cgNA+web : A Visual Interface to the cgNA+ Sequence-dependent Statistical Mechanics Model of Double-stranded Nucleic Acids.

The sequence-dependent statistical mechanics of double-stranded nucleic acid, or dsNA, is believed to be essential in its biological functions. In turn, the equilibrium statistical mechanics behaviour of dsNA depends strongly both on sequence-dependent perturbations in its ground state shape away from an idealised, uniform, double-helical configuration, and on its fluctuations as governed by its sequence-dependent stiffness. We here describe the cgNA+web browser-based interactive tool for visualising the sequence-dependent ground states of dsNA fragments of arbitrary sequences, as predicted by the underlying cgNA+ coarse-grain model. Parameter sets are provided to model dsDNA, including the possibility of epigenetically modified CpG dinucleotide steps, dsRNA, and DNA:RNA Hybrid double helical fragments. The cgNA+web interface is specifically designed to compare ground state shapes of different sequences of the same dsNA, or analogous sequences of different dsNAs. The cgNA+web server is freely available at cgDNAweb.epfl.ch without any login requirement.

Mechanics of two filaments in tight orthogonal contact.

Networks of flexible filaments often involve regions of tight contact. Predictively understanding the equilibrium configurations of these systems is challenging due to intricate couplings between topology, geometry, large nonlinear deformations, and friction. Here, we perform an in-depth study of a simple, yet canonical, problem that captures the essence of contact between filaments. In the orthogonal clasp, two filaments are brought into contact, with each centerline lying in one of a pair of orthogonal planes. Our data from X-ray tomography (µCT) and mechanical testing experiments are in excellent agreement with finite element method (FEM) simulations. Despite the apparent simplicity of the physical system, the data exhibit strikingly unintuitive behavior, even when the contact is frictionless. Specifically, we observe a curvilinear diamond-shaped ridge in the contact-pressure field between the two filaments, sometimes with an inner gap. When a relative displacement is imposed between the filaments, friction is activated, and a highly asymmetric pressure field develops. These findings contrast to the classic capstan analysis of a single filament wrapped around a rigid body. Both the µCT and FEM data indicate that the cross-sections of the filaments can deform significantly. Nonetheless, an idealized geometrical theory assuming undeformable tube cross-sections and neglecting elasticity rationalizes our observations qualitatively and highlights the central role of the small, but nonzero, tube radius of the filaments. We believe that our orthogonal clasp analysis provides a building block for future modeling efforts in frictional contact mechanics of more complex filamentary structures.

Sequence-dependent structural properties of B-DNA: what have we learned in 40 years?

The structure of B-DNA, the physiological form of the DNA molecule, has been a central topic in biology, chemistry and physics. Far from uniform and rigid, the double helix was revealed as a flexible and structurally polymorphic molecule. Conformational changes that lead to local and global changes in the helix geometry are mediated by a complex choreography of base and backbone rearrangements affecting the ability of the B-DNA to recognize ligands and consequently on its functionality. In this sense, the knowledge obtained from the sequence-dependent structural properties of B-DNA has always been thought crucial to rationalize how ligands and, most notably, proteins recognize B-DNA and modulate its activity, i.e. the structural basis of gene regulation. Honouring the anniversary of the first high-resolution X-ray structure of a B-DNA molecule, in this contribution, we present the most important discoveries of the last 40 years on the sequence-dependent structural and dynamical properties of B-DNA, from the early beginnings to the current frontiers in the field.

The static and dynamic structural heterogeneities of B-DNA: extending Calladine-Dickerson rules.

We present a multi-laboratory effort to describe the structural and dynamical properties of duplex B-DNA under physiological conditions. By processing a large amount of atomistic molecular dynamics simulations, we determine the sequence-dependent structural properties of DNA as expressed in the equilibrium distribution of its stochastic dynamics. Our analysis includes a study of first and second moments of the equilibrium distribution, which can be accurately captured by a harmonic model, but with nonlocal sequence-dependence. We characterize the sequence-dependent choreography of backbone and base movements modulating the non-Gaussian or anharmonic effects manifested in the higher moments of the dynamics of the duplex when sampling the equilibrium distribution. Contrary to prior assumptions, such anharmonic deformations are not rare in DNA and can play a significant role in determining DNA conformation within complexes. Polymorphisms in helical geometries are particularly prevalent for certain tetranucleotide sequence contexts and are always coupled to a complex network of coordinated changes in the backbone. The analysis of our simulations, which contain instances of all tetranucleotide sequences, allow us to extend Calladine-Dickerson rules used for decades to interpret the average geometry of DNA, leading to a set of rules with quantitative predictive power that encompass nonlocal sequence-dependence and anharmonic fluctuations.

cgDNAweb: a web interface to the cgDNA sequence-dependent coarse-grain model of double-stranded DNA.

