A statistical model of intra-chromosome contact maps

Rich Chromatin Structure Prediction from Hi-C Data

Recent studies involving the 3-dimensional conformation of chromatin have revealed the important role it has to play in different processes within the cell. These studies have also led to the discovery of densely interacting segments of the chromosome, called topologically associating domains. The accurate identification of these domains from Hi-C interaction data is an interesting and important computational problem for which numerous methods have been proposed. Unfortunately, most existing algorithms designed to identify these domains assume that they are non-overlapping whereas there is substantial evidence to believe a nested structure exists. We present a methodology to predict hierarchical chromatin domains using chromatin conformation capture data. Our method predicts domains at different resolutions, calculated using intrinsic properties of the chromatin data, and effectively clusters these to construct the hierarchy. At each individual level, the domains are non-overlapping in such a way that the intra-domain interaction frequencies are maximized. We show that our predicted structure is highly enriched for actively transcribing housekeeping genes and various chromatin markers, including CTCF, around the domain boundaries. We also show that large-scale domains, at multiple resolutions within our hierarchy, are conserved across cell types and species. We also provide comparisons against existing tools for extracting hierarchical domains. Our software, Matryoshka, is written in C++11 and licensed under GPL v3; it is available at https://github.com/COMBINE-lab/matryoshka.

The Rabl configuration limits topological entanglement of chromosomes in budding yeast

The three dimensional organization of genomes remains mostly unknown due to their high degree of condensation. Biophysical studies predict that condensation promotes the topological entanglement of chromatin fibers and the inhibition of function. How organisms balance between functionally active genomes and a high degree of condensation remains to be determined. Here we hypothesize that the Rabl configuration, characterized by the attachment of centromeres and telomeres to the nuclear envelope, helps to reduce the topological entanglement of chromosomes. To test this hypothesis we developed a novel method to quantify chromosome entanglement complexity in 3D reconstructions obtained from Chromosome Conformation Capture (CCC) data. Applying this method to published data of the yeast genome, we show that computational models implementing the attachment of telomeres or centromeres alone are not sufficient to obtain the reduced entanglement complexity observed in 3D reconstructions. It is only when the centromeres and telomeres are attached to the nuclear envelope (i.e. the Rabl configuration) that the complexity of entanglement of the genome is comparable to that of the 3D reconstructions. We therefore suggest that the Rabl configuration is an essential player in the simplification of the entanglement of chromatin fibers.

Effective Hamiltonian of topologically stabilized polymer states

Topologically stabilized polymer conformations in melts of nonconcatenated polymer rings and crumpled globules are considered to be a good candidate for the description of the spatial structure of mitotic chromosomes. Despite significant efforts, the microscopic Hamiltonian capable of describing such systems still remains unknown. We describe a polymer conformation by a Gaussian network - a system with a Hamiltonian quadratic in all coordinates - and show that by tuning interaction constants, one can obtain equilibrium conformations with any fractal dimension between 2 (an ideal polymer chain) and 3 (a crumpled globule). Monomer-to-monomer distances in topologically stabilized states, according to available numerical data, fit very well the Gaussian distribution, giving an additional argument in support of the quadratic Hamiltonian model. Mathematically, the polymer conformations are mapped onto the trajectories of a subdiffusive fractal Brownian particle. Moreover, we explicitly show that the quadratic Hamiltonian with a hierarchical set of coupling constants provides the microscopic background for the description of the path integral of the fractional Brownian motion with an algebraically decaying kernel.

Spontaneous domain formation in disordered copolymers as a mechanism for chromosome structuring

Motivated by the problem of domain formation in chromosomes, we studied a co-polymer model where only a subset of the monomers feel attractive interactions. These monomers are displaced randomly from a regularly-spaced pattern, thus introducing some quenched disorder in the system. Previous work has shown that in the case of regularly-spaced interacting monomers this chain can fold into structures characterized by multiple distinct domains of consecutive segments. In each domain, attractive interactions are balanced by the entropy cost of forming loops. We show by advanced replica-exchange simulations that adding disorder in the position of the interacting monomers further stabilizes these domains. The model suggests that the partitioning of the chain into well-defined domains of consecutive monomers is a spontaneous property of heteropolymers. In the case of chromosomes, evolution could have acted on the spacing of interacting monomers to modulate in a simple way the underlying domains for functional reasons.

