The elastic rod model for DNA and its application to the tertiary structure of DNA minicircles in mononucleosomes

DNA superhelicity

Closing each strand of a DNA duplex upon itself fixes its linking number L. This topological condition couples together the secondary and tertiary structures of the resulting ccDNA topoisomer, a constraint that is not present in otherwise identical nicked or linear DNAs. Fixing L has a range of structural, energetic and functional consequences. Here we consider how L having different integer values (that is, different superhelicities) affects ccDNA molecules. The approaches used are primarily theoretical, and are developed from a historical perspective. In brief, processes that either relax or increase superhelicity, or repartition what is there, may either release or require free energy. The energies involved can be substantial, sufficient to influence many events, directly or indirectly. Here two examples are developed. The changes of unconstrained superhelicity that occur during nucleosome attachment and release are examined. And a simple theoretical model of superhelically driven DNA structural transitions is described that calculates equilibrium distributions for populations of identical topoisomers. This model is used to examine how these distributions change with superhelicity and other factors, and applied to analyze several situations of biological interest.

Shapes of a filament on the surface of a bubble

The shape assumed by a slender elastic structure is a function both of the geometry of the space in which it exists and the forces it experiences. We explore, by experiments and theoretical analysis, the morphological phase space of a filament confined to the surface of a spherical bubble. The morphology is controlled by varying bending stiffness and weight of the filament, and its length relative to the bubble radius. When the dominant considerations are the geometry of confinement and elastic energy, the filament lies along a geodesic and when gravitational energy becomes significant, a bifurcation occurs, with a part of the filament occupying a longitude and the rest along a curve approximated by a latitude. Far beyond the transition, when the filament is much longer than the diameter, it coils around the selected latitudinal region. A simple model with filament shape as a composite of two arcs captures the transition well. For better quantitative agreement with the subcritical nature of bifurcation, we study the morphology by numerical energy minimization. Our analysis of the filament’s morphological space spanned by one geometric parameter, and one parameter that compares elastic energy with body forces, may provide guidance for packing slender structures on complex surfaces.

Surprising Twists in Nucleosomal DNA with Implication for Higher-order Folding

While nucleosomes are dynamic entities that must undergo structural deformations to perform their functions, the general view from available high-resolution structures is a largely static one. Even though numerous examples of twist defects have been documented, the DNA wrapped around the histone core is generally thought to be overtwisted. Analysis of available high-resolution structures from the Protein Data Bank reveals a heterogeneous distribution of twist along the nucleosomal DNA, with clear patterns that are consistent with the literature, and a significant fraction of structures that are undertwisted. The subtle differences in nucleosomal DNA folding, which extend beyond twist, have implications for nucleosome disassembly and modeled higher-order structures. Simulations of oligonucleosome arrays built with undertwisted models behave very differently from those constructed from overtwisted models, in terms of compaction and inter-nucleosome contacts, introducing configurational changes equivalent to those associated with 2-3 base-pair changes in nucleosome spacing. Differences in the nucleosomal DNA pathway, which underlie the way that DNA enters and exits the nucleosome, give rise to different nucleosome-decorated minicircles and affect the topological mix of configurational states.

An analytical method to connect open curves for modeling protein-bound DNA minicircles

We introduce an analytical method to generate the pathway of a closed protein-bound DNA minicircle. We develop an analytical equation to connect two open curves smoothly and use the derived expressions to join the ends of two helical pathways and form models of nucleosome-decorated DNA minicircles. We find that the simplest smooth connector which satisfies the boundary conditions at the end points and the length requirement for such connections to be a quartic function on the xy-plane and linear along the z-direction. This is a general method which can be used to connect any two open curves with well defined mathematical definitions as well as pairs of discrete systems found experimentally. We used this method to describe the configurations of torsionally relaxed, 360-base pair DNA rings with two evenly-spaced, ideal nucleosomes. We considered superhelical nucleosomal pathways with different levels of DNA wrapping and allowed for different inter-nucleosome orientations. We completed the DNA circles with the smooth connectors and studied the associated bending and electrostatic energies for different configurations in the absence and presence of salt. The predicted stable states bear close resemblance to reconstituted minicircles observed under low and high salt conditions.

