4 Dynamics of DNA Supercoiling 53formation (k∼n=k 0 exp((2Bn+1)/RT)and∼kn=k 0 exp(−(2Bn+1)/RT))
(Fig. 4.4b). After formation ofTgnE∼and∼TgnE, followed by reversible
cleavage of the gate segment, the direction of topoisomerisation (i.e. sign of
the linking number change) is strictly defined by the symmetry of complex
formed. The latter process uses ATP hydrolysis as an external source of energy.
The rate of this step is assumed to be topologically independent (thektvalue
is proportional to the DNA-induced ATP hydrolysis constant).
A system of differential equations that describes concentration change of
either the reaction mixture components can be written for both cases. Al-
though it could not be solved analytically, the numerical solution is available.
Then the〈σ〉and〈ρ〉values at any time can be calculated according to the
formula:
〈σ〉=ΣgσgTn/ΣTn, (4.16)
and
〈ρ〉=Σ(gρoσn)Tn, (4.17)
whereρ(σ) – as an empiric function calculated using the dependencies shown
in Fig. 4.3 andTndenotes the concentration of DNA withnsuperturns.
The two ultimate cases ofρ(t) during DNA relaxation mediated by topo-
isomerases were obtained in the couarse of the numerical simulation of the
relaxation process. This evaluation is dependent on the enzyme type and
reaction conditions:
- After binding of the enzyme to supercoiled DNA, a fast and virtually
complete relaxation of DNA takes place. Dissociation of resulting enzyme- relaxed DNA complex occurs. This process possibly takes place in the
case of eukaryotic topoisomerase I [29] and topoisomerases II under opti-
mal conditions. If excess DNA is present in reaction mixture, only initial
(supercoiled) and final (relaxed) topoisomers exist during the reaction
course (Fig. 4.5), and (4.13) can be rearranged in (4.14):
- relaxed DNA complex occurs. This process possibly takes place in the
ρ(t)=(1−λ(t))ρinit+λ(t)ρlot, (4.18)
whereλ(t) is the level of completion of the reaction.- For enzymes that reversibly nick DNA and change the ∆Lk strictly by
2 (for euckaryotic topoisomerases II, Scheme 2) the ligation of the gap
formed occurs in the next step. Significant amounts of intermediate topoi-
somers could be registered at some conditions, and the distribution func-
tion is bell shaped with a maximum drifting to zero during reaction course
(Fig. 4.6).
In both cases theρvalue correlates to〈σ〉and unequivocally reflects the
topoisomer distribution in the reaction mixture, so FLD technique could be
used for monitoring the topoisomerisation process.