Irwin Tobias

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

Finite element analysis of DNA supercoiling

A DNA polymer with hundreds or thousands of base pairs is modeled as a thin elastic rod. To find the equilibrium configurations and associated elastic energies as a function of linking number difference (ΔLk), a finite element scheme based on Kirchhoff’s rod theory is newly formulated so as to be able to treat self‐contact. The numerical results obtained here agree well with those already published, both analytical and numerical, but a much more detailed picture emerges of the several equilibrium states which can exist for a given ΔLk. Of particular interest is the discovery of interwound states having odd integral values of the writhing number and very small twist energy. The existence of a noncircular cross section, inhomogeneous elastic constants, intrinsic curvature, and sequence‐dependent bending and twisting can all be readily incorporated into the new formalism.

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.

Electrostatic effects in short superhelical DNA

We present Monte Carlo simulations of the equilibrium configurations of short closed circular DNA that obeys a combined elastic, hard-sphere, and electrostatic energy potential. We employ a B-spline representation to model chain configuration and simulate the effects of salt on chain folding by varying the Debye screening parameter. We obtain global equilibrium configurations of closed circular DNA, with several imposed linking number differences, at two salt concentrations (specifically at the extremes of no added salt and the high salt regime), and for different chain lengths. Minimization of the composite elastic/long-range potential energy under the constraints of ring closure and fixed chain length is found to produce structures that are consistent with the configurations of short supercoiled DNA observed experimentally. The structures generated under the constraints of an electrostatic potential are less compact than those subjected only to an elastic term and a hard-sphere constraint. For a fixed linking number difference greater than a critical value, the interwound structures obtained under the condition of high salt are more compact than those obtained under the condition of no added salt. In the case of no added salt, the electrostatic energy plays a dominant role over the elastic energy in dictating the shape of the closed circular DNA. The DNA supercoil opens up with increasing chain length at low salt concentration. A branched three-leaf rose structure with a fixed linking number difference is higher in energy than the interwound form at both salt concentrations employed here.

Modeling self-contact forces in the elastic theory of DNA supercoiling

A DNA polymer with thousands of base pairs is modeled as an elastic rod with the capability of treating each base pair independently. Elastic theory is used to develop a model of the double helix which incorporates intrinsic curvature as well as inhomogeneities in the bending, twisting, and stretching along the length of the polymer. Inhomogeneities in the elastic constants can also be dealt with; thus, sequence-dependent structure and deformability can be taken into account. Additionally, external forces have been included in the formalism, and since these forces can contain a repulsive force, DNA self-contact can be explicitly treated. Here the repulsive term takes the form of a modified Debye–Huckel force where screening can be varied to account for the effect of added salt. The supercoiling of a naturally straight, isotropic rod in 0.1M NaCl is investigated and compared with earlier treatments of supercoiled DNA modeled by a line of point charges subject to electrostatic interactions and an elastic po...

Potential Energy Curves for the X1Σg+ and B1Σu+ States of Hydrogen

The recently reported accurate data of Herzberg and Howe have been used to calculate the potential energy curves for the X 1Σg+ and B 1Σu+ states of hydrogen by the Rydberg‐Klein‐Rees (RKR) method. The results are compared with the results of the quantum calculations of Kolos and Roothaan and Dalgarno and Lynn. The agreement is good; in fact, the interaction energy obtained by Dalgarno and Lynn for the ground state agrees with the RKR values to within 2% in the region where the two curves overlap. The rotational predissociation observed in the ν″=14 level of the X 1Σg+ state is discussed. It appears as if there may exist here an unusual case of a sharp spectral line associated with a state which is very close in energy to the maximum of a potential barrier.

Flexing and folding double helical DNA.

DNA base sequence, once thought to be interesting only as a carrier of the genetic blueprint, is now recognized as playing a structural role in modulating the biological activity of genes. Primary sequences of nucleic acid bases describe real three-dimensional structures with properties reflecting those structures. Moreover, the structures are base sequence dependent with individual residues adopting characteristic spatial forms. As a consequence, the double helix can fold into tertiary arrangements, although the deformation is much more gradual and spread over a larger molecular scale than in proteins. As part of an effort to understand how local structural irregularities are translated at the macromolecular level in DNA and recognized by proteins, a series of calculations probing the structure and properties of the double helix have been performed. By combining several computational techniques, complementary information as well as a series of built-in checks and balances for assessing the significance of the findings are obtained. The known sequence dependent bending, twisting, and translation of simple dimeric fragments have been incorporated into computer models of long open DNAs of varying length and chemical composition as well as in closed double helical circles and loops. The extent to which the double helix can be forced to bend and twist is monitored with newly parameterized base sequence dependent elastic energy potentials based on the observed configurations of adjacent base pairs in the B-DNA crystallographic literature.

A class of exact dynamical solutions in the elastic rod model of DNA with implications for the theory of fluctuations in the torsional motion of plasmids

New explicit solutions are obtained for the nonlinear equations of Kirchhoff’s theory of the dynamics of inextensible elastic rods without neglect of rotatory inertia. These exact solutions describe a class of motions possible in closed circular rings possessing a uniform distribution of intrinsic curvature ku and intrinsic torsion. When ku≠0, the motions in this class are such that the axial curve of the ring remains stationary while the cross sections rotate about their centers in such a way that the angle ψ of rotation is independent of axial location and is governed by the nonlinear pendulum equation. When ku=0, such uniform rotation of cross sections can occur at an arbitrary steady rate. The methods of classical equilibrium statistical mechanics yield the following conclusions for canonical ensembles of rings for which the motion is this type of pure homogeneous torsion. When 1/ku=11.85 nm (i.e., when the intrinsic curvature ku is among the highest observed in naturally occurring, approximately unif...