Spatial confinement is a major determinant of the folding landscape of human chromosomes

The scales, mechanisms, and dynamics of the genome architecture

Even when split into several chromosomes, DNA molecules that make up our genome are too long to fit into the cell nuclei unless massively folded. Such folding must accommodate the need for timely access to selected parts of the genome by transcription factors, RNA polymerases, and DNA replication machinery. Here, we review our current understanding of the genome folding inside the interphase nuclei. We consider the resulting genome architecture at three scales with a particular focus on the intermediate (meso) scale and summarize the insights gained from recent experimental observations and diverse computational models.

Efficient Hi-C inversion facilitates chromatin folding mechanism discovery and structure prediction

Genome-wide chromosome conformation capture (Hi-C) experiments have revealed many structural features of chromatin across multiple length scales. Further understanding genome organization requires relating these discoveries to the mechanisms that establish chromatin structures and reconstructing these structures in three dimensions, but both objectives are difficult to achieve with existing algorithms that are often computationally expensive. To alleviate this challenge, we present an algorithm that efficiently converts Hi-C data into contact energies, which measure the interaction strength between genomic loci brought into proximity. Contact energies are local quantities unaffected by the topological constraints that correlate Hi-C contact probabilities. Thus, extracting contact energies from Hi-C contact probabilities distills the biologically unique information contained in the data. We show that contact energies reveal the location of chromatin loop anchors, support a phase separation mechanism for genome compartmentalization, and parameterize polymer simulations that predict three-dimensional chromatin structures. Therefore, we anticipate that contact energy extraction will unleash the full potential of Hi-C data and that our inversion algorithm will facilitate the widespread adoption of contact energy analysis.

Efficient Hi-C inversion facilitates chromatin folding mechanism discovery and structure prediction

Genome-wide chromosome conformation capture (Hi-C) experiments have revealed many structural features of chromatin across multiple length scales. Further understanding genome organization requires relating these discoveries to the mechanisms that establish chromatin structures and reconstructing these structures in three dimensions, but both objectives are difficult to achieve with existing algorithms that are often computationally expensive. To alleviate this challenge, we present an algorithm that efficiently converts Hi-C data into contact energies, which measure the interaction strength between genomic loci brought into proximity. Contact energies are local quantities unaffected by the topological constraints that correlate Hi-C contact probabilities. Thus, extracting contact energies from Hi-C contact probabilities distills the biologically unique information contained in the data. We show that contact energies reveal the location of chromatin loop anchors, support a phase separation mechanism for genome compartmentalization, and parameterize polymer simulations that predict three-dimensional chromatin structures. Therefore, we anticipate that contact energy extraction will unleash the full potential of Hi-C data and that our inversion algorithm will facilitate the widespread adoption of contact energy analysis.

Gradient Nanoconfinement Facilitates Binding of Transcriptional Factor NF-κB to Histone- and Protamine-DNA Complexes

Mechanically induced chromosome reorganization plays important roles in transcriptional regulation. However, the interplay between chromosome reorganization and transcription activities is complicated, such that it is difficult to decipher the regulatory effects of intranuclear geometrical cues. Here, we simplify the system by introducing DNA, packaging proteins (i.e., histone and protamine), and transcription factor NF-κB into a well-defined fluidic chip with changing spatical confinement ranging from 100 to 500 nm. It is uncovered that strong nanoconfinement suppresses higher-order folding of histone- and protamine-DNA complexes, the fracture of which exposes buried DNA segments and causes increased quantities of NF-κB binding to the DNA chain. Overall, these results reveal a pathway of how intranuclear geometrical cues alter the open/closed state of a DNA-protein complex and therefore affect transcription activities: i.e., NF-κB binding.

Multivalent binding proteins can drive collapse and reswelling of chromatin in confinement

Collapsed conformations of chromatin have been long suspected of being mediated by interactions with multivalent binding proteins, which can bring together distant sections of the chromatin fiber. In this study, we use Langevin dynamics simulation of a coarse grained chromatin polymer to show that the role of binding proteins can be more nuanced than previously suspected. In particular, for chromatin polymer in confinement, entropic forces can drive reswelling of collapsed chromatin with increasing binder concentrations, and this reswelling transition happens at physiologically relevant binder concentrations. Both the extent of collapse, and also of reswelling depends on the strength of confinement. We also study the kinetics of collapse and reswelling and show that both processes occur in similar timescales. We characterise this reswelling of chromatin in biologically relevant regimes and discuss the non-trivial role of multivalent binding proteins in mediating the spatial organisation of the genome.

