Self-avoiding Walks Research Articles

Comparing random walks in graph embedding and link prediction.

Random walks find extensive applications across various complex network domains, including embedding generation and link prediction. Despite the widespread utilization of random walks, the precise impact of distinct biases on embedding generation from sequence data and their subsequent effects on link prediction remain elusive. We conduct a comparative analysis of several random walk strategies, including the true self-avoiding random walk and the traditional random walk. We also analyze walks biased towards node degree and those with inverse node degree bias. Diverse adaptations of the node2vec algorithm to induce distinct exploratory behaviors were also investigated. Our empirical findings demonstrate that despite the varied behaviors inherent in these embeddings, only slight performance differences manifest in the context of link prediction. This implies the resilient recovery of network structure, regardless of the specific walk heuristic employed to traverse the network. Consequently, the results suggest that data generated from sequences governed by unknown mechanisms can be successfully reconstructed.

Self-Avoiding Walks on Cayley Graphs Through the Lens of Symbolic Dynamics

We study dynamical and computational properties of the set of bi-infinite self-avoiding walks on Cayley graphs, as well as ways to compute, approximate and bound their connective constant. To do this, we introduce the skeleton $X_{G,S}$ of a finitely generated group $G$ relative to a generating set $S$, which is a one-dimensional subshift made of configurations on $S$ that avoid all words that reduce to the identity. We provide a characterization of groups which have SFT skeletons and sofic skeletons: first, there exists a finite generating set $S$ such that $X_{G,S}$ is a subshift of finite type if and only if $G$ is a plain group; second, there exists $S$ such that $X_{G,S}$ is sofic if and only if $G$ is a plain group, $\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ or $\mathcal{D}_{\infty}\times\mathbb{Z}/2\mathbb{Z}$. We also characterize finitely generated torsion groups as groups whose skeletons are aperiodic. For connective constants, using graph height functions and bridges, we show that Cayley graphs of finitely generated torsion groups do not admit graph height functions, and that for groups that admit transitive graph height functions, the connective constant is equal to the growth rate of periodic points of the skeleton. Finally, we take a brief look at the set of bi-infinite geodesics and introduce an analog of the connective constant for the geodesic growth.

Exponential growth rate of lattice comb polymers

Abstract We investigate a lattice model of comb polymers and derive bounds on the exponential growth rate of the number of embeddings of
the comb. A comb is composed of a backbone that is a self-avoiding walk and a set of $t$ teeth, also modelled as mutually and self-avoiding walks, attached to the backbone at vertices or nodes of degree 3. Each tooth of the comb has $m_a$ edges and there are $m_b$ edges in the backbone between adjacent degree 3 vertices and between the first and last nodes of degree 3 and the end vertices of degree 1 of the backbone. We are interested in the exponential growth rate as $t \to \infty$ with $m_a$ and $m_b$ fixed. We prove upper bounds on this growth rate and show that for small values of $m_a$ the growth rate is strictly less than that of self-avoiding walks.

Lattice polymers near a permeable interface

Abstract We study the localisation of lattice polymer models near a permeable interface in two dimensions. Localisation can arise due to an interaction between the polymer and the interface, and can be altered by a preference for the bulk solvent on one side or by the application of a force to manipulate the polymer. Different combinations of these three effects give slightly different statistical mechanical behaviours. The canonical lattice model of polymers is the self-avoiding walk which we mainly study with Monte Carlo simulation to calculate the phase diagram and critical phenomena. For comparison, a solvable directed walk version is also defined and the phase diagrams are compared for each case. We find broad agreement between the two models, and most minor differences can be understood as due to the different entropic contributions. In the limit where the bulk solvent on one side is overwhelmingly preferred we see how the localisation transition transforms to the adsorption transition; the permeable interface becomes effectively an impermeable surface.

