A Polymer Model for the Quantitative Reconstruction of Chromosome Architecture from HiC and GAM Data

Hi-C-guided many-polymer model to decipher 3D genome organization

We propose a high-throughput chromosome conformation capture data-based many-polymer model that allows us to generate an ensemble of multi-scale genome structures. We demonstrate the efficacy of our model by validating the generated structures against experimental measurements and employ them to address key questions regarding genome organization. Our model first confirms a significant correlation between chromosome size and nuclear positioning. Specifically, smaller chromosomes are distributed at the core region, whereas larger chromosomes are at the periphery, interacting with the nuclear envelope. The spatial distribution of A- and B-type compartments, which is nontrivial to infer from the corresponding high-throughput chromosome conformation capture maps alone, can also be elucidated using our model, accounting for an issue such as the effect of chromatin-lamina interaction on the compartmentalization of conventional and inverted nuclei. In accordance with imaging data, the overall shape of the 3D genome structures generated from our model displays significant variation. As a case study, we apply our method to the yellow fever mosquito genome, finding that the predicted morphology displays, on average, a more globular shape than the previously suggested spindle-like organization and that our prediction better aligns with the fluorescence in situ hybridization data. Our model has great potential to be extended to investigate many outstanding issues concerning 3D genome organization.

Delayed Excitations Induce Polymer Looping and Coherent Motion

We consider inhomogeneous polymers driven by energy-consuming active processes which encode temporal patterns of athermal kicks. We find that such temporal excitation programs, propagated by tension along the polymer, can effectively couple distinct polymer loci. Consequently, distant loci exhibit correlated motions that fold the polymer into specific conformations, as set by the local actions of the active processes and their distribution along the polymer. Interestingly, active kicks that are canceled out by a time-delayed echo can induce strong compaction of the active polymer.

Footprints of loop extrusion in statistics of intra-chromosomal distances: An analytically solvable model

Active loop extrusion-the process of formation of dynamically growing chromatin loops due to the motor activity of DNA-binding protein complexes-is a firmly established mechanism responsible for chromatin spatial organization at different stages of a cell cycle in eukaryotes and bacteria. The theoretical insight into the effect of loop extrusion on the experimentally measured statistics of chromatin conformation can be gained with an appropriately chosen polymer model. Here, we consider the simplest analytically solvable model of an interphase chromosome, which is treated as an ideal chain with disorder of sufficiently sparse random loops whose conformations are sampled from the equilibrium ensemble. This framework allows us to arrive at the closed-form analytical expression for the mean-squared distance between pairs of genomic loci, which is valid beyond the one-loop approximation in diagrammatic representation. In addition, we analyze the loop-induced deviation of chain conformations from the Gaussian statistics by calculating kurtosis of probability density of the pairwise separation vector. The presented results suggest the possible ways of estimating the characteristics of the loop extrusion process based on the experimental data on the scale-dependent statistics of intra-chromosomal pair-wise distances.

LoopSage: An energy-based Monte Carlo approach for the loop extrusion modeling of chromatin

The connection between the patterns observed in 3C-type experiments and the modeling of polymers remains unresolved. This paper presents a simulation pipeline that generates thermodynamic ensembles of 3D structures for topologically associated domain (TAD) regions by loop extrusion model (LEM). The simulations consist of two main components: a stochastic simulation phase, employing a Monte Carlo approach to simulate the binding positions of cohesins, and a dynamical simulation phase, utilizing these cohesins' positions to create 3D structures. In this approach, the system's total energy is the combined result of the Monte Carlo energy and the molecular simulation energy, which are iteratively updated. The structural maintenance of chromosomes (SMC) protein complexes are represented as loop extruders, while the CCCTC-binding factor (CTCF) locations on DNA sequence are modeled as energy minima on the Monte Carlo energy landscape. Finally, the spatial distances between DNA segments from ChIA-PET experiments are compared with the computer simulations, and we observe significant Pearson correlations between predictions and the real data. LoopSage model offers a fresh perspective on chromatin loop dynamics, allowing us to observe phase transition between sparse and condensed states in chromatin.

