3 Monte Carlo Simulation of DNA Topological Properties 393.7.2 Simulation of DNA Conformations with Low Probability
of Appearance
The Chain of Conditional Probabilities
Although modern computers allow one to perform up to 10^9 moves for a model
chain corresponding to DNA molecule a few kb in length, this may not be suf-
ficient to evaluate probabilities of some rare conformations with a reasonable
statistical error. This can be the case for juxtapositions of specific sites or
DNA ends, for example. Two methods have been developed to overcome this
problem.
The first method enables one to calculate values ofj-factors for short DNA
fragments, about 200 base pairs, when these values are less than 10−^8 Mand
direct Monte Carlo simulation is inefficient. Suppose we want to estimate
P(r 0 ), the probability of the conformations with end-to-end distance,r, less
than a small valuer 0. We choose a sequence of distancesr 0 <r 1 <···<rn,
wherern is larger or equal to the chain contour length. LetP(ri)bethe
probability of conformations withr<ri. We can also define the conditional
probabilities,P(ri|ri+1), of conformations withr<riin the subset of confor-
mations withr<ri+1. SinceP(ri)=P(ri|ri+1)P(ri+1)andP(rn) = 1, the
value ofP(r 0 ) can be found as
P(r 0 )=n∏− 1i=0P(ri|ri+1). (3.13)The sequence of distancesr 0 <r 1 < ···<rncan be chosen so that
allP(ri|ri+1) values are relatively large. This can be always achieved since
P(ri|ri+1) approaches 1 whenri+1approachesri. The large values ofP(ri|ri+1)
can be efficiently and accurately calculated by the Metropolis procedure. Each
P(ri|ri+1) is calculated as the fraction of the conformations withr<riin the
subset of equilibrium conformations withr<ri+1. These subsets are gener-
ated in the Monte Carlo procedure by rejecting any trial conformation with
r>ri+1. The valuesP(ri|ri+1) are calculated sequentially fromP(r 0 |r 1 )to
P(rn− 1 |rn). The starting conformation for each subset is the last conforma-
tion from the previous subset. The calculation ofP(r 0 |r 1 ) is started from a
conformation withr= 0. The estimation shows that the best efficiency in
estimatingP(r 0 ) is achieved when the values ofP(ri|ri+1) are close to 0.2.
Using this approach, one can speed up the computations by a few orders of
magnitude compared to the direct Monte Carlo procedure [39, 52].
Umbrella Method
The umbrella method [53] addresses calculation of conformational distribu-
tions under the condition that specific sites are juxtaposed in a proper ori-
entation. It is based on introducing an artificial potential,U(x), wherex