DNA spools under tension

How a Soft Rod Wraps around a Rotating Cylinder

The unique characteristics of helical coils are utilized in nature, manufacturing processes, and daily life. These coils are also pivotal in the development of soft machines, such as artificial muscles and soft grippers. The stability of these helical coils is generally dependent on the mechanical properties of the rods and geometry of the supporting objects. In this Letter, the shapes formed by a flexible, heavy rod wrapping around a slowly rotating rigid cylinder are investigated through a combination of experimental and theoretical approaches. Three distinct morphologies-tight coiling, helical wrapping, and no wrapping-are identified experimentally. These findings are rationalized by numerical simulations and a geometrically nonlinear Kirchhoff rod theory. Despite the frictional contact present, the local shape of the rod is explained by the interplay between bending elasticity, gravity, and the geometry of the system. Our Letter provides a comprehensive physical understanding of the ordered morphology of soft threads and rods. Implications of this understanding are significant for a wide range of phenomena, from the recently discovered wrapping motility mode of bacterial flagella to the design of an octopus-inspired soft gripper.

Assignment of structural transitions during mechanical unwrapping of nucleosomes and their disassembly products

Nucleosomes, the fundamental structural unit of chromatin, consists of ∼147 DNA base pairs wrapped around a histone protein octamer. To characterize the strength of the nucleosomal barrier and its contribution as a mechanism of control of gene expression, it is essential to determine the forces required to unwrap the DNA from the core particle and the stepwise transitions involved. In this study, we performed combined optical tweezers and single-molecule fluorescence measurements to identify the specific DNA segments unwrapped during the force transitions observed in mechanical stretching of nucleosomes. Furthermore, we characterize the mechanical signatures of subnucleosomal hexasomes and tetrasomes. The characterization performed in this work is essential for the interpretation of ongoing studies of chromatin remodelers, polymerases, and histone chaperones.

Nucleosome DNA unwrapping and its disassembly into hexasomes and tetrasomes is necessary for genomic access and plays an important role in transcription regulation. Previous single-molecule mechanical nucleosome unwrapping revealed a low- and a high-force transitions, and force-FRET pulling experiments showed that DNA unwrapping is asymmetric, occurring always first from one side before the other. However, the assignment of DNA segments involved in these transitions remains controversial. Here, using high-resolution optical tweezers with simultaneous single-molecule FRET detection, we show that the low-force transition corresponds to the undoing of the outer wrap of one side of the nucleosome (∼27 bp), a process that can occur either cooperatively or noncooperatively, whereas the high-force transition corresponds to the simultaneous unwrapping of ∼76 bp from both sides. This process may give rise stochastically to the disassembly of nucleosomes into hexasomes and tetrasomes whose unwrapping/rewrapping trajectories we establish. In contrast, nucleosome rewrapping does not exhibit asymmetry. To rationalize all previous nucleosome unwrapping experiments, it is necessary to invoke that mechanical unwrapping involves two nucleosome reorientations: one that contributes to the change in extension at the low-force transition and another that coincides but does not contribute to the high-force transition.

Multiscale modeling of genome organization with maximum entropy optimization

Three-dimensional (3D) organization of the human genome plays an essential role in all DNA-templated processes, including gene transcription, gene regulation, and DNA replication. Computational modeling can be an effective way of building high-resolution genome structures and improving our understanding of these molecular processes. However, it faces significant challenges as the human genome consists of over 6 × 109 base pairs, a system size that exceeds the capacity of traditional modeling approaches. In this perspective, we review the progress that has been made in modeling the human genome. Coarse-grained models parameterized to reproduce experimental data via the maximum entropy optimization algorithm serve as effective means to study genome organization at various length scales. They have provided insight into the principles of whole-genome organization and enabled de novo predictions of chromosome structures from epigenetic modifications. Applications of these models at a near-atomistic resolution further revealed physicochemical interactions that drive the phase separation of disordered proteins and dictate chromatin stability in situ. We conclude with an outlook on the opportunities and challenges in studying chromosome dynamics.

Asymmetry in histone rotation in forced unwrapping and force quench rewrapping in a nucleosome

Single molecule pulling experiments have shown that DNA in the nucleosomes unwraps in two stages from the histone protein core (HPC). The first stage, attributed to the rupture of the outer DNA turn, occurs between 3 and 5 pNs, and is reversible. The inner DNA turn ruptures irreversibly at forces between 9 and 15 pNs (or higher) in the second stage. Molecular simulations using the Self-Organized Polymer model capture the experimental findings. The unwrapping of the outer DNA turn is independent of the pulling direction. The rupture of the DNA inner turn depends on the pulling direction and involves overcoming substantial energetic (most likely electrostatic in origin) and kinetic barriers. They arise because the mechanical force has to generate sufficient torque to rotate the HPC by 180°. On the other hand, during the rewrapping process, HPC rotation is stochastic, with force playing no role. The assembly of the outer DNA wrap upon force quench nearly coincides with the unwrapping process, confirming the reversibility of the outer turn rupture. The asymmetry in HPC rotation during unwrapping and rewrapping explains the observed hysteresis in the stretch-release cycles in experiments. We propose experiments to test the prediction that HPC rotation produces kinetic barriers in the unwrapping process.

