MOLECULAR DYNAMICS ,..., What is it? 2 = i i

Transcription

1 MOLECULAR DYNAMICS What s t? d d x t 2 m 2 = F ( x 1,..., x N ) =1,,N r ( x1 ( t),..., x ( t)) = v = ( x& 1 ( t ),..., x& ( t )) N N

2 What are some uses of molecular smulatons and modelng? Conformatonal searchng wth MD and mnmzaton Exploraton of molecules fluctuatons and dynamcs molecular move; correlaton functons; quas-harmoncal modes (prncpal components); MD as an ensemble sampler: thermodynamcs descrbes the drvng force for the process; knetcs descrbes the mechansm for the process; Free energy smulatons wth MD: estmates of bndng free energes; stablty of molecular assembles; MD s wdely used n expermental procedures such as X- ray crystallography and NMR structure determnaton

3 Some Hstory 1957 Hard spheres, Alder B., Wanwrght T ( J.Chem. Phys.) Observed phase transton 1964 Smple Lqud Rahman A. ( Phys. Rev. A) Calculated dffuson constant for lqud argon whch dffers from expermental value by less then 15% 1967 Smple Lqud Verlet N. ( J. Chem. Phys.) Calculated thermodynamc state equatons of noble gases whch were approved later n experments 1974 Lqud Water Stllnger F., Rahman A. (J. Chem. Phys) Par correlaton functon whch s close to the expermental one 1975 Polymer Chan Balabaev N., Grvtsov A., Shnol E. ( Proc.Acad.Sc.USSR) Pulsatons of freely-jonted chan Proten McCammon J., Geln B., Karplus M. (Nature) The structure of the bovne pancreatc trypsn nhbtor dd not move far from the X-ray structure durng 10 ps.

4 N=32,800 atoms t max = 10-9 s

5 N=11,200 atoms, t max = 2ns

6 What s the bass for the MD smulatons? Potental energy functon provdes relatonshp between structure and energy U ( r) F( r) = - U r Propagator allows to calculate the coordnates and veloctes at tme t+h from known coordnates and veloctes at tme t - equatons of motons; - ntegraton algorthm.

7 POTENTIAL ENERGY FUNCTION FOR BIOMOLECULES It s not yet feasble to treat large systems usng quantum mechancs Emprcal potental energy functons (force felds) provde reasonably good compromse between accuracy and computatonal effcency Most commonly used: AMBER, CHARMM, GROMOS, OPLS and DREIDING Mxed QM/MM force felds are under development (to allow events lke bond makng/breakng)

8 CHARMM force feld U (r) = E bonded + E non bonded E bonded - local, short-range nteractons; 2-, 3-, and 4- body nteractons whch reflect topology of chemcal bonds E non-bonded - non-local, long-range nteractons; par-wse nteractons between non-bonded atoms or atoms separates by 3 or more covalent bonds

9 1) Bond stretchng K b 500kcal / A 2 1,2-pars: atomc pars where atoms are separated by one covalent bond b 0 deal bond length K b force constant

10 2) Bond angle bendng θ K θ 50kcal / rad 2 1,3-pars: atoms are separated by 2 covalent bonds θ 0 - deal value of angle K θ - force constant

11 3) Torson angle potental models the presence of sterc barrers between atoms separated by 3 covalent bonds φ K 5kcal / θ mol 1,4-pars: atoms are separated by 3 covalent bonds n - coeffcent of symmetry K φ - force constant

12 4) The mproper dhedral term used to mantan chralty and planarty E mpr = K ( ω ω ) ω 0 2 torson angle ω

13 =,j pars nonbonded j j j j j Waals der van r r E σ σ ε σ ε

14 Calculaton of the nonbonded terms n the potental energy functon s the most tme consumng part of a molecular dynamcs smulaton Number of operatons : Local nteractons ~ N Non-bonded nteractons ~ N 2 To speed up the computaton: 1) the nteractons between two atoms separated by a dstance greater than a pre-defned dstance, the cutoff dstance, are gnored.

