article, we extend to a SARW subject to spatial constraints an analytical accurate method we have established for a SARW on an unbounded lattice. We now consider a square or cubic lattice extending without limit in the x direction but confined to D layers in the other directions. This is a model for the study of constrained macromolecules the behavior of which is important for instance in connection with the properties of thin polymeric films or fibers, or with the solubility of biopolymers in lipid bilayer membranes. We give recurrence relationships for calculating C_n, the total number of configurations, and x _n, the mean x projection of the end-to-end separation, for a chain length n. Our method, being analytical, allows a considerable economy of computational time with respect to other methods such as Monte Carlo calculations. It enables us to investigate the asymptotic behavior of x _n and to write the partition function of the system.