Uncovering the Principles of Genome Folding by 3D Chromatin Modeling

Abstract

Our understanding of how genomic DNA is tightly packed inside the nucleus, yet is still accessible for vital cellular processes, has grown dramatically over recent years with advances in microscopy and genomics technologies. Computational methods have played a pivotal role in the structural interpretation of experimental data, which helped unravel some organizational principles of genome folding. Here, we give an overview of current computational efforts in mechanistic and data-driven 3D chromatin structure modeling. We discuss strengths and limitations of different methods and evaluate the added value and benefits of computational approaches to infer the 3D structural and dynamic properties of the genome and its underlying mechanisms at different scales and resolution, ranging from the dynamic formation of chromatin loops and topological associated domains to nuclear compartmentalization of chromatin and nuclear bodies.

Over the last decade, a proliferation of improved sequencing-based genomics and microscopy technologies has led to new opportunities to unravel the intertwined relationship between the three-dimensional (3D) organization of genomes and key biological processes (Dekker et al. 2017; Kempfer and Pombo 2020). Experimental evidence points to a complex and polymorphic chromatin fiber at the intermediate folding level, which is in agreement with an irregular dynamic folding state for interphase chromatin (Ou et al. 2017; Bintu et al. 2018). A major advance in our understanding came from the realization of the ubiquitous nature of chromatin loops and topologically associating domains (TADs) from genome-wide mapping methods such as Hi-C (Dixon et al. 2012; Nora et al. 2012; Sexton et al. 2012; Rao et al. 2014), genome architecture mapping (GAM) (Beagrie et al. 2017), chromatin interaction analysis by paired-end tag sequencing (Chia-PET) (Fullwood et al. 2009), and by superresolution microscopy (Kempfer and Pombo 2020; McCord et al. 2020) (Fig. 1A,B). Chromatin loops are inherently dynamic structures and, together with TAD borders, emerge in ensemble data at binding site locations of insulator proteins from weak preferences of loop positions summed over many cells (Rao et al. 2014). Insulator proteins act as enhancer-blocking elements to prevent communication between enhancer and promoters and/or as barriers to prevent the spreading of heterochromatin. At the global level, mammalian genome folding is governed by affinity-driven spatial compartmentalization (Fig. 1A,B). Chromatin in different functional classes are spatially segregated by microphase separation (Hildebrand and Dekker 2020) and chromatin associations to nuclear bodies, including nuclear speckles and lamina domains (Steensel and Henikoff 2000; Guelen et al. 2008; Chen et al. 2018).

Figure 1.

Hallmarks of spatial genome organization. (A) Chromosomes form distinct territories in the nucleus (middle panel). Active and inactive chromatin are further segregated into nuclear compartments (right) and organized around nuclear bodies such as nuclear speckles, nucleoli, and nuclear lamina. At local and intermediate folding levels (40 kb–3 Mb), chromatin forms loops and topologically associating domains (TADs), the boundaries of which are enriched with CTCF and cohesin (left). (B) Schematic view of Hi-C contact map patterns characteristic of chromosome territories (middle), nuclear compartmentalization (right), and TADs and loops (left). TADs appear as blocks with enriched contacts along the diagonal; loops appear as dots of locally enriched contact frequencies, often at corners of chromatin domains. Domains can be nested with patterns of “sub-TADs” and nested loops. Nuclear compartmentalization is characterized by a checkerboard pattern for long-range and interchromosomal interactions. (C) Formation of TADs and loops by loop extrusion. Cohesin-loader protein NIPBL facilitates the initial association of cohesin with chromatin. Once cohesin is loaded, it starts extruding chromatin and stops when it encounters a bound CTCF protein with binding motif in the proper directional orientation. Other proteins and cohesin complexes may also terminate the extrusion process. Cohesin is released from chromatin by the cohesin release factor, WAPL. (D) Nuclear compartmentalization arises from phase separation of chromatin via favorable interactions between similar chromatin types.

Nuclear genome organization is therefore the result of a multitude of competing processes at various organizational levels (Misteli 2020). For instance, compartment and loop formation processes act antagonistically and are driven by distinct mechanisms (Fig. 1C,D), which prevents a common structural hierarchy for the description of chromatin folding across scales (Nuebler et al. 2018). Moreover, critical components and underlying physical mechanisms are often not yet fully understood. It is evident that the complexities of nuclear organization cannot be explained by a single experimental method and require a combination of complementary data and methods (Dekker et al. 2017). Computational simulations offer the unique opportunity to address some of these challenges: they provide a robust platform to integrate different data types, can assist in the interpretation of experimental data, and are capable to test hypotheses of physical processes that may be responsible for shaping genome organization. Over recent years, a multitude of different computational methods have been published, each with individual strengths and limitations. These complementary methods provide a renewed opportunity to gain better quantitative understanding of the nuclear organization. Here we provide a detailed overview of computational methods available for various applications in genome modeling.

MODELING APPROACHES

Computational simulations are uniquely suited to validate proposed mechanisms of genome folding if their outcome agrees with experimental observations. Simulating nuclear processes with all their complexities is a challenging task, and choosing the appropriate structural representation is a crucial component. An effective model requires the identification of descriptors, which are structural units that are most informative at the chosen level of resolution and whose interactions can be faithfully described. Such an approach is known as coarse-graining. For coarse-grained genome-scale models, the chromatin fiber is typically represented as a polymer (Fig. 2A), defined as a chain of connected beads, each representative of a specific chromatin region typically with hundreds to millions of base pairs depending on the chosen model of granularity (i.e., base-pair resolution) (Lin et al. 2019; Parmar et al. 2019; Brackey et al. 2020). Bead monomers can be assigned to different types (often referred to as colors), to account for epigenetic or biochemical identities.

