Segmental distribution properties of a polymer chain near an interacting barrier

The exact distribution of segments for self-avoiding walks of lengths N=4–14 bonds in the presence of an interacting barrier on the diamond lattice has been obtained by the method of direct enumeration. Behavior for the infinite chain was estimated and compared with Rubin's results for the normal random walk. It is shown that the onset of a well defined transition for the self-avoiding walk coincides with the location predicted for the normal random walk. It was found that for a self-avoiding walk a plot of θ₂ (the fraction of segments in level z) versus the interaction parameter φ is shifted to the right (higher φ), for all z, as compared with a similar plot for the normal random walk. Conditional probabilities for a self-avoiding walk having its t th segment in level z (when φ=0) are reported.     

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.     

Modeling diffusional transport in the interphase cell nucleus

In this paper a lattice model for the diffusional transport of particles in the interphase cell nucleus is proposed. Dense networks of chromatin fibers are created by three different methods: Randomly distributed, noninterconnected obstacles, a random walk chain model, and a self-avoiding random walk chain model with persistence length. By comparing a discrete and a continuous version of the random walk chain model, we demonstrate that lattice discretization does not alter particle diffusion. The influence of the three dimensional geometry of the fiber network on the particle diffusion is investigated in detail while varying the occupation volume, chain length, persistence length, and walker size. It is shown that adjacency of the monomers, the excluded volume effect incorporated in the self-avoiding random walk model, and, to a lesser extent, the persistence length affect particle diffusion. It is demonstrated how the introduction of the effective chain occupancy, which is a convolution of the geometric chain volume with the walker size, eliminates the conformational effects of the network on the diffusion, i.e., when plotting the diffusion coefficient as a function of the effective chain volume, the data fall onto a master curve.     

The role of chromatin conformations in diffusional transport of chromatin-binding proteins: Cartesian lattice simulations

In this paper, a lattice model for the diffusional transport of chromatin-binding particles in the interphase cell nucleus is proposed. Sliding effects are studied in dense networks of chromatin fibers created by three different methods: Randomly distributed, noninterconnected obstacles, a random walk chain model with an attractive step potential, and a self-avoiding random walk chain model with a hard repulsive core and attractive surroundings. By comparing a discrete and continuous version of the random walk chain model, we demonstrate that lattice discretization does not alter the diffusion of chromatin-binding particles. The influence of conformational properties of the fiber network on the particle sliding is investigated in detail while varying occupation volume, sliding probability, chain length, and persistence length. It is observed that adjacency of the monomers, the excluded volume effect incorporated in the self-avoiding random walk model, and the persistence length affect the chromatin-binding particle diffusion. It is demonstrated that sliding particles sense local chain structures. When plotting the diffusion coefficient as a function of the accessible volume for diffusing particles, the data fall onto master curves depending on the persistence length. However, once intersegment transfer is involved, chromatin-binding proteins no longer perceive local chain structures.     

Essential energy space random walk via energy space metadynamics method to accelerate molecular dynamics simulations

To overcome the possible pseudoergodicity problem, molecular dynamic simulation can be accelerated via the realization of an energy space random walk. To achieve this, a biased free energy function (BFEF) needs to be priori obtained. Although the quality of BFEF is essential for sampling efficiency, its generation is usually tedious and nontrivial. In this work, we present an energy space metadynamics algorithm to efficiently and robustly obtain BFEFs. Moreover, in order to deal with the associated diffusion sampling problem caused by the random walk in the total energy space, the idea in the original umbrella sampling method is generalized to be the random walk in the essential energy space, which only includes the energy terms determining the conformation of a region of interest. This essential energy space generalization allows the realization of efficient localized enhanced sampling and also offers the possibility of further sampling efficiency improvement when high frequency energy terms irrelevant to the target events are free of activation. The energy space metadynamics method and its generalization in the essential energy space for the molecular dynamics acceleration are demonstrated in the simulation of a pentanelike system, the blocked alanine dipeptide model, and the leucine model.     

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.     

