4 Dynamics of DNA Supercoiling 47∂n
∂t={H, n}, (4.9)whereHis the Hamiltonian of the system determined as the functional of
the energy densityn=∫Ed^3 r, whereEdepends on the states of the system
and{H, n}is the Poisson brackets. On the other hand, vectornin the case
of appropriate boundary conditions can be connected with the topological
invariant Lk
Lk(γ,γv)=
∫
(n,curln)dV. (4.10)Here dVis the volume element of the ball, embracing the band (γ,γν).
The combination of formulas (4.5) and (4.7) gives the final equation
〈∫
(n,curln)dV−q
〉
=
〈
Tw ̃〉
+
〈
̃Wr〉
. (4.11)
All these facts display the interrelation between the intrinsic properties of
DNA molecule, i.e. optical anisotropy, hydrodynamics and topology. This con-
clusion allows us to study the changes of DNA topology by monitoring the
optical properties of oriented DNA molecules.
4.2.1 Flow Linear Dichroism and Dynamics of DNA Supercoiling
The time-dependent topological transformations of supercoiled DNA can be
visualised using electrophoretic analysis of reaction mixture at different stages
of the process. However, this technique cannot provide instant and non-
disturbing quantitative kinetic analysis of reaction. A recently published pre-
cise approach based on the immobilisation of a single DNA molecule [14]
has demanded a state-of-the-art technique and cannot be utilised as a rou-
tine quantitative monitoring technique. Previously, we had first applied the
FLD method for the kinetic analysis of different nuclease reactions, using
flow-oriented supercoiled DNA molecule as a substrate [15–19]. This method
is based on the fact that oriented DNA possesses the property of optical
anisotropy [20, 21]. When the polymer molecule changes its topology, by low-
molecular weight effectors (as benzpyrene) [22] or biocatalysts [16, 23] this in-
stantly affects its hydrodynamics. This has an influence on the orientationO
and optic factorsSof DNA molecule, and leads to the alteration of the of
linear dichroism value. The principle of the method is displayed in Fig. 4.2.
The value of linear dichroismρis defined as the ratio between ∆A,dif-
ference in light absorption polarised parallel and perpendicular to the DNA
orientation axis, andA, optical density of a sample in non-polarised light [21].
For any sample containing optically anisotropic molecule,ρis a product of
orientationOand optic factorsS:
ρ=SO, (4.12)where the first one is the degree of alignment of the molecule along the ori-
entation axis, and the second one is the intrinsic optical anisotropy of the
molecules.