Enhanced sampling via molecular dynamics II: Unbiased approaches

Transcription

1 Enhanced sampling via molecular dynamics II: Unbiased approaches Mark E. Tuckerman Dept. of Chemistry and Courant Institute of Mathematical Sciences New York University, 100 Washington Square East, NY 10003

2 Spatial-Warping Transformations Z. Zhu, MET, S. O. Samuelson, G. J. Martyna, Phys. Rev. Lett (2002) P. Minary, MET, G. J. Martyna SIAM J. Sci. Comput. 30, 2055 (2007) Science, December 22, 2006

3 Molecular dynamics in a double well. V(x) p x = m dv p = = F( x) + heat bath dx x time Crossing the barrier is a rare event!

4 Spatial-warping transformations Z. Zhu, MET, S. O. Samuelson, G. J. Martyna, Phys. Rev. Lett (2002) V(x) Hamiltonian: 2 p H( x, p) = + V( x) 2m Canonical partition function: 2 p x= a x= a Q = dp dx exp β + V ( x) 2m Change the integration variables: u = f ( x) = a + dy e β a Transformed partition function: x V ( y) 1 βv ( f ( u)) dx = e du 2 p Q = dp du exp β + V f ( u) V f ( u) 2m ( 1 ) ( 1 )

5 V ( x ) Spatial-warping transformations (cont d) Choosing the transformation: V ( x) V( x) a< x< a = 0 Otherwise x = a x= a V(x) V ( x) V ( x ) V ( x ) 1 1 ( ( )) ( ( )) V f u V f u MD: p d u = p = Fu ( ) + heat bath F( u) = V f ( u) V f ( u) m du Reference potential spatial warping (REPSWA) 1 1 ( ) ( )

6 How is the space warped? Barrier: kt B Barrier: 5kT B Barrier: 10kT B

7 V No Transformation Transformation 5kT 10kT 10kT

8 How to do biomolecules and polymers P. Minary, MET, G. J. Martyna SIAM J. Sci. Comput. 30, 2055 (2007) Sampling hindered by barriers in the dihedral angles: V ( φ) Ramachandran dihedral angles (alanine dipeptide): φ ψ ψ φ

9 Dihedral angle transformations Transformed angle: φ = + d e u min φ φ φ χ min βv ( χ) V ( φ) φ min

10 P. Minary, G. J. Martyna and MET SIAM J. Sci. Comp. 30, 2055 (2008) r 6 V ( φ,{ r}) = V ( φ) S ( φ) + α V ( r ( φ,{ r}) r ) S ( r ( φ,{ r}) ri ) tors 1 n.b. 4 i 2 4 i nn

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12 A 400-mer alkane chain No nonbonded interactions 397 collective variables! Nonbonded interactions off

13 No Transformation REPSWA Transformations RIS Model value: 10

14 β-barrel Proteins

15 Honeycutt and Thirumalai PNAS 87, 3526 (1990)

16 MD REMC Transformations PT replicas = 16

17 Equations of motion: Integrators in molecular dynamics p q =, p = Fq ( ) m Time evolution: q( t) (0) (0) il t q q = e = φ t p( t) p(0) p(0) where il p = + F( q) m q p t p t = + 2 p m q 2 p 3 exp( ilt) exp Fq ( ) exp t exp Fq ( ) O( t) Pseudocode: p p+ 0.5* t* F; q q+ t* p; GetForce( q); p p+ 0.5* t* F;

18 Multiple time scale (r-respa) integration MET, G. J. Martyna and B. J. Berne, J. Chem. Phys. 97, 1990 (1992) p q = m p = F + F fast slow p il = + F + F = il + il m q p p fast slow ref slow

19 Multiple Time Scale Integration (cont d) Trotter factorized propagator: 3 exp( il t) = exp( ilslow t / 2)exp( ilref t)exp( ilslow t / 2) + O( t ) = exp( ilslow t / 2) exp( ilref δt) exp( ilslow t / 2) n t exp( il t) = exp Fslow 2 p δt p δt exp Ffast exp δt exp Ffast 2 p m q 2 p t exp F 2 slow p n