Replicating Chromosomes in Whole-Cell Models of Bacteria

Computational models of cells cannot be considered complete unless they include the most fundamental process of life, the replication of genetic material. In a recent study, we presented a computational framework to model systems of replicating bacterial chromosomes as polymers at 10 bp resolution with Brownian dynamics. This approach was used to investigate changes in chromosome organization during replication and extend the applicability of an existing whole-cell model (WCM) for a genetically minimal bacterium, JCVI-syn3A, to the entire cell cycle. To achieve cell-scale chromosome structures that are realistic, we modeled the chromosome as a self-avoiding homopolymer with bending and torsional stiffnesses that capture the essential mechanical properties of dsDNA in Syn3A. Additionally, the polymer interacts with ribosomes distributed according to cryo-electron tomograms of Syn3A. The polymer model was further augmented by computational models of loop extrusion by structural maintenance of chromosomes (SMC) protein complexes and topoisomerase action, and the modeling and analysis of multi-fork replication states.

Polymer folding through active processes recreates features of genome organization

From proteins to chromosomes, polymers fold into specific conformations that control their biological function. Polymer folding has long been studied with equilibrium thermodynamics, yet intracellular organization and regulation involve energy-consuming, active processes. Signatures of activity have been measured in the context of chromatin motion, which shows spatial correlations and enhanced subdiffusion only in the presence of adenosine triphosphate. Moreover, chromatin motion varies with genomic coordinate, pointing toward a heterogeneous pattern of active processes along the sequence. How do such patterns of activity affect the conformation of a polymer such as chromatin? We address this question by combining analytical theory and simulations to study a polymer subjected to sequence-dependent correlated active forces. Our analysis shows that a local increase in activity (larger active forces) can cause the polymer backbone to bend and expand, while less active segments straighten out and condense. Our simulations further predict that modest activity differences can drive compartmentalization of the polymer consistent with the patterns observed in chromosome conformation capture experiments. Moreover, segments of the polymer that show correlated active (sub)diffusion attract each other through effective long-ranged harmonic interactions, whereas anticorrelations lead to effective repulsions. Thus, our theory offers nonequilibrium mechanisms for forming genomic compartments, which cannot be distinguished from affinity-based folding using structural data alone. As a first step toward exploring whether active mechanisms contribute to shaping genome conformations, we discuss a data-driven approach.

A maximum-entropy model to predict 3D structural ensembles of chromatin from pairwise distances with applications to interphase chromosomes and structural variants

The principles that govern the organization of genomes, which are needed for an understanding of how chromosomes are packaged and function in eukaryotic cells, could be deciphered if the three-dimensional (3D) structures are known. Recently, single-cell imaging techniques have been developed to determine the 3D coordinates of genomic loci in vivo. Here, we introduce a computational method (Distance Matrix to Ensemble of Structures, DIMES), based on the maximum entropy principle, with experimental pairwise distances between loci as constraints, to generate a unique ensemble of 3D chromatin structures. Using the ensemble of structures, we quantitatively account for the distribution of pairwise distances, three-body co-localization, and higher-order interactions. The DIMES method can be applied to both small and chromosome-scale imaging data to quantify the extent of heterogeneity and fluctuations in the shapes across various length scales. We develop a perturbation method in conjunction with DIMES to predict the changes in 3D structures from structural variations. Our method also reveals quantitative differences between the 3D structures inferred from Hi-C and those measured in imaging experiments. Finally, the physical interpretation of the parameters extracted from DIMES provides insights into the origin of phase separation between euchromatin and heterochromatin domains.