Figure 2.

3D chromatin modeling with a mechanistic approach. (A) Different polymer models for simulating chromatin structures. A homopolymer chain models the chromatin fiber as a chain of beads connected by “springs” (i.e., harmonic potentials). Excluded volume potentials prevent spatial overlap of beads. Block copolymer models classify bead monomers into distinct types, which share similar physical properties. Beads of the same type experience favorable attractive interactions to each other. Some approaches include binder particles, which can mediate interactions between chromatin beads of the same type only if they are bound to the chain. The loop extrusion model imitates the loop extrusion mechanism. Extrusion factors are diffusive particles that can bind to regions on the fiber (orange links). The extrusion factor is represented by an additional bond between beads that form a loop base. This bond is iteratively shifted along the chain to gradually extrude the chromatin loop. The extrusion process is terminated when the extrusion factor encounters a boundary element with proper orientation (yellow beads) or another extrusion factor. (B) Schematic flowchart of a mechanistic modeling approach. The approach relies on prior knowledge about a folding hypothesis that can explain the experimental observations. The polymer model is defined to simulate the hypothesis. If the collected structures do not agree with experimental observations, the initial hypothesis and/or the model parameters are modified and new simulations are performed until the resulting structures are able to reproduce the experimental data. Finally, the collected models are used for quantitative structure analysis.

The chromatin bead chain dynamics is defined by a set of mathematical functions (also referred to as energy terms), which describe the energy of the model as a function of its coordinates. The complete set of energy functions is often referred to as the system Hamiltonian. A polymer Hamiltonian involves at least harmonic bonded terms (i.e., “springs” connecting the beads), which ensure chain connectivity, and excluded volume terms, which prevent spatial overlap between beads.

Additional terms are added to the Hamiltonian to account for model-specific interactions, which are either derived from physical principles or experimental data. For instance, a short-range attractive force between beads of the same type can reproduce phase separation of chromatin types and spatial constraints can account for chromatin tethering to the nuclear landmarks.

Genome structures show a high degree of cell-to-cell variation (Finn and Misteli 2019). Therefore, polymer models must sample large numbers of chromosome conformations, which are typically explored by standard computational techniques such as molecular dynamics (MD) (Allen and Tildesley 1987; Schlick 1996), simulated annealing (Kirkpatrick et al. 1983), or Monte Carlo simulations (Metropolis et al. 1953). Quantitative analysis of structural features is then performed on the ensemble of all sampled conformations.

Because of their versatility, polymer models have been highly successful in modeling structure and dynamics of biomolecules. Here, we distinguish between two classes of modeling approaches, namely, the mechanistic (bottom-up) and the data-driven (top-down) approach. In the mechanistic approach (Fig. 2B), a physical process is postulated based on empirical evidence or physical intuition, and is explicitly included in the Hamiltonian. This approach enables testing of folding mechanisms by estimating observables to be compared with experiments and predicting the response upon perturbations of the process. Parametrization has to be carefully calibrated for simulations to faithfully reproduce experimental evidence, which is far from trivial. Because genome structure is the result of a collective interplay between various processes across hierarchical scales, these models are not easily scalable and not all the molecular players nor potential mechanisms are known yet.

The second class of models are data-driven (Fig. 3B), for which the Hamiltonian calibration can be performed in an almost fully unsupervised fashion. Here, no prior knowledge about chromatin folding processes is required; instead, experimental data, such as Hi-C, are fed to a protocol that generates structures recapitulating the data. The Hamiltonian will include energy terms to model the effect of each data point. Special care has to be devoted to a faithful interpretation of experimental data and its representation in the Hamiltonian. Key components are adequate model assessment strategies, a description of model uncertainties, and detection of false data points, which could affect the outcome. Integration of data modalities from independent experiments can overcome some of the limitations to increase accuracy and structural coverage. Data-driven approaches resemble those in structural biology, in which 3D structures are examined to infer mechanistic insights and make quantitative structure–function predictions for specific genomic regions.

Figure 3.

3D chromatin modeling with a data-driven approach. (A) In the data-driven deconvolution approach, chromatin is modeled as a chain of beads with additional contacts derived from the Hi-C data. Each structure in the population contains a subset of chromatin contacts, the summation of which recapitulates the ensemble Hi-C data. (B) Data-driven methods use the experimental data explicitly to build the 3D models. They generate structures that are able to fully reproduce the experimental data. Adjustments of the initial models and/or data interpretation can be performed to ensure better agreement with experimental data. The collected models are finally used for quantitative structure analysis.

In the following sections, we will first review a number of mechanistic modeling approaches, which explain the formation of chromatin loops, TADs, and phase-separated compartments. Then, we focus on data-driven approaches, which use experimental data to generate genome structures to gain functional insights.