20 From: G. J. Martyna, MET, D. Tobias, and M. L. Klein Mol. Phys. 87, 1117 (1996).

21 Illustration of resonance A. Sandu and T. S. Schlick, J. Comput. Phys. 151, 74 (1999) F = ω x F = Ω x fast 2 2 slow t 2 p = p(0) Ω x(0) 2 p ' x( t) = x(0)cos( ω t) + sin( ω t) ω p = p(0)cos( ω t) ωx(0)sin( ω t) t p t = p Ω x t 2 2 ( ) ( ) x( t) x(0) = A( ω, Ω, t) p( t) p(0)

22 Illustration of resonance (cont d) 2 Ω t 1 cos( ω t) sin( ω t) sin( ω t) 2ω ω A( ω, Ω, t) = t Ω 2 Ω t ω sin( ω t) tω cos( ω t) cos( ω t) sin( ω t) 4ω 2ω Note: det(a) = 1 Depending on Δt, eigenvalues of A are either complex conjugate pairs 2 < Tr( A) < 2 or eigenvalues are both real Tr( A) 2 Leads to resonances ( Tr(A) 2) at Δt = nπ/ω

23 Isokinetic dynamics Constraint the kinetic energy of a system: 2 i 3N 1 p = kt 2m 2 Introduce constraint via a Lagrange multiplier: i i p r = p = F λp i i i i i m i Derivative of constraint yields multiplier: Fp i i / mi pi pi i p i = [ Fi λpi] = 0 λ = m m p p / m Partition function generated: i i i i i i i i p 3N 1 Ω= d p δ kt d r e i 2mi 2 Resonance-free but is a poor thermostat. 2 N i N βu( r)

24 Consider the equations of motion: Resonance-free stochastic isokinetic thermostat P. Minary, MET, G. J. Martyna PRL (2004); B. Leimkuhler, D. Margul, MET Mol. Phys. (in press) dq = vdt F dv = λv dt m dv = λv dt v v dt, k = 1,..., L 1, k 1, k 2, k 1, k Gv ( ) dv = dt γv dt + σdw 1, k 2, k 2, k k Q2 G( u) Isokinetic constraint: Multiplier: = = Qu β k= 1 λ L mv Q v v v v L L , k = Λ (, 1,1,..., 1, L) = β L + 1 k= 1 L L vf Q v v L + 1 Λ( vv,,..., v ) 1,1 1, L 2 1 1, k 2, k

25 Multiple time-step integrator Assume forces have fast and slow components: F = Ff + Fs il q = v q il = il + il + il + il + il ( f ) ( s) q v v N OU il F v v L ( f ) f ( f ) ( f ) v = λf F 1, k m λ v k = 1 v1, k F il v v L ( s) s ( s) ( s) v = λf λf 1, k m v k = 1 v1, k Gv ( ) il v v v v L L L 1, k N = λn λn 1, k 2, k 1, k + v k= 1 v1, k k= 1 v1, k k= 1 Q2 v2, k L L Qv v ( f ) vff ( s) vf s L + 1 k= 1 λf =, λf =, λ N = Λ Λ Λ 2 1 1, k 2, k

26 Multiple time-step integrator ( s) ( f ) /2 /2 ( f ) n ( s) il t iln t/2 ilv t/2 ilv δt/2 ilqδt il il OUδt qδt ilv δt/2 ilv t/2 iln t/2 e = e e e e e e e e e Solution of the Ornstein-Uhlenbeck process: e v = v (0) e + σ Rt ( ) ilouδ t γδ t 2, k 2, k 1 e 2γ 2γδ t

27 Numerical illustration of resonance Harmonic oscillator with quartic perturbation L 9 Uq ( ) = q q Nbins t 1 = 3, δt = ς () t = [ Pq ( i;) t Pexact ( qi) ] 100 N i= 1 bins P(q) q

