Publications

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.

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.

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  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 lp() attributable to intrinsic shape, from a part ld() 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 lp() has a standard deviation of 4 nm over sequence, whereas our dynamic persistence length ld() 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.

Maximum entropy procedures for estimating coarse-grain parameters from molecular dynamics (MD) simulation data are considered within the specific context of the sequence-dependent cgDNA rigid-base model of DNA. We describe a quite general approach that exploits a maximum absolute entropy principle to fit an observed matrix of covariances subject to the constraint of only allowing a prescribed sparsity pattern of nearest-neighbour interactions in the free energy. We also allow indefinite local stiffness-matrix parameter blocks, that nevertheless always generate a positive-definite model stiffness matrix. Beginning from a database of atomic-resolution MD simulations of a library of short DNA oligomers in explicit solvent, these procedures deliver a complete parameter set for the cgDNA model. Due to the intrinsic linear structure of DNA and the convergence characteristics of the MD time series data, the maximum absolute entropy parameter set yields significantly improved predictions of persistence lengths, when compared to a previous parameter set that was fit to the same MD data, but using a maximum relative entropy fitting principle and local stiffness-matrix parameter blocks that were constrained to be semi-definite.

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 overor 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.

How mechanical and biological processes are coordinated across cells, tissues, and organs to produce complex traits is a key question in biology. Cardamine hirsuta, a relative of Arabidopsis thaliana, uses an explosive mechanism to disperse its seeds. We show that this trait evolved through morphomechanical innovations at different spatial scales. At the organ scale, tension within the fruit wall generates the elastic energy required for explosion. This tension is produced by differential contraction of fruit wall tissues through an active mechanism involving turgor pressure, cell geometry, and wall properties of the epidermis. Explosive release of this tension is controlled at the cellular scale by asymmetric lignin deposition within endocarp b cells—a striking pattern that is strictly associated with explosive pod shatter across the Brassicaceae plant family. By bridging these different scales, we present an integrated mechanism for explosive seed dispersal that links evolutionary novelty with complex trait innovation.

Microsecond molecular dynamics simulations of BDNA 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.

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.

cgDNA is a package for the prediction of sequencedependent configuration-space free energies for Bform DNA at the coarse-grain level of rigid bases. For a fragment of any given length and sequence, cgDNA calculates the configuration of the associated free energy minimizer, i.e. the relative positions and orientations of each base, along with a stiffness matrix, which together govern differences in free energies. The model predicts non-local (i.e. beyond base-pair step) sequence dependence of the free energy minimizer. Configurations can be input or output in either the Curves+ definition of the usual helical DNA structural variables, or as a PDB file of coordinates of base atoms. We illustrate the cgDNA package by comparing predictions of free energy minimizers from (a) the cgDNA model, (b) time-averaged atomistic molecular dynamics (or MD) simulations, and (c) NMR or Xray experimental observation, for (i) the Dickerson– Drew dodecamer and (ii) three oligomers containing A-tracts. The cgDNA predictions are rather close to those of the MD simulations, but many orders of magnitude faster to compute. Both the cgDNA and MD predictions are in reasonable agreement with the available experimental data. Our conclusion is that cgDNA can serve as a highly efficient tool for studying structural variations in B-form DNA over a wide range of sequences.

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 sequencedependent 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, sequencespecific ion binding and coupling with DNA conformation.

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.

The problem of comparison of second-order (covariance) properties of two samples of random curves is considered. The work is motivated by the study of the mechanical properties of short strands of DNA. Our test is based on the common empirical Karhunen–Loève expansion and truncated approximation of the Hilbert–Schmidt distance of the empirical covariance operators.

Additional information: Editor F. Ferraty; DOI 10.1007/978-3-7908-2736-1_38

A novel hierarchy of coarse-grain, sequence-dependent, rigid-base models of B-form DNA in solution is introduced. The hierarchy depends on both the assumed range of energetic couplings, and the extent of sequence dependence of the model parameters. A significant feature of the models is that they exhibit the phenomenon of frustration: each base cannot simultaneously minimize the energy of all of its interactions. As a consequence, an arbitrary DNA oligomer has an intrinsic or pre-existing stress, with the level of this frustration dependent on the particular sequence of the oligomer. Attention is focussed on the particular model in the hierarchy that has nearest-neighbor interactions and dimer sequence dependence of the model parameters. For a Gaussian version of this model, a complete coarse-grain parameter set is estimated. The parameterized model allows, for an oligomer of arbitrary length and sequence, a simple and explicit construction of an approximation to the configuration-space equilibrium probability density function for the oligomer in solution. The training set leading to the coarse-grain parameter set is itself extracted from a recent and extensive database of a large number of independent, atomic-resolution molecular dynamics (MD) simulations of short DNA oligomers immersed in explicit solvent. The Kullback-Leibler divergence between probability density functions is used to make several quantitative assessments of our nearest-neighbor, dimer-dependent model, which is compared against others in the hierarchy to assess various assumptions pertaining both to the locality of the energetic couplings and to the level of sequence dependence of its parameters. It is also compared directly against all-atom MD simulation to assess its predictive capabilities. The results show that the nearest-neighbor, dimer-dependent model can successfully resolve sequence effects both within and between oligomers. For example, due to the presence of frustration, the model can successfully predict the nonlocal changes in the minimum energy configuration of an oligomer that are consequent upon a local change of sequence at the level of a single point mutation.

The most classic approach to the dynamics of an ndimensional mechanical system constrained by d independent holonomic constraints is to pick explicitly a new set of (n − d) curvilinear coordinates parametrizing the manifold of configurations satisfying the constraints, and to compute the Lagrangian generating the unconstrained dynamics in these (n − d) configuration coordinates. Starting from this Lagrangian an unconstrained Hamiltonian H(q,p) on 2(n−d) dimensional phase space can then typically be defined in the standard way via a Legendre transform. Furthermore, if the system is in contact with a heat bath, the associated Langevin and Fokker-Planck equations can be introduced. Provided that an appropriate fluctuation-dissipation condition is satisfied, there will be a canonical equilibrium distribution of the Gibbs form exp(−βH) with respect to the flat measure dqdp in these 2(n−d) dimensional curvilinear phase space coordinates. The existence of (n−d) coordinates satisfying the constraints is often guaranteed locally by an implicit function theorem. Nevertheless in many examples these coordinates cannot be constructed in any tractable form, even locally, so that other approaches are of interest. In ambient space formulations the dynamics are defined in the full original n-dimensional configuration space, and associated 2n-dimensional phase space, with some version of Lagrange multipliers introduced so that the 2(n − d) dimensional sub-manifold of phase space implied by the holonomic constraints and their time derivative, is invariant under the dynamics. In this article we review ambient space formulations, and explain that for constrained dynamics there is in fact considerable freedom in how a Hamiltonian form of the dynamics can be constructed. We then discuss and contrast the Langevin and Fokker-Planck equations and their equilibrium distributions for the different forms of ambient space dynamics.

Computer simulations of biomolecules such as molecular dynamics simulations are limited by the time scale of conformational rearrangements. Several sampling techniques are available to search the multi-minima free energy landscape but most efficient, time-dependent methods do generally not produce a canonical ensemble. A sampling algorithm based on a self-regulating ladder of searching copies in the dihedral subspace is developped in this paper. The learning process using short- and long-term memory functions allows an efficient search in phase space while combining a deterministic dynamics and stochastic swaps with the searching copies conserves a canonical limit. The sampling efficiency and accuracy are indicated by comparing the ansatz with conventional molecular dynamics and replica exchange simulations.

Keywords: Adaptive sampling, Convergence of molecular dynamics, Replica exchange, Dihedral angles

Numerical computations suggest that each point on a certain optimized shape called the ideal trefoil is in contact with two other points. We consider sequences of such contact points, such that each point is in contact with its predecessor and call it a billiard. Our numerics suggest that a particular billiard on the ideal trefoil closes to a periodic cycle after nine steps. This cycle also seems to be an attractor: all billiards converge to it.

Keywords: Ideal knots, contact chord, billiard, closed cycle

Additional information: MSC classes 53-04, 53A04, 65D18, 37E10

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.

Additional information: PACS numbers 87.10.Pq, 87.14.gk, 87.15.ad

Enforcing a specific symmetry group on a curve, knotted or not, is not trivial using standard interpolations such as polygons or splines. For a prescribed symmetry group we present a symmetrization process based on a Fourier description of a knot. The presence of symmetry groups implies a characteristic pattern in the Fourier coefficients. The relations between the coefficients are shown for five ideal knot shapes with their proposed symmetry groups.

Keywords: ideal knots, Fourier knots, symmetry of curves, Fourier coefficient pattern

What is the longest rope on the unit sphere? Intuition tells us that the answer to this packing problem depends on the rope's thickness. For a countably infinite number of prescribed thickness values we construct and classify all solution curves. The simplest ones are similar to the seamlines of a tennis ball, others exhibit a striking resemblance to Turing patterns in chemistry, or to ordered phases of long elastic rods stuffed into spherical shells.

Keywords: thick curves, tube packings

The stochastic equations of motion for a system of interacting rigid bodies in a solvent are formulated and studied. Three-dimensional bodies of arbitrary shape, with arbitrary couplings between translational and rotational degrees of freedom, as arise in coarse-grained models of polymers, are considered. Beginning from an Euler-Langevin form of the equations, two different, properly invariant, Hamilton-Langevin forms are derived and studied together with various associated measures. Under different conditions depending on the choice of rotational coordinates, the canonical measure is shown to be a stationary solution of an associated Fokker-Planck equation and to always factorize into independent measures on configuration and velocity spaces. Explicit expressions are given for these measures, along with a certain Jacobian factor associated with the three-dimensional rotation group. When specialized to a fully-coupled, quadratic model of a stiff polymer such as DNA, our results yield an explicit characterization of the complete set of model parameters.

Given two samples of continuous zero-mean iid Gaussian processes on [0,1], we consider the problem of testing whether they share the same covariance structure. Our study is motivated by the problem of determining whether the mechanical properties of short strands of DNA are significantly affected by their base-pair sequence; though expected to be true, this has not been observed in 3D electron microscopy data. The testing problem is seen to involve aspects of ill-posed inverse problems and a test based on a Karhunen-Loève approximation of the Hilbert-Schmidt distance of the empirical covariance operators is proposed and investigated. When applied to a dataset of DNA minicircles obtained through the electron microscope, our test suggests the existence of sequence effects on DNA shape.

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.

A method is described to extract a complete set of sequence-dependent material parameters for rigid base and basepair models of DNA in solution from atomistic molecular dynamics simulations. The method is properly consistent with equilibrium statistical mechanics, leads to effective shape, stiffness and mass parameters, and employs special procedures for treating spontaneous torsion angle flips and H-bond breaks, both of which can have a significant effect on the results. The method is accompanied by various analytical consistency checks that can be used to assess the equilibration of statistical averages, and different modeling assumptions pertaining to the rigidity of the bases and basepairs and the locality of the quadratic internal energy. The practicability of the approach is verified by estimating complete parameter sets for the 16-basepair palindromic oligomer G(TA)7C simulated in explicit water and counterions. Our results indicate that the method is capable of resolving sequence-dependent variations in each of the material parameters. Moreover, they show that the assumptions of rigidity and locality hold rather well for the base model, but not for the basepair model. For the latter, it is shown that the non-local nature of the internal energy can be understood in terms of a certain compatibility relation involving Schur complements.

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.