Dissecting the cosegregation probability from genome architecture mapping

Genome architecture mapping (GAM) is a recently developed methodology that offers the cosegregation probability of two genomic segments from an ensemble of thinly sliced nuclear profiles, enabling us to probe and decipher three-dimensional chromatin organization. The cosegregation probability from GAM binned at 1 Mb, which thus probes the length scale associated with the genomic separation greater than 1 Mb, is, however, not identical to the contact probability obtained from Hi-C, and its correlation with interlocus distance measured with fluorescence in situ hybridization is not so good as the contact probability. In this study, by using a polymer-based model of chromatins, we derive a theoretical expression of the cosegregation probability as well as that of the contact probability and carry out quantitative analyses of how they differ from each other. The results from our study, validated with in silico GAM analysis on three-dimensional genome structures from fluorescence in situ hybridization, suggest that to attain strong correlation with the interlocus distance, a properly normalized version of cosegregation probability needs to be calculated based on a large number of nuclear slices (n>103).

Uncovering the Quantitative Relationships Among Chromosome Fluctuations, Epigenetics, and Gene Expressions of Transdifferentiation on Waddington Landscape

The 3D spatial organization of the chromosomes appears to be linked to the gene function, which is cell type-specific. The chromosome structural ensemble switching model (CSESM) is developed by employing a heteropolymer model on different cell types and the important quantitative relationships among the chromosome ensemble, the epigenetic marks, and the gene expressions are uncovered, that both chromosome fluctuation and epigenetic marks have strong linear correlations with the gene expressions. The results support that the two compartments have different behaviors, corresponding to the relatively sparse and fluctuating phase (compartment A) and the relatively dense and stable phase (compartment B). Importantly, through the investigation of the transdifferentiation processes between the peripheral blood mononuclear cell (PBMC) and the bipolar neuron (BN), a quantitative description for the transdifferentiation is provided, which can be linked to the Waddington landscape. In addition, compared to the direct transdifferentiation between PBMC and BN, the transdifferentiation via the intermediate state neural progenitor cell (NPC) follows a different path (an "uphill" followed by a "downhill"). These theoretical studies bridge the gap among the chromosome fluctuations/ensembles, the epigenetics, and gene expressions in determining the cell fate.

Extracting multi-way chromatin contacts from Hi-C data

There is a growing realization that multi-way chromatin contacts formed in chromosome structures are fundamental units of gene regulation. However, due to the paucity and complexity of such contacts, it is challenging to detect and identify them using experiments. Based on an assumption that chromosome structures can be mapped onto a network of Gaussian polymer, here we derive analytic expressions for n-body contact probabilities (n > 2) among chromatin loci based on pairwise genomic contact frequencies available in Hi-C, and show that multi-way contact probability maps can in principle be extracted from Hi-C. The three-body (triplet) contact probabilities, calculated from our theory, are in good correlation with those from measurements including Tri-C, MC-4C and SPRITE. Maps of multi-way chromatin contacts calculated from our analytic expressions can not only complement experimental measurements, but also can offer better understanding of the related issues, such as cell-line dependent assemblies of multiple genes and enhancers to chromatin hubs, competition between long-range and short-range multi-way contacts, and condensates of multiple CTCF anchors.

The 3D Genome: From Structure to Function

The genome is the most functional part of a cell, and genomic contents are organized in a compact three-dimensional (3D) structure. The genome contains millions of nucleotide bases organized in its proper frame. Rapid development in genome sequencing and advanced microscopy techniques have enabled us to understand the 3D spatial organization of the genome. Chromosome capture methods using a ligation approach and the visualization tool of a 3D genome browser have facilitated detailed exploration of the genome. Topologically associated domains (TADs), lamin-associated domains, CCCTC-binding factor domains, cohesin, and chromatin structures are the prominent identified components that encode the 3D structure of the genome. Although TADs are the major contributors to 3D genome organization, they are absent in Arabidopsis. However, a few research groups have reported the presence of TAD-like structures in the plant kingdom.