Insights into the cell fate decision-making processes from chromosome structural reorganizations

The cell fate decision-making process, which provides the capability of a cell transition to a new cell type, involves the reorganizations of 3D genome structures. Currently, the high temporal resolution picture of how the chromosome structural rearrangements occur and further influence the gene activities during the cell-state transition is still challenging to acquire. Here, we study the chromosome structural reorganizations during the cell-state transitions among the pluripotent embryonic stem cell, the terminally differentiated normal cell, and the cancer cell using a nonequilibrium landscape-switching model implemented in the molecular dynamics simulation. We quantify the chromosome (de)compaction pathways during the cell-state transitions and find that the two pathways having the same destinations can merge prior to reaching the final states. The chromosomes at the merging states have similar structural geometries but can differ in long-range compartment segregation and spatial distribution of the chromosomal loci and genes, leading to cell-type-specific transition mechanisms. We identify the irreversible pathways of chromosome structural rearrangements during the forward and reverse transitions connecting the same pair of cell states, underscoring the critical roles of nonequilibrium dynamics in the cell-state transitions. Our results contribute to the understanding of the cell fate decision-making processes from the chromosome structural perspective.

Every gene everywhere all at once: High-precision measurement of 3D chromosome architecture with single-cell Hi-C

The three-dimensional (3D) structure of chromosomes influences essential biological processes such as gene expression, genome replication, and DNA damage repair and has been implicated in many developmental and degenerative diseases. In the past two centuries, two complementary genres of technology-microscopy, such as fluorescence in situ hybridization (FISH), and biochemistry, such as chromosome conformation capture (3C or Hi-C)-have revealed general principles of chromosome folding in the cell nucleus. However, the extraordinary complexity and cell-to-cell variability of the chromosome structure necessitate new tools with genome-wide coverage and single-cell precision. In the past decade, single-cell Hi-C emerges as a new approach that builds upon yet conceptually differs from bulk Hi-C assays. Instead of measuring population-averaged statistical properties of chromosome folding, single-cell Hi-C works as a proximity-based "biochemical microscope" that measures actual 3D structures of individual genomes, revealing features hidden in bulk Hi-C such as radial organization, multi-way interactions, and chromosome intermingling. Single-cell Hi-C has been used to study highly dynamic processes such as the cell cycle, cell-type-specific chromosome architecture ("structure types"), and structure-expression interplay, deepening our understanding of DNA organization and function.

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.

Modeling and simulation of cell nuclear architecture reorganization process

We develop a special phase field/diffusive interface method to model the nuclear architecture reorganization process. In particular, we use a Lagrange multiplier approach in the phase field model to preserve the specific physical and geometrical constraints for the biological events. We develop several efficient and robust linear and weakly nonlinear schemes for this new model. To validate the model and numerical methods, we present ample numerical simulations which in particular reproduce several processes of nuclear architecture reorganization from the experiment literature.

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.

Minimalistic 3D chromatin models: Sparse interactions in single cells drive the chromatin fold and form many-body units

Computational three-dimensional chromatin modeling has helped uncover principles of genome organization. Here, we discuss methods for modeling three-dimensional chromatin structures, with focus on a minimalistic polymer model which inverts population Hi-C into single-cell conformations. Utilizing only basic physical properties, this model reveals that a few specific Hi-C interactions can fold chromatin into conformations consistent with single-cell imaging, Dip-C, and FISH measurements. Aggregated single-cell chromatin conformations also reproduce Hi-C frequencies. This approach allows quantification of structural heterogeneity and discovery of many-body interaction units and has revealed additional insights, including (1) topologically associating domains as a byproduct of folding driven by specific interactions, (2) cell subpopulations with different structural scaffolds are developmental stage dependent, and (3) the functional landscape of many-body units within enhancer-rich regions. We also discuss these findings in relation to the genome structure-function relationship.

Multiscale modeling of genome organization with maximum entropy optimization

Three-dimensional (3D) organization of the human genome plays an essential role in all DNA-templated processes, including gene transcription, gene regulation, and DNA replication. Computational modeling can be an effective way of building high-resolution genome structures and improving our understanding of these molecular processes. However, it faces significant challenges as the human genome consists of over 6 × 109 base pairs, a system size that exceeds the capacity of traditional modeling approaches. In this perspective, we review the progress that has been made in modeling the human genome. Coarse-grained models parameterized to reproduce experimental data via the maximum entropy optimization algorithm serve as effective means to study genome organization at various length scales. They have provided insight into the principles of whole-genome organization and enabled de novo predictions of chromosome structures from epigenetic modifications. Applications of these models at a near-atomistic resolution further revealed physicochemical interactions that drive the phase separation of disordered proteins and dictate chromatin stability in situ. We conclude with an outlook on the opportunities and challenges in studying chromosome dynamics.

