Terminal Twist-Induced Writhe of DNA with Intrinsic Curvature

Supercoiling of a closed circular DNA rod may result from an application of terminal twist to the DNA rod by cutting the rod, rotating one of the cut faces as the other being fixed and then sealing the cut. According to White's formula, DNA supercoiling is probably accompanied by a writhe of the DNA axis. Deduced from the elastic rod model for DNA structure, an intrinsically straight closed circular DNA rod does not writhe as subject to a terminal twist, until the number of rotation exceeds a rod-dependent threshold. By contrast, a closed circular DNA rod with intrinsic curvature writhes instantly as subject to a terminal twist. This noteworthy character in fact belongs to many intrinsically curved DNA rods. By solving the dynamic equations, the linearization of the Euler–Lagrange equations governing intrinsically curved DNA rods, this paper shows that almost every clamped-end intrinsically curved DNA rod writhes instantly when subject to a terminal twist (clamped-end DNA rods include closed circular DNA rods and topological domains of open DNA rods). In terms of physical quantities, the exceptions are identified with points in ℝ⁶ whose projections onto ℝ⁵ (through ignoring the total energy density of a rod) form a subset of a quadratic hypersurface. This paper also suggests that the terminal twist induced writhe is due to the elasticity and the clamped-end boundary conditions of the DNA rods. To my sister for her 50th birthday.     

Twist and writhing in short circular DNAs according to first-order elasticity

The distribution of twist and writhing in a closed DNA shorter than its persistence length is examined. In this case, the only energy contribution is elastic. We have in tegrated the equations of elasticity for a homogeneous axially symmetric rod of undeformable infinitely small circular cross section with frictionless reactions, when there is no or only one self-contact. In the absence of self-contacts, the central line of the rod is drawn on a toroid. It makes ν turns around the axis of revolution of the toroid and m turns around its core. The integer, ν, is equal to one if the rod is unknotted. We prove that no infinitely thin rod with a positive Poisson ratio is stable in a toroidal conformation if there is no self-contact. However, m-leafed roses or rosettes, with but one multiple self-contact, are shown to actually be stable when their writhing is not too great. When the integer, m, is equal to two, we have figure-8 conformations. Buckling of the circle into a figure-8 conformation occurs for the constraint such that the figure-8 and the circular conformations have the same energy. This constraint is 1.845 turns for a bending-to-twisting elastic constants ratio of A/C = 1.5. For the same value of A/C, the figure-8 conformation is unstable for a constraint greater than 2.4 turns. Corrections caused by a finite value of the radius ratio, a/L, of the cross section to the length of the rod, are estimated. For instance, both the circular conformation and the infinitesimally warped circle are simultaneous solutions for particular values of the β twist. β = A/C (m² ? ν²)^½ [1 + (νπa/L)²/2]. The binding of ethidium to DNAs short enough to follow first-order elasticity has been studied. Buckling occurs at an apparent average constraint of about 0.6. How the DNA molecules are distributed in figure-8 conformations and circles has been determined as the ethidium concentration is varied.     

Terminal twist induced continuous writhe of a circular rod with intrinsic curvature

It is well known that a large linking number induces an abrupt writhing of a circular rod with zero intrinsic curvature, i.e., the stress-free state of the rod is straight. We show here that for any rod with a uniform natural curvature, no matter how small the intrinsic curvature is, a twist will induce a continuous writhing from the circular configuration and the abrupt writhing is only the limiting case when the intrinsic curvature is absolutely zero. The implication of this result on elastic models of circular DNA is discussed.     

The trajectory of a stiff rod in a curved potential energy trough. An initial result for short nucleosomal rods

The equilibrium trajectory of the axis of a rod subject to an externally imposed curved potential energy trough tends to conform to the shape of the curved trough, but also tends to be straight because of elastic resistance to bending. The actual path of the axis is a balance between the two extremes. We consider a potential energy trough centered along a circular arc of radius R. For a rod of small length compared to R, we show that the axis at equilibrium forms an arc of a circle of radius greater than R. The value of the radius of the axial path depends on the relative values of the Hooke's Law bending constant for the rod and the depth and width of the trough. Motivation for the calculation is provided by nucleosomal DNA, which conforms to the surface of a roughly cylindrical histone core at physiological ionic strength, but is observed to unwind into a partially extended conformation at very low ionic strength. We suggest that the rigidity to bending of short DNA segments becomes sufficiently great at low ionic strength to overcome attractive interactions with the histone surface. Alternately, of course, if during the cell cycle mutually attractive forces between DNA and histone core are weakened at constant ionic strength, the same type of unfolding would be expected to occur as the strength of the DNA-histone contacts drops below the level required to overcome elastic resistance to bending of the DNA rod.     