Insight into the unwrapping of the dinucleosome

Dynamics of nucleosomes, the building blocks of chromatin, has crucial effects on the expression, replication and repair of genomes in eukaryotes. Beside the constant movements of nucleosomes by thermal fluctuations, ATP-dependent chromatin remodelling complexes cause their active displacements. Here we propose a theoretical analysis of dinucleosome wrapping and unwrapping dynamics in the presence of an external force. We explore the energy landscape and configurations of a dinucleosome in different unwrapped states. Moreover, using a dynamical Monte-Carlo simulation algorithm, we demonstrate the dynamical features of the system such as the unwrapping force for partial and full wrapping processes. Furthermore, we show that in the short length of linker DNA (∼10-90 bp), asymmetric unwrapping occurs. These findings could shed some light on chromatin dynamics and gene accessibility.

Ensembles of Breathing Nucleosomes: A Computational Study

About three-fourths of the human DNA molecules are wrapped into nucleosomes, protein spools with DNA. Nucleosomes are highly dynamic, transiently exposing their DNA through spontaneous unspooling. Recent experiments allowed to observe the DNA of an ensemble of such breathing nucleosomes through x-ray diffraction with contrast matching between the solvent and the protein core. In this study, we calculate such an ensemble through a Monte Carlo simulation of a coarse-grained nucleosome model with sequence-dependent DNA mechanics. Our analysis gives detailed insights into the sequence dependence of nucleosome breathing observed in the experiment and allows us to determine the adsorption energy of the DNA bound to the protein core as a function of the ionic strength. Moreover, we predict the breathing behavior of other potentially interesting sequences and compare the findings to earlier related experiments.

Critical role of histone tail entropy in nucleosome unwinding

The nucleosome is the fundamental packaging unit for the genome. It must remain tightly wound to ensure genome stability while simultaneously being flexible enough to keep the DNA molecule accessible for genome function. The set of physicochemical interactions responsible for the delicate balance between these naturally opposed processes have not been determined due to challenges in resolving partially unwound nucleosome configurations at atomic resolution. Using a near atomistic protein-DNA model and advanced sampling techniques, we calculate the free energy cost of nucleosome DNA unwinding. Our simulations identify a large energetic barrier that decouples the outer and the inner DNA unwinding into two separate processes, occurring on different time scales. This dynamical decoupling allows the exposure of outer DNA at a modest cost to ensure accessibility while keeping the inner DNA and the histone core intact to maintain stability. We also reveal that this energetic barrier arises from a delayed loss of contacts between disordered histone tails and the DNA and is, surprisingly, largely offset by an entropic contribution from these tails. Implications of this enthalpy entropy compensation for the regulation of nucleosome stability and genome function are discussed.

Dynamics of Chromatin Fibers: Comparison of Monte Carlo Simulations with Force Spectroscopy

To elucidate conformational dynamics of chromatin fibers, we compared available force-spectroscopy measurements with extensive Monte Carlo simulations of nucleosome arrays under external force. Our coarse-grained model of chromatin includes phenomenological energy terms for the DNA-histone adhesion and the internucleosome stacking interactions. We found that the Monte Carlo fiber ensembles simulated with increasing degrees of DNA unwrapping and the stacking energy 8 kT can account for the intricate force-extension response observed experimentally. Our analysis shows that at low external forces (F < 3.0 picoNewtons), the DNA ends in nucleosomes breathe by ∼10 bp. Importantly, under these conditions, the fiber is highly dynamic, exhibiting continuous unstacking-restacking transitions, allowing accessibility of transcription factors to DNA, while maintaining a relatively compact conformation. Of note, changing the stacking interaction by a few kT, an in silico way to mimic histone modifications, is sufficient to transform an open chromatin state into a compact fiber. The fibers are mostly two-start zigzag folds with rare occurrences of three- to five-start morphologies. The internucleosome stacking is lost during the linear response regime. At the higher forces exceeding 4 picoNewtons, the nucleosome unwrapping becomes stochastic and asymmetric, with one DNA arm opened by ∼55 bp and the other arm only by ∼10 bp. Importantly, this asymmetric unwrapping occurs for any kind of sequence, including the symmetric ones. Our analysis brings new, to our knowledge, insights in dynamics of chromatin modulated by histone epigenetic modifications and molecular motors such as RNA polymerase.

Local DNA Sequence Controls Asymmetry of DNA Unwrapping from Nucleosome Core Particles

DNA is tightly wrapped around histone proteins in nucleosome core particles (NCPs) yet must become accessible for processing in the cell. This accessibility, a key component of transcription regulation, is influenced by the properties of both the histone proteins and the DNA itself. Small angle x-ray scattering with contrast variation is used to examine how sequence variations affect DNA unwrapping from NCPs at different salt concentrations. Salt destabilizes NCPs, populating multiple unwrapped states as many possible unwrapping pathways are explored by the complexes. We apply coarse-grained Monte Carlo methods to generate realistic sequence-dependent unwrapped structures for the nucleosomal DNA with thermal variations. An ensemble optimization method is employed to determine the composition of the overall ensemble as electrostatic interactions are weakened. Interesting DNA-sequence-dependent differences are revealed in the unwrapping paths and equilibrium constants. These differences are correlated with specific features within the nucleic acid sequences.

