Polymer models for interphase chromosomes

The chromosome folding problem and how cells solve it

Every cell must solve the problem of how to fold its genome. We describe how the folded state of chromosomes is the result of the combined activity of multiple conserved mechanisms. Homotypic affinity-driven interactions lead to spatial partitioning of active and inactive loci. Molecular motors fold chromosomes through loop extrusion. Topological features such as supercoiling and entanglements contribute to chromosome folding and its dynamics, and tethering loci to sub-nuclear structures adds additional constraints. Dramatically diverse chromosome conformations observed throughout the cell cycle and across the tree of life can be explained through differential regulation and implementation of these basic mechanisms. We propose that the first functions of chromosome folding are to mediate genome replication, compaction, and segregation and that mechanisms of folding have subsequently been co-opted for other roles, including long-range gene regulation, in different conditions, cell types, and species.

DNA packaging by molecular motors: from bacteriophage to human chromosomes

Dense packaging of genomic DNA is crucial for organismal survival, as DNA length always far exceeds the dimensions of the cells that contain it. Organisms, therefore, use sophisticated machineries to package their genomes. These systems range across kingdoms from a single ultra-powerful rotary motor that spools the DNA into a bacteriophage head, to hundreds of thousands of relatively weak molecular motors that coordinate the compaction of mitotic chromosomes in eukaryotic cells. Recent technological advances, such as DNA proximity-based sequencing approaches, polymer modelling and in vitro reconstitution of DNA loop extrusion, have shed light on the biological mechanisms driving DNA organization in different systems. Here, we discuss DNA packaging in bacteriophage, bacteria and eukaryotic cells, which, despite their extreme variation in size, structure and genomic content, all rely on the action of molecular motors to package their genomes.

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.

The Physical Behavior of Interphase Chromosomes: Polymer Theory and Coarse-Grain Computer Simulations

Fluorescence in situ hybridization and chromosome conformation capture methods point to the same conclusion: that chromosomes appear to the external observer as compact structures with a highly nonrandom three-dimensional organization. In this work, we recapitulate the efforts made by us and other groups to rationalize this behavior in terms of the mathematical language and tools of polymer physics. After a brief introduction dedicated to some crucial experiments dissecting the structure of interphase chromosomes, we discuss at a nonspecialistic level some fundamental aspects of theoretical and numerical polymer physics. Then, we inglobe biological and polymer aspects into a polymer model for interphase chromosomes which moves from the observation that mutual topological constraints, such as those typically present between polymer chains in ordinary melts, induce slow chain dynamics and "constraint" chromosomes to resemble double-folded randomly branched polymer conformations. By explicitly turning these ideas into a multi-scale numerical algorithm which is described here in full details, we can design accurate model polymer conformations for interphase chromosomes and offer them for systematic comparison to experiments. The review is concluded by discussing the limitations of our approach and pointing to promising perspectives for future work.

Interactions Between Nucleosomes: From Atomistic Simulation to Polymer Model

The organization of genomes in space and time dimension plays an important role in gene expression and regulation. Chromatin folding occurs in a dynamic, structured way that is subject to biophysical rules and biological processes. Nucleosomes are the basic unit of chromatin in living cells, and here we report on the effective interactions between two nucleosomes in physiological conditions using explicit-solvent all-atom simulations. Free energy landscapes derived from umbrella sampling simulations agree well with recent experimental and simulation results. Our simulations reveal the atomistic details of the interactions between nucleosomes in solution and can be used for constructing the coarse-grained model for chromatin in a bottom-up manner.

Predicting Genome Architecture: Challenges and Solutions

Genome architecture plays a pivotal role in gene regulation. The use of high-throughput methods for chromatin profiling and 3-D interaction mapping provide rich experimental data sets describing genome organization and dynamics. These data challenge development of new models and algorithms connecting genome architecture with epigenetic marks. In this review, we describe how chromatin architecture could be reconstructed from epigenetic data using biophysical or statistical approaches. We discuss the applicability and limitations of these methods for understanding the mechanisms of chromatin organization. We also highlight the emergence of new predictive approaches for scoring effects of structural variations in human cells.

A comparison of mechanism-inspired models for particle relative biological effectiveness (RBE)

BACKGROUND AND SIGNIFICANCE

The application of heavy ion beams in cancer therapy must account for the increasing relative biological effectiveness (RBE) with increasing penetration depth when determining dose prescriptions and organ at risk (OAR) constraints in treatment planning. Because RBE depends in a complex manner on factors such as the ion type, energy, cell and tissue radiosensitivity, physical dose, biological endpoint, and position within and outside treatment fields, biophysical models reflecting these dependencies are required for the personalization and optimization of treatment plans.

AIM

To review and compare three mechanism-inspired models which predict the complexities of particle RBE for various ion types, energies, linear energy transfer (LET) values and tissue radiation sensitivities.