High-resolution single-cell 3D-models of chromatin ensembles during Drosophila embryogenesis

Single-cell chromatin studies provide insights into how chromatin structure relates to functions of individual cells. However, balancing high-resolution and genome wide-coverage remains challenging. We describe a computational method for the reconstruction of large 3D-ensembles of single-cell (sc) chromatin conformations from population Hi-C that we apply to study embryogenesis in Drosophila. With minimal assumptions of physical properties and without adjustable parameters, our method generates large ensembles of chromatin conformations via deep-sampling. Our method identifies specific interactions, which constitute 5-6% of Hi-C frequencies, but surprisingly are sufficient to drive chromatin folding, giving rise to the observed Hi-C patterns. Modeled sc-chromatins quantify chromatin heterogeneity, revealing significant changes during embryogenesis. Furthermore, >50% of modeled sc-chromatin maintain topologically associating domains (TADs) in early embryos, when no population TADs are perceptible. Domain boundaries become fixated during development, with strong preference at binding-sites of insulator-complexes upon the midblastula transition. Overall, high-resolution 3D-ensembles of sc-chromatin conformations enable further in-depth interpretation of population Hi-C, improving understanding of the structure-function relationship of genome organization.

Data-Driven Polymer Model for Mechanistic Exploration of Diploid Genome Organization

Chromosomes are positioned nonrandomly inside the nucleus to coordinate with their transcriptional activity. The molecular mechanisms that dictate the global genome organization and the nuclear localization of individual chromosomes are not fully understood. We introduce a polymer model to study the organization of the diploid human genome. It is data-driven because all parameters can be derived from Hi-C data; it is also a mechanistic model because the energy function is explicitly written out based on a few biologically motivated hypotheses. These two features distinguish the model from existing approaches and make it useful both for reconstructing genome structures and for exploring the principles of genome organization. We carried out extensive validations to show that simulated genome structures reproduce a wide variety of experimental measurements, including chromosome radial positions and spatial distances between homologous pairs. Detailed mechanistic investigations support the importance of both specific interchromosomal interactions and centromere clustering for chromosome positioning. We anticipate the polymer model, when combined with Hi-C experiments, to be a powerful tool for investigating large-scale rearrangements in genome structure upon cell differentiation and tumor progression.

History-dependent nonequilibrium conformations of a highly confined polymer globule in a sphere

Chromatin undergoes condensation-decondensation processes repeatedly during its cell lifetime. The spatial organization of chromatin in nucleus resembles the fractal globule, of which structure significantly differs from an equilibrium polymer globule. There have been efforts to develop a polymer globule model to describe the fractal globulelike structure of tightly packed chromatin in nucleus. However, the transition pathway of a polymer toward a globular state has been often ignored. Because biological systems are intrinsically in nonequilibrium states, the transition pathway that the chromatin would take before reaching the densely packaged globule should be of importance. In this study, by employing a simple polymer model and Langevin dynamics simulations, we investigate the conformational transition of a single polymer from a swollen coil to a compact globule. We aim to elucidate the effect of transition pathways on the final globular structure. We show that a fast collapse induces a nonequilibrium structure even without a specific intramolecular interaction and that its relaxation toward an equilibrium globule is extremely slow. Due to a strong confinement, the fractal globule never relaxes into an equilibrium state during our simulations such that the globular structure becomes dependent on the transition pathway.

A Lamin-Associated Chromatin Model for Chromosome Organization

We propose a simple model for chromatin organization based on the interaction of the chromatin fibers with lamin proteins along the nuclear membrane. Lamin proteins are known to be a major factor that influences chromatin organization and hence gene expression in the cells. We provide a quantitative understanding of lamin-associated chromatin organization in a crowded macromolecular environment by systematically varying the heteropolymer segment distribution and the strength of the lamin-chromatin attractive interaction. Our minimal polymer model reproduces the formation of lamin-associated-domains and provides an in silico tool for quantifying domain length distributions for different distributions of heteropolymer segments. We show that a Gaussian distribution of heteropolymer segments, coupled with strong lamin-chromatin interactions, can qualitatively reproduce observed length distributions of lamin-associated-domains. Further, lamin-mediated interaction can enhance the formation of chromosome territories as well as the organization of chromatin into tightly packed heterochromatin and the loosely packed gene-rich euchromatin regions.

Braiding topology and the energy landscape of chromosome organization proteins

Assemblies of structural maintenance of chromosomes (SMC) proteins and kleisin subunits are essential to chromosome organization and segregation across all kingdoms of life. While structural data exist for parts of the SMC-kleisin complexes, complete structures of the entire complexes have yet to be determined, making mechanistic studies difficult. Using an integrative approach that combines crystallographic structural information about the globular subdomains, along with coevolutionary information and an energy landscape optimized force field (AWSEM), we predict atomic-scale structures for several tripartite SMC-kleisin complexes, including prokaryotic condensin, eukaryotic cohesin, and eukaryotic condensin. The molecular dynamics simulations of the SMC-kleisin protein complexes suggest that these complexes exist as a broad conformational ensemble that is made up of different topological isomers. The simulations suggest a critical role for the SMC coiled-coil regions, where the coils intertwine with various linking numbers. The twist and writhe of these braided coils are coupled with the motion of the SMC head domains, suggesting that the complexes may function as topological motors. Opening, closing, and translation along the DNA of the SMC-kleisin protein complexes would allow these motors to couple to the topology of DNA when DNA is entwined with the braided coils.