The trajectory of a stiff rod in a curved potential energy trough

The equilibrium trajectory of the axis of a rod subject to an externally imposed curved potential energy trough tends to conform to the shape of the curved trough, but also tends to be straight because of elastic resistance to bending. The actual path of the axis is a balance between the two extremes. We consider a potential energy trough centered along a circular arc of radiusR. For a rod of small length compared toR, we show that the axis at equilibrium forms an arc of a circle of radius greater thanR. The value of the radius of the axial path depends on the relative values of the Hooke’s Law bending constant for the rod and the depth and width of the trough. Motivation for the calculation is provided by nucleosomal DNA, which conforms to the surface of a roughtly cylindrical histone core at physiological ionic strength, but is observed to unwind into a partially extended conformation at very low ionic strength. We suggest that the rigidity to bending of short DNA segments becomes sufficiently great at low ionic strength to overcome attractive interactions with the histone surface. Alternately, of course, if during the cell cycle mutually attractive forces between DNA and histone core are weakened at constant ionic strength, the same type of unfolding would be expected to occur as the strength of the DNA-histone contacts drops below the level required to overcome elastic resistance to bending of the DNA rod.     

Intrinsic curvature of DNA influences LacR-mediated looping

Protein-mediated DNA looping is a common mechanism for regulating gene expression. Loops occur when a protein binds to two operators on the same DNA molecule. The probability of looping is controlled, in part, by the basepair sequence of inter-operator DNA, which influences its structural properties. One structural property is the intrinsic or stress-free curvature. In this article, we explore the influence of sequence-dependent intrinsic curvature by exercising a computational rod model for the inter-operator DNA as applied to looping of the LacR-DNA complex. Starting with known sequences for the inter-operator DNA, we first compute the intrinsic curvature of the helical axis as input to the rod model. The crystal structure of the LacR (with bound operators) then defines the requisite boundary conditions needed for the dynamic rod model that predicts the energetics and topology of the intervening DNA loop. A major contribution of this model is its ability to predict a broad range of published experimental data for highly bent (designed) sequences. The model successfully predicts the loop topologies known from fluorescence resonance energy transfer measurements, the linking number distribution known from cyclization assays with the LacR-DNA complex, the relative loop stability known from competition assays, and the relative loop size known from gel mobility assays. In addition, the computations reveal that highly curved sequences tend to lower the energetic cost of loop formation, widen the energy distribution among stable and meta-stable looped states, and substantially alter loop topology. The inclusion of sequence-dependent intrinsic curvature also leads to nonuniform twist and necessitates consideration of eight distinct binding topologies from the known crystal structure of the LacR-DNA complex.     

Studies on Circular Deoxyribonucleic Acid I. Isolation of Bovine Papilloma Virus and Characterization of Its Deoxyribonucleic Acid

A simple method is described for the isolation of bovine papilloma virus and its deoxyribonucleic acid (DNA). As found with other representatives of this virus group, this DNA preparation contains two components, I and II, as shown by sedimentation and electron microscopic studies. Component I is a fast-sedimenting, twisted, circular DNA molecule and represents usually 70 to 90% of the DNA in the mixture. The direction of the twist in the superstructure is right-handed. Component II originates from I by one or more single-strand breaks and is the "relaxed" circular from of the viral DNA.     

The effect of intrinsic curvature on supercoiling: Predictions of elasticity theory

Elasticity theory of naturally curved rods is employed to study the effects of intrinsic curvature on the properties of the equilibrium conformations of supercoiled DNA. The results stand in sharp contrast to those obtained when the molecule is viewed as being straight in its relaxed form. Starting from very fundamental principles of the theory, we show that the torsion of an open segment with a curved duplex axis can vary when the temperature, and along with it, the intrinsic twist is changed. Conversely, an imposed helicity, such as might be associated with binding to a histone, can change the intrinsic twist. It is also shown that another consequence of the presence of naturally curved sequences is that the twist density will, in general, vary with position along the chain in all equilibrium states. Then portions of the molecule will be more or less susceptible to interaction with other agents sensitive to such a variation. Finally, some closed equilibrium global structures uniquely associated with intrinsic curvature are discussed. © 1993 John Wiley & Sons, Inc.     

