Lecture 19: Free Energies in Modern Computational Statistical Thermodynamics: WHAM and Related Methods

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1 Statistical Thrmodynamics Lctur 19: Fr Enrgis in Modrn Computational Statistical Thrmodynamics: WHAM and Rlatd Mthods Dr. Ronald M. Lvy

2 Dfinitions Canonical nsmbl: A N, V,T = k B T ln Q N,V,T : systm paramtrs and/or constraints Q N, V,T = N a! 3N a 1 3N 3N N b! h dq a dq b b U q a, q b, focus on fr nrgy changs that do not involv changing th numbr of particls, or thir vlocitis, nor volum and tmpratur. Can work with configurational partition function: U q a, q b, Z N,V, T =Z = dq a dq b Z 2 A= A 2 A 1 = k B T ln Z 1 Or quival ntly: A Z 2 = Z 1

3 Exampl 1: Configurational Fr Enrgy Changs controls a conformational constraint (distanc btwn two particls, dihdral angl, tc.). Call th constraind proprty f(q): only conformations that satisfy th constraint f(q)= (within a tolranc of d ar allowd: Z (λ)= dq δ[ f (q) λ ] β U (q ) Now: Z d = dq d [ f q ] U q = dq U q =Z =1 So: intgration of th constraind Z ovr all th possibl constraints givs th unconstraind Z Fr nrgy work for imposing th constraint: Z (λ ) dq δ[ f (q) λ ] β A (λ) = = βu (q) Z dq βu (q) = δ[ f (q) λ ] Z

4 Fr nrgy work for imposing th constraint: β A (λ) = δ[ f (q) λ ] Z it's basically an avrag of a dlta function ovr th unconstraind nsmbl. Th dlta function is non-zro only xactly at. In actual numrical applications w would considr som finit intrval around a discrt st of i. Considr: x = { 1/ /2 x /2 0 othrwis } dx x = lim 0 x = x Fr nrgy for imposing constraint in intrval : β Δ A (λ i ) =Δ λ δδ λ [ f (q) λ i ] Z = θ[ f (q) λ i ] Z x = { 1 / 2 x / 2 0 othrwis 1 =1 }

5 β Δ A (λ i ) = θ[f (q) λ i ] Z = 1 N sampls 1 N sampls sampl k θ [f (q k ) λ i ]= ( )= ni N sampls = pi whr: ni (a.k.a. histogram) : numbr of tims f q is within of i pi : probability of finding f q within of i thrfor: Δ A (λ i )= k B T ln pi At th ->0 continuous limit: A (λ )= k B T ln [ p(λ) d λ ] probability dnsity

6 A( ) is th potntial of man forc along : Th zro of th fr nrgy is arbitrary: discrt : pj Δ A ji =Δ A (λ j ) Δ A (λ i )= k B T ln pi continuous : p d =d [ f q ] Z p(λ j )d λ p(λ j ) Δ A ji = A (λ j ) A (λ i )= k B T ln = k B T ln p(λ i )d λ p(λ i )

7 Lsson #1: configurational fr nrgis can b computd by counting that is by collcting histograms or probability dnsitis ovr unrstrictd trajctoris. Probably most usd mthod. Not ncssarily th most fficint. Achivabl rang of fr nrgis is limitd: N sampls, N-1 in bin 1 and on sampl in bin 2 p1 = N 1 1 N p 2= 1 N p2 1 A max = k B T ln k B T ln p1 N For N=10,000, Amax ~ 5 kcal/mol at room tmpratur But in practic nds many mor sampls than this minimum to achiv sufficint prcision

8 Mthod #2: Biasd Sampling umbrlla potntial w q =w [ x q ] x Any thrmodynamic obsrvabl can b unbiasd : U q w x U q w x dq O q dq [O q ] O w=0 = = = U q w x U q w x ] dq dq [ Z U w O q w x U w Z U w w x U w = O q w x U w w x U w