CHROMATIX: computing the functional landscape of many-body chromatin interactions in transcriptionally active loci from deconvolved single cells

Chromatin interactions are important for gene regulation and cellular specialization. Emerging evidence suggests many-body spatial interactions play important roles in condensing super-enhancer regions into a cohesive transcriptional apparatus. Chromosome conformation studies using Hi-C are limited to pairwise, population-averaged interactions; therefore unsuitable for direct assessment of many-body interactions. We describe a computational model, CHROMATIX, which reconstructs ensembles of single-cell chromatin structures by deconvolving Hi-C data and identifies significant many-body interactions. For a diverse set of highly active transcriptional loci with at least 2 super-enhancers, we detail the many-body functional landscape and show DNase accessibility, POLR2A binding, and decreased H3K27me3 are predictive of interaction-enriched regions.

Mesoscale Liquid Model of Chromatin Recapitulates Nuclear Order of Eukaryotes

The nuclear envelope segregates the genome of Eukaryota from the cytoplasm. Within the nucleus, chromatin is further compartmentalized into architectures that change throughout the lifetime of the cell. Epigenetic patterns along the chromatin polymer strongly correlate with chromatin compartmentalization and, accordingly, also change during the cell life cycle and at differentiation. Recently, it has been suggested that subnuclear chromatin compartmentalization might result from a process of liquid-liquid phase separation orchestrated by the epigenetic marking and operated by proteins that bind to chromatin. Here, we translate these observations into a diffuse interface model of chromatin, which we named the mesoscale liquid model of nucleus. Using this streamlined continuum model of the genome, we study the large-scale rearrangements of chromatin that happen at different stages of the growth and senescence of the cell and during nuclear inversion events. In particular, we investigate the role of droplet diffusion, fluctuations, and heterochromatin-lamina interactions during nuclear remodeling. Our results indicate that the physical process of liquid-liquid phase separation, together with surface effects, is sufficient to recapitulate much of the large-scale morphology and dynamics of chromatin along the life cycle of cells.

Heterogeneous Loop Model to Infer 3D Chromosome Structures from Hi-C

Adapting a well-established formalism in polymer physics, we develop a minimalist approach to infer three-dimensional folding of chromatin from Hi-C data. The three-dimensional chromosome structures generated from our heterogeneous loop model (HLM) are used to visualize chromosome organizations that can substantiate the measurements from fluorescence in situ hybridization, chromatin interaction analysis by paired-end tag sequencing, and RNA-seq signals. We demonstrate the utility of the HLM with several case studies. Specifically, the HLM-generated chromosome structures, which reproduce the spatial distribution of topologically associated domains from fluorescence in situ hybridization measurement, show the phase segregation between two types of topologically associated domains explicitly. We discuss the origin of cell-type-dependent gene-expression level by modeling the chromatin globules of α-globin and SOX2 gene loci for two different cell lines. We also use the HLM to discuss how the chromatin folding and gene-expression level of Pax6 loci, associated with mouse neural development, are modulated by interactions with two enhancers. Finally, HLM-generated structures of chromosome 19 of mouse embryonic stem cells, based on single-cell Hi-C data collected over each cell-cycle phase, visualize changes in chromosome conformation along the cell-cycle. Given a contact frequency map between chromatic loci supplied from Hi-C, HLM is a computationally efficient and versatile modeling tool to generate chromosome structures that can complement interpreting other experimental data.

Predicting three-dimensional genome organization with chromatin states

We introduce a computational model to simulate chromatin structure and dynamics. Starting from one-dimensional genomics and epigenomics data that are available for hundreds of cell types, this model enables de novo prediction of chromatin structures at five-kilo-base resolution. Simulated chromatin structures recapitulate known features of genome organization, including the formation of chromatin loops, topologically associating domains (TADs) and compartments, and are in quantitative agreement with chromosome conformation capture experiments and super-resolution microscopy measurements. Detailed characterization of the predicted structural ensemble reveals the dynamical flexibility of chromatin loops and the presence of cross-talk among neighboring TADs. Analysis of the model's energy function uncovers distinct mechanisms for chromatin folding at various length scales and suggests a need to go beyond simple A/B compartment types to predict specific contacts between regulatory elements using polymer simulations.