Eulerian formulation of elastic rods

In numerous biological, medical and engineering applications, elastic rods are constrained to deform inside or around tube-like surfaces. To solve efficiently this class of problems, the equations governing the deflection of elastic rods are reformulated within the Eulerian framework of this generic tubular constraint defined as a perfectly stiff normal ringed surface. This reformulation hinges on describing the rod-deformed configuration by means of its relative position with respect to a reference curve, defined as the axis or spine curve of the constraint, and on restating the rod local equilibrium in terms of the curvilinear coordinate parametrizing this curve. Associated with a segmentation strategy, which partitions the global problem into a sequence of rod segments either in continuous contact with the constraint or free of contact (except for their extremities), this re-parametrization not only trivializes the detection of new contacts but also transforms these free boundary problems into classic two-points boundary-value problems and suppresses the isoperimetric constraints resulting from the imposition of the rod position at the extremities of each rod segment.

Coarse-grained modeling of RNA 3D structure

Functional RNA molecules depend on three-dimensional (3D) structures to carry out their tasks within the cell. Understanding how these molecules interact to carry out their biological roles requires a detailed knowledge of RNA 3D structure and dynamics as well as thermodynamics, which strongly governs the folding of RNA and RNA-RNA interactions as well as a host of other interactions within the cellular environment. Experimental determination of these properties is difficult, and various computational methods have been developed to model the folding of RNA 3D structures and their interactions with other molecules. However, computational methods also have their limitations, especially when the biological effects demand computation of the dynamics beyond a few hundred nanoseconds. For the researcher confronted with such challenges, a more amenable approach is to resort to coarse-grained modeling to reduce the number of data points and computational demand to a more tractable size, while sacrificing as little critical information as possible. This review presents an introduction to the topic of coarse-grained modeling of RNA 3D structures and dynamics, covering both high- and low-resolution strategies. We discuss how physics-based approaches compare with knowledge based methods that rely on databases of information. In the course of this review, we discuss important aspects in the reasoning process behind building different models and the goals and pitfalls that can result.

Lac repressor mediated DNA looping: Monte Carlo simulation of constrained DNA molecules complemented with current experimental results

Tethered particle motion (TPM) experiments can be used to detect time-resolved loop formation in a single DNA molecule by measuring changes in the length of a DNA tether. Interpretation of such experiments is greatly aided by computer simulations of DNA looping which allow one to analyze the structure of the looped DNA and estimate DNA-protein binding constants specific for the loop formation process. We here present a new Monte Carlo scheme for accurate simulation of DNA configurations subject to geometric constraints and apply this method to Lac repressor mediated DNA looping, comparing the simulation results with new experimental data obtained by the TPM technique. Our simulations, taking into account the details of attachment of DNA ends and fluctuations of the looped subsegment of the DNA, reveal the origin of the double-peaked distribution of RMS values observed by TPM experiments by showing that the average RMS value for anti-parallel loop types is smaller than that of parallel loop types. The simulations also reveal that the looping probabilities for the anti-parallel loop types are significantly higher than those of the parallel loop types, even for loops of length 600 and 900 base pairs, and that the correct proportion between the heights of the peaks in the distribution can only be attained when loops with flexible Lac repressor conformation are taken into account. Comparison of the in silico and in vitro results yields estimates for the dissociation constants characterizing the binding affinity between O1 and Oid DNA operators and the dimeric arms of the Lac repressor.

Characterization of the geometry and topology of DNA pictured as a discrete collection of atoms

The structural and physical properties of DNA are closely related to its geometry and topology. The classical mathematical treatment of DNA geometry and topology in terms of ideal smooth space curves was not designed to characterize the spatial arrangements of atoms found in high-resolution and simulated double-helical structures. We present here new and rigorous numerical methods for the rapid and accurate assessment of the geometry and topology of double-helical DNA structures in terms of the constituent atoms. These methods are well designed for large DNA datasets obtained in detailed numerical simulations or determined experimentally at high-resolution. We illustrate the usefulness of our methodology by applying it to the analysis of three canonical double-helical DNA chains, a 65-bp minicircle obtained in recent molecular dynamics simulations, and a crystallographic array of protein-bound DNA duplexes. Although we focus on fully base-paired DNA structures, our methods can be extended to treat the geometry and topology of melted DNA structures as well as to characterize the folding of arbitrary molecules such as RNA and cyclic peptides.

Group theory and biomolecular conformation: I. Mathematical and computational models

Biological macromolecules, and the complexes that they form, can be described in a variety of ways ranging from quantum mechanical and atomic chemical models, to coarser grained models of secondary structure and domains, to continuum models. At each of these levels, group theory can be used to describe both geometric symmetries and conformational motion. In this survey, a detailed account is provided of how group theory has been applied across computational structural biology to analyze the conformational shape and motion of macromolecules and complexes.

