The Conformon - Molecular Theory of the Living Cell - page 235

J. H. White derived a mathematical formula (known as White's formula ; see pp.
795-796 in Stryer 1995) that specifies the relation among three parameters - (1) the
linking number , Lk , the number of times the two strands are intertwined, (2) twist,
Tw, a number determined by the local pitch of the helix, and (3) writhe, Wr, a
number determined by the degree of the twisting of the helical axis in space (Bauer
et al. 1980):

Lk ¼ Tw þ Wr (8.16)

A relaxed circular DNA duplex is characterized by the lack of any writhe, that is,
Wr 0, and nonzero values for the other two parameters. As described above,

writhe can be introduced into the circular DNA duplex by first cutting the two
strands of a relaxed form and by turning counter-clockwise n times before resealing
the ends to regenerate the circular form, which can be spontaneously converted into
supercoiled form. It is important to note that Lk can be altered only through the
cutting-twisting-resealing operation, which are efficiently carried out by ATP-
dependent enzymes known as topoisomerases or DNA gyrase, and that the
remaining two parameters, Tw and Wr , can change in a mutually compensating
manner. If the linking number of a relaxed circular DNA duplex is denoted as Lk 0
and the corresponding number for a supercoiled circular DNA duplex as Lk , then
the linking number difference (symbolized as

¼

a ) can be expressed as:

a ¼

Lk Lk 0 ¼ð Tw þ Wr Þð Tw 0 þ Wr 0 Þ

¼ð Tw Tw 0 Þþð Wr Wr 0 Þ

¼ D Tw þ D Wr (8.17)

Inside the cell, DNA molecules are commonly maintained by topoisomerases in
negatively supercoiled states, making their linking number Lk smaller than their
relaxed values Lk 0 so that

can be interpreted as a
quantitative measure of conformons embedded in circular DNA (Ji 2000).
Linking number difference a can be viewed as a quantitative measure of the free
energy stored in supercoiled DNA introduced by the nicking-twisting-resealing
operation on the circular DNA duplexes. Interestingly, this mechanical energy
can be distributed either in the twist (

a ¼

Lk - Lk 0 < 0 . Therefore, a

Wr ) of the supercoiled
DNA duplex as indicated in Eq. 8.17 . The former represents the mechanical energy
stored in local deformations, while the latter indicates the same energy distributed
over the whole circular DNA duplex, and these two different states of mechanical
energy distributions may actually fluctuate between them due to Brownian motions,
thus supporting the concept that conformons are mobile mechanical energy stored
in biopolymers. It can be imagined that such conformons will visit all possible local
sites within a circular DNA duplex and a transcription factor will bind to DNA if
and only if its resident conformons happen to “collide” with the transcription
factor. We will refer to this concept as the transcription factor-conformon collision
hypothesis (TFCCH) or mechanism (TFCCM) underlying the transcription factor

D Tw ) or write ( D