Writhing number

Fig. 6. The envelope of the writhing number of closed circular DNA subject to torsional stress as a function of 7, as computed from Langevin trajectories. Data are from [31].

Fuller, F.B. (1971) The writhing number of a space curve. Proc. Natl. Acad. Sci. USA 68, 815-819. Crick, F.H.C. (1976) Linking numbers and nucleosomes. Proc. Natl. Acad. Sci. USA 73,... 

Figure 5-19 Topological equivalence of toroidal (solenoidal) and plectonemically interwound forms of a circular DNA. These two forms have a constant value of the linking number Lk (or a), the twist Tw, and writhing number Wr.

Since L/c is constant in circularly closed DNA a change of one turn of B-DNA (Tw = 1) into a turn of Z-DNA (Tw = -1) will cause the writhing number Wr to change by -2. Conversely, if the writhing number is forced to change by -2, a turn of Z-DNA may develop somewhere in a suitable (G + C)-rich region of the DNA. 

Figure 27.19. Linking Number. The relations between the linking number (Lk), twisting number (Tw), and writhing number (Wr) of a circular DNA molecule revealed schematically. [After W. Saenger, Principles of Nucleic Acid Structure (Springer-Verlag, 1984), p. 452.]...

Fig. 4. The level of cellular DNA supercoiling is dependent on both DNA-tracking processes, such as transcription, and the activity of DNA topoisomerases [61-63]. fV is the writhing number W > 0 corresponds to positive supercoiling and JV <0 corresponds to negative supercoiling.

Figure 6. Typical results of Metropolis-Monte Carlo calculations on the dependence on the number of straight segments per Kuhn length, k, of a mean quantity (the mean writhing number, Wr, see Section 4.3.2.1, in the particular case) for closed polymer chain. The data are from Vologodskii and Frank-Kamenetskii (1992) [81].

The writhing number is a related shape descriptor derived from geometry and connectivity. It is normally used for closed curves, although it can be extended to strands. 

Note that detailed 3D folding features are somewhat overlooked in W. The writhing number involves a sum over handednesses of double points, and there are several combinations of them that can produce the same result. 

The linking number can also be used as a shape descriptor of a single loop of double-stranded ( duplex ) DNA. In this case, the DNA is approximated by a continuous flat ribbon,and one considers the linking between the two curves defined by the borders of the ribbon (i.e., the two strands). The parameter measures the number of times the two strands intertwine around each other. In the DNA ribbon model, the linking number is relatedto the writhing number °lf of the central axis of the ribbon as... 

Self-entanglements. The occurrence of twists, turns, and convolutions in space curves can be characterized by a number of descriptors using geometry and connectivity. These include the overcrossing probability distribution of a backbone of arbitrary architecture (and related parameters, such as the mean number of overcrossings N), the writhing number W of a curve, and the twist 9" of a ribbon model. 

Fuller, F. B., 1971, The writhing number of a space curve, Proc. Natl. Acad. Sci. USA 68 815. 

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