344 8.3  Mechanics of Biopolymers

(8.40)

〈

〉=

+

−

−

−(

)

−

(

)



^



^













R

nb

n

n

2

2 1

2

1

1

1

1

cos

cos

cos

cos

θ

θ

θ

θ

For high values of n, this equation can be reduced to the approximation

(8.41)

cos

cos

cos

cos

〈

〉≈

+

−



^



^

∴〈

〉≈

−





R

nb

R

b n

FJC

2

2

2

1

1

2

1

θ

θ

θ

θ





^≡

〈

〉( )

R

g

FJC

2

θ

This additional angular constraint often results in better agreement to experimental data, for

example, using light scattering measurements from many proteins (see Chapter 3) due to real

steric constraints due to the side groups of amino acid residues whose effective value of g is

in the range ~2–​3.

8.3.2  CONTINUUM MODEL FOR THE GAUSSIAN CHAIN

The GC is a continuum approximation of the FJC (which can also be adapted for the FRC)

for large n (>100), applying the central limit theorem to the case of a 1D random walk in

space. Considering n unitary steps taken on this random walk that are parallel to the x-​axis,

a general result for the probability p(n, x) for being a distance x away from the origin, which

emerges from Stirling’s approximation to the log of the binomial probability density per unit

length at large n, is

(8.42)

p n x

n

x

n

x

x pdx

n

,

(

) =

−



^



^

∴〈

〉=

=

−∞

+∞

∫

1

2

2

2

2

2

2

π

exp

This integral is evaluated using as a standard Gaussian integral. Thus, we can say

(8.43)

p n x

x

x

x

,

(

) =

〈

〉

−

〈

〉



^



^

1

2

2

2

2

2

π

exp

However, a FJC random walk is in 3D, and so the probability density per unit volume p(n, R)

is the product of the three independent probability distributions in x, y, and z, and each step

is of length b, leading to

(8.44)

p n R

R

R

R

nb

,

/

/

(

) =

〈

〉



^



^

−

〈

〉



^



^{= }

^



^

3

2

3

2

3

2

2

3 2

2

2

2

3 2

π

π

exp

exp ⁻



^



^

3

2

2

2

R

nb

For example, the probability of finding the end of the polymer in a spherical shell between

R and R +​ dR is then p(n, R) · 4πR²dR. This result can be used to predict the effective hydro­

dynamic (or Stokes) radius (RH). Here, the effective rate of diffusion of all arbitrary sections

between the ith and jth segments is additive, and since the diffusion coefficient is propor­

tional to the reciprocal of the effective radius of a diffusing particle from the Stokes–​Einstein

equation (Equation 2.12), this implies