Biased Walk Research Articles

Dynamics and processivity of 40S ribosome scanning on mRNA in yeast.

The eukaryotic 40S ribosomal subunit locates the translation initiation codon on an mRNA via the so-called scanning process that follows 40S binding to the capped 5' end. This key step in translation is required for the expression of almost all eukaryotic genes, yet the mechanism and dynamics of scanning are unknown. We have performed quantitative studies in vivo and in vitro of the movement of yeast 40S ribosomes along 5' untranslated regions (UTRs) of different lengths. 40S subunits perform cap-dependent scanning with high processivity for more than 1700 nucleotides in cells of Saccharomyces cerevisiae. Moreover, the observed rates of expression indicate that scanning is performed by an untethered 40S subunit that has been released from the 5' cap complex. Unexpectedly, the capability to maintain scanning competence on a long 5' UTR is more dependent on the Ded1/Dbp1 type of helicase than on eIF4A or eIF4B. In a yeast cell-free extract, scanning shows reduced processivity, with an estimated net 5'-->3' rate of approximately 10 nucleotides per second at 26 degrees C. We have developed a biased bidirectional walking model of ribosomal scanning that provides a framework for understanding the above observations as well as other known quantitative and qualitative features of this process.

Mechanism of Microgel Formation via Cross-Linking of Polymers in Their Dilute Solutions: Mathematical Explanation with Computer Simulations

The formation of poly(vinyl alcohol) microgels by γ-ray irradiation was previously studied by static and dynamic light scattering methods to obtain respectively the radius of gyration (Rg) and the hydrodynamic radius (R h ). It was proposed that the mechanism of this microgel formation was an intermolecular cross-linking step followed by an intramolecular cross-linking process. Presented herein is a mathematical model for interpreting the values of R g and R h , in which the Kirkwood theory of the friction factor was employed. To mimic the known extensions of brushes, a set of bias parameters were employed for the purpose of making a directional preference in generating the configurations of both parent and branch chains. The simulations with this simple model are in very good agreement with the proposed mechanism of microgel formation in the poly(vinyl alcohol) system. Justification of using the bias parameters is supported by computationally intensive Brownian dynamics simulations of a collection of 216 beads with a Velcro pair potential, i.e., a short-range attractive pair potential that acts as a formation of a chemical bond between the beads. The advantage of the bias walk model is that it allows one to explore different molecular configurations on a smaller computer and thus to select an appropriate set of parameters for more extended and time-consuming simulations on mainframe computers.

Biased Random Walk on Lamplighter Groups and Graphs

Lyons, Pemantle, and Peres(4) found an example of a group on which a simple random walk has sublinear speed but an inward biased walk has linear speed. We examine a similar example by introducing a martingale that allows us to compute the exact speed of the inward biased walk. We apply the martingale techniques to the example of Lyons et al., and improve on their speed estimates for small amounts of bias. Finally, we examine the speed on a related family of graphs and construct an example on which the speed of the walk has a second phase change.

Random walks and reactions on dendrimer structures

We perform model calculations of individual particle random walks on dendrimer structures, due to the emerging interest and wealth of recent experimental data. We use varying coordination numbers and generation orders. We find that it is practically impossible to verify numerically the customary walk properties with the existing asymptotic formalism, because of finite-size effects, even at short times and large system sizes. Several different sets of boundary conditions are employed, producing large differences in the observed behavior for the number of sites visited at least once, as a function of time, S N . We also investigate the case of reacting (annihilating) particles, using the well-known A+A and A+B models, in which case we find an anomalous scaling for the decay of the particle density, with the scaling exponents being f=0.80±0.03 and 0.50±0.03, respectively. We discuss how diffusion in dendrimers is analogous to biased random walks in one dimension, which demonstrates a breakdown of the usual scaling between the reaction progress (or reactant survival) and S N . An appendix gives results for biased walks in one dimension.

