The physics behind the larger scale organization of DNA in eukaryotes

Chromatin structure from high resolution microscopy: Scaling laws and microphase separation

Recent advances in experimental fluorescence microscopy allow high accuracy determination (resolution of 50 nm) of the three-dimensional physical location of multiple (up to ∼10^{2}) tagged regions of the chromosome. We investigate publicly available microscopy data for two loci of the human Chr21 obtained from multiplexed fluorescence in situ hybridization (FISH) methods for different cell lines and treatments. Inspired by polymer physics models, our analysis centers around distance distributions between different tags with the aim being to unravel the chromatin conformational arrangements. We show that for any specific genomic site, there are (at least) two different conformational arrangements of chromatin, implying coexisting distinct topologies which we refer to as phase α and phase β. These two phases show different scaling behaviors: the former is consistent with a crumpled globule, while the latter indicates a confined, but more extended conformation, such as a looped domain. The identification of these distinct phases sheds light on the coexistence of multiple chromatin topologies and provides insights into the effects of cellular context and/or treatments on chromatin structure.

Mesoscale, long-time mixing of chromosomes and its connection to polymer dynamics

Chromosomes are arranged in distinct territories within the nucleus of animal cells. Recent experiments have shown that these territories overlap at their edges, suggesting partial mixing during interphase. Experiments that knock-down of condensin II proteins during interphase indicate increased chromosome mixing, which demonstrates control of the mixing. In this study, we use a generic polymer simulation to quantify the dynamics of chromosome mixing over time. We introduce the chromosome mixing index, which quantifies the mixing of distinct chromosomes in the nucleus. We find that the chromosome mixing index in a small confinement volume (as a model of the nucleus), increases as a power-law of the time, with the scaling exponent varying non-monotonically with self-interaction and volume fraction. By comparing the chromosome mixing index with both monomer subdiffusion due to (non-topological) intermingling of chromosomes as well as even slower reptation, we show that for relatively large volume fractions, the scaling exponent of the chromosome mixing index is related to Rouse dynamics for relatively weak chromosome attractions and to reptation for strong attractions. In addition, we extend our model to more realistically account for the situation of the Drosophila chromosome by including the heterogeneity of the polymers and their lengths to account for microphase separation of euchromatin and heterochromatin and their interactions with the nuclear lamina. We find that the interaction with the lamina further impedes chromosome mixing.

Circuit topology analysis of cellular genome reveals signature motifs, conformational heterogeneity, and scaling

Reciprocal regulation of genome topology and function is a fundamental and enduring puzzle in biology. The wealth of data provided by Hi-C libraries offers the opportunity to unravel this relationship. However, there is a need for a comprehensive theoretical framework in order to extract topological information for genome characterization and comparison. Here, we develop a toolbox for topological analysis based on Circuit Topology, allowing for the quantification of inter- and intracellular genomic heterogeneity, at various levels of fold complexity: pairwise contact arrangement, higher-order contact arrangement, and topological fractal dimension. Single-cell Hi-C data were analyzed and characterized based on topological content, revealing not only a strong multiscale heterogeneity but also highly conserved features such as a characteristic topological length scale and topological signature motifs in the genome. We propose that these motifs inform on the topological state of the nucleus and indicate the presence of active loop extrusion.

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Circuit topology quantifies heterogeneity in genomic arrangement

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Scale analysis reveals a characteristic length scale of 10 Mb in genome topology

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We identify highly conserved topological structures related to loop extrusion

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We suggest a topological model of chromatin arrangement for loop extrusion, the L-loop

Macromolecule crowding effects on the phase separation of semi-flexible polymer in spherical confined space

Current works focus on detecting macromolecule crowding effects on the phase separation of the mixture between semi-flexible polymer and crowders (hydrophilic polymers) in confined space by Monte Carlo simulations. With the increasing addition of crowders into the spherical confined space, the semi-flexible polymer was first compressed into a condensed state from the initial coil state, and then the condensed conformation expanded and deposited on the inner surface of the spherical confined space with an extended state. The phase diagram in the phase space of the volume fraction of crowders and the scaled radius of spherical confined space by crowder diameter, and the direct conformation transition of semi-flexible polymer have validated the phase transition process successfully. In addition, the deposition of extended conformation on the inner surface of the spherical confined space was qualified by the vertex density, its curve shifted along the radial direction with the increasing volume fraction of crowder. During the phase separation process, the critical volume fraction φ^∗ relates to the crowder diameter approximately linearly and the relation between the critical volume fraction and the crowder diameter strongly depends on the size of the spherical confined space.

