Oscar Gonzalez

Mechanical systems subject to holonomic constraints: differential—algebraic formulations and conservative integration

Abstract 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.

On the parameterization of rigid base and basepair models of DNA from molecular dynamics simulations

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)(7)C 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.

A sequence-dependent rigid-base model of DNA

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.

Self-Interactions of Strands and Sheets

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.

EXISTENCE OF IDEAL KNOTS

Ideal knots are curves are that maximize the scale invariant ratio of thickness to length. Here we present a simple argument to establish the existence of ideal knots for each knot type and each isotopy class and show that they are C1,1 curves.

A First Course in Continuum Mechanics: Contents

A concise account of various classic theories of fluids and solids, this book is for courses in continuum mechanics for graduate students and advanced undergraduates. Thoroughly class-tested in courses at Stanford University and the University of Warwick, it is suitable for both applied mathematicians and engineers. The only prerequisites are an introductory undergraduate knowledge of basic linear algebra and differential equations. Unlike most existing works at this level, this book covers both isothermal and thermal theories. The theories are derived in a unified manner from the fundamental balance laws of continuum mechanics. Intended both for classroom use and for self-study, each chapter contains a wealth of exercises, with fully worked solutions to odd-numbered questions. A complete solutions manual is available to instructors upon request. Short bibliographies appear at the end of each chapter, pointing to material which underpins or expands upon the material discussed.

A First Course in Continuum Mechanics: Frontmatter

A concise account of various classic theories of fluids and solids, this book is for courses in continuum mechanics for graduate students and advanced undergraduates. Thoroughly class-tested in courses at Stanford University and the University of Warwick, it is suitable for both applied mathematicians and engineers. The only prerequisites are an introductory undergraduate knowledge of basic linear algebra and differential equations. Unlike most existing works at this level, this book covers both isothermal and thermal theories. The theories are derived in a unified manner from the fundamental balance laws of continuum mechanics. Intended both for classroom use and for self-study, each chapter contains a wealth of exercises, with fully worked solutions to odd-numbered questions. A complete solutions manual is available to instructors upon request. Short bibliographies appear at the end of each chapter, pointing to material which underpins or expands upon the material discussed.

On Stable, Complete, and Singularity-Free Boundary Integral Formulations of Exterior Stokes Flow

A new boundary integral formulation of the second kind for exterior Stokes flow is in- troduced. The formulation is stable, complete, singularity-free, and natural for bodies of complicated shape and topology. We prove an existence and uniqueness result for the formulation for arbitrary flows and illustrate its performance via several numerical examples using a Nystrom method with Gauss-Legendre quadrature rules of different order.

cgDNA: a software package for the prediction of sequence-dependent coarse-grain free energies of B-form DNA

cgDNA is a package for the prediction of sequence-dependent configuration-space free energies for B-form 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 X-ray 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.

Dynamics of a rigid body in a Stokes fluid

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