On the Cole-Cole relaxation function and related Mittag-Leffler distribution

In the framework of the one-dimensional fractal time random walk (FTRW) relaxation model, we rigorously show that the frequency domain response takes, in both nonbiased and biased walks, the only possible Cole-Cole form. The underlying reason for this is the specific form of the relaxation function (the survival probability of a relaxing system) determined in this model by the Mittag-Leffler distribution. We provide also analytical formulas for the propagators of the nonbiased and biased FTRWs.

Reflection principles for biased correlated walks. Simple applications

The new type of reflection principle applicable to the Laplace transform of the arrival probability for biased walks developed by Khantha and Balakrishnan is extended to the cases of biased correlated walks with absorbing and/or reflecting walls. An exact expression for the mean escape time <t≳ for the linear discrete walk bounded by absorbing walls at 0 and N is obtained as follows: <t≳=N(1−δ)[2(N−1)(a−d)]−1[(aN+dN)/ (aN−dN)−(1+δ)/(a−d)] +δ[(N−1)(a−d)]−1{1−[N(a−d)dN−1]/ (aN−dN)}, where a=1/2(a+δ+ε) and d=1/2(1+δ−ε), and δ and ε are the degrees of correlation and anisotropy, respectively.

Aggregates with biased random walks on a square lattice.

Irreversible stochastic models for the growth of clusters on a square lattice are formulated and studied by Monte Carlo simulation in which diffusing particles are biased globally and locally. In the case of global bias, the preferred directions of incoming particles are diagonal. Contrary to the biased walks on continuum space, the lattice anisotropy is enhanced and results in diagonally elongated clusters. With the local biased walk controlled by a parametric \ensuremath{\zeta}, it seems that there is a crossover length L(\ensuremath{\zeta}). For the region rL(\ensuremath{\zeta}), the cluster is more or less compact while for the region r>L(\ensuremath{\zeta}) the shape of the clusters becomes similar to that of the ordinary diffusion-limited aggregation. The comparison of clusters of our model is made with those of the model considered recently by Meakin, Feder, and Jo/ssang [Phys. Rev. A 43, 1952 (1991)].

Collisional energy transfer at high temperatures from the biased walk model

A new expression for the probability, P(E,E'), of collisional energy transfer for large polyatomics at high temperatures is deduced by using the assumptions of the biased random walk (BRW) model [Lim, K. F.; Gilbert, R. G. J. Chem. Phys. 1986, 84, 6124] and of a Gaussian equilibrium population distribution. The latter approximation is shown to be very accurate at sufficiently high temperatures. Simple analytic expressions are obtained for P(E,E'), which show the correct limiting weak- and strong-collision forms as the duration of a collision approaches zero and infinity, respectively. Expressions are found for the average energy-transfer quantities , , and , as functions of initial energy E'. The expression for explicitly shows the change of sign as E' is varied over a wide range of E'. The new BRW expression for P(E,E') can be accurately approximated as Gaussian. The derived expressions will be of use in modeling high-temperature processes such as combustion.

Disorder-induced roughening of diverse manifolds.

We propose a unified treatment of the roughening of manifolds by impurities in quenched correlated random media. Our perspective incorporates such apparently distinct problems as domain walls in dirty Ising magnets, biased walks upon random lattices, and flux creep in high-${\mathit{T}}_{\mathit{c}}$ materials. By means of generalized Imry-Ma arguments and a functional renormalization group (RG), we find new results, including the random-bond interfacial roughening exponent, ${\mathrm{\ensuremath{\zeta}}}_{\mathrm{RB}}$=2\ensuremath{\epsilon}/9, as well as estimates for the many-dimensional directed-polymer wandering index. This last quantity is also investigated via real-space RG methods, where we find, for example, ${\mathrm{\ensuremath{\zeta}}}_{2+1}$=0.602, in reasonable agreement with the functional RG value 3/5. Finally, since the Burgers equation permits translation of our directed-polymer results to the Eden cluster and ballistic deposition problems in higher dimensions, we can compare to the most recent computer simulations of these stochastic growth models. In particular, we address issues regarding the exponent conjectures that have been made and suggest the possibility of a finite upper critical dimension.

