Vibrational properties of proteins and nanoparticles

1
Vibrational properties of proteins and
nano-particles

• Francesco Piazza
• Laboratore de Biophysique Statistique
• EPF-Lausanne, Switzerland
• NBI2006

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Outline

• Modeling protein dynamics. Physics versus biology
  and the necessity of coarse-graining.
• Residue-based models the elastic network model
  and native-centric schemes.
• Worked examples
• Normal modes the dynamics of the PDZ binding
  domain.
• Langevin modes energy relaxation in proteins and
  metal
• nano-clusters.
• Conclusions

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Protein dynamics matters
IBM Announces 100 Million Research Initiative to
build World's Fastest Supercomputer"Blue Gene" to
Tackle Protein Folding Grand Challenge YORKTOWN
HEIGHTS, NY, December 6, 1999 -- IBM today
announced a new 100 million exploratory research
initiative to build a supercomputer 500 times
more powerful than the worlds fastest computers
today. The new computer - nicknamed "Blue Gene"
by IBM researchers - will be capable of more than
one quadrillion operations per second (one
peta-flop). This level of performance will make
Blue Gene 1,000 times more powerful than the Deep
Blue machine that beat world chess champion Garry
Kasparov in 1997, and about 2 million times more
powerful than today's top desktop PCs.
What can we really do with such super-computers?
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Time and length scales of protein dynamics
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The atomistic perspective
Paradigm currently adopted by computational
biologists the biological complexity is an
irreducible one - the more detail included the
better.
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Representative functional forms for
inter-particle interactions used in force fields
for atomistic simulations
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Crude workload estimates for all-atom MD
simulations on the fastest machines
The computational effort required to study
protein dynamics is enormous. For example,
following 100 ?s of a single trajectory
Petaflop (1015 f.p. operations/sec) 3.3 years
(2006?)
Blue gene 360 Teraflop (3.6 x1014 f.p.
operations/sec) 8.8 years
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The atomistic perspective is not necessarily the
most sensible paradigm for all aspects of protein
dynamics. What about coarse-graining?
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Coarse-grained models for the study of functional
motions
The masses of amino-acids are concentrated on the
corresponding ? carbons lying on the backbone
chain the number of interacting agents is
reduced by a factor of about 202

• Elastic network model
• Study of small fluctuations
• around the equilibrium structure
• Functional motions are collective
• ones, involving the concerted
• vibration of large sub-structures
• Normal modes
• Langevin modes

• Native-centric models
• Inter-residue contacts
• are divided between native
• and non-native
• Simple angular and bond-
• stretching potentials
• Study of large fluctuations
• and conformational changes

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Elastic network models
This is the good method to explore the effects of
the topology independently of the chemistry!
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The Hamiltonian in the harmonic approximation
becomes
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Normal modes
The normal modes of the proteins are the
eigenvectors of the mass-weighted Hessian matrix
Eigenvalues are squared frequencies
Low-frequency modes represent collective
displacement patterns of the entire protein.
Moving along a mode is a natural way to exploit
the spatial correlations embedded in the folded
structure.
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Example the binding dynamics of the PDZ domain
The PDZ is a widespread domain whose function is
to grab selected proteins by their
C-terminal. The sequence and structures of these
domains are highly conserved.
Loop L1

• Sequence conservation is dictated by the
  chemistry

• We propose that the fold has peculiar
• dynamical properties favorable to the
  binding

Helix ?B
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Cross-correlations within a given mode
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Involvement coefficients
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Thermal involvement coefficients
The relevant thermodynamic quantities are the
thermal averages
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Example the conformational change between free
and peptide-bound folds
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Coarse-grained native-centric models
The structure is coarse-grained at the amino-acid
level and inter-residue stretch and angular
potential introduced. Brownian dynamics
simulations.
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Example involvement coefficients as functions
of the temperature
Start from the native (relaxed) structure and
perform Brownian dynamics simulations at
different temperatures. The conformationalchanges
with respect to the equilibrium structure at
fixed temperature can be projected on the NM basis
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Results the NM that describes the opening
dynamics of the binding pocket gets increasing
spectral weight
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Proteins do not perform their functions in
vacuum Langevin dynamics
A simple tool to introduce the coupling with the
solvent in the normal mode calculations.
Particles displacements are governed by
stochastic equations of motion of the Langevin
type
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Equations of motion in matrix form
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Including solvent effects in the game Langevin
modes
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Relaxation dynamics of local or distributed
energy fluctuations
This broad topic encompasses some of the
fundamental processes of molecular biology, such
as the dynamics of relaxation and redistribution
of energy released at specific sites in a
protein structure after, e.g.

• absorption of electromagnetic radiation
  (conformational changes induced
• in rhodopsin after absorption of a visible
  photon),

• completion of an exothermic chemical reaction
  (hydrolysis of an ATP
• molecule into ADP, the basic fuelling
  mechanism for functioning of molecular
• motors).

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Fokker-Planck formulation of the problem
is the probability that the system is described
by theset of displacements and positions Y at
time t if itsinitial configuration at time t 0
was Y(0)
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The solution
where G is the propagator matrix and
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The evolution law for the correlation matrix

• C(0) describes the initial excitation.

the relaxation depends only on the temperature
difference
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The energy decay
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Relaxation in a metal nano-cluster

• Relaxation after excitation with laser light has
  twocharacteristic time scales
• fast (
• slow ( ps) dynamics of heat dissipation to the
• environment

• Heat dissipation from bio-functionalized
  particles used to selectively kill cells or to
  study protein denaturation

• Heat dissipation is also an important issue in
  laser-induced annealing and size and shape
  transformation of metal particles.

Experimental evidence for slow (stretched
exponential) relaxation(M. Hu and G. V.
Hartland, J. Phys. Chem. B. 106, 7029 (2002))
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Myoglobin
Sample nano-cluster
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Conclusions

• Coarse-grained models allow a great deal of
  dynamical processes in
• nano-metric systems to be studied
  quantitatively under reasonable time
• constraints.

• Normal modes may be calculated from the harmonic
  approximation of different force fields
• The long-range spatial correlation imprinted in
  the first low-frequency modes
• describe functional motions.
• One or a few selected low-frequency modes
  capture the thermal fluctuations even at
• working temperatures.
• These motion patterns are to a large extent
  independent of the microscopic details
• of the model IN NATURE, THE TOPOLOGY
  DICTATES THE FUNCTION. Can this
• perspective be adopted in designing synthetic
  nano-machines?

• The solvent effects may also be taken into
  account to describe a wealth
• of relaxation phenomena in nano-systems.
  Notably, phenomena of controlled
  storage/release of energy in a medium of choice.

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Co-workers

• Paolo De Los Rios, EPFL, Lausanne, CH
• Yves-Henri Sanejouand, ENS, Lyon, FR
• Fabio Cecconi, Università di Roma La Sapienza,
  IT

http//marie.epfl.ch/fpiazza/