Circuit Topology Analysis of Polymer Folding Reactions

Circuit topology is emerging as a versatile measure to classify the internal structures of folded linear polymers such as proteins and nucleic acids. The topology framework can be applied to a wide range of problems, most notably molecular folding reactions that are central to biology and molecular engineering. In this Outlook, we discuss the state-of-the art of the technology and elaborate on the opportunities and challenges that lie ahead.

Multiple particle tracking analysis in isolated nuclei reveals the mechanical phenotype of leukemia cells

The nucleus is fundamentally composed by lamina and nuclear membranes that enclose the chromatin, nucleoskeletal components and suspending nucleoplasm. The functional connections of this network integrate external stimuli into cell signals, including physical forces to mechanical responses of the nucleus. Canonically, the morphological characteristics of the nucleus, as shape and size, have served for pathologists to stratify and diagnose cancer patients; however, novel biophysical techniques must exploit physical parameters to improve cancer diagnosis. By using multiple particle tracking (MPT) technique on chromatin granules, we designed a SURF (Speeded Up Robust Features)-based algorithm to study the mechanical properties of isolated nuclei and in living cells. We have determined the apparent shear stiffness, viscosity and optical density of the nucleus, and how the chromatin structure influences on these biophysical values. Moreover, we used our MPT-SURF analysis to study the apparent mechanical properties of isolated nuclei from patients of acute lymphoblastic leukemia. We found that leukemia cells exhibited mechanical differences compared to normal lymphocytes. Interestingly, isolated nuclei from high-risk leukemia cells showed increased viscosity than their counterparts from normal lymphocytes, whilst nuclei from relapsed-patient's cells presented higher density than those from normal lymphocytes or standard- and high-risk leukemia cells. Taken together, here we presented how MPT-SURF analysis of nuclear chromatin granules defines nuclear mechanical phenotypic features, which might be clinically relevant.

Nucleosome conformational variability in solution and in interphase nuclei evidenced by cryo-electron microscopy of vitreous sections

In Eukaryotes, DNA is wound around the histone octamer forming the basic chromatin unit, the nucleosome. Atomic structures have been obtained from crystallography and single particle cryo-electron microscopy (cryoEM) of identical engineered particles. But native nucleosomes are dynamical entities with diverse DNA sequence and histone content, and little is known about their conformational variability, especially in the cellular context. Using cryoEM and tomography of vitreous sections we analyse native nucleosomes, both in vitro, using purified particles solubilized at physiologically relevant concentrations (25–50%), and in situ, within interphase nuclei. We visualize individual nucleosomes at a level of detail that allows us to measure the distance between the DNA gyres wrapped around. In concentrated solutions, we demonstrate a salt-dependent transition, with a high salt compact conformation resembling the canonical nucleosome and an open low salt one, closer to nuclear nucleosomes. Although further particle characterization and cartography are needed to understand the relationship between this conformational variability and chromatin functional states, this work opens a route to chromatin exploration in situ.

Challenges and guidelines toward 4D nucleome data and model standards

Due to recent advances in experimental and theoretical approaches, the dynamic three-dimensional organization (3D) of the nucleus has become a very active area of research in life sciences. We now understand that the linear genome is folded in ways that may modulate how genes are expressed during the basic functioning of cells. Importantly, it is now possible to build 3D models of how the genome folds within the nucleus and changes over time (4D). Because genome folding influences its function, this opens exciting new possibilities to broaden our understanding of the mechanisms that determine cell fate. However, the rapid evolution of methods and the increasing complexity of data can result in ambiguity and reproducibility challenges, which may hamper the progress of this field. Here, we describe such challenges ahead and provide guidelines to think about strategies for shared standardized validation of experimental 4D nucleome data sets and models.

Molecular Dynamics simulations of the Strings and Binders Switch model of chromatin

In recent years interest has grown on the applications of polymer physics to model chromatin folding in order to try to make sense of the complexity of experimental data emerging from new technologies such as Hi-C or GAM, in a principled way. Here we review the methods employed to efficiently implement Molecular Dynamics computer simulations of polymer models, focusing in particular on the String&Binders Switch (SBS) model. The constant improvement of such methods and computer power is returning increasingly more accurate insights on the structure and molecular mechanisms underlying the spatial organization of chromosomes in the cell nucleus. We aim to provide an account of the state of the art of computational techniques employed in this type of investigations and to review recent applications of such methods to the description of real genomic loci, such as the Sox9 locus in mESC.

The biology and polymer physics underlying large-scale chromosome organization

Chromosome large-scale organization is a beautiful example of the interplay between physics and biology. DNA molecules are polymers and thus belong to the class of molecules for which physicists have developed models and formulated testable hypotheses to understand their arrangement and dynamic properties in solution, based on the principles of polymer physics. Biologists documented and discovered the biochemical basis for the structure, function and dynamic spatial organization of chromosomes in cells. The underlying principles of chromosome organization have recently been revealed in unprecedented detail using high-resolution chromosome capture technology that can simultaneously detect chromosome contact sites throughout the genome. These independent lines of investigation have now converged on a model in which DNA loops, generated by the loop extrusion mechanism, are the basic organizational and functional units of the chromosome.