The sequence-dependent statistical mechanical properties of fragments of double-stranded DNA is believed to be pertinent to its biological function at length scales from a few base pairs (or bp) to a few hundreds of bp, e.g. indirect read-out protein binding sites, nucleosome positioning sequences, phased A-tracts, etc. In turn, the equilibrium statistical mechanics behaviour of DNA depends upon its ground state configuration, or minimum free energy shape, as well as on its fluctuations as governed by its stiffness (in an appropriate sense). We here present cgDNAweb, which provides browser-based interactive visualization of the sequence-dependent ground states of double-stranded DNA molecules, as predicted by the underlying cgDNA coarse-grain rigid-base model of fragments with arbitrary sequence. The cgDNAweb interface is specifically designed to facilitate comparison between ground state shapes of different sequences. The server is freely available at cgDNAweb.epfl.ch with no login requirement.

Analyzing DNA curvature and its impact on the ionic environment: application to molecular dynamics simulations of minicircles.

We propose a method for analyzing the magnitude and direction of curvature within nucleic acids, based on the curvilinear helical axis calculated by Curves+. The method is applied to analyzing curvature within minicircles constructed with varying degrees of over- or under-twisting. Using the molecular dynamics trajectories of three different minicircles, we are able to quantify how curvature varies locally both in space and in time. We also analyze how curvature influences the local environment of the minicircles, notably via increased heterogeneity in the ionic distributions surrounding the double helix. The approach we propose has been integrated into Curves+ and the utilities Canal (time trajectory analysis) and Canion (environmental analysis) and can be used to study a wide variety of static and dynamic structural data on nucleic acids.

Sequence-Dependent Persistence Lengths of DNA.

A Monte Carlo code applied to the cgDNA coarse-grain rigid-base model of B-form double-stranded DNA is used to predict a sequence-averaged persistence length of lF = 53.5 nm in the sense of Flory, and of lp = 160 bp or 53.5 nm in the sense of apparent tangent-tangent correlation decay. These estimates are slightly higher than the consensus experimental values of 150 bp or 50 nm, but we believe the agreement to be good given that the cgDNA model is itself parametrized from molecular dynamics simulations of short fragments of length 10-20 bp, with no explicit fit to persistence length. Our Monte Carlo simulations further predict that there can be substantial dependence of persistence lengths on the specific sequence [Formula: see text] of a fragment. We propose, and confirm the numerical accuracy of, a simple factorization that separates the part of the apparent tangent-tangent correlation decay [Formula: see text] attributable to intrinsic shape, from a part [Formula: see text] attributable purely to stiffness, i.e., a sequence-dependent version of what has been called sequence-averaged dynamic persistence length l̅d (=58.8 nm within the cgDNA model). For ensembles of both random and λ-phage fragments, the apparent persistence length [Formula: see text] has a standard deviation of 4 nm over sequence, whereas our dynamic persistence length [Formula: see text] has a standard deviation of only 1 nm. However, there are notable dynamic persistence length outliers, including poly(A) (exceptionally straight and stiff), poly(TA) (tightly coiled and exceptionally soft), and phased A-tract sequence motifs (exceptionally bent and stiff). The results of our numerical simulations agree reasonably well with both molecular dynamics simulation and diverse experimental data including minicircle cyclization rates and stereo cryo-electron microscopy images.

Analyzing ion distributions around DNA: sequence-dependence of potassium ion distributions from microsecond molecular dynamics.

Microsecond molecular dynamics simulations of B-DNA oligomers carried out in an aqueous environment with a physiological salt concentration enable us to perform a detailed analysis of how potassium ions interact with the double helix. The oligomers studied contain all 136 distinct tetranucleotides and we are thus able to make a comprehensive analysis of base sequence effects. Using a recently developed curvilinear helicoidal coordinate method we are able to analyze the details of ion populations and densities within the major and minor grooves and in the space surrounding DNA. The results show higher ion populations than have typically been observed in earlier studies and sequence effects that go beyond the nature of individual base pairs or base pair steps. We also show that, in some special cases, ion distributions converge very slowly and, on a microsecond timescale, do not reflect the symmetry of the corresponding base sequence.

µABC: a systematic microsecond molecular dynamics study of tetranucleotide sequence effects in B-DNA.