Designing nucleosomal force sensors

About three quarters of our DNA is wrapped into nucleosomes: DNA spools with a protein core. It is well known that the affinity of a given DNA stretch to be incorporated into a nucleosome depends on the geometry and elasticity of the basepair sequence involved, causing the positioning of nucleosomes. Here we show that DNA elasticity can have a much deeper effect on nucleosomes than just their positioning: it affects their "identities". Employing a recently developed computational algorithm, the mutation Monte Carlo method, we design nucleosomes with surprising physical characteristics. Unlike any other nucleosomes studied so far, these nucleosomes are short-lived when put under mechanical tension whereas other physical properties are largely unaffected. This suggests that the nucleosome, the most abundant DNA-protein complex in our cells, might more properly be considered a class of complexes with a wide array of physical properties, and raises the possibility that evolution has shaped various nucleosome species according to their genomic context.

Stretching an Elastic Loop: Crease, Helicoid, and Pop Out

Under geometric constraints, a thin structure can respond to an external loading in an unexpected way. A paper strip that is looped and pulled can be used for simple experimentation of such a process. Here, we study this seemingly very simple phenomenon in detail by combing experiments and theory. We identify the three types of shape transitions, i.e., crease, helicoid, and pop out, from a stretched loop, and classify them in terms of parameters characterizing a ribbon geometry. We establish a transition-type diagram by compiling our extensive experimental data. Numerical simulations based on the Kirchhoff rod theory and scaling argument reveal that the pop-out transition is governed by a single characteristic length ξ∼b^{2}/h, where b and h are the ribbon's width and thickness, respectively. We also reveal the key roles of other physical effects such as the anisotropy of the bending elasticity and plastic deformations upon the shape selection mechanisms of a constraint ribbon.

Tension-Dependent Free Energies of Nucleosome Unwrapping

Nucleosomes form the basic unit of compaction within eukaryotic genomes, and their locations represent an important, yet poorly understood, mechanism of genetic regulation. Quantifying the strength of interactions within the nucleosome is a central problem in biophysics and is critical to understanding how nucleosome positions influence gene expression. By comparing to single-molecule experiments, we demonstrate that a coarse-grained molecular model of the nucleosome can reproduce key aspects of nucleosome unwrapping. Using detailed simulations of DNA and histone proteins, we calculate the tension-dependent free energy surface corresponding to the unwrapping process. The model reproduces quantitatively the forces required to unwrap the nucleosome and reveals the role played by electrostatic interactions during this process. We then demonstrate that histone modifications and DNA sequence can have significant effects on the energies of nucleosome formation. Most notably, we show that histone tails contribute asymmetrically to the stability of the outer and inner turn of nucleosomal DNA and that depending on which histone tails are modified, the tension-dependent response is modulated differently.

Semiflexible Chains at Surfaces: Worm-Like Chains and beyond

We give an extended review of recent numerical and analytical studies on semiflexible chains near surfaces undertaken at Institut Charles Sadron (sometimes in collaboration) with a focus on static properties. The statistical physics of thin confined layers, strict two-dimensional (2D) layers and adsorption layers (both at equilibrium with the dilute bath and from irreversible chemisorption) are discussed for the well-known worm-like-chain (WLC) model. There is mounting evidence that biofilaments (except stable d-DNA) are not fully described by the WLC model. A number of augmented models, like the (super) helical WLC model, the polymorphic model of microtubules (MT) and a model with (strongly) nonlinear flexural elasticity are presented, and some aspects of their surface behavior are analyzed. In many cases, we use approaches different from those in our previous work, give additional results and try to adopt a more general point of view with the hope to shed some light on this complex field.

Nucleosome dynamics: Sequence matters

About three quarter of all eukaryotic DNA is wrapped around protein cylinders, forming nucleosomes. Even though the histone proteins that make up the core of nucleosomes are highly conserved in evolution, nucleosomes can be very different from each other due to posttranslational modifications of the histones. Another crucial factor in making nucleosomes unique has so far been underappreciated: the sequence of their DNA. This review provides an overview of the experimental and theoretical progress that increasingly points to the importance of the nucleosomal base pair sequence. Specifically, we discuss the role of the underlying base pair sequence in nucleosome positioning, sliding, breathing, force-induced unwrapping, dissociation and partial assembly and also how the sequence can influence higher-order structures. A new view emerges: the physical properties of nucleosomes, especially their dynamical properties, are determined to a large extent by the mechanical properties of their DNA, which in turn depends on DNA sequence.

Elastic model for dinucleosome structure and energy

The equilibrium structure of a dinucleosome is studied using an elastic model that takes into account the force and torque balance conditions. Using the proper boundary conditions, it is found that the conformational energy of the problem does not depend on the length of the linker DNA. In addition it is shown that the two histone octamers are almost perpendicular to each other, and the linker DNA in short lengths is almost straight. These findings could shed some light on the role of DNA elasticity in the chromatin structure.

Why Do Nucleosomes Unwrap Asymmetrically?

