3
Monte Carlo Simulation of DNA
Topological Properties
A. Vologodskii
Summary.In the scale of hundreds and thousands of base pairs, DNA double helix
is a very flexible polymer chain that adopts many different conformations in solution.
The properties of such molecules have to be analyzed in terms of statistical mechan-
ics. Now these properties can be simulated with very good accuracy. Here we review
this simulation technique, with emphasis on topological properties of circular DNA.
We describe the basic concepts related with DNA topological properties and illus-
trate, by comparing simulation results with the experimental data, how accurately
these properties can be computed. We consider DNA model used in the simulation,
methods of sampling of the statistical ensemble, simulation of DNA supercoiling,
and different problems, related with knots and links in circular DNA. To analyze
topological state of closed chain one needs to calculate a topological invariant. We
describe the algorithms that allow one to compute one of such invariants, Alexan-
der’s polynomial, which is especially suitable for the Monte Carlo simulation. At the
end, we consider special methods of sampling for rare DNA conformations.
3.1 Introduction
It became clear in the last few years that large-scale conformational properties
of DNA can be simulated with very good accuracy. These simulations repro-
duce experimental data on hydrodynamic properties of DNA molecules [1–3],
DNA cyclization [4–7], equilibrium distributions of topological states [8–13],
elasticity of the single molecules [14, 15] and light and neutron scattering data
on supercoiled DNA [16–19]. The simulations are based on the statistical–
mechanical treatment of a well-established model of the double helix. All
parameters of the model have been reliably determined for various solution
conditions. Thus, the simulations are capable of providing reliable quantita-
tive information on many DNA properties that are hardly measurable exper-
imentally. They became an important instrument in the studies of different
protein systems that interact simultaneously with two or more DNA sites. On
the other hand, DNA molecules represent an ideal object for polymer physics,
and especially for studying topological properties of polymer chains. They