Mathematical modelling of interwound DNA supercoils

Small loops of DNA are affected by a variety of enzymes which remove turns of twist relative to the underlying double-helical structure. The molecule adopts a complex three-dimensional shape known as a supercoil in order to relieve the resulting internal stresses. This article describes an approach to modelling the overall shape of the supercoiled structure using elastic rod theory, which leads to simple expressions for predicting the shape of the structure. Predictions on the number of crossings in the balanced ply and the length of the end loops are compared to data in the literature and show reasonable agreement. The effect of the charged phosphate groups along the backbone of the DNA on the resulting supercoiled shape are also examined, and it is shown that this shape is very sensitive to the ionic concentration of the solution.     

Mapping the phase diagram of the writhe of DNA nanocircles using atomistic molecular dynamics simulations

We have investigated the effects of duplex length, sequence, salt concentration and superhelical density on the conformation of DNA nanocircles containing up to 178 base pairs using atomistic molecular dynamics simulation. These calculations reveal that the partitioning of twist and writhe is governed by a delicate balance of competing energetic terms. We have identified conditions which favour circular, positively or negatively writhed and denatured DNA conformations. Our simulations show that AT-rich DNA is more prone to denaturation when subjected to torsional stress than the corresponding GC containing circles. In contrast to the behaviour expected for a simple elastic rod, there is a distinct asymmetry in the behaviour of over and under-wound DNA nanocircles. The most biologically relevant negatively writhed state is more elusive than the corresponding positively writhed conformation, and is only observed for larger circles under conditions of high electrostatic screening. The simulation results have been summarised by plotting a phase diagram describing the various conformational states of nanocircles over the range of circle sizes and experimental conditions explored during the study. The changes in DNA structure that accompany supercoiling suggest a number of mechanisms whereby changes in DNA topology in vivo might be used to influence gene expression.     

Remarks on discrete and continuous large-scale models of DNA dynamics.

We present a comparison of the continuous versus discrete models of large-scale DNA conformation, focusing on issues of relevance to molecular dynamics. Starting from conventional expressions for elastic potential energy, we derive elastic dynamic equations in terms of Cartesian coordinates of the helical axis curve, together with a twist function representing the helical or excess twist. It is noted that the conventional potential energies for the two models are not consistent. In addition, we derive expressions for random Brownian forcing for the nonlinear elastic dynamics and discuss the nature of such forces in a continuous system.     

An elastic model of the large-scale structure of duplex DNA.

A general model for the large-scale, time-independent structure of duplex DNA is developed based on elastic considerations. The general conditions of elastic equilibrium are given. These equations are solved for the equilibrium shape of stressed duplex DNA, based on the assumption that the double helix behaves mechanically as a symmetric, linearly elastic rod. It is shown that, in general, two orders of superhelicity will arise at equilibrium. Several possible applications of this approach to the supercoiling of closed circular DNA are described.     

The effect of the superhelicity on the double helix twist angle in DNA.

An attempt to estimate the relative contributions of twisting and bending to the free energy of superhelix formation from the relaxed DNA is undertaken. The extent of teritiary ordering (number of DNA axis turns tau) and that of secondary ordering (duplex twist angle beta) have been taken as thermodynamical parameters, which characterize the state of the supercoild DNA at the fixed linking number (Lk) value. Such a thermodynamical approach implies the phenomenological parameters of rigidities of twisting and supercoiling (Gbeta, Gtau). Gtau/Gbeta ratio is estimated from the presented experimental data on the winding of the double helix upon increasing the ionic strength when twist alterations are followed by circular dichroism method. The adequacy of such interpretation of CD spectra changes are discussed. The values of Gtau and Gbeta are estimated to be of the same order of magnitude.     

Configurational transitions in Fourier series-represented DNA supercoils.

A new Fourier series representation of supercoiled DNA is employed in Langevin dynamics simulations to study large-scale configurational motions of intermediate-length chains. The polymer is modeled as an ideal elastic rod subject to long-range van der Waals' interactions. The van der Waals' term prevents the self-contact of distant chain segments and also mimics attractive forces thought to stabilize the association of closely spaced charged rods. The finite Fourier series-derived polymer formulation is an alternative to the piecewise B-spline curves used in past work to describe the motion of smoothly deformed supercoiled DNA in terms of a limited number of independent variables. This study focuses on two large-scale configurational events: the interconversion between circular and figure-8 forms at a relatively low level of supercoiling, and the transformation between branched and interwound structures at a higher superhelical density.     