We present the results of microsecond molecular dynamics simulations carried out by the ABC group of laboratories on a set of B-DNA oligomers containing the 136 distinct tetranucleotide base sequences. We demonstrate that the resulting trajectories have extensively sampled the conformational space accessible to B-DNA at room temperature. We confirm that base sequence effects depend strongly not only on the specific base pair step, but also on the specific base pairs that flank each step. Beyond sequence effects on average helical parameters and conformational fluctuations, we also identify tetranucleotide sequences that oscillate between several distinct conformational substates. By analyzing the conformation of the phosphodiester backbones, it is possible to understand for which sequences these substates will arise, and what impact they will have on specific helical parameters.

Analyzing ion distributions around DNA.

We present a new method for analyzing ion, or molecule, distributions around helical nucleic acids and illustrate the approach by analyzing data derived from molecular dynamics simulations. The analysis is based on the use of curvilinear helicoidal coordinates and leads to highly localized ion densities compared to those obtained by simply superposing molecular dynamics snapshots in Cartesian space. The results identify highly populated and sequence-dependent regions where ions strongly interact with the nucleic and are coupled to its conformational fluctuations. The data from this approach is presented as ion populations or ion densities (in units of molarity) and can be analyzed in radial, angular and longitudinal coordinates using 1D or 2D graphics. It is also possible to regenerate 3D densities in Cartesian space. This approach makes it easy to understand and compare ion distributions and also allows the calculation of average ion populations in any desired zone surrounding a nucleic acid without requiring references to its constituent atoms. The method is illustrated using microsecond molecular dynamics simulations for two different DNA oligomers in the presence of 0.15 M potassium chloride. We discuss the results in terms of convergence, sequence-specific ion binding and coupling with DNA conformation.

Twist and stretch of helices explained via the Kirchhoff-Love rod model of elastic filaments.

In various single-molecule experiments, a chiral polymer, such as DNA, is simultaneously pulled and twisted. We address an elementary but fundamental question raised by various authors: does the molecule overwind or unwind under tension? We show that within the context of the classic Kirchhoff-Love rod model of elastic filaments, both behaviors are possible, depending on the precise constitutive relations of the polymer. More generally, our analysis provides an effective linear response theory for helical structures that relates axial force and axial torque to axial translation and rotation.

Looping probabilities of elastic chains: a path integral approach.

We consider an elastic chain at thermodynamic equilibrium with a heat bath, and derive an approximation to the probability density function, or pdf, governing the relative location and orientation of the two ends of the chain. Our motivation is to exploit continuum mechanics models for the computation of DNA looping probabilities, but here we focus on explaining the novel analytical aspects in the derivation of our approximation formula. Accordingly, and for simplicity, the current presentation is limited to the illustrative case of planar configurations. A path integral formalism is adopted, and, in the standard way, the first approximation to the looping pdf is obtained from a minimal energy configuration satisfying prescribed end conditions. Then we compute an additional factor in the pdf which encompasses the contributions of quadratic fluctuations about the minimum energy configuration along with a simultaneous evaluation of the partition function. The original aspects of our analysis are twofold. First, the quadratic Lagrangian describing the fluctuations has cross-terms that are linear in first derivatives. This, seemingly small, deviation from the structure of standard path integral examples complicates the necessary analysis significantly. Nevertheless, after a nonlinear change of variable of Riccati type, we show that the correction factor to the pdf can still be evaluated in terms of the solution to an initial value problem for the linear system of Jacobi ordinary differential equations associated with the second variation. The second novel aspect of our analysis is that we show that the Hamiltonian form of these linear Jacobi equations still provides the appropriate correction term in the inextensible, unshearable limit that is commonly adopted in polymer physics models of, e.g. DNA. Prior analyses of the inextensible case have had to introduce nonlinear and nonlocal integral constraints to express conditions on the relative displacement of the end points. Our approximation formula for the looping pdf is of quite general applicability as, in contrast to most prior approaches, no assumption is made of either uniformity of the elastic chain, nor of a straight intrinsic shape. If the chain is uniform the Jacobi system evaluated at certain minimum energy configurations has constant coefficients. In such cases our approximate pdf can be evaluated in an entirely explicit, closed form. We illustrate our analysis with a planar example of this type and compute an approximate probability of cyclization, i.e., of forming a closed loop, from a uniform elastic chain whose intrinsic shape is an open circular arc.

A systematic molecular dynamics study of nearest-neighbor effects on base pair and base pair step conformations and fluctuations in B-DNA.