Nucleosomes, DNA spools with a protein core, engage about three-quarters of eukaryotic DNA and play a critical role in chromosomal processes, ranging from gene regulation, recombination, and replication to chromosome condensation. For more than a decade, micromanipulation experiments where nucleosomes are put under tension, as well as the theoretical interpretations of these experiments, have deepened our understanding of the stability and dynamics of nucleosomes. Here we give a theoretical explanation for a surprising new experimental finding: nucleosomes wrapped onto the 601 positioning sequence (the sequence used in most laboratories) respond highly asymmetrically to external forces by always unwrapping from the same end. Using a computational nucleosome model, we show that this asymmetry can be explained by differences in the DNA mechanics of two very short stretches on the wrapped DNA portion. Our finding suggests that the physical properties of nucleosomes, here the response to forces, can be tuned locally by the choice of the underlying base-pair sequence. This leads to a new view of nucleosomes: a physically highly varied set of DNA-protein complexes whose properties can be tuned on evolutionary time scales to their specific function in the genomic context.

Free-energy landscape of mono- and dinucleosomes: Enhanced rotational flexibility of interconnected nucleosomes

The nucleosome represents the basic unit of eukaryotic genome organization, and its conformational fluctuations play a crucial role in various cellular processes. Here we provide insights into the flipping transition of a nucleosome by computing its free-energy landscape as a function of the linking number and nucleosome orientation using the density-of-states Monte Carlo approach. To investigate how the energy landscape is affected by the presence of neighboring nucleosomes in a chromatin fiber, we also compute the free-energy landscape for a dinucleosome array. We find that the mononucleosome is bistable between conformations with negatively and positively crossed linkers while the conformation with open linkers appears as a transition state. The dinucleosome exhibits a markedly different energy landscape in which the conformation with open linkers populates not only the transition state but also the global minimum. This enhanced stability of the open state is attributed to increased rotational flexibility of nucleosomes arising from their mechanical coupling with neighboring nucleosomes. Our results provide a possible mechanism by which chromatin may enhance the accessibility of its DNA and facilitate the propagation and mitigation of DNA torsional stresses.

Characterization of nucleosome unwrapping within chromatin fibers using magnetic tweezers

Nucleosomal arrays fold into chromatin fibers and the higher-order folding of chromatin plays a strong regulatory role in all processes involving DNA access, such as transcription and replication. A fundamental understanding of such regulation requires insight into the folding properties of the chromatin fiber in molecular detail. Despite this, the structure and the mechanics of chromatin fibers remain highly disputed. Single-molecule force spectroscopy experiments have the potential to provide such insight, but interpretation of the data has been hampered by the large variations in experimental force-extension traces. Here we explore the possibility that chromatin fibers are composed of both single-turn and fully wrapped histone octamers. By characterizing the force-dependent behavior of in vitro reconstituted chromatin fibers and reanalyzing existing data, we show the unwrapping of the outer turn of nucleosomal DNA at 3 pN. We present a model composed of two freely-jointed chains, which reveals that nucleosomes within the chromatin fiber show identical force-extension behavior to mononucleosomes, indicating that nucleosome-nucleosome interactions are orders-of-magnitude smaller than previously reported and therefore can be overcome by thermal fluctuations. We demonstrate that lowering the salt concentration externally increases the wrapping energy significantly, indicative of the electrostatic interaction between the wrapped DNA and the histone octamer surface. We propose that the weak interaction between nucleosomes could allow easy access to nucleosomal DNA, while DNA unwrapping from the histone core could provide a stable yet dynamic structure during DNA maintenance.

Routes to DNA accessibility: alternative pathways for nucleosome unwinding

The dynamic packaging of DNA into chromatin is a key determinant of eukaryotic gene regulation and epigenetic inheritance. Nucleosomes are the basic unit of chromatin, and therefore the accessible states of the nucleosome must be the starting point for mechanistic models regarding these essential processes. Although the existence of different unwound nucleosome states has been hypothesized, there have been few studies of these states. The consequences of multiple states are far reaching. These states will behave differently in all aspects, including their interactions with chromatin remodelers, histone variant exchange, and kinetic properties. Here, we demonstrate the existence of two distinct states of the unwound nucleosome, which are accessible at physiological forces and ionic strengths. Using optical tweezers, we measure the rates of unwinding and rewinding for these two states and show that the rewinding rates from each state are different. In addition, we show that the probability of unwinding into each state is dependent on the applied force and ionic strength. Our results demonstrate not only that multiple unwound states exist but that their accessibility can be differentially perturbed, suggesting possible roles for these states in gene regulation. For example, different histone variants or modifications may facilitate or suppress access to DNA by promoting unwinding into one state or the other. We anticipate that the two unwound states reported here will be the basis for future models of eukaryotic transcriptional control.

Complex kinetics of DNA condensation revealed through DNA twist tracing

Toroid formation is an important mechanism for DNA condensation in cells. The length change during DNA condensation was investigated in previous single-molecule experiments. However, DNA twist is key to understanding the topological kinetics of DNA condensation. In this study, DNA twist as well as DNA length was traced during the DNA condensation by the freely orbiting magnetic tweezers and the tilted magnetic tweezers combined with Brownian dynamics simulations. The experimental results disclose the complex relationship between DNA extension and backbone rotation. Brownian dynamics simulations show that the toroid formation follows a wiggling pathway which leads to the complex DNA backbone rotation as revealed in our experiments. These findings provide the complete description of multivalent cation-dependent DNA toroid formation under tension.