Volume Transitions of Isolated Cell Nuclei Induced by Rapid Temperature Increase

Understanding the physical mechanisms governing nuclear mechanics is important as it can impact gene expression and development. However, how cell nuclei respond to external cues such as heat is not well understood. Here, we studied the material properties of isolated nuclei in suspension using an optical stretcher. We demonstrate that isolated nuclei regulate their volume in a highly temperature-sensitive manner. At constant temperature, isolated nuclei behaved like passive, elastic and incompressible objects, whose volume depended on the pH and ionic conditions. When the temperature was increased suddenly by even a few degrees Kelvin, nuclei displayed a repeatable and reversible temperature-induced volume transition, whose sign depended on the valency of the solvent. Such phenomenon is not observed for nuclei subjected to slow heating. The transition temperature could be shifted by adiabatic changes of the ambient temperature, and the magnitude of temperature-induced volume transition could be modulated by modifying the chromatin compaction state and remodeling processes. Our findings reveal that the cell nucleus can be viewed as a highly charged polymer gel with intriguing thermoresponsive properties, which might play a role in nuclear volume regulation and thermosensing in living cells.

Physical properties of the chromosomes and implications for development

Remarkable progress has been made in understanding chromosome structures inside the cell nucleus. Recent advances in Hi-C technologies enable the detection of genome-wide chromatin interactions, providing insight into three-dimensional (3D) genome organization. Advancements in the spatial and temporal resolutions of imaging as well as in molecular biological techniques allow the tracking of specific chromosomal loci, improving our understanding of chromosome movements. From these data, we are beginning to understand how the intra-nuclear locations of chromatin loci and the 3D genome structure change during development and differentiation. This emerging field of genome structure and dynamics research requires an interdisciplinary approach including efficient collaborations between experimental biologists and physicists, informaticians, or engineers. Quantitative and mathematical analyses based on polymer physics are becoming increasingly important for processing and interpreting experimental data on 3D chromosome structures and dynamics. In this review, we aim to provide an overview of recent research on the physical aspects of chromosome structure and dynamics oriented for biologists. These studies have mainly focused on chromosomes at the cellular level, using unicellular organisms and cultured cells. However, physical parameters that change during development, such as nuclear size, may impact genome structure and dynamics. Here, we discuss how chromatin dynamics and genome structures in early embryos change during development, which we expect will be a hot topic in the field of chromatin dynamics in the near future. We hope this review helps developmental biologists to quantitatively investigate the physical natures of chromosomes in developmental biology research.

Double-Strand Breaks in Genome-Sized DNA Caused by Ultrasound

DNA double-strand breaks (DSBs) caused by ultrasound were evaluated in a quantitative manner by single-molecule fluorescence microscopy. We compared the effect of time-interval (or pulse) sonication to that of continuous wave (CW) sonication at a fixed frequency of 30 kHz. Pulses caused fewer DSBs than CW sonication under the same total input ultrasound energy when the pulse repetition period was above the order of a second. In contrast, pulses caused more DSBs than CW sonication for pulse widths shorter than a second. These effect of ultrasound on DNA were interpreted in terms of the time-dependent decay in the probability of breakage during the duration of a pulse. We propose a simple phenomenological model by considering a characteristic decay in the probability of DSBs during single-pulse sonication, which reproduces the essence of the experimental trend. In addition, a data analysis revealed a characteristic scaling behavior between the number of pulses and the number of DSBs.

Polymer models of the hierarchical folding of the Hox-B chromosomal locus

As revealed by novel technologies, chromosomes in the nucleus of mammalian cells have a complex spatial organization that serves vital functional purposes. Here we use models from polymer physics to identify the mechanisms that control their three-dimensional spatial organization. In particular, we investigate a model of the Hox-B locus, an important genomic region involved in embryo development, to expose the principles regulating chromatin folding and its complex behaviors in mouse embryonic stem cells. We reconstruct with high accuracy the pairwise contact matrix of the Hox-B locus as derived by Hi-C experiments and investigate its hierarchical folding dynamics. We trace back the observed behaviors to general scaling properties of polymer physics.