Alterations in Chromatin Folding Patterns in Cancer Variant-Enriched Loci

In this study, we focus on the following question: do genomic regions enriched in cancer variant mutations have significantly different chromatin folding patterns? We utilize publicly available Hi-C data to characterize chromatin folding patterns in healthy (GM12878) and cancer (K562) cells based on status of A/B compartmentalization and random vs non-random chromatin physical interactions. We then perform statistical testing to assess if chromatin folding patterns in cancer variant-enriched loci are significantly different from non-enriched loci. Our results indicate that loci with cancer variant status have significantly altered (FDR < 0.05) chromatin folding patterns.

Computational Structural Biology: Successes, Future Directions, and Challenges

Computational biology has made powerful advances. Among these, trends in human health have been uncovered through heterogeneous 'big data' integration, and disease-associated genes were identified and classified. Along a different front, the dynamic organization of chromatin is being elucidated to gain insight into the fundamental question of genome regulation. Powerful conformational sampling methods have also been developed to yield a detailed molecular view of cellular processes. when combining these methods with the advancements in the modeling of supramolecular assemblies, including those at the membrane, we are finally able to get a glimpse into how cells' actions are regulated. Perhaps most intriguingly, a major thrust is on to decipher the mystery of how the brain is coded. Here, we aim to provide a broad, yet concise, sketch of modern aspects of computational biology, with a special focus on computational structural biology. We attempt to forecast the areas that computational structural biology will embrace in the future and the challenges that it may face. We skirt details, highlight successes, note failures, and map directions.

Probabilistic control of HIV latency and transactivation by the Tat gene circuit

The reservoir of HIV latently infected cells is the major obstacle for eradication of HIV infection. The "shock-and-kill" strategy proposed earlier aims to reduce the reservoir by activating cells out of latency. While the intracellular HIV Tat gene circuit is known to play important roles in controlling latency and its transactivation in HIV-infected cells, the detailed control mechanisms are not well understood. Here we study the mechanism of probabilistic control of the latent and the transactivated cell phenotypes of HIV-infected cells. We reconstructed the probability landscape, which is the probability distribution of the Tat gene circuit states, by directly computing the exact solution of the underlying chemical master equation. Results show that the Tat circuit exhibits a clear bimodal probability landscape (i.e., there are two distinct probability peaks, one associated with the latent cell phenotype and the other with the transactivated cell phenotype). We explore potential modifications to reactions in the Tat gene circuit for more effective transactivation of latent cells (i.e., the shock-and-kill strategy). Our results suggest that enhancing Tat acetylation can dramatically increase Tat and viral production, while increasing the Tat-transactivation response binding affinity can transactivate latent cells more rapidly than other manipulations. Our results further explored the "block and lock" strategy toward a functional cure for HIV. Overall, our study demonstrates a general approach toward discovery of effective therapeutic strategies and druggable targets by examining control mechanisms of cell phenotype switching via exactly computed probability landscapes of reaction networks.

Crowding, Entropic Forces, and Confinement: Crucial Factors for Structures and Functions in the Cell Nucleus

The view of the cell nucleus as a crowded system of colloid particles and that chromosomes are giant self-avoiding polymers is stimulating rapid advances in our understanding of its structure and activities, thanks to concepts and experimental methods from colloid, polymer, soft matter, and nano sciences and to increased computational power for simulating macromolecules and polymers. This review summarizes current understanding of some characteristics of the molecular environment in the nucleus, of how intranuclear compartments are formed, and of how the genome is highly but precisely compacted, and underlines the crucial, subtle, and sometimes unintuitive effects on structures and reactions of entropic forces caused by the high concentration of macromolecules in the nucleus.

How epigenome drives chromatin folding and dynamics, insights from efficient coarse-grained models of chromosomes

The 3D organization of chromosomes is crucial for regulating gene expression and cell function. Many experimental and polymer modeling efforts are dedicated to deciphering the mechanistic principles behind chromosome folding. Chromosomes are long and densely packed-topologically constrained-polymers. The main challenges are therefore to develop adequate models and simulation methods to investigate properly the multi spatio-temporal scales of such macromolecules. Here, we proposed a generic strategy to develop efficient coarse-grained models for self-avoiding polymers on a lattice. Accounting accurately for the polymer entanglement length and the volumic density, we show that our simulation scheme not only captures the steady-state structural and dynamical properties of the system but also tracks the same dynamics at different coarse-graining. This strategy allows a strong power-law gain in numerical efficiency and offers a systematic way to define reliable coarse-grained null models for chromosomes and to go beyond the current limitations by studying long chromosomes during an extended time period with good statistics. We use our formalism to investigate in details the time evolution of the 3D organization of chromosome 3R (20 Mbp) in drosophila during one cell cycle (20 hours). We show that a combination of our coarse-graining strategy with a one-parameter block copolymer model integrating epigenomic-driven interactions quantitatively reproduce experimental data at the chromosome-scale and predict that chromatin motion is very dynamic during the cell cycle.