A Hi-C data-integrated model elucidates E. coli chromosome's multiscale organization at various replication stages

The chromosome of Escherichia coli is riddled with multi-faceted complexity. The emergence of chromosome conformation capture techniques are providing newer ways to explore chromosome organization. Here we combine a beads-on-a-spring polymer-based framework with recently reported Hi-C data for E. coli chromosome, in rich growth condition, to develop a comprehensive model of its chromosome at 5 kb resolution. The investigation focuses on a range of diverse chromosome architectures of E. coli at various replication states corresponding to a collection of cells, individually present in different stages of cell cycle. The Hi-C data-integrated model captures the self-organization of E. coli chromosome into multiple macrodomains within a ring-like architecture. The model demonstrates that the position of oriC is dependent on architecture and replication state of chromosomes. The distance profiles extracted from the model reconcile fluorescence microscopy and DNA-recombination assay experiments. Investigations into writhe of the chromosome model reveal that it adopts helix-like conformation with no net chirality, earlier hypothesized in experiments. A genome-wide radius of gyration map captures multiple chromosomal interaction domains and identifies the precise locations of rrn operons in the chromosome. We show that a model devoid of Hi-C encoded information would fail to recapitulate most genomic features unique to E. coli.

A unified framework for inferring the multi-scale organization of chromatin domains from Hi-C

Chromosomes are giant chain molecules organized into an ensemble of three-dimensional structures characterized with its genomic state and the corresponding biological functions. Despite the strong cell-to-cell heterogeneity, the cell-type specific pattern demonstrated in high-throughput chromosome conformation capture (Hi-C) data hints at a valuable link between structure and function, which makes inference of chromatin domains (CDs) from the pattern of Hi-C a central problem in genome research. Here we present a unified method for analyzing Hi-C data to determine spatial organization of CDs over multiple genomic scales. By applying statistical physics-based clustering analysis to a polymer physics model of the chromosome, our method identifies the CDs that best represent the global pattern of correlation manifested in Hi-C. The multi-scale intra-chromosomal structures compared across different cell types uncover the principles underlying the multi-scale organization of chromatin chain: (i) Sub-TADs, TADs, and meta-TADs constitute a robust hierarchical structure. (ii) The assemblies of compartments and TAD-based domains are governed by different organizational principles. (iii) Sub-TADs are the common building blocks of chromosome architecture. Our physically principled interpretation and analysis of Hi-C not only offer an accurate and quantitative view of multi-scale chromatin organization but also help decipher its connections with genome function.

An array of square blocks and checkerboard patterns characteristic to Hi-C data reflects the multi-scale organization of the chromatin chain. Deciphering the structures of chromatin domains from Hi-C and associating them with genome function are open problems of great importance in genome research. However, most existing methods are specialized in finding domains at different scales, making it difficult to integrate the solutions. Here we develop a unified framework for modeling and inferring domain structures over multiple scales, based on a physical model of the chromosome that reflects its nature as a three-dimensional object. Our method efficiently explores the space of domain solutions at different genomic scales, and systematically infers the chromatin domains over multiple scales from Hi-C data by employing a single tuning parameter. Our principled interpretation of Hi-C not only offers a quantitative view of multi-scale chromatin organization but also helps understand its connections with genome function.

Self assembly of model polymers into biological random networks

The properties of biological networks, such as those found in the ocular lens capsule, are difficult to study without simplified models. Model polymers are developed, inspired by "worm-like" curve models, that are shown to spontaneously self assemble to form networks similar to those observed experimentally in biological systems. These highly simplified coarse-grained models allow the self assembly process to be studied on near-realistic time-scales. Metrics are developed (using a polygon-based framework) which are useful for describing simulated networks and can also be applied to images of real networks. These metrics are used to show the range of control that the computational polymer model has over the networks, including the polygon structure and short range order. The structure of the simulated networks are compared to previous simulation work and microscope images of real networks. The network structure is shown to be a function of the interaction strengths, cooling rates and external pressure. In addition, "pre-tangled" network structures are introduced and shown to significantly influence the subsequent network structure. The network structures obtained fit into a region of the network landscape effectively inaccessible to random (entropically-driven) networks but which are occupied by experimentally-derived configurations.