Mesoscale Modeling Reveals Hierarchical Looping of Chromatin Fibers Near Gene Regulatory Elements

While it is well-recognized that chromatin loops play an important role in gene regulation, structural details regarding higher order chromatin loops are only emerging. Here we present a systematic study of restrained chromatin loops ranging from 25 to 427 nucleosomes (fibers of 5-80 Kb DNA in length), mimicking gene elements studied by 3C contact data. We find that hierarchical looping represents a stable configuration that can effectively bring distant regions of the GATA-4 gene together, satisfying connections reported by 3C experiments. Additionally, we find that restrained chromatin fibers larger than 100 nucleosomes (∼20Kb) form closed plectonemes, whereas fibers shorter than 100 nucleosomes form simple hairpin loops. By studying the dependence of loop structures on internal parameters, we show that loop features are sensitive to linker histone concentration, loop length, divalent ions, and DNA linker length. Specifically, increasing loop length, linker histone concentration, and divalent ion concentration are associated with increased persistence length (or decreased bending), while varying DNA linker length in a manner similar to experimentally observed "nucleosome free regions" (found near transcription start sites) disrupts intertwining and leads to loop opening and increased persistence length in linker histone depleted (-LH) fibers. Chromatin fiber structure sensitivity to these parameters, all of which vary throughout the cell cycle, tissue type, and species, suggests that caution is warranted when using uniform polymer models to fit chromatin conformation capture genome-wide data. Furthermore, the folding geometry we observe near the transcription initiation site of the GATA-4 gene suggests that hierarchical looping provides a structural mechanism for gene inhibition, and offers tunable parameters for design of gene regulation elements.

Confinement-Induced Glassy Dynamics in a Model for Chromosome Organization

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

Crumpled globule formation during collapse of a long flexible and semiflexible polymer in poor solvent

By introducing explicit solvent particles and hydrodynamic interactions we demonstrate that crumpled globules are formed after the collapse of long polymer chains (N = 10(4)) in a poor solvent. During the collapse crumples of all sizes form sequentially, but small crumples are not stable and convert to blobs with Gaussian statistics. The observed effective mean squared distance R(2)(n) ∼ n(0.38) at n > Ne and contact probability index p(n) ∼ n(-0.5) at n ≫ Ne, which is not following either the model of a fractal globule, or the predictions for an equilibrium globule. Polymer chain stiffness pushes the system to form globular crystallite, and this freezes crumpled structure with R(2)(n) ∼ n(0.33) at n > Ne as a stable state. We note that there is some similarity to crumple globule formation and crystallization of polymer melt.

Restraint-based three-dimensional modeling of genomes and genomic domains

Chromosomes are large polymer molecules composed of nucleotides. In some species, such as humans, this polymer can sum up to meters long and still be properly folded within the nuclear space of few microns in size. The exact mechanisms of how the meters long DNA is folded into the nucleus, as well as how the regulatory machinery can access it, is to a large extend still a mystery. However, and thanks to newly developed molecular, genomic and computational approaches based on the Chromosome Conformation Capture (3C) technology, we are now obtaining insight on how genomes are spatially organized. Here we review a new family of computational approaches that aim at using 3C-based data to obtain spatial restraints for modeling genomes and genomic domains.

Inferential modeling of 3D chromatin structure

For eukaryotic cells, the biological processes involving regulatory DNA elements play an important role in cell cycle. Understanding 3D spatial arrangements of chromosomes and revealing long-range chromatin interactions are critical to decipher these biological processes. In recent years, chromosome conformation capture (3C) related techniques have been developed to measure the interaction frequencies between long-range genome loci, which have provided a great opportunity to decode the 3D organization of the genome. In this paper, we develop a new Bayesian framework to derive the 3D architecture of a chromosome from 3C-based data. By modeling each chromosome as a polymer chain, we define the conformational energy based on our current knowledge on polymer physics and use it as prior information in the Bayesian framework. We also propose an expectation-maximization (EM) based algorithm to estimate the unknown parameters of the Bayesian model and infer an ensemble of chromatin structures based on interaction frequency data. We have validated our Bayesian inference approach through cross-validation and verified the computed chromatin conformations using the geometric constraints derived from fluorescence in situ hybridization (FISH) experiments. We have further confirmed the inferred chromatin structures using the known genetic interactions derived from other studies in the literature. Our test results have indicated that our Bayesian framework can compute an accurate ensemble of 3D chromatin conformations that best interpret the distance constraints derived from 3C-based data and also agree with other sources of geometric constraints derived from experimental evidence in the previous studies. The source code of our approach can be found in https://github.com/wangsy11/InfMod3DGen.

