Molecular Dynamics and Monte Carlo Methods

Transcription

1 May 8, 1 Molecula Modelng and Smulaton Molecula Dynamcs and Monte Calo Methods Agcultual Bonfomatcs Reseach Unt, Gaduate School of Agcultual and Lfe Scences, The Unvesty of Tokyo Tohu Teada 1

2 Contents Equatons of moton Molecula dynamcs (MD) method Execse 1 Isothemal sobac ensemble Monte Calo methods Execse Constant-tempeatue MD method

3 Classcal mechancs Newton s equaton of moton F ma F: foce, m: mass, a: acceleaton 3D vectos ae denoted by bold-face symbols. Scala values ae denoted by talc symbols. a s second-devatve of poston wth espect to tme t. d dx dy dz v,, dt dt dt dt a dv dt d dt d x, dt d y dt, d dt z 3

4 Soluton of equaton of moton (1) Fall of a body of mass m fom heght h h v = z F = mg d z m mg dt v gt C, v dz dt z z z t t v z h t 1 gt gt D h z h d z dt z t dvz g, dt C Two ntal condtons ae equed to solve second ode dffeental equatons. 4 1 gt D

5 Soluton of equaton of moton () Hamonc oscllato Mass: m Spng length: Length of unstaned spng: Foce constant: k F k F kq m d dt k q m d q dt kq 5

6 Soluton of equaton of moton () cos, cos,, sn cos, t q t q t q B q A v dt dq q q t B t A t q m k q q m k dt q d t Geneal soluton Intal condtons 6

7 Potental enegy and foce (1) Defnton of potental enegy E() at poston : E F d C Integate dotpoduct of F and d fom O to along C. Ogn O Path C Foce F Dsplacement d Poston 7

8 Potental enegy and foce () E E E E z z E y y E x x E z y x E z z y x E z z y x E z z y y x E z z y y x E z z y y x x E z y x E z z y y x x E E E F F,,,,,,,,,,,,,,,, Change n potental enegy by dsplacement. Snce we obtan Foce can be calculated fom potental enegy. 8

9 Enegy consevaton law (1) In an solated system, the sum (denoted by H) of potental enegy E and knetc enegy K emans constant. Knetc enegy 1 K m v Poof of the consevaton law dh dt dk de dv mv dt dt dt mv a v F v dx E dt x ma F 9

10 Enegy consevaton law () h z v = F = mg Lft W h mgh mg dz Wok nceases the potental enegy. E W mgh Release 1 z gt t K E h, g 1 mv h v mgh gh Potental enegy deceases. Knetc enegy nceases. The sum s unchanged. 1

11 Enegy consevaton law (3) v E t t H Hamonc oscllato t k d q d cost m mq dt v t t sn k q snt t kq k cos t 1 kq 11

12 Molecula dynamcs method Molecula dynamcs (MD) method calculates the tme vaaton of the postons and the veloctes of the atoms n a molecula system, evaluatng the foces fom the potental enegy functon and ntegatng Newton s equatons of moton. The equatons of moton fo a system composed of moe than two atoms cannot be ntegated analytcally. In ths case, they ae ntegated numecally, whee the whole calculaton s decomposed nto a sees of the calculatons fo a vey shot tme peod. Accuacy of the numecal ntegaton s evaluated by examnng the enegy consevaton. 1

13 Velocty Velet ntegato 1 x(t) f(x(t)) v(t) v(t+ t/) t f x t v t v t t vt t 3 x t t xt v t t m t 5 t vt 4 x(t+ t) f(x(t+ t)) v(t+ t) f x t t m t 1 Calculate foce Calculate veloctes at t+ t/ 3 Calculate coodnates at t+ t 4 Calculate foce 5 Calculate veloctes at t+ t 6 Retun to and epeat 13

14 Hamonc oscllato (1) Hamonc oscllato Mass: m Spng length: Length of unstaned spng: Foce constant: k F k F kq m d dt k q m d q dt kq 14

15 Hamonc oscllato () A Pel pogam (osc.pl) that numecally ntegates the equaton of moton of hamonc oscllato wth velocty Velet method Intal poston: q()=1, ntal velocty: v()= $q=1.;$v=.; $m=1.;$k=1.; $dt=.1;$nstep=1; sub calc_foce { my $q=$_[]; my $f=-$k*$q; my $e=.5*$k*$q**; etun ($e,$f); } open(out,">osc$dt.csv"); ($e,$f)=calc_foce($q); fo($=1;$<=$nstep;$++) { $v+=.5*$f/$m*$dt; $q+=$v*$dt; ($e,$f)=calc_foce($q); $v+=.5*$f/$m*$dt; $H=.5*$m*$v**+$e; pnt OUT $*$dt,",",$q, ",",$v,",",$h," n"; 15 }

