Vibrational properties of proteins and nano-particles Francesco Piazza Laboratore de Biophysique Statistique EPF-Lausanne, Switzerland NBI2006
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
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 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?
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. The atomistic perspective Paradigm currently adopted by computational biologists: the biological complexity is an irreducible one - the more detail included the better.
Representative functional forms for inter-particle interactions used in force fields for atomistic simulations
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
Bead-spring network • pairwise harmonic forces among residues within a fixed cutoff distance • No explicit angular forces • Backbone representation • Coarse-graining at the amino-acid level • Angular and bond-stretching potentials • Native contacts “privileged” (native-centric models) The atomistic perspective is not necessarily the most sensible paradigm for all aspects of protein dynamics. What about coarse-graining?
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
A suitable cutoff distance sets the interaction range. The pairs connected by springs are determined only by the topologyof the native fold Elastic network models This is the good method to explore the effects of the topology independently of the chemistry!
is the Hessian matrix are the residues’ fluctuations All masses are taken as equal The Hamiltonian in the harmonic approximation becomes
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.
Hydrophobic pocket 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
k = 2 Cross-correlations within a given mode
Quantify the weight of a given low-frequency NM in the spectral decomposition of a given conformational change . Any thermal fluctuation of the structure canbe spectrally decomposed on the NM basis Involvement coefficients
Frequency-weighted inv. coefficient Average thermodynamic overlap between the conformational change and the thermal fluctuations of the structure Thermal involvement coefficients The relevant thermodynamic quantities are the thermal averages
The second mode captures most of the deformation Example: the conformational change between free and peptide-bound folds
Native contacts Non-native contacts 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.
The quantity of interest is now 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
Results: the NM that describes the opening dynamics of the binding pocket gets increasing spectral weight
harmonic force damping stochastic force Fluctuation-dissipation theorem 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
The eigenvalues of the matrix have a real part that specifies the mode relaxation rate The vector of surface fractions exposedto the solvent fixes the damping rates where 0 < Si < 1 Including solvent effects in the game: Langevin modes
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. 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).
Example: redistribution of the energy releasedfollowing ATP hydrolysis ATP-binding domainof HSP-70 Fokker-Planck formulation of the problem is the probability that the system is described by theset of displacements and positionsYat timetif itsinitial configuration at timet= 0wasY(0)
The solution whereGis the propagator matrix and
where • For a uniform excitation,T(0) = T0 > T, The evolution law for the correlation matrix • C(0)describes the initial excitation. the relaxation depends only on the temperature difference
In log-log scaleDis straight linewith slope • one ifE(t)decays exponentially • if E(t)decays as a stretched exponential The energy decay
Relaxation in a metal nano-cluster • Relaxation after excitation with laser light has twocharacteristic 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))
Stretched exponentials dashed line Myoglobin Sample nano-cluster
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.
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/