Mechanistic (Bottom-Up) Modeling Methods

Studying the Origins of Chromatin Loops and TADs

A breakthrough in our understanding of chromatin folding came from the genome-wide detection of chromatin loops, contact domains (Rao et al. 2014), and TADs from Hi-C experiments (Dixon et al. 2012; Nora et al. 2012; Sexton et al. 2012), and subsequently other genomics (e.g., SPRITE [Quinodoz et al. 2018], GAM [Beagrie et al. 2017], and imaging data [Bintu et al. 2018; Nir et al. 2018]). Chromatin loops are identified as locally enriched peaks in the Hi-C maps (Rao et al. 2014), often located at domain borders. TADs are revealed in Hi-C maps as squared blocks of enriched frequencies along the diagonal (Fig. 1B), indicating preferred interactions within and depleted interactions between TAD regions (Dixon et al. 2012; Nora et al. 2012; Sexton et al. 2012). Larger TADs are often nested with smaller “sub-TADs” (i.e., contact domains) and shared loops at their boundaries (Shen et al. 2012; Phillips-Cremins et al. 2013; Rao et al. 2014). Loop and most TAD boundaries are enriched with CTCF (CCCTC-binding factor protein) with binding motifs in convergent orientation at the two loop anchors as well as cohesin-binding sites (Rao et al. 2014).

Loop formation was initially explained by random polymer collisions, which could form small loops and condense chromosomes efficiently (Marko and Siggia 1997; Sankararaman and Marko 2005), but would be ineffective for the reproducible formation of larger loops and could not explain the enrichment of convergent CTCF-binding motifs at TAD boundaries.

The Loop Extrusion Model. Over the last decade, loop extrusion has emerged as a potential mechanism for chromosome condensation and chromatin looping, initially proposed as a statistical mechanical model (Nasmyth 2001; Alipour and Marko 2012). Only recently, a series of landmark papers have introduced polymer simulations to manifest this process as a primary mechanism for TAD formation by recapitulating a wealth of experimental observations from Hi-C data and superresolution microscopy (Sanborn et al. 2015; Fudenberg et al. 2016, 2017; Goloborodko et al. 2016a,b). Loop extrusion is initiated when a loop-extruding factor selectively attaches to the chromatin fiber and actively reels in the chromatin fiber in both directions, thus leading to a progressively growing loop until the loop-extruder falls off or encounters either another extrusion factor or an extrusion barrier at specific boundary elements (Fig. 1C). Several loop extruder complexes have been identified, including structural maintenance of chromosomes (SMCs) complexes, condensin for compaction of mitotic chromosomes (Hirano 2002), and cohesin (Wood et al. 2010; Uhlmann 2016) for loop extrusion in interphase (Kagey et al. 2010). The insulator protein CTCF acts as an extrusion barrier only if the CTCF-binding motifs are oriented in a convergent sequence orientation at the loop anchors (Rao et al. 2014). Other proteins may also act as extrusion barriers, including Znf143 (Bailey et al. 2015) and YY1 (Weintraub et al. 2017).

To simulate loop extrusion, the chromatin polymer can be represented as a self-avoiding chain of beads (Fig. 2A), which are connected by harmonic potentials, and typically have a base-pair resolution of ∼0.1 kb (Gibcus et al. 2018) to ∼1 kb (Sanborn et al. 2015). The interaction between the extrusion factor and the polymer chain is modeled by an additional bond between two beads, which forms the loop base (Fudenberg et al. 2016); active extrusion is then implemented by iteratively shifting the bond at a given rate to increasingly separated pairs of beads, one bead at a time. The process stops when the extruder runs into either an extrusion barrier or another extruder, or stochastically dissociates from the fiber, in which case the bond is removed. Size and position of loops are controlled, among others, by the separation and permeability of the boundary elements, and by the lifetime and processivity of the extruding factors. These simulation parameters play a critical role in recapitulating experimental observations, and are either inferred from Chip-seq, Chia-PET, or Hi-C data or selected so that models best match the experiments (Fudenberg et al. 2017; Gassler et al. 2017). Polymer simulations of loop extrusion have been very successful at recapitulating a wide variety of chromosomal phenomena, including TAD and loop formation in interphase (Sanborn et al. 2015; Fudenberg et al. 2016), structure changes upon perturbations (Fudenberg et al. 2017; Nuebler et al. 2018), and compaction of mitotic chromatids (Alipour and Marko 2012; Goloborodko et al. 2016b; Gibcus et al. 2018). Simulations naturally explain CTCF directionality and predict almost all characteristic Hi-C patterns of nested sub-TADs, peaks at TAD corners, and stripes from unidirectional chromatin extrusion when cohesin lands near a CTCF site (Fudenberg et al. 2016). Models correctly predicted changes in Hi-C maps upon degradation of protein factors (Fudenberg et al. 2017; Nuebler et al. 2018), including the loss of peaks and TADs upon induced degradation of cohesin (Rao et al. 2017; Wutz et al. 2017) and cohesin-loading complexes (Schwarzer et al. 2017), together with an overall structure decompaction (Nozaki et al. 2017). Simulating CTCF depletion (Fudenberg et al. 2017; Nuebler et al. 2018) blurred TAD boundaries but maintained chromatin compaction from ongoing loop extrusion (Nora et al. 2017; Nozaki et al. 2017; Wutz et al. 2017), while depletion of the cohesin release factor WAPL correctly predicted a proliferation in the number of peaks (Haarhuis et al. 2017; Wutz et al. 2017), resulting in prophase-like elongated nuclear chromatin structures, also known as vermicelli chromatids (“vermicelli” is Italian for “small worms”) (Tedeschi et al. 2013).

A debate has been ongoing whether loop extrusion is an equilibrium or energy-driven process (Banigan and Mirny 2020). Whereas most works support an active, ATP-driven process, others (Brackley et al. 2017, 2018) proposed a process without motor activity, where cohesin complexes slide directionally biased by osmotic pressure from subsequently loaded cohesin complexes (also known as the slip-link model). Recently, single-molecule experiments imaged active DNA loop extrusion by condensin and cohesin in vitro, therefore showing evidence of a nonequilibrium process (Ganji et al. 2018; Davidson et al. 2019; Golfier et al. 2020; Kong et al. 2020).

Block Copolymer Models to Study Global Chromatin Compartmentalization

Loop extrusion is typically limited to local interactions of up to tens of Mb sequence distance, and cannot account for the characteristic checkerboard patterns of long-range and interchromosomal interactions in Hi-C maps (Fig. 1B). Those are a result of spatial segregation of euchromatin and heterochromatin compartments, driven by preferential affinity for chromatin in the same functional state (Lieberman-Aiden et al. 2009; Rao et al. 2014; Solovei et al. 2016; van Steensel and Belmont 2017; Falk et al. 2019; Hildebrand and Dekker 2020).

Phase separation has been extensively studied with a variety of block copolymer models (Leibler 1980), which describe the chromatin fiber as a self-avoiding chain of beads of distinct types (i.e., active and inactive chromatin) and short-range attractive forces acting only between beads of the same type (Jost et al. 2014; Chiariello et al. 2016; Di Pierro et al. 2016; Michieletto et al. 2016; Haddad et al. 2017; MacPherson et al. 2018). Because the chromatin fiber consists of alternating blocks of active and inactive chromatin, the covalent linkages of the chain prevent the complete separation into only two macrophases (Hildebrand and Dekker 2020). Instead, the two compartments spatially segregate (microphase separation) into a larger number of smaller clusters (Fig. 1D). Microphase separation is possibly driven by phase separation (also referred as condensate formation), which creates distinct phases from a single homogeneous mixture through multivalent weak interactions between proteins or nucleic acids that preferentially associate with a given chromatin type and form condensates once a critical concentration is reached (Erdel and Rippe 2018).

Several studies explored phase separation of chromatin compartments as a cause for the spatial segregation of transcriptionally active euchromatin and inactive heterochromatin, driven by preferential affinity of chromatin to their own kind. Jost et al. (2014) modeled chromatin regions in Drosophila melanogaster at 10 kb resolution with a block copolymer of four chromatin types derived from epigenetic profiles. MD simulations with weak homotypic interactions produce random coil structures, while strong interactions reproduce the checkerboard-like interaction patterns of phase-separated chromatin (Jost et al. 2014). At specific interaction strengths, metastable configurations emerge in which TAD-like domains exist and transiently interact with each other, while global chromatin compartments undergo microphase separation.

Phase separation can be driven by binding factors, which bridge chromatin through multivalent interactions. Heterochromatin protein 1 (HP1) binds methylated H3K9me3 histone tails and oligomerizes with HP1 at other chromatin regions, which then condenses into a heterochromatic phase (Larson et al. 2017; Strom et al. 2017). Some simulations imitate such a process. MacPherson et al. (2018) modeled human chromosome 16 as a worm-like chain of nucleosomes classified into eu- or heterochromatin based on H3K9me3 occupancy. Heterochromatin interaction strengths varied with the number of HP1 entities bound to nucleosomes. At specific HP1 concentrations, simulations led to phase separation of heterochromatin with chromatin densities similar to those in experiments. Besides appropriate interaction strengths and HP1 concentrations, sufficiently large copolymer block sizes (here ∼20 kb for given modeling parameters) were also crucial requirements to faithfully reproduce compartmental phase separation. Overall, these models reproduced some but not all aspects of the experiment, such as the scaling behavior of contacts in Hi-C maps, which emphasizes that additional physical processes must be considered.

Recently, a pivotal study investigated what type of homotypic interactions are the dominant factor in reproducing compartmental phase separation of rod cells from nocturnal mammalians (Falk et al. 2019). Falk et al. used a copolymer model with three chromatin types (euchromatin, heterochromatin, and pericentromeric heterochromatin) to simulate an artificial nucleus with eight chromosomes at a base-pair resolution of 40 kb per bead (Fig. 4A). Exhaustive screens across all combinations of interaction strengths indicated that heterochromatin attractive interactions played the primary role in phase separation in both inverted (i.e., interior heterochromatin) and conventional nuclei (i.e., peripheral heterochromatin), whereas euchromatic attractive interactions were dispensable. However, the nuclear position of heterochromatin in conventional nuclei cannot be reproduced by phase separation alone, and requires additional attractive forces to the nuclear periphery.

Figure 4.

Selected examples of modeling studies investigating chromatin organization. (A) Falk et al. (2019) used a mechanistic approach with a block copolymer chromatin model containing three chromatin types. The model predicted the inverted chromatin organization observed in microscopy. Predicting the chromatin organization in conventional nuclei required additional interactions of heterochromatin regions (B and C types) with the nuclear lamina, which highlights the crucial role of lamina interactions in maintaining the conventional nuclei organization. (Panel A is from Falk et al. 2019; adapted, with permission, from Springer Nature © 2019.) (B) This example showcases our data-driven deconvolution approach, where we simulated a population of diploid genome structures for GM12878 cells from Hi-C data (Yildirim et al. 2021). The models were generated from Hi-C data without explicit subcompartment notations or block copolymer chromatin classes. The resulting structures were able to show the segregation of different chromatin subcompartments and provided additional insights into structural features that distinguish different subcompartments.