15 Dfferent ways to termnate the nteracton between two atoms : SHIFT: modfes the entre potental energy surface equlbrum dstances are slghtly decreased SWITCH : modfes the nteracton potental over a predefned range of dstances. suffers from strong forces n the wtchng regon whch can slghtly perturb the equlbrum structure The SWITCH functon s not recommended when usng short cutoff regons

16 2) Calculatons of non-bonded lsts R NBlst R cut NB lst s regenerated only at each INBFRQ step.

17 TREARMENT OF SOLVENT IN MD SIMULATIONS Implct treatment of the solvent: effectve delectrc constant ( dstant-dependent) ε eff = ε 0 r j Implct solvent potental energy functons Explct treatment of the solvent: boundary condtons should be consdered

18 1. Perodc boundary condtons Perodc boundary condtons enable a smulaton to be performed usng a relatvely small number of partcles n such a way that the partcles experence forces as though they were n a bulk soluton

19 2. Stochastc boundary condtons: buffer zone thermostat

20 Propagators and Integraton Algorthms Propagator ( mcroscopc equatons of moton) models external condtons (macroscopc or thermodynamc state) : - Mcrocanoncal ensemble (NVE) : The thermodynamc state characterzed by a fxed number of atoms, N, a fxed volume, V, and a fxed energy, E. (solated system) - Canoncal Ensemble (NVT): Ths s a collecton of all systems whose thermodynamc state s characterzed by a fxed number of atoms, N, a fxed volume, V, and a fxed temperature, T. (system n thermostat) - Isobarc-Isothermal Ensemble (NPT): Ths ensemble s characterzed by a fxed number of atoms, N, a fxed pressure, P, and a fxed temperature, T. (system n barostat) - Grand canoncal Ensemble (mvt): The thermodynamc state for ths ensemble s characterzed by a fxed chemcal potental, m, a fxed volume, V, and a fxed temperature, T.

21 Isolated System Propagator: Newton s equaton of moton v = dv dt dx dt m = F ( x 1,..., x N ) Numercal algorthms for ntegratng the equatons of moton should satsfy the followng crtera: the algorthm should conserve energy and momentum t should be computatonally effcent t should permt a long tme step for ntegraton

22 a( t) = F( t) m Verlet algorthm The Verlet algorthm uses postons and acceleratons at tme t and the postons from tme t-δt to calculate new postons at tme t+δt. The advantages of the Verlet algorthm are: ) t s straghtforward, and ) the storage requrements are modest. The dsadvantages are: ) the algorthm s of moderate precson ) the Verlet algorthm uses no explct veloctes.

23 Leap-frog Verlet algorthm 1. the veloctes are frst calculated at tme t+1/2δt (the veloctes leap over the postons) 2. these are used to calculate the postons, r, at tme t+δt. (then the postons leap over the veloctes) The advantage of ths algorthm s that the veloctes are explctly calculated. The dsadvantage s that they are not calculated at the same tme as the postons. (The veloctes at tme t can be approxmated)

24 Velocty Verlet Algorthm The advantages: ths algorthm yelds postons, veloctes and acceleratons at tme t there s no compromse on precson stable and convenent

25 System n a Thermostat (constant temperature) ) ( T Nk t K B >= < ) ( 2 1 ) ( 1 m v 2 t t K N = = 1. Rescalng veloctes B Nk t K T t v t v 3 ) / ( 2 ) ( ) ( 0 = λ λ It s used only durng equlbraton Ths s an artfcal procedure

26 2. Nose-Hoover method = = = = N B T Nk m v Q m v r F m v v r ) ( ζ ζ & & & & 3. Andersen method For each atom: at tme moment t c the veloctes are changed to a new ones w c tme moments t c occures randomly w c s taken randomly from the Boltsman dstrbuton at temperature T 0