16 Execse 1 Calculate the aveages of the absolute dffeences of total eneges and ts ntal value < H-H >, changng the tme steps as $dt=.1,.,.5,.1,.,.5, and 1 n osc.pl. Plot < H H > aganst the tme step n the Excel sheet osc.xlsx. Befly dscuss the esult. 16

17 Eo depends on tme step Plots of $q aganst $v calculated wth $dt =.1 (black),.5 (ed). < H-H > y =.633x -.5x + E Tme step Plot of < H H > aganst $dt. mv kq H.5 v q 1 H H coesponds devaton fom the ccle. 17

18 Choce of appopate tme step The smalle tme step causes the smalle eo n the total enegy. In geneal, 1/1 1/ of the cycle of the fastest moton s used fo the tme step. In the case of a poten, the fastest moton s the bond-stetchng moton (3 cm 1 ; 1 fs) of X H bonds (X=C, N, O, o S). Theefoe,.5 1. fs s appopate. 18

19 A system wth many atoms (1) A system composed of atoms wth van de Waals nteactons (vdw.pl) $wdth $natom=3; $wdth=1.; $scale=1.; $fcap=1.; $sgma=1.; $epslon=1.; $mass=1.; $nstep=1; $nsave=1; $dt=.1; $seed=1161; # Numbe of patcles # Wdth of ntal patcle dstbuton # Scalng facto fo ntal velocty # Foce constant fo sphecal bounday # Atom adus # Well depth # Atomc mass # Numbe of MD steps # Fequency of savng tajectoy # Tme step # Random seed 19

20 A system wth many atoms () Intal aangement Atoms ae andomly placed wth n a cube wth the edge length of $wdth. Intal veloctes Randomly assgned. The magntude can be changed by $scale paamete. Potental enegy functon: E E cap N 1 j1 N 4 j f cap cut j N 1 E cut cut cap s dstance fom ogn. cut s set to the half of $wdth.

21 A system wth many atoms (3) Result can be vsualzed by usng UCSF Chmea. 1. Double-clck the con of Chmea Choose Tools MD/Ensemble Analyss MD Move. Set Tajectoy fomat to PDB, PDB fames contaned n to Sngle fle, and vdw.pdb to the fle. Then, clck OK. 3. Choose Actons Atoms/Bonds stck to show atoms. 4. Clck playback button to stat anmaton. Examne the effect on the dynamcs of the paametes. 1

22 Compason wth expemental data Integaton of Newton s equatons of moton coesponds to the smulaton of the dynamcs of an solated system. Expementally obseved data ae the aveages ove a huge numbe (say 1 3 ) of molecules. Ae the esults fom molecula smulatons compaable wth the expemental data?

23 Real system Poten An solated system (Constant-NVE) A constant-tempeatue and constant-volume (constant- NVT) system composed of 1 3 poten molecules 3

24 Constant-NVT system (1) The system s composed of many dentcal sub-systems. Each sub-system s composed of a poten molecule and ts suoundng wate molecules. Each sub-system can exchange heat wth ts neghbos. Numbe of the unt system and the total enegy of the whole system ae constant. 4

25 Constant-NVT system () Expementally obseved data ae the aveages of the obsevables of each state weghted by ts pobablty of exstence. A : pobablty of exstence Dstbuton wth maxmum entopy = canoncal dstbuton 1 Z exp e kbt e : enegy of state Z exp e k T Z: patton functon A, 1 B 5

26 Constant-NVT system (3) In a molecula smulaton, each state n the whole system s geneated sequentally. Sequentally geneated ρ 1 ρ ρ 3 ρ n Weghted aveage X Expemental data n 1 X 6

27 Ensemble aveage Evaluate the ensemble aveage of the total enegy of a hamonc oscllato H q, p q, q, p H H p m k q, pq, pdqdp k T H q, p exp k T B H q, p exp dqdp kbt Method 1: Numecal ntegaton wth gd Method : Monte Calo ntegaton Method 3: Impotance samplng B Exact value 7

