Self-avoiding Random Walks Research Articles

A growth model for the generation of particle aggregates with tunable fractal dimension

A new 3D off-lattice model is proposed to generate porous and controlled fractal aggregate structures. The method is a variation of the Eden Model with the additional capability of randomly inactivating particles with a given inactivation probability. As this probability increases from 0 to 1, the proposed model follows a continuous transition from an off-lattice Eden Model leading to a core-dense random structure to a self-avoiding random walk model leading to a linear chain with partially constrained growth directions. The resulting aggregates have been characterized in terms of their radius of gyration, shape factor, mass and surface fractal dimensions, pore volume fraction, specific surface area and pore size distribution. A large range of microstructural properties can be controlled by adjusting the inactivation probability and the number of particles (e.g. mass fractal dimension from 1.6 to 3). In addition the proposed approach is convenient to implement and highly computationally efficient. The morphology (radius of gyration and mass fractal dimension) of silica aggregates characterized by small angle X-ray scattering in the literature was successfully reproduced to demonstrate the capabilities of the method.

Large deviations of convex hulls of self-avoiding random walks.

A global picture of a random particle movement is given by the convex hull of the visited points. We obtained numerically the probability distributions of the volume and surface of the convex hulls of a selection of three types of self-avoiding random walks, namely, the classical self-avoiding walk, the smart-kinetic self-avoiding walk, and the loop-erased random walk. To obtain a comprehensive description of the measured random quantities, we applied sophisticated large-deviation techniques, which allowed us to obtain the distributions over a large range of support down to probabilities far smaller than P=10^{-100}. We give an approximate closed form of the so-called large-deviation rate function Φ which generalizes above the upper critical dimension to the previously studied case of the standard random walk. Further, we show correlations between the two observables also in the limits of atypical large or small values.

Prediction of kinetics of protein folding with non-redundant contact information.

The majority of the inter-residue distances in a protein structure are correlated given a fixed topology. Here, we investigate whether we are able to predict a structure's folding rate, which is known to depend on the complexity of its fold, while considering only a small, uncorrelated fraction of its contacts. We define an expression for the probabilistic information content associated to the relative position of a pair of amino acid residues in a protein structure. By means of fitting the protein chain to a self-avoiding random walk model, we derive a probability distribution for the distance between residues as a function of their separation along the sequence. We then show that the average information content for all residue pairs in a structure, considered as an estimate of its fold's complexity, is well correlated to the logarithm of its folding rate. Moreover, the same information content measure may be exploited to rank contacts and identify redundancies, allowing the prediction the structure's folding rate with similar accuracy while taking into account less than 5% of its contacts. An implementation of the described model and the experimental data are available at http://github.com/luciano-censoni/sarw-lnkf.

Community detection in networks using self-avoiding random walks

Different kinds of random walks have proven to be useful in the study of structural properties of complex networks. Among them, the restricted dynamics of self-avoiding random walks (SAW), which visit only at most once each vertex in the same walk, has been successfully used in network exploration. The detection of communities of strongly connected vertices in networks remains an open problem, despite its importance, due to the high computational complexity of the associated optimization problem and the lack of a unique formal definition of communities. In this work, we propose a SAW-based method to extract the community distribution of a network and show that it achieves high modularity scores, specially for real-world networks. We combine SAW with principal component analysis to define the dissimilarity measure to be used for agglomerative hierarchical clustering. To evaluate the performance of this method we compare it with four popular methods for community detection: Girvan–Newman, Fastgreedy, Walktrap and Infomap using two types of synthetic networks and six well-known real-world cases.

Target guiding self-avoiding random walk with intersection algorithm for minimum exposure path problem in wireless sensor networks

To solve minimum exposure path (MEP) problem in wireless sensor networks more efficiently, this work proposes an algorithm called target guiding self-avoiding random walk with intersection (TGSARWI), which mimics the behavior of a group of random walkers that seek path to their destinations in a strange area. Target guiding leads random walkers move toward their end points, while self-avoiding prevents them from taking roundabout routes. Route intersections further accelerate the speed of seeking connected paths. Dijkstra algorithm (DA) is applied to solve MEP problem in a sub-network formed by multiple connected paths that walkers generate (called TGSARWI DA). Simulations show that the path exposure found by TGSARWI DA is very close to that by DA in the global network (Global DA), whereas the time complexity of computation is much lower. Compared with existing heuristic algorithms such as physarum optimization algorithm (POA), our algorithm shows higher generality and efficiency. This algorithm also exhibits good robustness to the fluctuations of parameters. Our algorithm could be very useful for the solution to MEP problem in fields with large- or high-density sensors.

