Polymers in Confined Geometry.pdf

Chapter 4
Simulation methods
This chapter starts with a review of the Monte-Carlo method in general and how
to calculate partition sums in the canonical ensemble. The Metropolis algorithm
is introduced as a method to implement the idea of Markov chains in computer
programs. In this context the problem of correlated sampling and the importance
of the quality of random number generators are discussed.
Then we turn to the discussion of these methods in the context of the wormlike
chain model in bulk solution and in confined environments. Finally we give
a list of the important observables which are obtained in simulations and where
they are used.
4.1 Motivation
As in most cases in physics there is no way of exploiting all interesting parameter
values of a given problem analytically. So one has to resort to simulation
techniques [4, 13, 22]. We saw in the chapters 2 and 3 that analytical results are
limited to either the very stiff and flexible polymer, or the strongly confined and
totally free environment.
For problems from biotechnology it is very important to obtain quantitative
results particularly in the intermediate regime—where up to now only qualitative
scaling arguments were available (cf. section 3.2). The problem of flexible DNA
confined to nano-channels with lateral dimensions in the order of the persistence
length, as introduced in section 3.1, is just in this complicated parameter regime.
Also the properties of semi-flexible F-Actin filaments, with a contour-length of
about the persistence length, are in this parameter range.
We therefore have to rely on simulation techniques. The Monte-Carlo simulation
is the method of choice, since it allows to effectively sample the important
conformations of the polymers. The problem of other methods, e.g. molecular dynamics
(MD) simulations, is the large difference in time-scales. Typical changes
on the monomer level are much faster than conformational changes of the whole
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