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This IMA Volume in Mathematics and its Applications MATHEMATICAL APPROACHES TO BIOMOLECULAR STRUCTURE AND DYNAMICS is one of the two volumes based on the proceedings of the 1994 IMA Sum­ mer Program on "Molecular Biology" and comprises Weeks 3 and 4 of the four-week program. Weeks 1 and 2 appeared as Volume 81: Genetic Mapping and DNA Sequencing. We thank Jill P. Mesirov, Klaus Schulten, and De Witt Sumners for organizing Weeks 3 and 4 of the workshop and for editing the proceedings. We also take this opportunity to thank the National Institutes of Health (NIH) (National Center for Human Genome Research), the National Science Foundation (NSF) (Biological Instrumen­ tation and Resources), and the Department of Energy (DOE), whose fi­ nancial support made the summer program possible. A vner Friedman Robert Gulliver v PREFACE The revolutionary progress in molecular biology within the last 30 years opens the way to full understanding of the molecular structures and mech­ anisms of living organisms. Interdisciplinary research in mathematics and molecular biology is driven by ever growing experimental, theoretical and computational power. The mathematical sciences accompany and support much of the progress achieved by experiment and computation as well as provide insight into geometric and topological properties of biomolecular structure and processes. This volume consists of a representative sample of the papers presented during the last two weeks of the month-long Institute for Mathematics and Its Applications Summer 1994 Program in Molecular Biology.

Inhaltsverzeichnis

Frontmatter

Biomolecular Topology

Tangle Complexity and the Topology of the Chinese Rings

Abstract
This paper discusses a question in complexity of untanglement of graph embeddings in relation to the well-known Chinese Rings puzzle. A series of examples is described in which the complexity of untanglement is conjectured to rise exponentially as a function of the structural complexity of the associated graphs. Possible relations with biological and physical systems are discussed.
Louis H. Kauffman

Lattice Invariants for Knots

Abstract
The geometry of polygonal knots in the cubic lattice may be used to define some knot invariants. One such invariant is the minimal edge number, which is the minimum number of edges necessary (and sufficient) to construct a lattice knot of given type. In addition, one may also define the minimal (unfolded) surface number, and the minimal (unfolded) boundary number; these are the minimum number of 2-cells necessary to construct an unfolded lattice Seifert surface of a given knot type in the lattice, and the minimum number of edges necessary in a lattice knot to gaurantee the existence of an unfolded lattice Seifert surface. In addition, I derive some relations amongst these invariants.
E. J. Janse Van Rensburg

Topology and Geometry of Biopolymers

Abstract
This paper is concerned with some simple lattice models of the entanglement complexity of polymers in dilute solution, with special reference to biopolymers such as DNA. We review a number of rigorous results about the asymptotic behaviour of the knot probabihty, the entanglement complexity and the writhe of a lattice polygon (as a model of a ring polymer) and discuss Monte Carlo results for intermediate length polygons. In addition we discuss how this model can be augmented to include the effect of solvent quality and ionic strength. We also describe a lattice ribbon model which is able to capture the main properties of an oriented ribbon-like molecule (such as duplex DNA).
E. J. Janse van Rensburg, Enzo Orlandini, De Witt Sumners, M. Carla Tesi, Stuart G. Whittington

Energy Functions for Knots: Beginning to Predict Physical Behavior

Abstract
Several definitions have been proposed for the “energy” of a knot. The intuitive goal is to define a number u(K) that somehow measures how “tangled” or “crumpled” a knot K is. Typically, one starts with the idea that a small piece of the knot somehow repels other pieces, and then adds up the contributions from all the pieces. From a purely mathematical standpoint, one may hope to define new knot-type invariants, e.g by considering the minimum of u(K) as K ranges over all the knots of a given knot-type. We also are motivated by the desire to understand and predict how knot-type affects the behavior of physically real knots, in particular DNA loops in gel electrophoresis or random knotting experiments. Despite the physical naiveté of recently studied knot energies, there now is enough laboratory data on relative gel velocity, along with computer calculations of idealized knot energies, to justify the assertion that knot energies can predict relative knot behavior in physical systems. The relationships between random knot frequencies and either gel velocities or knot energies is less clear at this time.
Jonathan Simon

Biomolecular Structure and Dynamics: Theory

The Elastic Rod Provides a Model for DNA and Its Functions

Abstract
The processes of transcription and replication are catalysed by processive enzyme complexes which move translationally along the DNA helix, unwinding the DNA helix ahead of the complex and reforming a duplex helix behind the complex (Gamper and Hearst 1982; Cook et al. 1994). These processes are known to torsionally stress DNA.
John E. Hearst, Yaoming Shi

Hamiltonian Formulations and Symmetries in Rod Mechanics

Abstract
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.
Donald J. Dichmann, Yiwei Li, John H. Maddocks