Polarization selectivity of interaction of DNA molecules with X-ray radiation

The optimum form of a long helical molecule, which DNA is, has been calculated in terms of the classical electromagnetic theory. Three different methods of classical electrodynamics are used: the theory of dipole radiation of electromagnetic waves, the energetic power approach, and a helical model of molecules of chiral medium. In all three cases, an identical result for the optimum geometrical form of a long spiral molecule has been obtained. The lead angle between the tangent to the helix and the plane normal to the axis of the helix should be equal to 24.5 degrees. This condition imposes restrictions on the radius and the pitch of the helical molecule. The experimentally measured geometrical characteristics of the DNA molecule satisfy the theoretically calculated condition precisely enough. Having the optimum geometrical form, the DNA molecule is not influenced by a circularly right-polarized electromagnetic wave in the soft X-ray range λ = 7–8 nm. This wave, for which the right-handed DNA molecule is “transparent,” should propagate orthogonally to the helix axis and form a right-handed screw in space. The wave radiated by the right-handed DNA molecule orthogonally to helix axis in the range of λ ≈ 7–8 nm has, accordingly, the left-handed circular polarization. The polarization selectivity of the DNA molecule by the action of X-ray radiation is exhibited strongly enough in the wavelength range of λ ≈ 1–35 nm. The results obtained are valid for any distribution of electric currents in DNA, i.e., for any sequence of nitrogenous bases in DNA.     

A multiscale dynamic model of DNA supercoil relaxation by topoisomerase IB

In this study, we report what we believe to be the first multiscale simulation of the dynamic relaxation of DNA supercoils by human topoisomerase IB (topo IB). We leverage our previous molecular dynamics calculations of the free energy landscape describing the interaction between a short DNA fragment and topo IB. Herein, this landscape is used to prescribe boundary conditions for a computational, elastodynamic continuum rod model of a long length of supercoiled DNA. The rod model, which accounts for the nonlinear bending, twisting, and electrostatic interaction of the (negatively charged) DNA backbone, is extended to include the hydrodynamic drag induced by the surrounding physiological buffer. Simulations for a 200-bp-long DNA supercoil in complex with topo IB reveal a relaxation timescale of ∼0.1–1.0 μs. The relaxation follows a sequence of cascading reductions in the supercoil linking number (Lk), twist (Tw), and writhe (Wr) that follow companion cascading reductions in the supercoil elastic and electrostatic energies. The novel (to our knowledge) multiscale modeling method may enable simulations of the entire experimental setup that measures DNA supercoiling and relaxation via single molecule magnetic trapping.     

Effect of superhelical structure on the secondary structure of DNA rings

A quantity, called the linking number, is defined, which specifies the total number of twists in a circular helix. The linking number is invariant under continuous deformations of the ring and therefore enables one to calculate the influence of superhelical structures on the secondary helix of a circular molecule. The linking number can be determined by projecting the helix into a plane and counting strand crosses in the projection as described. For example, it has been shown that for each 180° twist in a left-handed superhelix, a right-handed 360° twist is removed from the secondary helix, thus allowing local unwinding.     

Structure of plectonemically supercoiled DNA

Using electron microscopy and topological methods, we have deduced an average structure for negatively supercoiled circular DNA in solution. Our data suggest that DNA has a branched plectonemic (interwound) form over the range of supercoiling tested. The length of the superhelix axis is constant at 41% of the DNA length, whereas the superhelix radius decreases essentially hyperbolically as supercoiling increases. The number of supercoils is 89% of the linking deficit. Both writhe and twist change with supercoiling, but the ratio of the change in writhe to the change in twist is fixed at 2.6:1. The extent of branching of the superhelix axis is proportional to the length of the plasmid, but is insensitive to superhelix density. The relationship between DNA flexibility constants for twisting and bending calculated using our structural data is similar to that deduced from previous studies. The extended thin form of plectonemically supercoiled DNA offers little compaction for cellular packaging, but promotes interaction between cis-acting sequence elements that may be distant in primary structure. We discuss additional biological implications of our structural data.