Independent sets of a given size and structure in the hypercube

We determine the asymptotics of the number of independent sets of size $\lfloor \beta 2^{d-1} \rfloor$ in the discrete hypercube $Q_d = \{0,1\}^d$ for any fixed $\beta \in (0,1)$ as $d \to \infty$ , extending a result of Galvin for $\beta \in (1-1/\sqrt{2},1)$ . Moreover, we prove a multivariate local central limit theorem for structural features of independent sets in $Q_d$ drawn according to the hard-core model at any fixed fugacity $\lambda>0$ . In proving these results we develop several general tools for performing combinatorial enumeration using polymer models and the cluster expansion from statistical physics along with local central limit theorems.

Effects of alignment activity on the collapse kinetics of a flexible polymer

The nonequilibrium kinetics of various biological filaments can be understood within the framework of active polymer models. Increasing the alignment activity of monomers changes the intermediates during collapse from pearl-necklace to dumbbell-like conformations.

Fast Algorithms for General Spin Systems on Bipartite Expanders

A spin system is a framework in which the vertices of a graph are assigned spins from a finite set. The interactions between neighbouring spins give rise to weights, so a spin assignment can also be viewed as a weighted graph homomorphism. The problem of approximating the partition function (the aggregate weight of spin assignments) or of sampling from the resulting probability distribution is typically intractable for general graphs. In this work, we consider arbitrary spin systems on bipartite expander Δ-regular graphs, including the canonical class of bipartite random Δ-regular graphs. We develop fast approximate sampling and counting algorithms for general spin systems whenever the degree and the spectral gap of the graph are sufficiently large. Roughly, this guarantees that the spin system is in the so-called low-temperature regime. Our approach generalises the techniques of Jenssen et al. and Chen et al. by showing that typical configurations on bipartite expanders correspond to “bicliques” of the spin system; then, using suitable polymer models, we show how to sample such configurations and approximate the partition function in Õ( n 2 ) time, where n is the size of the graph.

Perspectives for the reconstruction of 3D chromatin conformation using single cell Hi-C data

Construction of chromosomes 3D models based on single cell Hi-C data constitute an important challenge. We present a reconstruction approach, DPDchrom, that incorporates basic knowledge whether the reconstructed conformation should be coil-like or globular and spring relaxation at contact sites. In contrast to previously published protocols, DPDchrom can naturally form globular conformation due to the presence of explicit solvent. Benchmarking of this and several other methods on artificial polymer models reveals similar reconstruction accuracy at high contact density and DPDchrom advantage at low contact density. To compare 3D structures insensitively to spatial orientation and scale, we propose the Modified Jaccard Index. We analyzed two sources of the contact dropout: contact radius change and random contact sampling. We found that the reconstruction accuracy exponentially depends on the number of contacts per genomic bin allowing to estimate the reconstruction accuracy in advance. We applied DPDchrom to model chromosome configurations based on single-cell Hi-C data of mouse oocytes and found that these configurations differ significantly from a random one, that is consistent with other studies.

Estimation of effective concentrations enforced by complex linker architectures using conformational ensembles.

Proteins and protein assemblies often tether interaction partners to strengthen interactions, to regulate activity through auto-inhibition or -activation, or to boost enzyme catalysis. Tethered reactions are regulated by the architecture of tether, which define an effective concentration of the interactors. Effective concentrations can be estimated theoretically for simple linkers via polymer models, but there is currently no general method for estimating effective concentrations for complex linker architectures consisting of both flexible and folded domains. We describe how effective concentrations can be estimated computationally for any protein linker architecture by defining a realistic conformational ensemble. We benchmark against prediction from a worm-like chain and values measured by competition experiments, and find minor differences likely due to excluded volume effects. Systematic variation of the properties of flexible and folded segments show that the effective concentration is mainly determined by the combination of the total length of flexible segments and the distance between termini of the folded domains. We show that a folded domain in a disordered linker can increase the effective concentration beyond what can be achieved by a fully disordered linker by focusing the end-to-end distance at the appropriate spacing. This suggest that complex linker architecture may have advantages over simple flexible linker, and emphasize that annotation as a linker should depend on the molecular context.

Polymer models are a versatile tool to study chromatin 3D organization

The development of new experimental technologies is opening the way to a deeper investigation of the three-dimensional organization of chromosomes inside the cell nucleus. Genome architecture is linked to vital functional purposes, yet a full comprehension of the mechanisms behind DNA folding is still far from being accomplished. Theoretical approaches based on polymer physics have been employed to understand the complexity of chromatin architecture data and to unveil the basic mechanisms shaping its structure. Here, we review some recent advances in the field to discuss how Polymer Physics, combined with numerical Molecular Dynamics simulation and Machine Learning based inference, can capture important aspects of genome organization, including the description of tissue-specific structural rearrangements, the detection of novel, regulatory-linked architectural elements and the structural variability of chromatin at the single-cell level.

Molecular-Level Investigation of Cycloaliphatic Epoxidised Ionic Liquids as a New Generation of Monomers for Versatile Poly(Ionic Liquids)

Recently, a new generation of polymerised ionic liquids with high thermal stability and good mechanical performances has been designed through novel and versatile cycloaliphatic epoxy-functionalised ionic liquids (CEILs). From these first promising results and unexplored chemical structures in terms of final properties of the PILs, a computational approach based on molecular dynamics simulations has been developed to generate polymer models and predict the thermo–mechanical properties (e.g., glass transition temperature and Young’s modulus) of experimentally investigated CEILs for producing multi-functional polymer materials. Here, a completely reproducible and reliable computational protocol is provided to design, test and tune poly(ionic liquids) based on epoxidised ionic liquid monomers for future multi-functional thermoset polymers.