Optimal plane beams modelling elastic linear objects

This paper discusses analytical and deterministic models for a plane curve with minimum deformation that may be utilized in planning the motion of elastic linear objects and investigating the inverse kinematics of a hyper-redundant robot. It usually requires intensive computation to determine the configuration of elastic linear objects. In addition, conventional optimization-based numerical techniques that identify the shape of elastic linear objects in equilibrium involve non-deterministic aspects. Several analytical models that produce the configuration of elastic linear objects in an efficient and deterministic manner are presented in this paper. To develop the analytical expressions for elastic linear objects, we consider a cantilever beam where the deflections are determined according to the Euler–Bernoulli beam theory. The deflections of the cantilever beam are determined for prescribed constraints imposed on the deflections at the free end to replicate various elastic linear objects. Deflections of a cantilever beam with roller supports are explored to replicate elastic linear objects in contact with rigid objects. We verify the analytical models by comparing them with exact beam deflections. The analytical model is precisely accurate for beams with small deflections as it is developed on the basis of the Euler–Bernoulli beam theory. Although it is applied to beams undergoing large deflections, it is still reasonably accurate and at least as precise as the conventional pseudo-rigid-body model. The computational demand involved in using the analytical models is negligible. Therefore, efficient motion planning for elastic linear objects can be realized when the proposed analytical models are combined with conventional motion planning algorithms. We also demonstrate that the analytical model solves the inverse kinematics problem in an efficient and robust manner through numerical simulations.

A symplectic integration method for elastic filaments

A new method is proposed for integrating the equations of motion of an elastic filament. In the standard finite-difference and finite-element formulations the continuum equations of motion are discretized in space and time, but it is then difficult to ensure that the Hamiltonian structure of the exact equations is preserved. Here we discretize the Hamiltonian itself, expressed as a line integral over the contour of the filament. This discrete representation of the continuum filament can then be integrated by one of the explicit symplectic integrators frequently used in molecular dynamics. The model systematically approximates the continuum partial differential equations, but has the same level of computational complexity as molecular dynamics and is constraint-free. Numerical tests show that the algorithm is much more stable than a finite-difference formulation and can be used for high aspect ratio filaments, such as actin.

The Stochastic Elastica and Excluded-Volume Perturbations of DNA Conformational Ensembles

A coordinate-free Lie-group formulation for generating ensembles of DNA conformations in solution is presented. In this formulation, stochastic differential equations define sample paths on the Euclidean motion group. The ensemble of these paths exhibits the same behavior as solutions of the Fokker-Planck equation for the stochastically forced elastica. Longer chains for which the effects of excluded volume become important are handled by piecing together shorter chains and modeling their interactions. It is assumed that the final chain lengths of interest are long enough for excluded volume effects to become important, but not so long that the semi-flexible nature of the chain is lost. The effect of excluded volume is then taken into account by grouping short self-avoiding conformations into 'bundles' with common end constraints and computing average interaction effects between bundles. The accuracy of this approximation is shown to be good when using a numerically generated ensemble of self-avoiding sample paths as the baseline for comparison.

Mesoscale modeling of multi-protein-DNA assemblies: the role of the catabolic activator protein in Lac-repressor-mediated looping

DNA looping plays a key role in the regulation of the lac operon in Escherichia coli. The presence of a tightly bent loop (between sequentially distant sites of Lac repressor protein binding) purportedly hinders the binding of RNA polymerase and subsequent transcription of the genetic message. The unexpectedly favorable binding interaction of this protein-DNA assembly with the catabolic activator protein (CAP), a protein that also bends DNA and paradoxically facilitates the binding of RNA polymerase, stimulated extension of our base-pair level theory of DNA elasticity to the treatment of DNA loops formed in the presence of several proteins. Here we describe in detail a procedure to determine the structures and free energies of multi-protein-DNA assemblies and illustrate the predicted effects of CAP binding on the configurations of the wild-type 92-bp Lac repressor-mediated O3-O1 DNA loop. We show that the DNA loop adopts an antiparallel orientation in the most likely structure and that this loop accounts for the published experimental observation that, when CAP is bound to the loop, one of the arms of LacR binds to an alternative site that is displaced from the original site by 5 bp.