Saxs Confirms that FRET Dyes Promote Collapse of an Otherwise Fully Disordered Protein

Significance: The dimensions of intrinsically disordered proteins (IDPs) remains controversial. Using SAXS, we previously found that even relatively hydrophobic IDPs remain expanded in the complete absence of denaturant, contrary to conclusions drawn from FRET studies that foldable sequences collapse in water. Here we use our methodology to demonstrate that labeling with Alexa488 causes the IDP to undergo significant dye-induced contraction. These results demonstrate that the addition of dyes influences the properties of IDPs and is likely the primary origin of the discrepancy between prior SAXS and FRET results. ABSTRACT: The origins of the discrepancy between SAXS and FRET measurements of IDP dimensions remain controversial. To address this, we recently developed a robust analysis for extracting Rg and ν (the Flory exponent; Rg∼Nν) from single SAXS experiments using a molecular form factor derived from simulations. Performing this analysis on a 334 AA IDP, we find that the addition of a dye pair promotes collapse, decreasing Rg from 51 to 45 Å and ν from 0.54 to 0.50. This observation is consistent with prior studies on PEG, which found the addition of dyes induces similar contraction1 despite the fact that, as we show here, the polymer behaves as a self-avoiding random walk. Recent studies have suggested that FRET and SAXS results can be reconciled if SAXS reports on Rg while remaining relatively insensitive to Rend-to-end, essentially decoupling the two. To the contrary, our analysis finds that the entire SAXS profile is sensitive to conformational ensembles enriched in shortened end-to-end distances, such as those produced by dye-dye interactions. More broadly, our simulations define a route to characterize the deviations from homopolymeric behavior inherent in disordered proteins. 1Watkins et al, PNAS 2015.

Hierarchical structure of MWCNT reinforced semicrystalline HDPE composites: A contrast matching study by neutron and X-ray scattering

Combining neutron and X-ray scattering (SANS, SAXS, and WAXD) as the contrast matching probes, the hierarchical structures of the multi-wall carbon nanotubes/high density polyethylene (MWCNTs/HDPE) composites (0.1–2.0 wt%) have been studied. SANS measurement avoided the correlation peak associated to the semicrystalline nature of HDPE from SAXS, and the passible scattering from the stretch induced voids. Benefitting from SANS data, the power law (α) taking the flexible cylinder suggested that whole MWCNTs in the unstretched composites follow a self-avoiding random walk (α ≈ 1.67). SANS data suggest a hollow flexible cylinder structure with outer and inner diameters (core-shell-like cross section) of the scatterers as ca. 12.6 nm and 4.1 nm, while the Kuhn length is about 43.0 nm. Tensile test displayed improved mechanical properties. Interestingly, the Hermans orientation parameter (f) of HDPE crystals (fc) in the stretched composites are around a plateau ca. 0.96 initially, while decreased remarkably to 0.776 when the fraction is further increased to 2.0 wt%. f obtained by SANS showed a similar tendency (decreased from 0.756 to 0.572). Combining all the results along with differential scanning calorimetry (DSC) measurement, a mechanism of adhered crystals on the surface of MWCNTs and a reinforcement model are proposed.

Mapping of the Bak, Tang, and Wiesenfeld sandpile model on a two-dimensional Ising-correlated percolation lattice to the two-dimensional self-avoiding random walk.