We describe Curves+, a new nucleic acid conformational analysis program which is applicable to a wide range of nucleic acid structures, including those with up to four strands and with either canonical or modified bases and backbones. The program is algorithmically simpler and computationally much faster than the earlier Curves approach, although it still provides both helical and backbone parameters, including a curvilinear axis and parameters relating the position of the bases to this axis. It additionally provides a full analysis of groove widths and depths. Curves+ can also be used to analyse molecular dynamics trajectories. With the help of the accompanying program Canal, it is possible to produce a variety of graphical output including parameter variations along a given structure and time series or histograms of parameter variations during dynamics.

We consider the variational problem of finding the longest closed curves of given minimal thickness on the unit sphere. After establishing the existence of solutions for any given thickness between $0$ and $1$ we explicitly construct for each given thickness $\Theta_n:=\sin\pi/(2n),$ $n\in\N$, exactly $\varphi(n)$ solutions, where $\varphi$ is Euler's totient function from number theory. Then we prove that these solutions are unique, and also provide a complete characterisation of sphere filling curves on the unit sphere, i.e. of those curves whose spherical tubular neighbourhood completely covers the surface area of the unit sphere exactly once. All of these results carry over to open curves as well, as indicated in the last section.

Additional information: Mathematics Subject Classification (2000) 49Q10, 51M15, 51M25, 52C15, 53A04, 74K10

Ph. Caussignac Mathematical Models and Methods in Applied Sciences, 19 (2009), 911-937 Download pdf Abstract We adapt an existing asymptotic method to set up a one-dimensional model for the fall of a closed filament in an infinite fluid in the Stokes regime. Starting from the single-layer integral representation of the fluid velocity around the filament, we get, for a very slender filament, a Fredholm integral equation on the filament centerline. From this equation, we can compute the drag and force acting on the filament and consequently the resistance matrix. The integral equation is discretized with a collocation method. The study of a scalar model problem yields existence and uniqueness results together with an error estimate for the discretization scheme. Then, we compare the resistance matrix of thin ideal knots obtained from the discretization of the present model to a boundary element method; numerical convergence results and a good agreement of both methods validate our model.

Keywords: knots sedimentation; 1D-model; Fredholm integral equation

Abstract: Molecular dynamics (MD) simulations were employed to investigate the structure, dy- namics, and local base-pair step deformability of the free 16S ribosomal helix 44 from Thermus thermophilus and of a canonical A-RNA double helix. While helix 44 is bent in the crystal structure of the small ribosomal subunit, the simulated helix 44 is intrinsically straight. It shows, however, substantial instantaneous bends that are isotropic. The spontaneous motions seen in simulations achieve large degrees of bending seen in the X-ray structure and would be entirely sufficient to allow the dynamics of the upper part of helix 44 evidenced by cryo-electron microscopic studies. Analysis of local base-pair step deformability reveals a patch of flexible steps in the upper part of helix 44 and in the area proximal to the bulge bases, suggesting that the upper part of helix 44 has enhanced flexibility. The simulations identify two conformational substates of the second bulge area (bottom part of the helix) with distinct base pairing. In agreement with nuclear magnetic resonance (NMR) and X-ray studies, a flipped out conformational substate of conserved 1492A is seen in the first bulge area. Molecular dynamics (MD) simulations reveal a number of reversible a-g backbone flips that correspond to transitions between two known A-RNA backbone families. The flipped sub- states do not cumulate along the trajectory and lead to a modest transient reduction of helical twist with no significant influence on the overall geometry of the duplexes. Despite their considerable flexibility, the simulated structures are very stable with no indication of substantial force field inaccuracies.

Keywords: sequence-dependent RNA structure and deformability

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.

Keywords: buried area, Marginally compact, Protein structure, DNA structure, tube

This article is a News and Views editorial describing an article by E. L. Starostin, G. H. M. van der Heijden.

We set up a gauge finite element formulation of the 2D steady state incompressible Stokes problem. This formulation allows to decouple the two unknowns from the equations and to take account for the boundary conditions in a non iterative way. A finite element discretization is used to obtain numerical convergence results. The method is then applied to the Navier-Stokes equations for solving the lid-driven cavity problem and comparing different gauges.

Keywords: incompressible Navier-Stokes equation, Gauge method, finite element method

We derive the low-energy effective theory of gravity for a generalized Randall-Sundrum scenario, allowing for a third self-gravitating brane to live in the 5D bulk spacetime. At zero order the 5D spacetime is composed of two slices of anti-de Sitter spacetime, each with a different curvature scale, and the 5D Weyl tensor vanishes. Two boundary branes are at the fixed points of the orbifold whereas the third brane is free to move in the bulk. At first order, the third brane breaks the otherwise continuous evolution of the projection of the Weyl tensor normal to the branes. We derive a junction condition for the projected Weyl tensor across the bulk brane, and combining this constraint with the junction condition for the extrinsic curvature tensor, allows us to derive the first-order field equations on the middle brane. The effective theory is a generalized Brans-Dicke theory with two scalar fields. This is conformally equivalent to Einstein gravity and two scalar fields, minimally coupled to the geometry, but nonminimally coupled to matter on the three branes.

Keywords: cosmology with extra-dimensions, brane-world models, low-energy effective theory of gravity

The understanding of the long-range correlations (LRC) observed in DNA sequences is still an open and very challenging problem. In this paper, we start reviewing recent results obtained when exploring the scaling properties of eucaryotic, eubacterial and archaeal genomic sequences using the space-scale decomposition provided by the wavelet transform (WT). These results suggest that the existence of LRC up to distances ~ 20-30 kbp is the signature of the nucleosomal structure and dynamics of the chromatin fiber. Actually the LRC are mainly observed in the DNA bending profiles obtained when using some structural coding of the DNA sequences that accounts for the fluctuations of the local double-helix curvature within the nucleosome complex. Because of the approximate planarity of nucleosomal DNA loops, we then study the influence of the LRC structural disorder on the thermodynamical properties of 2D elastic chains submitted locally to mechanical/topological constraint as loops. The equilibrium properties of the one-loop system are derived numerically and analytically in the quite realistic weak-disorder limit. The LRC are shown to favor the spontaneous formation of small loops, the larger the LRC, the smaller the size of the loop. We further investigate the dynamical behavior of such a loop using the mean first passage time (MFPT) formalism. We show that the typical short-time loop dynamics is superdiffusive in the presence of LRC. For displacements larger than the loop size, we use large-deviation theory to derive a LRC-dependent anomalous-diffusion rule that accounts for the lack of disorder self-averaging. Potential biological implications on DNA loops involved in nucleosome positioning and dynamics in eucaryotic chromatin are discussed.

Recent single-molecule micromanipulation experiments on DNA subject to small distorsion revealed positive coupling between DNA stretching and twisting - for instance, DNA elongates when overtwisted. Here we propose a method to calculate the twist-stretch (TS) coupling constant specific to a DNA fragment of a given sequence. The method employs a sequence-dependent dinucleotide force field and is based on constrained minimization of the fragment's deformation energy. Using a force field inferred from atomistic molecular dynamics simulations, we obtain the twist-stretch coupling for random sequence to be 0.30 nm/turn, close to experimental values. An exhaustive calculation for all oligomers of 9 base pairs yields values between 0.14 and 0.45 nm/turn, positively correlated with the contents of pyrimidine-purine steps in the sequence. Our method is simple to use and allows one to explore the hypothesis that some sequences may be optimized for TS coupling.

Keywords: sequence-dependent DNA deformability, constitutive relations for DNA

We use cryo-electron microscopy to compare 3-D 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 CAP 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.

Keywords: DNA minicircles, cryo-electron microscopy

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.

Keywords: DNA cyclization, DNA deformability, AMBER

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.

Additional information: Description of FIgure 5 (see link below)

(C) A nonuniform helical equilibrium obtained for a nonisotropic rod with quadratic energy (Eq. A20) and register (Eq. A21). Such examples are highly exceptional.

This article is a survey of the present state of the transfer operator approach to the effective dynamics of metastable complex systems, and the variety of algorithms associated with it. We emphasize both conceptual foundations, and concrete application to the conformation dynamics of a biomolecular system. The algorithmic aspects are illustrated by means of several examples of various degrees of complexity, culminating in their application to a full-scale molecular dynamics simulation of a B-DNA oligomer.

We study the influence of a structural disorder on the thermodynamical properties of 2D-elastic chains submitted to mechanical/topological constraint as loops. The disorder is introduced via a spontaneous curvature whose distribution along the chain presents either no correlation or long-range correlations (LRC). The equilibrium properties of the one-loop system are derived numerically and analytically for weak disorder. LRC are shown to favor the formation of small loop, larger the LRC, smaller the loop size. We use the mean first passage time formalism to show that the typical short time loop dynamics is superdiffusive in the presence of LRC. Potential biological implications on nucleosome positioning and dynamics in eukaryotic chromatin are discussed. This paper is about an absolutely uninteresting mathematical theorem.

Keywords: Proof, Sharp thinking, Milestone theorem

The movements of specific chromosomal regions within living cells can be monitored using real-time fluorescence confocal microscopy. Such studies reveal that in healthy interphase nuclei the decondensed chromosomes show a dynamic behaviour that is, however, remarkably different from movements of extra-chromosomal DNA constructs. While the latter can freely diffuse and explore the entire nuclear volume, the chromosomal sites are confined to much smaller available volumes within the nucleus. Interestingly, different chromosomal sites enjoy different degrees of freedom during a certain period; they can explore larger or smaller portions of nuclear volume. In addition, in yeast, some specific sites are free to move in 3-D while others move preferentially along the inner surface of the nuclear membrane. Several earlier studies measured the volume accessible to freely diffusing chromosomal sites, but for chromosomal sites that show preferential attachment there has been no attempt to measure the surface of the nuclear membrane along which they can freely slide. Here we measure the accessible surface for two chromosomal sites (yeast telomeres Tel3R and Tel6R), that both exhibit strong preferential association with the nuclear membrane in galactose-containing media, but differ significantly in gene activity.

Keywords: chromatin, telomeres

We simulate the extension of spatially confined chromatin fibers and analyse the effect of the flexibility of the modeled fiber and its degree of freedom. We have applied the developed formalism to analyse experimental data of telomere-telomere distances in living yeast cells in the absence of confining factors as identified by the proteins Sir4 and yKu70. The analysis indicates that intrinsic properties of the chromatin fiber, in particular its elastic properties and flexibility, are involved in the juxtaposition of the telomeric ends of chromosomes, modeled here as polymer chains. However, measurements in intact yeast cells show that the telomeres of chromosomes 3 and 6 come even closer together than the parameters of constraint imposed on the simulations would predict. This attraction is specific to telomeres of same chromosomes and overrides the separating tendency imposed by anchoring.

Keywords: chromatin, Monte Carlo simulations

DNA 7-hydro-8-oxoguanine (8-oxoG) is implicated in frameshift formation in an G6 sequence of the HPRT gene in mismatch repair (MMR) defective cells. Using oligonucleotides based on this frameshift hotspot, we investigated how a single 8-oxoG modified the structural and dynamic properties of the G6 tract. A 30 ns molecular dynamics (MD) simulation indicated compression of the minor groove in the immediate vicinity of the lesion. Fluorescence polarization anisotropy (FPA) and MD demonstrated that 8-oxoG increases DNA torsional rigidity and also constrains the movement of the single-stranded region at the single/double stranded DNA junction of model DNA replication template/primer. These constraints influenced the efficiency of primer extension by Klenow (exo-) DNA polymerase.