9 Exampl: Unbiasd probability for bing nar xi: pi = x x i w x =0 = β w ( x ) w( x ) x x i w x w x w x w x β w( x ) βu (q) βw ( x) dq = = Z U+ w ZU βδ A = ZU+ w w Δ A w =fr nrgy of U + w (biasd) stat rlativ to th unbiasd stat pi = β Δ A w 1 N j θ[ x (q j ) xi ] β w [ x (q j )] β Δ A w β w( x i ) ni β Δ A β w (x ) = pi N w i Works only if biasd potntial dpnds only on ni =numbr of sampls in bin i in biasd nsmbl N =total numbr of sampls from biasd nsmbl pi =biasd probability at x i i p x ni β Δ Aw N β w ( x i )

10 How do w mrg data from multipl biasd simulations? distribution with biasing potntial 1 w1 (histogram probabilitis p1i) w2 distribution with biasing potntial 2 (histogram probabilitis p2i) w3 distribution with biasing potntial 3 (histogram probabilitis p3i) distribution with biasing potntial 1 (histogram probabilitis p2i) distribution with biasing potntial 3 Unbiasd distribution p2i Unbiasd distribution p3i w1 (histogram probabilitis p1i) distribution with biasing potntial 2 Unbiasd distribution p1i? w2 Unbiasd distribution pi w3 (histogram probabilitis p3i) p i = s u si p si Any st of wights usi givs th right answr, but what is th bst st of usi for a givn finit sampl siz?

11 Multipl biasd simulations to covr conformational spac Biasing potntials ws(x) x

12 Wightd Histogram Analysis Mthod (WHAM): Optimal way of combining multipl sourcs of biasd data Any st of wights usi givs th right answr, but what is th p i = s u si p si optimal (most accurat stimat) st of usi for a givn finit sampl siz? Lt's assum th unbiasd probability is known, w can prdict what would b th biasd distribution with biasd potntial ws: β Δ A s p i = biasd probability fr nrgy factor f s = β Δ As β w s (x i ) p si p si = p si = f s c si pi β Δ As β w s ( x i ) pi unbiasd probability bias of bin i in simulation s w s x i c si =

13 n si =histogram of x from simulation s Liklihood of histogram at s givn probabilitis at s (multinomial distribution): P n s1 n sm p s1 p sm = N s! i n si! n s1 n sm p s1 p sm N s = i n si = total numbr of sampls from simulation s In trms of unbiasd probabiltis: m P n s1 n sm p s1 p sm =const. f s c si pi n si i=1 Joint liklihood of th histograms from all simulations: P n s1 n sm ; ; n S1 n Sm p s1 p sm ; ; p S1 p Sm S m n f s c si p i s=1 i=1 si

14 Log liklihood: : S m n si ln P= ln f s c si pi const. s=1 i=1 Max liklihood principl: choos pi that maximiz th liklihood of th obsrvd histograms. Nd to xprss fs in trms of pi: f 1 s β Δ A s = Zs 1 βu (q) β w = = dq Z0 Z0 s [ x (q)] = 1 β U (q) β w [ x(q)] dq dx δ[ x (q) x ]= Z0 dx β w (x ) dq Z1 βu (q) δ [ x (q) x ]= dx β w ( x) δ[ x (q) x ] 0= 0 s s s dx β w (x) p0 ( x ) i β w (x ) Δ x i p0 ( x i )= i c si p i s s i

15 Log liklihood: : S m n si ln P= ln f s c si p i const= s=1 i=1 s ln f s i n si i ln pi s n si s i nsi ln c si = s N s ln f s i ni ln pi const ni =total numbr of sampls in bin i from all simulations N s = total numbr of sampls from simulation s ln P ln P ln P f s s = = pk pk f f s p pk nk Ns nk 2 s f s c sk = s N s f s c sk =0 p k fs pk s i

