Dominant Paths in Protein Folding

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1 Dominant Paths in Protein Folding Henri Orland SPhT, CEA-Saclay France work in collaboration with P. Faccioli, F. Pederiva, M. Sega University of Trento Henri Orland Annecy meeting 2006

2 Outline Basic notions on Proteins Langevin dynamics Dominant paths Example: Villin The Folding Path problem Path integral representation Hamilton-Jacobi representation

3 1. What is a Protein Biological Polymers (biopolymers): Proteins, Nucleic Acids (DNA and RNA), Polysaccharides! catalytic activity: enzymes! transport of ions: hemoglobin (O 2 ), ion channels! motor protein! shell of viruses (influenza, HIV, etc...)! prions! food, etc Proteins have an active site: biological activity

4 Polymers built with amino-acids! 20 types of amino acids! all left-handed! Ala, Ile, Leu, Met, Phe, Pro, Trp, Val, Asn, Cys, Gln, Gly, Ser, Thr, Tyr, Arg, His, Lys, Asp, Glu! 10! Number of Monomers! 500 H H O N C C H OH R residue

5 Among the 20 amino-acids:! 12 hydrophilic (polar)! 8 hydrophobic (non polar) 8 uncharged 4 charged In a typical protein:! polar! hydrophobic Examples of residues: H : glycine C H H : alanine H : phenylalanine

6 Polymerisation (polycondensation) NH2---CHR1---COOH + NH2---CHR2---COOH +... NH2---CH (CO NH)-- CH (CO NH)-- CH (CO NH)--- \ / R1 peptide R2 R3 bond + H2O! weakly branched polymer

7 ! Hard degrees of freedom: covalent bonds valence angles peptide bonds improper dihedrals! Soft degrees of freedom " torsion angles : #, $, % very small energies

8 Proteins exist under two states:! Denatured = Unfolded Random Coil (swollen) Molten Globule (compact) No biological activity! Native = Folded = Unique compact structure Biologically active Number of compact structures of a polymer : N ~! Puzzle: below folding transition temperature, the protein seems to exist under a unique conformation (zero conformational entropy). Folding transition: depends on temperature, ph, denaturant agent, salt, etc Time scales: Microscopic time : s Folding time: 10-2 to 1 s

9 Tertiary structure: 3d structure of the folded protein!compact packing of secondary structures.

10 HIV protease (199 residues)

11 The Chemist s Approach 1. Look for effective atom-atom interactions semi-empirical Hamiltonian 2. Molecular dynamics or Monte Carlo. What interactions are present? bonded -covalent bond -sulfur bridges (cysteins) non bonded solvent. -Coulomb (with partial charges) -Van der Waals (steric repulsion) -Hydrogen bonds : intra-molecular or with the The solvent is polar (Water) and induces hydrophobic interactions which might be responsible for the collapse transition.

12 Energy Scales 1 ev = 23 kcal/mole = 10000! K 300! K = 0.6 kcal /mole " Covalent bond: kcal /mole " Sulfur Bridge: 51 kcal/mole " Hydrogen bonds: 5-8 kcal/mole (non polar solvent) 1-2 kcal/mole (polar solvent) " Van der Waals: 1 kcal/mole " Coulomb: 1-2 kcal/mole Denaturation temperature! 1 kcal/mole Chemical sequence is frozen and only non-covalent interactions drive the folding.

13 Parametrization (CHARMM, AMBER, OPLS, ) E % " bonds k b ( b ) b 0 ) 2 $ $ " ( & ' )/ " ij " & 12 ij 6 4* ) # ij ( ) ( ) $ " i# j ij ij i# j $ % # $ (1 $ cos( n. ) - )) 2 k/ (/ 0) k. k, valence angles dihedrals impropers + r + r 332 * q q i r ij j $ " (, ), ) 0 2 Use Newton or Langevin dynamics m % E... i ri " $ i ri "!! i % ri ( t) where! i (t) is a Gaussian noise satisfying the fluctuation-dissipation theorem: $! ( t)! ( t) #! 2$ k T" " ( t t') i j i B ij,

14 Then, it is well known that P({ r }, t) i exp ) t, * ({ r}) Ek i - - / B T ( & ' T o discretize, one m ust use "t ~ s N um ber of degrees of freedom : N # 1000 L ongest available runs (w ith w ater) t ~ 10-8 s W e see that t < < folding tim e. Reason: system is trapped in an exponential number of metastable traps.

15 The protein folding problem is too complicated Simpler problem: how do proteins go from the unfolded state to the native state?

16 Denaturation curves [Fraction Native] [Denaturant] In given denaturant conditions, a fraction of the proteins are native, and the rest are denatured

17 This means that in given denaturant conditions, a protein spends a fraction of its time in the native state and a fraction of its time in a denatured state.

