Topology in Molecular Biology

3 Monte Carlo Simulation of DNA Topological Properties 37

Fig. 3.13.The simplest links and their Alexander polynomials, g∆(s,t). All links
that can be drawn with less than six intersections and one of three links with six
intersections are shown. For an unlinked contour ∆(s,t) = 0

curves (for unlinked curves ∆(s, t) = 0 equals 0)...[42]. In the most cases
calculation of ∆(s, t) = 0 also takes less computer time than that of the
Gauss integral. Four simplest links and the corresponding ∆(s, t)areshown
in Fig. 3.13. Complete table of links with less than 11 crossings can be found
in [42].
Checking topology of a circular chain may be the most time-consuming
part of the whole calculation. Therefore this part of the computer program
deserves maximum attention in terms of its rationality. In this connection a
procedure of reducing the order of the Alexander matrix before calculating it,
by eliminating trivial intersections, is very useful (for details, see [26, 47, 48].

3.6 Calculation of Writhe

For many problems related with circular DNA one needs to calculate writhe,
Wr, of a closed chain. It can be done by using definition of Wr through the
Gauss integral (see [49], for example). For straight segments this integral may
be presented in the form of a double sum of simple terms [50]. The method
allows natural extension to the case of linear chains where the integral can
serve as a measure of the chain chirality [32]. However, there are more conve-
nient and efficient methods to calculate Wr [48, 50]. In the method suggested
by Le Bret the total writhing value is presented in the form of two contribu-
tions, one of which is the directional writhing number, which can be calculated
simultaneously with calculating the Alexander polynomial, virtually without
additional computations. The directional writhing number in thezdirection
is the sum of +1 or−1 over all crossings. The sign of each term is determined
by the type of crossing (Fig. 3.11). The second contribution is a sum over all
elementary segmentsrmof the chain:

∑

m

{arcsin[sinξmsin(φm+1−χm)]−arcsin[sinξmsin(φm−χm)]}/ 2 π,