Biomolecular Spectroscopy

1
Biomolecular Spectroscopy
Quantum mechanics The particle in a
box Electronic states and energies Transitions
between states Absorption and emission Electronic
spectroscopy
2
Postulates of quantum mechanics are assumptions
found to be consistent with observation
The first postulate states that the state of a
system can be represented by a wavefunction
Y(q1, q2,.. q3n, t). The qi are coordinates of
the particles in the system and t is time. The
wavefunction can also be time-independent or
stationary, y(q1, q2,.. q3n).
3
Postulate 2. The probability of finding a
particle in a region of space is given by
Postulate 2. Assumptions 1. YY is real (Y is
Hermitian). 2. The wavefunction is normalized. 3.
We integrate over all relevant space.
4
Normalization is needed so that probabilities are
meaningful.
Normalization means that the integral of
the square of the wavefunction (probability
density) over all space is equal to one.
The significance of this equation is that
the probability of finding the particle
somewhere in the universe is one.
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Postulate 3. Every physical observable is
associated with a linear Hermitian operator
Observables are energy, momentum,
position, dipole moment, etc.
The fact that the operator is Hermitian
ensures that the observable will be real.
6
Postulate 4. The average value of a physical
property can be calculated by
Normalization
7
Postulate 5. The calculation of a physical
observable can be written as an eigenvalue
equation
This is an operator equation that returns
the same wavefunction multiplied by the constant
P. P is an eigenvalue. An eigenvalue is a number.
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Postulate 6. The form of the operators is
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Stationary State Wave Equation Quantum
Mechanical Description
Hamiltonian Energy Operator
Eigenvalue Energy value
Wavefunction
The Hamiltonian and wavefunction are
time-independent
10
The wavefunction is composed of electronic and
nuclear parts
Nuclear
Total
Electronic
The wavefunction represents the probability
amplitude of electrons and nuclei.
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The wave equation can be separated into
electronic and nuclear parts
Hamiltonian Energy Operator
Eigenvalue Energy value
Wavefunctions
12
The Hamiltonian contains both kinetic and
potential energy terms
Kinetic Energy
Potential Energy
13
The solution to the wave equation for the
hydrogen atom gives rise to energy levels that
depend on quantum numbers
R is called the Rydberg constant. R 13.6 eV
(electron volts) R 107,900 cm-1 (wavenumbers)
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The hydrogen atom has no nuclear part and the
electronic solutions give rise to atomic orbitals
d
s
p
These are the angular parts of the wavefunction.
The radial part decays exponentially with
distance from the nucleus of an atom.
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The Particle in a Box
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The kinetic energy operator for a
time-independent system
q is a generic coordinate x, y, or z m is the
mass P hat is the momentum
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The Schrödinger equation for a free particle
The solutions are
eikq
e-ikq
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The particle in a box has boundary conditions
Y(0) 0
Y(a) 0
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The solutions to the particle in a box
y2
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The boundary conditions determine the values for
the constants A and B
sin will vanish at 0 since x 0 and sin 0
0. sin will vanish at a if ka np. Therefore, k
np/a.
Not Normalized !
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ElectronicStates and Energies
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The molecular orbitals in a diatomic molecule are
formed from linear combinations of atomic orbitals
Bonding s
Anti-Bonding s
s
s
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Constructive overlap between two atomic orbitals
gives rise to a bonding state
NO NODES
Ys Ys gt 1/Ö2(Ys
Ys)
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Destructive overlap between two atomic orbitals
gives rise to an anti-bonding state
ONE NODE
Ys Ys gt 1/Ö2(Ys -
Ys)
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For diatomic hydrogen we consider the s and s
molecular orbitals in the following energy diagram
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The highest occupied molecular orbital of ethylene
p
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The lowest unoccupied molecular orbital of
ethylene
p
p
1 p node
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We can construct molecular orbitals of benzene
using the six electrons in p orbitals
Electrons are spin-paired
Benzene Structure
Electronic Energy Levels
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An electronic wavefunction corresponds to each
energy level
NODES
3 2 1 0
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Transitions between states
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A time-dependent formalism permits transitions
between stationary states
Electronic and vibrational wavefunctions
considered thus far are stationary. In order
to observe a transition, there must be a
time-dependent change in the system. We call
this change a perturbation. In order to drive
the system from state 1 to state 2
the perturbation must occur with angular
frequency w12.