Computational construction of 3D chromatin ensembles and prediction of functional interactions of alpha-globin locus from 5C data

Conformation capture technologies measure frequencies of interactions between chromatin regions. However, understanding gene-regulation require knowledge of detailed spatial structures of heterogeneous chromatin in cells. Here we describe the nC-SAC (n-Constrained-Self Avoiding Chromatin) method that transforms experimental interaction frequencies into 3D ensembles of chromatin chains. nC-SAC first distinguishes specific from non-specific interaction frequencies, then generates 3D chromatin ensembles using identified specific interactions as spatial constraints. Application to α-globin locus shows that these constraints (∼20%) drive the formation of ∼99% all experimentally captured interactions, in which ∼30% additional to the imposed constraints is found to be specific. Many novel specific spatial contacts not captured by experiments are also predicted. A subset, of which independent ChIA-PET data are available, is validated to be RNAPII-, CTCF-, and RAD21-mediated. Their positioning in the architectural context of imposed specific interactions from nC-SAC is highly important. Our results also suggest the presence of a many-body structural unit involving α-globin gene, its enhancers, and POL3RK gene for regulating the expression of α-globin in silent cells.

Topology of polymer chains under nanoscale confinement

Spatial confinement limits the conformational space accessible to biomolecules but the implications for bimolecular topology are not yet known. Folded linear biopolymers can be seen as molecular circuits formed by intramolecular contacts. The pairwise arrangement of intra-chain contacts can be categorized as parallel, series or cross, and has been identified as a topological property. Using molecular dynamics simulations, we determine the contact order distributions and topological circuits of short semi-flexible linear and ring polymer chains with a persistence length of l_p under a spherical confinement of radius R_c. At low values of l_p/R_c, the entropy of the linear chain leads to the formation of independent contacts along the chain and accordingly, increases the fraction of series topology with respect to other topologies. However, at high l_p/R_c, the fraction of cross and parallel topologies are enhanced in the chain topological circuits with cross becoming predominant. At an intermediate confining regime, we identify a critical value of l_p/R_c, at which all topological states have equal probability. Confinement thus equalizes the probability of more complex cross and parallel topologies to the level of the more simple, non-cooperative series topology. Moreover, our topology analysis reveals distinct behaviours for ring- and linear polymers under weak confinement; however, we find no difference between ring- and linear polymers under strong confinement. Under weak confinement, ring polymers adopt parallel and series topologies with equal likelihood, while linear polymers show a higher tendency for series arrangement. The radial distribution analysis of the topology reveals a non-uniform effect of confinement on the topology of polymer chains, thereby imposing more pronounced effects on the core region than on the confinement surface. Additionally, our results reveal that over a wide range of confining radii, loops arranged in parallel and cross topologies have nearly the same contact orders. Such degeneracy implies that the kinetics and transition rates between the topological states cannot be solely explained by contact order. We expect these findings to be of general importance in understanding chaperone assisted protein folding, chromosome architecture, and the evolution of molecular folds.

Spatial organization of the budding yeast genome in the cell nucleus and identification of specific chromatin interactions from multi-chromosome constrained chromatin model

Nuclear landmarks and biochemical factors play important roles in the organization of the yeast genome. The interaction pattern of budding yeast as measured from genome-wide 3C studies are largely recapitulated by model polymer genomes subject to landmark constraints. However, the origin of inter-chromosomal interactions, specific roles of individual landmarks, and the roles of biochemical factors in yeast genome organization remain unclear. Here we describe a multi-chromosome constrained self-avoiding chromatin model (mC-SAC) to gain understanding of the budding yeast genome organization. With significantly improved sampling of genome structures, both intra- and inter-chromosomal interaction patterns from genome-wide 3C studies are accurately captured in our model at higher resolution than previous studies. We show that nuclear confinement is a key determinant of the intra-chromosomal interactions, and centromere tethering is responsible for the inter-chromosomal interactions. In addition, important genomic elements such as fragile sites and tRNA genes are found to be clustered spatially, largely due to centromere tethering. We uncovered previously unknown interactions that were not captured by genome-wide 3C studies, which are found to be enriched with tRNA genes, RNAPIII and TFIIS binding. Moreover, we identified specific high-frequency genome-wide 3C interactions that are unaccounted for by polymer effects under landmark constraints. These interactions are enriched with important genes and likely play biological roles.