METHODS

The review of models and mechanisms focuses on the Local Effect Model (LEM), the Microdosimetric-Kinetic (MK) model, and the Repair-Misrepair-Fixation (RMF) model in combination with the Monte Carlo Damage Simulation (MCDS). These models relate the induction of potentially lethal double strand breaks (DSBs) to the subsequent interactions and biological processing of DSB into more lethal forms of damage. A key element to explain the increased biological effectiveness of high LET ions compared to MV x rays is the characterization of the number and local complexity (clustering) of the initial DSB produced within a cell. For high LET ions, the spatial density of DSB induction along an ion's trajectory is much greater than along the path of a low LET electron, such as the secondary electrons produced by the megavoltage (MV) x rays used in conventional radiation therapy. The main aspects of the three models are introduced and the conceptual similarities and differences are critiqued and highlighted. Model predictions are compared in terms of the RBE for DSB induction and for reproductive cell survival.

RESULTS AND CONCLUSIONS

Comparisons of the RBE for DSB induction and for cell survival are presented for proton (¹ H), helium (⁴ He), and carbon (¹² C) ions for the therapeutically most relevant range of ion beam energies. The reviewed models embody mechanisms of action acting over the spatial scales underlying the biological processing of potentially lethal DSB into more lethal forms of damage. Differences among the number and types of input parameters, relevant biological targets, and the computational approaches among the LEM, MK and RMF models are summarized and critiqued. Potential experiments to test some of the seemingly contradictory aspects of the models are discussed.

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.

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.

Quantified effects of chromosome-nuclear envelope attachments on 3D organization of chromosomes

We use a combined experimental and computational approach to study the effects of chromosome-nuclear envelope (Chr-NE) attachments on the 3D genome organization of Drosophila melanogaster (fruit fly) salivary gland nuclei. We consider 3 distinct models: a Null model - without specific Chr-NE attachments, a 15-attachment model - with 15 previously known Chr-NE attachments, and a 48-attachment model - with 15 original and 33 recently identified Chr-NE attachments. The radial densities of chromosomes in the models are compared to the densities observed in 100 experimental images of optically sectioned salivary gland nuclei forming "z-stacks." Most of the experimental z-stacks support the Chr-NE 48-attachment model suggesting that as many as 48 chromosome loci with appreciable affinity for the NE are necessary to reproduce the experimentally observed distribution of chromosome density in fruit fly nuclei. Next, we investigate if and how the presence and the number of Chr-NE attachments affect several key characteristics of 3D genome organization: chromosome territories and gene-gene contacts. This analysis leads to novel insight about the possible role of Chr-NE attachments in regulating the genome architecture. Specifically, we find that model nuclei with more numerous Chr-NE attachments form more distinct chromosome territories and their chromosomes intertwine less frequently. Intra-chromosome and intra-arm contacts are more common in model nuclei with Chr-NE attachments compared to the Null model (no specific attachments), while inter-chromosome and inter-arm contacts are less common in nuclei with Chr-NE attachments. We demonstrate that Chr-NE attachments increase the specificity of long-range inter-chromosome and inter-arm contacts. The predicted effects of Chr-NE attachments are rationalized by intuitive volume vs. surface accessibility arguments.

Chromosome dynamics and folding in eukaryotes: Insights from live cell microscopy

How chromosomes are folded and how this folding relates to function remain fundamental questions. Answering them is rendered difficult by the stochasticity of chromatin fiber motion which inevitably results in heterogeneity of the populations analyzed. Even if single cell analyses are beginning to yield precious insights, how can we determine whether a snapshot of position is related to function of the probed locus or cell-type? Fluorescence labeling of DNA at single or multiple loci allows determination of their position relative to nuclear landmarks and to each other, enabling us to derive physical parameters of the underlying chromatin fiber. Here I review the contribution of quantitative spatial and temporal analysis of labeled DNA to our understanding of chromosome conformation in different cell types, highlighting live cell imaging techniques and large scale geometrical analysis of multiple loci in 3D.

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.

Etiology and treatment of hematological neoplasms: stochastic mathematical models

Leukemias are driven by stemlike cancer cells (SLCC), whose initiation, growth, response to treatment, and posttreatment behavior are often "stochastic", i.e., differ substantially even among very similar patients for reasons not observable with present techniques. We review the probabilistic mathematical methods used to analyze stochastics and give two specific examples. The first example concerns a treatment protocol, e.g., for acute myeloid leukemia (AML), where intermittent cytotoxic drug dosing (e.g., once each weekday) is used with intent to cure. We argue mathematically that, if independent SLCC are growing stochastically during prolonged treatment, then, other things being equal, front-loading doses are more effective for tumor eradication than back loading. We also argue that the interacting SLCC dynamics during treatment is often best modeled by considering SLCC in microenvironmental niches, with SLCC-SLCC interactions occurring only among SLCC within the same niche, and we present a stochastic dynamics formalism, involving "Poissonization," applicable in such situations. Interactions at a distance due to partial control of total cell numbers are also considered. The second half of this chapter concerns chromosomal aberrations, lesions known to cause some leukemias. A specific example is the induction of a Philadelphia chromosome by ionizing radiation, subsequent development of chronic myeloid leukemia (CML), CML treatment, and treatment outcome. This time evolution involves a coordinated sequence of > 10 steps, each stochastic in its own way, at the subatomic, molecular, macromolecular, cellular, tissue, and population scales, with corresponding time scales ranging from picoseconds to decades. We discuss models of these steps and progress in integrating models across scales.

