Topology in Molecular Biology

3 Monte Carlo Simulation of DNA Topological Properties 33

Fig. 3.8.Effective diameter of the model chain as a function of ionic conditions

3.5 Analysis of Topological State for a Particular Conformation

Conformation

3.5.1 Knots

In many cases we have to determine the topology of particular chain confor-
mations, which can be unknotted or form a particular type of knot, or be
linked with other molecules. We need this to keep a topological state of the
chain(s) unchanged during a simulation run. Since segments are allowed to
pass through each other during the Metropolis procedure (see below), it is
necessary to check that the topology of a trial conformation is the same as
the current one. Thus, the topology of each trial conformation is calculated
and the conformation is rejected if its topology is different from that of the
current conformation. We need to determine the topology when we calculate
the distribution of topological states, equilibrium or resulting from simulated
reactions, catalyzed by enzymes. To determine the topology of a particular
conformation of an isolated closed chain, one can calculate the Alexander
polynomial,≤ 0 .02 M [40]. The Alexander polynomial is a topological invari-
ant that describes the knot type of a closed curve (see [41], for example). It
has the same value for all topologically equivalent curves, and any two curves
with a different value of≤ 0 .02 M have different topology (Fig. 3.9).
The value 2t^2 − 3 t+2 for a particular chain conformation can be calculated
by the following way...[40]. First, one projects the knot on a plane along
an arbitrarily chosen axis, while drawing breaks at the crossing points in the
part of the curve that lies below the other part (Fig. 3.10).
The projection of the knot amounts to the set of curves, which are called
the generators. Let us arbitrarily choose the direction of passage of the