Effects of the nucleoid protein HU on the structure, flexibility, and ring-closure properties of DNA deduced from Monte Carlo simulations

The histone-like HU (heat unstable) protein plays a key role in the organization and regulation of the Escherichia coli genome. The nonspecific nature of HU binding to DNA complicates analysis of the mechanism by which the protein contributes to the looping of DNA. Conventional models of the looping of HU-bound duplexes attribute the changes in biophysical properties of DNA brought about by the random binding of protein to changes in the effective parameters of an ideal helical wormlike chain. Here, we introduce a novel Monte Carlo approach to study the effects of nonspecific HU binding on the configurational properties of DNA directly. We randomly decorated segments of an ideal double-helical DNA with HU molecules that induce the bends and other structural distortions of the double helix find in currently available X-ray structures. We find that the presence of HU at levels approximating those found in the cell reduces the persistence length by roughly threefold compared with that of naked DNA. The binding of protein has particularly striking effects on the cyclization properties of short duplexes, altering the dependence of ring closure on chain length in a way that cannot be mimicked by a simple wormlike model and accumulating at higher-than-expected levels on successfully closed chains. Moreover, the uptake of protein on small minicircles depends on chain length, taking advantage of the HU-induced deformations of DNA structure to facilitate ligation. Circular duplexes with bound HU show much greater propensity than protein-free DNA to exist as negatively supercoiled topoisomers, suggesting a potential role of HU in organizing the bacterial nucleoid. The local bending and undertwisting of DNA by HU, in combination with the number of bound proteins, provide a structural rationale for the condensation of DNA and the observed expression levels of reporter genes in vivo.

Writhe formulas and antipodal points in plectonemic DNA configurations

The linking and writhing numbers are key quantities when characterizing the structure of a piece of supercoiled DNA. Defined as double integrals over the shape of the double helix, these numbers are not always straightforward to compute, though a simplified formula was established in a theorem by Fuller [Proc. Natl. Acad. Sci. U.S.A. 75, 3557 (1978)]. We examine the range of applicability of this widely used simplified formula, and show that it cannot be employed for plectonemic DNA. We show that inapplicability is due to a hypothesis of Fuller theorem that is not met. The hypothesis seems to have been overlooked in many works.

CENP-A-containing nucleosomes: easier disassembly versus exclusive centromeric localization

CENP-A is a histone variant that replaces conventional H3 in nucleosomes of functional centromeres. We report here, from reconstitutions of CENP-A- and H3-containing nucleosomes on linear DNA fragments and the comparison of their electrophoretic mobility, that CENP-A induces some positioning of its own and some unwrapping at the entry-exit relative to canonical nucleosomes on both 5 S DNA and the alpha-satellite sequence on which it is normally loaded. This steady-state unwrapping was quantified to 7(+/-2) bp by nucleosome reconstitutions on a series of DNA minicircles, followed by their relaxation with topoisomerase I. The unwrapping was found to ease nucleosome invasion by exonuclease III, to hinder the binding of a linker histone, and to promote the release of an H2A-H2B dimer by nucleosome assembly protein 1 (NAP-1). The (CENP-A-H4)2 tetramer was also more readily destabilized with heparin than the (H3-H4)2 tetramer, suggesting that CENP-A has evolved to confer its nucleosome a specific ability to disassemble. This dual relative instability is proposed to facilitate the progressive clearance of CENP-A nucleosomes that assemble promiscuously in euchromatin, especially as is seen following CENP-A transient over-expression.

LFM-Pro: a tool for detecting significant local structural sites in proteins

MOTIVATION

The rapidly growing protein structure repositories have opened up new opportunities for discovery and analysis of functional and evolutionary relationships among proteins. Detecting conserved structural sites that are unique to a protein family is of great value in identification of functionally important atoms and residues. Currently available methods are computationally expensive and fail to detect biologically significant local features.

RESULTS

We propose Local Feature Mining in Proteins (LFM-Pro) as a framework for automatically discovering family-specific local sites and the features associated with these sites. Our method uses the distance field to backbone atoms to detect geometrically significant structural centers of the protein. A feature vector is generated from the geometrical and biochemical environment around these centers. These features are then scored using a statistical measure, for their ability to distinguish a family of proteins from a background set of unrelated proteins, and successful features are combined into a representative set for the protein family. The utility and success of LFM-Pro are demonstrated on trypsin-like serine proteases family of proteins and on a challenging classification dataset via comparison with DALI. The results verify that our method is successful both in identifying the distinctive sites of a given family of proteins, and in classifying proteins using the extracted features.

AVAILABILITY

The software and the datasets are freely available for academic research use at http://bioinfo.ceng.metu.edu.tr/Pub/LFMPro.