Quantitative analysis of single-molecule force spectroscopy on folded chromatin fibers

Single-molecule techniques allow for picoNewton manipulation and nanometer accuracy measurements of single chromatin fibers. However, the complexity of the data, the heterogeneity of the composition of individual fibers and the relatively large fluctuations in extension of the fibers complicate a structural interpretation of such force-extension curves. Here we introduce a statistical mechanics model that quantitatively describes the extension of individual fibers in response to force on a per nucleosome basis. Four nucleosome conformations can be distinguished when pulling a chromatin fiber apart. A novel, transient conformation is introduced that coexists with single wrapped nucleosomes between 3 and 7 pN. Comparison of force-extension curves between single nucleosomes and chromatin fibers shows that embedding nucleosomes in a fiber stabilizes the nucleosome by 10 kBT. Chromatin fibers with 20- and 50-bp linker DNA follow a different unfolding pathway. These results have implications for accessibility of DNA in fully folded and partially unwrapped chromatin fibers and are vital for understanding force unfolding experiments on nucleosome arrays.

Condensation transition and forced unravelling of DNA-histone H1 toroids: a multi-state free energy landscape

DNA is known to condense with multivalent cations and positively charged proteins. However, the properties and energetics of DNA superstructures, such as chromatin, are poorly understood. As a model system, we investigate histone H1 condensation of DNA with tethered particle motion and force-extension measurements. We show that after the addition of H1 to DNA, a concentration dependent lag time is followed by the DNA spontaneously condensing. The trigger for this condensation phase transition can be modeled as sufficient H1s having bound to the DNA, providing insight into the 30 nm fiber condensation upon H1 binding. Furthermore, optical tweezers force-extension measurements of histone H1 condensed DNA reveals a sequence of state transitions corresponding to the unwinding of superhelical turns. We determine the complete, experimental, multi-state free energy landscape for the complex using Crooks fluctuation theorem. The measured force-versus-extension and free energy landscape are compared to predictions from a simple, theoretical model. This work encourages the theoretical description of DNA/protein structure and energetics and their role in chromatin and other, more complex, systems.

Wrapping/unwrapping transition of double-stranded DNA in DNA-nanosphere complexes induced by multivalent anions

Wrapping and unwrapping behaviors of double-stranded DNA around a positively charged nanosphere in solution are studied by using the coarse-grained molecular dynamics (CGMD) simulation method. When monovalent, divalent and trivalent anions are added to the DNA-nanosphere complex solution, double-stranded DNA binds with a nanosphere owing to strong electrostatic attraction. However, when tetravalent anions are added to the DNA-nanosphere complex solution, local charge inversion is observed for a high anion concentration of tetravalent anions and the double-stranded DNA can be unwrapped from the nanosphere because of the local charge inversion near the nanosphere. Moreover, the helical structure of DNA is damaged when double-stranded DNA wraps around the nanosphere and the helical structure can be rebuilt when the double-stranded DNA unwraps from the nanosphere. This study can help us understand how to control the release of DNA in DNA-nanosphere complexes.

Regulation of the nucleosome repeat length in vivo by the DNA sequence, protein concentrations and long-range interactions

The nucleosome repeat length (NRL) is an integral chromatin property important for its biological functions. Recent experiments revealed several conflicting trends of the NRL dependence on the concentrations of histones and other architectural chromatin proteins, both in vitro and in vivo, but a systematic theoretical description of NRL as a function of DNA sequence and epigenetic determinants is currently lacking. To address this problem, we have performed an integrative biophysical and bioinformatics analysis in species ranging from yeast to frog to mouse where NRL was studied as a function of various parameters. We show that in simple eukaryotes such as yeast, a lower limit for the NRL value exists, determined by internucleosome interactions and remodeler action. For higher eukaryotes, also the upper limit exists since NRL is an increasing but saturating function of the linker histone concentration. Counterintuitively, smaller H1 variants or non-histone architectural proteins can initiate larger effects on the NRL due to entropic reasons. Furthermore, we demonstrate that different regimes of the NRL dependence on histone concentrations exist depending on whether DNA sequence-specific effects dominate over boundary effects or vice versa. We consider several classes of genomic regions with apparently different regimes of the NRL variation. As one extreme, our analysis reveals that the period of oscillations of the nucleosome density around bound RNA polymerase coincides with the period of oscillations of positioning sites of the corresponding DNA sequence. At another extreme, we show that although mouse major satellite repeats intrinsically encode well-defined nucleosome preferences, they have no unique nucleosome arrangement and can undergo a switch between two distinct types of nucleosome positioning.

Taking into account nucleosomes for predicting gene expression

The eukaryotic genome is organized in a chain of nucleosomes that consist of 145-147 bp of DNA wrapped around a histone octamer protein core. Binding of transcription factors (TF) to nucleosomal DNA is frequently impeded, which makes it a challenging task to calculate TF occupancy at a given regulatory genomic site for predicting gene expression. Here, we review methods to calculate TF binding to DNA in the presence of nucleosomes. The main theoretical problems are (i) the computation speed that is becoming a bottleneck when partial unwrapping of DNA from the nucleosome is considered, (ii) the perturbation of the binding equilibrium by the activity of ATP-dependent chromatin remodelers, which translocate nucleosomes along the DNA, and (iii) the model parameterization from high-throughput sequencing data and fluorescence microscopy experiments in living cells. We discuss strategies that address these issues to efficiently compute transcription factor binding in chromatin.

