Anomalous diffusion, spatial coherence, and viscoelasticity from the energy landscape of human chromosomes

Organization and Dynamics of Chromosomes

How long thread-like eukaryotic chromosomes fit tidily in the small volume of the nucleus without significant entanglement is just beginning to be understood, thanks to major advances in experimental techniques. Several polymer models, which reproduce contact maps that measure the probabilities that two loci are in spatial contact, have predicted the 3D structures of interphase chromosomes. Data-driven approaches, using contact maps as input, predict that mitotic helical chromosomes are characterized by a switch in handedness, referred to as perversion. By using experimentally derived effective interactions between chromatin loci in simulations, structures of conventional and inverted nuclei have been accurately predicted. Polymer theory and simulations show that the dynamics of individual loci in chromatin exhibit subdiffusive behavior but the diffusion exponents are broadly distributed, which accords well with experiments. Although coarse-grained models are successful, many challenging problems remain, which require the creation of new experimental and computational tools to understand genome biology.

AB, Brahmachari S, Wolynes PG, Di Pierro M, Onuchic JN. Energy landscape analysis of the development of the chromosome structure across the cell cycle

During mitosis, there are significant structural changes in chromosomes. We used a maximum entropy approach to invert experimental Hi-C data to generate effective energy landscapes for chromosomal structures at different stages during the cell cycle. Modeled mitotic structures show a hierarchical organization of helices of helices. High-periodicity loops span hundreds of kilobases or less, while the other low-periodicity ones are larger in genomic separation, spanning several megabases. The structural ensembles reveal a progressive decrease in compartmentalization from interphase to mitosis, accompanied by the appearance of a second diagonal in prometaphase, indicating an organized array of loops. While there is a local tendency to form chiral helices, overall, no preferential left-handed or right-handed chirality appears to develop on the time scale of the cell cycle. Chromatin thus appears to be a liquid crystal containing numerous defects that anneal rather slowly.

Transport of partially active polymers in chemical gradients

The transport of molecules for chemical reactions is critically important in various cellular biological processes. Despite thermal diffusion being prevalent in many biochemical processes, it is unreliable for any sort of directed transport or preferential accumulation of molecules. In this paper, we propose a strategy for directed motion in which the molecules are transported by partially active polymeric structures. These polymers are assumed to be Rouse chains, in which the monomers are connected via harmonic springs and these chains are studied in environments that have activity varying spatially. The transport of such polymers is facilitated by these chemical/activity gradients which generate an effective drift. By marginalizing out the active degrees of freedom of the system, we obtain an effective Fokker-Planck equation for the Rouse modes of the polymer. In particular, we solve for the steady state distribution of the center of mass and its mean first passage time to reach an intended destination. We focus on how the arrangement of active units within the polymer affects its steady-state and dynamic behavior and how they can be optimized to achieve high accumulation or rapid motility.

Non-Gaussian behavior in fractional Laplace motion with drift

We study fractional Laplace motion (FLM) obtained from subordination of fractional Brownian motion (FBM) to a gamma process in the presence of an external drift that acts on the composite process or of an internal drift acting solely on the parental process. We derive the statistical properties of this FLM process and find that the external drift does not influence the mean-squared displacement, whereas the internal drift leads to normal diffusion, dominating at long times in the subdiffusive Hurst exponent regime. We also investigate the intricate properties of the probability density function (PDF), demonstrating that it possesses a central Gaussian region whose expansion in time is influenced by FBM's Hurst exponent. Outside of this region, the PDF follows a non-Gaussian pattern. The kurtosis of this FLM process converges toward the Gaussian limit at long times insensitive to the extreme non-Gaussian tails. Additionally, in the presence of the external drift, the PDF remains symmetric and centered at x=vt. In contrast, for the internal drift this symmetry is broken. The results of our computer simulations are fully consistent with the theoretical predictions. The FLM model is suitable for describing stochastic processes with a non-Gaussian PDF and long-ranged correlations of the motion.

Decoding biomolecular condensate dynamics: an energy landscape approach

Many eukaryotic proteins and RNAs contain low-complexity domains (LCDs) with a strong propensity for binding and driving phase separation into biomolecular condensates. Mutations in LCDs frequently disrupt condensate dynamics, resulting in pathological transitions to solid-like states. Understanding how the molecular sequence grammar of LCDs governs condensate dynamics is essential for uncovering their biological functions and the evolutionary forces that shape these sequences. To this end, we present an energy landscape framework that operates on a continuous 'stickiness' energy scale rather than relying on an explicit alphabet-based sequence. Sequences are characterized by Wasserstein distance relative to thoroughly shuffled or random counterparts. Armed with an energy landscape framework, map diagrams of material and dynamical properties governed by key energy landscape features modulated by the degree of complexity in LCD arrangements, including the periodicity and local disorder in LCDs. Highly periodic LCD patterns promote elasticity-dominated behavior, while random sequences exhibit viscosity-dominated properties. Our results reveal that minimum sticker periodicity is crucial for maintaining fluidity in condensates, thereby avoiding transitions to glassy or solid-like states. Moreover, we demonstrate that the energy landscape framework explains the recent experimental findings on prion domains and predicts systematic alterations in condensate viscoelasticity. Our work provides a unifying perspective on the sequence-encoded material properties whereby key features of energy landscapes are conserved while sequences are variable.

Chromatin folding through nonuniform motorization by responsive motor proteins

Chromatin is partially structured through the effects of biological motors. "Swimming motors" such as RNA polymerases and chromatin remodelers are thought to act differentially on the active parts of the genome and the stored inactive part. By systematically expanding the many-body master equation for chromosomes driven by swimming motors, we show that this nonuniform aspect of motorization leads to heterogeneously folded conformations, thereby contributing to chromosome compartmentalization.