Computational approaches for inferring 3D conformations of chromatin from chromosome conformation capture data

Chromosome conformation capture (3C) and its variants are powerful experimental techniques for probing intra- and inter-chromosomal interactions within cell nuclei at high resolution and in a high-throughput, quantitative manner. The contact maps derived from such experiments provide an avenue for inferring the 3D spatial organization of the genome. This review provides an overview of the various computational methods developed in the past decade for addressing the very important but challenging problem of deducing the detailed 3D structure or structure population of chromosomal domains, chromosomes, and even entire genomes from 3C contact maps.

Toward understanding the dynamic state of 3D genome

The three-dimensional (3D) genome organization and its role in biological activities have been investigated for over a decade in the field of cell biology. Recent studies using live-imaging and polymer simulation have suggested that the higher-order chromatin structures are dynamic; the stochastic fluctuations of nucleosomes and genomic loci cannot be captured by bulk-based chromosome conformation capture techniques (Hi-C). In this review, we focus on the physical nature of the 3D genome architecture. We first describe how to decode bulk Hi-C data with polymer modeling. We then introduce our recently developed PHi-C method, a computational tool for modeling the fluctuations of the 3D genome organization in the presence of stochastic thermal noise. We also present another new method that analyzes the dynamic rheology property (represented as microrheology spectra) as a measure of the flexibility and rigidity of genomic regions over time. By applying these methods to real Hi-C data, we highlighted a temporal hierarchy embedded in the 3D genome organization; chromatin interaction boundaries are more rigid than the boundary interior, while functional domains emerge as dynamic fluctuations within a particular time interval. Our methods may bridge the gap between live-cell imaging and Hi-C data and elucidate the nature of the dynamic 3D genome organization.

Microrheology for Hi-C Data Reveals the Spectrum of the Dynamic 3D Genome Organization

The one-dimensional information of genomic DNA is hierarchically packed inside the eukaryotic cell nucleus and organized in a three-dimensional (3D) space. Genome-wide chromosome conformation capture (Hi-C) methods have uncovered the 3D genome organization and revealed multiscale chromatin domains of compartments and topologically associating domains (TADs). Moreover, single-nucleosome live-cell imaging experiments have revealed the dynamic organization of chromatin domains caused by stochastic thermal fluctuations. However, the mechanism underlying the dynamic regulation of such hierarchical and structural chromatin units within the microscale thermal medium remains unclear. Microrheology is a way to measure dynamic viscoelastic properties coupling between thermal microenvironment and mechanical response. Here, we propose a new, to our knowledge, microrheology for Hi-C data to analyze the dynamic compliance property as a measure of rigidness and flexibility of genomic regions along with the time evolution. Our method allows the conversion of an Hi-C matrix into the spectrum of the dynamic rheological property along the genomic coordinate of a single chromosome. To demonstrate the power of the technique, we analyzed Hi-C data during the neural differentiation of mouse embryonic stem cells. We found that TAD boundaries behave as more rigid nodes than the intra-TAD regions. The spectrum clearly shows the dynamic viscoelasticity of chromatin domain formation at different timescales. Furthermore, we characterized the appearance of synchronous and liquid-like intercompartment interactions in differentiated cells. Together, our microrheology data derived from Hi-C data provide physical insights into the dynamics of the 3D genome organization.