Coagulation-flocculation process on a lattice: Monte Carlo simulations

Coagulation-flocculation, the physicochemical process widely used for purification a wastewater, is affected both by chemical details of involved polymers and by the statistics of their conformations on a large scale. The latter aspect is covered in this study by employing a coarse-grained modeling approach based on a combination of two paradigms of statistical mechanics. One is the self-avoiding walk (SAW) which generates a range of conformations for a linear polymer of N SAW monomers. Another one is a non-trivial diffusion limited aggregation (DLA) process of N DLA impurities (referred thereafter as ‘particles’) which describes their coagulation occurring with the probability 0<p⩽1 (p = 1 recovers a standard DLA). DLA of diffusive particles is complemented by their irreversible adsorption on the SAW monomers occurring with the probability equal to one, both processes resulting in formation of the DLA-SAW agglomerates. The dynamics of formation of such agglomerates, as well as their fractal dimensions and internal structure are of practical interest. We consider a range of related characteristics, such as: (i) absolute Na and relative na adsorbing efficiencies of SAW; (ii) effective gyration radius RgDLA-SAW of the DLA-SAW agglomerates; and (iii) the fractal dimension DDLA-SAW of these aggregates. These are studied within a wide range for each parameter from a set {p,NDLA,NSAW} .

IDRWalker: A Random Walk Based Tool for Generating Intrinsically Disordered Regions in Large Protein Complexes.

Intrinsically disordered regions (IDRs), which may be functionally important, are common in proteins. However, the structures of IDRs are often missing due to their highly dynamic nature. In the study of IDRs, integrative modeling combining computational simulations and experimental data is a common approach, for which initial structures of the IDRs need to be built. However, applying this method to large protein complexes is challenging because existing structure generation tools are sometimes unsuitable for IDRs in large systems. To facilitate convenient and rapid structure generation of IDRs in large protein complexes, we developed a computational tool named IDRWalker based on self-avoiding random walks. Three protein complexes were used to illustrate the efficiency of the tool, and it was found that IDRs in more than 800 chains of the nuclear pore complex could be generated in minutes. These structures of large protein complexes with added IDRs can be further used to run computational simulations for integrative modeling.

Snake Tilings, Skeletons Subshift and Self-Avoiding Walks

Snake Tilings, Skeletons Subshift and Self-Avoiding Walks

Thermodynamics of a confined macromolecule: An analytical approach

We consider a fully directed self-avoiding walk model on a cubic lattice to mimic the conformations of an infinitely long flexible polymer chain and also to mimic confirmations of a short flexible chain under confined conditions. The confinement conditions is achieved using two parallel impenetrable plates. The confined chain is under good solvent conditions and we revisit this problem to solve the real (self avoiding) polymer's model for any length of the chain and also for any given separation in between the confining plates. The equilibrium statistics of the confined polymer chain are derived using analytical approach of the generating function technique. The force of the confinement, the surface tension and the monomer density profile of the confined chain are obtained analytically. We propose that the methods of calculation are suitable to understand thermodynamics of an arbitrary length confined polymer chain under other possible conditions of the confinement.

Hydrodynamic Radii of Intrinsically Disordered Proteins: Fast Prediction by Minimum Dissipation Approximation and Experimental Validation.

The diffusion coefficients of globular and fully unfolded proteins can be predicted with high accuracy solely from their mass or chain length. However, this approach fails for intrinsically disordered proteins (IDPs) containing structural domains. We propose a rapid predictive methodology for estimating the diffusion coefficients of IDPs. The methodology uses accelerated conformational sampling based on self-avoiding random walks and includes hydrodynamic interactions between coarse-grained protein subunits, modeled using the generalized Rotne-Prager-Yamakawa approximation. To estimate the hydrodynamic radius, we rely on the minimum dissipation approximation recently introduced by Cichocki et al. Using a large set of experimentally measured hydrodynamic radii of IDPs over a wide range of chain lengths and domain contributions, we demonstrate that our predictions are more accurate than the Kirkwood approximation and phenomenological approaches. Our technique may prove to be valuable in predicting the hydrodynamic properties of both fully unstructured and multidomain disordered proteins.

On currents in the $O(n)$ loop model

Using methods from the conformal bootstrap, we study the properties of Noether currents in the critical O(n)O(n) loop model. We confirm that they do not give rise to a Kac-Moody algebra (for n≠ 2n≠2), a result expected from the underlying lack of unitarity. By studying four-point functions in detail, we fully determine the current-current OPEs, and thus obtain several structure constants with physical meaning. We find in particular that the terms :\!J\bar{J}\!::JJ‾: in the identity and adjoint channels vanish exactly, invalidating the argument made in [Nucl. Phys. B 419, 411 (1994)] that adding orientation-dependent interactions to the model should lead to continuously varying exponents in self-avoiding walks. We also determine the residue of the identity channel in the JJJJ two-point function, finding that it coincides both with the result of a transfer-matrix computation for an orientation-dependent correlation function in the lattice model, and with an earlier Coulomb gas computation of Cardy [Nucl. Phys. B 265, 409 (1986)]. This is, to our knowledge, one of the first instances where the Coulomb gas formalism and the bootstrap can be successfully compared.

