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03-871 Molecular Biophysics CD Spectroscopy September 25, 2009

Optical Rotary Dispersion and Circular Dichroism
Suggested reading: Chapter 10 van Holde et al., pp. 465-496.
: The Feynman Lectures on Physics. Vol 2, Chapter 32.
Summary:
• Circular dichroism (CD) arises because of the differential
interaction of circularly polarized light with molecules which
contain a chiral center or have coupled electric dipoles that are
chiral.
• CD is not observed unless an absorption band exists.
• CD spectra of proteins and nucleic acids are much more sensitive
to conformational changes than absorption spectra.
• CD spectra of proteins can be used to obtain useful information on
secondary structure.
Introduction: Until now we have been studying the interaction of the
transition electric dipole of molecules with the electric field of light. In
addition to this interaction, the magnetic field of light can also interact
with magnetic transition dipoles in molecules. In order for this interaction
to be productive it is necessary that the molecule possess a center of
asymmetry (such as α-carbons) or to have coupled electronic dipoles.
These centers will interact with polarized light. The consequences of this
interaction are:
λ >> λABS:
• Birefringence: (nL - nR)
• Optical Rotary Dispersion (ORD): φ = 180 l (nL - nR)/ λ
λ ≈ λABS:
• Circular dichroism (CD) differential absorption of circularly
polarized light: AL-AR
• Ellipticity: θ = 2.303 (AL-AR) 180/ 4π or ∆ε = εL – εR
Linear & Circularly Polarized Light: To understand how these effects
occur when polarized light is passed through a sample we will decompose
linearly polarized light into right and left circular-polarized light. For a
wave propagating along the y axis (j), the electric field at the origin is:
ˆ
E r = k cos ωt + iˆ sin ωt ˆ ˆ
E l = k cos ωt − i sin ωt
The sum of these two fields gives linearly polarized light.
For the case of Er the electric field vector will rotate in a right hand
direction because the x-component of the electric field grows after t=0. In
contrast El will rotate in a left hand direction.
Figure 1: Generation of circular polarized light.
Figure 1: Generation of circular-polarized light

The optical activity of a molecule is due to a differential interaction of the
molecule with right or left circularly polarized light. To help understand
the origin of optical activity consider the
following two 'molecules', which are mirror images of each other:

Figure 1: Some "lock-washer" molecules

1


Origin of Optical Activity: A flat molecule has no optical activity
because it possesses a plane of symmetry. In contrast, the helical
molecules shown below have optical activity. When a molecule is placed
in an electric field the electrons involved in the transition will oscillate
according to the direction of the electric field. In the case of the helical
molecule they can be driven in a helical path. This generates a current
loop in the helix and the current loop generates a magnetic dipole in the
same way an electromagnet generates a magnetic field. The induced
magnetic dipole is parallel with the z-component of the electric field that
generated it. The induced magnetic dipole can interact with the magnetic
component of the electric field.
• No interaction is possible with the magnetic field associated from
the z-component of the electric field. Why?
• An interaction between the induced magnetic dipole and the
electric field in the x-y plane is possible. Why?
The phase of the electric field oscillation affects the coupling between the
induced magnetic dipole and the magnetic field. For a productive
interaction between the induced magnetic dipole and the magnetic field
from the x-y electric vector the two fields must oscillate with the same
phase. Only one direction of the circularly polarized light (right or left)
will possess the correct phase, hence the reason why one of the
components of the polarized light (Er or El) interacts more strongly with
the chromophore.

2


Circular Birefringence & Optical Rotary Dispersion (ORD)
Optically active material will show a different refractive index for right
and left circular polarized light. The index of refraction of a material is a
consequence of the generation of an electric field by the dipoles within the
material which oscillate due to the applied light.
• At frequencies far removed from the absorption band the
oscillations of the applied field induce oscillations of charges in
the material which are in phase with the applied field.
• As the frequency of the applied light approaches the absorption
maximum there is an increasingly larger phase shift in the induced
oscillations.
• At higher frequencies, the induced oscillations change sign,
reversing the effects seen at low frequencies.
The oscillating electrons generate an electric field that contributes to the
field of the transmitted light. The induced field generates a difference in
refractive index for the two forms of light. The difference in refractive
index is called circular birefringence: nl - nr

Optical Rotary Dispersion (ORD) arises because of a difference in the
index of refraction for left and right circularly polarized light. Due to the
difference in refractive index the wavelength of the light for each direction
of circularly polarized light is different while the light is in the media.
Consequently, a phase shift will develop between the two circular
components. This phase shift will cause a rotation of the linearly polarized
light when it leaves the media.
180l (nl − nr )
φ= [degrees]
λ
The optical rotation as a function of wavelength is referred to the optical
rotary dispersion. The ORD curve changes sign at the absorption
maximum. This occurs because the phase of the oscillations of the
electrons becomes 180 degrees out of phase from the incident light. This
implies that the polarized light (right or left) which was retarded by the
material at longer wavelength now becomes advanced with respect to the
other polarized direction at shorter wavelengths.