Uncovering the lung cancer mechanisms through the chromosome structural ensemble characteristics and nucleation seeds

Lung cancer is one of the most common cancers in humans. However, there is still a need to understand the underlying mechanisms of a normal cell developing into a cancer cell. Here, we develop the chromosome dynamic structural model and quantify the important characteristics of the chromosome structural ensemble of the normal lung cell and the lung cancer A549 cell. Our results demonstrate the essential relationship among the chromosome ensemble, the epigenetic marks, and the gene expressions, which suggests the linkage between chromosome structure and function. The analysis reveals that the lung cancer cell may have a higher level of relative ensemble fluctuation (micro CFI) and a higher degree of phase separation between the two compartments than the normal lung cell. In addition, the significant conformational "switching off" events (from compartment A to B) are more than the significant conformational "switching on" events during the lung cancerization. We identify "nucleation seeds" or hot spots in chromosomes, which initiate the transitions and determine the mechanisms. The hot spots and interaction network results reveal that the lung cancerization process (from normal lung to A549) and the reversion process have different mechanisms. These investigations have revealed the cell fate determination mechanism of the lung cancer process, which will be helpful for the further prevention and control of cancers.

Fluctuating Chromatin Facilitates Enhancer-Promoter Communication by Regulating Transcriptional Clustering Dynamics

Enhancers regulate gene expression by forming contacts with distant promoters. Phase-separated condensates or clusters formed by transcription factors (TFs) and cofactors are thought to facilitate these enhancer-promoter (E-P) interactions. Using polymer physics, we developed distinct coarse-grained chromatin models that produce similar ensemble-averaged Hi-C maps but with "stable" and "dynamic" characteristics. Our findings, consistent with recent experiments, reveal a multistep E-P communication process. The dynamic model facilitates E-P proximity by enhancing TF clustering and subsequently promotes direct E-P interactions by destabilizing the TF clusters through chain flexibility. Our study promotes physical understanding of the molecular mechanisms governing E-P communication in transcriptional regulation.

Transcription-dependent mobility of single genes and genome-wide motions in live human cells

The human genome is highly dynamic across all scales. At the gene level, chromatin is persistently remodeled and rearranged during active processes such as transcription, replication and DNA repair. At the genome level, chromatin moves in micron-scale domains that break up and re-form over seconds, but the origin of these coherent motions is unknown. Here, we investigate the connection between genomic motions and gene-level activity. Simultaneous mapping of single-gene and genome-wide motions shows that the coupling of gene transcriptional activity to flows of the nearby genome is modulated by chromatin compaction. A motion correlation analysis suggests that a single active gene drives larger-scale motions in low-compaction regions, but high-compaction chromatin drives gene motion regardless of its activity state. By revealing unexpected connections among gene activity, spatial heterogeneities of chromatin and its emergent genome-wide motions, these findings uncover aspects of the genome's spatiotemporal organization that directly impact gene regulation and expression.

Decoding Biomolecular Condensate Dynamics: An Energy Landscape Approach

A significant fraction of eukaryotic proteins contain low-complexity sequence elements with unknown functions. Many of these sequences are prone to form biomolecular condensates with unique material and dynamic properties. Mutations in low-complexity regions often result in abnormal phase transitions into pathological solid-like states. Therefore, understanding how the low-complexity sequence patterns encode the material properties of condensates is crucial for uncovering the cellular functions and evolutionary forces behind the emergence of low-complexity regions in proteins. In this work, we employ an alphabet-free energy landscape framework of the stickers and spacers to dissect how the low complexity patterns of proteins encode the material properties of condensates. We find a broad phase diagram of material properties determined by distinct energy landscape features, showing that periodic repeat motifs promote elastic-dominated while random sequences are viscous-dominated properties. We find that a certain degree of sticker periodicity is necessary to maintain the fluidity of condensates, preventing them from forming glassy or solid-like states. Finally, we show that the energy landscape framework captures viscoelastic trends seen in the recent experiments on prion domains and makes predictions for systematic variation of protein condensate viscoelasticity via altering the periodicity and strength of sticker motifs.

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Mesoscale molecular assembly is favored by the active, crowded cytoplasm

The mesoscale organization of molecules into membraneless biomolecular condensates is emerging as a key mechanism of rapid spatiotemporal control in cells. Principles of biomolecular condensation have been revealed through in vitro reconstitution. However, intracellular environments are much more complex than test-tube environments: they are viscoelastic, highly crowded at the mesoscale, and are far from thermodynamic equilibrium due to the constant action of energy-consuming processes. We developed synDrops, a synthetic phase separation system, to study how the cellular environment affects condensate formation. Three key features enable physical analysis: synDrops are inducible, bioorthogonal, and have well-defined geometry. This design allows kinetic analysis of synDrop assembly and facilitates computational simulation of the process. We compared experiments and simulations to determine that macromolecular crowding promotes condensate nucleation but inhibits droplet growth through coalescence. ATP-dependent cellular activities help overcome the frustration of growth. In particular, stirring of the cytoplasm by actomyosin dynamics is the dominant mechanism that potentiates droplet growth in the mammalian cytoplasm by reducing confinement and elasticity. Our results demonstrate that mesoscale molecular assembly is favored by the combined effects of crowding and active matter in the cytoplasm. These results move toward a better predictive understanding of condensate formation in vivo.

Delayed Excitations Induce Polymer Looping and Coherent Motion

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

OpenNucleome for high-resolution nuclear structural and dynamical modeling

The intricate structural organization of the human nucleus is fundamental to cellular function and gene regulation. Recent advancements in experimental techniques, including high-throughput sequencing and microscopy, have provided valuable insights into nuclear organization. Computational modeling has played significant roles in interpreting experimental observations by reconstructing high-resolution structural ensembles and uncovering organization principles. However, the absence of standardized modeling tools poses challenges for furthering nuclear investigations. We present OpenNucleome-an open-source software designed for conducting GPU-accelerated molecular dynamics simulations of the human nucleus. OpenNucleome offers particle-based representations of chromosomes at a resolution of 100 KB, encompassing nuclear lamina, nucleoli, and speckles. This software furnishes highly accurate structural models of nuclear architecture, affording the means for dynamic simulations of condensate formation, fusion, and exploration of non-equilibrium effects. We applied OpenNucleome to uncover the mechanisms driving the emergence of 'fixed points' within the nucleus-signifying genomic loci robustly anchored in proximity to specific nuclear bodies for functional purposes. This anchoring remains resilient even amidst significant fluctuations in chromosome radial positions and nuclear shapes within individual cells. Our findings lend support to a nuclear zoning model that elucidates genome functionality. We anticipate OpenNucleome to serve as a valuable tool for nuclear investigations, streamlining mechanistic explorations and enhancing the interpretation of experimental observations.

