Rare Events. Transition state theory Bennett-Chandler Approach 16.2 Transition path sampling16.4

Transcription

1 Rare Events Transition state theory Bennett-Chandler Approach 16.2 Transition path sampling16.4

2 Outline Part 1 Rare event and reaction kinetics Linear Response theory Transition state theory Free energy methods Bennet Chandler approach Example zeolites Part 2 Two ended methods Transition path sampling Rate constants Reaction coordinate analysis Application to biomolecules

3 crystallisation catalysis solution reactions complex fluids Rare Events enzyme reactions solvent effects folding & binding isomerization

4 Rare events Interesting transitions in complex systems solution chemistry protein folding F A B enzymatic reactions complex surface reactions diffusion in porous media nucleation q q B These reactions happen on a long time scale compared to the molecular timescale A dominated by collective, rare events Straightforward MD very inefficient mol t stable

5 Example: Diffusion in porous material

6 Phenomenological reaction kinetics A rare event can be seen as a chemical reaction between reactant A and product B A B A B The change in population c(t) is (0<c<1) dc A ( t) dt = k A B c ( A t) + k B A c ( B t) dc B dt ( t)!# A B " $ = 0 dt Equilibrium: c A t Total number change in population d c ( t) + c ( t) ( ) ( ) =+ k c t k c t ( ) = c B ( t) = 0 A B A B A B This gives a relation between equilibrium population and reaction rates c A c B = k B A k A B

7 Let us make a perturbation of the equilibrium populations. The dynamics of such an equilibrium fluctuation gives the desired information on the response of the system to an external field ( ) = +Δ ( ) For state A c t c c t For state B: A A A We can rewrite the kinetics in terms of the perturbation: 1 dδc A dt ( t) ( ) ( ) = k Δc t k Δc t A B A B A A ( ) ( 0exp ) ( ) Δ ca t = ΔcA "& ka B+ kb A t# ' With τ A ( 0exp ) [ t τ ] = Δc ( k k ) 1 1 = + ( ) 1 A B B A Relaxation time population = k 1+ c c = A B A B c A ( t) + c B ( t) =1 ( ) = Δ ( ) c t c c t B B A C A (t) C B (t) k c B A B time

8 Microscopic theory Microscopic description of the progress of a reaction q Reaction coordinate: in this case the z-coordinate of the particle We need to write the kinetics of the reaction in terms of this microscopic reaction coordinate q

9 A B Reaction coordinate Reactant A: q< q* Product B: q > q* Transition state: q = q* Let us introduce the function g A : Heaviside θ-function # 0 q q* < 0 θ ( q q* ) = $ % 1 q q* > 0 g q q* = 1 θ q q* = θ q* q A ( ) ( ) ( ) With this function we write for the probability c A (t) the system is in state A: c A ( t) = g ( A t)

10 Microscopic theory Is going to give us the macroscopic relaxation in terms of a microscopic time correlation function exp [ t τ ] = A ( 0) Δ ( ) Δg g t c A c B A

11 Perturbed Hamiltonian Let us consider the effect of a static perturbation: H = H g q q 0 ε A * ( ) This external potential increases the concentration of A For the equilibrium concentration as a function of ε: Δ c = c c = g A ε g A 0 A A ε A 0 We need to compute the ensemble average in the form of : A 0 = dγaexp dγexp [ β H0 ] [ β H ] 0

12 Linear Response theory (static) The original Hamiltonian (H 0 ) is perturbed by εd: H = H 0 ε D This gives as change in the expectation value of A: Δ A = A A 0 with A = ( ) dγ Aexp β H 0 ε D dγ exp β ( H 0 ε D) A 0 = dγaexp dγexp [ β H0 ] [ β H ] 0 If the perturbation is small we can write A = A + A 0 ε 0 ε

13 For such a small perturbation ΔA = A 0 ε = ε A ε 0 ε with A ε ( ) ( ) = A ε = dγβ ADexp! " β H 0 εd # $ dγexp! " β H 0 εd { dγexp! " β ( H 0 εd # ) } 2 $ ( ) dγ Aexp! " β H 0 εd ( ) # $ dγβdexp! " β H 0 εd ( ) { # } 2 dγexp! " β H 0 εd $ # $ # $ Evaluated for ε= 0 A = dγ β ADexp β H 0 ε 0 { dγexp β H 0 } dγ Aexp β H 0 dγβ Dexp β H 0 dγexp β H 0 dγexp β H 0 Giving: A = β { AD ε 0 A 0 D } 0 0