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

The global architecture of the cell nucleus and the spatial organization of chromatin play important roles in gene expression and nuclear function. Single-cell imaging and chromosome conformation capture-based techniques provide a wealth of information on the spatial organization of chromosomes. However, a mechanistic model that can account for all observed scaling behaviors governing long-range chromatin interactions is missing. Here we describe a model called constrained self-avoiding chromatin (C-SAC) for studying spatial structures of chromosomes, as the available space is a key determinant of chromosome folding. We studied large ensembles of model chromatin chains with appropriate fiber diameter, persistence length and excluded volume under spatial confinement. We show that the equilibrium ensemble of randomly folded chromosomes in the confined nuclear volume gives rise to the experimentally observed higher-order architecture of human chromosomes, including average scaling properties of mean-square spatial distance, end-to-end distance, contact probability and their chromosome-to-chromosome variabilities. Our results indicate that the overall structure of a human chromosome is dictated by the spatial confinement of the nuclear space, which may undergo significant tissue- and developmental stage-specific size changes.

Investigation of the chromosome regions with significant affinity for the nuclear envelope in fruit fly--a model based approach

Three dimensional nuclear architecture is important for genome function, but is still poorly understood. In particular, little is known about the role of the “boundary conditions” – points of attachment between chromosomes and the nuclear envelope. We describe a method for modeling the 3D organization of the interphase nucleus, and its application to analysis of chromosome-nuclear envelope (Chr-NE) attachments of polytene (giant) chromosomes in Drosophila melanogaster salivary glands. The model represents chromosomes as self-avoiding polymer chains confined within the nucleus; parameters of the model are taken directly from experiment, no fitting parameters are introduced. Methods are developed to objectively quantify chromosome territories and intertwining, which are discussed in the context of corresponding experimental observations. In particular, a mathematically rigorous definition of a territory based on convex hull is proposed. The self-avoiding polymer model is used to re-analyze previous experimental data; the analysis suggests 33 additional Chr-NE attachments in addition to the 15 already explored Chr-NE attachments. Most of these new Chr-NE attachments correspond to intercalary heterochromatin – gene poor, dark staining, late replicating regions of the genome; however, three correspond to euchromatin – gene rich, light staining, early replicating regions of the genome. The analysis also suggests 5 regions of anti-contact, characterized by aversion for the NE, only two of these correspond to euchromatin. This composition of chromatin suggests that heterochromatin may not be necessary or sufficient for the formation of a Chr-NE attachment. To the extent that the proposed model represents reality, the confinement of the polytene chromosomes in a spherical nucleus alone does not favor the positioning of specific chromosome regions at the NE as seen in experiment; consequently, the 15 experimentally known Chr-NE attachment positions do not appear to arise due to non-specific (entropic) forces. Robustness of the key conclusions to model assumptions is thoroughly checked.

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.

Physical origin of the contact frequency in chromosome conformation capture data

Physical proximity between each pair of genomic loci in a nucleus is measured as a form of contact frequency in chromosome conformation capture-based methods. Complexity of chromosome structure in interphase can be characterized by measuring a statistical property of physical distance between genomic loci according to genomic separation along single chromatids. To find a relationship between the physical distance and the contact frequency, we propose a polymer model derived from the Langevin equation. The model is derived by considering a structure of a chromosome as a trajectory of a particle, where each consecutive segment in the chromosome corresponds to a transient position in the trajectory over time. Using chromosome conformation capture data, we demonstrate the functional relationship between the two quantities. The physical distances derived from the mean contact frequencies by the model show a good correlation with those from experimental data. From the model, we present that the mean contact frequency curve can be divided into three components that arise from different physical origins and show that the contact frequency is proportional to the contact surface area, not to the volume of segments suggested by the fractal globule model. The model explains both a decaying pattern of the contact frequency and the biphasic relationship between the physical distance and the genomic length.

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.

Lattice animal model of chromosome organization

Polymer models tied together by constraints of looping and confinement have been used to explain many of the observed organizational characteristics of interphase chromosomes. Here we introduce a simple lattice animal representation of interphase chromosomes that combines the features of looping and confinement constraints into a single framework. We show through Monte Carlo simulations that this model qualitatively captures both the leveling off in the spatial distance between genomic markers observed in fluorescent in situ hybridization experiments and the inverse decay in the looping probability as a function of genomic separation observed in chromosome conformation capture experiments. The model also suggests that the collapsed state of chromosomes and their segregation into territories with distinct looping activities might be a natural consequence of confinement.