Keywords: 7-hydro-8-oxoguanine, frameshift, DNA conformation, DNA dynamics, molecular dynamics

The scaling of the average gyration radius of polymers as a function of their length can be experimentally determined from ensemble measurements, such as light scattering, and agrees with analytical estimates. Ensemble techniques, yet, do not give access to the full probability distributions. Single molecule techniques, instead, can deliver information on both average quantities and distribution functions. Here we exploit the high resolution of atomic force microscopy over long DNA molecules adsorbed on a surface to measure the average end-to-end distance as a function of the DNA length, and its full distribution function. We find that all the scaling exponents are close to the predicted 3D values (
\nu 0.589 \pm 0.006 and \delta 2.58 \pm 0.77). These results suggest that the adsorption process is akin to a geometric projection from 3D to 2D, known to preserve the scaling properties of fractal objects of dimension d_f < 2.

Keywords: Single-Molecule experiments, DNA, critical exponents

Molecular dynamics (MD) simulations including water and counterions on B-DNA oligomers containing all 136 unique tetranucleotide base pair steps are reported. The objective is to obtain the calculated dynamical structure for at least two copies of each case, and use the results to examine issues with regard to convergence and dynamical stability of MD on DNA, and to determine the significance of sequence context effects on all unique dinucleotide steps. This information is essential to understand sequence effects on DNA structure and has implications on diverse problems in the structural biology of DNA. Calculations were carried out on the 136 cases embedded in 39 DNA oligomers with repeating tetranucleotide sequences, capped on both ends by GC pairs and each having a total length of 15 nucleotide pairs. All simulations were carried out using a well-defined state-of-the-art MD protocol, the AMBER suite of programs, and the parm94 force field. In a previous article (Beveridge et al., Biophysical Journal 87, 3799-3813), the research design, details of the simulation protocol, and informatics issues were described. Preliminary results from 15 ns MD trajectories were presented for the d(CpG) step in all ten unique sequence contexts. The results indicated the sequence context effects to be small for this step, but revealed that MD on DNA at this length of trajectory is subject to surprisingly persistent cooperative transitions of the sugar-phosphate backbone torsion angles ? and ?. In this article, we report detailed analysis of the entire trajectory database and occurrence of various conformational substates and its impact on studies of context effects. The analysis reveals a possible direct correspondence between the sequence dependent dynamical tendencies of DNA structure and the tendency to undergo transitions that "trap" them in non-standard conformational substates. The difference in mean of the observed base pair step helicoidal parameter distribution with different flanking sequence sometimes differs by as much as 1 standard deviation, indicating that the extent of sequence effects could be significant. The observation reveals that the impact of a flexible dinucleotide such as CpG could extend beyond the immediate base pair neighbors. The results in general provide new insight into MD on DNA and the sequence dependent dynamical structural characteristics of DNA.

Keywords: Sequence directed DNA structure, curvature, flexibility, DNA structure library, sequence context effects

Inspired by experiments that use single particle tracking to measure the regions of confinement of selected chromosomal regions within cell nuclei, we have developed an analytical approach that takes into account various possible positions and shapes of the confinement regions. We show, in particular, that a confinement of a particle into a subregion that is entirely enclosed within a spherical volume can lead to a higher limit of the mean radial square displacement value than one associated with a particle that can explore the entire spherical volume. Finally, we apply the theory to analyse the motion of extrachromosomal chromatin rings within nuclei of living yeast.

Keywords: general theory and mathematical aspects, chromatin, interdisciplinary applications of Physics

We introduce a 3-D parametric active contour algorithm for the shape estimation of DNA molecules from stereo cryo-electron micrographs. We estimate the shape by matching the projections of a 3-D global shape model with the micrographs; we choose the global model as a 3-D filament with a B-spline skeleton and a specified radial profile. The active contour algorithm iteratively updates the B-spline coefficients, which requires us to evaluate the projections and match them with the micrographs at every iteration. Since the evaluation of the projections of the global model is computationally expensive, we propose a fast algorithm based on locally approximating it by elongated blob-like templates. We introduce the concept of projection-steerability and derive a projection-steerable elongated template. Since the 2-D projections of such a blob at any 3-D orientation can be expressed as a linear combination of a few basis functions, matching the projections of such a 3-D template involves evaluating a weighted sum of inner-products between the basis functions and the micrographs. The weights are simple functions of the 3-D orientation and the inner-products are evaluated efficiently by separable filtering. We choose an internal energy term that penalizes the average curvature magnitude. Since the exact length of the DNA molecule is known a-priori, we introduce a constraint energy term that forces the curve to have this specified length. The sum of these energies along with the image energy derived from the matching process is minimized using conjugate-gradients algorithm. We validate the algorithm using real as well as simulated data, and show that it performs well.

Keywords: active contour, cryo, microscopy, ridge, spline, steerable, separable filtering

A new formula for the force vs extension relation is derived from the discrete version of the so called Worm-like chain model. This formula correctly fits some recent experimental data on polymer stretching. Moreover, we have compared our formula with a Monte Carlo simulation of a semiflexible polymer.

Keywords: single-molecule experiments, semiflexible polymers, worm-like chain model, Monte Carlo methods

An extensive analysis of structural databases is carried out to investigate the relative flexibility of B-DNA and A-RNA duplexes in crystal form. Our results show that the general anisotropic concept of flexibility is not very useful to compare the deformability of B-DNA and A-RNA duplexes, since the flexibility patterns of B-DNA and A-RNA are quite different. In other words, flexibility is a dangerous word for describing macromolecules, unless it is clearly defined. A few soft essential movements explain most of the natural flexibility of A-RNA, whereas many are necessary for B-DNA. Essential movements occurring in naked B-DNAs are identical to those necessary to deform DNA in DNA protein complexes, which suggest that evolution has designed DNA proteincomplexes so that B-DNAis deformed according to its natural tendency. DNA is generally more flexible, but for some distortions A-RNA is easier to deform. Local stiffness constants obtained for naked B-DNAs and DNA complexes are very close, demonstrating that global distortions in DNA necessary for binding to proteins are the result of the addition of small concerted deformations at the base-pair level. Finally, it is worth noting that in general the picture of the relative deformability of A-RNA and DNA derived from database analysis agrees very well with that derived from state of the art molecular dynamics (MD) simulations.

Keywords: DNA; RNA; flexibility; crystal structures; similarity indexes

State of the art molecular dynamics simulations are used to study the structure, dynamics, molecular interaction properties and flexibility of DNA and RNA duplexes in aqueous solution. Special attention is paid to the deformability of both types of structures, revisiting concepts on the relative flexibility of DNA and RNA duplexes. Our simulations strongly suggest that the concepts of flexibility, rigidity and deformability are much more complex than usually believed, and that it is not always true that DNA is more flexible than RNA.

Keywords: DNA; RNA; flexibility; molecular dynamics; similarity indexes

We describe herein a computationally intensive project aimed at carrying out molecular dynamics (MD) simulations including water and counterions on B-DNA oligomers containing all 136 unique tetranucleotide base sequences. This initiative was undertaken by an international collaborative effort involving nine research groups, the "Ascona B-DNA Consortium" (ABC). Calculations were carried out on the 136 cases imbedded in 39 DNA oligomers with repeating tetranucleotide sequences, capped on both ends by GC pairs and each having a total length of 15 nucleotide pairs. All MD simulations were carried out using a well defined protocol, the AMBER suite of programs, and the parm94 force field. Phase I of the ABC project involves a total of roughly 0.6 µs of simulation for systems containing roughly 24,000 atoms. The resulting trajectories involve 600,000 coordinate sets and represent roughly 400 GB of data. In this article, the research design, details of the simulation protocol, informatics issues, and the organization of the results into a web accessible data base are described. Preliminary results from 15 ns MD trajectories are presented for the d(CpG) step in its ten unique sequence contexts, and issues of stability and convergence, the extent of quasi-ergodic problems, and the possibility of long lived conformational substates are discussed.

Keywords: sequence directed DNA structure, curvature, flexibility, DNA structure library

We study a two-state statistical process with a non-Poisson distribution of sojourn times. In accordance with earlier work, we find that this process is characterized by aging and we study three different ways to define the correlation function of arbitrary age of the corresponding dichotomous fluctuation based respectively on the Generalized Master Equation formalism, on a Liouville-like approach and on a trajectory perspective.

It is demonstrated that a uniform and hyperelastic, but otherwise arbitrary, nonlinear Cosserat rod subject to appropriate end-loadings has equilibria whose centerlines form two-parameter families of helices. For inextensible, unshearable rods the two-parameters correspond to arbitrary values of the curvature and torsion of the helix. For non-isotropic rods, each member of the two-dimensional family of helical centerlines has at least two possible equilibrium orientations of the director frame. The possible orientations are characterized by a pair of finite-dimensional, dual variational principles involving point-wise values of the strain-energy density and its conjugate function. For isotropic rods, the characterization of possible equilibrium configurations degenerates, and in place of a discrete number of two-parameter families of helical equilibria, typically a single four-parameter family arises. The four continuous parameters correspond to the two of the helical centerlines, a one-parameter family of possible angular phases, and a one-parameter family of imposed excess twists.

Motivated by recent experimental data on DNA stretching in presence of polyvalent counterions, we study the force-induced unfolding of a homopolymer on and off lattice. In the fixed force ensemble the globule unravels via a series of steps due to surface effects which play an important role for finite size chains. This holds both for flexible and stiff polymers. We discuss in a qualitative way how this result may impact on the interpretation of DNA stretching experiments showing peaks in the characteristics curves, by extracting from the raw data the corresponding elongation versus force characteristic curves. Furthemore approximate analytical and numerical calculations, valid in a quasi-equilibrium fixed stretch ensemble, and if the initial low temperature state is ordered in a spool, show that the average force versus elongation displays peaks related to the geometry of the initial configuration. We finally argue how the proposed mechanisms identified for the arising of peaks may couple in the experiments, and comment on the role of dynamic effects.

Keywords: DNA stretching, flexible and stiff polymers, fixed force and fixed stretch ensembles

We study the response to perturbation of non-Poisson dichotomous fluctuations that generate super-diffusion. We adopt the Liouville perspective and with it a quantum-like approach based on splitting the density distribution into a symmetric and an anti-symmetric component. To accomodate the equilibrium condition behind the stationary correlation function, we study the time evolution of the anti-symmetric component, while keeping the symmetric component at equilibrium. For any realistic form of the perturbed distribution density we expect a breakdown of the Onsager principle, namely, of the property that the subsequent regression of the perturbation to equilibrium is identical to the corresponding equilibrium correlation function. We find the directions to follow for the calculation of higher-order correlation functions, an unsettled problem, which has been addressed in the past by means of approximations yielding quite different physical effects.

Keywords: stochastic processes, non-Poisson processes, Liouville and Liouville-like equations, correlation function, regression to equilibrium

A complete set of harmonic force constants describing the DNA deformation energetics at the base pair level was obtained using unrestrained atomic-resolution molecular dynamics simulations of selected duplex oligonucleotides and subsequent analysis of structural fluctuations from the simulated trajectories. The deformation was described by the six base pair conformational parameters (buckle, propeller, opening, shear, stretch, stagger). The results for 13 AT pairs and 11 GC pairs in different sequence contexts suggest that buckle and propeller are very flexible (more than roll in TA dinucleotide steps), while stretch is exceptionally stiff. Only stretch and opening stiffness were found to depend unambiguously on the base pair identity (AT vs GC). The relationship of the results to a simple plates-and-springs model of base- base interactions is discussed.

