Rouse model with transient intramolecular contacts on a timescale of seconds recapitulates folding and fluctuation of yeast chromosomes.

DNA folding and dynamics along with major nuclear functions are determined by chromosome structural properties, which remain, thus far, elusive in vivo. Here, we combine polymer modeling and single particle tracking experiments to determine the physico-chemical parameters of chromatin in vitro and in living yeast. We find that the motion of reconstituted chromatin fibers can be recapitulated by the Rouse model using mechanical parameters of nucleosome arrays deduced from structural simulations. Conversely, we report that the Rouse model shows some inconsistencies to analyze the motion and structural properties inferred from yeast chromosomes determined with chromosome conformation capture techniques (specifically, Hi-C). We hence introduce the Rouse model with Transient Internal Contacts (RouseTIC), in which random association and dissociation occurs along the chromosome contour. The parametrization of this model by fitting motion and Hi-C data allows us to measure the kinetic parameters of the contact formation reaction. Chromosome contacts appear to be transient; associated to a lifetime of seconds and characterized by an attractive energy of -0.3 to -0.5 kBT. We suggest attributing this energy to the occurrence of histone tail-DNA contacts and notice that its amplitude sets chromosomes in 'theta' conditions, in which they are poised for compartmentalization and phase separation.

High-throughput chromatin motion tracking in living yeast reveals the flexibility of the fiber throughout the genome.

Chromosome dynamics are recognized to be intimately linked to genomic transactions, yet the physical principles governing spatial fluctuations of chromatin are still a matter of debate. Using high-throughput single-particle tracking, we recorded the movements of nine fluorescently labeled chromosome loci located on chromosomes III, IV, XII, and XIV of Saccharomyces cerevisiae over an extended temporal range spanning more than four orders of magnitude (10(-2)-10(3) sec). Spatial fluctuations appear to be characterized by an anomalous diffusive behavior, which is homogeneous in the time domain, for all sites analyzed. We show that this response is consistent with the Rouse polymer model, and we confirm the relevance of the model with Brownian dynamics simulations and the analysis of the statistical properties of the trajectories. Moreover, the analysis of the amplitude of fluctuations by the Rouse model shows that yeast chromatin is highly flexible, its persistence length being qualitatively estimated to <30 nm. Finally, we show that the Rouse model is also relevant to analyze chromosome motion in mutant cells depleted of proteins that bind to or assemble chromatin, and suggest that it provides a consistent framework to study chromatin dynamics. We discuss the implications of our findings for yeast genome architecture and for target search mechanisms in the nucleus.

Mathematical models of radiation action on living cells: From the target theory to the modern approaches. A historical and critical review.

Cell survival is conventionally defined as the capability of irradiated cells to produce colonies. It is quantified by the clonogenic assays that consist in determining the number of colonies resulting from a known number of irradiated cells. Several mathematical models were proposed to describe the survival curves, notably from the target theory. The Linear-Quadratic (LQ) model, which is to date the most frequently used model in radiobiology and radiotherapy, dominates all the other models by its robustness and simplicity. Its usefulness is particularly important because the ratio of the values of the adjustable parameters, α and ß, on which it is based, predicts the occurrence of post-irradiation tissue reactions. However, the biological interpretation of these parameters is still unknown. Throughout this review, we revisit and discuss historically, mathematically and biologically, the different models of the radiation action by providing clues for resolving the enigma of the LQ model.

Distinct polymer physics principles govern chromatin dynamics in mouse and Drosophila topological domains.

BACKGROUND: In higher eukaryotes, the genome is partitioned into large "Topologically Associating Domains" (TADs) in which the chromatin displays favoured long-range contacts. While a crumpled/fractal globule organization has received experimental supports at higher-order levels, the organization principles that govern chromatin dynamics within these TADs remain unclear. Using simple polymer models, we previously showed that, in mouse liver cells, gene-rich domains tend to adopt a statistical helix shape when no significant locus-specific interaction takes place. RESULTS: Here, we use data from diverse 3C-derived methods to explore chromatin dynamics within mouse and Drosophila TADs. In mouse Embryonic Stem Cells (mESC), that possess large TADs (median size of 840 kb), we show that the statistical helix model, but not globule models, is relevant not only in gene-rich TADs, but also in gene-poor and gene-desert TADs. Interestingly, this statistical helix organization is considerably relaxed in mESC compared to liver cells, indicating that the impact of the constraints responsible for this organization is weaker in pluripotent cells. Finally, depletion of histone H1 in mESC alters local chromatin flexibility but not the statistical helix organization. In Drosophila, which possesses TADs of smaller sizes (median size of 70 kb), we show that, while chromatin compaction and flexibility are finely tuned according to the epigenetic landscape, chromatin dynamics within TADs is generally compatible with an unconstrained polymer configuration. CONCLUSIONS: Models issued from polymer physics can accurately describe the organization principles governing chromatin dynamics in both mouse and Drosophila TADs. However, constraints applied on this dynamics within mammalian TADs have a peculiar impact resulting in a statistical helix organization.