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.

Theoretical analysis of the role of chromatin interactions in long-range action of enhancers and insulators

Long-distance regulatory interactions between enhancers and their target genes are commonplace in higher eukaryotes. Interposed boundaries or insulators are able to block these long-distance regulatory interactions. The mechanistic basis for insulator activity and how it relates to enhancer action-at-a-distance remains unclear. Here we explore the idea that topological loops could simultaneously account for regulatory interactions of distal enhancers and the insulating activity of boundary elements. We show that while loop formation is not in itself sufficient to explain action at a distance, incorporating transient nonspecific and moderate attractive interactions between the chromatin fibers strongly enhances long-distance regulatory interactions and is sufficient to generate a euchromatin-like state. Under these same conditions, the subdivision of the loop into two topologically independent loops by insulators inhibits interdomain interactions. The underlying cause of this effect is a suppression of crossings in the contact map at intermediate distances. Thus our model simultaneously accounts for regulatory interactions at a distance and the insulator activity of boundary elements. This unified model of the regulatory roles of chromatin loops makes several testable predictions that could be confronted with in vitro experiments, as well as genomic chromatin conformation capture and fluorescent microscopic approaches.

Hierarchies in eukaryotic genome organization: Insights from polymer theory and simulations

Eukaryotic genomes possess an elaborate and dynamic higher-order structure within the limiting confines of the cell nucleus. Knowledge of the physical principles and the molecular machinery that govern the 3D organization of this structure and its regulation are key to understanding the relationship between genome structure and function. Elegant microscopy and chromosome conformation capture techniques supported by analysis based on polymer models are important steps in this direction. Here, we review results from these efforts and provide some additional insights that elucidate the relationship between structure and function at different hierarchical levels of genome organization.

Genome3D: a viewer-model framework for integrating and visualizing multi-scale epigenomic information within a three-dimensional genome

BACKGROUND

New technologies are enabling the measurement of many types of genomic and epigenomic information at scales ranging from the atomic to nuclear. Much of this new data is increasingly structural in nature, and is often difficult to coordinate with other data sets. There is a legitimate need for integrating and visualizing these disparate data sets to reveal structural relationships not apparent when looking at these data in isolation.

RESULTS

We have applied object-oriented technology to develop a downloadable visualization tool, Genome3D, for integrating and displaying epigenomic data within a prescribed three-dimensional physical model of the human genome. In order to integrate and visualize large volume of data, novel statistical and mathematical approaches have been developed to reduce the size of the data. To our knowledge, this is the first such tool developed that can visualize human genome in three-dimension. We describe here the major features of Genome3D and discuss our multi-scale data framework using a representative basic physical model. We then demonstrate many of the issues and benefits of multi-resolution data integration.

CONCLUSIONS

Genome3D is a software visualization tool that explores a wide range of structural genomic and epigenetic data. Data from various sources of differing scales can be integrated within a hierarchical framework that is easily adapted to new developments concerning the structure of the physical genome. In addition, our tool has a simple annotation mechanism to incorporate non-structural information. Genome3D is unique is its ability to manipulate large amounts of multi-resolution data from diverse sources to uncover complex and new structural relationships within the genome.

Diffusion-driven looping provides a consistent framework for chromatin organization

Chromatin folding inside the interphase nucleus of eukaryotic cells is done on multiple scales of length and time. Despite recent progress in understanding the folding motifs of chromatin, the higher-order structure still remains elusive. Various experimental studies reveal a tight connection between genome folding and function. Chromosomes fold into a confined subspace of the nucleus and form distinct territories. Chromatin looping seems to play a dominant role both in transcriptional regulation as well as in chromatin organization and has been assumed to be mediated by long-range interactions in many polymer models. However, it remains a crucial question which mechanisms are necessary to make two chromatin regions become co-located, i.e. have them in spatial proximity. We demonstrate that the formation of loops can be accomplished solely on the basis of diffusional motion. The probabilistic nature of temporary contacts mimics the effects of proteins, e.g. transcription factors, in the solvent. We establish testable quantitative predictions by deriving scale-independent measures for comparison to experimental data. In this Dynamic Loop (DL) model, the co-localization probability of distant elements is strongly increased compared to linear non-looping chains. The model correctly describes folding into a confined space as well as the experimentally observed cell-to-cell variation. Most importantly, at biological densities, model chromosomes occupy distinct territories showing less inter-chromosomal contacts than linear chains. Thus, dynamic diffusion-based looping, i.e. gene co-localization, provides a consistent framework for chromatin organization in eukaryotic interphase nuclei.