Motorized chain models of the ideal chromosome

An array of motor proteins consumes chemical energy in setting up the architectures of chromosomes. Here, we explore how the structure of ideal polymer chains is influenced by two classes of motors. The first class which we call "swimming motors" acts to propel the chromatin fiber through three-dimensional space. They represent a caricature of motors such as RNA polymerases. Previously, they have often been described by adding a persistent flow onto Brownian diffusion of the chain. The second class of motors, which we call "grappling motors" caricatures the loop extrusion processes in which segments of chromatin fibers some distance apart are brought together. We analyze these models using a self-consistent variational phonon approximation to a many-body Master equation incorporating motor activities. We show that whether the swimming motors lead to contraction or expansion depends on the susceptibility of the motors, that is, how their activity depends on the forces they must exert. Grappling motors in contrast to swimming motors lead to long-ranged correlations that resemble those first suggested for fractal globules and that are consistent with the effective interactions inferred by energy landscape analyses of Hi-C data on the interphase chromosome.

Intermittent subdiffusion of short nuclear actin rods due to interactions with chromatin

The interior of cellular nuclei, the nucleoplasm, is a crowded fluid that is pervaded by protein-decorated DNA polymers, the chromatin. Due to the complex architecture of chromatin and a multitude of associated nonequilibrium processes, e.g., DNA repair, the nucleoplasm can be expected to feature nontrivial material properties and hence anomalous transport phenomena. Here, we have used single-particle tracking on nuclear actin rods to probe such transport phenomena. Our analysis reveals that short actin rods in the nucleus show an intermittent, antipersistent subdiffusion with clear signatures of fractional Brownian motion. Moreover, the diffusive motion is heterogeneous with clear signatures of an intermittent switching of trajectories between at least two different mobilities, most likely due to transient associations with chromatin. In line with this interpretation, hyperosmotic stress is seen to stall the motion of nuclear actin rods, whereas hypo-osmotic conditions yield a reptationlike motion. Our data highlights the heterogeneity of transport in the nucleoplasm that needs to be taken into account for an understanding of nucleoplasmic organization and the mechanobiology of nuclei.

Modelling Heterogeneous Anomalous Dynamics of Radiation-Induced Double-Strand Breaks in DNA during Non-Homologous End-Joining Pathway

The process of end-joining during nonhomologous repair of DNA double-strand breaks (DSBs) after radiation damage is considered. Experimental evidence has revealed that the dynamics of DSB ends exhibit subdiffusive motion rather than simple diffusion with rare directional movement. Traditional models often overlook the rare long-range directed motion. To address this limitation, we present a heterogeneous anomalous diffusion model consisting of subdiffusive fractional Brownian motion interchanged with short periods of long-range movement. Our model sheds light on the underlying mechanisms of heterogeneous diffusion in DSB repair and could be used to quantify the DSB dynamics on a time scale inaccessible to single particle tracking analysis. The model predicts that the long-range movement of DSB ends is responsible for the misrepair of DSBs in the form of dicentric chromosome lesions.

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.

Nucleosome spacing controls chromatin spatial structure and accessibility

Recent research highlights the significance of the three-dimensional structure of chromatin in regulating various cellular processes, particularly transcription. This is achieved through dynamic chromatin structures that facilitate long-range contacts and control spatial accessibility. Chromatin consists of DNA and a variety of proteins, of which histones play an essential structural role by forming nucleosomes. Extensive experimental and theoretical research in recent decades has yielded conflicting results about key factors that regulate the spatial structure of chromatin, which remains enigmatic. By using a computer model that allows us to simulate chromatin volumes containing physiological nucleosome concentrations, we investigated whether nucleosome spacing or nucleosome density is fundamental for three-dimensional chromatin accessibility. Unexpectedly, the regularity of the nucleosome spacing is crucial for determining the accessibility of the chromatin network to diffusive processes, whereas variation in nucleosome concentrations has only minor effects. Using only the basic physical properties of DNA and nucleosomes was sufficient to generate chromatin structures consistent with published electron microscopy data. Contrary to other work, we found that nucleosome density did not substantially alter the properties of chromatin fibers or contact probabilities of genomic loci. No breakup of fiber-like structures was observed at high molar density. These findings challenge previous assumptions and highlight the importance of nucleosome spacing as a key driver of chromatin organization. These results identified changes in nucleosome spacing as a tentative mechanism for altering the spatial chromatin structure and thus genomic functions.

Effect of loops on the mean-square displacement of Rouse-model chromatin

Chromatin polymer dynamics are commonly described using the classical Rouse model. The subsequent discovery, however, of intermediate-scale chromatin organization known as topologically associating domains (TADs) in experimental Hi-C contact maps for chromosomes across the tree of life, together with the success of loop extrusion factor (LEF) model in explaining TAD formation, motivates efforts to understand the effect of loops and loop extrusion on chromatin dynamics. This paper seeks to fulfill this need by combining LEF-model simulations with extended Rouse-model polymer simulations to investigate the dynamics of chromatin with loops and dynamic loop extrusion. We show that loops significantly suppress the averaged mean-square displacement (MSD) of a gene locus, consistent with recent experiments that track fluorescently labeled chromatin loci. We also find that loops reduce the MSD's stretching exponent from the classical Rouse-model value of 1/2 to a loop-density-dependent value in the 0.45-0.40 range. Remarkably, stretching exponent values in this range have also been observed in recent experiments [Weber et al., Phys. Rev. Lett. 104, 238102 (2010)0031-900710.1103/PhysRevLett.104.238102; Bailey et al., Mol. Biol. Cell 34, ar78 (2023)1059-152410.1091/mbc.E23-04-0119]. We also show that the dynamics of loop extrusion itself negligibly affects chromatin mobility. By studying static "rosette" loop configurations, we also demonstrate that chromatin MSDs and stretching exponents depend on the location of the locus in question relative to the position of the loops and on the local friction environment.