Statistical Mechanics of Supercoiled DNA

Abstract
The conformation of long supercoiled DNA loops under near physiological conditions has been the subject of a long series of laboratory and Monte Carlo experiments, and impressive agreement between theory and experiment has been achieved [1–2]. A closed unnicked DNA loop has fixed linking number: this constraint leads to a competition between twisting and bending (writhing) elastic energy which drives plectonemic supercoiling. However, entropy, as manifest through a fluctuation induced repulsive force, overwhelms this elastic effect for fractional linking number perturbations ∣σ∣ < 0.02, causing a chiral random-coil conformation. For ∣σ∣ > 0.02, plectonemic supercoils are stable, but thermal fluctuations continue to play an important role, competing with bending and twisting elastic energy to set the superhelical radius. We have constructed an analytic theory for this effect that agrees well with experiment [3], using methods from polymer statistical mechanics.
John F. Marko, Eric D. Siggia

Determination of the DNA Helical Repeat and of the Structure of Supercoiled DNA by Cryo-Electron Microscopy

Abstract
Cryo-electron microscopy provides an unique possibility to directly observe shape of individual DNA molecules freely suspended in cryo-vitrifled liquid media. We used this technique to characterize the superhelical trajectory adopted by linear DNA molecules composed of directly repeated intrinsically bent DNA segments which have 10 or 11 base pair each. Although the DNA helix is not directly discerned by this method, the measured values of diameter, pitch and handedness of the formed superhelices allow to determine the chirality and the number of base-pairs per turn of the constituting DNA. We also used cryo-electron microscopy to study the response of supercoiled DNA molecules to increasing counterions’ concentration. We observed that upon substantial neutralization of the negative charge of the DNA, the supercoiled molecules have a tendency to adopt a so called “tight” configuration, whereby the opposing segments of interwound molecules directly approach each other. Metropolis Monte Carlo simulations of shapes of supercoiled DNA molecules revealed that some short range attractive interactions between DNA segments would be needed to compensate for the entropy loss during transition from a loose toward a “tight” configuration of supercoiled DNA molecules. Since earlier studies of DNA helical repeat and DNA supercoiling are not familiar to non-biologists we included in to this chapter a short chronological overview of these studies.
Andrzej Stasiak, Jan Bednar, Patrick Furrer, Vsevolod Katritch, Jacques Dubochet

Dynamics of Twist and Writhe and the Modeling of Bacterial Fibers

Abstract
We discuss a range of issues associated with the dynamics of twist and writhe including some new theoretical and numerical results and techniques. A precise understanding of twist and writhe is important in a variety of physical and biological processes and, in particular, we describe how these ideas can be used to model the dynamics of the self-assembling bacterial fiber, bacilus subtilis.
Michael Tabor, Isaac Klapper

Biomolecular Structure and Dynamics: Computation

Integration Methods for Molecular Dynamics

Abstract
Classical molecular dynamics simulation of a macromolecule requires the use of an efficient time-stepping scheme that can faithfully approximate the dynamics over many thousands of timesteps. Because these problems are highly nonlinear, accurate approximation of a particular solution trajectory on meaningful time intervals is neither obtainable nor desired, but some restrictions, such as symplecticness, can be imposed on the discretization which tend to imply good long term behavior. The presence of a variety of types and strengths of interatom potentials in standard molecular models places severe restrictions on the timestep for numerical integration used in explicit integration schemes, so much recent research has concentrated on the search for alternatives that possess (1) proper dynamical properties, and (2) a relative insensitivity to the fastest components of the dynamics. We survey several recent approaches.
Benedict J. Leimkuhler, Sebastian Reich, Robert D. Skeel

On the Parallelization of CHARMM on the CM-5/5E

Abstract
We describe a new port of CHARMM to the CM-5 supercomputer. This port is based on the original B-H parallelization of CHARMM[2] for MIMD machines but it has an improved communications library.
Jill P. Mesirov, Pablo Tamayo, Robert J. Nagle

Computational Studies of Spatially Constrained DNA

Abstract
Closed loops of double stranded DNA are ubiquitous in nature, occurring in systems ranging from plasmids, bacterial chromosomes, and many viral genomes, which form single closed loops, to eu-karyotic chromosomes and other linear DNAs, which appear to be organized into topologically constrained domains by DNA-binding proteins [1,2]. The topological constraints in the latter systems are determined by the spacing of the bound proteins along the contour of the double helix along with the imposed turns and twists of DNA in the intermolecular complexes [3,4]. As long as the duplex remains unbroken, the linking number Lk, or number of times the two strands of the DNA wrap around one another, is conserved [5–8]. If one of the strands is nicked and later re-sealed, the change in overall folding that accompanies DNA-protein interactions leads to a change in Lk. The supercoiling brought about by such protein action, in turn, determines a number of key biological events, including replication, transcription, and recombination [9].
Wilma K. Olson, Timothy P. Westcott, Jennifer A. Martino, Guo-Hua Liu

Pursuing Laplace’s Vision on Modern Computers

Abstract
This contribution is an informal essay based on a talk delivered at the Institute for Mathematics and its Applications (IMA) in Minneapolis, under the summer program in molecular biology, July 18–22, 1994. I exclude many technical details, which can be found elsewhere, and instead focus on the basic ideas of molecular dynamics simulations, with the goal of conveying to students and non-specialists the key concepts of the theory and practice of large-scale simulations. Following a description of the basic idea in molecular dynamics, I discuss some of the practical details involved in simulations of large biological molecules, the numerical timestep problem, and approaches to this problem based on implicit-integration techniques. I end with a perspective of open challenges in the field and directions for future research.
Tamar Schlick

Backmatter

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