Thermodynamic pathways to genome spatial organization in the cell nucleus

The architecture of the eukaryotic genome is characterized by a high degree of spatial organization. Chromosomes occupy preferred territories correlated to their state of activity and, yet, displace their genes to interact with remote sites in complex patterns requiring the orchestration of a huge number of DNA loci and molecular regulators. Far from random, this organization serves crucial functional purposes, but its governing principles remain elusive. By computer simulations of a statistical mechanics model, we show how architectural patterns spontaneously arise from the physical interaction between soluble binding molecules and chromosomes via collective thermodynamics mechanisms. Chromosomes colocalize, loops and territories form, and find their relative positions as stable thermodynamic states. These are selected by thermodynamic switches, which are regulated by concentrations/affinity of soluble mediators and by number/location of their attachment sites along chromosomes. Our thermodynamic switch model of nuclear architecture, thus, explains on quantitative grounds how well-known cell strategies of upregulation of DNA binding proteins or modification of chromatin structure can dynamically shape the organization of the nucleus.

Spatially confined folding of chromatin in the interphase nucleus

Genome function in higher eukaryotes involves major changes in the spatial organization of the chromatin fiber. Nevertheless, our understanding of chromatin folding is remarkably limited. Polymer models have been used to describe chromatin folding. However, none of the proposed models gives a satisfactory explanation of experimental data. In particularly, they ignore that each chromosome occupies a confined space, i.e., the chromosome territory. Here, we present a polymer model that is able to describe key properties of chromatin over length scales ranging from 0.5 to 75 Mb. This random loop (RL) model assumes a self-avoiding random walk folding of the polymer backbone and defines a probability P for 2 monomers to interact, creating loops of a broad size range. Model predictions are compared with systematic measurements of chromatin folding of the q-arms of chromosomes 1 and 11. The RL model can explain our observed data and suggests that on the tens-of-megabases length scale P is small, i.e., 10-30 loops per 100 Mb. This is sufficient to enforce folding inside the confined space of a chromosome territory. On the 0.5- to 3-Mb length scale chromatin compaction differs in different subchromosomal domains. This aspect of chromatin structure is incorporated in the RL model by introducing heterogeneity along the fiber contour length due to different local looping probabilities. The RL model creates a quantitative and predictive framework for the identification of nuclear components that are responsible for chromatin-chromatin interactions and determine the 3-dimensional organization of the chromatin fiber.

Random loop model for long polymers

Remarkably little is known about the higher-order folding motifs of the chromatin fiber inside the cell nucleus. Folding depends among others on local gene density and transcriptional activity and plays an important role in gene regulation. Strikingly, at fiber lengths above 5 to 10 Mb the measured mean square distance <R2> between any two points on the chromatin fiber is independent of polymer length. We propose a polymer model that can explain this leveling-off by means of random looping. We derive an analytical expression for the mean square displacement between two arbitrary beads. Here the average is taken over the thermal ensemble with a fixed but random loop configuration, while quenched averaging over the ensemble of different loop configurations--which turns out to be equivalent to averaging over an ensemble of random matrices--is performed numerically. A detailed investigation of this model shows that loops on all scales are necessary to fit experimental data.

Three-dimensional genome organization in interphase and its relation to genome function

Higher order chromatin structure, i.e. the three-dimensional (3D) organization of the genome in the interphase nucleus, is an important component in the orchestration of gene expression in the mammalian genome. In this review we describe principles of higher order chromatin structure discussing three organizational parameters, i.e. chromatin folding, chromatin compaction and the nuclear position of the chromatin fibre. We argue that principles of 3D genome organization are probabilistic traits, reflected in a considerable cell-to-cell variation in 3D genome structure. It will be essential to understand how such higher order organizational aspects contribute to genome function to unveil global genome regulation.

Chromatin dynamics of unfolding and refolding controlled by the nucleosome repeat length and the linker and core histones

Chromatin is composed of genomic DNA and histones, forming a hierarchical architecture in the nucleus. The chromatin hierarchy is common among eukaryotes despite different intrinsic properties of the genome. To investigate an effect of the differences in genome organization, chromatin unfolding processes were comparatively analyzed using Schizosaccaromyces pombe, Saccharomyces cerevisiae, and chicken erythrocyte. NaCl titration showed dynamic changes of the chromatin. 400-1000 mM NaCl facilitated beads with approximately 115 nm in diameter in S. pombe chromatin. A similar transition was also observed in S. cerevisiae chromatin. This process did not involve core histone dissociation from the chromatin, and the persistence length after the transition was approximately 26 nm for S. pombe and approximately 28 nm for S. cerevisiae, indicating a salt-induced unfolding to "beads-on-a-string" fibers. Reduced salt concentration recovered the original structure, suggesting that electrostatic interaction would regulate this discrete folding-unfolding process. On the other hand, the linker histone was extracted from chicken chromatin at 400 mM NaCl, and AFM observed the "beads-on-a-string" fibers around a nucleus. Unlike yeast chromatin, therefore, this unfolding was irreversible because of linker histone dissociation. These results indicate that the chromatin unfolding and refolding depend on the presence and absence of the linker histone, and the length of the linker DNA.

Dynamic alterations in the conformation of the Ifng gene region during T helper cell differentiation

Gene expression and silencing in eukaryotic systems can be controlled by regulatory elements acting over a distance. Here, we analyze chromatin conformation of the 24-kb region of the Ifng gene during CD4(+) T helper (Th) cell differentiation. We find that chromatin within this region is a highly flexible structure that undergoes dynamic changes during the course of transcriptional activation and silencing of the Ifng gene. Each Th subset displays a common core conformation in this gene region and unique features that distinguish neutral and effector Th1 and Th2 lineages. This chromatin configuration brings distal regions into close proximity to the gene. Th1 cells that produce high levels of IFN-gamma display the most open conformation. In contrast, IFN-gamma silent Th2 cells have a tightly closed conformation. Therefore, we postulate that there is a direct structure-function relationship between the spatial organization of the chromatin around the Ifng gene and its transcriptional potential.

Capturing chromosome conformation

We describe an approach to detect the frequency of interaction between any two genomic loci. Generation of a matrix of interaction frequencies between sites on the same or different chromosomes reveals their relative spatial disposition and provides information about the physical properties of the chromatin fiber. This methodology can be applied to the spatial organization of entire genomes in organisms from bacteria to human. Using the yeast Saccharomyces cerevisiae, we could confirm known qualitative features of chromosome organization within the nucleus and dynamic changes in that organization during meiosis. We also analyzed yeast chromosome III at the G1 stage of the cell cycle. We found that chromatin is highly flexible throughout. Furthermore, functionally distinct AT- and GC-rich domains were found to exhibit different conformations, and a population-average 3D model of chromosome III could be determined. Chromosome III emerges as a contorted ring.

Making contacts on a nucleic acid polymer

The interaction of proteins bound at distant sites on a nucleic acid chain plays an important role in many molecular biological processes. Contact between the proteins is established by looping of the intervening polymer, which can comprise either double- or single-stranded DNA or RNA, or interphase or metaphase chromatin. The effectiveness of this process, as well as the optimal separation distance, is highly dependent on the flexibility and conformation of the linker. This article reviews how the probability of looping-mediated interactions is calculated for different nucleic acid polymers. In addition, the application of the equations to the analysis of experimental data is illustrated.

Higher-order structure of mammalian chromatin deduced from viscoelastometry data

The results of viscoelastometry (VE) for mammalian DNA have been puzzling because they have two orders of magnitude smaller measured viscoelastic relaxation times for mammalian chromosomes than that expected for DNA linear coils of chromosomal size. In an attempt to resolve this discrepancy, we have applied a recent model of G1 chromosome structure (J.Y. Ostashevsky, Mol Biol. Cell 9, 3031-3040, 1998) in which the 30 nm chromatin fiber of each chromosome forms a string of loop clusters (micelles). This model has two parameters: the number of loops per micelle (f) and the average loop size (Mf), which can be estimated independently from VE data. Using our VE data for plateau phase V79 Chinese hamster cells (unirradiated and X-irradiated with doses up to 40 Gy) we show that f approximately 13 , which is close to other estimates made using the model (f ranges from 10-20), and Mf approximately 2 Mbp, which is similar to estimates made from our nucleoid data (1.3 Mbp) and to estimates made in the literature using a variety of techniques (1-3 Mbp).

Locations of radiation-produced DNA double strand breaks along chromosomes: a stochastic cluster process formalism

Ionizing radiation produces DNA double strand breaks (DSBs) in chromosomes. For densely ionizing radiation, the DSBs are not spaced randomly along a chromosome: recent data for size distributions of DNA fragments indicate break clustering on kbp-Mbp scales. Different DSB clusters on a chromosome are typically made by different, statistically independent, stochastically structured radiation tracks, and the average number of tracks involved can be small. We therefore model DSB positions along a chromosome as a stationary Poisson cluster process, i.e. a stochastic process consisting of secondary point processes whose locations are determined by a primary point process that is Poisson. Each secondary process represents a break cluster, typically consisting of 1-10 DSBs in a comparatively localized stochastic pattern determined by chromatin geometry and radiation track structure. Using this Poisson cluster process model, which we call the randomly located clusters (RLC) formalism, theorems are derived for how the DNA fragment-size distribution depends on radiation dose. The RLC dose-response relations become non-linear when the dose becomes so high that DSB clusters from different tracks overlap or adjoin closely. The RLC formalism generalizes previous models, fits current data adequately and facilitates mechanistically based extrapolations from high-dose experiments to the much lower doses of interest for most applications.