28 1: Numecal ntegaton wth gd p q Evaluate exp( H/k B T) at each gd pont and compute ts sum and the sum of the poduct wth H. Only the gd ponts wthn 1 q 1 and 1 p 1 ae consdeed. Plot the ato of the sums (.e. <H>) aganst the numbe of gd ponts. Evaluate H and exp( H/k B T). 8

29 : Monte Calo ntegaton p q Daw q and p fom a unfom dstbuton wthn 1 q 1 and 1 p 1. Evaluate exp( H/k B T) at each pont and compute ts sum and the sum of the poduct wth H. Plot the ato of the sums (.e. <H>) aganst the numbe of the sample ponts. 9

30 Compason of the esults (1) Black: Gd Red: Monte Calo k B T = 1. Monte Calo ntegaton s slow to convege. 3

31 Poblem of Monte Calo ntegaton Eo s nvesely popotonal to the squae oot of the numbe of samples To decease the eo by a facto of 1, 1 tmes lage samples ae equed. Howeve, gd appoach s not applcable to bomacomolecules due to the lage ntenal degees of feedom. A 1-esdue poten has moe than dffeent confomatons. It s necessay to mpove the accuacy of the Monte Calo method. mpotance samplng 31

32 Impotance samplng The Monte Calo ntegaton calculates the weghted sum of H wth the weghtng factos of exp( H/k B T). At (q, p) = (, ), the weghtng facto s one, wheeas at (q, p) = (1, 1), t s The contbutons to the aveage ae dffeent between sample ponts, whch deceases computatonal effcency. The effcency s maxmzed when the numbe of sample ponts fom a egon s popotonal to the weghtng facto, exp( H/k B T), of the egon. mpotance samplng 3

33 3: Impotance samplng Daw samples fom a dstbuton popotonal to the weght exp( H/k B T). Evaluate H at each sample pont and calculate the aveage. Plot the aveage <H> aganst the numbe of samples. 33

34 A Pel pogam $kt=1.; $p=atan(1.,1.)*4.; $max_npt=1; fo($npt=;$npt<=$max_npt;$npt+=) { $val1=.; fo($=;$<$npt**;++$) { $x1=and; $x=and; #Convet unfom dstbuton nto nomal dstbuton. $q=sqt(-.*$kt*log($x1))*cos(.*$p*$x); $p=sqt(-.*$kt*log($x1))*sn(.*$p*$x); } $H=.5*$q**+.5*$p**; $val1+=$h; #Sum of total enegy } pntf("%d %f n",$npt**,$val1/($npt**)); 34

35 Compason of the esults () Black: gd Red: Monte Calo Geen: Impotance samplng k B T = 1. Accuacy and effcency ae mpoved. 35

36 Sample geneaton (1) Unfom dstbuton y Cumulatve dstbuton functon g x x x dx Nomal dstbuton x Conveson nto nomal dstbuton g 1 x y, x g y 36

37 Sample geneaton () Ths method s possble only when the cumulatve dstbuton functon (CDF) can be calculated. It s mpossble to obtan an analytcal fom of the CDF fo a system of bomacomolecules, because the elaton between bonded and non-bonded nteactons s qute complcated. It s also mpossble to calculate t numecally due to the huge ntenal degees of feedom. Use Makov chan 37

38 Makov chan Tanston pobablty π j π kj π lj k l j π jj Let the pobablty of tanston fom state to state j be π j. Let the pobablty of exstence of state befoe tanston be ρ, the pobablty of exstence of state j afte tanston s gven by, 1 j j Tanston pobablty satsfes the followng: 1 j j 38

39 Example of Makov chan (1) Consde two states. Let tanston pobabltes be: π 11 =.6 π 1 =.4 π 1 =.3 π =.7 Stat wth state 1 Step State 1 State Pobablty State 1 State Step 39

40 Example of Makov chan () 1 When statng.8 wth state, the.6.4 pobabltes. convege to the same values. Step Afte convegence, ρ = ρπ. ρ s a egenvecto of matx π. ρ s unquely detemned by π. Pobablty State 1 State

41 Metopols method (1) We want to deve tanston matx fom pobablty dstbuton. The detaled balance condton s the suffcent condton fo ρ = ρπ. j j Metopols method: j j j j j j, j j j j f f 1, j j j 1 and and j j j j j j j j and j j j 1 j j 41

