An introduction to continuum electrostatics

Transcription

1 Chapter An ntroducton to contnuum electrostatcs For both major subjects of ths work, foldng and ttraton, contnuum electrostatcs s of crucal mportance. Therefore, I wll gve n ths chapter a basc ntroducton to the concepts and methods of contnuum electrostatcs that were appled here. Solvent wth onc strength I Molecule I=0) low ) solvent accessble surface Charges wthn the molecule mmoble) hgh ) Ions moble) Ion excluson layer, hgh, but I=0 Fgure.: A molecule n a heterogenous delectrc medum. Fgure. shows a large molecule n an onc soluton. In the electrostatc contnuum approach, the solvent s not represented by explct solvent molecules, but mplctly by a medum wth a hgh delectrc constant. The nteror of the molecule, separated from the solvent by the solvent accessble surface, s assgned a low delectrc constant. At a frst glance, one should expect ths constant to be unty as n vacuum) snce the atoms of the molecule are all represented explctly n ths model. However, electronc and nuclear polarzaton effects not consdered explctly by the model may cause a hgher delectrc constant to be more approprate. The value used for the delectrc constant wthn the molecule s subject of ntense dscusson Warshel & Russel, 984; Warshel & Åqvst, 99; Hong & Ncholls, 995; Warshel et al., 997; Rabensten et al., 998a; Ullmann & Knapp, 999). I wll provde more detals of ths dscusson n connecton wth the applcatons descrbed n the followng chapters see especally secton ).

2 CHAPTER. AN INTRODUCTION TO CONTINUUM ELECTROSTATICS. The Posson-Boltzmann equaton.. Dervaton To descrbe the electrostatc nteracton of the molecular system depcted n Fgure. mathematcally, the Posson-Boltzmann equaton s used. It s derved from the Coulomb potental: φr q r r where the sum runs over all pont charges q at the poston r. s the delectrc constant. The Coulomb potental can also be formulated for the charge densty ρ nstead of pont charges: φr ρ r dr r r Applyng the Laplace operator to both sdes of ths equaton results after a few transformatons n the Posson equaton: φr 4π ρr.3) If s not constant but depends on the poston r, the Posson equaton adopts the followng form:.).) r φr!" 4πρr.4) To descrbe the effect of the ons n the solvent, the Debye-Hückel theory s needed, whch I wll not descrbe n detal here. The moble ons n the soluton arrange themselves Boltzmann dstrbuted n the electrc feld smlar to gas atoms n a gravtaton potental. The resultng charge densty ρ on r of the ons s added to the charge densty ρr wthout ons: ρ on r βq c s r q s e# s φ$ %r&.5) s The sum runs over all knd of ons s. c s s the orgnal concentraton of on s, q s s the charge of on s. β equals k B T #. Addng of the onc charge densty to the Posson equaton leads to the Posson-Boltzmann equaton PBE): ' r φr ) The Debye-Hückel theory requres overall electroneutralty: βq 4πc s r q s e# s φ$*%r& 4πρr.6) s c s r q s 0.7) s In the specal case of only monovalent ons, the Posson-Boltzmann equaton can therefore be wrtten elegantly wth snh snh x s defned as ex e# x ): r φr 8πc s r snh βeφr," 4πρr.8) where e s the unt elementary charge. To smplfy the numercal soluton, the Posson-Boltzmann equaton can be lnearzed by expandng the exponentals: βq c s r q s e# s φ$*%r&.- s c s r q s s βc s r q s φr.9) s

