46 A. Gabibov et al.
where vectorυ⊥is in the frame (t,υgυ⊥), a right-hand system, andtis the
unit tangent vector to the curveγ. The twist of the curve is a continuous
quantity. The writhing number is the integral:
Wr =1
4 π∫
γ∫
γ([dr 1 dr 2 ],r 1 −r 2 )
(r 1 −r 2 )^3, (4.4)
Wr is determined directly by the curveγ and is merely evaluated in the
experiment. It is also a continuous quantity.
The left side of formula (4.1) is the Gauss linking number:
Lk =1
4 π∫∫
γγv([dr 1 dr 2 ],r 1 −r 2 )
(r 1 −r 2 )^3, (4.5)
whereris the radius vector of the curve’s point, and [ ] and ( ) are the vector
and scalar products, respectively. The two main questions to be considered
when the real-time kinetics of scDNA biocatalytic conversion is studied are:
(1) how the equations, suitable for the “closed ribbon” DNA model, could
be applied to the more realistic “ladder” model and (2) how to estimate the
changes of the ensemble of topoisomers during the reaction time course.
Let us assume that the process of removal of supercoils takes place exclu-
sively by cutting the edges of the DNA ribbon, twisting and sewing the band.
In this case, formula (4.2) can be rearranged as:
(Lk−q)=Tw + ̃ ̃Wr. (4.6)Hereqis the number of cutting–twisting–sewing events, and Tw and Wr are
modified parameters of formula (4.2).
As (4.4) is purely topological, it is also valid for the ladder DNA model, and
all parameters can be described in terms of simplicial divisions [12]. Obviously,
we deal with the ensemble of cuts and sews, which requires the averaging over
all states:
〈(Lk−q)〉=
〈
Tw ̃〉
+
〈
( ̃Wr〉
. (4.7)
Finally this explains the applicability of topological approach for the real-
time course of the scDNA relaxation, caused by enzymatic activity of topo-
isomerases. It can be shown that intrinsic hydrodynamic behaviour of DNA
is closely related with this process. In fact the tensor parameter of order,
biaxial in general, can be reduced for the free-rotating DNA to the uniaxial
one. Then, the parameter of order as in the case of nematic liquid crystal [13]
can be described as follows:
Aik=A 0 (nink− 1 / 3 δik), (4.8)wheren=(n 1 ,n 2 ,n 3 ) is the unit vector, andδikis the Kronecker delta func-
tion. Topological characteristics of DNA are connected with hydrodynamic
equations and this equation can be presented as: