Lecture 13: Conformational Sampling: MC and MD

Transcription

1 Statistical Thrmodynamics Lctur 13: Conformational Sampling: MC and MD Dr. Ronald M. Lvy Contributions from Mik Andrc and Danil Winstock

2 Importanc Sampling and Mont Carlo Mthods Enrgy functions ar uslss without sampling mthods Knowing th nrgy of vry point in a high-dimnsional phas spac is ssntial but not trribly informativ Thrmodynamic quantitis ar avrags nsmbl ovr th ntir phas spac ovr an W ar oftn intrstd in th distribution of crtain quantitis (.g. radius of gyration) avragd ovr all of th unintrsting dgrs of frdom (aka potntials of man forc )

3 Thrmodynamic quantitis ar avrags ovr an nsmbl ovr th ntir phas spac This could b valuatd by numrical intgration ovr a grid, or by gnrating random points uniformly ovr phas spac and stimating th intgral by MC intgration: In fact, th complt phas spac may not b ndd, sinc th vlocity contributions can oftn b accountd for analytically. Thn, w only nd to considr th potntial nrgy of conformational dgrs of frdom.

4 Ths mthods ar gnrally hoplss for molcular systms: N is hug, Q is unknown, and most points in a uniform sampling hav a vry small valu of th intgrand. Frnkl & Smit (2002) Undrstanding Molcular Simulations, 2nd Ed., Acadmic Prss

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6 Importanc Sampling by Markov Chain Mont Carlo Want to produc conformations distributd according to Givn a currnt point i in configuration spac, choos a subsqunt point i+1 with transition probability ( i, i+1) that dpnds only on i. To gt th corrct sampling, it is sufficint that th transition probabilitis satisfy microscopic rvrsibility: ( i) ( i, i+1) = ( i+1) ( i+1, i), or i ij = j ji (flux i to j = flux j to i = quilibrium) j i

7 πij =qij Pij Proposal probability (probability of picking mov) If q ji =q ij Accptanc probability (probability of accpting mov onc it has bn slctd) (symmtric MC schm) j U =min 1, on valid choic: P ij =min 1, i j i ρi π ij =ρi q ij =ρi q ji i ij = j ji i j ji = j q ji = i q ji j ij j i

8 Th classic Mtropolis algorithm Pick a dgr of frdom x Displac x by a uniformly distributd random numbr in rang ± Calculat th potntial nrgy diffrnc btwn th currnt stat i and th proposd displacd stat j Accpt th mov if U j U i Othrwis draw random numbr 0 1 and accpt if U j U i Rjct dos not man omit j i

9 Molcular Dynamics (MD) Connctions btwn microscopic information such as atomic positions and vlocitis and macroscopic obsrvabls is through statistical mchanics. In statistical mchanics, avrags ar dfind as nsmbl avrags Howvr, in MD simulations, w calculat tim avrags Ergodic Hypothsis A nsmbl= A tim

10 Historical Background 1956: Aldr and Wainwright MD mthod dvlopd to study intractions of hard sphrs 1964: Rahman first simulation with ralistic potntial - liquid Argon 1971: Stillingr and Rahman first simulation of ralistic systm - liquid watr 1977: McCammon, Glin, and Karplus first protin simulation - BPTI

11 Systm of N particls Positions of th N particls, Vlocitis of th N particls Enrgy(E) of th systm Kintic Enrgy Potntial Enrgy, V( r) Systm Tmpratur Microcanonical nsmbl (constant NVE) closd systm - no nrgy ntrs or lavs nrgy consrvation usd to chck MD algorithm

12 Nwton s Equations of Motion Th forcs ar complicatd functions of th coordinats, non-linar functions of position, so th st of 3N coupld diffrntial quations cannot b solvd analytically.

