Chapter 9 – Emerging Biophysics Techniques 429
interactions between the CCGGGC sequence at the 3′ end and the GCCCGG
sequence at the 5′ end, such that the 3′ and 5′ ends are base paired directly
opposite to each other. This leaves the AAAAAAAAA segment that has nothing
to base pair with and so will form a single-stranded loop. The single-stranded
persistence length of 0.7 nm is equivalent to roughly two base pair separation;
thus, it is likely that the time-averaged structure will have the single-stranded
nine adenine base pairs as a circular loop.
b
The most stable lowest free energy structure has the 3′ and 5′ ends separated by
the width of the DNA double helix, which for B-DNA is 2 nm (see Chapter 2). The
normalized blue dye fluorescence will have a similar distance dependence as for
a FRET transition between a donor and acceptor FRET dye pair (see Chapter 4):
1
1 1 1
2 3 3
0 047
6
−
= −
+(
) =
εmax
/ (
/ . )
.
However, inspection of the DNA sequence suggests that it may be possible to gen
erate several structures that have fewer paired nucleotides bases, for example, by
relative transposition at integer base pair intervals from the lowest energy struc
ture of part (a) of up to five bases. The axial separation of individual base pairs is
0.34 nm. These imply separations of the dye–quencher pair of ~2.03, 2.11, 2.25,
2.42, and 2.62 nm, which is turn imply normalized blue dye intensity values of
~0.051, 0.064, 0.091, 0.135, and 0.200 by the same aforementioned argument,
and thus a sensible plot might suggest transient transitions between these states
before the stable 0.047 lowest energy state is reached.
c
If the quencher molecule is placed on the central adenine base, then the distance
between it and the blue dye for the time-average stable structure assuming a cir
cular ring of eight adenine nucleotides and six Watson–Crick base pairs is, from
simple trigonometry,
d = √
+
×
(
)+
×
(
)
(
) =
(
.
.
)
.
2
6
0 34
8
0 34 2
3 18
2
2
/
nm
π
The normalized blue dye intensity for this separation is thus given by
1 1 1
3 183 3
0 445
6
−
+(
) =
/(
.
. )
.
Once this lowest free energy structure is formed, the circular ring is likely to be
held under reasonably high tension since the persistence length is equivalent to
a two base pair separation and so large fluctuations in the circular shape, and
hence the normalized blue dye intensity, are unlikely.
Worked Case Example 9.3: Structural Coloration
When a light wave is reflected at a boundary with an optical medium of a higher refractive
index, it will undergo a phase change equivalent to 180°.
a Derive the condition 2n1d1cosθ1=(m – 1/2)λ for constructive interference between the
back-reflected light from the upper and lower surfaces of a thin film of thickness d,
assuming the light has propagated first through air and n1>1.
b A section of a butterfly wing scale was found to contain just a single thin layer of the
biopolymer protein chitin. Although the average scale size was measured at approxi
mately 170 × 50 µm with a thickness varying from 1 to 3 µm using scanning electron