It is well recognized that base sequence exerts a significant influence on the properties of DNA and plays a significant role in protein-DNA interactions vital for cellular processes. Understanding and predicting base sequence effects requires an extensive structural and dynamic dataset which is currently unavailable from experiment. A consortium of laboratories was consequently formed to obtain this information using molecular simulations. This article describes results providing information not only on all 10 unique base pair steps, but also on all possible nearest-neighbor effects on these steps. These results are derived from simulations of 50-100 ns on 39 different DNA oligomers in explicit solvent and using a physiological salt concentration. We demonstrate that the simulations are converged in terms of helical and backbone parameters. The results show that nearest-neighbor effects on base pair steps are very significant, implying that dinucleotide models are insufficient for predicting sequence-dependent behavior. Flanking base sequences can notably lead to base pair step parameters in dynamic equilibrium between two conformational sub-states. Although this study only provides limited data on next-nearest-neighbor effects, we suggest that such effects should be analyzed before attempting to predict the sequence-dependent behavior of DNA.

Bending modes of DNA directly addressed by cryo-electron microscopy of DNA minicircles.

We use cryo-electron microscopy (cryo-EM) to study the 3D shapes of 94-bp-long DNA minicircles and address the question of whether cyclization of such short DNA molecules necessitates the formation of sharp, localized kinks in DNA or whether the necessary bending can be redistributed and accomplished within the limits of the elastic, standard model of DNA flexibility. By comparing the shapes of covalently closed, nicked and gapped DNA minicircles, we conclude that 94-bp-long covalently closed and nicked DNA minicircles do not show sharp kinks while gapped DNA molecules, containing very flexible single-stranded regions, do show sharp kinks. We corroborate the results of cryo-EM studies by using Bal31 nuclease to probe for the existence of kinks in 94-bp-long minicircles.

Structural motifs of biomolecules.

Biomolecular structures are assemblies of emergent anisotropic building modules such as uniaxial helices or biaxial strands. We provide an approach to understanding a marginally compact phase of matter that is occupied by proteins and DNA. This phase, which is in some respects analogous to the liquid crystal phase for chain molecules, stabilizes a range of shapes that can be obtained by sequence-independent interactions occurring intra- and intermolecularly between polymeric molecules. We present a singularity-free self-interaction for a tube in the continuum limit and show that this results in the tube being positioned in the marginally compact phase. Our work provides a unified framework for understanding the building blocks of biomolecules.

Kinking occurs during molecular dynamics simulations of small DNA minicircles.

Recent experiments on minicircle formation suggest that a conformational mechanism other than smooth deformation may be playing a role in enhancing DNA flexibility. Both local base unpairing and kink formation have been suggested as possible explanations. Although kinks within isolated DNA were proposed 30 years ago, they have, until now, only been observed within DNA complexed with proteins. In order to test how DNA behaves in the strong bending regime, we have carried out molecular dynamics simulations of a 94 base pair minicircle in explicit solvent with two different linking numbers, corresponding to a torsionally relaxed state and a positively supercoiled state. The simulations suggest that sharp kinks can indeed arise in small minicircles. The relaxed minicircle is generally associated with a single kink, while two kinks occur with the supercoiled state. No evidence is seen of base unpaired regions.

3D reconstruction and comparison of shapes of DNA minicircles observed by cryo-electron microscopy.

We use cryo-electron microscopy to compare 3D shapes of 158 bp long DNA minicircles that differ only in the sequence within an 18 bp block containing either a TATA box or a catabolite activator protein binding site. We present a sorting algorithm that correlates the reconstructed shapes and groups them into distinct categories. We conclude that the presence of the TATA box sequence, which is believed to be easily bent, does not significantly affect the observed shapes.

Helices.

Helices are among the simplest shapes that are observed in the filamentary and molecular structures of nature. The local mechanical properties of such structures are often modeled by a uniform elastic potential energy dependent on bending and twist, which is what we term a rod model. Our first result is to complete the semi-inverse classification, initiated by Kirchhoff, of all infinite, helical equilibria of inextensible, unshearable uniform rods with elastic energies that are a general quadratic function of the flexures and twist. Specifically, we demonstrate that all uniform helical equilibria can be found by means of an explicit planar construction in terms of the intersections of certain circles and hyperbolas. Second, we demonstrate that the same helical centerlines persist as equilibria in the presence of realistic distributed forces modeling nonlocal interactions as those that arise, for example, for charged linear molecules and for filaments of finite thickness exhibiting self-contact. Third, in the absence of any external loading, we demonstrate how to construct explicitly two helical equilibria, precisely one of each handedness, that are the only local energy minimizers subject to a nonconvex constraint of self-avoidance.