The architecture of the cell nucleus and the spatial organization of the genome are important in determining nuclear functions. Single-cell imaging techniques and chromosome conformation capture (3C) based methods have provided a wealth of information on the spatial organization of chromosomes. Here we describe a multi-chromosome ensemble model of chromatin chains for understanding the folding principles of budding yeast genome. By overcoming severe challenges in sampling self-avoiding chromatin chains in nuclear confinement, we succeed in generating a large number of model genomes of budding yeast. Our model predicts chromatin interactions that have good correlation with experimental measurements. Our results showed that the spatial confinement of cell nucleus and excluded-volume effect are key determinants of the folding behavior of yeast chromosomes, and largely account for the observed intra-chromosomal interactions. Furthermore, we determined the specific roles of individual nuclear landmarks and biochemical factors, and our analysis showed that centromere tethering largely determines inter-chromosomal interactions. In addition, we were able to infer biological properties from the organization of modeled genomes. We found that the spatial locations of important elements such as fragile sites and tRNA genes are largely determined by the tethering of centromeres to the Spindle Pole Body. We further showed that many of these spatial locations can be predicted by using the genomic distances to the centromeres. Overall, our results revealed important insight into the organizational principles of the budding yeast genome and predicted a number of important biological findings that are fully experimentally testable.

Genomic Energy Landscapes

Energy landscape theory, developed in the context of protein folding, provides, to our knowledge, a new perspective on chromosome architecture. We review what has been learned concerning the topology and structure of both the interphase and mitotic chromosomes from effective energy landscapes constructed using Hi-C data. Energy landscape thinking raises new questions about the nonequilibrium dynamics of the chromosome and gene regulation.

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.

Transferable model for chromosome architecture

In vivo, the human genome folds into a characteristic ensemble of 3D structures. The mechanism driving the folding process remains unknown. We report a theoretical model for chromatin (Minimal Chromatin Model) that explains the folding of interphase chromosomes and generates chromosome conformations consistent with experimental data. The energy landscape of the model was derived by using the maximum entropy principle and relies on two experimentally derived inputs: a classification of loci into chromatin types and a catalog of the positions of chromatin loops. First, we trained our energy function using the Hi-C contact map of chromosome 10 from human GM12878 lymphoblastoid cells. Then, we used the model to perform molecular dynamics simulations producing an ensemble of 3D structures for all GM12878 autosomes. Finally, we used these 3D structures to generate contact maps. We found that simulated contact maps closely agree with experimental results for all GM12878 autosomes. The ensemble of structures resulting from these simulations exhibited unknotted chromosomes, phase separation of chromatin types, and a tendency for open chromatin to lie at the periphery of chromosome territories.

Challenges in structural approaches to cell modeling

Computational modeling is essential for structural characterization of biomolecular mechanisms across the broad spectrum of scales. Adequate understanding of biomolecular mechanisms inherently involves our ability to model them. Structural modeling of individual biomolecules and their interactions has been rapidly progressing. However, in terms of the broader picture, the focus is shifting toward larger systems, up to the level of a cell. Such modeling involves a more dynamic and realistic representation of the interactomes in vivo, in a crowded cellular environment, as well as membranes and membrane proteins, and other cellular components. Structural modeling of a cell complements computational approaches to cellular mechanisms based on differential equations, graph models, and other techniques to model biological networks, imaging data, etc. Structural modeling along with other computational and experimental approaches will provide a fundamental understanding of life at the molecular level and lead to important applications to biology and medicine. A cross section of diverse approaches presented in this review illustrates the developing shift from the structural modeling of individual molecules to that of cell biology. Studies in several related areas are covered: biological networks; automated construction of three-dimensional cell models using experimental data; modeling of protein complexes; prediction of non-specific and transient protein interactions; thermodynamic and kinetic effects of crowding; cellular membrane modeling; and modeling of chromosomes. The review presents an expert opinion on the current state-of-the-art in these various aspects of structural modeling in cellular biology, and the prospects of future developments in this emerging field.

Shape Transitions and Chiral Symmetry Breaking in the Energy Landscape of the Mitotic Chromosome

We derive an unbiased information theoretic energy landscape for chromosomes at metaphase using a maximum entropy approach that accurately reproduces the details of the experimentally measured pairwise contact probabilities between genomic loci. Dynamical simulations using this landscape lead to cylindrical, helically twisted structures reflecting liquid crystalline order. These structures are similar to those arising from a generic ideal homogenized chromosome energy landscape. The helical twist can be either right or left handed so chiral symmetry is broken spontaneously. The ideal chromosome landscape when augmented by interactions like those leading to topologically associating domain formation in the interphase chromosome reproduces these behaviors. The phase diagram of this landscape shows that the helical fiber order and the cylindrical shape persist at temperatures above the onset of chiral symmetry breaking, which is limited by the topologically associating domain interaction strength.