Keywords: DNA flexibility, atomic-resolution molecular dynamics, AMBER

The article reviews some recent developments in studying DNA sequence-dependent deformability, with emphasis on computer modeling. After a brief outline of available experimental techniques, we proceed to computational methods and focus on atomic-resolution molecular dynamics (MD) simulations. A sequence-dependent local (base-pair step) force field inferred from MD is compared with force fields obtained by other techniques. Various methods for establishing global (flexible-rod) DNA elastic constants are reviewed, including an approach based on atomic resolution MD. The problem of defining the global deformation variables, as well as the question of anisotropy and nonlocal effects, are discussed. As an example, both local and global deformability calculations from atomic-resolution MD of EcoRI dodecamer are presented.

Keywords: DNA flexibility, computer simulations, EcoRI dodecamer, AMBER

Keywords: DNA flexibility, atomic-resolution molecular dynamics, AMBER

We use an elastic rod model with contact to study the extension versus rotation diagrams of single supercoiled DNA molecules. We reproduce quantitatively the supercoiling response of overtwisted DNA and, using experimental data, we get an estimation of the effective supercoiling radius and of the twist rigidity of B-DNA. We find that unlike the bending rigidity, the twist rigidity of DNA seems to vary widely with the nature and concentration of the salt buffer in which it is immerged.

Keywords: elastic twisted rods, contact, numerical path following, biomechanics.

Analyses of genomic DNA sequences have shown in previous works that base pairs are correlated at large distances with scale-invariant statistical properties. We show in the present study that these correlations between nucleotides (letters) result in fact from long-range correlations (LRC) between sequence-dependent DNA structural elements (words) involved in the packaging of DNA in chromatin. Using the wavelet transform technique, we perform a comparative analysis of the DNA text and of the corresponding bending profiles generated with curvature tables based on nucleosome positioning data. This exploration through the optics of the so-called "wavelet transform microscope" reveals a characteristic scale of 100 - 200 bp that separates two regimes of different LRC. We focus here on the existence of LRC in the small-scale regime (< 200 bp). Analysis of genomes in the three kingdoms reveals that this regime is specifically associated to the presence of nucleosomes. Indeed, small scale LRC are observed in eukaryotic genomes and to a less extent in archaeal genomes, in contrast with their absence in eubacterial genomes. Similarly, this regime is observed in eukaryotic but not in bacterial viral DNA genomes. There is one exception for genomes of Poxviruses, the only animal DNA viruses that do not replicate in the cell nucleus and do not present small scale LRC. Furthermore, no small scale LRC are detected in the genomes of all examined RNA viruses, with one exception in the case of retroviruses. Altogether, these results strongly suggest that small-scale LRC are a signature of the nucleosomal structure. Finally, we discuss possible interpretations of these small-scale LRC in terms of the mechanisms that govern the positioning, the stability and the dynamics of the nucleosomes along the DNA chain. This paper is maily devoted to a pedagogical presentation of the theoretical concepts and physical methods which are well suited to perform a statistical analysis of genomic sequences. We review the results obtained with the so-called waveletbased multifractal analysis when investigating the DNA sequences of various organisms in the three kingdoms. Some of these results have been announced in B. Audit et al. [1, 2].

Keywords: chromatin, DNA bending profile, fractals, genomic DNA sequence, long-range correlations, nucleosome, scale-invariance, wavelet transform

The packaging of the eucaryotic genomic DNA involves the wrapping around the histone proteins followed by the successive foldings of higher order structured nucleoprotein complexes. The bending properties of DNA play an essential role in these compaction processes. This hierarchically organized pathway is likely to be reflected in the fractal behavior of DNA bending signals in eucaryotic genomes, but the challenge is to somehow extract this structural information by a clever reading of the DNA sequences. We show that when using an adapted mathematical tool, the "wavelet transform microscope", to explore the fluctuations of bending profiles, one reveals a characteristic scale of 100-200bp that separates two different regimes of (long-range) power-law correlations (PLC) that are common to eucaryotic as well as eubacterial and archaeal genomes. The same analysis of the DNA text yields strikingly similar results to those obtained with bending profiles, and this for all three kingdoms. In the small-scale regime, PLC are observed in eucaryotic genomes, in nuclear replicating DNA viruses and in archaeal genomes, which contrasts with their total absence in the genomes of eubacteria and their viruses, thus indicating that small-scale PLC are likely to be related to the mechanisms underlying the wrapping of DNA around histone proteins. These results together with the observation of PLC between particular sequence motifs known to participate in the formation of nucleosomes (e.g. AA dinucleotides) show that the 10-200~bp PLC provide a very efficient diagnostic of the nucleosomal structure and this in coding as well as in noncoding regions. We discuss possible interpretations of these PLC in terms of the physical mechanisms that might govern the positioning and dynamics of the nucleosomes along the DNA chain through cooperative processes. We further speculate that the large-scale PLC are the signature of the higher-order structure and dynamics of chromatin.

Keywords: sequence analysis, long range correlations, nucleosomes

Additional information: Paper presented at 24th Int. Coll. Group Theoretical Methods in Physics, Paris, France, 15-20 July 2002

We revisit the results of single molecule DNA stretching experiments using a Rod-Like Chain model that explicitly includes some intrinsic structural disorder induced by the sequence. The investigation of artificial and real genomic sequences shows that the Worm-Like Chain model reproduces quite well the data but with an effective bend stiffness A_{eff} which underestimates the true elastic bend stiffness A, independently of the elastic twist stiffness C. Mainly dominated by the amplitude of the structural disorder, this correction seems rather insensitive to the presence of long-range correlations. This RLC model is shown to remarkably fit the experimental data for the Phage Lambda-DNA when considering A ~ 70 +/- 10~nm ( A_{eff} > 50 nm), in good agreement with previous experimental estimates of the ``dynamic'' persistent length. From the analysis of large human contigs, we speculate about the possible dependence of A_{eff} and/or A upon the (G+C) content of the considered sequence.

A variational approach is used to find the shortest curves connecting two pairs of points. The curves are to be separated by a constant distance and they are confined to lie in two orthogonal planes. The method is applicable to constructing tight shapes of linked structures each component of which is known to be planar. Two particular examples are considered and explicit solutions are presented for the Borromean rings, and for two clasped pieces of a rope that provide the minimum of the centreline length for a fixed diameter of the cross-section. A concept of tight periodic structures is introduced and discussed.

Keywords: ideal knot and link, Borromean rings, tight clasp

We combine the global radius of curvature characterisation of knot thickness, the biarc discretisation of space curves, and simulated annealing code to compute approximations to the ideal shapes of the trefoil and figure-eight knots. The computations contain no discretisation error, and give rigorous lower bounds on thickness of the true ideal shapes. The introduction of a precise definition of a contact set for an approximately ideal shape allows us to resolve previously unobserved features. For example, in our approximations of both the ideal trefoil and figure-eight knots, local curvature is within a rather small tolerance of being active, i.e. achieving thickness, at several points along the knot.

Additional information: Eds. J. Calvo, K. Millett, E. Rawdon, and A. Stasiak, published by World Scientific.

The study of DNA minicircles, i.e. closed loops of the double helix with lengths of the order of a few hundred base pairs, is a commonly used experimental technique to probe the sequence-dependent mechanical properties of DNA, such as stiffnesses and intrinsic shape. This article reviews how the mathematical methods of bifurcation theory and symmetry breaking can be used to compute the sequence-dependent equilibrium shapes of mini-circles. The computations yield quite good comparison with experimental data, despite the fact that they assume an isotropic bending law for the DNA, and that at the single base-pair scale the bending response of DNA is almost certainly strongly anisotropic. The effective isotropic behavior can be explained via a two-scale expansion involving the high intrinsic twist parameter of the DNA double helix (one turn per 10.5 base pairs). The effective isotropic stiffnesses can be found in terms of the local anisotropic stiffnesses via a study of the lowest order term in this expansion. However to understand the first correction, symmetry breaking techniques are again required.

Keywords: DNA minicircles, elastic rods, cyclization, Poincare-Melnikov functions, continuous symmetry, isotropy

Additional information: Eds. D. Givoli, M.J. Grote, G. Papanicolaou, Kluwer Science Publishers

Motivated by applications in the modeling of deformations of the DNA double-helix, we construct a continuum mechanics model of two elastically interacting elastic strands. The two strands are described in terms of averaged, or macroscopic, variables plus an additional small, internal or microscopic, perturbation. We call this composite structure a birod. The balance laws for the macroscopic configuration variables of the birod can be cast in the form of a classic Cosserat rod model with coupling to the internal balance laws through the constitutive relations. The internal balance laws for the microstructure variables also take a mathematical form analogous to that for a Cosserat rod, but with coupling to the macroscopic system through terms corresponding to distributed force and couple loads.

Keywords: elastic rods, Cosserat rod, Kirchhoff rod, double strand, microstructure, constitutive relations, mixture theory

We study the mechanics of uniform n-plies, that is, structures formed by n helical strands plied together. We first show that the well-known lock-up phenomenon for n = 2, described by a pitchfork bifurcation, gets unfolded for higher n. Geometrically, n-plies with n > 2 are all found to behave qualitatively the same. Next we consider the mechanics of n-plies, allowing for axial end forces and end moments. An expression for the interstrand pressure force is derived and used to investigate the birdcage failure mode of plied structures.

We demonstrate that the dynamics of a rigid body falling in an infinite, viscous fluid can, in the Stokes limit, be reduced to the study of a three-dimensional system of ordinary differential equations ${\dot\Grav} = \Grav\times M_2\Grav$ where $M_2\in\Rtt$ is a generally non-symmetric matrix containing certain hydrodynamic mobility coefficients. We further show that all steady states and their stability properties can be classified in terms of the Schur form of $M_2$. Steady states correspond to screw motions (or limits thereof) in which the center of mass traces a helical path, while the body spins uniformly about the vertical. All rigid bodies have at least one such stable, screw motion. Bodies for which $M_2$ has exactly one real eigenvalue have a unique, globally attracting, asymptotically stable screw motion, while other bodies can have multiple, stable and unstable steady motions. One application of our theory is to the case of rigid filaments, which in turn is a first step in modeling the sedimentation rate of flexible polymers such as DNA. For rigid filaments the matrix $M_2$ can be approximated using the Rotne-Prager theory, and we present various examples corresponding to certain {\it ideal\/} shapes of knots which illustrate the various possible multiplicities of steady states. Our simulations of rigid ideal knots in a Stokes fluid predict an approximate linear relation between sedimentation speed and average crossing number, as has been observed experimentally for the much more complicated system of real DNA knots in gel electrophoresis.

We consider equilibrium configurations of inextensible, unshearable, isotropic, uniform and naturally straight and prismatic rods when subject to end loads and clamped boundary conditions. In a first paper, we discussed symmetry properties of the equilibrium configurations of the centre line of the rod. Here we are interested in the set of all parameter values that yield equilibrium configurations that fulfill clamped boundary conditions. We call this set the solution manifold and we compute it unsing a recently introduced continuation algorithm. We then describe the topology of this manifold and how it comprises different interconneted layers. We show that the border set of the different layers is the well known solution set of buckled rings.

Keywords: numerical continuation, boundary value problem for twisted rods, surface following algorithm

Additional information: Mathematics Subject Classifications (2000) : 74B20, 74K10, 74G60, 65P30, 65L10

The thin elastic rod is a conventional model for the mesoscale structure of DNA. The symmetric multi-leafed closed equilibria with one multiple self-contact point are examined. The contact force is assumed to be pointwise and frictionless. The bifurcation diagram is presented for first branches of two- and three-leafed families.

