Implicit Models of Solvation

Transcription

1 Implct Models of Solvaton hvodan bedde påvke klmaet? Goal: Appoxmate (confomaton-dependent) solvaton fee eneges wthout explct solvent Cauton: Ths feld s faught wth contovesy and vaous goups of eseaches wth dogmatc ponts of vew. On the othe hand, mplct solvent models have poven vey useful fo a wde ange of applcatons to bomolecula systems. Focus wll be on models of solvaton based on contnuum electostatcs; nonpola aspects coveed pevously. ε= ε=

2 Heachy of Solvent Models Polazable explct solvent Computatonal expense Fxed chage explct solvent Nonlnea Posson-Boltzmann Lnea Posson-Boltzmann Genealzed Bon Physcal ealty Dstance-dependent delectc Suface aea based models

3 Intoducton to PB Basc dea: Teat solvent as a delectc contnuum. Essentally exact wthn the constants of ths appoxmaton. Often teated as the gold standad fo mplct solvent models. Howeve, the contnuum appoxmaton smply cannot account fo a numbe of mpotant fst-shell solvaton effects elated to the sze of a wate molecule. [Thee s absolutely no nfomaton about the sze of a wate molecule wth PB teatments.] Numecal soluton of the contnuum electostatcs ntoduces addtonal appoxmatons. Bay Hong was one of the fst people to ecognze the usefulness of PB fo macomolecules, and developed the most mpotant pogam, Delph. Thee ae othe methods that ae fomally smla, e.g., the Langevn dpole method of Washel. But PB has had the lagest mpact by fa.

4 ϕ( )= ε ( )= ρ( )= The Posson Pat of PB ε 4 ( ) ϕ( ) = πρ( ) electostatc potental delectc constant (small nsde poten; 80 outsde) chage densty (patal chages nsde; ons outsde) Ths s one of the fundamental equatons of classcal electostatcs. In fact, Coulomb s law can be deved as a specal case whee the delectc s constant. The teatment of chage densty due to ons n soluton needs specal consdeaton. Can buld a model based on the assumptons of Debye-Huckel theoy; that s how we get the Boltzmann pat...

5 Ionc Contbutons to Chage Densty Debye-Huckel theoy gves the densty of ons as ρ ( ) = ρ 0 e q ϕ Ionc densty n bulk soluton ( ) kt Ths just gves a model fo the enchment of, e.g., negatve ons n places whee the potental s postve. So, fo a 1:1 salt soluton, we have ρ onc ( ) = ρ ( ) + ρ ( ) + = ρ 0 e ϕ ( ) + ϕ ( ) ( ) kt ρ 0 e kt = ϕ ρ snh kt 2 0 The geneal esult s wtten as...

6 The Boltzmann Pat of PB ε 4 ( ) ϕ( ) κ snh[ ϕ( )] = πρ( ) Ths s the nonlnea PB equaton. Note that the chage densty on the ght sde s now just the patal chages n the cavty. In many applcatons, the onc stength tem s lneazed, by expandng t n a Taylo sees and takng the fst tem: 3 ( ) ϕ snh[ ϕ( )] = ϕ( ) + + L ϕ 6 ε ϕ κϕ = 4πρ ( ) ( ) ( ) ( ) ( ) Ths s easonable when the onc stength of the soluton s not too hgh. The lnea PB equaton s computatonally smple to solve, but the nonlnea PB equaton s pobably safe fo thngs lke DNA/RNA whee onc stength effects ae known to be mpotant.

7 Fnte Dffeence Solutons fo PB Delph uses ths method. Lay down a 3D gd aound the poten. The PB equaton must be satsfed at evey gd pont. Devatves n the PB equaton eplaced by fnte dffeences ove ths cube. Patal chages of poten must be dstbuted onto gd (smla to PME). Delectc constant smple to assgn. The value of the potental at each gd pont depends on the neghbong gd ponts. Thus you need an teatve soluton. The accuacy of ths method depends ctcally on the fneness of the gd. Can pefom a sees of calculatons wth nceasng gd sze ( focusng ). The PB equaton can also be solved by a bounday element method. The two appoaches ae fomally equvalent. My undestandng s that FD s moe effcent fo nonlnea PB, but BE may be moe effcent fo lnea PB.

8 What s the ntenal delectc constant of a poten? Nonpola oganc molecules have delectc constant of ~2-4. Ths s due pmaly to electonc polazablty. Ths s a easonable estmate fo the ntenal delectc of a poten. Some computatonal wok, e.g., pedctng the pka s of goups nsde potens, has shown bette ageement wth expement usng lage values of the ntenal delectc, such as 20. Some expements have also suggested lage ntenal delectcs. Howeve, these lage values may eflect effects that should not be consdeed as pat of the delectc constant, specfcally 1) Confomatonal elaxaton. When the chage state of a goup changes, the est of the macomolecule can espond to help stablze t. Howeve, most PB calculatons ae done wth a fxed stuctue; usng a hgh delectc ntoduces an atfcal sceenng that can cudely mmc the effects of elaxaton. 2) Potonaton state changes. Changng the chage state of a goup can nduce othe potonaton states to change. If ths s not explctly taken nto account, the effectve delectc agan appeas to be bgge.

