Chapter 8 – Theoretical Biophysics 343
(8.35)
R
R
R
R
R
R
G
j
j
n
G
i
G
i
n
i
j
j
n
i
n
n
R
n
n
n
=
∴
=
−
(
)
=
−
=
=
=
=
∑
∑
∑
1
1
1
1
1
2
2
1
1
2
1∑
∑
∑
∑
=
−
⋅
+
⋅
=
=
=
=
1
2
1
2
2
1
1
1
2
1
n
n
n
i
i
j
j
k
n
k
j
n
j
n
i
n
R
R
R
R
R
∑
∑
∑
∑
=
−
⋅
=
−
(
)
=
=
=
1
1
1
2
2
1
2
1
1
2
n
n
n
i
i
n
j
k
n
k
j
n
j
k
R
R
R
R
R
2
1
1 k
n
j
n
=
= ∑
∑
Using the result of Equation 8.34, we can then say
(8.36)
R
n
j
k b
b n n
n
j
n
k
n
G
2
2
1
1
2
2
1
2
2
6
1
=
−
=
−
(
)
−
(
)
=
=
∑∑
A valuable general approximation comes from using high values of n in this discrete equation
or by approximating the discrete summations as continuum integrals, which come to the
same result of
(8.37)
R
n
dj dk j
k b
b
n
dj dk j
k
nb
R
R
G
n
n
n
j
FJC
G
2
2
0
0
2
2
2
0
0
2
2
1
2
6
6
≈
−
=
−
(
) =
=
∴
∫∫
∫∫
2
2
2
6
0 41
≈
≈
R
R
FJC
FJC
.
Another useful result that emerges from similar analysis is the case of the radius of gyration
of a branched polymer, where the branches are of equal length and joined at a central node
such that if there are f such branches, the system is an f arm star polymer. This result is valu
able for modeling biopolymers that form oligomers by binding together at one specific end,
which can be reduced to
(8.38)
R
b
n
j
k j
k f
f
fb
n
j
k
G f
nif
nif
nif
j
,
−
=
+
(
)
+
(
)+
=
∫∫
∫∫
arm
d
d
d d
2
2
0
0
2
2
0
0
2
1
jj
k
−
(
)
The first and second terms represent inter- and intra-arm contributions respectively, which
can be evaluated as
(8.39)
R
nb
f
f
f
R
f
G
arm
G
–(
,
)
−
=
−
(
) =
〈
〉
2
2
2
6
3
2
Thus, at small f (= 1 or 2), α is 1, for much larger f, α is ~3/f and RG f-arm decreases roughly as
~3RG/f The FRC has similar identical stiff segment assumptions as for the FJC; however, here
the angle θ between position vectors of neighboring segments is fixed but the torsional angle
ψ (i.e., the angle of twist of one segment around the axis of a neighboring segment) is free to
rotate. A similar though slightly more involved analysis to that of the FJC, which considers
the recursive relation between adjacent segments, leads to