Chapter 4 – Making Light Work Harder in Biology  147

end-​to-​end length distribution is dependent on a Gaussian probability function, a Gaussian

polymer coil (see Chapter 8), and can be used to approximate P for relatively small Q:

(4.30)

KC

R

M

BC

R

Q

w

G

(θ) ^≈

+



^



^

+



^



^

1

2

1

3

2

2

The scattering vector in the case of elastic scattering can be calculated precisely as

(4.31)

Q =



^

^

4

2

π

λ

θ

sin

Therefore, a plot of KC/​R versus C at high values of θ would have a gradient approaching

~2B, which would allow the second virial coefficient to be estimated. Similarly, the gradient

at small values of θ approaches ~ 2

1

/3

B

R

Q

G

+

(

)

2

2 and therefore the radius of gyration can

be estimated (in practice, no gradients are manually determined as such since the B and RG

parameters are outputted directly from least-​squares fitting analysis). Typically, the range of

C explored varies from the equivalent of ~0.1 up to a few mg mL⁻¹.

A mixed/​polydisperse population of different types of molecules can also be monitored

using SLS. The results of Mw and RG estimates from SLS will be manifested with either multi­

modal distributions or apparent large widths to unimodal distributions (which hides under­

lying multimodality). This can be characterized by comparison with definitively pure samples.

The technique is commonly applied as a purity check in advance of other more involved bio­

physical techniques, which require ultrahigh purity of samples, for example, the formation of

crystals for use in x-​ray crystallography (see Chapter 7).

4.6.2  DYNAMIC LIGHT SCATTERING

Dynamic light scattering (DLS), also referred to as photon correlation spectroscopy (or

quasielastic light scattering), is a complementary technique to SLS, which uses the time-​

resolved fluctuations of scattered intensity signals, and is therefore described as “dynamic.”

These fluctuations result from molecular diffusion, which is dependent on molecular size. It

can therefore be used to determine the characteristic hydrodynamic radius, also known as

the Stokes radius RS, using an in vitro solution of biomolecules, as well as estimating the dis­

tribution of the molecular sizes in a polydisperse solution.

As incident light is scattered from diffusing biomolecules in solution, the motion results in

randomizing the phase of the scattered light. Therefore, the scattered light from a population

of molecules will interfere, both destructively and constructively, leading to fluctuations in

measured intensity at a given scatter angle θ as a function of time t. Fluctuations are usually

quantified by a normalized second-​order autocorrelation function g, similar to the analysis

performed in FCS discussed previously in this chapter:

(4.32)

g

I

I

t

I t

(

(

( ,

τ θ

τ θ

τ

θ

θ

, )=

, )

,

)

〈

+

(

)〉

〈

〉²

A monodispersed molecular solution can be modeled as gm:

(4.33)

g

g

DQ

m

m

(

,

τ θ

θ

β

τ

, )

)

exp(

2

=

∞

+

−

(

)

2

where gm(∞, θ) is the baseline of the autocorrelation function at “infinite” time delay. In prac­

tice, autocorrelations are performed using time delays τ in the range ~10⁻⁶ s⁻¹, and so gm(∞,