9 Gasp Ths s a geneal pogam fo vsualzng vaous popetes on the suface of a macomolecule (developed by Ncholls and Hong), but the most mpotant applcaton s vsualzng the electostatc feld (.e., as the feld lnes ntesect the suface). Red = negatve Blue = postve

10 Electostatc foces speed the bndng of the postvelychaged substate to acetylcholnestease by a facto of moe than 100. Applcatons of PB. 1. Role of Electostatcs n Lgand Bndng AChE and Fascculn 2 bnd wth electostatcally-steeed, dffuson-contolled knetcs. Hong goup, Columba U.

11 Applcatons of PB. 2. Role of electostatcs n on channels Contnuum electostatcs calculatons based on the fnte-dffeence Posson- Boltzman equaton ae used to show that the cavty and the poe helces of the KcsA channel ae "tuned" to be pefeably occuped by a monovalent caton. Note that the ole of helx dpole n othe potens s contovesal, but n membanes (low delectc), seems to be faly stong. B. Roux and R. MacKnnon, "The cavty and poes helces n the KcsA K+ channel: electostatc stablzaton of monovalent catons", Scence 285, (1999).

12 Applcatons of PB. 3. pka calculatons The effectve pka of a goup (e.g., caboxylate) can be stongly petubed by ts envonment n a macomolecule. Fo example, a caboxylate that foms a salt bdge wth Ag o Lys s moe lkely to be negatvely chaged, whle a caboxylate bued n a hydophobc cavty s moe lkely to be neutal (desolvaton of a unt chage costs about 60 kcal/mol!). PB calculatons can quantfy ths effect, and pedct how the effectve pka (.e., the ph at whch the goup s 50% potonated) shfts fom the nomnal value. model system (neutal) e.g., acetc acd model system (chaged) pka maco = pka model n macomolecule (neutal) e.g., Asp n macomolecule (chaged) ( maco model G G ) 2.303RT The tcky pat s that the pka of the goup wll depend on potonaton states of all othe ttatable sde chans; need some sot of optmzaton stategy.

13 Othe Applcatons of PB Molecula dynamcs. It s tough to get devatves of the solvaton fee enegy, whch ae fomally needed fo ntegatng Newton s equatons of moton. But t can be done. Scong functon fo modelng o lgand dockng. We ll have a lectue late on MM-PBSA, whch can povde easonably accuate estmates of bndng fee eneges. Identfyng esdues ctcal fo bndng, o explan mutageness esults. It s staghtfowad to quantfy the electostatc consequences of mutatons, e.g., n a poten-poten nteface.

14 Intoducton to GB Ognally developed by Clak Stll. Ognal pape s vtually mpossble to undestand; GB appoxmaton emans a bt mysteous. My own vewpont: GB can be undestood as an envonment-specfc dstance dependent delectc model. In othe wods, t coectly pedcts (at least qualtatvely) dffeences n effectve electostatc nteactons n poten nteo vs. solvent-exposed exteo. Ease to obtan devatves of solvaton fee eneges than n PB methods; vey useful fo MD and mnmzaton.

15 Basc Idea of GB The Bon model of on solvaton: PB equaton fo a sngle chage at the cente of a sphee has an analytcal soluton... G solv 1 1 = 1 2 ε ext 2 q R Now consde a chage nsde an abtay cavty (e.g., macomolecule). Assumng we can compute the assocated solvaton fee enegy, equate ths wth the Bon equaton to detemne the sphee sze that would gve the same esult. Ths effectve Bon adus s usually epesented by the symbol α G solv q 2 ε = ext α If the atom s solvent exposed, the Bon adus wll be close to (but lage than) the atomc adus. Fo a bued atom, the Bon adus can become qute lage. Now, consde seveal chages nsde an abtay cavty. Each one has ts own Bon adus. The GB method povdes a method fo estmatng the total fee enegy of solvaton, based on these Bon ad.

16 The GB Pa Tem G pa j = ε nt 1 ε ext 2 j q q + α α j exp j 2 j 4α α j Q: How dd Stll come up wth ths tem? A: No one knows. Howeve, notce the followng... 2 self pa q 1. If =j, then get back the Bon equaton: G = G = 2 ε ε nt ext α 2. In the lmt of lage ntenuclea dstances, get complete sceenng. That s, the pa tem almost exactly cancels the Coulomb s law attacton: pa pa 1 1 qq j Gj + G j = ε nt ε ext j 3. You can also show that the pa tem leads to coect behavo n the lmt of a dpole nsde a sphee, whch s also analytcal (.e., usng PB equaton). 4. Othe than the lmtng behavos, I know of no physcal bass fo the eqn. solv pa self G = G = G + 2 j j j> G pa j

17 How do we get Bon α s? Fom solvng PB equaton: In pactce, t would be enomously neffcent to solve the PB equaton fo each patal chage. Howeve, an nteestng ecent pape... Effectve Bon Rad n the Genealzed Bon Appoxmaton: The Impotance of Beng Pefect. ALEXEY ONUFRIEV, DAVID A. CASE, DONALD BASHFORD. J. Comp. Chem. 23 (2002) Ognal volume ntegal fomulaton: E k = π ε nt 1 ε ext V 2 qk R k 4 d 3 R k = poston of the chage R = vecto of ntegaton Ths s not an exact soluton fo the solvaton fee enegy of the chage n the cavty, but can be shown to be a fst-ode appoxmaton. Specfcally, n the bounday element fomulaton of PB, t epesents the nteacton of the pont chage wth the suface chage t nduces, but neglects hghe-ode effects due to the electc feld nduced by the suface chage.