Rigid-body molecular dynamics of DNA inside a nucleosome

The majority of eukaryotic DNA, about three quarter, is wrapped around histone proteins forming so-called nucleosomes. To study nucleosomal DNA we introduce a coarse-grained molecular dynamics model based on sequence-dependent harmonic rigid base pair step parameters of DNA and nucleosomal binding sites. Mixed parametrization based on all-atom molecular dynamics and crystallographic data of protein-DNA structures is used for the base pair step parameters. The binding site parameters are adjusted by experimental B-factor values of the nucleosome crystal structure. The model is then used to determine the energy cost for placing a twist defect into the nucleosomal DNA which allows us to use Kramers theory to calculate nucleosome sliding caused by such defects. It is shown that the twist defect scenario together with the sequence-dependent elasticity of DNA can explain the slow time scales observed for nucleosome mobility along DNA. With this method we also show how the twist defect mechanism leads to a higher mobility of DNA in the presence of sin mutations near the dyad axis. Finally, by performing simulations on 5s rDNA, 601, and telomeric base pair sequences, it is demonstrated that the current model is a powerful tool to predict nucleosome positioning.

Dynamics of forced nucleosome unraveling and role of nonuniform histone-DNA interactions

A coarse-grained model of the nucleosome is introduced to investigate the dynamics of force-induced unwrapping of DNA from histone octamers. In this model, the DNA is treated as a charged, discrete worm-like chain, and the octamer is treated as a rigid cylinder carrying a positively charged superhelical groove that accommodates 1.7 turns of DNA. The groove charges are parameterized to reproduce the nonuniform histone/DNA interaction free energy profile and the loading rate-dependent unwrapping forces, both obtained from single-molecule experiments. Brownian dynamics simulations of the model under constant loading conditions reveal that nucleosome unraveling occurs in three distinct stages. At small extensions, the flanking DNA exhibits rapid unwrapping-rewrapping (breathing) dynamics and the octamer flips ∼180° and moves toward the pulling axis. At intermediate extensions, the outer turn of DNA unwraps gradually and the octamer swivels about the taut linkers and flips a further ∼90° to orient its superhelical axis almost parallel to the pulling axis. At large extensions, a portion of the inner turn unwraps abruptly with a notable rip in the force-extension plot and a >90° flip of the octamer. The remaining inner turn unwraps reversibly to leave a small portion of DNA attached to the octamer despite extended pulling. Our simulations further reveal that the nonuniform histone/DNA interactions in canonical nucleosomes serve to: stabilize the inner turn against unraveling while enhancing the breathing dynamics of the nucleosome and prevent dissociation of the octamer from the DNA while facilitating its mobility along the DNA. Thus, the modulation of the histone/DNA interactions could constitute one possible mechanism for regulating the accessibility of the nucleosome-wound DNA sequences.

Unwinding and rewinding the nucleosome inner turn: force dependence of the kinetic rate constants

A simple model for the force-dependent unwinding and rewinding rates of the nucleosome inner turn is constructed and quantitatively compared to the results of recent measurements [A. H. Mack et al., J. Mol. Biol. 423, 687 (2012)]. First, a coarse-grained model for the histone-DNA free-energy landscape that incorporates both an elastic free-energy barrier and specific histone-DNA bonds is developed. Next, a theoretical expression for the rate of transitions across a piecewise linear free-energy landscape with multiple minima and maxima is presented. Then, the model free-energy landscape, approximated as a piecewise linear function, and the theoretical expression for the transition rates are combined to construct a model for the force-dependent unwinding and rewinding rates of the nucleosome inner turn. Least-mean-squares fitting of the model rates to the rates observed in recent experiments rates demonstrates that this model is able to well describe the force-dependent unwinding and rewinding rates of the nucleosome inner turn, observed in the recent experiments, except at the highest forces studied, where an additional ad hoc term is required to describe the data, which may be interpreted as an indication of an alternate high-force nucleosome disassembly pathway, that bypasses simple unwinding. The good agreement between the measurements and the model at lower forces demonstrates that both specific histone-DNA contacts and an elastic free-energy barrier play essential roles for nucleosome winding and unwinding, and quantifies their relative contributions.

Impact of DNA twist accumulation on progressive helical wrapping of torsionally constrained DNA

DNA wrapping is an important mechanism for chromosomal DNA packaging in cells and viruses. Previous studies of DNA wrapping have been performed mostly on torsionally unconstrained DNA, while in vivo DNA is often under torsional constraint. In this study, we extend a previously proposed theoretical model for wrapping of torsionally unconstrained DNA to a new model including the contribution of DNA twist energy, which influences DNA wrapping drastically. In particular, due to accumulation of twist energy during DNA wrapping, it predicts a finite amount of DNA that can be wrapped on a helical spool. The predictions of the new model are tested by single-molecule study of DNA wrapping under torsional constraint using magnetic tweezers. The theoretical predictions and the experimental results are consistent with each other and their implications are discussed.