The self-organized criticality on the random fractal networks has many motivations, like the movement pattern of fluid in the porous media. In addition to the randomness, introducing correlation between the neighboring portions of the porous media has some nontrivial effects. In this paper, we consider the Ising-like interactions between the active sites as the simplest method to bring correlations in the porous media, and we investigate the statistics of the BTW model in it. These correlations are controlled by the artificial "temperature" T and the sign of the Ising coupling. Based on our numerical results, we propose that at the Ising critical temperature T_{c} the model is compatible with the universality class of two-dimensional (2D) self-avoiding walk (SAW). Especially the fractal dimension of the loops, which are defined as the external frontier of the avalanches, is very close to D_{f}^{SAW}=4/3. Also, the corresponding open curves has conformal invariance with the root-mean-square distance R_{rms}∼t^{3/4} (t being the parametrization of the curve) in accordance with the 2D SAW. In the finite-size study, we observe that at T=T_{c} the model has some aspects compatible with the 2D BTW model (e.g., the 1/log(L)-dependence of the exponents of the distribution functions) and some in accordance with the Ising model (e.g., the 1/L-dependence of the fractal dimensions). The finite-size scaling theory is tested and shown to be fulfilled for all statistical observables in T=T_{c}. In the off-critical temperatures in the close vicinity of T_{c} the exponents show some additional power-law behaviors in terms of T-T_{c} with some exponents that are reported in the text. The spanning cluster probability at the critical temperature also scales with L^{1/2}, which is different from the regular 2D BTW model.

The ABC (in any D) of logarithmic CFT

Logarithmic conformal field theories have a vast range of applications, from critical percolation to systems with quenched disorder. In this paper we thoroughly examine the structure of these theories based on their symmetry properties. Our analysis is model-independent and holds for any spacetime dimension. Our results include a determination of the general form of correlation functions and conformal block decompositions, clearing the path for future bootstrap applications. Several examples are discussed in detail, including logarithmic generalized free fields, holographic models, self-avoiding random walks and critical percolation.

Instant Construction and Visualization of Crowded Biological Environments.

We present the first approach to integrative structural modeling of the biological mesoscale within an interactive visual environment. These complex models can comprise up to millions of molecules with defined atomic structures, locations, and interactions. Their construction has previously been attempted only within a non-visual and non-interactive environment. Our solution unites the modeling and visualization aspect, enabling interactive construction of atomic resolution mesoscale models of large portions of a cell. We present a novel set of GPU algorithms that build the basis for the rapid construction of complex biological structures. These structures consist of multiple membrane-enclosed compartments including both soluble molecules and fibrous structures. The compartments are defined using volume voxelization of triangulated meshes. For membranes, we present an extension of the Wang Tile concept that populates the bilayer with individual lipids. Soluble molecules are populated within compartments distributed according to a Halton sequence. Fibrous structures, such as RNA or actin filaments, are created by self-avoiding random walks. Resulting overlaps of molecules are resolved by a forced-based system. Our approach opens new possibilities to the world of interactive construction of cellular compartments. We demonstrate its effectiveness by showcasing scenes of different scale and complexity that comprise blood plasma, mycoplasma, and HIV.

A DNA Walker as a Fluorescence Signal Amplifier

Sensing nucleic acids typically involves the recognition of a specific sequence and reporting by, for example, a fluorogenic reaction yielding one activated dye molecule per detected nucleic acid. Here, we show that after binding to a DNA origami track a bound DNA target (a "DNA walker") can release the fluorescence of many molecules by acting as the catalyst of an enzymatic nicking reaction. As the walking kinetics sensitively depends on the walker sequence, the resulting brightness distribution of DNA origamis is a sequence fingerprint with single-nucleotide sensitivity. Using Monte Carlo simulations, we rationalize that the random self-avoiding walk is mainly terminated when steps to nearest neighbors are exhausted. Finally, we demonstrate that the DNA walker is also active in a plasmonic hotspot for fluorescence enhancement, indicating the potential of combining different amplification mechanisms enabled by the modularity of DNA nanotechnology.

Compaction of quasi-one-dimensional elastoplastic materials

Insight into crumpling or compaction of one-dimensional objects is important for understanding biopolymer packaging and designing innovative technological devices. By compacting various types of wires in rigid confinements and characterizing the morphology of the resulting crumpled structures, here, we report how friction, plasticity and torsion enhance disorder, leading to a transition from coiled to folded morphologies. In the latter case, where folding dominates the crumpling process, we find that reducing the relative wire thickness counter-intuitively causes the maximum packing density to decrease. The segment size distribution gradually becomes more asymmetric during compaction, reflecting an increase of spatial correlations. We introduce a self-avoiding random walk model and verify that the cumulative injected wire length follows a universal dependence on segment size, allowing for the prediction of the efficiency of compaction as a function of material properties, container size and injection force.