PHi-C: deciphering Hi-C data into polymer dynamics

Genomes are spatiotemporally organized within the cell nucleus. Genome-wide chromosome conformation capture (Hi-C) technologies have uncovered the 3D genome organization. Furthermore, live-cell imaging experiments have revealed that genomes are functional in 4D. Although computational modeling methods can convert 2D Hi-C data into population-averaged static 3D genome models, exploring 4D genome nature based on 2D Hi-C data remains lacking. Here, we describe a 4D simulation method, PHi-C (polymer dynamics deciphered from Hi-C data), that depicts 4D genome features from 2D Hi-C data by polymer modeling. PHi-C allows users to interpret 2D Hi-C data as physical interaction parameters within single chromosomes. The physical interaction parameters can then be used in the simulations and analyses to demonstrate dynamic characteristics of genomic loci and chromosomes as observed in live-cell imaging experiments. PHi-C is available at https://github.com/soyashinkai/PHi-C.

The Functional 3D Organization of Unicellular Genomes

Genome conformation capture techniques permit a systematic investigation into the functional spatial organization of genomes, including functional aspects like assessing the co-localization of sets of genomic elements. For example, the co-localization of genes targeted by a transcription factor (TF) within a transcription factory. We quantify spatial co-localization using a rigorous statistical model that measures the enrichment of a subset of elements in neighbourhoods inferred from Hi-C data. We also control for co-localization that can be attributed to genomic order. We systematically apply our open-sourced framework, spatial-mHG, to search for spatial co-localization phenomena in multiple unicellular Hi-C datasets with corresponding genomic annotations. Our biological findings shed new light on the functional spatial organization of genomes, including: In C. crescentus, DNA replication genes reside in two genomic clusters that are spatially co-localized. Furthermore, these clusters contain similar gene copies and lay in genomic vicinity to the ori and ter sequences. In S. cerevisae, Ty5 retrotransposon family element spatially co-localize at a spatially adjacent subset of telomeres. In N. crassa, both Proteasome lid subcomplex genes and protein refolding genes jointly spatially co-localize at a shared location. An implementation of our algorithms is available online.

Inference of chromosome 3D structures from GAM data by a physics computational approach

The combination of modelling and experimental advances can provide deep insights for understanding chromatin 3D organization and ultimately its underlying mechanisms. In particular, models of polymer physics can help comprehend the complexity of genomic contact maps, as those emerging from technologies such as Hi-C, GAM or SPRITE. Here we discuss a method to reconstruct 3D structures from Genome Architecture Mapping (GAM) data, based on PRISMR, a computational approach introduced to find the minimal polymer model best describing Hi-C input data from only polymer physics. After recapitulating the PRISMR procedure, we describe how we extended it for treating GAM data. We successfully test the method on a 6 Mb region around the Sox9 gene and, at a lower resolution, on the whole chromosome 7 in mouse embryonic stem cells. The PRISMR derived 3D structures from GAM co-segregation data are finally validated against independent Hi-C contact maps. The method results to be versatile and robust, hinting that it can be similarly applied to different experimental data, such as SPRITE or microscopy distance data.

Many-body contacts in fractal polymer chains and fractional Brownian trajectories

We calculate the probabilities that a trajectory of a fractional Brownian motion with arbitrary fractal dimension d_{f} visits the same spot n≥3 times, at given moments t_{1},...,t_{n}, and obtain a determinant expression for these probabilities in terms of a displacement-displacement covariance matrix. Except for the standard Brownian trajectories with d_{f}=2, the resulting many-body contact probabilities cannot be factorized into a product of single-loop contributions. Within a Gaussian network model of a self-interacting polymer chain, which we suggested recently [K. Polovnikov et al., Soft Matter 14, 6561 (2018)1744-683X10.1039/C8SM00785C], the probabilities we calculate here can be interpreted as probabilities of multibody contacts in a fractal polymer conformation with the same fractal dimension d_{f}. This Gaussian approach, which implies a mapping from fractional Brownian motion trajectories to polymer conformations, can be used as a semiquantitative model of polymer chains in topologically stabilized conformations, e.g., in melts of unconcatenated rings or in the chromatin fiber, which is the material medium containing genetic information. The model presented here can be used, therefore, as a benchmark for interpretation of the data of many-body contacts in genomes, which we expect to be available soon in, e.g., Hi-C experiments.