Kinetics and thermodynamics of phenotype: unwinding and rewinding the nucleosome

Chromatin "remodeling" is widely accepted as the mechanism that permits access to DNA by the transcription machinery. To date, however, there has been no experimental measurement of the changes in the kinetics and thermodynamics of the DNA-histone octamer association that are required to remodel chromatin so that transcription may occur. Here, we present the results of optical tweezer measurements that compare the kinetic and thermodynamic properties of nucleosomes composed of unmodified histones with those of nucleosomes that contain a mutant histone H4 (H4-R45H), which has been shown to allow SWI/SNF remodeling factor-independent transcription from the yeast HO promoter in vivo. Our measurements, carried out in a force-clamp mode, determine the force-dependent unwinding and rewinding rates of the nucleosome inner turn. At each force studied, nucleosomes containing H4-R45H unwind more rapidly and rewind more slowly than nucleosomes containing unmodified H4, indicating that the latter are the more stable. Extrapolation to forces at which the winding and unwinding rates are equal determines the absolute free energy of the nucleosome inner turn to be -32k(B)T for nucleosomes containing unmodified H4 and -27k(B)T for nucleosomes containing H4-R45H. Thus, the "loosening" or "remodeling" caused by this point mutation, which is demonstrated to be sufficient to allow transcriptional machinery access to the HO promoter (in the absence of other remodeling factors), is 5k(B)T. The correlation between the free energy of the nucleosome inner turn and the sin (SWI/SNF-independent) transcription suggests that, beyond partial unwinding, complete histone unwinding may play a role in transcriptional activation.

Length-Dependent Assembly of a Stiff Polymer Chain at the Interface of a Carbon Nanotube

We perform computer simulations to explore the suprastructures and their formation mechanism in the length-dependent assembly of a stiff polymer chain on the carbon nanotube surface. Three types of local conformations, that is, helical wrapping along the nanotube threadline, nonhelical loop, and straight extension along the nanotube, are identified in the very stiff polymer, depending on its length. It is revealed that the high elastic energy penalty and the large length of a long stiff polymer hinder its conformation transition on the nanotube, which impairs the match between the polymer beads and the structural details of the underlying nanotube surface and thereby weakens the polymer-nanotube interaction. A preferred chain length with an energy minimum is documented for the first time in the self-assemble of a stiff polymer at the nanotube interface. These data significantly advance our understanding of the superstructure formation by self-assembly of various chain-like molecules (e.g., polymer, surfactants, DNA, peptides, etc.) on carbon nanotube.

Nucleosome dynamics between tension-induced states

We studied the dynamical behavior of a mononucleosome under tension using a theoretical model that takes into account the nucleosomal geometry, DNA elasticity, nonspecific DNA-protein binding, and effective repulsion between the two DNA turns. Using a dynamical Monte-Carlo simulation algorithm, we demonstrate that this model shows a behavior that for an appropriate set of parameters is in quantitative agreement with data from micromanipulation experiments on individual nucleosomes. All of the parameters of the model follow from the data obtained from two types of pulling experiments, namely, constant force and constant loading rate ensembles.

Modelling chromatin structure and dynamics: status and prospects

The packaging of genomic DNA into chromatin in the eukaryotic cell nucleus demands extensive compaction. This requires attractive nucleosome-nucleosome interactions to overcome repulsion between the negatively charged DNA segments as well as other constraints. At the same time, DNA must be dynamically accessible to the cellular machinery that operates on it. Recent progress in the experimental characterisation of the higher order structure and dynamics of well-defined chromatin fibres has stimulated the attempts at theoretical description of chromatin and the nucleosome. Here we review the present status of chromatin modelling, with particular emphasis on coarse-grained computer simulation models, the role of electrostatic interactions, and discuss future perspectives in the field.

Dissecting DNA-histone interactions in the nucleosome by molecular dynamics simulations of DNA unwrapping

The nucleosome complex of DNA wrapped around a histone protein octamer organizes the genome of eukaryotes and regulates the access of protein factors to the DNA. We performed molecular dynamics simulations of the nucleosome in explicit water to study the dynamics of its histone-DNA interactions. A high-resolution histone-DNA interaction map was derived that revealed a five-nucleotide periodicity, in which the two DNA strands of the double helix made alternating contacts. On the 100-ns timescale, the histone tails mostly maintained their initial positions relative to the DNA, and the spontaneous unwrapping of DNA was limited to 1-2 basepairs. In steered molecular dynamics simulations, external forces were applied to the linker DNA to investigate the unwrapping pathway of the nucleosomal DNA. In comparison with a nucleosome without the unstructured N-terminal histone tails, the following findings were obtained: 1), Two main barriers during unwrapping were identified at DNA position ±70 and ±45 basepairs relative to the central DNA basepair at the dyad axis. 2), DNA interactions of the histone H3 N-terminus and the histone H2A C-terminus opposed the initiation of unwrapping. 3), The N-terminal tails of H2A, H2B, and H4 counteracted the unwrapping process at later stages and were essential determinants of nucleosome dynamics. Our detailed analysis of DNA-histone interactions revealed molecular mechanisms for modulating access to nucleosomal DNA via conformational rearrangements of its structure.