14 If we apply this result for c A : A = β { AD ε 0 A 0 D } 0 0 with H = H g q q 0 ε A * ( ) Δ c = g g A A ε A ( ) ( ) 2 2 ga ga 0 Δ ca = β 0 ε 0 Since g A =0 or 1: g A (x) g A (x) =g A (x) ( g ( 1 )) A ga = β 0 0 ( ( 1 )) = β c c = β c c A 0 A 0 A 0 B 0 Giving: Δc A = β c A 0 c B 0 ε

15 Linear Response theory (dynamic) Let us now switch off the perturbation at t=0 H = H 0 ε D H = H 0 at t>0: Let us see how the system relaxes to equilibrium (dynamical perturbation) We take <A> 0 =0 ΔA( t) = A( t) A 0 = A( t) Similar as for the static case for small values of ε, we have A( t) ε 0 = dγ β A t ( ) Dexp β H 0 { } dγ exp β H 0 = β D( 0) A( t) Giving: ΔA( t) = βε D( 0) A( t)

16 ΔA( t) = βε D( 0) A( t) If we apply this result to D = Δg A and A = Δg A We obtain: Δc A From static perturbation: Δc A ( t) = βε Δg ( A 0)Δg ( A t) ( t) = Δc ( A 0) Δg 0 A ( )Δg A t c A c B ( ) ( ) βε = Δc A 0 c A c B Compare linear response expression with the macroscopic expression Δc A ( t) = Δc A 0 ( )exp t τ

17 exp Derivative Microscopic rate expression [ t τ ] = A ( 0) Δ ( ) Δg g t 1 τ exp t τ = g 0 A c A c B A ( ) g ( A t) c A c B Stationary (t is arbitrary, only depends on τ) Δ has disappeared because of the derivative = g ( 0 A )g A t c A c B ( ) d dt A( t)b( t + τ ) = 0 A( t) B ( t + τ ) + A ( t) B( t + τ ) = 0 A( 0) B ( τ ) = A ( 0) B( τ )

18 We have 1 τ exp " # t τ $ % = g 0 A ( ) g ( A t) c A c B Using 1 τ = k A B 1+ c A c B ( ) 1 = c B k A B For sufficiently short t, we obtain Using the definition of g A we can write k A B ( t) = g ( 0 A )g A t c A ( ) ( ) = q g ( q q* ) A g A q q* q = q g ( q q* ) B q k A B ( t) = ( ) g q ( 0 ) q* B q 0 ( ) q c A g B ( t) We now have an expression that relates the macroscopic reaction rate to microscopic properties

19 k A B g B ( t) = ( ) g q ( 0 ) q* B q 0 ( t) = θ q t) ( ( ) q * ) g ( B q( 0) q *) q ( ) q c A = Θ q 0 g B ( t) ( ( ) q *) q ( ) = δ q( 0) q * Let us look at the different terms in this equation Only when the system is in the product state we get a contribution to the ensemble average Only when the system starts at the transition state, we get a contribution to the ensemble average ( ) Velocity at t=0 q 0 c A = Θ( q * q) Concentration of A k A B ( t) = q 0 ( )δ ( q( 0) q *)θ q t ( ( ) q *) ( ) θ q * q

20 k A B Transition state theory ( t) = q 0 ( )δ ( q( 0) q *)θ q t ( ( ) q *) ( ) θ q * q Let us consider the limit: t 0 + lim t 0 + ( ) = θ ( q ( t) ) = θ q( t) q* This gives: k TST A B = q ( 0)δ ( q( 0) q* )θ q θ ( q* q) ( ) Eyring s transition state theory

21 C(t) k A B k TST Decay of rate expression ( t) = q 0 ( )δ ( q( 0) q *)θ q t ( ( ) q *) ( ) θ q * q lower value because of recrossings t k(t) k AB ~exp(-t/! rxn )! mol t