Thermodynamic model of heterochromatin formation through epigenetic regulation

Gene regulation in eukaryotes requires the segregation of silenced genomic regions into densely packed heterochromatin, leaving the active genes in euchromatin regions more accessible. We introduce a model that connects the presence of epigenetically inherited histone marks, methylation at histone 3 lysine-9, to the physical compaction of chromatin fibers via the binding of heterochromatin protein 1 (HP1). Our model demonstrates some of the key physical features that are necessary to explain experimental observations. In particular, we demonstrate that strong cooperative interactions among the HP1 proteins are necessary to see the phase segregation of heterochromatin and euchromatin regions. We also explore how the cell can use the concentration of HP1 to control condensation and under what circumstances there is a threshold of methylation over which the fibers will compact. Finally, we consider how different potential in vivo fiber structures as well as the flexibility of the histone 3 tail can affect the bridging of HP1. Many of the observations that we make about the HP1 system are guided by general thermodynamics principles and thus could play a role in other DNA organizational processes such as the binding of linker histones.

Chiral symmetry breaking of a double-stranded helical chain through bend-writhe coupling

This paper explores asymmetric elasticity of a double-stranded helical chain, which serves as a minimal model of biopolymers. The model consists of two elastic chains that mutually intertwine in a right-handed manner, forming a double-stranded helix. A simple numerical experiment for structural relaxation, which reduces the total elastic energy of the model monotonically without thermal fluctuations, reveals possible asymmetric elasticity inherent in the helical chain. It is first shown that a short segment of the double-stranded helical chain has a tendency to unwind when it is bent. It is also shown that a short segment of the helical chain has a tendency to writhe in the left direction upon bending. This tendency gives rise to a propensity for a longer segment of the chain to form a left-handed superhelix spontaneously upon bending. Finally, this propensity of the helical chain to form a left-handed superhelix is proposed to be a possible origin of the uniform left-handed wrapping of DNA around nucleosome core particles in nature. The results presented here could provide deeper insights into the roles and significance of helical chirality of biopolymers.

Influence of Rigidity and Knot Complexity on the Knotting of Confined Polymers

We employ computer simulations and thermodynamic integration to analyze the effects of bending rigidity and slit confinement on the free energy cost of tying knots, ΔF_knotting, on polymer chains under tension. A tension-dependent, nonzero optimal stiffness κ_min exists, for which ΔF_knotting is minimal. For a polymer chain with several stiffness domains, each containing a large amount of monomers, the domain with stiffness κ_min will be preferred by the knot. A local analysis of the bending in the interior of the knot reveals that local stretching of chains at the braid region is responsible for the fact that the tension-dependent optimal stiffness has a nonzero value. The reduction in ΔF_knotting for a chain with optimal stiffness relative to the flexible chain can be enhanced by tuning the slit width of the 2D confinement and increasing the knot complexity. The optimal stiffness itself is independent of the knot types we considered, while confinement shifts it toward lower values.

Models of chromosome structure

Understanding the mechanisms that control chromosome folding in the nucleus of eukaryotes and their contribution to gene regulation is a key open issue in molecular biology. Microscopy and chromatin-capture techniques have shown that chromatin has a complex organization, which dynamically changes across organisms and cell types. The need to make sense of such a fascinating complexity has prompted the development of quantitative models from physics, to find the principles of chromosome folding, its origin and function. Here, we concisely review recent advances in chromosome modeling, focusing on a recently proposed framework, the Strings & Binders Switch (SBS) model, which recapitulates key features of chromosome organization in space and time.

Information measure for long-range correlated sequences: the case of the 24 human chromosomes

A new approach to estimate the Shannon entropy of a long-range correlated sequence is proposed. The entropy is written as the sum of two terms corresponding respectively to power-law (ordered) and exponentially (disordered) distributed blocks (clusters). The approach is illustrated on the 24 human chromosome sequences by taking the nucleotide composition as the relevant information to be encoded/decoded. Interestingly, the nucleotide composition of the ordered clusters is found, on the average, comparable to the one of the whole analyzed sequence, while that of the disordered clusters fluctuates. From the information theory standpoint, this means that the power-law correlated clusters carry the same information of the whole analysed sequence. Furthermore, the fluctuations of the nucleotide composition of the disordered clusters are linked to relevant biological properties, such as segmental duplications and gene density.

Relevance and limitations of crowding, fractal, and polymer models to describe nuclear architecture

Chromosome architecture plays an essential role for all nuclear functions, and its physical description has attracted considerable interest over the last few years among the biophysics community. These researches at the frontiers of physics and biology have been stimulated by the demand for quantitative analysis of molecular biology experiments, which provide comprehensive data on chromosome folding, or of live cell imaging experiments that enable researchers to visualize selected chromosome loci in living or fixed cells. In this review our goal is to survey several nonmutually exclusive models that have emerged to describe the folding of DNA in the nucleus, the dynamics of proteins in the nucleoplasm, or the movements of chromosome loci. We focus on three classes of models, namely molecular crowding, fractal, and polymer models, draw comparisons, and discuss their merits and limitations in the context of chromosome structure and dynamics, or nuclear protein navigation in the nucleoplasm. Finally, we identify future challenges in the roadmap to a unified model of the nuclear environment.