Circular Dichroism occurs as the wavelength of the incident light
approaches that of the absorption band. In this case the oscillation of
charges in the material is damped as energy is removed from the field by
the absorption process. If the absorption is different for right and left
handed circular-polarized light then the linearly polarized light will
become elliptically polarized. The ellipticity (θ) of the light is defined by
the arc tangent of the ratio of the major axis to the minor axis of the
transmitted light. Usually, the actual absorption of each component of
light is measured and the difference in absorption is called the circular
dichroism (CD):

CD = Al – Ar

Which is related to the ellipticity by:
2.303( Al − Ar )180
θ= = 32.98 × ( Al − Ar )
4π

3


ER − EL
tan θ =
ER + EL

I R/ 2 − I L/ 2
1 1 Small angle approximation.
θ = 1/ 2 1/ 2
IR + IL
I = I 0 e − A ln10 Beer’s law A=-log(I/Io)
− AR − AL ∆A
ln x = ln10 log10 x
ln 10 ln 10 ln 10
e 2
−e 2
e 2
−1
θ= − AR − AL
= ∆A
+ AL ln 10
ln10 ln10 ln 10 2
+1 Multiply by e
e 2
+e 2
e 2

∆A
ln 10 ∆A
e2 ≈1+ ln 10
2 Series expansion of ex ≈1 + x

∆A Apply series expansion
ln 10
θ≈ 2
∆A
ln 10 + 2
2
∆A Assume ∆A <<1
ln 10
θ≈ 2
2
Convert to degrees.
∆A  360 
θ≈ ln 10 ×  
4  2π 

The molar ellipticity of the sample is given by:
[θ ] = 100θ
Cl
Where C is the concentration in moles/liter and l is the path-length in cm,
and θ is the measured ellipticity. The historical units of [θ] are deg cm2
dmole-1 instead of deg M-1 cm-1. Although these units seem odd, they can
be easily derived:
L mol 1000cm 3 mol
deg M −1cm −1 = deg × cm −1 = deg × = deg100cm 2 dmol −1
mol 10dmol mol 10dmol

The molar circular dichroism can also be related to the difference in the
extinction coefficient for right and left circularly polarized light:
θ = 32.98 × (ε l − ε r ) deg M −1cm
[θ ] = 3,298∆ε = 3,298 (ε l − ε r ) deg cm2 dmol-1
It is becoming more common to express CD spectra in terms of the much
more straight-forward difference in molar extinction coefficient:
∆ε = ε l − ε r

4


Applications of Circular Dichroism:
Determining Secondary Structure by CD: Various secondary
structures have characteristic CD spectra (Figure 3).

helix

beta

Figure 3: CD spectra of polypeptides (left) and CD spectra extracted from proteins.
The scale on the left has been multiplied by 10-3, i.e. 80 = 80,000.
Note that the CD spectrum (or optical activity) of α-helical residues is
significantly greater than either random coil or β-sheet. This is a
consequence of the interaction between transition dipoles oriented by the
helical structure of peptide. This interaction is weaker in β-sheet and
random coil. A similar interaction will also occur with nucleic acids (see
below).

The determination of secondary structure by CD is based on the fact that
the CD spectra of a protein is well approximated by the CD of each
peptide linkage:
N
θ λ = ∑θ iλ
i =1

Where θiλ is the CD of a single peptide bond.
There are three methods are currently used to determine the secondary
structure of a protein from it's CD spectrum.
Method 1: The simplest method is to use the CD spectra of polyamino
acids. The CD spectrum of a protein can be taken to be the linear sum of
the CD spectra from these various secondary structures:

θ λ = fαθ λ + f βθ λβ + f cθ λc
α

Where fα are the fraction of the number of residues in the protein that are
in an α-helical conformation. fβ is the fraction in the β-sheet, etc. The
experimental data are fit to the reference spectra to determine the amounts
of each secondary structure in the protein. A significant problem with this
approach is that the model compounds are usually infinite in length, thus
they do not mimic secondary structures in proteins, which are of finite

5


length. In addition, the following problems also occur with any method of
using CD to obtain secondary structure:

• random coils are seldom random
• Phe, Tyr, His, and Trp can contribute to peptide CD spectra
• Left handed helical structures can occur
• Disulfide bonds are very active
• Prosthetic groups are also very active

Method 2: A more reliable approach is to use proteins of known structure
(and CD spectra) to define the basis set. For each of the known proteins
the following is assumed:
3
θ λ = ∑ χ jθ λj
j =1

For each of the known proteins, the fractions of residues in various
secondary structures are known ( e.g. χ α ). From these data it is possible
to obtain basis spectra for the three types of secondary structure. These
basis spectra should now reflect the influence of the protein structure of
the CD spectra of residues in various secondary structures. As with the
first method, the CD spectra of the unknown protein is the weighted sum
of the reference CD spectra:
θ λ = fαθ λ + f βθ λβ + f cθ λc
α

The hope here is that the new reference spectra will be 'more accurate' than
those based on homo-polymers.

A flaw in the above approach is the assumption that there are only three
classes of secondary structure and that each class has a unique CD
spectrum.

Method 3 [Hennessey & Johnson, Biochemistry, 20, 1085.]: A more
unbiased method of approaching this problem is to extract, using
mathematical methods, a generalized basis set from the CD spectra of
proteins with known structures without regard to the actual secondary
structure of the protein. That is, we now assume the CD spectra of a
protein is:
N
θ λ = ∑ aiθ λi
i

Where N are the total number of basis spectra, ai is the weight of the ith
spectra to the total CD and θλi is the CD of the ith basis spectra at λ.
Determining the basis spectra and the coefficients, ai is accomplished by a
general technique called singular value decomposition.

SVD: Singular Value Decomposition: This is a general mathematical
technique that can be used to extract out the principle components of a
complex mixture of different spectra. Principle components are analogous
to the three basis vectors that can be used to define any vector in a three-
dimensional space.

6


SVD can be used in many different situations. As an example, say CD
spectra (or IR spectra, or UV-Vis spectra...) were acquired at 3 different
ligand concentrations and at five different wavelengths and we wanted to
extract out the 'part' of the spectra that was affected by ligand binding.
The raw experimental data can be written in matrix form (each column
represents a different ligand concentration):

 aλ 1 aλ 1 aλ 1 
a aλ 2 aλ 2 
 λ2 
A =  aλ 3 aλ 3 aλ 3 
 
 aλ 4 aλ 4 aλ 4 
 aλ 5
 aλ 5 aλ 5 


This matrix can be decomposed into a product of three matrices:
A = USV T
U: The columns of this matrix are the 'basis spectra', ui. A weighted linear
sum of these spectra can be used to generate the original raw data.
 u1 1
λ
2
uλ 1 uλ1 
3

 1 2 3 
uλ 2 uλ 2 uλ 2 
U =  uλ 3
1 2
uλ 3 uλ 3 
3

 1 2 3 
uλ 4 uλ 4 uλ 4 
 u1 2
uλ 5 uλ 5 
3
 λ5 
S: This is a diagonal matrix:
 s1 0 0
S = 0
 s2 0 
0
 0 s3 

Each entry in this matrix is associated with a basis spectrum in the U
matrix (e.g. s2 is associated with u2. The size of each si determines the
importance, or size of contribution, of this basis to the experimental data.
This can be seen by taking the product of U and S:
 uλ 1 u λ 1
1 2
uλ 1 
3
 s1uλ1
1 2
s 2 uλ 1 s3uλ1 
3

 1 3   1 3 
2
uλ 2 uλ 2 uλ 2   s1 0 0   s1uλ 2 2
s 2 uλ 2 s3uλ 2 
U × S =  uλ 3 u λ 3  0  =  s1uλ 3
1 2
uλ 3  ×  0
3
s2  
1 2
s 2 uλ 3 s3uλ 3 
3

 1 3  3 
2
uλ 4 uλ 4 uλ 4   0
 0 s3   s1uλ 4

1 2
s 2 uλ 4 s3uλ 4 
u1 u 2 3 
uλ 5   s u1 2
s 2 uλ 5 s3uλ 5 
3
 λ5 λ5  1 λ5 
VT Is a square matrix, each entry gives the contribution of each basis
vector to the signal at a particular wavelength:
 s1uλ1
1
s2uλ12
s3uλ1 
3

 1 3 
 s1uλ 2
2
s 2 uλ 2 s3uλ 2  V11 V12 V13 
U × S × V =  s1uλ 3  
s3uλ 3  × V21 V22 V23 
T 1 2 3
s 2 uλ 3
 1 3 
 s1uλ 4
2
s 2 uλ 4 s3uλ 4  V31 V32 V33 
 
 s u1 2
s2uλ 5 3 
s3uλ 5 
 1 λ5

7


For example, the measured spectra at λ1 for the 2nd ligand concentration is:
 • 
 
 
  A C2 = s1uλ1V12 + s2uλ1V22 + s3uλ1V33
λ1
1 2 31

 
 

 


Figure 4: Basis sets used to fit CD data (left). Fit to CD spectrum of papain (right).