Keywords: thin elastic rod, equilibrium, boundary value problem, self-contact

We investigate the configurations of twisted elastic rods under applied end loads and clamped boundary conditions. We classify all the possible equilibrium states of inextensible, unshearable, isotropic, uniform and naturally straight and prismatic rods. The Kirchhoff equations which describe the equilibria of these rods are integrated in a formal way which enable us to describe the boundary conditions in terms of 2 closed form equations involving 4 free parameters. We show that only reversible solutions of the Kirchhoff equations fulfil the clamped boundary conditions and we sort them according to their period in the phase plane. We show how planar untwisted configurations as well as circularly closed configurations play an important role in the classification

We describe how the stability properties of DNA minicircles can be directly read from plots of various biologically intuitive quantities along families of equilibrium configurations. Our conclusions follow from extensions of the mathematical theory of distinguished bifurcation diagrams that are applied within the specific context of an elastic rod model of minicircles. Families of equilibria arise as a twisting angle $\alpha$ is varied. This angle is intimately related to the continuously varying linking number Lk for nicked DNA configurations that is defined as the sum of Twist and Writhe. We present several examples of such distinguished bifurcation diagrams involving plots of the energy E, linking number Lk, and a twist moment m_3, along families of cyclized equilibria of both intrinsically straight, and intrinsically curved, DNA fragments.

In this paper we introduce metric-based means for the space of positive-definite matrices. The mean associated with the Euclidean metric of the ambient space is the usual arithmetic mean. The mean associated with the Riemannian metric corresponds to the geometric mean. We discuss some invariance properties of the Riemannian mean and we use differential geometric tools to give a characterization of this mean.

Keywords: geometric mean, positive-definite symmetric matrices, Riemannian distance, geodesics

The writhe of a space curve fragment is considered for various boundary conditions. An expression for the writhe as a function of arclength for an arbitrary space curve is obtained. The formula is built on the base of closing the tangent indicatrix with a geodesic. The corresponding closure of a curve in 3-space is explicitly constructed. The addition rule for writhe is formulated. A relationship connecting the writhe with the Gauss integral over the open curve is presented. The single and double regular helical shapes are examined as examples.

Keywords: the writhe, the twist, the linking number

Additional information: Eds. J. Calvo, K. Millett, E. Rawdon, and A. Stasiak, published by World Scientific

We use the Cosserat rod theory to present a unified picture of jump phenomena, associated with looping, snap-through, pop-out, etc., in twisted clamped rods undergoing large deections. Both contact-free rods and rods with isolated points of self-contact are considered. Taking proper account of the symmetries of the problem we find that an arbitrary contact-free solution is fully characterised by four parameters; each point contact adds another two. A shooting method is used for solving the boundary value problem. An intricate bifurcation picture emerges with a strong interplay between planar and spatial rod configurations. We find new jump phenomena by treating the ratio of torsional to bending stiffness of the rod as a bifurcation parameter. Load-deection curves are computed and compared with results from carefully conducted experiments on contact- free as well as self-contacting metal-alloy rods.

Keywords: elastic rods, clamped boundary conditions, set of equilibria, contact

We show how an energy analysis can be used to derive the equilibrium equation and boundary conditions for an end-loaded variable ply much more efficiently than was done in previous works. Numerical results are then presented for a clamped balanced ply approaching lock up. We also use the energy method to derive the equations for a more general ply made of imperfect anisotropic rods and briefly consider their helical solutions.

Keywords: clamped boundary conditions, contact, elastic rods

Spatial equilibria of a closed thin isotropic elastic rod are considered. The thin elastic rod is a classical model for the large-scale structure of relatively long DNA molecules. Particular attention is paid to the shapes with self-contacts which are assembled from the elementary loops.

Keywords: hin elastic rod, equilibrium, DNA, self-contact

It is shown that the proof of the Hannay formula, given by him, is based on an improper argument. An improved consideration is suggested.

Keywords: ribbon, linking number, writhe, twist, turning number, path in SO(3)

{In this paper we address the mechanics of ply formation in DNA supercoils. We extend the variable ply formulation of Coleman and Swigon to include end loads, and the derived constitutive relations of this generalised ply are shown to be in excellent agreement with experiments. We make a careful physical examination of the uniform ply in which two strands coil around one another in the form of a helix. We next address the problem of determining the link (Lk), twist (Tw) and writhe (Wr) of a closed DNA plasmid from an inspection of its electron micrograph. Previous work has made use of the topological relation

Keywords: supercoiling of DNA, DNA molecule, writhing, ply mechanics, electron micrograph, rod theory

We use three different approaches to describe the static spatial configurations of a twisted rod as well as its stability during rigid loading experiments. The first approach considers the rod as infinite in length and predicts an instability causing a jump to self-contact at a certain point of the experiment. Semi-finite corrections, taken into account as a second approach, reveal some possible experiments in which the configuration of a very long rod will be stable through out. Finally, in a third approach, we consider a rod of real finite length and we show that another type of instability may occur, leading to possible hysteresis behavior. As we go from infinite to finite length, we compare the different information given by the three approaches on the possible equilibrium configurations of the rod and their stability. These finite size effects studied here in a 1D elasticity problem could help us guess what are the stability features of other more complicated (2D elastic shells for example) problems for which only the infinite length approach is understood.

Keywords: stability and bifurcations, buckling, finite deflections, elastic material, beams and columns

The standard radius of curvature at a point q(s) on a smooth curve can be defined as the limiting radius of circles through three points that all coalesce to q(s). In the study of ideal knot shapes it has recently proven useful to consider a global radius of curvature of the curve at q(s) defined as the smallest possible radius amongst all circles passing through this point and any two other points on the curve, coalescent or not. In particular, the minimum value of the global radius of curvature gives a convenient measure of curve thickness. Given the utility of the construction inherent to global curvature, it is also natural to consider variants of global radii of curvature defined in related ways. For example multi-point radius functions can be introduced as the radius of a sphere through four points on the curve, circles that are tangent at one point of the curve and intersect at another, etc. Then single argument, global radius of curvature functions can be constructed by minimizing over all but one argument. In this article we describe the interrelations between all possible global radius of curvature functions of this type, and show that there are two of particular interest. Properties of the divers global radius of curvature functions are illustrated with the simple examples of ellipses and helices, including certain critical helices that arise in the optimal shapes of compact filaments, in alpha-helical proteins, and in B-form DNA.

Keywords: global curvature, curve thickness, multi-point distances

Additional information: Edited by: Jorge Alberto Calvo, Kenneth C. Millett, and Eric J. Rawdon

ISSN 0271-4132

Nous établissons l'équation d'équilibre d'un câble composé de n brins enroulés en hélice de pas constant. Nous montrons d'abord que les brins se touchent d'une façon particulière : le domaine de contact pour chacun des brins est une double hélice et non plus une ligne droite comme c'est le cas pour un câble à deux brins (A.D.N. par exemple). Cette géométrie nouvelle implique l'existence d'un jour entre les brins le long de l'axe. Nous montrons ensuite que la force de pression exercée sur un brin par les deux brins qui lui sont edjacents est perpendiculaire à l'axe d'enroulement. Finalement nous appliquons ces calculs à la triple hélice du collagène.

Keywords: nonlinear elasticity, biochemistry, molecular biology

In this paper we give precise definitions of different, properly invariant, notions of mean or average rotation. Each mean is associated with a metric in SO(3). The metric induced from the Frobenius inner product gives rise to a mean rotation that is given by the closest special orthogonal matrix to the usual arithmetic mean of the given rotation matrices. The mean rotation associated with the intrinsic metric on SO(3) is the Riemannian center of mass of the given rotation matrices. We show that the Riemannian mean rotation shares many common features with the geometric mean of positive numbers and the geometric mean of positive Hermitian operators. We give some examples with closed-form solutions of both notions of mean.

Keywords: special orthogonal group, rotation, geodesics, operator means, averaging

Physical strands or sheets that can be modelled as curves or surfaces embedded in three dimensions are ubiquitous in nature, and are of fundamental importance in mathematics, physics, biology and engineering. Often the physical interpretation dictates that self-avoidance should be enforced in the continuum model, i.e. finite energy configurations should not self-intersect. Current continuum models with self-avoidance frequently employ pairwise repulsive potentials, which are of necessity singular. Moreover the potentials do not have an intrinsic length scale appropriate for modelling the finite thickness of the physical systems. Here we develop a framework for modelling self-avoiding strands and sheets which avoids singularities, and which provides a way to introduce a thickness length scale. In our approach pairwise interaction potentials are replaced by many-body potentials involving three or more points, and the radii of certain associated circles or spheres. Self-interaction energies based on these many-body potentials can be used to describe the statistical mechanics of self-interacting strands and sheets of finite thickness.

Keywords: polymer, membrane, surface, protein, self-avoidance

Many different physical systems, e.g. super-coiled DNA molecules, have been successfully modelled as elastic curves, ribbons or rods. We will describe all such systems as framed curves, and will consider problems in which a three dimensional framed curve has an associated energy that is to be minimized subject to the constraint of there being no self-intersection. For closed curves the knot type may therefore be specified a priori. Depending on the precise form of the energy and imposed boundary conditions, local minima of both open and closed framed curves often appear to involve regions or self-contact, that is, regions in which points that are distant along the curve are close in space. While this phenomenon of self-contact is familiar through every day experience with string, rope and wire, the idea is surprisingly difficult to define in a way that is simultaneously physically reasonable, mathematically precise, and analytically tractable. Here we use the notion of global radius of curvature of a space curve in a new formulation of the self-contact constraint, and exploit our formulation to derive existence results for minimizers, in the presence of self-contact, of a range of elastic energies that define various framed curve models. As a special case we establish the existence of ideal shapes of knots.

Keywords: existence theories, curves in Euclidean space, knots and links in S^3, nonlinear elasticity, biochemistry, molecular biology

The thin elastic rod is a traditional model for the large-scale structure of long DNA molecules. The solutions for closed equilibria are considered. Particular attention is paid to the shapes with self-contacts. A new class of analytical solutions of the corresponding boundary value problem is presented. Its relation to the known multi-leafed "rose-like" symmetric shapes is discussed.

Keywords: thin elastic rod, equilibrium, loop, BVP, DNA

Additional information: Eds. Michel Deville and Robert Owens

ISBN 3-9522075-1-9

Three different existing steady-state models with quantum correction for simulating the resonant tunnelling diode are summarized. Numerical methods are briefly described and also a theoretical argument for one of the models. Results of simulation are focused on the capability of reproducing the negative differential resistivity.

This article outlines the capabilities of a scientific visualization toolkit called DataViewer, and compares it to analogous software. DataViewer was originally designed for the construction of the visualization part of certain computational steering packages, and consequently it is particularly straightforward to closely couple DataViewer with numerical calculations. Rendering is performed through a high-level scene graph which facilitates the easy construction of complex visualizations. DataViewer differs from other such libraries by allowing complex geometrical objects, which efficiently encapsulate large amounts of data, to be used as nodes in the scene graph. Graphics hardware access is through the OpenGL API.

Keywords: scene graphs, scientific visualization, steered computations

Additional information: The extended abstract (link below) from: The 20th Eurographics UK Conference , IEEE Computer Society Publications, 147-148

This article is a News and Views editorial describing an article by Maritan et al (same issue of Nature). Maritan et al use the idea of global radius of curvature introduced by Gonzalez and Maddocks (see link below) to discuss certain optimal packing problems for tubes or strings, and identify a critical pitch to radius ratio for single helices which just happens to arise for crystal structures of many alpha-helical proteins. We generalize the observation of Maritan et al to remark that the classic B-form double helix for DNA also satisfies a related critical value of pitch to radius ratio.