28 Flexible SPC/E water U() r = U () r + U () r = U () r + U () r + U () r intra nb intra short long qq i jerfc( αr) ij Ushort () r = ULJ ( rij ) + ij rij 1 4π 2 2 α U ( r) = e S( k) q long 2 V kk, < k k S( k) = RESPA schemes i qe i ikr F. Paesani et al. J. Chem. Phys. 125, (2006) max k /4α 2 2 i π i qq U () r = U (; r α, k ) + 1 δ θ( r r ) ( ) ij (1) i j ij long recip res ij cut i> j rij ( ) ij erf ( αr) qq U () r = U (; r α, k ) 1 δ 1 θ( r r ) erf ( αr) (2) i j ij long recip max ij cut i> j rij

29 δ t = 0.5 fs t = 3 fs T =? RESPA 1: L = 4, r cut = 6 Å RESPA 2: L = 4, r cut = 8 Å

30 RESPA 1: L = 4, r cut = 6 Å L = 4 (colors) L = 1 (dashed) L = 2 (circles)

31 Parallel Tempering Swendsen and Wang, Phys. Rev. Lett. 57, 2607 (1986) T M TM 1 T 1 f r ( K ) K ( ) = ( K ) βk U ( r ) e Z( NVT,, ) K F M (1) ( M) ( K) ( r,..., r ) f( r ) = K = 1 ( K+ 1) ( K) ( K+ 1) ( K) ( K) ( K+ 1) fk( r ) fk+ 1( r ) A( r, r r, r ) = min 1, = min 1, e fk( r ) fk+ 1( r ) K, K+ 1 ( K) ( K+ 1) ( β β ) KK, + 1 = K K+ 1 U r ( K) ( K+ 1) ( ) U( ) r

32 Comparison for a 50-mer chain using CHARMM22 REMC/PT replicas = 10; 300 < T < 1000, 10 replicas 20% acceptance Nonbonded interactions off

33 Comparison for 50-mer using CHARMM22 all interactions

34 Comparison for 50-mer with all interactions PT replicas = 10; REPSWA α = 0.8; Every 10 th dihedral not transformed

35 Replica exchange with solute tempering P. Liu, B. Kim, R. A. Friesner, B. J. Berne PNAS (2005); L. Wang, R. A. Friesner, B. J. Berne J. Phys. Chem. B (2011) A downside of replica-exchange is that U ~ N, therefore, the energy differences between neighboring replica need to shrink as the system size grows, which means more and more replica are needed in order for the acceptance probability to remain approximately fixed. Replica exchange with solute tempering (REST) was developed to allow just part of the system (the solute ) to be subject to attempted exchanges. It relies on the idea that different replicas can be subject to different potentials. f K () r = βkuk( r) e Z( NVT,, ) ( ( K ) ( K 1) ) ( ( K ( ) ( ) ( 1) ) ( ( K U r U r + β U r + U r ) )) = β + KK, + 1 K K K K+ 1 K+ 1 K+ 1 If the potential energy of a solute-solvent system is: U () r = U () r + U () r + U () r K s sv v The latest version of REST (called REST2) uses the following ladder of potentials: U () β β r = U () r + U () r + U () r K K K s sv v β0 β0 K

36 The ladder of potentials Replica exchange with solute tempering allows each replica to be run at a single temperature T 0 so that the solute probability e = e β ( β / β ) U ( r) β U ( r) 0 K 0 s K s For full potential ladder: βu () r = βu () r + ββ U () r + βu () r 0 K K s 0 K sv 0 v Acceptance probability determined by β K, K + 1 = K K + 1 s s + sv βk + βk+ 1 ( K) ( K+ 1) ( ) ( 1) ( ) ( ) ( ) 0 K K+ β β U r U r ( U ( r ) U( r ))

37 Solute potential energy Convergence of alanine dipeptide in solution