Diffusion and trapping in dendrimer structures

We investigate diffusion on models of large molecules with dendrimer structure. The models we use are topological Cayley trees. We focus on diffusion properties, such as the excursion distance, the mean square displacement of the diffusing particles, and the area probed, as given by the random walk parameter S_N, the number of the distinct sites visited. Finally we look at the trapping kinetics for randomly distributed stationary traps of low concentrations. We use different coordination numbers, z, and different generation orders g of a dendrimer structure. We show that for calculations on dendrimers where we simulate the entire structure the finite size effects dominate making the results worthless. We, therefore, implemented two algorithms that do not retain the underlying structure, and thus effectively the random walk is performed on an equivalent infinite structure. We find a universal linear behavior for all the properties examined, thus differing from the regular lattices. The average displacement R grows linearly with time, implying that R² grows as t² instead of t. The S_N property also grows linearly with time, which resembles the random walk case in three-dimensional lattices, but with a different prefactor which strongly depends on z. Trapping is dependent on S_N, as expected, and follows exactly its behavior. Due to the nature of these structures the random walk is indeed a type of a “biased” walk. We devise a model in which we take out this “bias” factor, and in this model we show that the system becomes diffusive in nature, i.e. the mean square displacement is linear with time. These results could be utilized if one is to design a dendrimer system with specifically desired properties.     

Self avoiding walk trees and laces

This paper is divided into two sections. In the first section we describe the use depth first search trees and breadth first search trees in understanding self avoiding walks and restricted random walks. While we derive our results for Euclidean lattices the method is totally general and can be used for other lattices as well. In the second section derive an expression for the number of laces in the lace expansion of a memory four restricted random walk.     

Self-Diffusion in Simple Liquids as a Random Walk Process

It is demonstrated that self-diffusion in dense liquids can be considered a random walk process; its characteristic length and time scales are identified. This represents an alternative to the often assumed hopping mechanism of diffusion in the liquid state. The approach is illustrated using the one-component plasma model.     

Energy difference space random walk to achieve fast free energy calculations

A method is proposed to efficiently obtain free energy differences. In the present algorithm, free energy calculations proceed by the realization of an energy difference space random walk. Thereby, this algorithm can greatly improve the sampling of the regions in phase space where target states overlap.     

Time series analysis of collective motions in proteins

The dynamics of alpha-amylase inhibitor tendamistat around its native state is investigated using time series analysis of the principal components of the C(alpha) atomic displacements obtained from molecular dynamics trajectories. Collective motion along a principal component is modeled as a homogeneous nonstationary process, which is the result of the damped oscillations in local minima superimposed on a random walk. The motion in local minima is described by a stationary autoregressive moving average model, consisting of the frequency, damping factor, moving average parameters and random shock terms. Frequencies for the first 50 principal components are found to be in the 3-25 cm(-1) range, which are well correlated with the principal component indices and also with atomistic normal mode analysis results. Damping factors, though their correlation is less pronounced, decrease as principal component indices increase, indicating that low frequency motions are less affected by friction. The existence of a positive moving average parameter indicates that the stochastic force term is likely to disturb the mode in opposite directions for two successive sampling times, showing the modes tendency to stay close to minimum. All these four parameters affect the mean square fluctuations of a principal mode within a single minimum. The inter-minima transitions are described by a random walk model, which is driven by a random shock term considerably smaller than that for the intra-minimum motion. The principal modes are classified into three subspaces based on their dynamics: essential, semiconstrained, and constrained, at least in partial consistency with previous studies. The Gaussian-type distributions of the intermediate modes, called "semiconstrained" modes, are explained by asserting that this random walk behavior is not completely free but between energy barriers.     

Random walks and chemical graph theory

Simple random walks probabilistically grown step by step on a graph are distinguished from walk enumerations and associated equipoise random walks. Substructure characteristics and graph invariants correspondingly defined for the two types of random walks are then also distinct, though there often are analogous relations. It is noted that the connectivity index as well as some resistance-distance-related invariants make natural appearances among the invariants defined from the simple random walks.     

Sequential injection: a new concept for chemical sensors,process analysis and laboratory assays

Established flow-injection techniques allow advanced solution handling for laboratory purposes, but chemical sensing and continuous monitoring of chemical processes require dramatically simplified flow schemes and instrumentation with the potential for miniaturization and an inherent ruggedness. Considerations based on the random walk model have led to the concept of sensor injection and sequential injection analysis. This new approach to automated analysis is designed to fill a gap in present flow-injection methodology.     

