1

Theoretical Studies of

Protein Circular Dichroism

Jonathan Hirst

School of Chemistry

jonathan.hirst@nottingham.ac.uk

http://comp.chem.nottingham.ac.uk

Theory of CD

Spectroscopy

Outline

• Light + dichroism

• Electronic structure:

– 1

– 2

– Many

• Applications

• Charge transfer (deep-UV)

Let there be light

• Beam of unpolarized

light (viewed end-on)

• Beam of polarized

light (viewed end-on)

To put it planely

• plane polarized light - an

oscillation of the electric field

vector intensity in a plane that

has the same direction of the

beam (red light).

• the intensity of the electric

field vector (magenta) varies

from a positive maximum (t1)

to a negative minimum (t3)

over time.

On the other hand

• Decompose the magenta vector in two green vectors of equal intensity.

• Change in intensity of the magenta electric field vector causes these two green vectorsto rotate in opposite directions round the circumference.

• A beam of polarized light is described by the set of each of the two green vectors along the original plane polarized light direction.

• a plane polarized light can be seen as resulting from a right (clockwise) and left (anticlockwise) circularly polarized beams of light.

2

Refractive index

• Passage of light through any medium is characterized by medium’s refractive index, η

• Optically active compounds have different refractive indices for left and right circularly polarized light

• ηL ≠ ηR

• Leads to a rotation of the plane of polarization

Refractive index

• Passage of light through any medium is characterized by medium’s refractive index, η

• Optically active compounds have different refractive indices for left and right circularly polarized light

• ηL ≠ ηR

• Leads to a rotation of the plane of polarization

Dichroism – two colours (Gk.)

• Linear polarized light

– superposition of opposite

circular polarized light of

equal amplitude and

phase.

• different absorption of

the left- and right-hand

polarized component

– ellipticity (CD)

– optical rotation (OR)

• Actual effect is minute

Ellipticity

• The ratio of the minor to the major axis of the ellipse is the tangent of the angle θ (blue). This angle is called ellipticity; angle OR is called optical rotation.

• θ = 32.98 ∆ε

• To remove the dependence on cuvette path length and solute concentration, use molar ellipticity, defined as:

• [θ] = 100 θ / C l

• where C is the solute molarity and l the cuvette path length. The factor 100 as l in cm.

• Dimensions of molar ellipticity are:

• 100 deg dm3 mol-1 cm-1 = 100 deg cm3/1000 mol-1

cm-1 = deg cm2/10 mol-1 = deg cm2 dmol-1

Traditional electronic absorption spectroscopy

Selection rule: electric dipole allowed and magnetic dipole forbidden or magnetic dipole allowed and electric dipole forbidden

Intensity (oscillator strength, f )

Circular dichroism – differential absorption of left and right circularly polarised light

Selection rule: transitions are electric and magnetic dipole allowed

Intensity (rotational strength, R)

Electric dipole allowed = translation of charge

Magnetic dipole allowed = rotation of charge

Translation + rotation = helix

Circular Dichroism (CD)

2µ∝

mµ∝

Electronic excited states of amides

3

Amide electronic structure

πb

πnb

n’

n

π*

σ*

NO

NO

NO

NO

NO

nπ* transition @ 220 nm

πb

πnb

n’

n

π*

σ*

NO

NO

ππ* transition @ 190 nm

πb

πnb

n’

n

π*

σ*

NO

NO

ππ* transition

• Electric dipole allowed

• Electric transition moment

• Translation of charge

O O O+

+

µ

Molecular orbital diagram

monoamides

Gas phase Condensed phase

Pre-1996

πb

πnb

n’

n

π*σ*

Current

n’πb

πnb

n

Rydberg s/p

π*

Rydberg d

πb

πnb

n

Rydberg

π*

n’

Molecular orbital diagrams

n

πnb

π*

n

πnb

π*

Monomer 1 Monomer 2Dimer

Eππ∗ + V12

Eππ∗ - V12

πb

πnb

n

Rydberg

π*

n’

Monomer

4

Diamide Hamiltonian matrix

=

2

*

22

**

21

**

21

**

22

**

2

*

21

**

21

**

12

**

12

**

1

*

11

**

12

**

12

**

11

**

1

*

ππππππππππππ

πππππππππ

ππππππππππππ

πππππππππ

EVVV

VEVV

VVEV

VVVE

H

nn

nnnnn

nn

nnnnn

Moffitt’s dipole-dipole interaction

5

21

3

21 )()(3

r

rr

rV

••−

•=

µµµµ

∑ ∑∫∫ ≈=s t

st

ts

r

qq

rV

12

2

*

1

* ππππ ρρ

Dimer of interacting monomers

• Hamiltonian: kinetic + potential energies

• H = H1 + H2 + V12

• V12 - potential energy of interaction

5

21

3

21 )()(3

r

rr

rV

••−

•=

µµµµ

Dimer of interacting monomers

5

21

3

21 )()(3

r

rr

rV

••−

•=

µµµµ

µµµµ1 µµµµ2r

mu dot r ?