Map conformational landscapes of intrinsically disordered proteins with polymer physics quantities

Disordered proteins are conformationally flexible proteins that are biologically important and have been implicated in devastating diseases such as Alzheimer’s disease and cancer. Unlike stably folded structured proteins, disordered proteins sample a range of different conformations that needs to be accounted for. Here, we treat disordered proteins as polymer chains, and compute a dimensionless quantity called instantaneous shape ratio (Rs), as Rs = Ree2/Rg2, where Ree is end-to-end distance and Rg is radius of gyration. Extended protein conformations tend to have high Ree compared with Rg, and thus have high Rs values, whereas compact conformations have smaller Rs values. We use a scatter plot of Rs (representing shape) against Rg (representing size) as a simple map of conformational landscapes. We first examine the conformational landscape of simple polymer models such as Random Walk, Self-Avoiding Walk, and Gaussian Walk (GW), and we notice that all protein/polymer maps lie within the boundaries of the GW map. We thus use the GW map as a reference and, to assess conformational diversity, we compute the fraction of the GW conformations (fC) covered by each protein/polymer. Disordered proteins all have high fC scores, consistent with their disordered nature. Each disordered protein accesses a different region of the reference map, revealing differences in their conformational ensembles. We additionally examine the conformational maps of the nonviral gene delivery vector polyethyleneimine at various protonation states, and find that they resemble disordered proteins, with coverage of the reference map decreasing with increasing protonation state, indicating decreasing conformational diversity. We propose that our method of combining Rs and Rg in a scatter plot generates a simple, meaningful map of the conformational landscape of a disordered protein, which in turn can be used to assess conformational diversity of disordered proteins.

How hydrophobicity, side chains, and salt affect the dimensions of disordered proteins.

Despite the generally accepted role of the hydrophobic effect as the driving force for folding, many intrinsically disordered proteins (IDPs), including those with hydrophobic content typical of foldable proteins, behave nearly as self-avoiding random walks (SARWs) under physiological conditions. Here, we tested how temperature and ionic conditions influence the dimensions of the N-terminal domain of pertactin (PNt), an IDP with an amino acid composition typical of folded proteins. While PNt contracts somewhat with temperature, it nevertheless remains expanded over 10-58°C, with a Flory exponent, ν, >0.50. Both low and high ionic strength also produce contraction in PNt, but this contraction is mitigated by reducing charge segregation. With 46% glycine and low hydrophobicity, the reduced form of snow flea anti-freeze protein (red-sfAFP) is unaffected by temperature and ionic strength and persists as a near-SARW, ν ~ 0.54, arguing that the thermal contraction of PNt is due to stronger interactions between hydrophobic side chains. Additionally, red-sfAFP is a proxy for the polypeptide backbone, which has been thought to collapse in water. Increasing the glycine segregation in red-sfAFP had minimal effect on ν. Water remained a good solvent even with 21 consecutive glycine residues (ν > 0.5), and red-sfAFP variants lacked stable backbone hydrogen bonds according to hydrogen exchange. Similarly, changing glycine segregation has little impact on ν in other glycine-rich proteins. These findings underscore the generality that many disordered states can be expanded and unstructured, and that the hydrophobic effect alone is insufficient to drive significant chain collapse for typical protein sequences.

Force-induced desorption of copolymeric comb polymers

We investigate a lattice model of comb copolymers that can adsorb at a surface and that are subject to a force causing desorption. The teeth (the comb’s side chains) and the backbone of the comb are chemically distinct and can interact differently with the surface. That is, the strength of the surface interaction can be different for the monomers in the teeth and in the backbone. We consider several cases including (i) the uniform case where the number of teeth is fixed and the lengths of the branches in the backbone and the lengths of the teeth are all identical, (ii) the case where the teeth are short compared to the branches in the backbone, (iii) the situation where the teeth are long compared to the backbone, and (iv) the case where the number of teeth approaches infinity. We obtain expressions for the free energies in the thermodynamic limit in terms of those for self-avoiding walks and discuss the nature of the phase diagrams of the model.