In silico single-molecule manipulation of DNA with rigid body dynamics.

We develop a new powerful method to reproduce in silico single-molecule manipulation experiments. We demonstrate that flexible polymers such as DNA can be simulated using rigid body dynamics thanks to an original implementation of Langevin dynamics in an open source library called Open Dynamics Engine. We moreover implement a global thermostat which accelerates the simulation sampling by two orders of magnitude. We reproduce force-extension as well as rotation-extension curves of reference experimental studies. Finally, we extend the model to simulations where the control parameter is no longer the torsional strain but instead the torque, and predict the expected behavior for this case which is particularly challenging theoretically and experimentally.

Theory and simulations of toroidal and rod-like structures in single-molecule DNA condensation.

DNA condensation by multivalent cations plays a crucial role in genome packaging in viruses and sperm heads, and has been extensively studied using single-molecule experimental methods. In those experiments, the values of the critical condensation forces have been used to estimate the amplitude of the attractive DNA-DNA interactions. Here, to describe these experiments, we developed an analytical model and a rigid body Langevin dynamics assay to investigate the behavior of a polymer with self-interactions, in the presence of a traction force applied at its extremities. We model self-interactions using a pairwise attractive potential, thereby treating the counterions implicitly. The analytical model allows to accurately predict the equilibrium structures of toroidal and rod-like condensed structures, and the dependence of the critical condensation force on the DNA length. We find that the critical condensation force depends strongly on the length of the DNA, and finite-size effects are important for molecules of length up to 10(5)µm. Our Langevin dynamics simulations show that the force-extension behavior of the rod-like structures is very different from the toroidal ones, so that their presence in experiments should be easily detectable. In double-stranded DNA condensation experiments, the signature of the presence of rod-like structures was not unambiguously detected, suggesting that the polyamines used to condense DNA may protect it from bending sharply as needed in the rod-like structures.

DNA topology in chromosomes: a quantitative survey and its physiological implications.

Using a simple geometric model, we propose a general method for computing the linking number of the DNA embedded in chromatin fibers. The relevance of the method is reviewed through the single molecule experiments that have been performed in vitro with magnetic tweezers. We compute the linking number of the DNA in the manifold conformational states of the nucleosome which have been evidenced in these experiments and discuss the functional dynamics of chromosomes in the light of these manifold states.

A single formula to describe radiation-induced protein relocalization: towards a mathematical definition of individual radiosensitivity.

Immunofluorescence with antibodies against DNA damage repair and signaling protein is revolutionarising the estimation of the genotoxic risk. Indeed, a number of stress response proteins relocalize in nucleus as identifiable foci whose number, pattern and appearance/disappearance rate depend on several parameters such as the stress nature, dose, time and individual factor. Few authors proposed biomathematical tools to describe them in a unified formula that would be relevant for all the relocalizable proteins. Based on our two previous reports in this Journal (Foray et al., 2005; Gastaldo et al., 2008), we considered that foci response to stress is composed of a recognition and a repair phase, both described by an inverse power function provided from a Euler's Gamma distribution. The resulting unified formula called "Bodgi's function" is able to describe appearance/disappearance kinetics of nuclear foci after any condition of genotoxic stress. By applying the Bodgi's formula to DNA damage repair data from 45 patients treated with radiotherapy, we deduced a classification of human radiosensitivity based on objective molecular criteria, notably like the number of unrepaired DNA double-strand breaks and the radiation-induced nucleo-shuttling of the ATM kinase.

The condensed chromatin fiber: an allosteric chemo-mechanical machine for signal transduction and genome processing.

Allostery is a key concept of molecular biology which refers to the control of an enzyme activity by an effector molecule binding the enzyme at another site rather than the active site (allos = other in Greek). We revisit here allostery in the context of chromatin and argue that allosteric principles underlie and explain the functional architecture required for spacetime coordination of gene expression at all scales from DNA to the whole chromosome. We further suggest that this functional architecture is provided by the chromatin fiber itself. The structural, mechanical and topological features of the chromatin fiber endow chromosomes with a tunable signal transduction from specific (or nonspecific) effectors to specific (or nonspecific) active sites. Mechanical constraints can travel along the fiber all the better since the fiber is more compact and regular, which speaks in favor of the actual existence of the (so-called 30 nm) chromatin fiber. Chromatin fiber allostery reconciles both the physical and biochemical approaches of chromatin. We illustrate this view with two supporting specific examples. Moreover, from a methodological point of view, we suggest that the notion of chromatin fiber allostery is particularly relevant for systemic approaches. Finally we discuss the evolutionary power of allostery in the context of chromatin and its relation to modularity.

