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, the nucleosome considered here, for example, we may be able to determine how the binding of various proteins contributes to chiral structural changes of the DNA.
Deep Dive into Two Perspectives on the Twist of DNA.
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
nucleosome considered here, for example, we may be able to determine how the binding of various proteins contributes to chiral structural changes of the DNA.
The structure of a DNA molecule is often described as a succession of base pairs, each represented as a rectangular plane. 1 A knowledge of the relative locations of origins positioned within these planes and the relative orientations of the short and long axes of the rectangles allows one to determine for each pair of adjacent base pairs in the molecule -a so-called base-pair step -the numerical values of six rigid-body parameters, three translational: shift, slide, and rise, and three angular: tilt, roll, and twist. [2][3][4][5][6][7] Some forty years ago mathematicians defined a “twist” which, shortly after its introduction, was applied to DNA and used, like the step-parameter twist mentioned above, to characterize the secondary structure of the molecule. [8][9][10] This twist was defined as the value of a certain integral involving two continuous space curves. In the application to DNA, the structure of which at that time was often depicted in terms of space curves, 11 one of the curves was taken to be the axis of the double helix, and the other one of the strands winding about this axis. One then went on to compute the twist of the DNA, a unitless scalar representing the number of times the strand wound about the helical axis.
Here we are concerned with differences in the properties of these two twists, the step-parameter twist, and the twist of the preceding paragraph, which, because of its connection with the global shape of the helical axis of a closed DNA molecule, a plasmid, for example, we shall refer to as the twist of supercoiling. In the next section we begin by reviewing the definition of the twist of supercoiling, and its well-known connection with the writhing number and the linking number. 12,13 Then, after characterizing two space curves consistent with today’s picture of DNA as a succession of discrete rectangular planes, we go on to describe a method for the computation of the twist of supercoiling for a single base-pair step. We also point out how easy it is to compute the writhe for the case of a closed molecule with an axial curve envisioned as a succession of line segments connecting the origins.
Of particular interest is the difference in properties between the step-parameter twist and the twist of supercoiling. We note that in a relaxed, undeformed configuration of a DNA molecule, the two twists are expected to be close in value for all base-pair steps. However, we find that although translations of the base pairs leave the stepparameter twist unchanged, that is not generally the case for the twist of supercoiling. It, instead, is sensitive to translational distortions that are chiral in nature. To illustrate the point, we compare the values of the two types of twist for certain base-pair steps in DNA wrapped around the core of eight histone proteins in a nucleosome. 14 We find for the chosen base-pair steps an instance in which the local protein-induced deformation of the DNA structure preserves the value of the step-parameter twist but alters the twist of supercoiling.
As we pointed out above, in much of the early theoretical work describing the equilibrium configurations of DNA, the atomistic details of the molecule were ignored, and instead, the structure of the molecule was, in fact, described in terms of two space curves. The tertiary structure of the molecule was represented, as shown in Fig. 1, by the shape of a smooth space curve C mirroring the shape of the axis of the double helix.
For the case of the tangent, Eq. ( 1) leads to the equation
One of the Frenet-Serret equations, the three equations relating the tangent to the
The first term on the right- hand-side of Eq. ( 2) then becomes
The second term in that equation, the one containing the component of dΩ along t C (s C ) , is proportional to the twist density. Replacing t C (s C ) by d(s C )in Eq. ( 2) and then taking the projection of the resulting expression for dΩ along the tangent shows that
The twist ( , )
where
If the curves C, given by r C (s C ) , and D, given by r D (s D ) , are closed, the conventional twist is simply related to two other integrals, 12,13 so-called Gauss integrals, the linking number ( , )
and the writhing number ( ), WC or writhe for short,
The linking number is an integer, a topological invariant, equal to the number of times that the curve D passes through a surface bounded by C. 15 (In computing this sum each pass-through is assigned a value of either +1 or -1 according to a convention consistent with the form of the Gauss integral.) This integer remains unchanged for all distortions in shape of the curves C and D as long as the curves do not intersect each other during the distortions. The writhe, a property of closed curve C alone, is a measure of the chiral distortion of the curve from planarity. Fuller pointed out
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