Combining Loop Extrusion with Block Copolymer Models

Phase separation describes an equilibrium state and generally does not recapitulate loops from energy-driven loop extrusions. Nuebler et al. (2018) jointly simulated loop extrusion and copolymer phase separation and demonstrated that active loop extrusion shifts local chromosome structures to a nonequilibrium state and overrides locally the fine-scale compartment patterns that would otherwise be visible at longer-range interactions. Genome compartments and loops are driven by distinct mechanisms and a common hierarchical organization with TADs as building blocks of compartments does not likely exist. For instance, simulating depletion of loop extrusion factor cohesin reduced TADs and revealed finer compartments, while increased processivity of cohesin strengthened large TADs and reduced compartmentalization (Fudenberg et al. 2017; Nuebler et al. 2018). Depletion of extrusion barrier protein CTCF weakened TADs, while leaving compartments and chromatin compaction unaffected. There is growing experimental evidence that TADs can exist without compartments and vice versa (Nuebler et al. 2018; Mirny et al. 2019). A comprehensive discussion of phase separation and its role in genome organization is provided in a recent review (Hildebrand and Dekker 2020).

The minimal chromatin model (MiChroM) (Di Pierro et al. 2016) introduces a transferable force field based on predefined chromatin types. The method combines loops with block copolymer modeling and does not rely on a specific mechanism for loop formation. The chromosome is modeled as a self-avoiding polymer chain at 50 kb resolution, while loop locations and chromatin types are derived from Hi-C data. The energy function contains 27 adjustable parameters, which are determined by maximizing the agreement of contact frequencies from simulated models and Hi-C data by means of Lagrange multipliers. Lagrange multipliers are used to find the extrema of a function that is subject to constraints. They are coefficients in the potential energy function to maximize the agreement between the contact probabilities from models and Hi-C experiments. Parameters trained on one chromosome can be used to model other chromosomes, in good agreement with Hi-C contact frequencies. Chromosome compaction is driven by a generic energy function, mimicking the behavior of a liquid crystal (Di Pierro et al. 2016).

An ensemble of chromosome structures is collected by MD simulations, which recapitulates the characteristic features of chromatin organization, including chromosome territories, phase separation of chromatin types, loops between specific anchors, and knot-free chromatin conformations (Di Pierro et al. 2016). MiChroM was later extended by a machine learning algorithm, MEGABASE, to infer chromatin types from Chip-seq data (Di Pierro et al. 2017). A recent application on six human cell types studied the structural heterogeneity of chromosomes across cell types (Cheng et al. 2020). When MiChroM is combined with Langevin dynamics (i.e., standard MD with additional stochastic and viscous energy terms allowing a thermal equilibration with the environment), models reproduce some dynamic behavior of chromosomes, including chromatin subdiffusion, viscoelasticity, and spatial coherence of chromatin as well as dynamically associated domains (DADs) from phase separation at longer time intervals (Di Pierro et al. 2018).

Qi and Zhang (2019) expanded MiChroM with an energy function for chromatin at 5 kb resolution. The Hamiltonian includes attractive forces for 15 chromatin states from histone modifications, and interaction potentials for intra-TAD chromatin, for a total of ∼1800 parameters, which are adjusted to maximize agreement with Hi-C data. Parameters are transferable and, when trained on specific chromosome segments in one cell, produce structures of about 20 Mb in length for other chromosomes with good agreement to Hi-C data. A minimum of six chromatin types were sufficient to reproduce chromatin compartmentalization, while short-range chromatin interactions within TADs required a more detailed energy function. Recently, a generalized energy function was introduced to simulate diploid genome structures at 1 Mb resolution (Qi et al. 2020) and considered intra- and interchromosomal interaction potentials, centromere clustering, A/B compartmentalization, and inactive X-chromosome condensation. Overall, the structures captured global structural features such as chromosome locations and territories, and phase separation of A/B compartments. Their work suggested that correct chromosome positioning requires specific interchromosomal interactions and centromere clustering, and is not driven by phase separation alone.

Shi et al. (2018) developed a transferable chromosome copolymer model (CCM) with one free parameter for chromatin type and loop anchor interactions to reproduce TAD and compartment organization. Brownian dynamics simulations of relatively small chromosomal regions at 1.2 kb resolution revealed a hierarchical folding where chromosome droplets (CDs) similar in size to TADs formed first, which then coalesced to a more compact state.

The strings and binder switch (SBS) model (Nicodemi and Prisco 2009; Barbieri et al. 2012) is an approach for jointly modeling loop formation and chromatin compartmentalization. The model is inspired by transcription factors (TFs) that bridge cis-regulatory elements with distal promoters upon chromatin binding (Barbieri et al. 2017; Kundu et al. 2017). The diffusive-bridge model (Brackley et al. 2013, 2016; Buckle et al. 2018) also uses a similar strategy: the model introduces diffusive particles representing protein complexes that form polymer-protein-polymer bridges, which subsequently aggregate and induce local polymer compaction. In the SBS model, chromatin is modeled as a self-avoiding chain of beads surrounded by a cloud of diffusing “binder” particles, which can bind to chromatin and only then mediate interactions between distal chromatin beads (Fig. 2A). Beads and binders are classified into types (i.e., colors), and binders experience an attraction only to cognate beads of the same color. Multivalent binder interactions can generate a variety of loop patterns and TAD boundaries, which depend on the number of binder types, the sequence locations of binding sites, and the concentrations of binder particles. To express binder-mediated chromatin interactions, a truncated finite-range Lennard-Jones potential is added to a traditional polymer chain Hamiltonian. The Lennard-Jones potential describes an interaction between particles and consists of both a repulsive term at very small particle distances (to prevent particle overlap) and a weak attractive term at larger distances (which eventually converges to zero at very large distances). The interaction strength and equilibrium distance can be adjusted by choosing adequate parameter settings. MD simulations will then induce a classic coil-to-globule phase-separation of chromatin at specific binder concentrations, binding affinities, and diffusive properties of binders and chromatin (Annunziatella et al. 2016, 2018; Chiariello et al. 2016; Conte et al. 2020). Binder concentrations and affinities are unknown beforehand, and simulations over a range of values allow selection of those that maximize the agreement between models and experiment (Chiariello et al. 2016; Bianco et al. 2017, 2018).