Keywords: DNA, alpha-helical protein, helix, optimal packing

A method is described to extract a complete set of sequence-dependent energy parameters for a rigid base-pair model of DNA from Molecular Dynamics (MD) simulations. The method is properly consistent with equilibrium statistical mechanics, and leads to effective inertia parameters for the base-pair units as well as stacking and stiffness parameters for the base-pair junctions. We give explicit formulas that yield a complete set of base-pair model parameters in terms of equilibrium averages that can be estimated from a time series generated in an MD simulation. The expressions to be averaged depend strongly both on the choice of coordinates used to describe rigid body orientations, and on the choice of strain measures at each junction.

Keywords: DNA, base pair parameters, statistical mechanics, molecular dynamics

Motivated by the application of modelling DNA by an elastic rod, we use the technique of two-scale homogenization on a two-point boundary value problem (BVP) involving buckling of a rod with a high intrinsic twist. The basic question is to understand the errors generated in replacing a rod with a high, but finite, intrinsic twist, by an effective rod in the infinite intrinsic twist limit. For our buckling problem the effective problem is isotropic. As a consequence there is a continuous family of buckled planar solutions. We show that the buckled configurations of the rod with high intrinsic twist are close to certain special planar buckled configurations of the isotropic rod that are selected from among all the solutions to the effective problem. The selected planes are independent of the values of the (bending and twisting) stiffnesses and of the load.

Keywords: elastic rod, high twist, homogenization, effective constitutive relation, strut, symmetry breaking

Additional information: Eds. Michel Deville and Robert Owens

ISBN 3-9522075-1-9

Motivated by applications to continuum models of tertiary structures of DNA, we use averaging theory to determine effective isotropic bending laws for non-isotropic elastic rods with high intrinsic twist.

Keywords: elastic rod, high twist, homogenization, effective constitutive relation

Additional information: Eds. Michel Deville and Robert Owens

ISBN 3-9522075-1-9

It is shown that the index of a constrained critical point in the isoperimetric calculus of variations is simply related to the number of negative eigenvalues of a certain bordered operator associated with the second variation. A conjugate point theory for this bordered operator is then established.

Keywords: index, constrained critical point, constrained variational principle

Various formulations of the equations of motion for both finite- and infinite-dimensional constrained Lagrangian dynamical systems are studied. The different formulations correspond to different ways of enforcing constraints through multiplier fields. All the formulations considered are posed on ambient spaces whose members are not restricted to satisfy constraint equations, but each formulation is shown to possess an invariant set on which the constraint equations and physical balance laws are satisfied. The stability properties of the invariant set within its ambient space differ in each of the cases. For the model problem of linearized incompressible elastodynamics, we study three formulations and establish the well-posedness of one formulation corresponding to a homogeneous, isotropic material body with a specified traction on its boundary.

Keywords: holonomic constraints, invariant sets, stability, incompressible elastodynamics

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 basepairs. Further, 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.

Keywords: symmetry breaking, elastic rod model, DNA minicircles, DNA intrinsic curvature, topoisomer link distribution, Metropolis Monte Carlo

Searching within the context of information retrieval may be viewed as a communication process between the users and the indexers (or the authors). It is known that in expressing the same concept or idea, different people tend to use different words or phrases, and also that the meaning of words attached to document surrogates tends to change over time. To overcome these phenomena, various learning schemes have been designed so as to automatically infer knowledge about document content from the relevance assessments of past queries. Thus, in contrast to most retrieval models that represent the semantic content of documents as static entities, these adaptive search models might change the descriptions of documents through an inductive learning scheme. The evaluation of such dynamic document space strategies may be based on retrospective tests within which the same set of queries is applied to train and test the system. Based on cross-validation principles, this paper suggests a more "honest" evaluation methodology that demonstrates a radically different perspective relative to retrieval effectiveness measures used by current information retrieval learning schemes. Moreover, this paper demonstrates that the size of the test-collection might also lead to misleading conclusions about the search performance of a retrieval scheme.

Keywords: information retrieval, relevance feedback, learning

Additional information: The authors do not agree with the editor about the required revisions and decided not to rewrite this paper according to the requirements. This paper must be viewed as a technical paper.

Like other learning paradigms, the performance of the genetic algorithms (GAs) is dependent on the parameter choice, on the problem representation, and on the fitness landscape. Accordingly, a GA can show good or weak results even when applied on the same problem. Following this idea, the crossover operator plays an important role, and its study is the object of the present paper. A mathematical analysis has led us to construct a new form of crossover operator inspired from genetic programming (GP) that we have already applied in field of information retrieval. In this paper we extend the previous results and compare the new operator with several known crossover operators under various experimental conditions.

Keywords: genetic algorithms, crossover operators, test functions

The numerical integration in time of the equations of motion for mechanical systems subject to holonomic constraints is considered. Schemes are introduced for the direct treatment of a differential-algebraic form of the equations that preserve the constraints, the total energy, and other integrals such as linear and angular momentum arising from affine symmetries. Moreover, the schemes can be shown to preserve the property of time-reversibility in an appropriate sense. An example is given to illustrate various aspects of the proposed methods.

Keywords: numerical integration, holonomic constraints, Hamiltonian systems, differential-algebraic equations

Suppose that a consistent one-step numerical method of order r is applied to a smooth system of ordinary differential equations. Given any integer m >= 1, the method may be shown to be of order r+m as an approximation to a certain modified equation. If the method and the system have a particular qualitative property then it is important to determine whether the modified equations inherit this property. In this article, a technique is introduced for proving that the modified equations inherit qualitative properties from the method and the underlying system. The technique uses a straightforward contradiction argument applicable to arbitrary one-step methods and does not rely on the detailed structure of associated power series expansions. Hence the conclusions apply, but are not restricted, to the case of Runge-Kutta methods. The new approach unifies and extends results of this type that have been derived by other means: results are presented for integral preservation, reversibility, inheritance of fixed points, Hamiltonian problems and volume preservation. The technique also applies when the system has an integral that the method preserves not exactly, but to order greater than r. Finally, a negative result is obtained by considering a gradient system and gradient numerical method possessing a global property that is not shared by the associated modified equations.

Keywords: modified equations, backward error analysis, numerical integration

A stationary 3D energy-transport model valid for semiconductor heterostructure devices is derived from a semiclassical Boltzmann equation by the moment method. In addition to the well-known conservation equations, we obtain original interface conditions, which are essential to have a mathematically well-posed problem. An appropriate modelling of the physical parameters appearing in the system of equations is proposed for gallium arsenide. The model being written and its particularities mentioned, we present a novel numerical algorithm to solve it. The discretization of the equations is achieved by means of standard and mixed finite element methods. We apply the model and numerical algorithm to simulate a 2D AlGaAs/GaAs MODFET. Comparisons between experimental measurements and calculations are carried out. The influence of the modelling of the physical parameters, especially the electron mobility and the energy relaxation time, is noted. The results show the satisfactory behaviour of our model and numerical algorithm when applied to GaAs heterostructure devices.

Keywords: energy-transport, finite element, semiconductors, simulation, transport

Chemically modified DNA oligonucleotides have been crucial to the success of antisense therapeutics. Although such modifications are ubiquitous in the clinic high-resolution structural studies of pharmaceutically relevant derivatives have been limited to only a few molecules. We have completed a high-resolution NMR structural study of three DNA\RNA hybrids with the sequence d(CCTATAATCC)r(GGAUUAUAGG). All hybrids contain an unmodified RNA strand, whereas the DNA strand of each hybrid contains one of three different sugar- phosphate backbone linkages at each nucleotide: 1) phosphate, 2) [Rp]- phosphorothioate, or 3) phosphorodithioate. The UV and NMR melting profiles revealed that the normal hybrid was more stable than the [Rp]-phosphorothioate, which in turn was more stable than the phosphorodithioate. Homonuclear two- dimensional nuclear Overhauser effect spectroscopy and double quantum-filtered correlation spectroscopy afforded nearly complete nonlabile proton assignments. The three molecules show nearly equivalent chemical shifts, with the exception of H3' protons, which are shifted downfield in a manner that appears correlated with the degree of sulfur substitution at phosphate. All three hybrids exhibit unusually broad linewidths for deoxyribose protons H2' and H2".

Distance restraints were calculated from NOE cross-peak intensities via a complete relaxation matrix approach using the program RANDMARDI. Detailed comparison of interproton distances from each hybrid indicates that the three molecules share a common structure, with neither strand in canonical A or B form. Correlation of R factors, calculated using the program CORMA with DNA H2'-base and H3'-base distances, revealed a relative increase in the population of B-type sugar conformations for deoxyriboses in the A+T-rich center of the hybrid sequence. It is widely known that the activity of enzymes which act upon DNA.RNA hybrid substrates (e.g. ribonuclease H) is impacted when the hybrids contain phosphorothioate or phosphorodithioate substitutions. The structural similarity of the three hybrids examined here suggests that factors other than global structure may mediate the activity of these enzymes.

Keywords: antisense, RNA.DNA hybrid structure, phosphorothioate, phosphorodithioate, RNase H

The global radius of curvature of a space curve is introduced. This function is related to, but distinct from, the standard local radius of curvature, and is connected to various physically appealing properties of a curve. In particular, the global radius of curvature function provides a concise characterization of the thickness of a curve, and of certain ideal shapes of knots as have been investigated within the context of DNA.

Keywords: global curvature, curve thickness, ideal knots, knot energy, writhe

Symmetry breaking is considered in the context of a parameter-dependent two-point boundary value problem describing a twisted elastic ring that arises as a model of DNA minicircles. We explicitly determine, via a perturbation expansion, those representatives of a manifold of solutions to a symmetric BVP that persist when the symmetry is broken. Conditions generating a generic splitting are identified, and some degenerate cases are studied. In both the generic and degenerate cases, the results of the perturbation expansion are used to improve the efficiency of existing algorithms for computing symmetry-broken bifurcation diagrams.

Information retrieval proposes solutions for searching, in a given set of objects, for those replying to a given description. Often, these objects are documents, however other forms like image, video, sounds can be considered. Besides statistical approaches, artificial intelligence models present an attractive paradigm to improve performance in IR systems, and the genetic algorithm (GA) represents one of them. How can the GA be used in IR and under which conditions? What are the reasons of its failures? What can be done to improve its performance ? In this paper we try to answer these questions.

Keywords: recherche d'information, algorithmes génétiques, apprentissage

Additional information: Eds. G. Ritschard, A. Berthold, F. Duc, D. A. Zighed

Hermès, Paris

A method called PARSE (Probability Assessment via Relaxation rates of a Structural Ensemble) is described for determination of ensembles of structures from NMR data. The problem is approached in two separate steps: (1) generation of a pool of potential conformers, and (2) determination of the conformers' probabilities which best account for the experimental data. The probabilities are calculated by a global constrained optimization of a quadratic objective function meaureing the agreement between observed NMR parmaters and those calculated for the ensemble. The performance of the method is tested on synthetic data sets simulated for various structural ensembles of the complementary dinucleotide d(CA)d(TG).

Keywords: NMR refinement, structural ensembles, relaxation rates

Genetic algorithms (GAs) search for good solutions to a problem by operations inspired from the natural selection of living beings. Among their many uses, we can count information retrieval (IR). In this field, the aim of the GA is to help an IR system to find, in a huge documents text collection, a good reply to a query expressed by the user. The analysis of phenomena seen during the implementation of a GA for IR has brought us to a new crossover operation. This article introduces this new operation and compares it with other learning methods.