From a melt of rings to chromosome territories: the role of topological constraints in genome folding

We review pro and contra of the hypothesis that generic polymer properties of topological constraints are behind many aspects of chromatin folding in eukaryotic cells. For that purpose, we review, first, recent theoretical and computational findings in polymer physics related to concentrated, topologically simple (unknotted and unlinked) chains or a system of chains. Second, we review recent experimental discoveries related to genome folding. Understanding in these fields is far from complete, but we show how looking at them in parallel sheds new light on both.

Structure-driven homology pairing of chromatin fibers: the role of electrostatics and protein-induced bridging

Chromatin domains formed in vivo are characterized by different types of 3D organization of interconnected nucleosomes and architectural proteins. Here, we quantitatively test a hypothesis that the similarities in the structure of chromatin fibers (which we call "structural homology") can affect their mutual electrostatic and protein-mediated bridging interactions. For example, highly repetitive DNA sequences in heterochromatic regions can position nucleosomes so that preferred inter-nucleosomal distances are preserved on the surfaces of neighboring fibers. On the contrary, the segments of chromatin fiber formed on unrelated DNA sequences have different geometrical parameters and lack structural complementarity pivotal for stable association and cohesion. Furthermore, specific functional elements such as insulator regions, transcription start and termination sites, and replication origins are characterized by strong nucleosome ordering that might induce structure-driven iterations of chromatin fibers. We propose that shape-specific protein-bridging interactions facilitate long-range pairing of chromatin fragments, while for closely-juxtaposed fibers electrostatic forces can in addition yield fine-tuned structure-specific recognition and pairing. These pairing effects can account for some features observed for mitotic and inter-phase chromatins.

Exploring the three-dimensional organization of genomes: interpreting chromatin interaction data

How DNA is organized in three dimensions inside the cell nucleus and how this affects the ways in which cells access, read and interpret genetic information are among the longest standing questions in cell biology. Using newly developed molecular, genomic and computational approaches based on the chromosome conformation capture technology (such as 3C, 4C, 5C and Hi-C), the spatial organization of genomes is being explored at unprecedented resolution. Interpreting the increasingly large chromatin interaction data sets is now posing novel challenges. Here we describe several types of statistical and computational approaches that have recently been developed to analyse chromatin interaction data.

Toward a unified physical model of nucleosome patterns flanking transcription start sites

Recent genome-wide maps of nucleosome positions in different eukaryotes revealed patterns around transcription start sites featuring a nucleosome-free region flanked by a periodic modulation of the nucleosome density. For Saccharomyces cerevisiae, the average in vivo pattern was previously shown to be quantitatively described by a "nucleosome gas" model based on the statistical positioning mechanism. However, this simple physical description is challenged by the fact that the pattern differs quantitatively between species and by recent experiments that appear incompatible with statistical positioning, indicating important roles for chromatin remodelers. We undertake a data-driven search for a unified physical model to describe the nucleosome patterns of 12 yeast species and also consider an extension of the model to capture remodeling effects. We are led to a nucleosome gas that takes into account nucleosome breathing, i.e., transient unwrapping of nucleosomal DNA segments. This known biophysical property of nucleosomes rationalizes a "pressure"-induced dependence of the effective nucleosome size that is suggested by the data. By fitting this model to the data, we find an average energy cost for DNA unwrapping consistent with previous biophysical experiments. Although the available data are not sufficient to reconstruct chromatin remodeling mechanisms, a minimal model extension by one mechanism yields an "active nucleosome gas" that can rationalize the behavior of systems with reduced histone-DNA ratio and remodeler knockouts. We therefore establish a basis for a physical description of nucleosome patterns that can serve as a null model for sequence-specific effects at individual genes and in models of transcription regulation.

Weakly charged cationic nanoparticles induce DNA bending and strand separation

Weakly charged cationic nanoparticles cause structural changes including local denaturing and compaction to DNA under mild conditions. The charged ligands bind to the phosphate backbone of DNA and the uncharged ligands penetrate the helix and disrupt base pairing. Mobility shifts in electrophoresis, molecular dynamics, and UV-vis spectrophotometry give clues to the details of the interactions.

A fractal model for nuclear organization: current evidence and biological implications

Chromatin is a multiscale structure on which transcription, replication, recombination and repair of the genome occur. To fully understand any of these processes at the molecular level under physiological conditions, a clear picture of the polymorphic and dynamic organization of chromatin in the eukaryotic nucleus is required. Recent studies indicate that a fractal model of chromatin architecture is consistent with both the reaction-diffusion properties of chromatin interacting proteins and with structural data on chromatin interminglement. In this study, we provide a critical overview of the experimental evidence that support a fractal organization of chromatin. On this basis, we discuss the functional implications of a fractal chromatin model for biological processes and propose future experiments to probe chromatin organization further that should allow to strongly support or invalidate the fractal hypothesis.