Non-Markovian Diffusion and Adsorption-Desorption Dynamics: Analytical and Numerical Results

The interplay of diffusion with phenomena like stochastic adsorption-desorption, absorption, and reaction-diffusion is essential for life and manifests in diverse natural contexts. Many factors must be considered, including geometry, dimensionality, and the interplay of diffusion across bulk and surfaces. To address this complexity, we investigate the diffusion process in heterogeneous media, focusing on non-Markovian diffusion. This process is limited by a surface interaction with the bulk, described by a specific boundary condition relevant to systems such as living cells and biomaterials. The surface can adsorb and desorb particles, and the adsorbed particles may undergo lateral diffusion before returning to the bulk. Different behaviors of the system are identified through analytical and numerical approaches.

Transcription-induced active forces suppress chromatin motion

The organization of interphase chromosomes in a number of species is starting to emerge thanks to advances in a variety of experimental techniques. However, much less is known about the dynamics, especially in the functional states of chromatin. Some experiments have shown that the motility of individual loci in human interphase chromosome decreases during transcription and increases upon inhibiting transcription. This is a counterintuitive finding because it is thought that the active mechanical force (F) on the order of ten piconewtons, generated by RNA polymerase II (RNAPII) that is presumably transmitted to the gene-rich region of the chromatin, would render it more open, thus enhancing the mobility. We developed a minimal active copolymer model for interphase chromosomes to investigate how F affects the dynamical properties of chromatin. The movements of the loci in the gene-rich region are suppressed in an intermediate range of F and are enhanced at small F values, which has also been observed in experiments. In the intermediate F, the bond length between consecutive loci increases, becoming commensurate with the distance at the minimum of the attractive interaction between nonbonded loci. This results in a transient disorder-to-order transition, leading to a decreased mobility during transcription. Strikingly, the F-dependent change in the locus dynamics preserves the organization of the chromosome at [Formula: see text]. Transient ordering of the loci, which is not found in the polymers with random epigenetic profiles, in the gene-rich region might be a plausible mechanism for nucleating a dynamic network involving transcription factors, RNAPII, and chromatin.

Prediction and Control in DNA Nanotechnology

DNA nanotechnology is a rapidly developing field that uses DNA as a building material for nanoscale structures. Key to the field's development has been the ability to accurately describe the behavior of DNA nanostructures using simulations and other modeling techniques. In this Review, we present various aspects of prediction and control in DNA nanotechnology, including the various scales of molecular simulation, statistical mechanics, kinetic modeling, continuum mechanics, and other prediction methods. We also address the current uses of artificial intelligence and machine learning in DNA nanotechnology. We discuss how experiments and modeling are synergistically combined to provide control over device behavior, allowing scientists to design molecular structures and dynamic devices with confidence that they will function as intended. Finally, we identify processes and scenarios where DNA nanotechnology lacks sufficient prediction ability and suggest possible solutions to these weak areas.

Anomalous Transport of Small Polarons Arises from Transient Lattice Relaxation or Immovable Boundaries

Elucidating transport mechanisms is crucial for advancing material design, yet state-of-the-art theory is restricted to exact simulations of small lattices with severe finite-size effects or approximate ones that assume the nature of transport. We leverage algorithmic advances to tame finite-size effects and exactly simulate small polaron formation and transport in the Holstein model. We further analyze the applicability of the ubiquitously used equilibrium-based Green-Kubo relations and nonequilibrium methods to predict charge mobility. We find that these methods can converge to different values and track this disparity to finite-size dependence and the sensitivity of Green-Kubo relations to the system's topology. Contrary to standard perturbative calculations, our results demonstrate that small polarons exhibit anomalous transport that manifests transiently due to nonequilibrium lattice relaxation or permanently as a signature of immovable boundaries. These findings can offer new interpretations of transport experiments on polymers and transition metal oxides.

Chromatin Remodeling Due to Transient-Link-and-Pass Activity Enhances Subnuclear Dynamics

Spatiotemporal coordination of chromatin and subnuclear compartments is crucial for cells. Numerous enzymes act inside nucleus-some of those transiently link and pass two chromatin segments. Here, we study how such an active perturbation affects fluctuating dynamics of an inclusion in the chromatic medium. Using numerical simulations and a versatile effective model, we categorize inclusion dynamics into three distinct modes. The transient-link-and-pass activity speeds up inclusion dynamics by affecting a slow mode related to chromatin remodeling, viz., size and shape of the chromatin meshes.

Four-Dimensional Mesoscale Liquid Model of Nucleus Resolves Chromatin's Radial Organization

Recent advances chromatin capture, imaging techniques, and polymer modeling have dramatically enhanced quantitative understanding of chromosomal folding. However, the dynamism inherent in genome architectures due to physical and biochemical forces and their impact on nuclear architecture and cellular functions remains elusive. While imaging of chromatin in four dimensions is becoming more common, there is a conspicuous lack of physics-based computational tools appropriate for revealing the forces that shape nuclear architecture and dynamics. To this end, we have developed a multiphase liquid model of the nucleus, which can resolve chromosomal territories, compartments, and nuclear lamina using a physics-based and data-informed free-energy function. The model enables rapid hypothesis-driven prototyping of nuclear dynamics in four dimensions, thereby facilitating comparison with whole nucleus imaging experiments. As an application, we model the Drosophila nucleus and map phase diagram of various possible nuclear morphologies. We shed light on the interplay of adhesive and cohesive interactions which give rise to distinct radial organization seen in conventional, inverted, and senescent nuclear architectures. The results also show the highly dynamic nature of the radial organization, the disruption of which leads to significant variability in domain coarsening dynamics and consequently variability of chromatin architecture. The model also highlights the impact of oblate nuclear geometry and heterochromatin-subtype interactions on the global chromatin architecture and local asymmetry of chromatin compartments.