Compartmentalization of interphase chromosomes observed in simulation and experiment

Human interphase chromosomes were simulated as a flexible fiber with excluded volume interaction, which represents the chromatin fiber of each chromosome. For the higher-order structures, we assumed a folding into 120 kb loops and an arrangement of these loops into rosette-like subcompartments. Chromosomes consist of subcompartments connected by small fragments of chromatin. Number and size of subcompartments correspond with chromosome bands in early prophase. We observed essentially separated chromosome arms in both our model calculations and confocal laser scanning microscopy, and measured the same overlap in simulation and experiment. Overlap, number and size of chromosome 15 subcompartments of our model chromosomes agree with subchromosomal foci composed of either early or late replicating chromatin, which were observed at all stages of the cell cycle and possibly provide a functionally relevant unit of chromosome territory compartmentalization. Computed distances of chromosome specific markers both on Mb and 10-100 Mb scale agree with fluorescent in situ hybridization measurements under different preparation conditions.

A polymer model for the structural organization of chromatin loops and minibands in interphase chromosomes

A quantitative model of interphase chromosome higher-order structure is presented based on the isochore model of the genome and results obtained in the field of copolymer research. G1 chromosomes are approximated in the model as multiblock copolymers of the 30-nm chromatin fiber, which alternately contain two types of 0.5- to 1-Mbp blocks (R and G minibands) differing in GC content and DNA-bound proteins. A G1 chromosome forms a single-chain string of loop clusters (micelles), with each loop approximately 1-2 Mbp in size. The number of approximately 20 loops per micelle was estimated from the dependence of geometrical versus genomic distances between two points on a G1 chromosome. The greater degree of chromatin extension in R versus G minibands and a difference in the replication time for these minibands (early S phase for R versus late S phase for G) are explained in this model as a result of the location of R minibands at micelle cores and G minibands at loop apices. The estimated number of micelles per nucleus is close to the observed number of replication clusters at the onset of S phase. A relationship between chromosomal and nuclear sizes for several types of higher eukaryotic cells (insects, plants, and mammals) is well described through the micelle structure of interphase chromosomes. For yeast cells, this relationship is described by a linear coil configuration of chromosomes.

Polymer models of meiotic and mitotic chromosomes

Polymers tied together by constraints exhibit an internal pressure; this idea is used to analyze physical properties of the bottle-brush-like chromosomes of meiotic prophase that consist of polymer-like flexible chromatin loops, attached to a central axis. Using a minimal number of experimental parameters, semiquantitative predictions are made for the bending rigidity, radius, and axial tension of such brushes, and the repulsion acting between brushes whose bristles are forced to overlap. The retraction of lampbrush loops when the nascent transcripts are stripped away, the oval shape of diplotene bivalents between chiasmata, and the rigidity of pachytene chromosomes are all manifestations of chromatin pressure. This two-phase (chromatin plus buffer) picture that suffices for meiotic chromosomes has to be supplemented by a third constituent, a chromatin glue to understand mitotic chromosomes, and explain how condensation can drive the resolution of entanglements. This process resembles a thermal annealing in that a parameter (the affinity of the glue for chromatin and/or the affinity of the chromatin for buffer) has to be tuned to achieve optimal results. Mechanical measurements to characterize this protein-chromatin matrix are proposed. Finally, the propensity for even slightly chemically dissimilar polymers to phase separate (cluster like with like) can explain the apparent segregation of the chromatin into A + T- and G + C-rich regions revealed by chromosome banding.

Regional differences in the compaction of chromatin in human G0/G1 interphase nuclei

The large-scale structure of chromatin corresponding to G- and R-bands in human G0/G1 interphase nuclei was compared. Fluorescence in situ hybridization (FISH) was used to measure the interphase distance between 42 pairs of probes separated by 0.1-1.5 Mbp. The probe pairs were derived from 21q22.2 and Xp21.3, G-band positive regions, and from 4p16.3, 6p21.3, and Xq28, R-band positive regions. Distributions of measured interphase distances in all regions approximated a Rayleigh distribution, suggesting that the chromatin follows a random-walk path over this range. A linear correlation of mean-square interphase distance and genomic separation, also indicative of random-walk folding, was observed in all regions. The slope of the correlation observed using probes from G-band regions was systematically lower than that from R-band regions. The difference in the slope between Xp21.3 and Xq28 was particularly striking and was observed in normal fibroblast cells, fixed alternatively with methanol and acetic acid or paraformaldehyde, and HeLa cells. These results demonstrate regional differences in large-scale chromosome structure during interphase, with the more openly configured chromatin corresponding to R-bands.

