38 A. Vologodskii
whereφmis the angle of the projection ofrmon thex, yplane with thex
axis. The perpendicular to the plane formed by vectorsrmandrm+1makes
the angleξm(0≤ξm≤π/2) with thezaxis, and its projection on thex, y
plane makes the angleχmwith thexaxis.
3.7 Simulation Procedure
3.7.1 General Approach
The Metropolis–Monte Carlo procedure [51] is usually used for statistical sam-
pling of the chain conformations. The procedure consists of consecutive steps
that include displacements of certain parts of the chain. One may use different
displacements in the procedure. Usually the displacement consists of the ro-
tation of an arbitrary number of adjacent segments by a random angle within
the interval (−φ 0 ,+φ 0 ) around the straight line connecting two randomly
chosen vertices,m 1 andm 2 (Fig. 3.14) [30].
Theφ 0 value is adjusted during the simulation in such a way that about
one-half of the steps would be successful. The rate of exchange between differ-
ent conformations of a supercoiled chain can be further increased ifφ 0 depends
on the distance between verticesm 1 andm 2. To increase the rate of sampling
in the simulation of supercoiled DNA we also introduced more complex type
of motion, the so-called reptation motion [32]. Additional type of motion must
be introduced for linear chains [14]. In general, one can introduce any type of
motion to increase the rate of sampling as long as it does not interrupt the
principle of microscopic equilibrium [51].
Whether the new conformation is accepted is determined by applying the
rules of the procedure: (a) If the energy of the new conformationEnewis lower
than the energy of the previous conformation,Eold, the new conformation is
accepted, (b) If the energy of the new conformation is greater than the energy
of the previous conformation, then the new conformation is accepted with the
probabilityp=exp[(Eold−Enew)/kBT].The starting conformation is chosen
arbitrarily.
Fig. 3.14.The major type of displacement in the course of the Metropolis procedure
described in the text