42 Metopols method () Randomly move an atom wthn a cube centeed at the atom wth the edge length of to geneate a new state. 1 j Wthn the cube N Outsde the cube j The move s accepted f the enegy of the new state, e j, s lowe than that of the ognal state, e. Othewse, the move s accepted wth the followng pobablty: j exp e e k T exp e k T j If not accepted, the atom does not move. 1 j j j B j j j 1 j j and j j B 4

43 An applcaton A hamonc oscllato Intal condton: (q, p) = (, ) <H> Numbe of samples 43

44 A Pel pogam $nstep=1; ($q,$p)=(.,.); $delta=1.; $kt=1.;$m=1.;$k=1.; $delta_q=$delta/sqt($k); $delta_p=$delta*sqt($m); open(out,">metopols$delta.csv"); #Numbe of steps #Intal states #Maxmum dsplacement #kt, mass, foce constant #Output fle $ave=.; $H=&calc_H($q,$p); #Intal enegy fo($=1;$<=$nstep;$++) { $q_new=$q+.*$delta_q*(and()-.5); #Tal move $p_new=$p+.*$delta_p*(and()-.5); #Tal move $H_new=&calc_H($q_new,$p_new); $pobablty=exp(($h-$h_new)/$kt); f($pobablty >= 1. $pobablty >= and()) { #Metopols cteon $q=$q_new;$p=$p_new;$h=$h_new; } $ave+=$h; pntf(out "%d,%f n",$,$ave/$) f($ % 1 == ); } close(out); sub calc_h { my ($q,$p)=@_; etun.5*$p*$p/$m+.5*$k*$q*$q; } #Enegy functon 44

45 Execse Download metopols.pl fom the web page of ths lectue and double-clck the con of the downloaded fle to execute t. Plot <H> aganst the sample numbe. Check whethe <H> conveges to 1. Check the convegence changng the value fo $delta. Ty $delta=.1. Dscuss why the convegence depends on $delta. 45

46 Applcaton to bomacomolecules Metopls method can be ealzed by choosng an atom andomly and movng the atom to a andom poston. Howeve, such a move changes the bond length and causes ncease of enegy. Pobablty of acceptance s vey small. To avod ths poblem, only dhedal angles ae changed. But, ths has followng dawbacks. It s dffcult to handle multple molecules. In the egon whee atoms ae closely packed, such as poten coes, change n the dhedal angle wll cause stec clashes. 46

47 Constant-tempeatue MD (1) Constant-tempeatue MD can geneate a canoncal ensemble. Ths can be moe easly appled to the systems of bomacomolecules than Monte Calo method. The ensemble aveage s gven by the tme aveage. Tempeatue s egulated by modfyng the velocty. N 3 v NkT 1 m 47

48 Constant-tempeatue MD () Nosé method Nosé-Hoove chan method Constant method Only the coodnate pat follows the canoncal dstbuton. Langevn dynamcs Tempeatue s egulated by fcton and andom foce. Beendsen weak-couplng method Does not geneate a canoncal ensemble. Smple and easy to use. Degees of feedom of the heat bath ae explctly consdeed. Heat bath 48

49 Langevn dynamcs The physcal system exchanges heat wth the heat bath though collsons wth fcttous patcles of the heat bath. Equatons of moton ma Fx v Rt Random foce R satsfes the followng: R Fcton Random foce caused by the collsons t, Rt Rt 6k T t t Aveage Vaance and covaance B 49

50 An applcaton A system composed of atoms wth van de Waals nteactons (vdw_langevn.pl) $natom=4; $wdth=1.; $fcap=1.; $sgma=1.; $epslon=1.; $mass=1.; $nstep=1; $nsave=1; $dt=.1; $seed=158; $gamma=1.; $kt=1.; # Numbe of patcles # Wdth of ntal patcle dstbuton # Foce constant fo sphecal bounday # Atom adus # Well depth # Atomc mass # Numbe of MD steps # Fequency of savng tajectoy # Tme step # Random seed # fcton coeffcent Knetc enegy Tme Aveage: 5.91 (exact value: 6) 5

51 Beendsen weak-couplng method 1. Calculate nstantaneous tempeatue T' at evey step of velocty Velet. N 3 m v NkT 1. Scale veloctes by a facto of χ. Constant τ contols the speed of adjustment. 1 t T 1 T 1 51

52 How to send you epot Use PowePont to ceate you epot. Repot should nclude the esults and dscusson of execses 1 and. Send the PowePont fle to tteada@u.a.u-tokyo.ac.jp. Subject of the e-mal should be Molecula modelng and wte you name and ID cad numbe n the body of the e-mal. 5