1. Vibrational properties of proteins and nano-particles
Francesco Piazza
Laboratore de Biophysique Statistique
EPF-Lausanne, Switzerland
NBI2006

2. 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

3. 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, IBM today announced a new $100 million exploratory research initiative to build a supercomputer 500 times more powerful than the world’s 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?

4. Time and length scales of protein dynamics
Local Motions
Atomic fluctuations
Side-chain Motions
Loop Motions
0.01 to 5 �, to 10-1 s
Rigid Body Motions
Helix Motions
Domain Motions (hinge bending)
Subunit motions
1 to 10� Å, 10-9 to 1 s
Large-Scale Motions
Helix-coil transitions
Dissociation/Association
Folding and Unfolding
> 5 �, 10-7 to 104 s

5. The atomistic perspective
Paradigm currently adopted by computational biologists: the biological complexity is an irreducible one - the more detail included the better.
All-atom representation
Empirical inter-atomic potentials
Mixed classical/quantum treatment
Atomistic description of the solvent: simulations performed in a box filled with thousands of water molecules. +

6. Representative functional forms for inter-particle interactions used in force fields for atomistic simulations

7. 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…
Physical time for simulation
10-4 seconds
Typical time-step size
10-15 seconds
Number of MD time steps
1011
Atoms in a typical protein and water simulation
32000
Approximate number of interactions in force calculation
1000
Machine instructions per force calculation
109
Total number of machine instructions
1023
Petaflop (1015 f.p. operations/sec)
~ 3.3 years (2006?)
Blue gene:
360 Teraflop (3.6 x1014 f.p. operations/sec)
~ 8.8 years

8. The atomistic perspective is not necessarily the most sensible paradigm for all aspects of protein dynamics. What about coarse-graining?
Backbone representation
Coarse-graining at the amino-acid level
Angular and bond-stretching potentials
Native contacts “privileged” (native-centric models)
Bead-spring network
pairwise harmonic forces
among residues within a
fixed cutoff distance
No explicit angular forces

9. 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
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
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

10. Elastic network models
A suitable cutoff distance sets the interaction range. The pairs connected by springs are determined only by the topology of the native fold
This is the good method to explore the effects of the topology independently of the chemistry!

11. The Hamiltonian in the harmonic approximation becomes
is the Hessian matrix
are the residues’ fluctuations
All masses are taken as equal

12. 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.

13. Example: the binding dynamics of the PDZ domain
Hydrophobic pocket
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

14. Cross-correlations within a given mode
k = 2

15. Involvement coefficients
Quantify the weight of a given low-frequency NM in the spectral decomposition of a given conformational change
Any thermal fluctuation of the structure can be spectrally decomposed on the NM basis

16. Thermal involvement coefficients
The relevant thermodynamic quantities are the thermal averages
Frequency-weighted inv. coefficient
Average thermodynamic overlap between the conformational change and the thermal fluctuations of the structure

17. Example: the conformational change between free and peptide-bound folds
The second mode captures most of the
deformation

18. 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.
Native contacts
Non-native contacts

19. Example: involvement coefficients as functions of the temperature
Start from the native (relaxed) structure and perform Brownian dynamics simulations at different temperatures. The conformational changes with respect to the equilibrium structure at fixed temperature can be projected on the NM basis
The quantity of interest is now

20. Results: the NM that describes the opening dynamics of the binding pocket gets increasing spectral weight

21. 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
harmonic force
damping
stochastic force
Fluctuation-dissipation theorem

22. Equations of motion in matrix form

23. Including solvent effects in the game: Langevin modes
The vector of surface fractions exposed to the solvent fixes the damping rates
where 0 < Si < 1
Including solvent effects in the game: Langevin modes
The eigenvalues of the matrix have a real part that specifies the
mode relaxation rate

24. 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).
Such phenomena of relaxation dynamics and related ones can be studied analytically by solving the Fokker-Planck equation associated with the Langevin elastic network model of the system.

25. Fokker-Planck formulation of the problem
is the probability that the system is described by the set of displacements and positions Y at time t if its initial configuration at time t = 0 was Y(0)
Example:
redistribution of
the energy released following ATP
hydrolysis
ATP-binding domain of HSP-70

26. The solution
where G is the propagator matrix and

27. The evolution law for the correlation matrix
where
C(0) describes the initial excitation.
For a uniform excitation, T(0) = T0 > T,
the relaxation depends only on the temperature difference

28. The energy decay In log-log scale D is straight line with slope
one if E(t) decays exponentially
 if E(t) decays as a stretched exponential

29. Relaxation in a metal nano-cluster
Relaxation after excitation with laser light has two characteristic time scales:
fast (< ps): dynamics of e-e equilibration
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))

30. Stretched
exponentials
Myoglobin
dashed line
Sample nano-cluster

31. 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.

32. Co-workers Paolo De Los Rios, EPFL, Lausanne, CH
Yves-Henri Sanejouand, ENS, Lyon, FR
Fabio Cecconi, Università di Roma “La Sapienza”, IT