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Time Dependent Wave Equation Quantum Mechanical
Description
Hamiltonian Static/Time-dependent
Time-dependent Energy operator
The Hamiltonian and wavefunction are
time-dependent
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There is a relationship between time-dependent
and static wave functions
The total Hamiltonian is composed of two parts
Total Static Time-dependent
We can use the time-dependent Hamiltonian
to connect static wavefunctions of different
states.
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The transition probability calculated using the
Fermi Golden Rule
This expression is derived using
time-dependent perturbation theory. It is valid
for a number of time-dependent processes that
involve transitions between states 1 and 2. The
rate constant k12 P12/t.
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The Fermi Golden Rule can be used to calculate
many types of transitions
Transition H(t) dependence 1. Optical
transitions Electric field 2. NMR
transitions Magnetic field 3. Electron
transfer Non-adiabaticity 4. Energy
transfer Dipole-dipole 5. Atom
transfer Non-adiabaticity 6. Internal
conversion Non-adiabaticity 7. Intersystem
crossing Spin-orbit coupling
36
Optical electromagnetic radiation permits
transitions among electronic states
where m is the dipole operator and the
dot represents the dot product. If the dipole m
is aligned with the electric vector E(t) then
H(t) - mE(t). If they are perpendicular
then H(t) 0.
where e is the charge on an electron and q is the
distance.
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The time-dependent perturbation has the form of
an time-varying electric field
where w is the angular frequency. The electric
field oscillation drives a polarization in an
atom or molecule. A polarization is a coherent
oscillation between two electronic states. The
symmetry of the states must be correct in order
for the polarization to be created. The
orientation average and time average over the
square of the field is -m.E(t)2 is m2E02/6.
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Absorption of visible or ultraviolet radiation
leads to electronic transitions
Polarization of Radiation
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Absorption of visible or ultraviolet radiation
leads to electronic transitions
Transition moment
The change in nodal structure also implies a
change in orbital angular momentum.
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The interaction of electromagnetic radiation with
a transition moment
The electromagnetic wave has an angular momentum
of 1. Therefore, an atom or molecule must have a
change of 1 in its orbital angular momentum to
conserve this quantity. This can be seen for
hydrogen atom
Electric vector of radiation
l 0
l 1
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The Fermi Golden Rule for optical electronic
transitions
The rate constant is proportional to the
square of the matrix element elt Y1q Y2gt times a
delta function. The delta function is an
energy matching function d(w - w12) 1 if w
w12 d(w - w12) 0 if w ¹ w12.
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Separation of electronic and nuclear parts of the
transition moment
The transition moment, -eltY1qY2gt can
be separated into the electronic wavefunction
y that depends on q and the nuclear
wavefunction c that does not. These enter the
rate expression as the square
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The electronic transition moment
The electronic transition moment is M12
-elty1qy2gt The importance of the transition
moment in absorption spectroscopy is that it can
be used to about the conformation of
macromolecules. Light will be absorbed when the
electric vector is aligned with the transition
moment. Light will not be absorbed when the
electric vector is perpendicular to the
transition moment.
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Transition moment of lowest p-p transition of
ethylene
p
p
The transition moment is perpendicular to the
change in nodal structure. Electromagnetic
polarized along this direction will give the
maximum transition probability.
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The electronic transition moment determines the
intensity of an absorptive transition
The p and p states are
The transition moment is
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The magnitude of the transition moment for C2H4
can be calculated from a simple model
where x1 - x2 is the CC bond length. One charge
displaced through 1 Å has a dipole moment of 4.8
D. Mp-p 3.24 D for C2H4.
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The Franck-Condon factor determines the envelop
of the absorption lineshape
D
D
D
S D2/2 S is electron-phonon coupling D
is nuclear displacement
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The Franck-Condon factor is due to the overlap of
ground state v0 with excited state v0, 1,
etc.
Excited state
0-0
Ground state
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The Franck-Condon factor is due to the overlap of
ground state v0 with excited state v0, 1,
etc.
