3 Monte Carlo Simulation of DNA Topological Properties 27Fig. 3.3.Probing conformational properties of supercoiled DNA by formation of
topological links [11]. The diagram shows formation of links between supercoiled
and cyclizing linear molecules. The cyclization occurred via long cohesive ends and
resulted in nicked circular molecules. The simplest links shown here comprised at
least 90% of all links formed
properties of supercoiled DNAs, a form that is characterized by frequent close
approaches between distal DNA segments. Because these close approaches are
rare in linear and open circular DNA molecules, the accurate description of
intersegment interaction is not so crucial for the prediction of most of their
properties. Thus, it was important to test how well the model can describe
conformational properties of supercoiled DNA. We completed this task by
comparing computed and measured equilibrium linkage between supercoiled
DNA and cyclizing linear molecules as illustrated in Fig. 3.3.
The probabilityP that a given open circular DNA will be linked with
supercoiled molecules of concentration Wr can be expressed as
P=
∫∞
0p(R)4πr^2 cNA
M
dr, (3.5)where Wr is the probability of linking of these two molecules if their centers
of mass are separated by distanceR,NAis Avogadro’s number, andMis the
molecular weight of DNA. The equation has a simple interpretation because
the term Wr is the probability of finding a supercoiled molecule in the vol-
ume element Wr. Equation (3.5) assumes that the concentration Wr is small
enough so that we can ignore the formation of three or more linked molecules.
The probability Wr must be averaged over all possible conformations and ori-
entations of the two chains. We used the Monte Carlo method, described in
detail below, to calculate Wr. Figure 3.4 shows Wr computed for two circular
DNA molecules.
For a comparison with experimental data, it is convenient to introduce the
constantB:
B=NA
M
∫∞
0p(R)4πr^2 dr. (3.6)This allows us to expressPas
P=Bc. (3.7)