18 The Folding Pathway Problem The problem: Assume a protein can go from state A to state B. Which pathway (or family of pathways) does the protein take? Is there a transition state ensemble? Examples: from denatured to native in native conditions Allosteric transition between A and B

19 Langevin dynamics The case of one particle in a potential at temperature Use Langevin dynamics T U(x) where γ is the friction and ζ(t) is a random noise m d2 x dt 2 + γ dx dt + U x = ζ(t)

20 Overdamped Langevin dynamics At large enough time scale, mass term negligible mω 2 γω τ 2π m γ γ = k BT D τ s D = 10 5 cm 2 /s m kg

21 Take overdamped Langevin (Brownian) dynamics x t = D k B T U x + η(t) Gaussian noise with zero av with Gaussian noise: s a Gaussian noise with zer η(t)η(t ) = 2Dδ(t t ) constant of the particle in

22 Equation of motion is a stochastic equation The Probability to find the particle at point x at time t is given by a Fokker-Planck equation t P(x,t) = D x ( 1 k B T U(x) x P(x,t) ) + D 2 x 2 P(x,t) P (x, 0) = δ(x x i )

23 Fokker-Planck equation looks very much like a Schrödinger equation, except for 1st order derivative. Define P (x, t) = e βu(x) 2 Q(x, t) The function Q(x, t) equation with a Hamiltonian H satisfies a Schrödinger

24 Using the notations of Quantum Mechanics P (x f, t f x i, t i ) = e U(x f ) U(x i) 2k B T < x f e (t f t i )H x i > where H is a quantum Hamiltonian given by H = D( 2 x Spectral decomposition U(x) (β x )2 β 2 U(x) x 2 ) < x f e (t f t i )H x i >= α e (t f t i )E α Ψ α (x f )Ψ α (x i )

25 At large time, the matrix element is dominated by the ground state with so that Ψ 0 (x) = e βu(x)/2 Z Z = e βu(x) HΨ 0 = 0 P (x f, t f x i, t i ) e βu(x) Z + e β U(x f ) U(x i ) 2 e (t f t i )E 1 Ψ 1 (x f )Ψ 1 (x i )

26 Stationary distribution: the Boltzmann distribution lim t + ( ) General form: Path Integral Boundary conditions: at the stationary solution o P (x, t) = P(x) exp( U(x)/k B T ) he boundary conditions x P(x f,t f x i,t i ) = e U(x f ) U(x i ) 2k B T Z x f Dx(τ)e S e f f [x]/2d, x i R ( ) ) x(t i ) = x i ntegral: x(t f ) = x f

27 The effective action is given by S e f f [x] = R t t i d τ ( ẋ2 (τ) Z and the effective R ( ) potential is given by Z ) 2 +V e f f [x(τ)] ( ) V e f f (x) = D2 2 ( 1 k B T ) U(x) 2 D2 x k B T 2 U(x) x 2.

28 U(x) = x 2 (5(x 1) 2 0.5) -0.5 V eff (x) = U (x) 2 /2 T U (x) N x V eff (x) = U (x) 2 /2 T U (x) T = 0 T = Henri Orland Annecy meeting 2006 N N

29 Effective Native States and Transition States It seems natural to define the native state as the minimum of anharmonicity V eff (x). Shift due to x N (T ) x N (0) + T U 0 U 2 0

30 V eff (x) 4 3 T = x Denatured state Henri Orland Annecy meeting 2006 Native

31 Dominant trajectories: classical trajectories with correct boundary conditions. Problem: one does not know the transition time. Solution: go from time-dependent Newtonian dynamics to energy-dependent Hamilton-Jacobi description. d 2 x dt 2 = ( V eff [x]) x

32 N N T = 0.5 T = E eff = ẋ2 2 V eff (x)

33 The method: minimize the Hamilton-Jacobi action S HJ = Z x f over all paths joining to x i dl 2(E e f f +V e f f [x(l)]), The total time is determined by t f t i = Z x f x i dl x i x f dl is an infinitesimal displacement along the path y. E e E f f is ais free a free parameter parameter which determines the to psed during the transition, Z 1 2(E e f f +V e f f [x(l)]). ld be stressed that the conserved quantity E

34 S e f f [x] = R t t i d τ ( ) ẋ2 (τ) 2 +V e f f [x(τ)] For classical trajectories One obtains ( ) E eff = ẋ2 2 V eff (x) S eff [x] = E eff (t f t i ) + x f x i dx 2(E eff + V eff (x))

35 E e f f ing tr is not the true energy of the system If the final state is an equilibrium state, then imulations). In the E e f f = V e f f (x f ), time. However, w

36 The HJ method is much more efficient than Newtonian mechanics because proteins spend most of their time trying to overcome energy barriers. No waiting-times in HJ: work with fixed interval length dl

37 For a Protein, minimize S HJ = N 1 n 2(E e f f +V e f f (n)) l n,n+1 + λp, where e P = N 1 i ( l i,i+1 l ) 2 and λ is a Lagrange multiplier to fix the interval length V e f f (n) = i ( l) 2 n,n+1 D2 D2 2(k B T) 2 k B T j ( ( j u(x i (n),x j (n)) j ] 2 ju(x i (n),x j (n)) = (x i (n + 1) x i (n)) 2, i ) 2

38 Go potential U X u x i ;x j X 1 2 j;i 1 i<j i<j 2 K b jx i x j j a i;j R0 12 2R 0 6 Rr 12 r ij r ij R 0 6 2R i;j ; r ij Initial conditions: 6 high temperature denatured states from MD the Villin Headpiece Subdomain

39 Results for the Villin Headpiece Go Model 2 60 Gyration radius (nm 2 ) Percentage of configurational steps

40 Percentage of monomers in alpha helix conformation Percentage of configurational steps Number of contacts Percentage of configurational steps

41 Conclusions Natural definition of Folding pathways No need for reaction coordinate Transition states Calculation of rates Working on using atomic potentials and including solvent.