13 Numrical Intgration of th Equations of Motion Th intgrator is th hart of an MD algorithm Givn molcular position, vlocitis and othr dynamic information at tim t, w attmpt to obtain th positions, vlocitis, tc. at a latr tim t+δt to a sufficint dgr of accuracy Finit diffrnc mthod: 1 2 r t t =r t v t t a t t 2 v t t =v t a t t 1 a t t = F t t m Not vry accurat, lads to divrgncs unlss δt is mad vry small.

14 Bttr O(δt2) algorithms: vlocity Vrlt, position Vrlt, lap frog, prdictor-corrctor,...? Alln and Tildsly. Computr Simulations of Liquids

15 Th Liouvill oprator H H il= {, H }= p q q p For any proprty A: il= A t = i L t A t =0 q~r p~m v p2 H= V q 2m p F q =il q i L p m q p q-componnt propagats coordinats in tim: i Lq t p p q t 1 t q t =q t t m q m p-componnt propagats momnta in tim: i Lp t p t 1 t F q p t = p t t F q p

16 Trottr xpansion q t i L t q 0 = p t p 0 t=p t q t i L t P q 0 i L t i L t q 0 =[ ] = p t p 0 p 0 In gnral i L q L p t i Lq t i Lpt Bcaus Lq and Lp don't commut Howvr if δt is sufficintly small: i L q L p t i L p t /2 i Lq t i Lp t / 2 Trottr q t i L t P q 0 il =[ ] = p t p 0 p t /2 i L p t /2 i Lp t/2 i Lq t i L p t /2 3 O t q 0 p 0

17 rrespa q t i L t P q 0 il =[ ] = p t p 0 p t /2 i L p t /2 i Lp t/2 i Lq t i L p t /2 q 0 p 0 Dfins intgrator mad of P stps ach mad of 3 oprations: 1. Propagat momnta to δt/2: i Lp t /2 q 0 q 0 p 0 p t / 2 = p 0 F q 0 t / 2 1. Propagat momnta to δt/2: i Lq t q 0 q t =q 0 t p t / 2 / m p t / 2 p t / 2 1. Propagat momnta to δt: i Lp t /2 q t q t p t / 2 p t = p t / 2 F q t t / 2

18 rrespa quivalnt to vlocity Vrlt Most othr intgrators can b drivd using diffrnt forms of short tim xpansions of th Liouvill propagator

19 Choic of Tim Stp Tim Stp should b small nough for trajctoris to b clos to xact Enrgy Consrvation is usd as a critria for choosing th tim stp signals good nrgy consrvation W want to us a tim stp that minimizs computational tim, whil maintaining Enrgy Consrvation Th fastr tim scals control th tim stp δt to us. To intgrat vibrations nd to choos δt much smallr than 1fs

20 Multipl tim stp rrespa Dcompos total forc in a fast componnt (covalnt intractions inxpnsiv) and a slow componnt (non-bondd intractions xpnsiv): il= p F f q F s q =il f i L s m q p p Short tim propagator: i L t i Ls t / 2 i Lf t i L s t /2 Thn brak up innr fast propagator using a shortr tim stp: t / N i L t i Ls t / 2 [ i L t / N ] f N i Ls t / 2 Fast forcs ar applid N tims in th innr loop allowing largr tim stp in outr loop.

21 Constant Tmpratur Molcular Dynamics Canonical nsmbl (constant N,V,T) Thr ar diffrnt mthods for constant tmpratur MD Andrsn vlocity rsampling P v x, v y, v z m v x v y v z 2 Nosé-Hoovr thrmostat (xtndd systm) th systm is coupld to xtra dgrs of frdom which simulat hat bath. Original systm is canonical, xtndd systm is microcanonical Langvin thrmostat. Priodically atomic vlocitis ar stochastically prturbd basd on friction cofficint and rlaxation tim. Brndsn thrmostat (vlocity rscaling non canonical) T targt t s= 1 1 T instantanus

22 Typical Organization of a MD simulation 1) 2) 3) 4) 5) Enrgy minimization Thrmalization Equilibration Production Trajctory Analysis