3 .. THE POISSON-BOLTZMANN EQUATION 3 s c s r q s s zero due to electroneutralty. s c s r q s s twce the onc strength I r/ s c s r q s. Together wth eq.6, ths results n the lnearzed PBE: ' r φr 0 8πβI r φr 4πρr.0) For a more detaled descrpton of the devaton of the PBE and the Debye-Hückel theory, see chapter 5- pp ) of McQuarre 976) or chapter 8 pp 3 339) of Hll 986)... Numercal soluton Analytcal solutons of the PBE exst only for smple geometres Krkwood, 934; Tanford & Krkwood, 957; Daune, 997). For complex geometres, solutons can be obtaned by numercal methods. Most often, the PBE s solved by fnte dfference methods Warwcker & Watson, 98; Ncholls & Hong, 99; Hong & Ncholls, 995). A fnte dfference method s also appled n ths work. However, there are also more elegant numercal methods such as boundary element methods Sklenar et al., 990; Zauhar & arnek, 996) or multgrd-based methods Holst et al., 994; Holst & Saed, 995) to solve the PBE. The reason for nevertheless preferrng a fnte-dfference method here s merely the practcal avalablty of a sutable sute of programs Bashford & Gerwert, 99; Bashford, 997) that has been thoroughly tested and proven to yeld relable results n numerous applcatons. So far, there have been only a few applcatons of boundary element methods to calculate protonaton behavor of protens Rpoll et al., 996; Juffer et al., 997; la et al., 998).... Fnte dfference method To apply the fnte dfference method, all relevant physcal quanttes charge q, delectrc constant, onc strength I, electrostatc potental φ) are mapped on a cubc grd wth grd constant l Fg..). φ 5 5 φ 3 φ φ 4 4 I 0 φ 0 q 0 3 φ 6 l φ 6 Fgure.: Part of the grd used to solve the PBE.

4 4 CHAPTER. AN INTRODUCTION TO CONTINUUM ELECTROSTATICS The lnearzed PBE eq.0) s ntegrated over the volume of one cubc grd element: r φr dr 8πβI r φr dr 4πρr dr.) Whle the second and thrd ntegrals are easly resolvable, the frst one s more dffcult. In a frst step, t s transformed nto a surface ntegral usng Gauß theorem: A r φr da 8πβI 0 φ 0 l 3 " 4πq 0.) The surface ntegral s now calculated separately for all sx sdes of the cubc grd element. In dong so, the gradent of the electrostatc potental φ s substtuted by ts fnte dfference form n the respectve drecton: 6 φ φ 0 l 8πβI 0 φ 0 l 3 " 4πq 0.3) l Eq..3 s smplfed and rearranged to yeld a fnte dfference expresson for φ 0 : φ 0 6 4πq 0 l φ.4) 6, 8πβI 0 l Startng from arbtrary values, the electrostatc potental s teratvely calculated for each grd pont accordng to eq.4 untl a convergence crteron s met. For the detals of ths procedure see Ncholls and Hong 99) and chapter 7.3 pp ) of Press et al. 99).... Focusng There s a problem at the borders of the grd, snce the grd ponts at the border have less than sx neghbor ponts. If the grd s much larger than the molecule, so that the border grd ponts are far away from the charges n the molecule, φ outsde of the grd can be set to zero or better to a value accordng to the Debye-Hückel sum Klapper et al., 986)). Due to computer lmtatons, the resoluton of a large grd has to be poor. To enhance resoluton, addtonal calculatons wth a smaller hgh-resoluton grd can be performed Klapper et al., 986). Ths grd s centered on the regon of nterest. On ts boundares, the electrostatc potental s set to a value nterpolated from the calculaton wth the larger grd. Such a focusng step can be repeated f necessary....3 Grd artfact An electrostatc energy can be calculated from the electrostatc potental φ: q φ.5) The sum runs over all atoms. q s the atomc partal charge of atom and φ the electrostatc potental at the poston of atom. The electrostatc energy of a classcal pont charge n ts own electrostatc potental, the so-called self energy, s nfnte large. These sngulartes are avoded f the PBE s solved on a grd, snce a pont charge s smeared over the cubc grd element where t s localzed. However, arbtrary values,