Excited state
0-1
Ground state
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The Franck-Condon factor is due to the overlap of
ground state v0 with excited state v0, 1,
etc.
Excited state
0-2
Ground state
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The Franck-Condon factor is due to the overlap of
ground state v0 with excited state v0, 1,
etc.
Excited state
0-3
Ground state
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The Franck-Condon factor is due to the overlap of
ground state v0 with excited state v0, 1,
etc.
Excited state
0-4
Ground state
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Based on the FC factors we can construct a
stick spectrum
0-1
0-2
0-3
0-0
0-4
Calculated assuming E(0-0) 8000 cm-1 and
vibrational mode of 1000 cm -1. 1 eV 8065.6
cm-1.
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The broadening of the stick lineshapes may be
either Gaussian or Lorentzian
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The Lorentzian lineshape function
The delta function implies that the
transition linewidth is infinitely narrow and
this is clearly not physical. We can replace the
delta function with a Lorentzian function The
levelwidth G is due to lifetime broadening and
pure dephasing whose combined times are called
T2. This is called homogeneous broadening.
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A Lorentzian is the Fourier transform of an
exponential
The origin of the Lorentzian form arises from
the fact that the excited state has a finite
lifetime and dephasing time. In analogy with NMR
these are T1 and T2, respectively. The overall
exponential decay time is T2 and the form is
e-t/T2.
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The Gaussian lineshape function
We can represent site broadening by a
Gaussian function. Site broadening means that
different molecules in a sample have different
environ- ments. This kind of broadening is also
known as inhomogeneous broadening. The
additional broadening s represents the spread in
energy due to different solvent and protein
configurations that molecules experience.
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The finite T2 time resultsin a spectral
broadening
Finite excited state lifetime and dephasing
Infinite excited state lifetime and dephasing
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Transition rate constants can be calculated using
the Fermi Golden Rule
Since experiments are not reported in
angular frequency we can also express this
transition rate constant as (w 2pn)
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Expression of the rate constant in terms of
intrinsic molecular properties
The rate constant depends on the intensity
of radiation since The integrated FC factor is
equal to one. Thus, the rate constant can be
expressed as
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Connection with experiment
Beers law states that
e(n) is the molar extinction coefficient. C is
the concentration. (NOTE c is speed of light) x
is the pathlength. In differential form this is
written
A comparable expression in terms of the
individual transition rates is given by
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Beer-Lambert Law
A is the absorbance. D is the pathlength. The
exponential attenuation of the intensity is shown
in the Figure. The absorption cross section for
an individual molecule is s. s hnk12.
x
x
dx
I0
I
I
IdI
d
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Experimental determination of the transition
moment by absorption spectroscopy
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Absorption and emission
The intrinsic coefficient for absorption B12 is
related to k12 NB12r, where r is the energy
density Einstein showed that the rate of
absorption and stimulated emission are equal.
The spontaneous emission rate has a definite
relation to the stimulated emission rate
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Spontaneous emission is fluorescenceStimulated
emission is used for lasers
spontaneous
stimulated
N1B12r
N2A21
N2B21r
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Fluorescence usually occurs after vibrational
relaxation
1. Absorption 2. Vibrational
relaxation 3.Fluorescence
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The Franck-Condon principleTransitions are
vertical in both absorption and emission
1. Absorption 2. Vibrational
relaxation 3.Fluorescence
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The Franck-Condon factor is the same for
absorbance and fluorescence
1. Absorption 2. Vibrational
relaxation 3.Fluorescence
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This leads to a mirror image relationship
between absorption and fluorescence bands
Fluorescence
Absorption
Energy
0-1
0-2
1-0
2-0
0-3
3-0
0-0
4-0
0-4
Wavelength
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Biopolymers
71
The carbonyl group has n - p and p - p
transitions
The p system of the carbonyl group can be
compared to that of ethylene. The non- bonding
orbitals on O consist of n and n. The lowest
electronic transition is n p. The next
important transition is p p.
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Determination of the direction of the transition
moment in the molecule reference frame
Single crystal polarized absorption spectroscopy
is used to determine the transition
moment direction. This is important since the
electric vector of the light must be aligned
with the transition moment to maximize an
absorption signal.