22 k A B ( t) = Transition state theory q ( 0)δ ( q( 0) q* )θ q t ( ( ) q* ) ( ) θ q* q We can rewrite this expression as a product k A B ( t) = q ( 0)δ ( q( 0) q* )θ q t ( ) δ q( 0) q* ( ( ) q* ) ( ( ) q* ) ( ) δ q 0 θ q* q k A B Conditional probability to find a particle on the top of the barrier with a positive velocity ( t) = q 0 ( )θ q t ( ( ) q* ) q=q* Ratio of probabilities to find particle on top of the barrier and in the state A δ ( q( 0) q* ) θ ( q* q)

23 Free energy barrier ( ) δ q( 0) q * ( ) θ q * q Ratio of the probabilities to find a particle on top of the barrier and in the state A Probability to be on top of the barrier: ( ) = C dqδ q q* δ q* q Probability to be in state A: Θ q * q ( ) ( ( )) ( )exp βf ( q) = C exp βf q* ( ) = C dqθ q q * This gives: ( ) ( ( )) ( )exp β F ( q) = C dqexp β F q ( ) δ q( 0) q* ( ) θ q* q We need to determine the free energy as a function of the order parameter = q<q* q<q* ( ( )) ( ) exp βf q* ( ) dqexp βf q

24 q ( 0)θ q t ( ( ) q* ) q=q* Conditional probability to find a particle on the top of the barrier with a positive velocity q ( 0) Assume that on top of the barrier the particle is in equilibrium: use Maxwell-Boltzmann distribution to generate this velocity q ( 0)θ q t ( ( ) q* ) Only particles with a positive velocity end up in the lim t 0 + q ( 0)θ q t product state. We assume that once in the product state they stay there. ( ( ) q* ) = q 0 ( ( )) = 0.5 q ( 0) ( )θ q 0 k TST = lim q ( 0 A B t 0 )θ q t + k TST A B = 0.5 q ( 0) q<q* ( ( ) q* ) q=q* ( ) ( ( )) exp βf ( q* ) dqexp βf q δ ( q( 0) q* ) θ ( q* q) Eyring s TST

25 1-D ideal gas particle on a hill k TST A B = 0.5 q ( 0) q<q* ( ) ( ( )) exp βf ( q* ) dqexp βf q Maxwell-Boltzmann: q ( 0) = 2k T B π m This gives for the hopping rate k TST A B = k B T 2πm q<q* ( ) ( ( )) exp βu ( q* ) dqexp βu q

26 Ideal gas particle on a not-so-ideal hill q * is the true transition state q 1 is the estimated transition state

27 For this case transition state theory will overestimate the hopping rate

28 Transition state theory One has to know the free energy accurately (MC/MD) C(t) Gives only an upper bound to the reaction rate Assumptions underlying transition theory should hold: no recrossings k A B k TST k(t) ( t) = q 0 ( )δ ( q( 0) q *)θ q t ( ( ) q *) ( ) θ q * q k AB t lower value because of recrossings ~exp(-t/! rxn )! mol t

29 Outline Part 1 Rare event and reaction kinetics Linear Response theory Transition state theory Free energy methods Bennet Chandler approach Example zeolites Part 2 Two ended methods Transition path sampling Rate constants Reaction coordinate analysis Application to biomolecules

30 Free energy barriers in complex systems Straightforward MD or MC and then use is highly inefficient for high barriers βf(q) = ln δ ( q(r) q) Many tricks have been proposed to overcome and sample barriers Temperature enhanced sampling: simulated tempering, parallel tempering, Temperature accelerated molecular dynamics ) Constraint dynamics: thermodynamic integration, blue moon sampling... Flat histogram sampling: umbrella sampling, hyperdynamics,... history dependent search: Wang-Landau, adaptive biasing force, metadynamics, non-equilibrium methods: steered MD, targeted MD,... trajectory-based methods: nudged elastic band, action minimization, string method, transition path sampling, forward flux sampling,...