Centric rings, acentric rings and excess acentric fragments based on a random-walk interphase chromosome model

Excess acentric fragments, consisting of acentric rings and acentric linear fragments, are among the most frequent kinds of chromosome-type aberrations produced by radiation. The frequency of acentric rings cannot be obtained directly by experiment but is estimated here from the ratio of acentric to centric rings, evaluated using a random-walk model for the organization of chromatin during interphase and an assumption that the probability of an exchange formation is proportional to the rate of collision between two DSB. This ratio is calculated to be 2.5 in low-LET irradiated human fibroblasts, significantly greater than the ratio if proximity effects are not considered. The calculated frequency of acentric rings is insufficient to account for all the observed excess acentric fragments. Assuming that the rest of the excess acentric fragments are due to incomplete exchanges, all possible recombinations between two DSB that result in acentric rings and acentric linear fragments have been identified. From the chromosome aberration data, the incompleteness parameter has been estimated. Intra-arm chromosome exchanges, either complete or incomplete, were estimated to account for more than 50% of the excess acentric fragments in human fibroblasts.

A two-backbone polymer model for interphase chromosome geometry

A polymer model for the overall geometric structure of a human chromosome during the G0/G1 portion of cell-cycle interphase is constructed, based on fluorescence in situ hybridization data on distances between defined genomic sequences. The model consists of flexible giant loops, averaging about 6 million base pairs, with two random-walk backbones; it involves essentially three parameters. Numerical results based on properly selected values of parameters fit the data well.

Formation of two double-strand breaks in the same DNA molecule by a single high-energy photon or ionizing particle

I calculate the probability that a single high-energy ionizing particle or photon makes two widely spaced double-strand breaks in the same DNA molecule. Deletions (or inversions) between two breaks formed by the same incident particle are linear in radiation dose and occur even at extremely low dose-rates; deletions between breaks induced by separate particles are quadratic in dose and are much fewer at very low dose-rate. The calculations show that for a few grays of sparsely ionizing radiations such as fast electrons, X-rays of gamma-rays, the formation of two double-strand breaks in a DNA molecule 1 megabase in size should be nearly entirely quadratic in dose. For heavily ionizing particles such as alpha particles from radon products, the linear and quadratic terms are comparable in size. These conclusions are robust and insensitive to details of the calculations. The results are essentially the same for DNA in a random coil configuration and for DNA uniformly and randomly distributed within a sphere.

Evidence for the organization of chromatin in megabase pair-sized loops arranged along a random walk path in the human G0/G1 interphase nucleus

We determined the folding of chromosomes in interphase nuclei by measuring the distance between points on the same chromosome. Over 25,000 measurements were made in G0/G1 nuclei between DNA sequences separated by 0.15-190 megabase pairs (Mbp) on three human chromosomes. The DNA sequences were specifically labeled by fluorescence in situ hybridization. The relationship between mean-square interphase distance and genomic separation has two linear phases, with a transition at approximately 2 Mbp. This biphasic relationship indicates the existence of two organizational levels at scales > 100 kbp. On one level, chromatin appears to be arranged in large loops several Mbp in size. Within each loop, chromatin is randomly folded. On the second level, specific loop-attachment sites are arranged to form a supple, backbonelike structure, which also shows characteristic random walk behavior. This random walk/giant loop model is the simplest model that fully describes the observed large-scale spatial relationships. Additional evidence for large loops comes from measurements among probes in Xq28, where interphase distance increases and then locally decreases with increasing genomic separation.

A random-walk/giant-loop model for interphase chromosomes

Fluorescence in situ hybridization data on distances between defined genomic sequences are used to construct a quantitative model for the overall geometric structure of a human chromosome. We suggest that the large-scale geometry during the G0/G1 part of the cell cycle may consist of flexible chromatin loops, averaging approximately 3 million bp, with a random-walk backbone. A fully explicit, three-parametric polymer model of this random-walk/giant-loop structure can account well for the data. More general models consistent with the data are briefly discussed.

Treatment of cells with alkaline borate buffer extends the capability of interphase FISH mapping

Interphase fluorescence in situ hybridization (FISH) has been shown to be a means to map DNA sequences relative to each other in the 100 kb to 1-2 Mb genomic-separation range. At distances below 0.1 Mb, probe sites are infrequently resolved in interphase chromatin. In the 0.1- to 1-Mb range, interphase chromatin can be modeled as a freely flexible chain. The mean square interphase distance between two probes is proportional to the genomic separation between the probes on the linear DNA molecule. Above 1-2 Mb, the relationship between interphase distance and genomic separation changes abruptly and appears to level off. We have used alkaline-borate treatment to expand the capability of interphase FISH mapping. We show here that alkaline-borate treatment increases nuclear diameter, the interphase distance between probes on homologous chromosomes, and the distance between probes on the same chromosome. We also show that the mean square distance between hybridization sites in borate-treated nuclei is proportional to genomic separation up to 4 Mb. Thus, alkaline-borate treatment enhances the capability of interphase FISH mapping by increasing the absolute distance between probes and extending the range of the simple relationship between interphase distance and genomic separation.