Perspectives: using polymer modeling to understand the formation and function of nuclear compartments

Compartmentalization is a ubiquitous feature of cellular function. In the nucleus, early observations revealed a non-random spatial organization of the genome with a large-scale segregation between transcriptionally active—euchromatin—and silenced—heterochromatin—parts of the genome. Recent advances in genome-wide mapping and imaging techniques have strikingly improved the resolution at which nuclear genome folding can be analyzed and have revealed a multiscale spatial compartmentalization with increasing evidences that such compartment may indeed result from and participate to genome function. Understanding the underlying mechanisms of genome folding and in particular the link to gene regulation requires a cross-disciplinary approach that combines the new high-resolution techniques with computational modeling of chromatin and chromosomes. In this perspective article, we first present how the copolymer theoretical framework can account for the genome compartmentalization. We then suggest, in a second part, that compartments may act as a “nanoreactor,” increasing the robustness of either activation or repression by enhancing the local concentration of regulators. We conclude with the need to develop a new framework, namely the “living chromatin” model that will allow to explicitly investigate the coupling between spatial compartmentalization and gene regulation.

Coupling 1D modifications and 3D nuclear organization: data, models and function

Over the past decade, advances in molecular methods have strikingly improved the resolution at which nuclear genome folding can be analyzed. This revealed a wealth of conserved features organizing the one dimensional DNA molecule into tridimensional nuclear domains. In this review, we briefly summarize the main findings and highlight how models based on polymer physics shed light on the principles underlying the formation of these domains. Finally, we discuss the mechanistic similarities allowing self-organization of these structures and the functional importance of these in the maintenance of transcriptional programs.

The folding landscape of the epigenome

The role of the spatial organization of chromatin in gene regulation is a long-standing but still open question. Experimentally it has been shown that the genome is segmented into epigenomic chromatin domains that are organized into hierarchical sub-nuclear spatial compartments. However, whether this non-random spatial organization only reflects or indeed contributes-and how-to the regulation of genome function remains to be elucidated. To address this question, we recently proposed a quantitative description of the folding properties of the fly genome as a function of its epigenomic landscape using a polymer model with epigenomic-driven attractions. We propose in this article, to characterize more deeply the physical properties of the 3D epigenome folding. Using an efficient lattice version of the original block copolymer model, we study the structural and dynamical properties of chromatin and show that the size of epigenomic domains and asymmetries in sizes and in interaction strengths play a critical role in the chromatin organization. Finally, we discuss the biological implications of our findings. In particular, our predictions are quantitatively compatible with experimental data and suggest a different mean of self-interaction in euchromatin versus heterochromatin domains.

Hierarchical block matrices as efficient representations of chromosome topologies and their application for 3C data integration

MOTIVATION

Recent advancements in molecular methods have made it possible to capture physical contacts between multiple chromatin fragments. The resulting association matrices provide a noisy estimate for average spatial proximity that can be used to gain insights into the genome organization inside the nucleus. However, extracting topological information from these data is challenging and their integration across resolutions is still poorly addressed. Recent findings suggest that a hierarchical approach could be advantageous for addressing these challenges.

RESULTS

We present an algorithmic framework, which is based on hierarchical block matrices (HBMs), for topological analysis and integration of chromosome conformation capture (3C) data. We first describe chromoHBM, an algorithm that compresses high-throughput 3C (HiT-3C) data into topological features that are efficiently summarized with an HBM representation. We suggest that instead of directly combining HiT-3C datasets across resolutions, which is a difficult task, we can integrate their HBM representations, and describe chromoHBM-3C, an algorithm which merges HBMs. Since three-dimensional (3D) reconstruction can also benefit from topological information, we further present chromoHBM-3D, an algorithm which exploits the HBM representation in order to gradually introduce topological constraints to the reconstruction process. We evaluate our approach in light of previous image microscopy findings and epigenetic data, and show that it can relate multiple spatial scales and provide a more complete view of the 3D genome architecture.

AVAILABILITY AND IMPLEMENTATION

The presented algorithms are available from: https://github.com/yolish/hbm

CONTACT

ys388@cam.ac.uk or pl219@cam.ac.uk

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Coalescence Model for Crumpled Globules Formed in Polymer Collapse

The rapid collapse of a polymer, due to external forces or changes in solvent, yields a long-lived "crumpled globule." The conjectured fractal structure shaped by hierarchical collapse dynamics has proved difficult to establish, even with large simulations. To unravel this puzzle, we study a coarse-grained model of in-falling spherical blobs that coalesce upon contact. Distances between pairs of monomers are assigned upon their initial coalescence, and do not "equilibrate" subsequently. Surprisingly, the model reproduces quantitatively the dependence of distance on segment length, suggesting that the slow approach to scaling is related to the wide distribution of blob sizes.