Modeling DNA loops using the theory of elasticity

An elastic rod model of a protein-bound DNA loop is adapted for application in multi-scale simulations of protein-DNA complexes. The classical Kirchhoff system of equations which describes the equilibrium structure of the elastic loop is modified to account for the intrinsic twist and curvature, anisotropic bending properties, and electrostatic charge of DNA. The effects of bending anisotropy and electrostatics are studied for the DNA loop clamped by the lac repressor protein. For two possible lengths of the loop, several topologically different conformations are predicted and extensively analyzed over the broad range of model parameters describing DNA bending and electrostatic properties. The scope and applications of the model in already accomplished and in future multi-scale studies of protein-DNA complexes are discussed.

Conformational Analysis of Stiff Chiral Polymers with End-Constraints

We present a Lie-group-theoretic method for the kinematic and dynamic analysis of chiral semi-flexible polymers with end constraints. The first is to determine the minimum energy conformations of semi-flexible polymers with end constraints, and the second is to perform normal mode analysis based on the determined minimum energy conformations. In this paper, we use concepts from the theory of Lie groups and principles of variational calculus to model such polymers as inextensible or extensible chiral elastic rods with coupling between twisting and bending stiffnesses, and/or between twisting and extension stiffnesses. This method is general enough to include any stiffness and chirality parameters in the context of elastic filament models with the quadratic elastic potential energy function. As an application of this formulation, the analysis of DNA conformations is discussed. We demonstrate our method with examples of DNA conformations in which topological properties such as writhe, twist, and linking number are calculated from the results of the proposed method. Given these minimum energy conformations, we describe how to perform the normal mode analysis. The results presented here build both on recent experimental work in which DNA mechanical properties have been measured, and theoretical work in which the mechanics of non-chiral elastic rods has been studied.

Theory of self-contact in Kirchhoff rods with applications to supercoiling of knotted and unknotted DNA plasmids

There are circumstances under which it is useful to model a molecule of duplex DNA as a homogeneous, inextensible, intrinsically straight, impenetrable elastic rod of circular cross-section obeying the theory of Kirchhoff. For such rods recent research has yielded exact analytical solutions of Kirchhoff's equations of mechanical equilibrium with the effects of impenetrability taken into account, and criteria have been derived for determining whether an equilibrium configuration is stable in the sense that it gives a strict local minimum to the elastic energy. This paper contains a summary of published results on equilibrium configurations for the case in which a rod has been pre-twisted and closed to form a knot-free ring. Emphasis is placed on the way the writhe Wr of the ring, the number of its discrete points of self-contact, and the presence or absence of lines of contact, depend on the excess link, Delta Lk, which is a measure of the amount the rod was twisted before its ends were joined. Bifurcation diagrams are presented and a summary is given of the properties of the primary, secondary and tertiary branches that arise by successive bifurcations from the 'trivial branch' comprised of configurations for which the axial curve is a circle. New results are presented in the theory of equilibrium configurations of closed rods with the topology of torus knots. It is remarked that examples of equilibrium configurations of closed rods of one knot type can be obtained from examples of other knot types using methods previously employed to calculate isolas of equilibrium configurations of knot-free rings. Bifurcation diagrams are shown for supercoiled (2,3) torus knots (trefoil knots). It is observed that for sufficiently large and sufficiently small Delta Lk the minimum elastic energy configuration of a trefoil knot contains plectonemic loops with straight contact lines, although the configuration that minimizes the elastic energy of a general (2,q) torus knot over the entire range of Delta Lk has self-contact along a closed curve. As the ratio of the diameter of the rod to its length approaches zero, that contact curve becomes a circle, and there is an open interval of values of Delta Lk for which stable equilibrium configurations with such circular contact curves exist. Examples of minimum energy configurations are presented for both torus knots and catenates formed by linking two unknots.

Nucleosome conformational flexibility and implications for chromatin dynamics

The active role of chromatin in the regulation of gene activity seems to imply a conformational flexibility of the basic chromatin structural unit, the nucleosome. This review is devoted to our recent results pertaining to this subject, using an original approach based on the topology of single particles reconstituted on DNA minicircles, combined with their theoretical simulation. Three types of chromatin particles have been studied so far: a subnucleosome, that is, the (H3-H4)(2) histone tetramer-containing particle, now known as the tetrasome; the nucleosome; and the linker histone H5/H1-bearing nucleosome (the chromatosome). All the particles were found to exist in two to three conformational states, which differ by their topological and mechanical properties. Our approach unveiled the molecular mechanisms of nucleosome conformational dynamics and will help to understand its functional relevance. A most surprising conclusion of the work was perhaps that DNA overall flexibility increases considerably upon particle formation, which might indeed be a requirement of genome function.

