Topology in Molecular Biology

4 Dynamics of DNA Supercoiling 45

Topoisomerases are much smaller than the polymeric substrates in vivo and
their interaction with DNA is restricted within a small area (1–2 turns of
DNA helix). The enzyme cannot recognise a conformation of a whole DNA
molecule and proceeds only with a short segment of double helix (or with
single crossover formed by the helix itself).
The Gibbs energy function describing conformational transitions of scDNA
depends on the superhelical density and determines the direction of topological
transformation and its rate [6]:

∆∆G=BRTσ, (4.1)

whereσis the superhelical density (σ=∆Lk/k 0 )andBis the rigidity index
and Lkfin,Lkinitare the final and initial linking numbers. For topoisomerases
I and II Lk changes by 1 and 2 correspondingly, during every step of topoiso-
merisation (Fig. 4.1). ∆∆G becomes proportional toσ, which is independent
of DNA length.
The above-mentioned statements mean that the parameter reflecting a
degree of DNA conversion should be a parameter that has no relation to the
concentration of either of the topoisomers averaged to the topological state of
DNA under the study〈σ〉.
The introduction of〈σ〉function provides DNA topoisomerisation stud-
ies with additional opportunities of quantitative estimations. The intrinsic
links between〈σ〉and physical property of the DNA molecule, namely linear
dichroism (LD), may provide a serious experimental basis for the field.

4.2 Theory

A continuous method of statistical mechanics of biopolymers was successfully
applied to the theoretical description of the processes of DNA supercoiling
[7, 8]. For the DNA described by the “closed ribbon” model the Calugareanu–
Fuller–White formula can be written:

Lk = Tw + Wr, (4.2)

where Lk is the Gauss linking number, Tw is the twist and Wr is the writhing
number [3, 9–11]. All definitions in (4.2) are well known topologically. Let us
regardγas a closed smooth curve embedded inR^3 (Euclidean space). Then
υis a normal vector field onγ. Let us assume that the magnitude of a vector
υ(t) is so small thatυ(t) intersectsγonly at one point. The endpoint ofυ
sweeps a curveγυ, which inherits the orientation ofγ, whileυitself inherits
a strip embedded inR^3. Then the twist ofυcan be defined as follows:

Tw =

1

2 π

∫

γ

υ⊥dυ, (4.3)