Canonical ensemble of particles in a self-avoiding random walk

We consider an ensemble of particles not interacting with each other and randomly walking in the d-dimensional Euclidean space ℝ d . The individual moves of each particle are governed by the same distribution, but after the completion of each such move of a particle, its position in the medium is “marked” as a region in the form of a ball of diameter r 0, which is not available for subsequent visits by this particle. As a result, we obtain the corresponding ensemble in ℝ d of marked trajectories in each of which the distance between the centers of any pair of these balls is greater than r 0. We describe a method for computing the asymptotic form of the probability density W n (r) of the distance r between the centers of the initial and final balls of a trajectory consisting of n individual moves of a particle of the ensemble. The number n, the trajectory modulus, is a random variable in this model in addition to the distance r. This makes it necessary to determine the distribution of n, for which we use the canonical distribution obtained from the most probable distribution of particles in the ensemble over the moduli of their trajectories. Averaging the density W n (r) over the canonical distribution of the modulus n allows finding the asymptotic behavior of the probability density of the distance r between the ends of the paths of the canonical ensemble of particles in a self-avoiding random walk in ℝ d for 2 ≤ d < 4.

Canonical ensemble of particles in a self-avoiding random walk

Рассматривается ансамбль не взаимодействующих между собой частиц, случайно блуждающих в $d$-мерном евклидовом пространстве $\mathbb R^d$. Отдельные перемещения каждой частицы подчинены одному и тому же распределению, но после завершения каждого такого перемещения ее положение фиксируется "меткой" - областью в виде шара диаметра $r_0$, недоступной для последующих посещений этой частицей. В результате в пространстве $\mathbb R^d$ получен соответствующий ансамбль "меченых" траекторий, в каждой из которых расстояние между центрами любой пары указанных шаров больше $r_0$. Изложен метод вычисления асимптотики плотности вероятности $W_n(\mathbf r)$ расстояния $r$ между центрами начального и конечного шаров траектории, состоящей из $n$ отдельных перемещений некоторой частицы ансамбля. Помимо расстояния $r$, случайной величиной в рассматриваемой модели является и число $n$ - модуль траектории, что обусловливает необходимость определения распределения для $n$, в качестве которого используется каноническое распределение, полученное при наиболее вероятном распределении частиц в ансамбле по модулям их траекторий. Усреднение плотности $W_n(\mathbf r)$ по каноническому распределению модуля $n$ позволяет найти асимптотику плотности вероятности расстояния $r$ между концами траекторий частиц канонического ансамбля, блуждающих без самопересечений в $\mathbb R^d$, когда $2\leq d&lt;4$.

Measuring the (Good) Solvent Quality of Water for Disordered Proteins from a Single SAXS Measurement

Significance: Proteins adopt non-native geometries during their synthesis, folding, transport, and turnover and a significant fraction of the proteome is intrinsically disordered. The characterization of disordered proteins is challenging in part due to difficulties in the interpretation of techniques including SAXS and FRET. We have developed a method to extract the dimensions and solvent quality from a single SAXS measurement. We apply it to intrinsically disordered proteins (IDPs), finding them to be expanded in water, which behaves as a good solvent.Abstract: SAXS has been used to measure the properties of denatured proteins and IDPs. Many ensembles are expanded in water, with little dependence on denaturant. These findings appear at odds with FRET studies, suggesting issues with data interpretation. We conduct SAXS measurements on IDPs up to 334 residues using in-line size-exclusion chromatography. The IDPs are expanded and the Flory exponent, describing the solvent quality and a chain's collapse propensity, is ν∼0.53. This value is between the ideal self-avoiding random walk (ν=0.59) and the theta solvent limit (ν=½) where intra-chain attraction exactly counterbalances excluded volume terms. Addition of denaturants results in a mild expansion, most noticeable in longer sequences, and an increase in ν to 0.59. Simulations generate a molecular form factor applicable to IDPs that can be used to determine the Rg and ν from a single measurement. The methodology is robust (hetero- or homopolymers, energy function, Ramachandran maps). This new analysis procedure and the dependence on chain length, along with recent FRET studies, largely resolves the controversy concerning the dimensions disordered proteins in water. These studies reaffirm that an early stable collapse often is not the first step in folding of single domain proteins upon dilution of denaturant.

Critical phenomena on k-booklets.