Large-scale data-driven and physics-based models offer insights into the relationships among the structures, dynamics, and functions of chromosomes

The organized three-dimensional chromosome architecture in the cell nucleus provides scaffolding for precise regulation of gene expression. When the cell changes its identity in the cell-fate decision-making process, extensive rearrangements of chromosome structures occur accompanied by large-scale adaptations of gene expression, underscoring the importance of chromosome dynamics in shaping genome function. Over the last two decades, rapid development of experimental methods has provided unprecedented data to characterize the hierarchical structures and dynamic properties of chromosomes. In parallel, these enormous data offer valuable opportunities for developing quantitative computational models. Here, we review a variety of large-scale polymer models developed to investigate the structures and dynamics of chromosomes. Different from the underlying modeling strategies, these approaches can be classified into data-driven ('top-down') and physics-based ('bottom-up') categories. We discuss their contributions to offering valuable insights into the relationships among the structures, dynamics, and functions of chromosomes and propose the perspective of developing data integration approaches from different experimental technologies and multidisciplinary theoretical/simulation methods combined with different modeling strategies.

Transcription Factor Dynamics: One Molecule at a Time

Cells must tightly regulate their gene expression programs and yet rapidly respond to acute biochemical and biophysical cues within their environment. This information is transmitted to the nucleus through various signaling cascades, culminating in the activation or repression of target genes. Transcription factors (TFs) are key mediators of these signals, binding to specific regulatory elements within chromatin. While live-cell imaging has conclusively proven that TF-chromatin interactions are highly dynamic, how such transient interactions can have long-term impacts on developmental trajectories and disease progression is still largely unclear. In this review, we summarize our current understanding of the dynamic nature of TF functions, starting with a historical overview of early live-cell experiments. We highlight key factors that govern TF dynamics and how TF dynamics, in turn, affect downstream transcriptional bursting. Finally, we conclude with open challenges and emerging technologies that will further our understanding of transcriptional regulation.

Mesoscale molecular assembly is favored by the active, crowded cytoplasm

The mesoscale organization of molecules into membraneless biomolecular condensates is emerging as a key mechanism of rapid spatiotemporal control in cells1. Principles of biomolecular condensation have been revealed through in vitro reconstitution2. However, intracellular environments are much more complex than test-tube environments: They are viscoelastic, highly crowded at the mesoscale, and are far from thermodynamic equilibrium due to the constant action of energy-consuming processes3. We developed synDrops, a synthetic phase separation system, to study how the cellular environment affects condensate formation. Three key features enable physical analysis: synDrops are inducible, bioorthogonal, and have well-defined geometry. This design allows kinetic analysis of synDrop assembly and facilitates computational simulation of the process. We compared experiments and simulations to determine that macromolecular crowding promotes condensate nucleation but inhibits droplet growth through coalescence. ATP-dependent cellular activities help overcome the frustration of growth. In particular, actomyosin dynamics potentiate droplet growth by reducing confinement and elasticity in the mammalian cytoplasm, thereby enabling synDrop coarsening. Our results demonstrate that mesoscale molecular assembly is favored by the combined effects of crowding and active matter in the cytoplasm. These results move toward a better predictive understanding of condensate formation in vivo.

Leveraging polymer modeling to reconstruct chromatin connectivity from live images

Chromosomal dynamics plays a central role in a number of critical biological processes, such as transcriptional regulation, genetic recombination, and DNA replication. However, visualization of chromatin is generally limited to live imaging of a few fluorescently labeled chromosomal loci or high-resolution reconstruction of multiple loci from a single time frame. To aid in mapping the underlying chromosomal structure based on parsimonious experimental measurements, we present an exact analytical expression for the evolution of the polymer configuration based on a flexible-polymer model, and we propose an algorithm that tracks the polymer configuration from live images of chromatin marked with several fluorescent marks. Our theory identifies the resolution of microscopy needed to achieve high-accuracy tracking for a given spacing of markers, establishing the statistical confidence in the assignment of genome identity to the visualized marks. We then leverage experimental data of locus-tracking measurements to demonstrate the validity of our modeling approach and to establish a basis for the design of experiments with a desired resolution. Altogether, this work provides a computational approach founded on polymer physics that vastly improves the interpretation of in vivo measurements of biopolymer dynamics.

Anomalous diffusion, nonergodicity, non-Gaussianity, and aging of fractional Brownian motion with nonlinear clocks

How do nonlinear clocks in time and/or space affect the fundamental properties of a stochastic process? Specifically, how precisely may ergodic processes such as fractional Brownian motion (FBM) acquire predictable nonergodic and aging features being subjected to such conditions? We address these questions in the current study. To describe different types of non-Brownian motion of particles-including power-law anomalous, ultraslow or logarithmic, as well as superfast or exponential diffusion-we here develop and analyze a generalized stochastic process of scaled-fractional Brownian motion (SFBM). The time- and space-SFBM processes are, respectively, constructed based on FBM running with nonlinear time and space clocks. The fundamental statistical characteristics such as non-Gaussianity of particle displacements, nonergodicity, as well as aging are quantified for time- and space-SFBM by selecting different clocks. The latter parametrize power-law anomalous, ultraslow, and superfast diffusion. The results of our computer simulations are fully consistent with the analytical predictions for several functional forms of clocks. We thoroughly examine the behaviors of the probability-density function, the mean-squared displacement, the time-averaged mean-squared displacement, as well as the aging factor. Our results are applicable for rationalizing the impact of nonlinear time and space properties superimposed onto the FBM-type dynamics. SFBM offers a general framework for a universal and more precise model-based description of anomalous, nonergodic, non-Gaussian, and aging diffusion in single-molecule-tracking observations.