31 Replica exchange/parallel tempering F! room temperature! Q! F! high temperature! Q!

32 Two replicas T = 360K 2 e 2U 2 (r N ) Total Boltzmann weight! e 1U 1 (r N ) e 2U 2 (r N ) T = 290K 1 e 1U 1 (r N )

33 Switching temperatures T = 290K 2 e 1U 2 (r N ) Total Boltzmann weight! e 1U 2 (r N ) e 2U 1 (r N ) T = 360K 1 e 2U 1 (r N )

34 The ratio of the new Boltzmann factor over the old one is: N (n) N (o) = e( 2 1)[U 2 (r N ) U 2 (r N )] the rule for switching temperatures should obey detailed balance Metropolis Monte Carlo scheme acc(1 2) = min 1,e ( 2 1)[U 2 (r N ) U 1 (r N )]

35 Set of replicas T = 360K R T = 293K 2 e 2U 2 (r N ) T = 290K 1 e 1U 1 (r N )

36 Overlap in potential energy

37 N Replica Exchange MD (REMD) 2 e 2U 2 (r N ) 1 e 1U 1 (r N ) Hansmann Chem Phys Lett 1997! Sugita & Okamoto Chem Phys Lett 1999!

38 Replica Exchange Exchange as a function of time. Free energy follows from Advantage: no order parameters needed Disadvantage: convergence of free energy landscape can be still slow, especially around phase transition: many replicas needed.

39 Metadynamics method to obtain free energy in a single simulation similar idea as Wang Landau sampling: add history dependent biasing potential to forcefield Laio and Parrinello, PNAS (2002)! s = predefined order parameters w = height of hills σ = width of gaussians F(s) w is reduced every cycle F (s) = lim V (s; t) t more controlled version: well tempered MetaD!! Barducci, Bussi, Parrinello, PRL, (2008).! s

40 Link to bernds animation

41 SN2 reaction between Cl - and CH3Cl Reactant Transition Product Complex State Complex S(R) = r C-Cl r C-Cl` S 1 (R) = r C-Cl S 2 (R) = r C-Cl` Meta-dynamics can relax the requirement of choosing a good reaction coordinate Bernd Ensing, Alessandro Laio, Michele Parrinello and Michael L. Klein, J. Phys. Chem. B 109 (2005),

42 Outline Part 1 Rare event and reaction kinetics Linear Response theory Transition state theory Free energy methods Bennet Chandler approach Example zeolites Part 2 Two ended methods Transition path sampling Rate constants Reaction coordinate analysis Application to biomolecules

43 Problem with TST There are recrossings that cause overestimation of the rate constant trajectories that seem to overcome the barrier but in fact bounce back

44 Bennett-Chandler approach k A B k A B ( t) = ( t) = q ( 0)δ ( q( 0) q* )θ q t ( ( ) q* ) ( ) θ q* q q ( 0)δ ( q( 0) q* )θ q t ( ) δ q( 0) q* ( ( ) q* ) ( ( ) q* ) ( ) δ q 0 θ q* q Computational scheme: 1. Determine the probability from the free energy using MC or MD, e.g. by umbrella sampling, thermodynamic integration or other free energy methods 2. Compute the conditional average from a MD simulation

45 TST k A B k A B ( t) = ( t) = q ( 0)δ ( q( 0) q 1 )θ q δ q 0 Transmission coefficient ( ) k ( t ) A B κ t Bennett-Chandler approach = ( ) q 1 ( ) q 1 ( ) q ( 0)δ ( q 0 )θ q t TST k A B ( ) δ q( 0) q 1 q ( 0)δ ( q( 0) q 1 )θ q t 0.5 q 0 ( ) δ ( q ( 0 ) q ) 1 θ ( q 1 q) ( ( ) q ) 1 δ ( q ( 0 ) q ) 1 θ ( q 1 q) ( ( ) q ) 1 ( ) MD simulation to correct the transition state result! MD simulation: 1. At t=0 q=q 1 2. Determine fraction at product state weighted with initial velocity

46 Example diffusion in zeolite Zeolites important class of materials Diffusion of alkanes in matrix is poorly described Approach molecular simulation of alkanes in fixed zeolite frame Unified atom FF by Dubbeldam et al. D. Dubbeldam, et al., J. Phys. Chem. B, 108, 12301, 2004

47 t : θ=1 For both q 0 ( ) and q 0 ( ) q ( 0)δ ( q( 0) q* )θ q t ( ( ) q* ) Low value of κ

48 Reaction coordinate cage window cage q * cage window cage q * βf(q) βf(q) q q

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50

51 Outline Part 1 Rare event and reaction kinetics Linear Response theory Transition state theory Free energy methods Bennet Chandler approach Example zeolites Part 2 Two ended methods Transition path sampling Rate constants Reaction coordinate analysis Application to biomolecules