Linker histone-dependent organization and dynamics of nucleosome entry/exit DNAs

A DNA sequence-dependent nucleosome structural and dynamic polymorphism was recently uncovered through topoisomerase I relaxation of mononucleosomes on two homologous approximately 350-370 bp DNA minicircle series, one originating from pBR322, the other from the 5S nucleosome positioning sequence. Whereas both pBR and 5S nucleosomes had access to the closed, negatively crossed conformation, only the pBR nucleosome had access to the positively crossed conformation. Simulation suggested this discrepancy was the result of a reorientation of entry/exit DNAs, itself proposed to be the consequence of specific DNA untwistings occurring in pBR nucleosome where H2B N-terminal tails pass between the two gyres. The present work investigates the behavior of the same two nucleosomes after binding of linker histone H5, its globular domain, GH5, and engineered H5 C-tail deletion mutants. Nucleosome access to the open uncrossed conformation was suppressed and, more surprisingly, the ability of 5S nucleosome to positively cross was largely restored. This, together with the paradoxical observation of a less extensive crossing in the negative conformation with GH5 than without, favored an asymmetrical location of the globular domain in interaction with the central gyre and only entry (or exit) DNA, and raised the possibility of the domain physical rotation as a mechanism assisting nucleosome fluctuation from one conformation to the other. Moreover, both negative and positive conformations showed a high degree of loop conformational flexibility in the presence of the full-length H5 C-tail, which the simulation suggested to reflect the unique feature of the resulting stem to bring entry/exit DNAs in contact and parallel. The results point to the stem being a fundamental structural motif directing chromatin higher order folding, as well as a major player in its dynamics.

Nucleosome repositioning via loop formation

Active (catalyzed) and passive (intrinsic) nucleosome repositioning is known to be a crucial event during the transcriptional activation of certain eukaryotic genes. Here we consider theoretically the intrinsic mechanism and study in detail the energetics and dynamics of DNA-loop-mediated nucleosome repositioning, as previously proposed by earlier works. The surprising outcome of the present study is the inherent nonlocality of nucleosome motion within this model-being a direct physical consequence of the loop mechanism. On long enough DNA templates the longer jumps dominate over the previously predicted local motion, a fact that contrasts simple diffusive mechanisms considered before. The possible experimental outcome resulting from the considered mechanism is predicted, discussed, and compared to existing experimental findings.

Sequence-dependent nucleosome structural and dynamic polymorphism. Potential involvement of histone H2B N-terminal tail proximal domain

Relaxation of nucleosomes on an homologous series (pBR) of ca 350-370 bp DNA minicircles originating from plasmid pBR322 was recently used as a tool to study their structure and dynamics. These nucleosomes thermally fluctuated between three distinct DNA conformations within a histone N-terminal tail-modulated equilibrium: one conformation was canonical, with 1.75 turn wrapping and negatively crossed entering and exiting DNAs; another was also "closed", but with these DNAs positively crossed; and the third was "open", with a lower than 1.5 turn wrapping and uncrossed DNAs. In this work, a new minicircle series (5S) of similar size was used, which contained the 5S nucleosome positioning sequence. Results showed that DNA in pBR nucleosomes was untwisted by approximately 0.2 turn relative to 5S nucleosomes, which DNase I footprinting confirmed in revealing a approximately 1 bp untwisting at each of the two dyad-distal sites where H2B N-terminal tails pass between the two gyres. In contrast, both nucleosomes showed untwistings at the dyad-proximal sites, i.e. on the other gyre, which were also observed in the high-resolution crystal structure. 5S nucleosomes also differ with respect to their dynamics: they hardly accessed the positively crossed conformation, but had an easier access to the negatively crossed conformation. Simulation showed that such reverse effects on the conformational free energies could be simply achieved by slightly altering the trajectories of entering and exiting DNAs. We propose that this is accomplished by H2B tail untwisting at the distal sites through action at a distance ( approximately 20 bp) on H3-tail interactions with the small groove at the nucleosome entry-exit. These results may help to gain a first glimpse into the two perhaps most intriguing features of the high-resolution structure: the alignment of the grooves on the two gyres and the passage of H2B and H3 N-terminal tails between them.