OpenABC enables flexible, simplified, and efficient GPU accelerated simulations of biomolecular condensates

Biomolecular condensates are important structures in various cellular processes but are challenging to study using traditional experimental techniques. In silico simulations with residue-level coarse-grained models strike a balance between computational efficiency and chemical accuracy. They could offer valuable insights by connecting the emergent properties of these complex systems with molecular sequences. However, existing coarse-grained models often lack easy-to-follow tutorials and are implemented in software that is not optimal for condensate simulations. To address these issues, we introduce OpenABC, a software package that greatly simplifies the setup and execution of coarse-grained condensate simulations with multiple force fields using Python scripting. OpenABC seamlessly integrates with the OpenMM molecular dynamics engine, enabling efficient simulations with performance on a single GPU that rivals the speed achieved by hundreds of CPUs. We also provide tools that convert coarse-grained configurations to all-atom structures for atomistic simulations. We anticipate that OpenABC will significantly facilitate the adoption of in silico simulations by a broader community to investigate the structural and dynamical properties of condensates.

Model chromatin flows: numerical analysis of linear and nonlinear hydrodynamics inside a sphere

We solve a hydrodynamic model of active chromatin dynamics, within a confined geometry simulating the cell nucleus. Using both analytical and numerical methods, we describe the behavior of the chromatin polymer driven by the activity of motors having polar symmetry, both in the linear response regime as well as in the long-term, fully nonlinear regime of the flows. The introduction of a boundary induces a particular geometry in the flows of chromatin, which we describe using vector spherical harmonics, a tool which greatly simplifies both our analytical and numerical approaches. We find that the long-term behavior of this model in confinement is dominated by steady, transverse flows of chromatin which circulate around the spherical domain. These circulating flows are found to be robust to perturbations, and their characteristic size is set by the size of the domain. This gives us further insight into active chromatin dynamics in the cell nucleus, and provides a foundation for development of further, more complex models of active chromatin dynamics.

Activity-Driven Phase Transition Causes Coherent Flows of Chromatin

We discover a new type of nonequilibrium phase transition in a model of chromatin dynamics, which accounts for the coherent motions that have been observed in experiment. The coherent motion is due to the long-range cooperation of molecular motors tethered to chromatin. Cooperation occurs if each motor acts simultaneously on the polymer and the surrounding solvent, exerting on them equal and opposite forces. This drives the flow of solvent past the polymer, which in turn affects the orientation of nearby motors and, if the drive is strong enough, an active polar ("ferromagnetic") phase of motors can spontaneously form. Depending on boundary conditions, either transverse flows or sustained longitudinal oscillations and waves are possible. Predicted length scales are consistent with experiments. We now have in hand a coarse-grained description of chromatin dynamics which reproduces the directed coherent flows of chromatin seen in experiments. This field-theoretic description can be analytically coupled to other features of the nuclear environment such as fluctuating or porous boundaries, local heterogeneities in the distribution of chromatin or its activity, leading to insights on the effects of activity on the cell nucleus and its contents.

Dynamic switching of transcriptional regulators between two distinct low-mobility chromatin states

How chromatin dynamics relate to transcriptional activity remains poorly understood. Using single-molecule tracking, coupled with machine learning, we show that histone H2B and multiple chromatin-bound transcriptional regulators display two distinct low-mobility states. Ligand activation results in a marked increase in the propensity of steroid receptors to bind in the lowest-mobility state. Mutational analysis revealed that interactions with chromatin in the lowest-mobility state require an intact DNA binding domain and oligomerization domains. These states are not spatially separated as previously believed, but individual H2B and bound-TF molecules can dynamically switch between them on time scales of seconds. Single bound-TF molecules with different mobilities exhibit different dwell time distributions, suggesting that the mobility of TFs is intimately coupled with their binding dynamics. Together, our results identify two unique and distinct low-mobility states that appear to represent common pathways for transcription activation in mammalian cells.

Polymer folding through active processes recreates features of genome organization

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

Structural reorganization and relaxation dynamics of axially stressed chromosomes

Chromosomes endure mechanical stresses throughout the cell cycle; for example, resulting from the pulling of chromosomes by spindle fibers during mitosis or deformation of the nucleus during cell migration. The response to physical stress is closely related to chromosome structure and function. Micromechanical studies of mitotic chromosomes have revealed them to be remarkably extensible objects and informed early models of mitotic chromosome organization. We use a data-driven, coarse-grained polymer modeling approach to explore the relationship between the spatial organization of individual chromosomes and their emergent mechanical properties. In particular, we investigate the mechanical properties of our model chromosomes by axially stretching them. Simulated stretching led to a linear force-extension curve for small strain, with mitotic chromosomes behaving about 10-fold stiffer than interphase chromosomes. Studying their relaxation dynamics, we found that chromosomes are viscoelastic solids with a highly liquid-like, viscous behavior in interphase that becomes solid-like in mitosis. This emergent mechanical stiffness originates from lengthwise compaction, an effective potential capturing the activity of loop-extruding SMC complexes. Chromosomes denature under large strains via unraveling, which is characterized by opening of large-scale folding patterns. By quantifying the effect of mechanical perturbations on the chromosome's structural features, our model provides a nuanced understanding of in vivo mechanics of chromosomes.

OpenABC Enables Flexible, Simplified, and Efficient GPU Accelerated Simulations of Biomolecular Condensates

Biomolecular condensates are important structures in various cellular processes but are challenging to study using traditional experimental techniques. In silico simulations with residue-level coarse-grained models strike a balance between computational efficiency and chemical accuracy. They could offer valuable insights by connecting the emergent properties of these complex systems with molecular sequences. However, existing coarse-grained models often lack easy-to-follow tutorials and are implemented in software that is not optimal for condensate simulations. To address these issues, we introduce OpenABC, a software package that greatly simplifies the setup and execution of coarse-grained condensate simulations with multiple force fields using Python scripting. OpenABC seamlessly integrates with the OpenMM molecular dynamics engine, enabling efficient simulations with performances on a single GPU that rival the speed achieved by hundreds of CPUs. We also provide tools that convert coarse-grained configurations to all-atom structures for atomistic simulations. We anticipate that Open-ABC will significantly facilitate the adoption of in silico simulations by a broader community to investigate the structural and dynamical properties of condensates. Open-ABC is available at https://github.com/ZhangGroup-MITChemistry/OpenABC.

