^{1}, Wilma K. Olson

^{1}and Irwin Tobias

^{1,a)}

### Abstract

Because of the double-helical structure of DNA, in which two strands of complementary nucleotides intertwine around each other, a covalently closed DNA molecule with no interruptions in either strand can be viewed as two interlocked single-stranded rings. Two closed space curves have long been known by mathematicians to exhibit a property called the linking number, a topologically invariant integer, expressible as the sum of two other quantities, the twist of one of the curves about the other, and the writhing number, or writhe, a measure of the chiral distortion from planarity of one of the two closed curves. We here derive expressions for the twist of supercoiled DNA and the writhe of a closed molecule consistent with the modern view of DNA as a sequence of base-pair steps. Structural biologists commonly characterize the spatial disposition of each step in terms of six rigid-body parameters, one of which, coincidentally, is also called the twist. Of interest is the difference in the mathematical properties between this step-parameter twist and the twist of supercoiling associated with a given base-pair step. For example, it turns out that the latter twist, unlike the former, is sensitive to certain translational shearing distortions of the molecule that are chiral in nature. Thus, by comparing the values for the two twists for each step of a high-resolution structure of a protein-DNA complex, we may be able to determine how the binding of various proteins contributes to chiralstructural changes of the DNA.

We thank Luke Czapla and Nicolas Clauvelin for valuable discussions. The U.S. Public Health Service under research Grant No. GM34809 and instrumentation Grant No. RR022375 has generously supported this work.

I. INTRODUCTION

II. THE TWIST OF SUPERCOILING

III. THE TWIST OF SUPERCOILING FOR THE MULTISTEP DNA MOLECULE

IV. THE WRITHE OF THE CLOSED MULTISTEP DNA MOLECULE

V. COMPARISON OF THE TWIST OF SUPERCOILING AND THE STEP-PARAMETER TWIST

VI. SUMMARY

## Figures

Schematic representation of DNA. The double-helical axis is given by curve and one of the helical strands by curve . For the purpose of calculation of the twist of about , is to be thought of as being traced out by the head of a vector everywhere normal to the tangent vector .

Schematic representation of DNA. The double-helical axis is given by curve and one of the helical strands by curve . For the purpose of calculation of the twist of about , is to be thought of as being traced out by the head of a vector everywhere normal to the tangent vector .

The vectors associated with a base-pair plane: an origin and a mutually orthogonal triad of unit vectors, the short axis , the long axis , and the normal .

The vectors associated with a base-pair plane: an origin and a mutually orthogonal triad of unit vectors, the short axis , the long axis , and the normal .

The passage from a smooth curve with circular segments of curvature to the entirely linear segments that connect the origins of the base-pair planes consistent with a high-resolution structure of DNA.

The passage from a smooth curve with circular segments of curvature to the entirely linear segments that connect the origins of the base-pair planes consistent with a high-resolution structure of DNA.

The vectors involved in the calculation of the twist of supercoiling of the DNA base-pair step bounded by the and the planes. Shown in (a), the four base-pair plane origins needed for the determination of the three vectors depicted in (b), and , which, in turn, are needed for specifying and . These last two are normal to the two planes seen in (c). Each of these planes contains a vector and a vector. Two of the angles needed for the twist calculation as given by Eq. (16) are denoted in (d), an enlargement of (c).

The vectors involved in the calculation of the twist of supercoiling of the DNA base-pair step bounded by the and the planes. Shown in (a), the four base-pair plane origins needed for the determination of the three vectors depicted in (b), and , which, in turn, are needed for specifying and . These last two are normal to the two planes seen in (c). Each of these planes contains a vector and a vector. Two of the angles needed for the twist calculation as given by Eq. (16) are denoted in (d), an enlargement of (c).

The vectors needed for the calculation of the step-parameter twist of the same step shown in the previous figure. Here knowledge of the direction of the two normals allows the determination of the single vector , which lies in each of the two base-pair planes. Also indicated are the two angles needed for the use of Eq. (18) for the twist calculation.

The vectors needed for the calculation of the step-parameter twist of the same step shown in the previous figure. Here knowledge of the direction of the two normals allows the determination of the single vector , which lies in each of the two base-pair planes. Also indicated are the two angles needed for the use of Eq. (18) for the twist calculation.

Construction of a model DNA structure characterized by a chiral deformation. Image labeled (a) shows four equally spaced and parallel base-pair planes having their origins lying on a line. The sequence of base-pair planes in (b) depicts the structure after the bend described in the text is introduced. The four origins are still coplanar, and the viewing direction is chosen to be normal to this plane. A translation of base pairs 3 and 4 as a single unit along the viewing direction, depending on the direction of the motion, results either in (c), a structure with a right-handed jog, or (d), one with a left-handed jog.

Construction of a model DNA structure characterized by a chiral deformation. Image labeled (a) shows four equally spaced and parallel base-pair planes having their origins lying on a line. The sequence of base-pair planes in (b) depicts the structure after the bend described in the text is introduced. The four origins are still coplanar, and the viewing direction is chosen to be normal to this plane. A translation of base pairs 3 and 4 as a single unit along the viewing direction, depending on the direction of the motion, results either in (c), a structure with a right-handed jog, or (d), one with a left-handed jog.

Structural deformation of DNA leading to a twist of supercoiling change dependent on the method used to determine the position of the origin, the method used here, or that in which the origin lies on a line connecting two carbon atoms on the bases. In structure (a), the same as structure (d) in Fig. 6, both methods lead to an origin for all four base pairs located in the same position. However when base pair 3 is buckled as shown to form structure (b), the origin determined by our method (blue dot) moves, but that of the other method (red dot) does not. One then observes a method-dependent twist of supercoiling for the middle step.

Structural deformation of DNA leading to a twist of supercoiling change dependent on the method used to determine the position of the origin, the method used here, or that in which the origin lies on a line connecting two carbon atoms on the bases. In structure (a), the same as structure (d) in Fig. 6, both methods lead to an origin for all four base pairs located in the same position. However when base pair 3 is buckled as shown to form structure (b), the origin determined by our method (blue dot) moves, but that of the other method (red dot) does not. One then observes a method-dependent twist of supercoiling for the middle step.

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