2 Topology in Biology: From DNA Mechanics to Enzymology 11Fig. 2.5.Interconversion of idealized branched and unbranched plectonemic super-
helices. (a) An unbranched plectonemic superhelix is in dynamic equilibrium with
branched forms, a process that is largely entropy driven. (b) Conformation of a neg-
atively supercoiled 4,600 base-pair plasmid simulated by the algorithm described
in [21]. The structure of the plasmid is clearly plectonemic and branched; the ability
of the algorithm to reproduce this property accounts for its strong predictive value
in computing equilibrium properties of random-sequence plasmids
the torsional rigidity, and an effective excluded-volume diameter for the double
helix. The Monte Carlo simulation has been extremely successful in accounting
for the bulk of available experimental data on superhelical DNA.
In addition to supercoiling, knotting, and catenation are other biologically
important topological states of circular DNA. DNA molecules can become
knotted or catenated through the action of topoisomerases and recombinases
(reviewed in [22]) and catenated DNAs are obligate intermediates in the repli-
cation of circular genomes. A major question in DNA enzymology is therefore
how cellular systems acting at the local DNA level sense the topological state
of DNA molecules, a global property, and use this information to resolve unfa-
vorable entanglements. Both knotting and catenation of a genome are serious
obstacles to normal biological function and fatal to the cell. Hence, all self-
knotting and linkage between individual genomes must be completely elimi-
nated. Even a distribution of topological states centered about the unlinked
state cannot be tolerated if a species is to be successfully propagated.
Knots and catenanes are characterized by their number and arrangement
of minimal or irreducible crossings. For knots, the number of irreducible cross-
ings is denoted Knand for catenanes the corresponding quantity is Ca.Ina