3 Monte Carlo Simulation of DNA Topological Properties 31It is therefore necessary to choose a value of
〈cosθ〉=∫π
0 cosθsinθexp(−gθ(^2) )dθ
∫π
0 sinθexp(−gθ
(^2) )dθ
that is large enough to ensure reliable results but small enough to keep the
computational time reasonable. The minimal value ofk, which provides the
limiting properties of the wormlike chain, depends on a property of interest.
Figure 3.7a shows dependence of the average〈Wr〉/∆Lk for highly supercoil-
ing DNA as a function ofk. Clearly, the results fork≥10 are nearly inde-
pendent ofk.So,k= 10 can be used for modeling DNA supercoiling. In this
case one straight segment of the model chain corresponds to≈30 bp (Kuhn
statistical length of the double helix corresponds to≈300 bp). For this value
ofkthe bending rigidity,〈Wr〉/∆Lk, equals 2.403. For another property of
circular chains, equilibrium probability of trefoil knots, the results fork=1
andk= 10 are hardly distinguishable (Fig. 3.7b). There are some properties,
however, which require much larger values ofk[14].
The excluded volume effect and electrostatic interactions between DNA
segments are taken into account in the model via the concept of effective
diameter,〈Wr〉/∆Lk. This is the diameter of impenetrable uncharged cylin-
drical segments of the model chain. The quantitative definition of〈Wr〉/∆Lk
is based on the concept of the second virial coefficient [33]. It was shown
that approximation of the electrostatic interaction by this hard core potential
Fig. 3.7.Simulation results as function ofk, the number of straight segments
per Kuhn length of the model chain. (a) Calculated value of〈Wr〉/∆Lk for highly
supercoiled DNA (3,500 base pairs in length, superhelix density of –0.05, effective
diameter of the chain equals 5 nm). (b) Probability of trefoil calculated for freely
jointed chain,k=1,(♦)andk=10(©)