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Peptide absorbance
____ random _ _ _ b-sheet - - - a-helix
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Interactions between the neighboring amide
groups result in changes in the absorption
spectrum.Exciton interactionb-sheet
a-helix
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Exciton interactions
For example, if we consider two neighboring amide
groups The singly excited wavefunctions are
f1 c1c20 and f2 c10c2. It is these
wavefunctions that interact to give excitonic
transitions.
Consider the excitation of each dipole c10 c1
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Exciton energies and wavefunctions
The exciton wavefunctions arise from linear
combinations of the excited state wavefunctions.
In a random coil peptide there is little
order and the amide transitions are
independent. In an a-helix, the periodic
arrangement of amides gives rise to two excitonic
bands.
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Dipole-dipole coupling
The dipole-dipole interactions of transition
moments are known as exciton interactions. There
is no exchange of electrons between neighboring
groups, but there is a through space Coulombic
dipole-dipole interaction
f1 and f2 are excited state wavefunctions.
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Exciton transition moments
We can form linear combinations of the transition
moments.
Since these add as vectors the direction of m- is
nearly orthgonal to that of m. The
examples shown here for two moments can be
extended to n moments, but the basic principle is
the same.
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Aromatic amino acids
The amino acids phenylalanine, tyrosine, and
tryptophan have p-p transitions.
Note the pattern of weak bands from 240 - 300
nm and much more intense bands between 190 - 220
nm. The weak bands are allowed by vibronic
coupling (L).
Wavelength (nm)
80
Phenylalanine and tyrosine
The amino acid phenylalanine has an absorption
spectrum that resembles toluene. Tyrosine
resembles p-methylphenol. Using benzene as a
model we can explain the the weak and strong
transitions. There is configuration interaction
that gives rise to a splitting of the lowest
transition.
B
L
Splitting due to CI
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Configuration interaction
Configuration interaction is the mixing
and splitting of two or more electronic
transitions that have the same symmetry on a
single molecule. It has a resemblance to
excitonic coupling, but they are not the same
thing. Configuration interaction is very
important for understanding the spectroscopy of
polyenes such as b-carotene, retinal etc. and
porphyrins, such as heme, chlorophyll, etc.
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Tryptophan
The two lowest absorption bands in tryptophan are
ascribed to the long and short axes of the indole
ring. A particle on a circle model can be used
for spectra of aromatic systems. The weak band at
280 nm is 1Lb and the strong band at 220 nm is
1Ba,b. The weak band has a large orbital
angular momentum change. The strong band has Dm
1.
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DNA base absorption spectra
The important electronic transitions of A, G, C,
and T are mostly p-p in nature although
there are N and O lone pairs that contribute to
n-p transitions that are buried under the p-p
. Single crystal polarized absorption
spectroscopy is used to determine the transition
moment direction. Knowledge of these is
important since the electric vector of the light
must be aligned with the transition moment to
maximize an absorption signal.
84
DNA base absorption spectra
Adenine
Guanine
Cytosine
Thymine
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DNA hypochromism
- - - Bases ___ DNA
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Applications and mechanism of DNA hypochromism
DNA hypochromism is useful for determining the
hybridization/melting of DNA. While the origin
of the effect is at least partly excitonic base
stacking also contributes by creating
a hydrophobic environment for the bases.
Since water is excluded the dielectric
environment is quite different in DNA and this
may have an effect on the absorption spectrum as
well.
87
Circular dichrosim
At a given wavelength, ?A AL AR where ?A is
the difference between absorbance of left
circularly polarized (LCP) and right circularly
polarized (RCP) light. It can also be expressed,
by applying Beers law, as ?A (eL
eR)Cl Where eL and eR are the molar extinction
coefficients for RCP and LCP light, C is the
molar concentration l is the path length in
centimeters (cm). Then, ?e (eL eR) is the
molar circular dichroism. This is what is usually
meant by the circular dichroism of the
substance. Although ?A is usually measured, for
historical reasons most measurements are reported
in degrees of ellipticity. Molar circular
dichroism and molar ellipticity, ?, are
readily interconverted by the equation ?
3298.2 ?e
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Circularly Polarized Light Circularly
polarised light can be described in terms of
electric (e) and magnetic (m) wave components.
Linearly and circularly polarized light are
contrasted below.
