Bernard D. Coleman

Theory of sequence-dependent DNA elasticity

The elastic properties of a molecule of duplex DNA are strongly dependent on nucleotide sequence. In the theory developed here the contribution ψn of the nth base-pair step to the elastic energy is assumed to be given by a function ψn of six kinematical variables, called tilt, roll, twist, shift, slide, and rise, that describe the relative orientation and displacement of the nth and (n+1)th base pairs. The sequence dependence of elastic properties is determined when one specifies the way ψn depends on the nucleotides of the two base pairs of the nth step. Among the items discussed are the symmetry relations imposed on ψn by the complementarity of bases, i.e., of A to T and C to G, the antiparallel nature of the DNA sugar–phosphate chains, and the requirement that ψn be independent of the choice of the direction of increasing n. Variational equations of mechanical equilibrium are here derived without special assumptions about the form of the functions ψn, and numerical solutions of those equations are...

On the dynamics of rods in the theory of Kirchhoff and Clebsch

We discuss here the dynamical equations of a theory of elastic rods that is due to Kmcm-ioFF [1, 2] and CL~Bsc~ [3, 4]. This properly invariant theory is applicable to motions in which the strains relative to an undistorted configuration remain small, although rotations may be large. It is constructed to be a first-order theory, i.e., a theory that is complete to within an error of order two in an appropriate dimensionless measure of thickness, curvature, twist, and extension. In a first-order theory of thin rods, one can treat the rod as inextensible, and we do so here at the outset. Thus, at each time t, the arc-length parameter s for the axial curve ~(t) is employed as a material coordinate, i.e., a parameter whose value at a material point is constant in time, and not only the resultant of the shearing forces on a cross section, but also the tension in the rod, are reactive quantities not given by constitutive equations. Consider for a moment a rod that is naturally prismatic and dynamically symmetric, i.e., a rod that in an undistorted stress-free configuration is a cylinder whose directrix, although not necessarily a circle, bounds a figure with equal principal moments of inertia. A motion of the rod is said to be planar and twist-flee, i.e,, a motion of pure flexure, if it is such that ~(t) lies at all times in a fixed plane ~ which contains a principal axis of inertia of each cross section. For such a motion we employ a fixed Cartesian coordinate system on ~, and, for consistency with a discussion of more general motions to be given later in this paper, we call the abscissa z and the ordinate x. We may write FZ(s, t), FX(s, t) for the z- and x-components of the resultant force F at time t on the cross section with arc-length coordinate s. The motion of the rod may be described by giving the (z, x)-coordinates of the points on ~(t) as functions of s and t. With O(s, t) the counterclockwise angle from the z-axis to the tangent of ~(t) at s, we have

Theory of Supercoiled Elastic Rings with Self-Contact and Its Application to DNA Plasmids

Methods are presented for obtaining exact analytical representations of supercoiled equilibrium configurations of impenetrable elastic rods of circular cross-section that have been pretwisted and closed to form rings, and a discussion is given of applications in the theory of the elastic rod model for DNA. When, as here, self-contact is taken into account, and the rod is assumed to be inextensible, intrinsically straight, transversely isotropic, and homogeneous, the important parameters in the theory are the excess link Δℒ (a measure of the amount the rod was twisted before its ends were joined), the ratio ω of the coefficients of torsional and flexural rigidity, and the ratio d of cross-sectional diameter to the length of the axial curve C. Solutions of the equations of equilibrium are given for cases in which self-contact occurs at isolated points and along intervals. Bifurcation diagrams are presented as graphs of Δℒ versus the writhe of C and are employed for analysis of the stability of equilibrium configurations. It is shown that, in addition to primary, secondary, and tertiary branches that arise by successive bifurcations from the trivial branch made up of configurations for which the axial curve is a circle, there are families of equilibrium configurations that are isolas in the sense that they are not connected to bifurcation branches by paths of equilibrium configurations compatible with the assumed impenetrability of the rod. Each of the isolas found to date is connected to a bifurcation branch by a path which, although made up of solutions of the governing equations, contains regions on which the condition of impenetrability does not hold.

On the thermodynamics of periodic phases

A computer system has been developed to handle archiving and analysis of data acquired during operations of the Continuous Wave Deuterium Demonstrator (CWDD). Data files generated by the CWDD Instrumentation and Control system are transferred across a local area network to the CWDD Archive system where they are enlisted into the archive and stored on removeable media optical disk drives. A relational database management system maintains an on-line database catalog of all archived files. This database contains information about file contents and formats, and holds signal parameter configuration tables needed to extract and interpret data from the files. Software has been developed to assist the selection and retrieval of data on demand based upon references in the catalog. Data retrieved from the archive is transferred to commercial data visualization applications for viewing, plotting and analysis.