Linker histones incorporation maintains chromatin fiber plasticity.

Genomic DNA in eukaryotic cells is organized in supercoiled chromatin fibers, which undergo dynamic changes during such DNA metabolic processes as transcription or replication. Indeed, DNA-translocating enzymes like polymerases produce physical constraints in vivo. We used single-molecule micromanipulation by magnetic tweezers to study the response of chromatin to mechanical constraints in the same range as those encountered in vivo. We had previously shown that under positive torsional constraints, nucleosomes can undergo a reversible chiral transition toward a state of positive topology. We demonstrate here that chromatin fibers comprising linker histones present a torsional plasticity similar to that of naked nucleosome arrays. Chromatosomes can undergo a reversible chiral transition toward a state of positive torsion (reverse chromatosome) without loss of linker histones.

Structural plasticity of single chromatin fibers revealed by torsional manipulation.

Magnetic tweezers were used to study the mechanical response under torsion of single nucleosome arrays reconstituted on tandem repeats of 5S positioning sequences. Regular arrays are extremely resilient and can reversibly accommodate a large amount of supercoiling without much change in length. This behavior is quantitatively described by a molecular model of the chromatin three-dimensional architecture. In this model, we assume the existence of a dynamic equilibrium between three conformations of the nucleosome, corresponding to different crossing statuses of the entry/exit DNAs (positive, null or negative, respectively). Torsional strain displaces that equilibrium, leading to an extensive reorganization of the fiber's architecture. The model explains a number of long-standing topological questions regarding DNA in chromatin and may provide the basis to better understand the dynamic binding of chromatin-associated proteins.Note: In the supplementary information initially published online to accompany this article, Supplementary Figure 2 was mistakenly replaced by Supplementary Equation 2. The error has been corrected online.

Effective interaction between charged nanoparticles and DNA.

We investigate the effective interaction mediated by salt ions between charged nanoparticles (NPs) and DNA. DNA is modeled as an infinite cylinder with a constant surface charge in an implicit solvent. Monte Carlo simulations are used to compute the free energy of the system described in the framework of the primitive model of electrolytes, which accounts for excluded volumes of salt ions. A mean-field Poisson-Boltzmann theory also allows us to compute the free energy and provides us with explicit formulae for its main characteristics (position and depth of the minimum). We intend here to identify the physical parameters that have a major impact on the NP-DNA interaction, in an attempt to evaluate physico-chemical properties which could play a role in genotoxicity or, which could be exploited for therapeutic use. Thus, we investigate the influence on the effective interaction of: the shape of the nanoparticle, the magnitude of the nanoparticle charge and its distribution, the value of the pH of the solution, the magnitude of Van der Waals interactions depending on the nature of the constitutive material of the NP (metal vs. dielectric). We show that for positively charged concave NPs the effective interaction is repulsive at short distance, so that it presents a minimum at distance from the DNA. This short-range repulsion is specific to indented particles and is a robust property that holds for a large range of materials and charge densities.

The nucleosome family: dynamic and growing.

Ever since the discovery of the nucleosome in 1974, scientists have stumbled upon discrete particles in which DNA is wrapped around histone complexes of different stoichiometries: octasomes, hexasomes, tetrasomes, "split" half-nucleosomes, and, recently, bona fide hemisomes. Do all these particles exist in vivo? Under what conditions? What is their physiological significance in the complex DNA transactions in the eukaryotic nucleus? What are their dynamics? This review summarizes research spanning more than three decades and provides a new meaning to the term "nucleosome." The nucleosome can no longer be viewed as a single static entity: rather, it is a family of particles differing in their structural and dynamic properties, leading to different functionalities.

Transcription within condensed chromatin: Steric hindrance facilitates elongation.

During eukaryotic transcription, RNA-polymerase activity generates torsional stress in DNA, having a negative impact on the elongation process. Using our previous studies of chromatin fiber structure and conformational transitions, we suggest that this torsional stress can be alleviated, thanks to a tradeoff between the fiber twist and nucleosome conformational transitions into an activated state named "reversome". Our model enlightens the origin of polymerase pauses, and leads to the counterintuitive conclusion that chromatin-organized compaction might facilitate polymerase progression. Indeed, in a compact and well-structured chromatin loop, steric hindrance between nucleosomes enforces sequential transitions, thus ensuring that the polymerase always meets a permissive nucleosomal state.