89
Elliptical polarized light (purple) is composed
of unequal contributions of right (blue) and
left (red) circular polarized light. This
relationship is derived by defining the
ellipticity of the polarization as where ER
and EL are the magnitudes of the electric field
vectors of the right-circularly and
left-circularly polarized light,
respectively. When ER equals EL (when there is
no difference in the absorbance of right- and
left-circular polarized light), ? is 0 and the
light is linearly polarized. When either ER or EL
is equal to zero (when there is complete
absorbance of the circular polarized light in
one direction), ? is 45 and the light is
circularly polarized.
90
Generally, the circular dichroism effect is
small, so tan? is small and can be approximated
as ? in radians. Since the intensity or
irradiance, I, of light is proportional to the
square of the electric-field vector, the
ellipticity becomes
(q
in radians) Then by substituting for I using
Beers Law in natural logarithm form I
I0e-Aln10 The ellipticity can now be written
as
(q in
radians) Since ?Altlt1, this expression can be
approximated by expanding the exponentials in a
Taylor series to first-order and then discarding
terms of ?A in comparison with unity and
converting from radians to degrees
91
The linear dependence of solute concentration and
pathlength is removed by defining molar
ellipticity as, Then combining the last two
expression with Beers law, molar ellipticity
becomes
92
Circular Dichroism Units There are
several different units of measurement for
circular dichroism. Molar ellipticity, mean
residue ellipticity and delta epsilons are all
mentioned in the literature. Ellipticity is
defined as the tangent of the ratio of the minor
to major elliptical axes. More modern CD
instruments measure the difference in absorption
of right and left circularly polarized light as
a function of wavelength. In accordance with the
BeerLambert law, wavelength is equal to the
difference in molar extinction coefficients
divided by the product of path length and
concentration. Mean residue ellipticity is the
most common unit (degree cm2 dmol1) and delta
epsilons are the new machine unit, often
referred to as molar circular dichroism (liter
mol1 cm1), not to be confused with molar
ellipticity (degrees decilitres mol1
decimeter1).
93
Application to biological molecules In
general, this phenomenon will be exhibited in
absorption bands of any optically active
molecule. As a consequence, circular dichroism
is exhibited by biological molecules, because of
the dextrorotary (e.g. some sugars) and
levorotary (e.g. some amino acids) molecules they
contain. Noteworthy as well is that a secondary
structure will also impart a distinct CD to its
respective molecules. Therefore, the alpha Helix
of proteins and the double helix of nucleic acids
have CD spectral signatures representative of
their structures. The ultraviolet CD spectrum of
proteins can predict important characteristics
of their secondary structure. CD spectra can be
readily used to estimate the fraction of a
molecule that is in the alpha-helix conformation,
the beta-sheet conformation, the beta-turn
conformation, or some other (e.g. random coil)
conformation.
94
These fractional assignments place important
constraints on the possible secondary
conformations that the protein can be in. CD
cannot, in general, say where the alpha helices
that are detected are located within the
molecule or even completely predict how many
there are. Despite this, CD is a valuable tool,
especially for showing changes in conformation.
It can, for instance, be used to study how the
secondary structure of a molecule changes as a
function of temperature or of the concentration
of denaturing agents. In this way it can reveal
important thermodynamic information about the
molecule that cannot otherwise be easily
obtained. CD spectroscopy is a quick method,
that does not require large amounts of proteins
and extensive data processing. Thus CD can be
used to survey a large number of solvent
conditions, varying temperature, pH, salinity
and presence of various cofactors.
95
Protein secondary structure from UVCD
The most important signatures are those of the
a-helix, b-sheet and random coil, shown below.
R
b
a
96
Protein secondary structure from UVCD

• Far UV-CD of random coil (RC)
• positive at 212 nm (p-gtp)
• negative at 195 nm (n-gtp)
• Far UV-CD of b-sheet
• negative at 218 nm (p-gtp)
• positive at 196 nm (n-gtp)
• Far UV-CD of a-helix exiton coupling of the p-gtp
  transitions
• leads to positive (p-gtp)perpendicular at 192
  nm and negative
• (p-gtp)parallel at 209 nm negative at 222 nm is
  red shifted (n-gtp)