SBS models are typically applied to chromosomal regions of a few ∼Mb in size (Brackley et al. 2013) at a base-pair resolution between ∼0.1 kb (Barbieri et al. 2012; Brackley et al. 2016; Bianco et al. 2018) to tens of kb per bead (Chiariello et al. 2016). The optimal number of binder types and binding sites and their sequence locations can be inferred from knowledge about the regulatory landscape (i.e., locations of genes, enhancers, CTCFs, and TFs) (Brackley et al. 2016; Barbieri et al. 2017) or from Hi-C data by a recently developed machine learning approach (PRISMS) (Bianco et al. 2018; Conte et al. 2020).

The strings and binders approach has been used in a broad range of applications, in which chromatin structures recapitulated contact probabilities and scaling from Hi-C (Brackley et al. 2016; Chiariello et al. 2016) and GAM data (Beagrie et al. 2017), and predicted spatial distances from FISH experiments (Barbieri et al. 2012; Nicodemi and Pombo 2014; Fraser et al. 2015; Chiariello et al. 2016; Bianco et al. 2017, 2018). SBS simulations reproduced loops and TADs as well as A/B compartments with a two-color copolymer model (Barbieri et al. 2012). However, more chromatin types are required to reproduce specific details of the Hi-C map at higher resolution (Chiariello et al. 2016; Bianco et al. 2017; Conte et al. 2020). SBS models studied the folding of the Sox9 and HoxB loci in mESC (Chiariello et al. 2016; Barbieri et al. 2017) and characterized the effects of pathogenic variants on the folding of EPHA4 locus in human fibroblasts (Bianco et al. 2017). The structure of the healthy locus was predicted with a model trained on Hi-C data. Then, models of the pathogenic mutants correctly predicted most of the ectopic interactions. SBS models also explored the tissue-specific architecture of the mouse Pitx1 gene, a regulator of hindlimb development (Kragesteen et al. 2018). Observed chromatin refolding in forelimb and hindlimb provided a rationale to explain expression data. Recently, models of human HTC116 and IMB90 loci in wild-type and cohesin depleted cells were validated against superresolution imaging (Bintu et al. 2018; Conte et al. 2020).

Data-Driven Approaches

Mechanistic models simulate the time evolution of chromatin-folding processes. Often, the underlying Hamiltonian uses predefined chromatin classes to generalize chromatin interactions, assuming the same chromatin types share identical physical properties. These generalized energy terms are parameterized so that models best agree with experiments (Fig. 2B).

Data-driven approaches use a different strategy. They do not generalize chromatin into predefined classes and do not require prior knowledge of folding mechanisms. Instead, they use all data points explicitly, assume an appropriate representation of experimental errors and uncertainties, and relate all data to an ensemble of 3D genome structures that are statistically consistent with it (Fig. 3B). These 3D structures are then examined to derive structure–function correlations and make quantitative predictions of structural features for specific genomic regions and study cell-to-cell variabilities of chromosome conformations. There are several data-driven modeling strategies, which differ in the functional interpretation of experimental data and sampling strategies to generate genome structures. We will first focus on data deconvolution methods, which attempt to de-multiplex ensemble data, and then discuss resampling methods.

Data Deconvolution Methods

The population-based genome structure modeling approach (PGS) is a probabilistic framework to model fully diploid genomes from Hi-C data at base-pair resolutions of tens to hundreds of kb and is available as a software package (Kalhor et al. 2012; Tjong et al. 2016; Hua et al. 2018). The approach performs a structure-based deconvolution of ensemble Hi-C data into a population of individual structures, in which the cumulated physical contacts across all structures recapitulate the Hi-C data. Chromosomes are modeled as polymer chains subject to chain connectivity, chromatin contacts, excluded volume and nuclear volume restraints (Fig. 3A). The key step is to infer those chromatin contacts likely to co-occur in the same structure. This problem is formulated as a maximum likelihood estimation problem, which is solved iteratively with a variant of the expectation-maximization algorithm and optimization strategies for efficient and scalable model estimation (Tjong et al. 2016; Li et al. 2017). Each iteration involves two steps: first, finding the optimal allocations of chromatin contacts across all structures by maximizing the log-likelihood over all contact assignments, given the optimized structures from the previous iteration, and second, generating genome structures by imposing physical contact restraints for all contact allocations using a combination of MD simulated annealing (Kirkpatrick et al. 1983) and conjugate gradient minimizations (Hestenes and Stiefel 1952). Each individual structure is described by a unique Hamiltonian, which expresses only a subset of contact restraints according to the optimized contact allocations. At each iteration, contact allocations are reevaluated and the process is repeated until convergence is reached. In addition, Hi-C contacts are gradually added to the iterative process, starting with the most frequent interactions. Gradually fitting an increasing number of contacts can effectively guide the search for the best solution and facilitates the detection of cooperative chromatin interactions. Population-based modeling produces structures of entire diploid genomes with high predictive value. For instance, genome models for GM12878 cells at 200 kb resolution allowed a detailed analysis of the spatial partitioning of chromatin subcompartments (Yildirim et al. 2021), as defined by Rao et al. (2014). Because subcompartment notations were not included as input information, the models allowed an independent characterization of structural features that distinguish chromatin in each subcompartment. Chromatin in the two active subcompartments (A1 and A2) differ in the variability of their nuclear locations: while A1 chromatin is localized in the nuclear interior in most cells, A2 chromatin shows large cell-to-cell variability (Fig. 4B). Models also revealed a relationship between micro-partitions of subcompartment chromatin and nuclear bodies, which made it possible to predict the locations of nuclear speckles in individual models, and predict with good accuracy data from SON tyramide signal amplification sequencing (TSA-seq) experiments (Chen et al. 2018). SON-TSA-seq estimates mean cytological distances of chromatin to nuclear speckles. The SON protein, an mRNA splicing cofactor, is a highly specific marker for nuclear speckles. SON-TSA produces a gradient of diffusible tyramide free radicals, instigated at regions of highest SON concentrations (i.e., the speckle locations), for distance-dependent biotin labeling of DNA. The models also predicted, with good accuracy, other omics data (e.g., laminB1 TSA-seq [Chen et al. 2018] and laminB1 DNA adenine methyltransferase identification [DamID], which maps genome-wide nuclear lamina interactions [Leemans et al. 2019]). The models also indicated a connection between a gene's association frequency to nuclear speckles and its transcript count in single-cell RNA-seq experiments (Osorio et al. 2019) and provided insights into the cell-to-cell variability of TADs (Yildirim et al. 2021).

