342 8.3  Mechanics of Biopolymers

without impediment) such that the directional vector ri of the ith segment is uncorrelated

with all other segments (Figure 8.5a). The end-​to-​end distance of the chain R is therefore

equivalent to a 3D random walk. We can trivially deduce the first moment of the equivalent

vector R (the time average over a long time) as

(8.31)

〈〉= 〈

〉=

−^∑

R

r

i

n

i

1

0

Since there is no directional bias to ri. The second moment of R (the time average of the dot-​

product R² over long time) is similarly given by

(8.32)

R

R R

r

r

r r

2

2

1

2

2

1

1

2

=

⋅

=

=



^



^

=

+

⋅

=

=

=

−

=

∑

∑

∑

∑

R

i

i

n

i

i

n

i

i

j

j i

n i

i

n

_

The last term equates to zero since ri · rj is given by rirj cos θij where θij is the angle between

the segment vectors, and 〈

〉=

cosθij

0 ^{over long time for uncorrelated segments. Thus, the}

mean square end-​to-​end distance is given simply by

(8.33)

R

n

nb

i

2

2

2

=

=

r

So the root mean square (rms) end-​to-​end distance scales simply with

n :

(8.34)

〈

〉=

R

nb

FJC

2

Another valuable straightforward result also emerges for the radius of gyration of the FJC,

which is the mean square distance between all of the segments in the chain. This is identical

to the mean distance of the chain from its center of mass, denoted by vector RG:

FIGURE 8.5  Modeling biopolymer mechanics. (a) Schematic of freely jointed and wormlike

chains (WLCs) indicating Kuhn (b) and persistence lengths (lp), respectively. (b) Typical fits of

WLC and freely jointed chain models to real experimental data obtained using optical tweezers

stretching of a single molecule of the muscle protein titin (dots), which (c) can undergo hyster­

esis due to the unfolding of domains in each structure. (Data courtesy of Mark Leake, University

of York, York, UK).