SVD of Protein CD spectra:

Determining Secondary Structure by SVD:

We would like to obtain the secondary structure of a protein from its CD
spectrum. In matrix form, this relationship is:
F= XA
where F is a vector that is the fraction of each secondary structure, A is the
CD spectrum of the unknown protein, and X relates the two. Since we
know from SVD that there are only five significant components to the CD
spectrum of a protein we will assume that we can represent a protein in
terms of five secondary structures:

8


θ λ 1 
 f H   X 11 . . . . . 
f   θλ 2 
 ⊥β   . . . . . . 
 θ 

λ3
 f||β  =  . . . . . . ×  
    θλ4 
 fT   . . . . . . 
θ λ 5 
 fO   .
   . . . . . 
 
θ λ 6 

If we can determine X, then the fraction of each secondary structure for an
unknown protein can be found.

If we begin with a large set of proteins with known secondary structure (F)
and CD spectra (A) we can obtain X, using SVD. For a set of 3 proteins
and CD data collected at six wavelengths, the matrices would look like:
 f1 θ 1 θ 2 θ 3 
fH 2
fH   X
3
. . . . .  λ 1 λ1 λ1
3 
H
 1 3  
11 1 2
2
 f⊥ β f⊥ β f ⊥ β  . . . . . .  θ λ 2 θ λ 2 θ λ 2 
 1 2 3 
  θ 1 θ 2 θ 3 
 f ||β f f = . . . . . . ×  λ 31
λ3
2
λ3
3
|| β ||β   θ θλ 4 θ λ 4 
 1 3   . . . . . .  λ 4
θ 1 θλ 5 θ λ 5 
2
 fT fT fT  2 3
 fO 1 2 3   .
 . . . . .  λ 5
 
 fO fO  1 2 3
θ λ 6 θ λ 6 θ λ 6 
 

For this set of known proteins:

F = XA = X (USV T ) , where A has been subject to SVD, i.e.
A=USVT

X can be solved by multiplying F by V, S-1, and U:

F × V × S −1 × U T = X (USV T ) × V × S −1 × U T
= X (US ) × S −1 × U T = X (U ) × U T = X
Note:
U×UT = 1 since all U vectors are orthonormal.
V×VT = 1 from the theory of SVD.
1 / s 0 
S × S-1 =1 if we define S-1 =  1
 0 1 / s2 

Secondary Structure of Membrane Proteins:
CD can also be used to determine the secondary structure of
membrane proteins (or other aggregated systems). However, the
analysis is far from simple. Two optical effects occur in
suspensions of particles which distort CD spectra. First, the
overall absorption of the system decreases because proteins in
the particle can be "shaded" by others in the same particle.
Second, the scattering of right and left circularly polarized light
is not equal. The former problem affects the overall amplitude of
the spectrum, while the latter affects the shape of the CD
Figure 5: Effect of particle size on CD Spectrum
spectrum. These effects can be corrected to produce rather
9


reasonable CD spectra of membrane bound proteins. However, it is clear
that IR is the preferred technique for obtaining information on the
secondary structure membrane proteins.

Other Application of CD Spectroscopy to Proteins.
CD can also be used in an empirical manner to determine protein
unfolding, ligand binding, etc. CD is much more sensitive to changes in
secondary structure than absorption spectroscopy and is thus an excellent
method of following protein denaturation.

Application of CD to Nucleic acids:
The major application of CD to the study of nucleic acids is to determine
the degree of base stacking. The CD of a dimer is very dependent on the
interaction of the monomers. For example: poly C has the following
spectral properties:

Solvent Ellipticity A260
Water 35,000 1.0
Ethylene glycol 7,000 1.3

In this case both the CD and the hyper-chromicity show that polyC is a
helix in water and that this helix is due to base stacking.

DNA-Protein Interactions:

Since proteins have weak CD bands at 250nm, CD is well suited for
following protein induced changes in nucleic acid structure. These changes
can be convenient for monitoring binding of proteins to nucleic acids.

10

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