Stretching of a Fractal Polymer around a Disc Reveals Kardar-Parisi-Zhang Scaling.

While stretching of a polymer along a flat surface is hardly different from the classical Pincus problem of pulling chain ends in free space, the role of curved geometry in conformational statistics of the stretched chain is an exciting open question. We use scaling analysis and computer simulations to examine stretching of a fractal polymer chain around a disc in 2D (or a cylinder in 3D) of radius R. We reveal that the typical excursions of the polymer away from the surface and curvature-induced correlation length scale as Δ∼R^{ß} and S^{*}∼R^{1/z}, respectively, with the Kardar-Parisi-Zhang (KPZ) growth ß=1/3 and dynamic exponents z=3/2. Although probability distribution of excursions does not belong to KPZ universality class, the KPZ scaling is independent of the fractal dimension of the polymer and, thus, is universal across classical polymer models, e.g., SAW, randomly branching polymers, crumpled unknotted rings. Additionally, our Letter establishes a mapping between stretched polymers in curved geometry and the Balagurov-Vaks model of random walks among traps.

Effects of topological constraints on globular polymers.

Topological constraints can affect both equilibrium and dynamic properties of polymer systems and can play a role in the organization of chromosomes. Despite many theoretical studies, the effects of topological constraints on the equilibrium state of a single compact polymer have not been systematically studied. Here we use simulations to address this longstanding problem. We find that sufficiently long unknotted polymers differ from knotted ones in the spatial and topological states of their subchains. The unknotted globule has subchains that are mostly unknotted and form asymptotically compact RG(s)∼s1/3 crumples. However, crumples display a high fractal dimension of the surface db=2.8, forming excessive contacts and interpenetrating each other. We conclude that this topologically constrained equilibrium state resembles a conjectured crumpled globule [Grosberg et al., Journal de Physique, 1988, 49, 2095], but differs from its idealized hierarchy of self-similar, isolated and compact crumples.

A statistical model of intra-chromosome contact maps.

A statistical model describing a fine structure of the intra-chromosome maps obtained by a genome-wide chromosome conformation capture method (Hi-C) is proposed. The model combines hierarchical chain folding with a quenched heteropolymer structure of primary chromatin sequences. It is conjectured that the observed Hi-C maps are statistical averages over many different ways of hierarchical genome folding. It is shown that the existence of a quenched primary structure coupled with hierarchical folding induces a full range of features observed in experimental Hi-C maps: hierarchical elements, chess-board intermittency and large-scale compartmentalization.

Fractal globules: a new approach to artificial molecular machines.

The over-damped relaxation of elastic networks constructed by contact maps of hierarchically folded fractal (crumpled) polymer globules was investigated in detail. It was found that the relaxation dynamics of an anisotropic fractal globule is very similar to the behavior of biological molecular machines like motor proteins. When it is perturbed, the system quickly relaxes to a low-dimensional manifold, M, with a large basin of attraction and then slowly approaches equilibrium, not escaping M. Taking these properties into account, it is suggested that fractal globules, even those made by synthetic polymers, are artificial molecular machines that can transform perturbations into directed quasimechanical motion along a defined path.

On predicting regulatory genes by analysis of functional networks in C. elegans.

BACKGROUND: Connectivity networks, which reflect multiple interactions between genes and proteins, possess not only a descriptive but also a predictive value, as new connections can be extrapolated and tested by means of computational analysis. Integration of different types of connectivity data (such as co-expression and genetic interactions) in one network has proven to benefit 'guilt by association' analysis. However predictive values of connectives of different types, that had their specific functional meaning and topological characteristics were not obvious, and have been addressed in this analysis. METHODS: eQTL data for 3 experimental C.elegans age groups were retrieved from WormQTL. WormNet has been used to obtain pair-wise gene interactions. The Shortest Path Function (SPF) has been adopted for statistical validation of the co-expressed gene clusters and for computational prediction of their potential gene expression regulators from a network context. A new SPF-based algorithm has been applied to genetic interactions sub-networks adjacent to the clusters of co-expressed genes for ranking the most likely gene expression regulators causal to eQTLs. RESULTS: We have demonstrated that known co-expression and genetic interactions between C. elegans genes can be complementary in predicting gene expression regulators. Several algorithms were compared in respect to their predictive potential in different network connectivity contexts. We found that genes associated with eQTLs are highly clustered in a C. elegans co-expression sub-network, and their adjacent genetic interactions provide the optimal functional connectivity environment for application of the new SPF-based algorithm. It was successfully tested in the reverse-prediction analysis on groups of genes with known regulators and applied to co-expressed genes and experimentally observed expression quantitative trait loci (eQTLs). CONCLUSIONS: This analysis demonstrates differences in topology and connectivity of co-expression and genetic interactions sub-networks in WormNet. The modularity of less continuous genetic interaction network does not correspond to modularity of the dense network comprised by gene co-expression interactions. However the genetic interaction network can be used much more efficiently with the SPF method in prediction of potential regulators of gene expression. The developed method can be used for validation of functional significance of suggested eQTLs and a discovery of new regulatory modules.

Phase transition in random planar diagrams and RNA-type matching.

We study the planar matching problem, defined by a symmetric random matrix with independent identically distributed entries, taking values zero and one. We show that the existence of a perfect planar matching structure is possible only above a certain critical density, p(c), of allowed contacts (i.e., of ones). Using a formulation of the problem in terms of Dyck paths and a matrix model of planar contact structures, we provide an analytical estimation for the value of the transition point, p(c), in the thermodynamic limit. This estimation is close to the critical value, p(c)≈0.379, obtained in numerical simulations based on an exact dynamical programming algorithm. We characterize the corresponding critical behavior of the model and discuss the relation of the perfect-imperfect matching transition to the known molten-glass transition in the context of random RNA secondary structure formation. In particular, we provide strong evidence supporting the conjecture that the molten-glass transition at T=0 occurs at p(c).