Title: Inelastic XRay Scattering Studies of Phonon Propagation and Damping in Biomolecular Assemblies
1Inelastic X-Ray Scattering Studies of Phonon
Propagation and Damping in Biomolecular Assemblies
Sow-Hsin Chen (sowhsin_at_mit.edu)
Department of Nuclear Science and Engineering,
MIT, Cambridge, MA, USA
Collaborators Y. Liu (MIT), D. Liu (MIT), D.
Berti (U, Florence), E. Fratini (U, Florence),
P. Baglioni (U, Florence), H. Sinn (APS), A.
Alatas (APS), A. Said (APS), E. Alp (APS)
Research supported by Material Sciences Program,
BES, US DOE
An invited lecture given at NSLS-ll Workshop on
IXS, BNL, Feb. 7th-8th, 2008
2Advantages of Inelastic X-Ray Scattering for
Investigation of Collective Modes in Bio-Materials
3Advantages of Inelastic X-Ray Scattering for
Investigation of Collective Dynamics in
Bio-Materials
4Three Effective Eigenmode (TEE) Model
The dynamic structure factor is given as,
with and
zu(k),fuT(k),zT(k) all real numbers.
Hydrodynamic limit (k?0)
The three eigenvalues, to the order of O(k2), are
where is the sound
damping.
5For finite k, zu(k),fuT(k),zT(k) become arbitary
functions of k. However, in most cases, the
eigenvalues of the matrix H consist of one real
number zh and a couple of conjugate complex
numbers ?s i?s. One can therefore write the DSF
in general in the hydrodynamic-like form
Damped Harmonic Oscillator limit (fuT(k)0)
The amplitude of the central peak of S(k,?) is
zero, the side peaks can be written as
The practical DHO model
6Generalized Dynamic Structure Factor for
Polyatomic Molecules
The Dynamic Structure Factor can be written as a
weighted sum of the partial dynamic structure
factor
where is the atomic form factor of
the ?-species of atoms .
S(k,?) is given by GTEE model.
The structure factor measured by an X-rays
diffraction experiment is thus
where the partial structure factor
is given by
The second moment sum rule can be shown to be
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12SUMMARY (1)
- Geralized Three Effective Eigenmode Theory is an
appropriate theory for analyses of IXS spectra
from lipid bilayers. It allows one to extract
the dynamic structure factor, the phonon
dispersion relation, the phonon damping, and
thermal diffusion coefficients of the bilayers. - The phonon dispersion relation of a bilayer shows
a deep minimum at the peak of the structure
factor. This minimum frequency is lower in the LC
phase than that in the gel phase. - Given a chain length, the high-frequency sound
speed is higher in the gel phase than in the LC
phase. For DMPC it is 2230 m/s vs 1936 m/s.
13SUMMARY (2)
- The phonon frequency at the peak of the
dispersion relation increases with the chain
length while that at the valley decreases with
the chain length. - Addition of cholesterol to DMPC, originally in
the LC phase, has an effect of changing the
dynamic structure factor and the phonon
dispersion relation closer to that of the gel
phase. This can be interpreted to mean that an
addition of cholesterol to bilayer at LC phase
temperature has a tendency to increase the order
of lipid molecules in the bilayer and makes the
bilayer more solid-like. - Addition of cholesterol to a bilayer hardly
changes the thermal diffusion coefficient DT of
the bilayer.
14DNA A molecular model 1 hydrogen, 2 oxygen, 3
carbon in the helical phosphate ester chains, 4
carbon and nitrogen in the cross-linked purine
and pyrimidine bases, 5 phosphorus B double
helix
15F. Livolant, A. Leforestier, Progress in Polymer
Science, 21, 1115, 1996
linear charge spacing between phosphate groups
along DNA axis D 1.7 ?
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17Important Effects of Counter-Ion Atmosphere on
Properties of DNA
- Counter-Ion Condensation.
- Counter-ions neutralize DNA molecules and
cause DNA to aggregate. - Counter-ions can cause charge inversion of
DNA molecules, which - introduces DNA aggregation even in dilute
solutions. - 2. DNA Structural Stability.
- Alkaline earth metal ions can stabilize B-DNA
against thermal denaturation. - Transition metal ions preferentially bind to
the nucleic bases, - disrupting hydrogen bonding between base
pairs and destabilizing - the B form DNA.
- 3. DNA Conformational Change.
- Mg2 can influence B-Z conformation
transition DNA A-B transition - is also dependent on counter-ions.
18Theory of Counterion Condensation
(G. S. Manning, The molecular theory of
polyelectrolyte solutions with applications to
the electrostatic properties of polynucleotides,
Quartely Reviews of Biophysics II, 2, 179, 1978 )
In a linear polyelectrolyte of sufficiently high
charge density, a certain fraction of its charge
is neutralized due to the territorial binding of
counterions in the immediate environment. The
condensed counterions are free to diffuse along
the polymer axis, but inhibited from diffusing
away radially.
The fraction of groups neutralized by the
counterion condensation,
b linear charge spacing between phosphate
groups along DNA axis 1.7 ?
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22Phonon Damping of 40 wt Na-DNA with Different
Counterions
23The Intermediate Scattering Function, F(Q,t)
40 wt Na-DNA in H2O
40 wt Na-DNA in 0.083 M CaCl2
24Summary
1. Generalized Three Effective Eigenmode Theory
is an appropriate theory for analyses of IXS
spectra of DNA samples. It allows one to extract
the dynamic structure factor, the phonon
dispersion relation, the phonon damping and the
relaxation rate of the central peak.