52 Barriers on smooth and rough energy landscapes Clearly, barrier is most important for rare event But how to obtain this barrier? In multidimensional energy landscapes barrier is saddle point # saddle points limited determined by potential energy use eigenvectors or Hessian to find them # saddle points uncountable entropy important, many pathways determined by free energy exploring requires sampling schemes Dellago logo TM

53 Breakdown of BC approach kappa can become immeasurable low if the reaction coordinate is at the wrong value the reaction coordinate is wrongly chosen W ( q) = kt ln dq" exp{ βe( q, q" )} If the reaction coordinate is not known, the wrong order parameter can lead to wrong transition states, mechanism and rates

54 Two ended methods Methods that take the entire path and fix the begin and end point Many methods proposed: Action minimization Nudged elastic band String method Path metadynamics Milestoning Transition path sampling...

55 Transition path sampling Samples the path ensemble: all trajectories that lead over barrier Sampling by Monte Carlo Requires definition of stable states A,B only Results in ensemble of pathways Reaction coordinate is a result of simulation not an input Allows for calculation of rate constants C. Dellago, P.G. Bolhuis, P.L. Geissler Adv. Chem. Phys. 123, Apply when process of interest is a rare event is complex and reaction coordinate is not known Examples: nucleation, reactions in solution, protein folding

56 Path probability density Path = Sequence of states x iδt

57 Transition path ensemble h A =1 h B =1

58 Metropolis MC of pathways 1. Generate new path from old one 2. Accept new path according to detailed balance: 3. Satisfy detailed balance with the Metropolis rule:

59 Shooting moves ) ( ) ( )] ( ) ( [ ) ( ) ( 0 ) ( ) ( n T B n A n o acc x h x h T x T x P = accept reject! " # = A x if A x if t h t t A 0 1 ) (

60 Standard TPS algorithm take existing path choose random time slice t change momenta slightly at t integrate forward and backward in time to create new path of length L accept if A and B are connected, otherwise reject and retain old path calculate averages repeat

61 Definition of the stable states

62 Classical nucleation (1926) ΔG 0 R Liquid R * R Crystal nucleus surface bulk ΔG = 4πR 2 γ 4 3 πr3 ρδµ ls How does the crystal form? What is the structure of the critical nucleus Is classical nucleation theory correct? What is the barrier? Rate constant γ : surface tension Δµ : chem. pot difference ρ: density

63 Path sampling of nucleation TIS in NPH ensemble, as density and temperature change N=10000, P=5.68 H=1.41 (25 % undercooling) D. Moroni, P. R. ten Wolde, and P. G. Bolhuis, Phys. Rev. Lett. 94, (2005)

64 Sampling paths is only the beginning Eugene Wigner: "It is nice to know that the computer understands the problem. But I would like to understand it too. Path ensemble needs to be further explored to obtain: Rate constants Free energy Transition state ensembles Mechanistic picture Reaction coordinate Illustrative example: crystal nucleation

65 Transition interface sampling B A Overall states in phase space: directly coming from A directly coming from B T. S. van Erp, D. Moroni and P. G. Bolhuis, J. Chem. Phys. 118, 7762 (2003) T. S. van Erp and P. G. Bolhuis, J. Comp. Phys. 205, 157 (2005)

66 A B λ i-1 λ i λ i+1 P A (λ i+1 λ i ) = probability that path crossing i for first time after leaving A reaches i+1 before A k TIS AB = φ AB h A n 1 = φ AB P A (λ i+1 λ i ) = Φ A P A (λ i+1 λ i ) h A i=1 n 1 i=1

67 TIS results for nucleation Free energy follows directly Moroni, van Erp, Bolhuis, PRE, 2005 Structural analysis?

68 Committor (aka p-fold, splitting probability) Probability that a trajectory initiated at r relaxes into B r B A L. Onsager, Phys. Rev. 54, 554 (1938). M. M. Klosek, B. J. Matkowsky, Z. Schuss, Ber. Bunsenges. Phys. Chem. 95, 331 (1991) V. Pande, A. Y. Grosberg, T. Tanaka, E. I. Shaknovich, J. Chem. Phys. 108, 334 (1998)

