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Inelastic XRay Scattering Studies of Phonon Propagation and Damping in Biomolecular Assemblies

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Title: Inelastic XRay Scattering Studies of Phonon Propagation and Damping in Biomolecular Assemblies


1
Inelastic 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
2
Advantages of Inelastic X-Ray Scattering for
Investigation of Collective Modes in Bio-Materials
3
Advantages of Inelastic X-Ray Scattering for
Investigation of Collective Dynamics in
Bio-Materials
4
Three 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.
5
For 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
6
Generalized 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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SUMMARY (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.

13
SUMMARY (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.

14
DNA 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
15
F. 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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Important 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.

18
Theory 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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Phonon Damping of 40 wt Na-DNA with Different
Counterions
23
The Intermediate Scattering Function, F(Q,t)
40 wt Na-DNA in H2O
40 wt Na-DNA in 0.083 M CaCl2
24
Summary
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.
25
Calculated 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
26
The Practical DHO Model Analysis Works with
Protein Data
27
Phonon Energies of Lysozyme and BSA Below and
Above the Glass Transition Temperature
28
The Structural Effect on Phonon Dispersion in
Lysozyme and BSA
29
Phonon Dispersion and Brillouin-to-central-peak
Intensity Ratio of Lysozyme and BSA
D. Liu and S.-H. Chen, et al, to be published
30
Preliminary 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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References
  • 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)

33
Advantages 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.

34
Possibility 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
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