4D epigenomics: deciphering the coupling between genome folding and epigenomic regulation with biophysical modeling

Recent experimental observations suggest a strong coupling between the 3D nuclear chromosome organization and epigenomics. However, the mechanistic and functional bases of such interplay remain elusive. In this review, we describe how biophysical modeling has been instrumental in characterizing how genome folding may impact the formation of epigenomic domains and, conversely, how epigenomic marks may affect chromosome conformation. Finally, we discuss how this mutual feedback loop between chromatin organization and epigenome regulation, via the formation of physicochemical nanoreactors, may represent a key functional role of 3D compartmentalization in the assembly and maintenance of stable - but yet plastic - epigenomic landscapes.

A polymer chain with dipolar active forces in connection to spatial organization of chromatin

A living cell is an active environment where the organization and dynamics of chromatin are affected by different forms of activity. Optical experiments report that loci show subdiffusive dynamics and the chromatin fiber is seen to be coherent over micrometer-scale regions. Using a bead-spring polymer chain with dipolar active forces, we study how the subdiffusive motion of the loci generate large-scale coherent motion of the chromatin. We show that in the presence of extensile (contractile) activity, the dynamics of the loci grows faster (slower) and the spatial correlation length increases (decreases) compared to the case with no dipolar forces. Hence, both the dipolar active forces modify the elasticity of the chain. Interestingly in our model, the dynamics and organization of such dipolar active chains largely differ from the passive chain with renormalized elasticity.

Diffusion in a two-dimensional energy landscape in the presence of dynamical correlations and validity of random walk model

Diffusion in a multidimensional energy surface with minima and barriers is a problem of importance in statistical mechanics and it also has wide applications, such as protein folding. To understand it in such a system, we carry out theory and simulations of a tagged particle moving on a two-dimensional periodic potential energy surface, both in the presence and absence of noise. Langevin dynamics simulations at multiple temperatures are carried out to obtain the diffusion coefficient of a solute particle. Friction is varied from zero to large values. Diffusive motion emerges in the limit of a long time, even in the absence of noise. Noise destroys the correlations and increases diffusion at small friction. Diffusion thus exhibits a nonmonotonic friction dependence at the intermediate value of the damping, ultimately converging to our theoretically predicted value. The latter is obtained using the well-established relationship between diffusion and random walk. An excellent agreement is obtained between theory and simulations in the high-friction limit but not so in the intermediate regime. We explain the deviation in the low- to intermediate-friction regime using the modified random walk theory. The rate of escape from one cell to another is obtained from the multidimensional rate theory of Langer. We find that enhanced dimensionality plays an important role. To quantify the effects of noise on the potential-imposed coherence on the trajectories, we calculate the Lyapunov exponent. At small friction values, the Lyapunov exponent mimics the friction dependence of the rate.

Interphase chromosomes of the Aedes aegypti mosquito are liquid crystalline and can sense mechanical cues

We use data-driven physical simulations to study the three-dimensional architecture of the Aedes aegypti genome. Hi-C maps exhibit both a broad diagonal and compartmentalization with telomeres and centromeres clustering together. Physical modeling reveals that these observations correspond to an ensemble of 3D chromosomal structures that are folded over and partially condensed. Clustering of the centromeres and telomeres near the nuclear lamina appears to be a necessary condition for the formation of the observed structures. Further analysis of the mechanical properties of the genome reveals that the chromosomes of Aedes aegypti, by virtue of their atypical structural organization, are highly sensitive to the deformation of the nuclei. This last finding provides a possible physical mechanism linking mechanical cues to gene regulation.

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.

Spatially coherent diffusion of human RNA Pol II depends on transcriptional state rather than chromatin motion

Gene transcription by RNA polymerase II (RNAPol II) is a tightly regulated process in the genomic, temporal, and spatial context. Recently, we have shown that chromatin exhibits spatially coherently moving regions over the entire nucleus, which is enhanced by transcription. Yet, it remains unclear how the mobility of RNA Pol II molecules is affected by transcription regulation and whether this response depends on the coordinated chromatin movement. We applied our Dense Flow reConstruction and Correlation method to analyze nucleus-wide coherent movements of RNA Pol II in living human cancer cells. We observe a spatially coherent movement of RNA Pol II molecules over ≈ 1 μm, which depends on transcriptional activity. Inducing transcription in quiescent cells decreased the coherent motion of RNA Pol II. We then quantify the spatial correlation length of RNA Pol II in the context of DNA motion. RNA Pol II and chromatin spatially coherent motions respond oppositely to transcriptional activities. Our study holds the potential of studying the chromatin environment in different nuclear processes.

Symmetry-based classification of forces driving chromatin dynamics

Chromatin - the functional form of DNA in the cell - exists in the form of a polymer immersed in a nucleoplasmic fluid inside the cell nucleus. Both chromatin and nucleoplasm are subject to active forces resulting from local biological processes. This activity leads to non-equilibrium phenomena, affecting chromatin organization and dynamics, yet the underlying physics is far from understood. Here, we expand upon a previously developed two-fluid model of chromatin and nucleoplasm by considering three types of activity in the form of force dipoles - two with both forces of the dipole acting on the same fluid (either polymer or nucleoplasm) and a third, with two forces pushing chromatin and solvent in opposite directions. We find that this latter type results in the most significant flows, dominating over most length scales of interest. Due to the friction between the fluids and their viscosity, we observe emergent screening length scales in the active flows of this system. We predict that the presence of different activity types and their relative strengths can be inferred from observing the power spectra of hydrodynamic fluctuations in the chromatin and the nucleoplasm.

Active and thermal fluctuations in multi-scale polymer structure and dynamics

The presence of athermal noise or biological fluctuations control and maintain crucial life-processes. In this work, we present an exact analytical treatment of the dynamic behavior of a flexible polymer chain that is subjected to both thermal and active forces. Our model for active forces incorporates temporal correlation associated with the characteristic time scale and processivity of enzymatic function (driven by ATP hydrolysis), leading to an active-force time scale that competes with relaxation processes within the polymer chain. We analyze the structure and dynamics of an active-Brownian polymer using our exact results for the dynamic structure factor and the looping time for the chain ends. The spectrum of relaxation times within a polymer chain implies two different behaviors at small and large length scales. Small length-scale relaxation is faster than the active-force time scale, and the dynamic and structural behavior at these scales are oblivious to active forces and, are thus governed by the true thermal temperature. Large length-scale behavior is governed by relaxation times that are much longer than the active-force time scale, resulting in an effective active-Brownian temperature that dramatically alters structural and dynamic behavior. These complex multi-scale effects imply a time-dependent temperature that governs living and non-equilibrium systems, serving as a unifying concept for interpreting and predicting their physical behavior.