69 Transition state ensemble r is a transition state (TS) if p B (r) = p A (r) =0.5 A B A B TSE: Intersections of transition pathways with the p B =1/2 surface

70 Committor distributions

71 Committor distribution N=243 Clearly, n is not entire story

72 Small and structured Structure Big and unstructured Committor analysis gives valuable insight

73 Outline Part 1 Rare event and reaction kinetics Linear Response theory Transition state theory Free energy methods Bennet Chandler approach Example zeolites Part 2 Two ended methods Transition path sampling Rate constants Reaction coordinate analysis Application to biomolecules

74 All-atom force fields for biomolecules Potential energy for protein V (r) = k r (r r eq ) 2 + k θ (θ θ eq ) i< j ' ) ( bonds a ij r ij 12 b ij r ij 6 + q i q j εr ij *, + angles 1 2 v (1+ cos(nφ φ ) n 0 dihedrals θ( 3 r 2,3 φ( 4 4 vdw interactions only between non-bonded i-j >4

75 Currently available empirical force fields CHARMm (MacKerrel et 96) AMBER (Cornell et al. 95) GROMOS (Berendsen et al 87) OPLS-AA (Jorgensen et al 95) ENCAD (Levitt et al 83) Subtle differences in improper torsions, scale factors 1-4 bonds, united atom rep. Partial charges based on empirical fits to small molecular systems Amber & Charmm also include ab-initio calculations Not clear which FF is best : top 4 mostly used Water models also included in description TIP3P, TIP4P SPC/E Current limit: 10 6 atoms, microseconds ( with Anton ms)

76 Folding of Trp-cage 20-residue protein NLYIQ WLKDG GPSSG RPPPS 2-state folder, experimental rate 4 µs (Andersen et al, Nature 2002, Zhou et al. PNAS 2004, others) System: 1L2Y in 2800 waters OPLSAA, PME, Nose-Hoover, GROMACS one time per 4 s! What is folding mechanism and kinetics in explicit water at 300K? Strategy: - Stable states by replica exchange MD - Mechanism by path sampling - Rate calculation Free! Energy! about equal population of unfolded and folded!

77 TPS of Trp-cage folding loop-helix 80% Rates by TIS! folding! helix-loop 20% unfolding! Jarek Juraszek! J. Juraszek, PGB PNAS 2006! Biophys. J (2008)! Rates! TIS! exp! k fol! 0.2 s -1! 0.24 s -1!! k unf! 0.8 s -1! 0.08 s -1!

78 Advanced path sampling results Very recent single replica multiple state transition interface sampling revealed much more complex network with many short-lived intermediates 1.0" U" Lo".2" Commi5or" other" SN" LSN" I".1" Pd" LN".001" meta" 0" N" W.Du & PGB, J. Chem. Phys 2014

79 Photoactive Yellow Protein bacterial blue-light sensor Tyr42 pca Glu46 Cys69 N-terminal domain PAS domain Thr50 Arg52 Absorption of a blue-light photon triggers the photo cycle J. Vreede et al. Biophys. J. 2005

80 Partial unfolding Loss of α-helical structure Exposure of hydrophobic groups Increased flexibility in parts of the protein backbone cis-chroh + Glu46- Force field MD (gromacs) Gromos96 - SPC water - PME Replica Exchange H 2 O H 2 O - H 2 O H 2 O H 2 O H 2 O

81 Partial unfolding: REMD free energy Unfolding happens on millisecond timescale (1000 times longer than possible by direct MD) Helix unfolded! Chromophore and Glu46 are solvent exp Agreement with experiment! Vreede et al. Biophys. J. 2005! Vreede et al. Proteins 2008! Bernard et al. Structure 2005! Jocelyne Vreede!

82 Partial unfolding: REMD free energy Unfolding happens on sub-millisecond timescale (1000 times longer than possible by direct MD) Helix unfolded! Chromophore and Glu46 are solvent exp Agreement with experiment! Vreede et al. Biophys. J. 2005! Vreede et al. Proteins 2008! Bernard et al. Structure 2005! Jocelyne Vreede!

83 Transitions in the partial unfolding

84 Solvent exposure transitions Vreede, Juraszek, PGB, PNAS 2010!

85 Juraszek, Vreede, PGB, Chem Phys 2011!

86 The end