16 Thus (WHAM quations): k p = f nk s N s f s c sk 1 s = i c si p i Solvd by itration until convrgnc. Compar with singl simulation cas drivd arlir: i p ni β Δ Aw N β w ( x i ) WHAM givs both probabilitis (PMFs) and stat fr nrgis Frrnbrg & Swndsn (1989) Kumar, Kollman t al. (1992) Bartls & Karplus (1997) Gallicchio, Lvy t al. (2005)

17 What about thos optimal combining wights w talkd about? p i = s u si p si i p= s n si s ' N s' p si = f s' xp [ w s ' x i ] : WHAM solution n si n si = p si {N s f s xp [ w s x i ]} N s f s xp [ w s x i ] Substituting... p i = s psi N s f s xp[ w s x i ] s ' N s ' f s ' xp [ w s ' x i ] Thrfor... N s f s xp[ β w s ( x i )] N s xp { β [ w s ( x i ) Δ A s ]} u si = = s ' N s ' f s ' xp [ β w s ' ( x i )] s ' N s ' xp { β[ w s ' ( x i) Δ A s ' ]} WHAM optimal combining wights A simulation maks a larg contribution at bin i if: 1. It provids many sampls (larg Ns) 2. Its bias is small compard to its fr nrgy rlativ to th unbiasd stat.

18 WHAM: gtting unbiasd avrags Computing th avrag of an obsrvabl that dpnds only on x is straightforward: O x 0 = dx O x p0 x bin i x i O x i p0 x i = bin i O i pi From WHAM For a proprty that dpnds on som othr coordinat y=y(q). Solv for p0(x,y) no bias on y and thn intgrat out x: O y 0= dx dy O y p 0 x, y = dy O y p 0 y whr: p 0 y = dx p0 x, y =marginal probability

19 Som WHAM Applications for PMFs PMF of Alanin Dipptid kf kf 2 w s, = s s Chkmarv, Ishida, & Lvy (2004) J. Phys. Chm. B 108:

20 2D Potntial. bias = tmpratur w s q ~ 0 s U q Gallicchio, Andrc, Flts & Lvy (2005) J. Phys. Chm. B 109:

21 -hairpin pptid. Bias = tmpratur Gallicchio, Andrc, Flts & Lvy (2005) J. Phys. Chm. B 109:

22 Th Concpt of WHAM wights Considr th simpl avrag of O(x): O x 0 = bin i O i p = bin i i O i ni s N s f s xp[ w s x i ] Lt's sum ovr individual sampls rathr than bins and st: x i : cntr of bin i w s xi w s x k O x 0 = sampl k Whr: W k = s x k : position of sampl k O k 1 s N s f s xp [ w s x k ] 1 N s f s xp [ w s x k ] = sampl k O k W k WHAM wight of sampl k. Masurs liklihood of ncountring xk in unbiasd simulation.

23 WHAM wights xampl: probability distribution O x 0 = sampl k O k W k Apply prvious avraging formula to pi sampl k pi = [ x q x i ] 0= x k x i W k = sampl k bin i W k That is th unbiasd probability at xi is th sum of th WHAM wights of th sampl blonging to that bin.

24 Latst dvlopmnt: No binning WHAM = MBAR WHAM quations: i p= ni f s N s f s c si 1 s = i c si p i substitut Gt: f 1 s = i ni c si s ' N s ' f s' cs ' i f Sum ovr sampls 1 s = sampl k c sk s ' N s ' f s' cs' k MBAR quation Solvd itrativly to convrgnc to gt th f's Distributions obtaind by binning th corrsponding WHAM wights. Shirts & Chodra J. Chm. Phys. (2008). Tan, Gallicchio, Laplosa, Lvy JCTC (2012).

CE 530 Molecular Simulation

CE 530 Molecular Simulation CE 53 Molcular Simulation Lctur 8 Fr-nrgy calculations David A. Kofk Dpartmnt of Chmical Enginring SUNY Buffalo kofk@ng.buffalo.du 2 Fr-Enrgy Calculations Uss of fr nrgy Phas quilibria Raction quilibria