Revisiting higher-order and large-scale chromatin organization

The past several years has seen increasing appreciation for plasticity of higher-level chromatin folding. Four distinct '30nm' chromatin fiber structures have been identified, while new in situ imaging approaches have questioned the universality of 30nm chromatin fibers as building blocks for chromosome folding in vivo. 3C-based approaches have provided a non-microscopic, genomic approach to investigating chromosome folding while uncovering a plethora of long-distance cis interactions difficult to accommodate in traditional hierarchical chromatin folding models. Recent microscopy based studies have suggested complex topologies co-existing within linear interphase chromosome structures. These results call for a reappraisal of traditional models of higher-level chromatin folding.

Higher-order chromatin structure: bridging physics and biology

Advances in microscopy and genomic techniques have provided new insight into spatial chromatin organization inside of the nucleus. In particular, chromosome conformation capture data has highlighted the relevance of polymer physics for high-order chromatin organization. In this context, we review basic polymer states, discuss how an appropriate polymer model can be determined from experimental data, and examine the success and limitations of various polymer models of higher-order interphase chromatin organization. By taking into account topological constraints acting on the chromatin fiber, recently developed polymer models of interphase chromatin can reproduce the observed scaling of distances between genomic loci, chromosomal territories, and probabilities of contacts between loci measured by chromosome conformation capture methods. Polymer models provide a framework for the interpretation of experimental data as ensembles of conformations rather than collections of loops, and will be crucial for untangling functional implications of chromosomal organization.

Nuclear organization and chromatin dynamics in yeast: biophysical models or biologically driven interactions?

Over the past decade, tremendous progress has been made in understanding the spatial organization of genes and chromosomes. Nuclear organization can be thought of as information that is not encoded in DNA, but which nevertheless impacts gene expression. Nuclear organizational influences can be cell-specific and are potentially heritable. Thus, nuclear organization fulfills all the criteria necessary for it to be considered an authentic level of epigenetic information. Chromosomal nuclear organization is primarily dictated by the biophysical properties of chromatin. Diffusion models of polymers confined in the crowded nuclear space accurately recapitulate experimental observation. Diffusion is a Brownian process, which implies that the positions of chromosomes and genes are not defined deterministically but are likely to be dictated by the laws of probability. Despite the small size of their nuclei, budding yeast have been instrumental in discovering how epigenetic information is encoded in the spatial organization of the genome. The relatively simple organization of the yeast nucleus and the very high number of genetically identical cells that can be observed under fluorescent microscopy allow statistically robust definitions of the gene and chromosome positions in the nuclear space to be constructed. In this review, we will focus on how the spatial organization of the chromatin in the yeast nucleus might impact transcription. This article is part of a Special Issue entitled: Nuclear Transport and RNA Processing.

A model for segregation of chromatin after replication: segregation of identical flexible chains in solution

We study the segregation of two long chains from parallel but randomly twisted start conformations under good solvent conditions using Monte Carlo simulations to mimic chromatin segregation after replication in eukaryotic cells in the end of prophase. To measure the segregation process, we consider the center-of-mass separation between the two chains and the average square distance between the monomers which were connected before segregation starts. We argue that segregation is dominated by free diffusion of the chains, assuming that untwisting can be achieved by Rouse-like fluctuations on the length scale of a twisted loop. Using scaling analysis, we find that chain dynamics is in very good agreement with the free diffusion hypothesis, and segregation dynamics follows this scaling nearly. Long chains, however, show retardation effects that can be described by a new (to us) dynamical exponent, which is slightly larger than the dynamical exponent for Rouse-like diffusion. Our results indicate that nearly free diffusion of chains during a timescale of a few Rouse-times can lead to segregation of chains. A main obstacle during segregation by free diffusion is random twists between daughter strands. We have calculated the number of twists formed by the daughter strands in the start conformations, which turns out to be rather low and increases only with the square-root of the chain length.

Confinement causes opposite effects on the folding transition of a single polymer chain depending on its stiffness

We investigated the folding transition between an elongated coil state and a compact state on a single polymer chain confined in a small space with different stiffness with the aid of Monte Carlo simulation. In a flexible polymer, the folding transition is retarded in a confined space. In contrast, the transition is promoted for a semiflexible chain, in which the discontinuity of the volume change occupied by a single chain is diminished by confinement. This unique confinement effect is interpreted in terms of conformational entropy and self-avoiding repulsive interaction.