Continuous-time random-walk theory of interfering diffusion and chemical reaction with an application to electrochemical impedance spectra of oxidized Zr-1%Nb

A microscopic theory is developed for the interplay of diffusion and chemical reaction and the results are compared with electrode impedance measurements on an oxide electrode. The theory is based on the ideas of continuous-time random walk and accounts for the interference of diffusion and recombination of the charge carriers in the oxide. The treatment results in a dispersive diffusivity with two time constants, one of them corresponding to the random walk, the other to the reaction. Combining this diffusivity with the Warburg electrode admittance expression, which refers to cases where the rate-limiting step is diffusion in a semi-infinite medium bounded by a plane, an admittance function is obtained. The phase angle is found to be higher than 45 degrees distinguishing it from the Gerischer impedance which was developed for a related problem. The oxides were produced by hydrothermal oxidation of Zr-l%Nb alloy, a metal used as cladding material for nuclear fuel elements. The electrode impedance spectra of Zr/Zr-oxide electrodes in aqueous SO(3) (2-) solutions were taken at various anodic voltages between 1 Hz and 100 kHz and temperatures between 278 and 333 K. The theoretical admittance functions could be successfully compared with the observed spectra. Both the functional forms and the fitted parameter values support our theory which is also in keeping with Macdonald's point-defect model.     

A comparison between the bond-strength-coordination number fluctuation model and the random walk model of viscosity

We have studied the temperature dependence of the viscosity of some polymeric materials by using both, the bond-strength-coordination number fluctuation model and the random walk model. The results reveal that both models show an excellent agreement with the experimental data. For the random walk model, two equations corresponding to two temperature regimes (low-T and high-T) separated by the critical temperature T _c, which is difficult to determine, are needed to describe the temperature dependence of the viscosity of a fragile system, whereas for the bond-strength-coordination number fluctuation model, a single equation with clear physical meaning describes the temperature dependence of the viscosity of both, the fragile and strong systems. We have also studied the relationship between the normalized temperature range of cooperativity and the fragility index. A theoretical expression for the relationship has been derived based on the bond-strength-coordination number fluctuation model. The comparison with the experimental data shows a good agreement, leading to the conclusion that the kinetic properties of glass forming liquids and the cooperativity of molecular relaxations are correlated.     

表面扩散的Monte Carlo初探

利用ＭｏｎｔｅＣａｒｌｏ方法模拟了理想表面和分形表面上的扩散过程；通过模拟可以发现，表面扩散系数不仅与表面浓度有关，而且还与扩散的时间、表面的几何形貌等有关。在表面覆盖度比较高时，表面扩散系数有一极大值。与理想表面相比，分形表面会使扩散系数减小。     

大分子固液界面吸附行为的数字模拟研究方法

介绍了随机数字模拟方法在研究链状大分子吸附行为的一般原理,以立方格子模型的自避行走为例,分析了模型的建立、初始位形的产生以及分子链运动的处理方法.回顾了随机数字模拟方法在研究大分子界面吸附行为方面的发展状况.预测了随机数字模拟方法在大分子吸附行为研究方面的应用前景.     

Strong correlations and Fickian water diffusion in narrow carbon nanotubes

The authors have used atomistic molecular dynamics (MD) simulations to study the structure and dynamics of water molecules inside an open ended carbon nanotube placed in a bath of water molecules. The size of the nanotube allows only a single file of water molecules inside the nanotube. The water molecules inside the nanotube show solidlike ordering at room temperature, which they quantify by calculating the pair correlation function. It is shown that even for the longest observation times, the mode of diffusion of the water molecules inside the nanotube is Fickian and not subdiffusive. They also propose a one-dimensional random walk model for the diffusion of the water molecules inside the nanotube. They find good agreement between the mean-square displacements calculated from the random walk model and from MD simulations, thereby confirming that the water molecules undergo normal mode diffusion inside the nanotube. They attribute this behavior to strong positional correlations that cause all the water molecules inside the nanotube to move collectively as a single object. The average residence time of the water molecules inside the nanotube is shown to scale quadratically with the nanotube length.     

Counting oriented rectangles and the propagation of waves

We propose a simple counting problem involving chains of rectangles on a planar lattice. The boundaries of the chains form a type of random walk with a finite inner scale. With orientation neglected, the continuum limit of the walk densities obeys the Telegraph equation, a form of diffusion equation with a finite signal velocity. Taking into account the orientation of the rectangles, the same continuum limit yields the Dirac equation. This provides an interesting context in which the Dirac equation is phenomenological rather than fundamental. A talk by G. N. Ord given at “Lattices and Trajectories: A Symposium of Mathematical Chemistry in honour of Ray Kapral and Stu Whittington”, Fields Institute, May 2007. Preprint submitted to Elsevier—13 March 2008.