Biopolymer Chain Elasticity: A novel concept and a least deformation energy principle predicts backbone and overall folding of DNA TTT hairpins in agreement with NMR distances

A new molecular modelling methodology is presented and shown to apply to all published solution structures of DNA hairpins with TTT in the loop. It is based on the theory of elasticity of thin rods and on the assumption that single-stranded B-DNA behaves as a continuous, unshearable, unstretchable and flexible thin rod. It requires four construction steps: (i) computation of the tri-dimensional trajectory of the elastic line, (ii) global deformation of single-stranded helical DNA onto the elastic line, (iii) optimisation of the nucleoside rotations about the elastic line, (iv) energy minimisation to restore backbone bond lengths and bond angles. This theoretical approach called 'Biopolymer Chain Elasticity' (BCE) is capable of reproducing the tri-dimensional course of the sugar-phosphate chain and, using NMR-derived distances, of reproducing models close to published solution structures. This is shown by computing three different types of distance criteria. The natural description provided by the elastic line and by the new parameter, Omega, which corresponds to the rotation angles of nucleosides about the elastic line, offers a considerable simplification of molecular modelling of hairpin loops. They can be varied independently from each other, since the global shape of the hairpin loop is preserved in all cases.

Monte Carlo simulations of supercoiled DNAs confined to a plane

Recent advances in atomic force microscopy (AFM) have enabled researchers to obtain images of supercoiled DNAs deposited on mica surfaces in buffered aqueous milieux. Confining a supercoiled DNA to a plane greatly restricts its configurational freedom, and could conceivably alter certain structural properties, such as its twist and writhe. A program that was originally written to perform Monte Carlo simulations of supercoiled DNAs in solution was modified to include a surface potential. This potential flattens the DNAs to simulate the effect of deposition on a surface. We have simulated transfers of a 3760-basepair supercoiled DNA from solution to a surface in both 161 and 10 mM ionic strength. In both cases, the geometric and thermodynamic properties of the supercoiled DNAs on the surface differ significantly from the corresponding quantities in solution. At 161 mM ionic strength, the writhe/twist ratio is 1.20-1.33 times larger for DNAs on the surface than for DNAs in solution and significant differences in the radii of gyration are also observed. Simulated surface structures in 161 mM ionic strength closely resemble those observed by AFM. Simulated surface structures in 10 mM ionic strength are similar to a minority of the structures observed by AFM, but differ from the majority of such structures for unknown reasons. In 161 mM ionic strength, the internal energy (excluding the surface potential) decreases substantially as the DNA is confined to the surface. Evidently, supercoiled DNAs in solution are typically deformed farther from the minimum energy configuration than are the corresponding surface-confined DNAs. Nevertheless, the work (Delta A(int)) done on the internal coordinates, which include uniform rotations at constant configuration, during the transfer is positive and 2.6-fold larger than the decrease in internal energy. The corresponding entropy change is negative, and its contribution to Delta A(int) is positive and exceeds the decrease in internal energy by 3.6 fold. The work done on the internal coordinates during the solution-to-surface transfer is directed primarily toward reducing their entropy. Evidently, the number of configurations available to the more deformed solution DNA is vastly greater than for the less deformed surface-confined DNA.

DNA rings with multiple energy minima

Within the context of DNA rings, we analyze the relationship between intrinsic shape and the existence of multiple stable equilibria, either nicked or cyclized with the same link. A simple test, based on a perturbation expansion of symmetry breaking within a continuum elastic rod model, provides good predictions of the occurrence of such multiple equilibria. The reliability of these predictions is verified by direct computation of nicked and cyclized equilibria for several thousand DNA minicircles with lengths of 200 and 900 bp. Furthermore, our computations of equilibria for nicked rings predict properties of the equilibrium distribution of link, as calculated by much more computationally intensive Monte Carlo simulations.