Keywords: genetic algorithms, information retrieval, crossover operators

This case study describes two software packages, VBM (Visualization of Bifurcation Manifolds) and MCCC (Multi-parameter Computation and Continuation Code) which provide an object oriented (OO) framework for visualizing, steering and implementing multi-parameter continuation algorithms.

Additional information: Eds. Michael E. Henderson, Christopher R. Anderson, and Stephen L. Lyons

ISBN: 0-89871-445-1

Minimal distance paths between prescribed initial and final configurations of a wheeled mobile robot are calculated, both analytically and numerically, using methods of optimal control. The construction of these trajectories can be considered as part of an analysis of obstacle avoidance.

Keywords: wheeled mobile robots, minimization, paths, route, obstacle avoidance, Reeds-Shepp

A conjugate point test determining an index of the constrained second variation in one-dimensional isoperimetric calculus of variations problems is described. The test is then implemented numerically to determine stability properties of equilibria within a continuum mechanics model of DNA minicircles.

Keywords: elastic rods, bifurcation diagrams, two-point boundary conditions, constrained second variation, index

An arbitrary consistent one-step approximation of an ordinary differential equation is studied. If the scheme is assumed to be accurate of $\sO(\Dt^r)$, then it may be shown to be $\sO(\Dt^{r+m})$ accurate as an approximation to a modified equation for any integer $m \ge 1$. A technique is introduced for proving that the modified equations inherit qualitative properties from the numerical method. For Hamiltonian problems the modified equation is shown to be Hamiltonian if the numerical method is symplectic, and for general problems the modified equation is shown to possess integrals shared between the numerical method and the underlying system. The technique for proving these results does not use the well-known theory for power series expansions of particular methods such as Runge-Kutta schemes, but instead uses a simple contradiction argument based on the approximation properties of the numerical method. Although the results presented are known, the method and generality of the proofs are new and may be of independent interest.

Keywords: modified equations, backward error analysis, numerical integration

Additional information: Eds. F. Cucker, M. Shub

Springer Verlag, Berlin

Using cryo-electron microscopy we reconstructed the three-dimensional trajectories adopted in cryovitrified solutions by double-stranded DNA molecules in which the backbone of one strand lacked a phosphate at regular intervals of 20 nucleotides. The shape of such nicked DNA molecules was compared with that of DNA molecules with exactly the same sequences but without any single-stranded scissions. Upon changing the salt concentration we observed opposite effects of charge neutralization on nicked and non-nicked DNA. In low salt solutions (10 mM Tris-HCl, 10 mM NaCl) the applied dense nicking caused ca 3.5-fold reduction of the DNA persistence legnth as compared with non-nicked DNA. Upon increasing the salt concentration, (to 150 mM NaCl and 10 mM MgCl2) the persistence length of non-nicked DNA appreciably decreased while that of nicked DNA molecules increased by a factor of 2.

Keywords: cryo-electron microscopy; DNA flexibility; DNA nicking; DNA persistence length; DNA structure

An interpretation of the CSM model as a one-parameter family of multi-dimensional soluble quantum billiards with potential is proposed. This interpretation is based on a specific ordering of the particles, on the separation of the centre-of-mass motion of the system and on the introduction of a multi-dimensional space generated by normal coordinates. The eigenvalues and eigenfunctions of the billiard Hamiltonian are given in terms of modified Sogo's solutions. Detailed calculations are presented for the particular case of the two-dimensional equilateral triangular billiard with potential.

♦♦♦

On propose une interprétation du modèle de CSM comme une famille à un paramètre de billiards avec potentiel, multidimensionnels, quantiques et solubles. Cette interprétation est basée sur un ordre spécifique des par ticules, sur la séparation du mouvement du centre de masse du système et sur l'introduction d'un espace multidimensionnel généré par des coordonnées normales. On donne les valeurs propres et les fonctions propres de l'Hamiltonien du billiard en termes de solutions de Sogo modifiées. On présente des calculs détaillés dans le cas particulier du billiard triangulaire équilatéral à deux dimensions avec potentiel.

An incompletely parabolic system of partial differential equations consist of one parabolic subsystem coupled to a hyperbolic subsystem. For the initial-boundary value problem, it has been shown, by requiring that a solution remains at every time bounded by the data, that boundary conditions which make both subsystems well-posed render the global system well-posed too. In this paper, we establish the same result with the help of the notion of semi-admissible boundary conditions in the theory of Friedrichs positive systems of differential equations. The theory is illustrated by the two examples of the compressible Navier-Stokes equations and of the hydrodynamic model for semiconductor devices, both in two space dimensions.

The Kirchhoff Kinetic Analogy relates the governing equations for the statics of elastic rods and the dynamics of rigid bodies. We discuss the Analogy in light of several different Hamiltonian formulations, including a non-canonical description of rod equilibria. We focus on the last three quadratures that are required to reconstruct the rod centerline from the frame variables, which form the complete configuration space in the rigid body interpretation. In particular we demonstrate that if the frame evolution is formulated as a canonical Hamiltonian system involving quaternions (or Euler parameters), then the last quadratures can all be computed explicitly in terms of algebraic relations involving invariants (or integrals) of the evolution, independent of whet her or not the entire system is completely integrable.

Nonlinear problems arising in modelling applications are frequently parameter dependent, so that families of solutions are of interest. Such problems naturally lend themselves to interactive computation that exploits parameter continuation methods combined with visualization techniques. Visualization provides both understanding of the solution set and feedback for computational steering. We describe various issues that have arisen in our investigations of problems of this general type.

Additional information: Be warned that there are many ray-traced figures in this paper. The compressed postscript is 5 MB, while the uncompressed postscript is 85 MB. Pages 60, 62 and 63 are color postscript.

This article provides a survey of contemporary rod mechanics, including both dynamic and static theories. Much of what we discuss is regarded as classic material within the mechanics community, but the objective here is to provide a self-contained account accessible to workers interested in modelling DNA. We also describe a number of recent results and computations involving rod mechanics that have been obtained by our group at the University of Maryland. This work was largely motivated by applications to modelling DNA, but our approach reflects a background of research in continuum mechanics. In particular, we emphasize the role that Hamiltonian formulations and symmetries play in the effective computation of special solutions, conservation laws of dynamics and integrals of statics.

Additional information: Eds. . P. Mesirov, K. Schulten, De Witt Sumners

Springer Verlag

Experimentally motivated parameters from a base-pair-level discrete DNA model are averaged to yield parameters for a continuum elastic rod with a curved unstressed shape reflecting the local DNA geometry. The continuum model permits computations with discretization lengths longer than the intrinsic discretization of the base-pair model, and, for this and other reasons, yields an efficient computational formulation. Obtaining continuum stiffnesses is straightforward, but obtaining a continuum unstressed shape is hindered by the "noisy" small-scale structure and rapid helix twist of the discrete unstressed shape. Filtering of the discrete data and an analytic transformation from the true normal-vector field to a natural (untwisted) frame allows a stable continuum fit. Equilibrium energies of closed rings predicted by the continuum model are found to match the energies of the underlying discrete model to within 0.5%. The model is applied to a set of 11 short DNA molecules (approximately 150 bp) and properly distinguishes their cyclization probabilities (J factors) when compared both to experimental cyclization rates and to Monte Carlo simulations. The continuum model does not include entropic contributions to the free energy. However, because of its rapid and accurate computation of internal energy, the continuum model should, when combined with further work on entropic effects, be a useful method for computing experimental DNA free energies.

This article concerns the three-dimensional, large deformation dynamics of an inextensible, unshearable rod. To enforce the conditions of inextensibility and unshearability, a technique we call the impetus-striction method is exploited to reformulate the constrained Lagrangian dynamics as an unconstrained Hamiltonian system in which the constraints appear as integrals of the evolution. We here show that this impetus-striction formulation naturally leads to a numerical scheme which respects the constraints and conservation laws of the continuous system. Simulations of the dynamics of a rod that is fixed at one end and free at the other are presented.

A method is developed for the derivation of unconstrained Hamiltonian systems of partial differential equations equivalent to given constrained Lagrangian systems, the motivation for this work being the study of an elastica and the stability of solitary waves. The Hamiltonian formulation is used to analyze the stability properties of the solitary waves, after characterization of the wave profiles at critical points of an appropriate time-invariant functional. It is shown that for a certain range of wave speeds the solitary wave profiles are nonisolated minimizers of the functional, and attention is drawn to the implications of this fact when considering nonlinear stability.

Additional information: see link below

The authors consider the system of differential equations $$dx/dt= J(z)\nabla H(z),\tag1$$ where $z\in \bbfR\sp n$ ($n$ need not be even), $J(z)$ is skew-symmetric and singular. Constrained variational principles are used to study the stability properties of equilibrium and relative equilibrium solutions of (1). In particular the variational characterization of the steady motions of a heavy asymmetric rigid body tumbling about a fixed point is described. In the last part of the article the variational principles are applied to the analysis of an even-dimensional system (1) with standard symplectic matrix $J$ and $r< n/2$ integrals in involution.

Keywords: Hamiltonian system, constrained variational principles, stability

Additional information: Eds. H.S. Dumas, K. R. Meyer and D.S. Schmidt

It is shown that for an appropriate class of dissipatively perturbed Hamiltonian systems, the number of unstable modes of the dynamics linearized at a nondegenerate equilibrium is determined solely by the index of the equilibrium regarded as a critical point of the Hamiltonian. In addition, the movement of the associated eigenvalues in the limit of vanishing dissipation is analyzed.

Keywords: index of equilibrium, unstable modes, nondegenerate equilibrium,critical point, eigenvalues, vanishing dissipation

The equations governing the time evolution of an ideal fluid in material coordinates are expressed as an unconstrained canonical Hamiltonian system. The incompressibility of the flow is consequent upon certain first integrals of the motion. The variable conjugate to the configuration field is not the usual linear momentum, but is instead a quantity that is related to linear momentum through an auxiliary scalar field whose time derivative is the pressure. The definition of the Hamiltonian involves a minimization with respect to this auxiliary field. The method of derivation may be generally applied to obtain unconstrained Hamiltonian descriptions of Lagrangian field equations subject to pointwise constraints.

Keywords: material coordinates, first integrals, linear momentum, minimization, Lagrangian field equations, pointwise constraints

In this article we present a novel procedure for the systematic analysis of the forward kinematics of a class of parallel manipulators that generalize the well-known Stewart platform. The designs comprise a movable platform connected to a fixed base by a set of legs, the lengths of which can be controlled. The legs are connected to the base and the platform by unactuated spherical joints. The forward kinematics problem is to determine all possible manipulator configurations ({\it i.e.} all possible positions and orientations of the platform) when a set of leg lengths is prescribed. The new formulation of the forward kinematics problem is general in the sense that it can, in principle, be applied to any manipulator design ({\it i.e.} to any fixed geometry of base and platform joints, with associated leg connectivities). One interesting aspect of our procedure is that it splits the forward kinematics problem into two parts, the first being a simple, design-dependent, inversion of a linear system of equations, and the second being a matrix-completion problem involving the solution of certain nonlinear, but design-independent, closure equations. The general formulation provides some insight into which manipulator designs have simple forward kinematics. We analyze four specific manipulator designs from the new point of view, one of which is a design with previously unrecognized closed-form forward kinematics.

Conservation laws and associated integrals of motion for the dynamics of rods are derived. The classic conservation laws are those of total linear and angular momentum, and, for hyperelastic rods, conservation of energy. It will here be shown that an additional conservation law arises in each of two cases. The first case is that of uniform, hyperelastic rods, the second is that of a class of transversely isotropic rods.

