Topology in Molecular Biology

2 Topology in Biology: From DNA Mechanics to Enzymology 9

static value of the persistence length,Ps. For DNA fragments that represent
a random assortment of DNA sequence elements, Schellman and Harvey [15]
derived an expression that relates the measured valueP to its component
dynamic and static contributions

1

P

=

1

Pd

+

1

Ps

. (2.6)

2.1.4 Topology of Circular DNA Molecules

The topological organization of double-stranded DNA is intimately connected
with the biology of the molecule. Most bacterial cells have circular genomes
consisting of a covalently closed double-stranded chromosome. Because the
two DNA strands are linked, scission of at least one, but usually both, of
the strands is essential for segregation of daughter chromosomes during cell
division. In addition, many aspects of DNA metabolism such as DNA synthe-
sis, transcription, and recombination generate torsional stress that leads to
underwinding or overwinding of the double helix. This torsional stress parti-
tions into local changes in the twist number of DNA (number of base pairs
per helix turn) and also global winding of the DNA helix axis, termedwrithe.
Although the genomes of eukaryotic cells consist of linear DNA molecules,
similar topological constraints apply due to the binding of architectural pro-
teins that maintain a scaffold structure within chromosomes (see below). In
eukaryotic genomes overall organization is highly complex with multiple lev-
els of DNA winding mediated by histone proteins, nucleosomal association,
and chromatin condensation. However, with approximately one gene located
in every 50,000 base pairs in the human genome [16], it is not unreasonable
to expect that activating accessibility of genes to the cell’s transcriptosome,
which involves remodeling of chromatin structure, would require maintaining
topologically independent domains every 50,000 base pairs, on average.
The topology of a covalently closed DNA molecule is described in terms
of a mathematical quantity calledthe linking number, Lk. Formally, Lk is
one-half the sum of signed crossings of the DNA single strands (see Fig. 2.4).
Lk is a topological invariant; no distortion of DNA structure short of breaking
one or both DNA strands alters Lk. Based on the work of Calugareanu [17],
White [18] and Pohl [19] showed that Lk is related to the local and global
geometry of a pair of linked space curves through the formula

Lk = Tw + Wr, (2.7)

where Tw is the total twist of the space curves about the central axis and
Wr is the self-linking number or writhe of the central axis. Because Lk is a
topological invariant, thermal fluctuations in Tw and Wr occur subject to
the constraint in (2.7). Relations between the geometry of a particular DNA
conformation and Tw and Wr, the latter in terms of the famous Gauss integral,
can be found in [20].