5 .. THE ANALYTICAL CONTINUUM SOLENT ACS) 5 dependng only on the grd resoluton and poston, reman from the nfnte self energy of the sngulartes. Ths grd artfact s called grd energy. There are several possbltes to get rd of the grd energy. The easest one s not to try to calculate absolute electrostatc energes but energy dfferences between systems where the grd energy s equal, so that the grd energy vanshes n the dfference. In chapter 3, ths method has been appled, and t wll be dscussed there n more detal. Another approach s to use a sngularty free charge dstrbuton or to use Green s theorem Beroza & Fredkn, 996). The boundary element methods reported earler do not need a grd and therefore are not afflcted wth the grd artfact at all.. The Analytcal Contnuum Solvent ACS) If Posson-Boltzmann electrostatcs s used for molecular dynamcs MD) or Monte Carlo MC) smulaton of large molecules, even the fastest methods to solve the PBE Hoffmann et al., 998) are too slow for solvng problems lke long-term proten dynamcs or even proten foldng. To overcome ths problem, Schaefer and Karplus 996) developed the so-called analytcal contnuum electrostatcs ACE) potental, whch approxmates the potental gven by solvng the Posson equaton, but needs much less computatonal effort. Smlar to other analytcal contnuum electrostatcs methods Sklenar et al., 990; Stll et al., 990; Schaefer & Froemmel, 990; Glson & Hong, 99), the self-energy part of ACE s based on the ntegrated feld concept, whose basc dea dates back to the work of Born 90). ACE combnes the self-energy potental wth the generalzed Born equaton for charge charge nteracton Klopman, 967; Constancel & Contreras, 984; Stll et al., 990). By addng a non-polar free energy of solvaton term, ACE s extended to the analytcal contnuum solvent ACS) Schaefer et al., 998). The followng s an ntroducton nto the bascs of ACE/ACS accordng to Schaefer and Karplus 996). Applcatons of ACS wll be descrbed n the last part of chapter... Integral formulaton of the electrostatc energy The startng pont of the ACE approach s the descrpton of the electrostatc energy n terms of the energy densty ur rather than of the electrostatc potental as the soluton of the Posson equaton. The energy densty ur of an electrostatc feld generated by a charge dstrbuton ρr can be expressed n terms of the electrc dsplacement vector as follows: Dr r E r.6) ur 8πr D r.7) By ntegratng the energy densty over the full space, the electrostatc energy s obtaned Born, 90): ur dr 8π r D r dr.8) Ths expresson can be derved from the well known expresson for the electrostatc energy of a charge dstrbuton ρr n the electrostatc potental φr : ρr φr dr.9)

6 6 CHAPTER. AN INTRODUCTION TO CONTINUUM ELECTROSTATICS The dscrete form of eq.9 was already ntroduced as eq.5. Accordng to the Posson equaton.4, the charge densty ρr n eq.9 s substtuted by 4π ' r φr : Integraton by parts leads to " 8π φr ' r φr dr.0) " 8π φr r φr 8π r φr φr dr.) The frst term vanshes due to the vanshng electrostatc potental φr at the nfnte far away) boundares of the volume. Wth E r34 φr and wth eq.6, the followng expresson s obtaned: 8π r E r E r dr 8π r Dr Dr dr.) Ths matches the expresson n eq.8. Correspondng to the stuaton descrbed before Fgure.), I splt now the ntegral expresson of the electrostatc energy eq.8) n two parts, one s the ntegral over the volume of the solute. e. the proten) p wth the low delectrc constant p, the other s the ntegral over the remanng volume of the solvent s wth the hgh delectrc constant s. 8π s s D r dr 8π p p D r dr.3) The so-called reduced delectrc constant s defned by the followng expresson: p s.4) Snce p 5 s, the reduced delectrc constant s always postve. Usng the reduced delectrc constant, eq.3 can be rewrtten extendng the frst ntegral over the full space : 8π s D r dr 8π p D r dr.5) Ths expresson s stll exact. An approxmaton s ntroduced by assumng that the frst term corresponds to the stuaton n a homogenous delectrc medum, where the delectrc dsplacement s descrbed by the smple Coulomb feld. Ths assumpton s not exact snce the delectrc dsplacement D n both ntegrals must satsfy the boundary condtons on the electrc feld at the nterface between solute and solvent,. e. the tangental component of the electrc feld E D6 and the normal component of the delectrc dsplacement D do not change when passng through the solute solvent boundary. Ths means that the feld lnes are somewhat dstorted at the delectrc boundary. Ths dstorton s not represented n the Coulomb feld n a homogenous delectrc medum. The relatve error ntroduced by the approxmaton s, however, not larger than a few percent Schaefer & Froemmel, 990). Such a smplfyng approxmaton s not possble for the second ntegral, whch s much more dffcult to evaluate. Gven the stuaton of dscrete atoms that bear pont charges as atomc partal charges, the nteracton terms of atoms wth themselves G sel f can be dstngushed from nteracton terms between dfferent atoms G nt j :