New spin model for the biased self-avoiding walk

We present a new spin model for the biased self-avoiding walk on the square lattice. Unlike a previous model containing both vector and Ising spins, our model contains only n-vector spins, evaluated in the n\ensuremath{\rightarrow}0 limit, and the spin Hamiltonian is Hermitian. Integration over one of the three sets of spins leads to a rigorous proof that the crossover exponent from straight chain to random walk behavior is 1. An analysis of the scaling behavior of the Hamiltonian in the small gauche-bond probability limit enables us to conclude that self-interaction effects in the biased walk will be irrelevant in three dimensions, but not in two dimensions, in agreement with some earlier numerical results.

Cell renormalisation study of biased self-avoiding walks

The authors present a cell renormalisation approach for the biased self-avoiding walks and show that a stiff-to-isotropic crossover exponent is exactly one in all dimensions and for all cell sizes. They also show, by use of renormalisation flow diagrams, a substantial difference between two and three dimensions in the crossover from stiff limit to isotropic limit as the length of walk N to infinity for fixed gauche weight p. In three dimensions, a crossover seems to occur first to the random walk limit and then to the self-avoiding walk limit, while, in two dimensions, it seems to occur directly to the self-avoiding walk limit in agreement with recent observations based on Monte Carlo simulations.

Crossover scaling in biased self-avoiding walks.

We discuss biased self-avoiding walks defined as self-avoiding random walks with anisotropic or different weights for steps in the same direction as the previous step and those in a different direction. This problem is a discrete analog of the stiff ``wormlike'' chain in polymer physics but with the excluded-volume effect included. Our approach extends that of Halley, Nakanishi, and Sundararajan to include three dimensions. The scaling analyses of our Monte Carlo simulations for the simple cubic and square lattices suggest qualitative differences in the effect of excluded volume on the chain conformation in the stiff limit betwen two and three dimensions in a manner similar to a suggestion made by Petschek. In the limit of gauche weight p\ensuremath{\rightarrow}0 and contour length N\ensuremath{\rightarrow}\ensuremath{\infty}, we find scaling for the mean-square end-to-end distance 〈${R}^{2}$〉 with the crossover exponent one as before; however, the scaling function in three dimensions closely matches the random stiff-chain result with no excluded volume while that in two dimensions exhibits marked deviation from the random limit.

Reflection principles for biased random walks and application to escape time distributions

We present a reflection principle for an arbitrarybiased continuous time random walk (comprising both Markovian and non-Markovian processes) in the presence of areflecting barrier on semi-infinite and finite chains. For biased walks in the presence of a reflecting barrier this principle (which cannot be derived from combinatorics) is completely different from its familiar form in the presence of an absorbing barrier. The result enables us to obtain closed-form solutions for the Laplace transform of the conditional probability for biased walks on finite chains for all three combinations of absorbing and reflecting barriers at the two ends. An important application of these solutions is the calculation of various first-passage-time and escape-time distributions. We obtain exact results for the characteristic functions of various kinds of escape time distributions for biased random walks on finite chains. For processes governed by a long-tailed event-time distribution we show that the mean time of escape from bounded regions diverges even in the presence of a bias—suggesting, in a sense, the absence of true long-range diffusion in such “frozen” processes.

Scaling in biased random walks.

We discuss the scaling of the statistics of a biased random walk. In the well-known model, a walk has probability p per step of stepping away from the direction of its last step. For a walk of N steps, such chains have persistence lengths ${p}^{\mathrm{\ensuremath{-}}1}$. We are interested in the crossover from straight-chain to random-walk behavior in the neighborhood of the point p=0, ${N}^{\mathrm{\ensuremath{-}}1}$=0. A scaling function has been computed for the problem of a random walk which avoids immediate returns by Schroll et al. Here we consider the self-avoiding case. A simple argument suggests a scaling function with crossover exponent 1. We present a proof that the crossover exponent is indeed one and numerical results for the scaling function on the square lattice which are consistent with this value. We also present a spin model which is asymptotically equivalent to the biased self-avoiding walk on the square lattice.