Population-based modeling has been successfully applied to a variety of cell types and organisms, including human GM12878 lymphocytes (Dai et al. 2016; Tjong et al. 2016; Hua et al. 2018; Yildirim et al. 2021), mouse neutrophils (Zhu et al. 2017), cardiac myocytes, liver tissue (Chapski et al. 2019), and D. melanogaster (Li et al. 2017) at base-pair resolutions from 200 kb to ∼3 Mb. Genome models in lymphoblastoid cells predicted chromosome-specific centromere clusters toward the nuclear interior, which were confirmed by cryo-soft X-ray tomography and play a pivotal role in chromosome positioning and stabilizing interchromosomal interactions (Tjong et al. 2016). The models also detected hundreds of frequently occurring multivalent chromatin clusters, which were enriched for the same regulatory factors (Dai et al. 2016). Another study characterized structural reorganizations during mouse neutrophil differentiation, leading to the discovery of chromosome supercontraction, driven by long-range heterochromatic interactions, combined with repositioning of centromeres and nucleoli (Zhu et al. 2017).

The scope of population-based modeling has recently been expanded to accommodate additional data modalities that can be cast into a polymer energy term. The resulting implementation is referred to as the Integrated Genome Modeling (IGM) platform (Li et al. 2017; Polles et al. 2019). The method was recently employed to generate diploid genome structures of human HFFc6 cells at 200 kb resolution from Hi-C, laminB1 DamID, SPRITE, and 3D HIPMAp FISH data which demonstrated that heterogeneous data sources can uncover structural features that may not be accessible to Hi-C alone (L. Boninsegna, unpubl.).

By integrating Hi-C and lamina DamID data, models of the D. melanogaster genome predicted location preferences for heterochromatin regions of each chromosome in the heterochromatic phase along preferred locations of the nucleolus, which were confirmed by FISH experiments (Li et al. 2017). Even though unphased Hi-C data cannot reveal interactions between chromosome copies, the models correctly show an anticorrelation between predicted pairing frequencies for homolog chromatin regions and the enrichment of binding sites for the mortality factor 4–like protein 1 (Mrg15), which is known to cause homolog unpairing.

Another approach by Giorgetti et al. (2014) models chromatin as a self-avoiding chain of beads, which interact via spherical well potentials. For each contact pair, the strength of the interaction potential is optimized to reproduce 5C contact frequencies. A study of the Xic locus on the inactive X chromosome in mouse embryonic stem (ES) cells at 3 kb resolution showed high structural variability, highlighted the role of cohesin/CTCF in shaping TAD conformations, and revealed insights into the relationship between conformational fluctuations and transcriptional variability. Further studies indicated that structural variations within TADs occur on timescales shorter than the cell cycle (Tiana et al. 2016). A study on ∼2500 TADs in mouse ES cells provided further information on the correlations between TAD structures and gene activity (Zhan et al. 2017).

In another approach, Zhang and Wolynes (2015) developed an iterative algorithm to approximate an energy landscape for an ensemble of chromosome conformations that is consistent with the maximum entropy principle and reproduces Hi-C contact frequencies. Chromosome 12 of human ES cells and fibroblasts were modeled as self-avoiding polymer at 40 kb resolution with a potential energy function involving terms for chain connectivity, chromosome confinement, hard/soft core repulsive interactions, and chromatin interaction terms with Lagrangian multipliers. The Lagrangian multipliers were determined through an iterative optimization scheme. Chromosome structures were obtained via a series of independent MD simulations from which tens of thousands of conformations are collected. The resulting structures agreed with Hi-C data and revealed highly variable chromosome configurations in which TADs play a key role to locally rigidify the chain. A subset of TADs showed two-state transitions, possibly to modulate transcriptional activity.