18 How do we get Bon α s? Suface ntegal fomulaton: Fesne ponted out that, by Geen s theoem, volume ntegal can be tansfomed to a suface ntegal (bette computatonal scalng) E k = 8 π ε nt ε ext S 2 qk R k 4 2 ( R ) n d R k k = poston of the chage R = vecto of ntegaton n = suface nomal Analytcal appoxmatons: Calculaton of the Bon ad by suface o volume ntegaton can be slow. These methods bascally assgn Bon ad based on local envonment of atom, usng analytcal, dffeentable fomulas. In my opnon, these methods do not captue dffeences n bual nealy as well as ntegaton methods, but clealy they ae much faste.

19 A Note About Sufaces All contnuum electostatcs calculatons must defne whee the suface of the macomolecule somehow,.e., whee does the hgh delectc end and the low delectc begn. Thee ae multple possble defntons, all of whch have the own pathologes. Molecula suface : Ths s the most commonly used suface fo PB. Imagne ollng a sphee (sze of a wate) on the suface of the macomolecule. All ponts of contact between the sphee and the macomolecule defne the suface. An analytcal defnton has been povded by Connolly. People sometmes efe to ths as the coect suface, but that t meanngless. Thee ae poblems wth shap cusps, and t s vtually mpossble to get devatves. Dffcult to fnd ntenal cavtes. Van de Waals suface : Smply take the ntesecton of the VDW sphees defned by each atom. Ths suface s poblematc because t leads to lage amounts of hgh delectc n the nteo of macomolecules, due lagely to small cavtes (too small to actually accommodate wates). Gaussan suface : Cente a 3D Gaussan aound each atom (somethng lke an s obtal. Add togethe all the Gaussans. Can then ethe 1) teat the suface as a contnuum, o 2) daw an socontou. Ths has the advantage of much ease devatves.

20 PB vs. GB Thee ae some athe dogmatc vewponts out thee on ths ssue. Anthony Ncholls gave an nfamous talk at ACS called GB and othe bad solvaton models. Two types of studes have been pefomed: 1) Teat PB as the coect answe (dangeous!) and see how well GB (and othe smplfed models) epoduce t. 2) Compae PB and GB at epoducng solvaton fee eneges of small molecules. Ths s fne, but small molecules (all atoms solvent exposed) ae vey dffeent than macomolecules (bued coe; moe complcated sufaces). Computatonal expense of eneges and devatves needs to be consdeed.

21 Implct Solvent vs. Explct Solvent Implct solvent geneally cannot be used when tyng to lean about dynamcs o knetcs, whch ae obvously nfluenced by the ganulaty of wate. Dynamcs ae geneally much faste n mplct solvent. Keep n mnd that the key ssue s not the cost of calculatng an enegy of nteacton between the poten and explct solvent, but gettng a solvaton fee enegy (.e., ensemble aveagng ove solvent confguatons). Explct solvent models, and peodc bounday condtons, of couse ae not fee fom atfacts. Fom my eseach and othes, t s clea that mplct solvent models can have a tough tme wth chaged systems. Wate n confned spaces s anothe tcky ssue.

22 When does mplct solvent beak down? An example fom my own wok. PBF SGB DelPh FEP Dvegence fom FEP (kcal/mol) FEP: Explct solvent PBF/Delph: 2 dffeent PB solves N-O dstance (Å) Conclusons: Devatons between mplct and explct solvent ae epoducble acoss seveal mplct models, ncludng both GB and PB. Peak devaton at 4-5 Å eadly explaned n tems of fst shell solvaton effect that we efe to as bdgng wates.

23 Dstance Dependent Delectc Heustc Models Use Coulomb s law, but make delectc constant vay wth dstance between chages. Most commonly, delectc nceases lnealy wth dstance. ε( ) Ths was used vey fequently n the ealy days of molecula dynamcs. It s bette than vacuum (ε=1), but ε= stll gossly oveestmates electostatc nteactons. ε=4 woks a lttle bette. Howeve, whateve the detals, note that DDD teats all pas of chages n a macomolecule dentcally, egadless of whethe they ae on the suface o n the nteo. [Ths s the bg advantage of GB; although the sceenng tem s pobably not pefect, t does teat coe and suface atoms dffeently.] Suface Aea Based Methods G solv = σ A Not used much n MD (devatves ae tcky), but shows up n othe contexts n combnaton wth foce felds, e.g., homology modelng. No eal physcal bass.