The Elastic Rod Model for DNA and Its Application to the Tertiary Structure of DNA Minicircles in Mononucleosomes

Explicit solutions to the equations of equilibrium in the theory of the elastic rod model for DNA are employed to develop a procedure for finding the configuration that minimizes the elastic energy of a minicircle in a mononucleosome with specified values of the minicircle size N in base pairs, the extent w of wrapping of DNA about the histone core particle, the helical repeat h(0)b of the bound DNA, and the linking number Lk of the minicircle. The procedure permits a determination of the set Y(N, w, h(0)b) of integral values of Lk for which the minimum energy configuration does not involve self-contact, and graphs of writhe versus w are presented for such values of Lk. For the range of N of interest here, 330 < N < 370, the set Y(N, w, h(0)b) is of primary importance: when Lk is not in Y(N, w, h(0)b), the configurations compatible with Lk have elastic energies high enough to preclude the occurrence of an observable concentration of topoisomer Lk in an equilibrium distribution of topoisomers. Equilibrium distributions of Lk, calculated by setting differences in the free energy of the extranucleosomal loop equal to differences in equilibrium elastic energy, are found to be very close to Gaussian when computed under the assumption that w is fixed, but far from Gaussian when it is assumed that w fluctuates between two values. The theoretical results given suggest a method by which one may calculate DNA-histone binding energies from measured equilibrium distributions of Lk.

The dependence of DNA tertiary structure on end conditions: Theory and implications for topological transitions

Explicit expressions are derived for the equilibrium configurations of long segments of a DNA double helix subject to boundary conditions of the type imposed by DNA‐bending proteins at the ends of otherwise free segments. The expressions, which are exact within the framework of Kirchhoff’s theory of elastic rods, show that, in appropriate ranges of parameters, small changes in end conditions can result in large changes in tertiary structure. A discussion is given of the implications of this observation for understanding the action of bending proteins and of proteins that induce topological transitions that change the linking number of closed loops of DNA.

Theory of the influence of end conditions on self‐contact in DNA loops

Explicit solutions of the equations of Kirchhoff’s theory of elastic rods are employed to derive properties of the tertiary structure of a looped segment of DNA that is subject to geometric constraints imposed at its end points by bound proteins. In appropriate circumstances small changes in such boundary data cause a nearly planar loop to undergo a continuous and reversible transition that can be described as a 180° rotation taking the loop from an uncrossed to a singly crossed structure in which sequentially separated base pairs are brought into proximity. Expressions are derived relating points and angles of crossing to end conditions, and results are presented that facilitate the calculation of changes in elastic energy during such transitions.

Implications of the dependence of the elastic properties of DNA on nucleotide sequence

Recent advances in structural biochemistry have provided evidence that not only the geometric properties but also the elastic moduli of duplex DNA are strongly dependent on nucleotide sequence in a way that is not accounted for by classical rod models of the Kirchhoff type. A theory of sequence–dependent DNA elasticity is employed here to calculate the dependence of the equilibrium configurations of circular DNA on the binding of ligands that can induce changes in intrinsic twist at a single base–pair step. Calculations are presented of the influence on configurations of the assumed values and distribution along the DNA of intrinsic roll and twist and a modulus coupling roll to twist. Among the results obtained are the following. For minicircles formed from intrinsically straight DNA, the distribution of roll–twist coupling strongly affects the dependence of the total elastic energy Ψ on the amount α of imposed untwisting, and that dependence can be far from quadratic. (In fact, for a periodic distribution of roll–twist coupling with a period equal to the intrinsic helical repeat length, Ψ can be essentially independent of α for –90° < α <90°.) When the minicircle is homogeneous and without roll–twist coupling, but with uniform positive intrinsic roll, the point at which Ψ attains its minimum value shifts towards negative values of α. It is remarked that there are cases in which one can relate graphs of Ψ versus α to the ‘effective values’ of bending and twisting moduli and helical repeat length obtained from measurements of equilibrium distributions of topoisomers and probabilities of ring closure. For a minicircle formed from DNA that has an ‘S’ shape when stress–free, the graphs of Ψ versus α have maxima at α = 0. As the binding of a twisting agent to such a minicircle results in a net decrease in Ψ, the affinity of the twisting agent for binding to the minicircle is greater than its affinity for binding to unconstrained DNA with the same sequence.

Stability of cylindrical bodies in the theory of surface diffusion

Abstract Perturbation arguments and finite element calculations are employed to study the nonlinear partial differential equations governing morphological changes induced by curvature-driven diffusion of mass in the surface of an axially symmetric body. Isotropy of surface properties is assumed. Second- and higher-order perturbation analyses indicate that the familiar result of the linear theory of small amplitude longitudinal perturbations of a cylinder to the effect that a long cylinder is stable against all perturbations with spatial Fourier spectra containing only wavelengths less than the circumference of the cylinder does not hold in the full nonlinear theory. The perturbation analyses yield criteria for determining when longitudinal perturbations with high wave-number spectra grow in amplitude, after an initial decay followed by an incubation time, and result in break-up of the body into a necklace of beads. The principal conclusions of the formal perturbation analyses are found to be in good accord with numerical solutions obtained by finite element methods.

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

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