From crystal and NMR structures, footprints and cryo-electron-micrographs to large and soft structures: nanoscale modeling of the nucleosomal stem

The interaction of histone H1 with linker DNA results in the formation of the nucleosomal stem structure, with considerable influence on chromatin organization. In a recent paper [Syed,S.H., Goutte-Gattat,D., Becker,N., Meyer,S., Shukla,M.S., Hayes,J.J., Everaers,R., Angelov,D., Bednar,J. and Dimitrov,S. (2010) Single-base resolution mapping of H1-nucleosome interactions and 3D organization of the nucleosome. Proc. Natl Acad. Sci. USA, 107, 9620-9625], we published results of biochemical footprinting and cryo-electron-micrographs of reconstituted mono-, di- and tri-nucleosomes, for H1 variants with different lengths of the cationic C-terminus. Here, we present a detailed account of the analysis of the experimental data and we include thermal fluctuations into our nano-scale model of the stem structure. By combining (i) crystal and NMR structures of the nucleosome core particle and H1, (ii) the known nano-scale structure and elasticity of DNA, (iii) footprinting information on the location of protected sites on the DNA backbone and (iv) cryo-electron micrographs of reconstituted tri-nucleosomes, we arrive at a description of a polymorphic, hierarchically organized stem with a typical length of 20 ± 2 base pairs. A comparison to linker conformations inferred for poly-601 fibers with different linker lengths suggests, that intra-stem interactions stabilize and facilitate the formation of dense chromatin fibers.

The dynamics of the nucleosome: thermal effects, external forces and ATP

With nucleosomes being tightly associated with the majority of eukaryotic DNA, it is essential that mechanisms are in place that can move nucleosomes 'out of the way'. A focus of current research comprises chromatin remodeling complexes, which are ATP-consuming protein complexes that, for example, pull or push nucleosomes along DNA. The precise mechanisms used by those complexes are not yet understood. Hints for possible mechanisms might be found among the various spontaneous fluctuations that nucleosomes show in the absence of remodelers. Thermal fluctuations induce the partial unwrapping of DNA from the nucleosomes and introduce twist or loop defects in the wrapped DNA, leading to nucleosome sliding along DNA. In this minireview, we discuss nucleosome dynamics from two angles. First, we describe the dynamical modes of nucleosomes in the absence of remodelers that are experimentally fairly well characterized and theoretically understood. Then, we discuss remodelers and describe recent insights about the possible schemes that they might use.

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.

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.

A lattice model for transcription factor access to nucleosomal DNA

Nucleosomes, the basic repeating unit of chromatin, consist of 147 basepairs of DNA that are wrapped in almost two turns around a histone protein octamer core. Because ∼3/4 of the human genomic DNA is found within nucleosomes, their position and DNA interaction is an essential determinant for the DNA access of gene-specific transcription factors and other proteins. Here, a DNA lattice model was developed for describing ligand binding in the presence of a nucleosome. The model takes into account intermediate states, in which DNA is partially unwrapped from the histone octamer. This facilitates access of transcription factors to up to 60 DNA basepairs located in the outer turn of nucleosomal DNA, while the inner DNA turn was found to be more resistant to competitive ligand binding. As deduced from quantitative comparisons with recently published experimental data, our model provides a better description than the previously used all-or-none lattice-binding model. Importantly, nucleosome-occupancy maps predicted by the nucleosome-unwrapping model also differed significantly when partial unwrapping of nucleosomal DNA was considered. In addition, large effects on the cooperative binding of transcription factors to multiple binding sites occluded by the nucleosome were apparent. These findings indicate that partial unwrapping of DNA from the histone octamer needs to be taken into account in quantitative models of gene regulation in chromatin.

Tension-dependent structural deformation alters single-molecule transition kinetics

We analyze the response of a single nucleosome to tension, which serves as a prototypical biophysical measurement where tension-dependent deformation alters transition kinetics. We develop a statistical-mechanics model of a nucleosome as a wormlike chain bound to a spool, incorporating fluctuations in the number of bases bound, the spool orientation, and the conformations of the unbound polymer segments. With the resulting free-energy surface, we perform dynamic simulations that permit a direct comparison with experiments. This simple approach demonstrates that the experimentally observed structural states at nonzero tension are a consequence of the tension and that these tension-induced states cease to exist at zero tension. The transitions between states exhibit substantial deformation of the unbound polymer segments. The associated deformation energy increases with tension; thus, the application of tension alters the kinetics due to tension-induced deformation of the transition states. This mechanism would arise in any system where the tether molecule is deformed in the transition state under the influence of tension.

Torque-induced deformations of charged elastic DNA rods: thin helices, loops, and precursors of DNA supercoiling

We study the deformations of charged elastic rods under applied end forces and torques. For neutral filaments, we analyze the energetics of initial helical deformations and loop formation. We supplement this elastic approach with electrostatic energies of bent filaments and find critical conditions for buckling depending on the ionic strength of the solution. We also study force-induced loop opening, for parameters relevant for DNA. Finally, some applications of this nano-mechanical DNA model to salt-dependent onset of the DNA supercoiling are discussed.