Nucleosome dynamics. VI. Histone tail regulation of tetrasome chiral transition. A relaxation study of tetrasomes on DNA minicircles

We have recently described the relaxation of mononucleosomes on an homologous series of 351-366 bp DNA minicircles, as a tool to study nucleosome structure and dynamics in vitro. Nucleosomes were found to have a tail-regulated access to three distinct DNA conformations, depending on the crossing between the entering and exiting DNAs, and its polarity. This approach was now used to explore tetrasome chiral transition, and the influence of the histone tails. The data confirmed the existence of two states, with linking number differences DeltaLk(t)=-0.74(+/-0.01) and +0.51(+/-0.06). As expected, the particle free energy is higher in the right-handed state (DeltaG(t)=1.9(+/-0.I) kT), but it decreased (to 1.3(+/-0.1) kT) upon histone acetylation and the addition of phosphate, a potent tail destabilizer. Removal of the tails with trypsin further decreased DeltaG(t) (to 0.6 kT), and also induced a loss of supercoiling in both states, to DeltaLk(t)=-0.64(+/-0.03) and +0. 35(+/-0.05). The loop end-conditions, and hence the parameters of the DNA superhelix, were then calculated for both states using the explicit solutions to the equations of the mechanical equilibrium in the theory of elastic rod model for DNA. Whereas the pitch of the DNA superhelix may be approximately equal and opposite in the two conformations, its radius (r) was 20% larger in the right-handed conformation, confirming previous observations by electron microscopy of a tetrasome lateral opening in that conformation. The above supercoiling losses were found to reflect a further 3 % increase in r (to 23 %) upon removal of the tails in the right-handed conformation, and a 14 % increase in the left-handed conformation. The use of composite tetramers with one histone tail intact and the other removed showed these effects to be essentially due to the H3 tails. Altogether, these results show that the H3 tails oppose the tetrasome opening which is expected to be required to relieve the clash between the entering and exiting DNAs in the course of the transition, but which also appears to be intrinsic to the protein reorientation mechanism. We propose that the block against opening results from the H3 tails intercalating into the small groove of the double helix at +/-10 bp from the dyad, and acting as wedges against local DNA straightening. The tails (especially H3) may therefore regulate tetrasome chiral transition in vivo.

Elastic stability of DNA configurations. I. General theory

Results are presented in the theory of the elastic rod model for DNA, among which are criteria enabling one to determine whether a calculated equilibrium configuration of a DNA segment is stable in the sense that it gives a local minimum to the sum of the segment's elastic energy and the potential of forces acting on it. The derived stability criteria are applicable to plasmids and to linear segments subject to strong anchoring end conditions. Their utility is illustrated with an example from the theory of configurations of the extranucleosomal loop of a DNA miniplasmid in a mononucleosome, with emphasis placed on the influence that nicking and ligation on one hand, and changes in the ratio of elastic coefficients on the other, have on the stability of equilibrium configurations. In that example, the configurations studied are calculated using an extension of the method of explicit solutions to cases in which the elastic rod modeling a DNA segment is considered impenetrable, and hence excluded volume effects and forces arising from self-contact are taken into account.

Elastic stability of DNA configurations. II. Supercoiled plasmids with self-contact

Configurations of protein-free DNA miniplasmids are calculated with the effects of impenetrability and self-contact forces taken into account by using exact solutions of Kirchhoff's equations of equilibrium for elastic rods of circular cross section. Bifurcation diagrams are presented as graphs of excess link, DeltaL, versus writhe, W, and the stability criteria derived in paper I of this series are employed in a search for regions of such diagrams that correspond to configurations that are stable, in the sense that they give local minima to elastic energy. Primary bifurcation branches that originate at circular configurations are composed of configurations with D(m) symmetry (m=2,3,...). Among the results obtained are the following. (i) There are configurations with C2 symmetry forming secondary bifurcation branches which emerge from the primary branch with m=3, and bifurcation of such secondary branches gives rise to tertiary branches of configurations without symmetry. (ii) Whether or not self-contact occurs, a noncircular configuration in the primary branch with m=2, called branch alpha, is stable when for it the derivative dDeltaL/dW, computed along that branch, is strictly positive. (iii) For configurations not in alpha, the condition dDeltaL/dW>0 is not sufficient for stability; in fact, each nonplanar contact-free configuration that is in a branch other than alpha is unstable. A rule relating the number of points of self-contact and the occurrence of intervals of such contact to the magnitude of DeltaL, which in paper I was found to hold for segments of DNA subject to strong anchoring end conditions, is here observed to hold for computed configurations of protein-free miniplasmids.

Collective variable modelling of nucleic acids

Collective variable models continue to contribute to our knowledge of nucleic acids. The past year has seen considerable progress both in modelling sequence-dependent effects on nucleic acid conformation and in understanding how proteins or external stresses influence nucleic acid structure. Algorithmic developments have also allowed collective models to be applied to studies of thermal fluctuations and dynamics. For larger systems, models with varying degrees of resolution are being refined and applied to nucleic acids containing hundreds or thousands of nucleotides.