We define a "k-booklet" to be a set of k semi-infinite planes with -∞<x<∞ and y≥0, glued together at the edges (the "spine") y=0. On such booklets we study three critical phenomena: self-avoiding random walks, the Ising model, and percolation. For k=2, a booklet is equivalent to a single infinite lattice, and for k=1 to a semi-infinite lattice. In both these cases the systems show standard critical phenomena. This is not so for k≥3. Self-avoiding walks starting at y=0 show a first-order transition at a shifted critical point, with no power-behaved scaling laws. The Ising model and percolation show hybrid transitions, i.e., the scaling laws of the standard models coexist with discontinuities of the order parameter at y≈0, and the critical points are not shifted. In the case of the Ising model, ergodicity is already broken at T=T_{c}, and not only for T<T_{c} as in the standard geometry. In all three models, correlations (as measured by walk and cluster shapes) are highly anisotropic for small y.

Thermal properties of a two-dimensional intrinsically curved semiflexible biopolymer**Project supported by the Minister of Science and Technology of China

We study the behaviors of mean end-to-end distance and specific heat of a two-dimensional intrinsically curved semiflexible biopolymer with a hard-core excluded volume interaction. We find the mean square end-to-end distance at large N, with N being the number of monomers. Both β and proportional constant are dependent on the reduced bending rigidity κ and intrinsic curvature c. The larger the c, the smaller the proportional constant, and 1.5 ≥ β ≥ 1. Up to a moderate κ = κc, or down to a moderate temperature T = Tc, β = 1.5, the same as that of a self-avoiding random walk, and the larger the intrinsic curvature, the smaller the κc. However, at a large κ or a low temperature, β is close to 1, and the conformation of the biopolymer can be more compact than that of a random walk. There is an intermediate regime with 1.5 > β > 1 and the transition from β = 1.5 to β = 1 is smooth. The specific heat of the system increases smoothly with increasing κ or there is no peak in the specific heat. Therefore, a nonvanishing intrinsic curvature seriously affects the thermal properties of a semiflexible biopolymer, but there is no phase transition in the system.

A family of self-avoiding random walks interpolating the loop-erased random walk and a self-avoiding walk on the Sierpiński gasket

We show that the ‘erasing-larger-loops-first’ (ELLF) method, which was first introduced for erasing loops from the simple random walk on the Sierpinski gasket, does work also for non-Markov random walks, in particular, self-repelling walks to construct a new family of self-avoiding walks on the Sierpinski gasket. The one-parameter family constructed in this method continuously connects the loop-erased random walk and a self-avoiding walk which has the same asymptotic behavior as the ‘standard’ self-avoiding walk. We prove the existence of the scaling limit and study some path properties: The exponent \begin{document}$ν$\end{document} governing the short-time behavior of the scaling limit varies continuously in the parameter. The limit process is almost surely self-avoiding, while it has path Hausdorff dimension \begin{document}$1/ν $\end{document} , which is strictly greater than \begin{document}$1$\end{document} .

Scaling Laws in Spatial Network Formation.

Geometric constraints impact the formation of a broad range of spatial networks, from amino acid chains folding to proteins structures to rearranging particle aggregates. How the network of interactions dynamically self-organizes in such systems is far from fully understood. Here, we analyze a class of spatial network formation processes by introducing a mapping from geometric to graph-theoretic constraints. Combining stochastic and mean field analyses yields an algebraic scaling law for the extent (graph diameter) of the resulting networks with system size, in contrast to logarithmic scaling known for networks without constraints. Intriguingly, the exponent falls between that of self-avoiding random walks and that of space filling arrangements, consistent with experimentally observed scaling of the radius of gyration of protein tertiary structures with their chain length.

Random Graph Approach to Multifunctional Molecular Networks

Formation of a molecular network from multifunctional precursors is modelled with a random graph process. The process does not account for spatial positions of the monomers explicitly, yet the Euclidean distances between the monomers are derived from the topological information by applying self-avoiding random walks. This allows favoring reactivity of monomers that are close to each other, and to disfavor the reactivity for monomers obscured by the surrounding. As a result, the model is applicable to large time scales. The phenomena of conversion-dependent reaction rates, gelation, microgelation, and structural inhomogeneity are predicted by the model. Resulting nonhomogeneous network topologies are analyzed to extract such descriptors as: size distribution, crosslink distances, and gel-point conversion. Furthermore, new to the molecular simulation community descriptors are suggested that are especially useful when explaining evolution of the gel as being a single molecule: local clustering coefficient, network modularity, cluster size distribution.