Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology (1995)

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 163 Figure 6.8a Examples of pairs of curves C and A with different values of twist. (a) Simple examples in which the axis A is a straight line and the twist is the number of times that C winds around A, being positive for right-handed twist and negative for left-handed twist. (b) An example in which the axis A is a helix winding around a linear axis and the curve C is a superhelix winding around A. In this case the twist of C around A is the number of times that C winds around A (in this case approximately 3.5) plus nsinγ where n is the number of times that A winds around the linear axis and γ is the pitch angle of the helix A. Here n is approximately 1.5 and γ is approximately 40°. Thus, Tw ≈ 4.46. APPLICATIONS TO DNA TOPOISOMERASE REACTIONS For a relaxed closed DNA that lies in a plane and has no self-crossings, we have seen that the writhing number is equal to 0. Therefore, by the fundamental formula, Tw = Lk. Thus both Tw and Lk are equal to the number of times that the backbone strands wind around the axis, or more precisely the number of times the backbone curve rises above and falls below the plane in which the axis lies. For such a DNA in the B-form, there are about 10.5 base pairs per turn of the backbone. This linking number is usually denoted Lk0. As we noted above, a relaxed DNA molecule of monkey virus SV40 with about 5,250 base pairs has Lk0 = Tw = 500. However, the linking number of most closed circular DNA is not that of the relaxed state. The actual linking number 

WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 164 Lk in most cases is less than Lk0. In the electron microscope, these DNA are supercoiled and appear to be contorted or coiled-up rings with many self-crossings. The quantity Lk − Lk0 = ∆Lk is called the linking number difference and is a measure of this supercoiling. By the fundamental theorem, a change in linking must consist of a change in twist and a change in writhe: ∆Lk = ∆Tw + ∆Wr. Because the writhe of a planar curve is 0, ∆Wr becomes simply Wr. Recent work (Boles et al., 1990) shows the ratio of the change in writhe to the change in twist is approximately 2.6:1; that is, ∆Wr = 0.72 ∆Lk and ∆Tw = 0.28 ∆Lk. Thus, for each change of 1 in the linking difference, there is a change of 0.72 in the writhe. Large changes in linking will therefore result in large changes in Wr. Because of this, there are negative crossovers introduced in many views of the DNA. In fact, it has been shown that the interwound coils shown in Figure 6.1c model negatively supercoiled DNA well. Such DNA are also called underwound because the twist is also reduced. Some time after the discovery of closed supercoiled DNA (Bauer, 1978), enzymes were found that can actually change the linking number difference and in fact change a highly supercoiled DNA into a relaxed open circular DNA (Wang, 1985). These enzymes, called topoisomerases, because they change the topology of the DNA, are divided into two classes according to their operational function: Type I topoisomerases, which introduce single-stranded breaks, and Type II topoisomerases, which introduce double-stranded breaks. An intuitive description of the action of a Type I topoisomerase can be given as follows. The first step in a Type 1 topoisomerase reaction is to break the backbone curve C of an underwound DNA at a point c. Then the backbone curve is allowed to rotate in a clockwise fashion to increase the twist in the direction to that preferred by B-form DNA. (This rotation is a natural process for DNA with smaller ∆Lk in absolute value and is energetically more favorable.) Finally, the break is again sealed at the point c. The twist increases by an integral amount depending on the number of rotations. Because the axis is virtually unchanged in the immediate process, the writhe remains unchanged. Therefore the linking number increases by the number of times the C strand is rotated around the axis A. Energetically, following the completion of the sealing, each change in linking of +1 will be 

WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 165 distributed to change the average twist by +0.28 and the average writhe by +0.72 in accordance with the results mentioned above (Boles et al., 1990). Eventually, in the presence of topoisomerases of Type I, the linking number difference of a supercoiled DNA can be reduced to 0. In reality, an entire population of supercoiled DNA with a specific linking number difference can be treated with a topoisomerase of Type 1. The result will be a mixture of the same DNA with linking number differences that vary all the way from the original number to 0. By way of example, the native state of the monkey virus SV40 is not a relaxed circle, but rather a supercoiled DNA with linking number difference of −25. After treatment with Type I topoisomerase, the same population exists with linking number differences −25, −24, −23, . . ., −1, 0, depending on the amount of change introduced by the enzyme. Interestingly enough, the same DNA with different linking number differences can be separated by means of gel electrophoresis (Wang and Bauer, 1979). Separation occurs due to the fact that such DNA travel through the tangled molecular matrix of a gel at different speeds because one has more Wr and is more compact than the other. This is one way in which the topology of a DNA can be used to characterize DNA's physical properties. Topoisomerases of Type II also have the function of reducing the amount of linking difference. In the reaction of Type I topoisomerases described above, the twist was increased to increase the linking number. In the case of Type II topoisomerases, the increase in linking is due to an increase in writhing, which is obtained by a self-passage of the entire DNA molecule. The first step is to bring distant segments of the DNA into close proximity. Because DNA is found mostly in the interwound form, it already has distant segments reasonably close, as shown in Figure 6.1 c. The node so created contributes a −1 to the writhing number at this point. Next a complete break of the DNA at the cross section of one of the neighboring segments takes place. This is essentially accomplished by breaks of both backbone chains in the cross section. The other segment is now passed through the double-stranded break. Finally, the original break is resealed. In this process of self-passage, the writhing number has been increased by +2. This is due to the fact that a negative crossing has been replaced by a positive crossing. Because the twist is virtually unaltered in the process, the linking number also increases by +2. After the process is over, the change in linking of +2