Keywords: integrals of motion, total linear and angular momentum, hyperelastic rods, energy, transversely isotropic rods

We consider the stability of multi- or $n$-soliton solutions to the Korteweg-de Vries equation (KdV) posed on the real line. It is shown that in the standard variational characterization of KdV multi-solitons as critical points, the $n$-solitons actually realize non-isolated constrained minimizers. (The case $n=1$ was already known to Benjamin, 1972.) From this fact a precise dynamic stability result for multi- solitons follows, namely that initial data close to a given $n$-soliton evolves in time so as to remain close (in the $H\sp n(\text{\bbfR})$ Sobolev norm) to the $n$-dimensional manifold of all $n$-solitons with appropriate wave speeds, i.e. to the set of constrained minimizers.\par Our techniques are also applicable to other Hamiltonian systems with several conserved quantities. In particular the inverse scattering formalism of KdV is not explicitly exploited.

Keywords: variational characterization, multi-solitons, critical points, constrained minimizers

We consider the friction-dominated, or quasi-static, motion of a rigid body being pushed over a horizontal plane in situations where the precise frictional interaction cannot be determined. Consequently, the complete set of motions corresponding to some friction distribution must be found. We assume the contact region between the body and the supporting plane has at most two components and demonstrate that the set of all possible motions coincides with the set of motions arising for two-point contact in the boundary of the contact region. The two-point problem is solved analytically. Various examples are presented and an effective numerical scheme based upon a dissipation function is implemented.

This paper classifies and constructs similarity solutions of the system of partial differential equations $x\sb t/\vert x\sb t\vert = (Tx\sb s)\sb s$, $\vert x\sb s\vert\sp 2 = 1$, where $x = x(s,t) \in {\germ R}\sp 2$ is a local contraction of a completely flexible inextensible chain which is dragging over a rough horizontal plane (with friction) as it is pulled at one or both ends. The solutions include translations, rotations, and travelling waves as well as more general motions.

Keywords: local Lie group; chain; similarity solutions; partial differential equations; travelling waves

In recent work, the exact dynamic equations for the motion of a finite rigid body in a central gravitational field were shown to be of Hamiltonian form with a noncanonical structure. In this paper, the notion of relative equilibrium is introduced based upon this exact model. In relative equilibrium, the orbit of the center of mass of the rigid body is a circle, but the center of attraction may or may not lie at the center of the orbit. This feature is used to classify great-circle and non-great-circle orbits. The existence of non-great-circle relative equilibria for the exact model is proved by means of various variational principles. While the orbital offset of the non-great-circle solutions is necessarily small, a numerical study reveals that there can be significant changes in orientation away from the classic Lagrange relative equilibria, which are solutions of an approximate model. Accordingly our analysis raises questions concerning the design and control of Earth pointing satellites.

This paper concerns the construction and stability properties of steady- state solutions of a system of partial differential equations that model simple shearing of a slab of thermo-plastic material. The class of constitutive laws that give rise to a variational formulation of the steady-state problem is identified, and a phase-plane argument is used to construct time-independent solutions that may be interpreted as steady- state shear-bands. Our variational framework captures several commonly adopted constitutive laws. Techniques from bifurcation theory for variational problems are applied to classify stable and unstable solutions.

Keywords: symmetric eigenvalue problem; Lyapunov functional; existence theorems; stability; slab; variational formulation; phase-plane argument; bifurcation

This paper concerns the dynamics of a rigid body of finite extent moving under the influence of a central gravitational field. A principal motivation behind this paper is to reveal the Hamiltonian structure of the $n$-body problem for masses of finite extent and to understand the approximation inherent to modeling the system as the motion of point masses. To this end, explicit account is taken of effects arising because of the finite extent of the moving body. In the spirit of Arnold and Smale, exact models of spin-orbit coupling are formulated, with particular attention given to the underlying Lie group framework.

Hamiltonian structures associated with such models are carefully constructed and shown to be noncanonical. Special motions, namely relative equilibria, are investigated in detail and the notion of a non-great circle relative equilibrium is introduced. Non-great circle motions cannot arise in the point mass model. In our analysis, a variational characterization of relative equilibria is found to be very useful.

The reduced Hamiltonian formulation introduced in this paper suggests a systematic approach to approximation of the underlying dynamics based on series expansion of the reduced Hamiltonian. The latter part of the paper is concerned with rigorous derivations of nonlinear stability results for certain families of relative equilibria. Here Arnold's energy-Casimir method and Lagrange multiplier methods prove useful.

This work concerns the variational characterization of stability properties of relative equilibria. In particular, tests bearing upon the second variation in constrained variational principles are applied to prove stability of certain symmetry-related solutions of noncanonical Hamiltonian systems. The new approach is essentially an appropriate extension of the principle of exchange of stability to conditional variational principles in which the Lagrange multipliers are viewed as bifurcation parameters. The technique is introduced in the context of the classic problem of a heavy asymmetric rigid-body moving about a fixed point. In this example, the special solutions are known as the axes of [J. fuer Math. CXIII, 318-334 (1894)]. The example is also used to elucidate issues concerning the necessity of stability conditions obtained by variational means. Finally, a discussion is presented of the behaviour of variational stability estimates as the symmetry class of the problem is altered.

Keywords: stability properties; relative equilibria; second variation; constrained variational principles; symmetry-related solutions; noncanonical Hamiltonian systems; principle of exchange of stability; Lagrange multipliers; bifurcation parameters; heavy asymmetric rigid-body

A wheeled mobile robot is here modelled as a planar rigid body that rides on an arbitrary number of wheels. The connections between the rigid body motion of the robot, and the steering and driving controls of wheels are developed. In particular, conditions are obtained that guarantee that rolling without skidding or sliding can occur. Explicit differential equations are derived to describe the rigid body motions that arise from rolling trajectories. The simplest wheel configuration that permits control of arbitrary rigid body motions is determined. The question of slippage due to misalignment of the wheels is investigated based on a physical model of friction. Examples are presented to illustrate the models.

The starting point of this investigation is the properties of restricted quadratic forms xT Ax, x S Rm, where A is an m x m real symmetric matrix, and S is a subspace. The index theory of Hestenes (1951) and Maddocks (1985) that treats the more general Hilbert-space version of this problem is first specialized to the finite-dimensional context, and appropriate extensions, valid only in finite dimensions, are made. The theory is then applied to obtain various inertia theorems for matrices and positivity tests for quadratic forms. Expressions for the inertias of divers symmetrically partitioned matrices are described. In particular, an inertia theorem for the generalized Schur complement is given. The investigation recovers, links, and extends several, formerly disparate, results in the general area of inertia theorems.

Equations are derived to govern the motion of vehicles which move on rolling wheels. A relation between the centers of curvature of the trajectories of the wheels and the center of rotation of the vehicle is established. From this relation the general kinematic laws of motion are derived. Applications to questions of offtracking (the difference between the trajectories of the front and back wheels of the vehicle) and optimal steering (how to steer around a tight corner) are considered.

Keywords: Euler-Savary formulae, offtracking, optimal steering

The mechanical equilibrium of a rope wrapped around a solid body or around another rope is investigated, with friction and tension being taken into account. Various examples are treated to illustrate the theory. Its application to knots and hitches is indicated.

Keywords: ropes, knots, hitches, friction

Following the approach of Ericksen, it is shown that the addition of an order-parameter to the Frank-Oseen model of nematic liquid crystals allows the construction of finite energy equilibria with a line singularity or disclination.

Additional information: Eds J.L. Ericksen and D. Kinderlehrer

It is known that when one branch of a simple fold in a bifurcation diagram represents (linearly) stable solutions, the other branch represents unstable solutions. The theory developed here can predict instability of some branches close to folds, without knowledge of stability of the adjacent branch, provided that the underlying problem has a variational structure. First, one particular bifurcation diagram is identified as playing a special role, the relevant diagram being specified by the choice of functional plotted as ordiante. The results are then stated in terms of a shape of the solution branch in this distinguished bifurcation diagram. In many problems arising in elasticity the preferred bifurcation diagram is the load-displacement graph. The theory is particularly useful in applications where a solution branch has a succession of folds.

The theory is illustrated with applications to simple models of thermal self-ignition and of a chemical reactor, both of which systems are of Emden-Fowler type. An analysis concerning an elastic rod is also presented.

The modelling of twins in crystals with strain gradient theories provides interesting problems both in thermodynamics and in the calculus of variations. Here, Dunn and Serrin's thermomechanical theory of interstitial working is used to obtain a variational principle that governs the equilibria of materials with non-convex Helmholtz free energy. In some geometries, this principle reduces to a novel calculus of variations problem; an example is described in which symmetry-related uniform equilibrium states can be connected by nonconstant extremals which realise local minima of the free energy. Certain implications of different definitions of local minima are also discussed.

Sorry, no postscript available Abstract The subjects of this investigation are the abstract properties and applications of restricted quadratic forms. The first part of the presentation resolves the following question : if L is a self-adjoint linear operator mapping a Hilbert space H into itself, and S is a subspace of H, when is the quadratic form positive for any nonzero u in S? In the second part of the presentation, restricted quadratic forms are further examined in the specific context of constrained variational principles; and the general theory is applied to obtain information on stability and bifurcation. Two examples are the solved: one is finite-dimensional and of an illustrative nature; the other is a longstanding problem in elasticity concerning the stability of a buckled rod. In addition to being a valuable analytical tool for isoperimetric problems in the calculus of variations, the tests described are amenable to numerical treatment.

Additional information: Errata : SIAM J. Math. Anal., 19 (1988), p. 1256 - 1257 (see link below)

The dynamical behaviour of a slender rod is analyzed here in terms of generalization of Euler's elastica theory. The model includes a linear stress-strain relation but nonlinear geometric terms. Properties of the rod may vary along its length and various boundary conditions are considered. A rotational inertia term that is neglected in many theories is retained, and is essential to the analysis. By use of the equivalence of an energy and a Sobolev norm, and by reformulation of the equations as a semilinear system, global existence of solutions is proved for any smooth initial data. Equilibrium solutions that are stable in the static sense of minimizing the potential energy are then proved to be stable in the dynamic sense due to Liapounov.

This paper describes previously unknown stabilites and instabilities of planar equilibrium configurations of a nonlinearly elastic rod that is buckled under the action of a dead-load. The governing equations are derived from variational principles, including ones of isoperimetric type. Properties of stability are accordingly determined by study of the second variation. Stabilities to deformations both in the plane and out of the plane are considered.

Among the newly discovered properties are: secondary bifurcation from the first buckled mode, marked differences between stability to two-dimensional and to three-dimensional variations, and the stabilizing influence of resistance to twist. In the isoperimetric examples, the analysis makes crucial use of a novel device to account for the dependence of the second variation on constraints.

Systems of Nonlinear Partial Differential Equations - Proc. of NATO/London Math. Soc. Advanced Study Institute (1983) Eds. J.M. Ball, Reidel, 311-322 Sorry, no postscript available Abstract This paper describes previously unknown stabilites and instabilities of planar equilibrium configurations of a nonlinearly elastic rod that is buckled under the action of a dead-load. The governing equations are derived from variational principles, including ones of isoperimetric type. Properties of stability are accordingly determined by study of the second variation. Stabilities to deformations both in the plane and out of the plane are considered. Among the newly discovered properties are: secondary bifurcation from the first buckled mode, marked differences between stability to two-dimensional and to three-dimensional variations, and the stabilizing influence of resistance to twist. In the isoperimetric examples, the analysis makes crucial use of a novel device to account for the dependence of the second variation on constraints.