Resampling Methods

Resampling approaches differ from deconvolution methods in the interpretation of the data. They generally express the agreement between data and structures by a solitary scoring function, in which Hi-C contact frequencies are typically expressed as distance restraints, representing either chromatin contacts or mean distances. However, because Hi-C data are accumulated over millions of cells with considerable variability in chromosome structures, they may contain conflicting information when imposed in a single scoring function. This may cause unresolved violations of restraints during optimization and could limit realistic descriptions of structural cell-to-cell variability. This problem is addressed in some approaches by considering only the most significant subsets of interactions, likely to be present in a dominant structural state (Baù et al. 2011; Umbarger et al. 2011; Gehlen et al. 2012; Trieu and Cheng 2014, 2016; Di Stefano et al. 2016; Paulsen et al. 2017; Serra et al. 2017; Yildirim and Feig 2018). To generate representative structures, resampling methods repeat independent optimizations of the solitary scoring function by Monte Carlo sampling, simulated annealing, MD simulations, expectation maximization or Bayesian optimization.

There is a variety of tools available to model small chromatin regions (Baù et al. 2011; Rousseau et al. 2011; Junier et al. 2012; Meluzzi and Arya 2013; Serra et al. 2017), chromosomes (Trieu and Cheng 2014, 2016; Wang et al. 2015; Zhu et al. 2018), whole-genomes (Gehlen et al. 2012; Di Stefano et al. 2016; Paulsen et al. 2017, 2018; Trieu and Cheng 2017), or bacterial chromosomes (Umbarger et al. 2011; Yildirim and Feig 2018). Comprehensive lists of data-driven modeling tools can be found elsewhere (Oluwadare et al. 2019; MacKay and Kusalik 2020). One of the most commonly applied methods, TADbit, uses significant contacts/noncontacts to study the α-globin locus in human K562 and GM12878 hematopoietic cells (Baù et al. 2011), 1 Mb genomic regions in D. melanogaster (Serra et al. 2017), and the Caulobacter crescentus genome (Umbarger et al. 2011). Resampling methods have also been expanded to incorporate Hi-C with other data sources, including lamin contacts from lamin Chip-seq data (Chrom3D) (Paulsen et al. 2017, 2018) and 3D FISH distances (Zhu et al. 2018; Abbas et al. 2019). Nir et al. (2018) combined Hi-C data with superresolution microscopy by generating 3D chromosome structures with TADbit, which were subsequently fitted to superresolution images from OligoSTORM and Oligo-DNA-PAINT microscopy. It was possible to generate structures for an 8 Mb region of Chromosome 19 in PGP1f cells at 10 kb resolution. These structures shed light on enhancer-promoter clusters, and 3D localization patterns of active and inactive regions.

Resampling methods are well suited to build genome structures from single-cell Hi-C data. The NucDynamics package (Stevens et al. 2017) and most other methods use simulated annealing optimizations of chromatin interaction restraints, polymer chain connectivity, and a noninteracting particle repulsion (Nagano et al. 2013). Most studies comprise 8–15 cells at 50–500 kb resolution and showed considerable structure variations, while chromosome territories and A/B compartment segregation is conserved (Nagano et al. 2013; Stevens et al. 2017; Tan et al. 2018). A study of mouse ES cells at 100 kb resolution revealed changes in chromosome conformations during cell cycle together with the timing of chromatin compartment and TAD formation (Nagano et al. 2017). There are conflicting reports about the existence of TADs in single cells. Nagano et al. (2013) observed them, whereas other studies reached the opposite conclusion (Flyamer et al. 2017; Stevens et al. 2017; Tan et al. 2018). Chromatin loops are dynamic structures and TADs emerge in ensemble Hi-C from weak preferences of loop positions summed over many cells. It is therefore not surprising that in any given cell loops vary. However, recent superresolution microscopy confirmed the existence of chromatin domains in single cells and supported their distinct structural features in individual cells (Bintu et al. 2018; Su et al. 2020).

Other single-cell modeling methods exist, including a manifold-based optimization (Paulsen et al. 2015), Bayesian estimation combined with gradient-based optimization (SIMBA3D) (Rosenthal et al. 2019), and Bayesian inferential structure determination combined with Markov chain Monte Carlo sampling (Carstens et al. 2016). Others use a cubic lattice representation of structures and 2D Gaussian imputation of contact matrices for 3D structure reconstructions (Zhu et al. 2019).

Several challenges remain related to sparsity of the single cell data, presence of false-positive contacts, and distinction of homologous chromosomes along with the unknown nuclear morphology, which could influence the outcome of single-cell modeling.

CONCLUDING REMARKS

Recent developments in experimental technologies provide new opportunities to probe the structure and dynamics of the nuclear genome and its structure–function relationships. However, inferring 3D structures, their dynamic attributes, and the physical processes that establish them from experimental observations alone is not a straightforward task. Consequently, computational methodologies have become a fundamental component in bridging the gap between experiments and their 3D structural interpretation. In this review, we summarized a growing number of state-of-the-art computational strategies for 3D chromatin structure modeling; these approaches are applied at various scopes and structural scales, ranging from simulations of individual gene loci, to chromosomes, to entire genomes. Both mechanistic and data-driven modeling tools have provided critical insights into 3D chromatin organization and underlying mechanisms, which inspired new hypotheses and triggered further experiments.

Despite the successes, major challenges still need to be addressed. Improving accuracy and coverage of structural models requires integration of complementary data modalities: in particular, further efforts are required to integrate microscopy with genomics technologies to overcome limitations of individual techniques. Moreover, it is crucial to develop standardized strategies for model assessment and define an accurate description of model uncertainties. It is also important to advance data sharing efforts, to allow a standardized access to structural models for the community. This would likely require a community effort to develop databanks, common file formats with standardized information about input data and parameters settings to ensure maximal reproducibility.