Parallel self-avoiding walks for a low-autocorrelation binary sequences problem

A low-autocorrelation binary sequences problem with a high figure of merit factor represents a formidable computational challenge. An efficient parallel computing algorithm is required to reach the new best-known solutions for this problem. Therefore, we developed the sokolskew solver for the skew-symmetric search space. The developed solver takes the advantage of parallel computing on graphics processing units. The solver organized the search process as a sequence of parallel and contiguous self-avoiding walks and achieved a speedup factor of 387 compared with lssOrel, its predecessor. The sokolskew solver belongs to stochastic solvers and cannot guarantee the optimality of solutions. To mitigate this problem, we established the predictive model of stopping conditions according to the small instances for which the optimal skew-symmetric solutions are known. With its help and 99% probability, the sokolskew solver found all the known and seven new best-known skew-symmetric sequences for odd instances from L=121 to L=223. For larger instances, the solver cannot reach 99% probability within our limitations, but it still found several new best-known binary sequences. We also analyzed the trend of the best merit factor values, and it shows that as sequence size increases, the value of the merit factor also increases, and this trend is flatter for larger instances.

Solvable models for 2+1D quantum critical points: Loop soups of 1+1D conformal field theories

We construct a class of solvable models for 2+1D quantum critical points by attaching 1+1D conformal field theories (CFTs) to fluctuating domain walls forming a “loop soup”. Specifically, our local Hamiltonian attaches gapless spin chains to the domain walls of a triangular lattice Ising antiferromagnet. The macroscopic degeneracy between antiferromagnetic configurations is split by the Casimir energy of each decorating CFT, which is usually negative and thus favors a short loop phase with a finite gap. However, we found a set of 1D CFT Hamiltonians for which the Casimir energy is effectively positive, making it favorable for domain walls to coalesce into a single “snake” which is macroscopically long and thus hosts a CFT with a vanishing gap. The snake configurations are geometrical objects also known as fully-packed self-avoiding walks or Hamiltonian walks which are described by an \mathrm{O}O(n=0) loop ensemble with a non-unitary 2+0D CFT description. Combining this description with the 1+1D decoration CFT, we obtain a 2+1D theory with unusual critical exponents and entanglement properties. Regarding the latter, we show that the log contributions from the decoration CFTs conspire with the spatial distribution of loops crossing the entanglement cut to generate a “non-local area law”. Our predictions are verified by Monte Carlo simulations.

Polymer translocation: Effects of confinement.

We investigate the influence of varying confinement on the dynamics of polymer translocation through a cone-shaped channel. For this, a linear polymer chain is modeled using self-avoiding walks on a square lattice. The cis side of a cone-shaped channel has a finite volume, while the trans side has a semi-infinite space. The confining environment is varied either by changing the position of the back wall while keeping the apex angle fixed or altering the apex angle while keeping the position of the back wall fixed. In both cases, the effective space ϕ, which represents the number of monomers in a chain relative to the total number of accessible sites within the cone, is reduced due to the imposed confinement. Consequently, the translocation dynamics are affected. We analyze the entropy of the confined system as a function of ϕ, which exhibits nonmonotonic behavior. We also calculate the free energy associated with the confinement as a function of a virtual coordinate for different positions of the back wall (base of the cone) along the conical axis for various apex angles. Employing the Fokker-Planck equation, we calculate the translocation time as a function of ϕ for different solvent conditions across the channel. Our findings indicate that the translocation time decreases as ϕ increases, but it eventually reaches a saturation point at a certain value of ϕ. Moreover, we highlight the possibility of controlling the translocation dynamics by manipulating the solvent quality across the channel. Furthermore, our investigation delves into the intricacies of polymer translocation through a cone-shaped channel, considering both repulsive and neutral interactions with the channel wall. This exploration unveils nuanced dynamics and sheds light on the factors that significantly impact translocation within confined channels.