Bridging the resolution gap in structural modeling of 3D genome organization

Over the last decade, and especially after the advent of fluorescent in situ hybridization imaging and chromosome conformation capture methods, the availability of experimental data on genome three-dimensional organization has dramatically increased. We now have access to unprecedented details of how genomes organize within the interphase nucleus. Development of new computational approaches to leverage this data has already resulted in the first three-dimensional structures of genomic domains and genomes. Such approaches expand our knowledge of the chromatin folding principles, which has been classically studied using polymer physics and molecular simulations. Our outlook describes computational approaches for integrating experimental data with polymer physics, thereby bridging the resolution gap for structural determination of genomes and genomic domains.

Salt-modulated structure of polyelectrolyte-macroion complex fibers

The structure and stability of strongly charged complex fibers, formed by complexation of a single long semi-flexible polyelectrolyte chain and many oppositely charged spherical macroions, are investigated numerically at the ground-state level using a chain-sphere cell model. The model takes into account chain elasticity as well as electrostatic interactions between charged spheres and chain segments. Using a numerical optimization method based on a periodically repeated unit cell, we obtain fiber configurations that minimize the total energy. The optimal fiber configurations exhibit a variety of helical structures for the arrangement of macroions including zig-zag, solenoidal and beads-on-a-string patterns. These structures result from the competition between attraction between spheres and the polyelectrolyte chain (which favors chain wrapping around the spheres), chain bending rigidity and electrostatic repulsion between chain segments (which favor unwrapping of the chain), and the interactions between neighboring sphere-chain complexes which can be attractive or repulsive depending on the system parameters such as salt concentration, macroion charge and chain length per macroion (linker size). At about physiological salt concentration, dense zig-zag patterns are found to be energetically most stable when parameters appropriate for the DNA-histone system in the chromatin fiber are adopted. In fact, the predicted fiber diameter in this regime is found to be around 30 nanometers, which roughly agrees with the thickness observed in in vitro experiments on chromatin. We also find a macroion (histone) density of 5-6 per 11nm which agrees with results from the zig-zag or cross-linker models of chromatin. Since our study deals primarily with a generic chain-sphere model, these findings suggest that structures similar to those found for chromatin should also be observable for polyelectrolyte-macroion complexes formed in solutions of DNA and synthetic nano-colloids of opposite charge. In the ensemble where the mean linear density of spheres on the chain is fixed, the present model predicts a phase separation at intermediate salt concentrations into a densely packed complex phase and a dilute phase.

Chromatin folding – from biology to polymer models and back

There is rapidly growing evidence that folding of the chromatin fibre inside the interphase nucleus has an important role in the regulation of gene expression. In particular, the formation of loops mediated by the interaction between specific regulatory elements, for instance enhancers and promoters, is crucial in gene control. Biochemical studies that were based on the chromosome conformation capture (3C) technology have confirmed that eukaryotic genomes are highly looped. Insight into the underlying principles comes from polymer models that explore the properties of the chromatin fibre inside the nucleus. Recent models indicate that chromatin looping can explain various properties of interphase chromatin, including chromatin compaction and compartmentalisation of chromosomes. Entropic effects have a key role in these models. In this Commentary, we give an overview of the recent conjunction of ideas regarding chromatin looping in the fields of biology and polymer physics. Starting from simple linear polymer models, we explain how specific folding properties emerge upon introducing loops and how this explains a variety of experimental observations. We also discuss different polymer models that describe chromatin folding and compare them to experimental data. Experimentally testing the predictions of such polymer models and their subsequent improvement on the basis of measurements provides a solid framework to begin to understand how our genome is folded and how folding relates to function.

The fractal globule as a model of chromatin architecture in the cell

The fractal globule is a compact polymer state that emerges during polymer condensation as a result of topological constraints which prevent one region of the chain from passing across another one. This long-lived intermediate state was introduced in 1988 (Grosberg et al. 1988) and has not been observed in experiments or simulations until recently (Lieberman-Aiden et al. 2009). Recent characterization of human chromatin using a novel chromosome conformational capture technique brought the fractal globule into the spotlight as a structural model of human chromosome on the scale of up to 10 Mb (Lieberman-Aiden et al. 2009). Here, we present the concept of the fractal globule, comparing it to other states of a polymer and focusing on its properties relevant for the biophysics of chromatin. We then discuss properties of the fractal globule that make it an attractive model for chromatin organization inside a cell. Next, we connect the fractal globule to recent studies that emphasize topological constraints as a primary factor driving formation of chromosomal territories. We discuss how theoretical predictions, made on the basis of the fractal globule model, can be tested experimentally. Finally, we discuss whether fractal globule architecture can be relevant for chromatin packing in other organisms such as yeast and bacteria.