Computational predictions of structures of multichromosomes of budding yeast

Knowledge of the global architecture of the cell nucleus and the spatial organization of genome is critical for understanding gene expression and nuclear function. Single-cell imaging techniques provide a wealth of information on the spatial organization of chromosomes. Computational tools for modelling chromosome structure have broad implications in studying the effect of cell nucleus on higher-order genome organization. Here we describe a multichromosome constrained self-avoiding chromatin model for studying ensembles of genome structural models of budding yeast nucleus. We successfully generated a large number of model genomes of yeast with appropriate chromatin fiber diameter, persistence length, and excluded volume under spatial confinement. By incorporating details of the constraints from single-cell imaging studies, our method can model the budding yeast genome realistically. The model developed here provides a general computational framework for studying the overall architecture of budding yeast genome.

Confinement-Induced Glassy Dynamics in a Model for Chromosome Organization

Recent experiments showing scaling of the intrachromosomal contact probability, P(s)∼s(-1) with the genomic distance s, are interpreted to mean a self-similar fractal-like chromosome organization. However, scaling of P(s) varies across organisms, requiring an explanation. We illustrate dynamical arrest in a highly confined space as a discriminating marker for genome organization, by modeling chromosomes inside a nucleus as a homopolymer confined to a sphere of varying sizes. Brownian dynamics simulations show that the chain dynamics slows down as the polymer volume fraction (ϕ) inside the confinement approaches a critical value ϕ(c). The universal value of ϕ(c)(∞)≈0.44 for a sufficiently long polymer (N≫1) allows us to discuss genome dynamics using ϕ as the sole parameter. Our study shows that the onset of glassy dynamics is the reason for the segregated chromosome organization in humans (N≈3×10(9), ϕ≳ϕ(c)(∞)), whereas chromosomes of budding yeast (N≈10(8), ϕ<ϕ(c)(∞)) are equilibrated with no clear signature of such organization.

Topology, structures, and energy landscapes of human chromosomes

Chromosome conformation capture experiments provide a rich set of data concerning the spatial organization of the genome. We use these data along with a maximum entropy approach to derive a least-biased effective energy landscape for the chromosome. Simulations of the ensemble of chromosome conformations based on the resulting information theoretic landscape not only accurately reproduce experimental contact probabilities, but also provide a picture of chromosome dynamics and topology. The topology of the simulated chromosomes is probed by computing the distribution of their knot invariants. The simulated chromosome structures are largely free of knots. Topologically associating domains are shown to be crucial for establishing these knotless structures. The simulated chromosome conformations exhibit a tendency to form fibril-like structures like those observed via light microscopy. The topologically associating domains of the interphase chromosome exhibit multistability with varying liquid crystalline ordering that may allow discrete unfolding events and the landscape is locally funneled toward "ideal" chromosome structures that represent hierarchical fibrils of fibrils.

Functional assembly of protein fragments induced by spatial confinement

Natural proteins are often confined within their local microenvironments, such as three-dimensional confinement in organelles or two-dimensional confinement in lipid rafts on cytoplasmic membrane. Spatial confinement restricts proteins' entropic freedom, forces their lateral interaction, and induces new properties that the same proteins lack at the soluble state. So far, the phenomenon of environment-induced protein functional alteration still lacks a full illustration. We demonstrate here that engineered protein fragments, although being non-functional in solution, can be re-assembled within the nanometer space to give the full activity of the whole protein. Specific interaction between hexahistidine-tag (His-tag) and NiO surface immobilizes protein fragments on NiO nanoparticles to form a self-assembled protein "corona" on the particles inside the nanopores of mesoporous silica. Site-specific assembly forces a shoulder-by-shoulder orientation and promotes fragment-fragment interaction; this interaction together with spatial confinement of the mesopores results in functional re-assembly of the protein half fragments. To our surprise, a single half fragment of luciferase (non-catalytic in solution) exhibited luciferase activity when immobilized on NiO in the mesopores, in the absence of the complimentary half. This shows for the first time that spatial confinement can induce the folding of a half fragment, reconstitute the enzyme active site, and re-gain the catalytic capability of the whole protein. Our work thereby highlights the under-documented notion that aside from the chemical composition such as primary sequence, physical environment of a protein also determines its function.

Multiscale Modeling of Cellular Epigenetic States: Stochasticity in Molecular Networks, Chromatin Folding in Cell Nuclei, and Tissue Pattern Formation of Cells

Genome sequences provide the overall genetic blueprint of cells, but cells possessing the same genome can exhibit diverse phenotypes. There is a multitude of mechanisms controlling cellular epigenetic states and that dictate the behavior of cells. Among these, networks of interacting molecules, often under stochastic control, depending on the specific wirings of molecular components and the physiological conditions, can have a different landscape of cellular states. In addition, chromosome folding in three-dimensional space provides another important control mechanism for selective activation and repression of gene expression. Fully differentiated cells with different properties grow, divide, and interact through mechanical forces and communicate through signal transduction, resulting in the formation of complex tissue patterns. Developing quantitative models to study these multi-scale phenomena and to identify opportunities for improving human health requires development of theoretical models, algorithms, and computational tools. Here we review recent progress made in these important directions.