The local persistence length of semi-flexible self-avoiding walks on the square lattice

By applying the pruned-enriched Rosenbluth Monte Carlo simulation method, we have studied the local persistence length of semi-flexible linear polymers presented by self-avoiding walks (SAWs) on the square lattice, where the stiffness property is characterised by the weight s assigned to each bend of the walk. In this model, the local persistence length λN(k) of N-step SAWs is formulated as an ensemble average of the projection of the end-to-end position vector onto the oriented kth step of the SAW path. We have found that the persistence length λN(k)(s) decreases with s and increases with k and N. Regarding the chain position parameter k, we have scrutinised two particular cases: the first case is when k has a constant (fixed) value, independent of the chain length, and the second one is the situation when k is comparable with the SAW length k∼N . For fixed k, we have learnt that in the limit N→∞ , the persistence length λN(k)(s) tends towards a constant value λp(k)(s) behaving as k1/2 . In the other analysed case k∼N , in the same limit, λN(k)(s) diverges as N ϕ , with φ=1/2 . We have also examined the dependence of the studied quantity on s and discussed our results in the context of previous findings and predictions related to the persistence length behaviour of SAWs in Euclidean spaces.

Two-point functions of random-length random walk on high-dimensional boxes

We study the two-point functions of a general class of random-length random walks (RLRWs) on finite boxes in Zd with d⩾3 , and provide precise asymptotics for their behaviour. We show that in a finite box of side length L, the two-point function is asymptotic to the infinite-lattice two-point function when the typical walk length is o(L2) , but develops a plateau when the typical walk length is Ω(L2) . We also numerically study walk length moments and limiting distributions of the self-avoiding walk and Ising model on five-dimensional tori, and find that they agree asymptotically with the known results for the self-avoiding walk on the complete graph, both at the critical point and also for a broad class of scaling windows/pseudocritical points. Furthermore, we show that the two-point function of the finite-box RLRW, with walk length chosen via the complete graph self-avoiding walk, agrees numerically with the two-point functions of the self-avoiding walk and Ising model on five-dimensional tori. We conjecture that these observations in five dimensions should also hold in all higher dimensions.

A single carbon nanotube-entangled high-performance buckypaper with tunable fracture mode.

It is well known that the traditional buckypaper (BP) is composed of a certain number of short carbon nanotubes (CNTs) intertwined with each other and sliding always happens when the BP is under tensile and impact loading, which results in inferior mechanical properties compared to single CNTs. In this work, a highly-entangled single-wire BP (SWBP) structure is constructed by a modified self-avoiding random walk approach. The in-plane mechanical properties and impacting behaviors of the SWBPs with different entanglement degrees and interface frictions are systematically investigated via newly developed coarse-grained molecular dynamics (CGMD) simulation. A coarse-grained method can effectively reflect the inter-tube van der Waals (vdW) interactions and the mechanical behaviors of CNTs, including tension, bending and adhesion. In this work, from the tensile simulations of the SWBP, the results showed that the self-locking mechanism between entangled CNTs could significantly enhance the tensile resistance of the film. Besides, the mechanical properties of the SWBP are highly dependent on the entanglement degree and the interface friction between CNTs. Furthermore, two distinct fracture modes, ductile fracture and brittle fracture, are revealed, which can be efficiently controlled by changing the related friction between CNTs. From the impacting simulations, it is found that the impacting performance can be effectively tuned by adjusting the entanglement degree of the film. In addition, the kinetic energy of the projectile could be rapidly dissipated through the stretching and bending of CNTs in the SWBP. This work provides an in-depth understanding of the effect of interface friction and entanglement degree on the mechanical properties of the buckypaper and provides a reference for the preparation of strong CNT-based micromaterials.

Interacting semi-flexible self-avoiding walks studied on a fractal lattice

Self-avoiding walks are studied on the 3-simplex fractal lattice as a model of linear polymer conformations in a dilute, nonhomogeneous solution. The model is supplemented with bending energies and attractive-interaction energies between nonconsecutively visited pairs of nearest-neighboring sites (contacts). It captures the main features of a semi-flexible polymer subjected to variable solvent conditions. A hierarchical structure of the fractal lattice enabled the determination of the exact recurrence equations for the generating function, through which universal and local properties of the model were studied. An analysis of the recurrence equations showed that for all finite values of the considered energies and nonzero temperatures, the polymer resides in an expanded phase. The critical exponents of the expanded phase are universal and the same as those for ordinary self-avoiding walks on the same lattice found earlier. As a measure of local properties, the mean number of contacts per mean number of steps as well as the persistence length, are calculated as functions of Boltzmann weights associated with bending energies and attractive interactions between contacts. Both quantities are monotonic functions of stiffness weights for fixed interaction, and in the limit of infinite stiffness, the number of contacts decreases to zero, while the persistence length increases unboundedly.