Chromatin fiber dynamics under tension and torsion.

Genetic and epigenetic information in eukaryotic cells is carried on chromosomes, basically consisting of large compact supercoiled chromatin fibers. Micromanipulations have recently led to great advances in the knowledge of the complex mechanisms underlying the regulation of DNA transaction events by nucleosome and chromatin structural changes. Indeed, magnetic and optical tweezers have allowed opportunities to handle single nucleosomal particles or nucleosomal arrays and measure their response to forces and torques, mimicking the molecular constraints imposed in vivo by various molecular motors acting on the DNA. These challenging technical approaches provide us with deeper understanding of the way chromatin dynamically packages our genome and participates in the regulation of cellular metabolism.

Polymer coil-globule phase transition is a universal folding principle of Drosophila epigenetic domains.

BACKGROUND: Localized functional domains within chromosomes, known as topologically associating domains (TADs), have been recently highlighted. In Drosophila, TADs are biochemically defined by epigenetic marks, this suggesting that the 3D arrangement may be the "missing link" between epigenetics and gene activity. Recent observations (Boettiger et al. in Nature 529(7586):418-422, 2016) provide access to structural features of these domains with unprecedented resolution thanks to super-resolution experiments. In particular, they give access to the distribution of the radii of gyration for domains of different linear length and associated with different transcriptional activity states: active, inactive or repressed. Intriguingly, the observed scaling laws lack consistent interpretation in polymer physics. RESULTS: We develop a new methodology conceived to extract the best information from such super-resolution data by exploiting the whole distribution of gyration radii, and to place these experimental results on a theoretical framework. We show that the experimental data are compatible with the finite-size behavior of a self-attracting polymer. The same generic polymer model leads to quantitative differences between active, inactive and repressed domains. Active domains behave as pure polymer coils, while inactive and repressed domains both lie at the coil-globule crossover. For the first time, the "color-specificity" of both the persistence length and the mean interaction energy are estimated, leading to important differences between epigenetic states. CONCLUSION: These results point toward a crucial role of criticality to enhance the system responsivity, resulting in both energy transitions and structural rearrangements. We get strong indications that epigenetically induced changes in nucleosome-nucleosome interaction can cause chromatin to shift between different activity states.

The Role of Supercoiling in the Motor Activity of RNA Polymerases.

RNA polymerase (RNAP) is, in its elongation phase, an emblematic example of a molecular motor whose activity is highly sensitive to DNA supercoiling. After a review of DNA supercoiling basic features, we discuss how supercoiling controls polymerase velocity, while being itself modified by polymerase activity. This coupling is supported by single-molecule measurements. Physical modeling allows us to describe quantitatively how supercoiling and torsional constraints mediate a mechanical coupling between adjacent polymerases. On this basis, we obtain a description that may explain the existence and functioning of RNAP convoys.

A physical model for the condensation and decondensation of eukaryotic chromosomes.

During the eukaryotic cell cycle, chromatin undergoes several conformational changes, which are believed to play key roles in gene expression regulation during interphase, and in genome replication and division during mitosis. In this paper, we propose a scenario for chromatin structural reorganization during mitosis, which bridges all the different scales involved in chromatin architecture, from nucleosomes to chromatin loops. We build a model for chromatin, based on available data, taking into account both physical and topological constraints DNA has to deal with. Our results suggest that the mitotic chromosome condensation/decondensation process is induced by a structural change at the level of the nucleosome itself.

Network Modeling of Crohn's Disease Incidence.

BACKGROUND: Numerous genetic and environmental risk factors play a role in human complex genetic disorders (CGD). However, their complex interplay remains to be modelled and explained in terms of disease mechanisms. METHODS AND FINDINGS: Crohn's Disease (CD) was modeled as a modular network of patho-physiological functions, each summarizing multiple gene-gene and gene-environment interactions. The disease resulted from one or few specific combinations of module functional states. Network aging dynamics was able to reproduce age-specific CD incidence curves as well as their variations over the past century in Western countries. Within the model, we translated the odds ratios (OR) associated to at-risk alleles in terms of disease propensities of the functional modules. Finally, the model was successfully applied to other CGD including ulcerative colitis, ankylosing spondylitis, multiple sclerosis and schizophrenia. CONCLUSION: Modeling disease incidence may help to understand disease causative chains, to delineate the potential of personalized medicine, and to monitor epidemiological changes in CGD.