µµµµ r

θ

00cos)(90

1cos)(0

cos

22

=•⇒=⇒=

=•⇒=⇒=

+=

=•

rlarperpendicu

rrparallel

rrr

rr

yx

µθθ

µµθθ

θµµ

Dimer energies & intensities

• E+ = E0 + V12

• E- = E0 - V12

π* π*

Eππ∗ + V12

Eππ∗ - V12

• D+ = D0 + D0cosθ

• D- = D0 - D0cosθ

• D0 = µ2

Angle

between

dipoles

5

Parallel dimers

N

N

5

21

3

21 )()(3

r

rr

rV

••−

•=

µµµµ

V = µ1µ2/r3 – 3µ1rµ2r/r5

V = µ1µ2/r3 – 3µ1µ2/r

3

V = –2µ1µ2/r3 = –2µ2/r3 = –2D0/r

3

D+ = D0 + D0cos0 = 2D0

D- = D0 - D0cos0 = 0

Shift

not split

Stacked dimers

N

N

5

21

3

21 )()(3

r

rr

rV

••−

•=

µµµµ

V = µ1µ2/r3 = D0/r

3

D+ = D0 + D0cos0 = 2D0

D- = D0 - D0cos0 = 0

Anti-parallel dimers

N

5

21

3

21 )()(3

r

rr

rV

••−

•=

µµµµ

V = –µ1µ2/r3 + 3µ1rµ2r/r5

V = –µ1µ2/r3 + 3µ1µ2/r

3

V = 2µ1µ2/r3 = 2µ2/r3 = 2D0/r

3

D+ = D0 + D0cos180 = D0 - D0 = 0

D- = D0 - D0cos180 = D0 + D0 = 2D0

Anti-parallel,

stacked dimers

N

Orthogonal,

stacked dimers

N

θ = 90°

Orthogonal,

in-plane dimers

N

Angle between

dipoles is 90°; angle

between dipole and r

is 45°.

6

Combinations of monomer ππ* transitions

• In phase combination

– parallel polarisation

• Out-of-phase combination

– perpendicular polarisation

1

2

34

1

Idealised α-helix (1968)

Reproduced in Cantor &

Schimmel

Textbook calculation

Ribonuclease A, Kurapkat et al. (1997)

Don’t trust textbooks

7

CD – probe of protein 2° structure

Kemp – peptides with >100% helicity!

Previous

Ab initio

side chains

Errors over 30 proteins

JACS, 1999; JPCB, 2003

Angew Chemie, 2001; Biochemistry, 2004

Protein ECD

Protein ECD

47 proteins

Near-quantitative correlation between calculated & exptl intensity at 222 nm

- Key wavelength, reflecting helix content

- Backbone only

Highly helical, e.g. myoglobin

No helix, e.g. plastocyanin

Kemp – peptides with >100% helicity!

Previous

Ab initio

side chains

Errors over 30 proteins

JACS, 1999; JPCB, 2003

Angew Chemie, 2001; Biochemistry, 2004

Protein ECD

r ~ 0.9

Protein ECD126 excised helices – only possible in silico

“Cooperativity” – residues in longer helices contribute more per residue to [θ]222

End effect – k ~ 3;

Infinite helix [θ]222 ~-40,000

Unfolding a helix by fraying ends reduces [θ]222 less than breaking in the middle

Kemp – peptides with >100% helicity!

Previous

Ab initio

side chains

Errors over 30 proteins

JACS, 1999; JPCB, 2003

Angew Chemie, 2001; Biochemistry, 2004

Protein ECDk ~ 3;

[θ]-inf ~-40000r ~ 0.9

8

Kemp – peptides with >100% helicity!

Protein ECD

[θ]222 for infinite helix ~ -40,000

Kemp peptides show -55,000 !?

Exptl conditions (solvent, low T) favour short h-bonds?

Test in silico –build models, compute CD

Variation of computed [θ]222

for different regular helices (310, α and π)

Kemp – peptides with >100% helicity!

Previous

Ab initio

side chains

Errors over 30 proteins

JACS, 1999; JPCB, 2003

Angew Chemie, 2001; Biochemistry, 2004

Protein ECDk ~ 3;

[θ]-inf ~-40000r ~ 0.9

H-bond distance important

Previous

Ab initio

side chains

Errors over 30 proteinsCASSCF/CASPT2 studies of aromatic side chains �parameters for matrix method, which improve accuracy in near-UV

Protein near-UV ECD

Origin of band @165 nm?

• πσ*? nσ*? n’π*?

Matsuo et al. J Biochem (2004) Oakley & Hirst J Am Chem Soc 128 12414 (2006)

Bulheller et al. J Phys Chem B 112 1866 (2008)

Charge Transfer

φ ψN

O

NO

NO

CASSCF/CASPT2 calculations on dipeptides with 10 important geometries from across the Ramachandran plot

Protein deep-UV ECD

Serrano-Andres & Fulscher J Am Chem Soc 1998120, 10912; Tozer et al. Mol Phys 1999, 97, 859

Protein deep-UV ECD

9

Charge transfer in the matrix method

Protein deep-UV ECD

Gilbert ATB & Hirst JD J Mol Struct (THEOCHEM) 2004, 675, 53

Protein deep-UV ECD

Correlation between computed and exptl intensity for a set of 71

structurally diverse proteins;

Exptal data from Bonnie Wallace

(Birkbeck)

πnb1-π*2 excitation important in the β-sheet / PP-II region.

N-lobe of the πnb1 orbital and C-lobe of the π*2 orbital are in close proximity.

Key transitions

The n1-π*2 transition occurs at low energy and is fairly strong in the α-helical region.

The n1 orbital and the C-lobe of the π*2 orbital are in close proximity and good alignment.

Protein CD - Summary

• Semi-quantitative – more to

do, but can …

• Probe structure-spectra

relationships computationally

– Helices

– Conformational dynamics

• Sidechains – near UV

• Charge transfer – deep UV

http://comp.chem.nottingham.ac.uk/dichrocalc

Bulheller BM, Rodger A & Hirst JD. Phys Chem Chem Phys, 9, 2020 (2007).

Further reading

• Fasman Circular dichroism and the

conformational analysis of biomolecules

• Charney The molecular basis of optical

activity

• Cantor & Schimmel Biophysical Chemistry

Part II Techniques for the study of

biological structure and function

Acknowledgments

Ben BulhellerLilianna Bryjko