Characterizing the variation in chromosome structure ensembles in the context of the nuclear microenvironment

Inside the nucleus, chromosomes are subjected to direct physical interaction between different components, active forces, and thermal noise, leading to the formation of an ensemble of three-dimensional structures. However, it is still not well understood to what extent and how the structural ensemble varies from one chromosome region or cell-type to another. We designed a statistical analysis technique and applied it to single-cell chromosome imaging data to reveal the heterogeneity of individual chromosome structures. By analyzing the resulting structural landscape, we find that the largest dynamic variation is the overall radius of gyration of the chromatin region, followed by domain reorganization within the region. By comparing different human cell-lines and experimental perturbation data using this statistical analysis technique and a network-based similarity quantification approach, we identify both cell-type and condition-specific features of the structural landscapes. We identify a relationship between epigenetic state and the properties of chromosome structure fluctuation and validate this relationship through polymer simulations. Overall, our study suggests that the types of variation in a chromosome structure ensemble are cell-type as well as region-specific and can be attributed to constraints placed on the structure by factors such as variation in epigenetic state.

Recent work has revealed principles of how chromosomes are folded and structured inside the human nucleus. It is now even possible to microscopically trace the path of chromosomes in 3D in individual cells. With this data, we can start to examine how much variation exists in chromosome structure and what biological factors may restrict or enhance this variation. Are chromosomes stuck in just a few possible positions or do they move around more freely, sampling many configurations? Here, we use a mathematical approach to compare chromosome structure variation in different cell types, at different locations along the genome, and when key structural proteins are removed. Through these comparisons and dynamic simulations of chromosome behavior, we identify factors that may constrain or promote variation in chromosome structure.

Quantifying Chromosome Structural Reorganizations during Differentiation, Reprogramming, and Transdifferentiation

We developed a nonequilibrium model to study chromosome structural reorganizations within a simplified cell developmental system. From the chromosome structural perspective, we predicted that the neural progenitor cell is on the neural developmental path and very close to the transdifferentiation path from the fibroblast to the neuron cell. We identified an early bifurcation of stem cell differentiation processes and the cell-of-origin-specific reprogramming pathways. Our theoretical results are in good agreement with available experimental evidence, promoting future applications of our approach.

Phase Separation and Correlated Motions in Motorized Genome

The human genome is arranged in the cell nucleus nonrandomly, and phase separation has been proposed as an important driving force for genome organization. However, the cell nucleus is an active system, and the contribution of nonequilibrium activities to phase separation and genome structure and dynamics remains to be explored. We simulated the genome using an energy function parametrized with chromosome conformation capture (Hi-C) data with the presence of active, nondirectional forces that break the detailed balance. We found that active forces that may arise from transcription and chromatin remodeling can dramatically impact the spatial localization of heterochromatin. When applied to euchromatin, active forces can drive heterochromatin to the nuclear envelope and compete with passive interactions among heterochromatin that tend to pull them in opposite directions. Furthermore, active forces induce long-range spatial correlations among genomic loci beyond single chromosome territories. We further showed that the impact of active forces could be understood from the effective temperature defined as the fluctuation-dissipation ratio. Our study suggests that nonequilibrium activities can significantly impact genome structure and dynamics, producing unexpected collective phenomena.

Heterogeneous interactions and polymer entropy decide organization and dynamics of chromatin domains

Chromatin is known to be organized into multiple domains of varying sizes and compaction. While these domains are often imagined as static structures, they are highly dynamic and show cell-to-cell variability. Since processes such as gene regulation and DNA replication occur in the context of these domains, it is important to understand their organization, fluctuation, and dynamics. To simulate chromatin domains, one requires knowledge of interaction strengths among chromatin segments. Here, we derive interaction-strength parameters from experimentally known contact maps and use them to predict chromatin organization and dynamics. Taking two domains on the human chromosome as examples, we investigate its three-dimensional organization, size/shape fluctuations, and dynamics of different segments within a domain, accounting for hydrodynamic effects. Considering different cell types, we quantify changes in interaction strengths and chromatin shape fluctuations in different epigenetic states. Perturbing the interaction strengths systematically, we further investigate how epigenetic-like changes can alter the spatio-temporal nature of the domains. Our results show that heterogeneous weak interactions are crucial in determining the organization of the domains. Computing effective stiffness and relaxation times, we investigate how perturbations in interactions affect the solid- and liquid-like nature of chromatin domains. Quantifying dynamics of chromatin segments within a domain, we show how the competition between polymer entropy and interaction energy influence the timescales of loop formation and maintenance of stable loops.

Uncovering the statistical physics of 3D chromosomal organization using data-driven modeling

In recent years, much effort has been devoted to understanding the three-dimensional (3D) organization of the genome and how genomic structure mediates nuclear function. The development of experimental techniques that combine DNA proximity ligation with high-throughput sequencing, such as Hi-C, have substantially improved our knowledge about chromatin organization. Numerous experimental advancements, not only utilizing DNA proximity ligation but also high-resolution genome imaging (DNA tracing), have required theoretical modeling to determine the structural ensembles consistent with such data. These 3D polymer models of the genome provide an understanding of the physical mechanisms governing genome architecture. Here, we present an overview of the recent advances in modeling the ensemble of 3D chromosomal structures by employing the maximum entropy approach combined with polymer physics. Particularly, we discuss the minimal chromatin model (MiChroM) along with the "maximum entropy genomic annotations from biomarkers associated with structural ensembles" (MEGABASE) model, which have been remarkably successful in the accurate modeling of chromosomes consistent with both Hi-C and DNA-tracing data.