2. The phonon dispersion relation of 40 wt
calf-thymus Na-DNA in water is constructed. At
low Q, it is linearly dependent on Q with the
sound speed similar to that of water (about 3100
m/s). It shows an oscillatory behavior in Q range
of 2 nm-1 to 30 nm-1, similar to that obtained
from lipid bilayers, and it extends to a
considerably higher Q than that observed in
molecular liquids and liquid metals.
3. In the interval of 10 nm-1 lt Q lt 22.5 nm-1,
the phonon peaks in the 40 wt calf-thymus Na-DNA
in solutions with divalent and trivalent
counter-ions are completely dampened out. We
conjecture that this is due to an increased
phonon damping caused by the increased
multivalent counter-ion condensation, which
change the water mobility around DNA. This leads
to a concept of the counter-ion friction.
25Calculated and Measured Average Structure Factors
of Globular Proteins
P. Etchegoin, Glassylike low-frequency dynamics
of globular proteins,PRE, 58, 845, (1998)
Intermediate range order of 2-6 ?
Structure factor of Lysozyme measured by x-ray
scattering
26The Practical DHO Model Analysis Works with
Protein Data
27Phonon Energies of Lysozyme and BSA Below and
Above the Glass Transition Temperature
28The Structural Effect on Phonon Dispersion in
Lysozyme and BSA
29Phonon Dispersion and Brillouin-to-central-peak
Intensity Ratio of Lysozyme and BSA
D. Liu and S.-H. Chen, et al, to be published
30Preliminary Observation of Phonons in Globular
Proteins
- We analyzed the IXS spectra by the practical DHO
model. The agreement with experimental data is
satisfactory. - All globular proteins have a universal average
structure factor showing a peak at Q 15 nm-1. - Phonons can only be observed for Q values at and
larger than this peak. - The peak phonon energy changes significantly at
the temperature of the so-called glass transition
of protein at Tg 225 K.
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32References
- C. Y. Liao, S. H. Chen, and F. Sette, Analysis
of inelastic X-ray scattering spectra of low
temperature water, Phys. Rev. E, 611518, 2000 - C. Y. Liao, and S. H. Chen, Theory of the
generalized dynamic structure factor of
polyatomic molecular fluids measured by inelastic
x-ray scattering, Phys. Rev. E, 64021205, 2001 - S. H. Chen, C. Y. Liao, H. W. Huang, T. M. Weiss,
M. C. Bellissent- Funel, and F. Sette, The
collective dynamics in fully hydrated
phospholipid bilayers studied by inelastic X-ray
scattering, Phys. Rev. Lett. 86, 740 (2001) - M. Terek, D. J. Tobias, S. H. Chen and M. L.
Klein, Short wavelength collective dynamics in
phospholipid bilayers a Molecular dynamics
study, Phys. Rev. Lett. 87, 238181 (2001) - T. M. Weiss, P.-J. Chen, H. Sinn, E. E. Alp,
S.-H. Chen, and H. W. Huang, Collective chain
dynamics in lipid bilayers by inelastic X-ray
scattering, Biophys. J. 84, 3767 (2003) - P.-J. Chen, Y. Liu, T. M. Weiss, H. Huang, H.
Sinn, E. E. Alp, A. Alatas, A. Said, and S.-H.
Chen, Studies of short-wavelength collective
molecular motions in lipid bilayers using high
resolution inelastic X-ray scattering,
Biophysical Chemistry, 105, 721 (2003) - Y. Liu et al,Inelastic X-ray scattering studies
of phonons in liquid crystalline DNA, Phys.
Chem.Chem. Phys. 6, 1499-1503 (2004) - Y. Liu et al, Inelastic X-ray scattering studies
of phonons propagating along the axial direction
of a DNA molecule having different counter-ion
atmosphere, J. Phys. Chem. Solids, 66, 2235-2245
(2005) - Y. Liu et al, Effects of counterion valency on
the damping of phonons propagating along the
axial direction of liquid-crystalline DNA, J.
Chem. Phys.,123, 214909 (2005)
33Advantages of IXS with ?E0.1meV for
Investigation of Phonons in Bio-Materials
By S.-H. Chen
- IXS is a coherent scattering, irrespectively of
atomic composition of a bio-molecule. There is no
need to deuterate the bio-molecule as in an INS
experiment. For coherent INS at thermal energies
?E is at best 0.25 meV and the Q range is not as
broad as in IXS. - While the number of x-ray photons or neutrons at
the sample are comparable - ( 3?108 photons or neutrons per sec), an IXS
beam size is much smaller (200 ? 100 ?m) than
that in an INS spectrometer. Thus the required
biomaterial is in mg quantity. - 3. Even if one deuterates a bio-molecule, the
dynamic structure factor as measured in an INS
experiment is a weighted sum of contributions
from partial structure factors of deuterium,
carbon, oxygen, nitrogen and phosphor, etc..
Fortunately, the dynamic structure factor
measured in an IXS is practically dominated by
the phosphate-phosphate (high Z atoms) PSF coming
from the backbones of the DNA molecules. So the
interpretation of the nature of the collective
modes measured is simpler. - 4. An IXS spectrum measured with an energy
resolution of ?E 0.1 meV can be calculated from
the corresponding intermediate scattering
function (ISF) computed with a time window of 15
ps. This allows the investigation of the alpha
relaxation at high Q in the ISF. Furthermore,
modern MD simulations of bio-macromolecules are
routinely pursued up to 100 ps. Thus measured IXS
spectra of bio-macromolecules can be used to
critically test the force field models of
proteins, lipid bilayers and DNAs.
34Possibility of Studying the Alpha Relaxation of
Biomolecules above and below Tg
By S.-H. Chen
40 wt Na-DNA in H2O
40 wt Na-DNA in 0.083 M CaCl2
?E0.1meV
?E0.1meV