7 THE ANALYTICAL CONTINUUM SOLENT ACS) 7 G sel f G nt j 9.6) j8 where both sums run up to the total number of atoms. G nt j and G sel f can be calculated accordng to eq.5 applyng the approxmaton of the delectrc dsplacement n the frst ntegral by the Coulomb feld): G nt j q q j s r j 8π p D r D j r dr.7) G sel f q s R 8π p D r dr.8) The self energy G sel f of a pont charge yelds a dvergng energy contrbuton. To avod ths sngularty, a trck s appled n eq.8: The atom s not longer consdered as a pont, but as a sphere wth radus R, e. g. the van-der-waals radus. The pont charge q s dstrbuted over the surface of ths sphere. The frst term n eq.8 s obtaned for ths specal dstrbuton of the charge q and s called the Born energy term Born, 90)... Solvaton energy The energy of placng a molecule from a homogenous delectrc medum wth the same delectrc constant p for solvent and solute nto a heterogenous delectrc medum wth p for the solute and a hgher delectrc constant s for the solvent, s called here solvaton energy. Ths energy s dentcal to the electrostatc part of the conventonal solvaton energy f p s unty, whch s the delectrc constant of vacuum. Ths energy, however, does not nclude the non-polar parts of the solvaton energy.) s calculated by takng the dfference between the electrostatc energy n the heterogenous delectrc medum accordng to eq.6) and the electrostatc energy n the homogenous delectrc medum G hom : G hom.9) If the Born formula s also appled for the calculaton of self energes n the homogenous delectrc medum, G hom s calculated as follows: G hom q p R Agan, the energy s spltted nto a self energy and an nteracton term: G sel f G nt Takng nto account eq.7 and eq.8, G nt j G nt j q q j s r j 8π p and G sel f q q j j8 p r j 9.30) G sel f G nt j 9.3) j8 resolve to D r D j r dr q q j p r j.3) G sel f q s R 8π p D r dr q p R.33)

8 7 8 CHAPTER. AN INTRODUCTION TO CONTINUUM ELECTROSTATICS Wth the expresson for the reduced delectrc constant eq.4, ths can be wrtten shorter as G nt j q q j r j 8π p D r D j r dr.34) G sel f : q R 8π p D r dr.35)..3 Generalzed Born approxmaton In the sprt of the Born energy term, G sel f can be formally wrtten as an analogous Born energy term wth a yet unknown effectve Born radus b, whch accounts for the volume covered by the solute surroundng atom : G sel f " q b.36) If G sel f s already calculated e. g. by solvng the PBE or by applyng the analytcal approxmaton descrbed n the next sectons), the effectve Born radus b s gven by the followng equaton: b " q G sel f.37) In the same way, G nt j can be wrtten wth an effectve nteracton dstance R solv j accountng for the nfluence of the solute volume: G nt j : q q j R solv j.38) Puttng together eq.37 and eq.38 wth eq.3 yelds the so-called generalzed Born equaton Stll et al., 990): : q b j8 q q j R solv j 9.39) What I call generalzed Born approxmaton here, s the approxmaton of R solv j by a functon of b and b j and the real dstance between q and q j, r j. To derve such a functon, we wll carry out a Gedankenexperment. Imagne a charge q k wth the effectve Born radus b k. Ths charge has a self energy of G sel f k " q k b k.40) Now we dvde the charge q k nto two smaller ones, q and q j, wth effectve Born rad b and b j, and separate the two new charges by an nfntesmal small dstance. The self energy of the two new charges plus ther nteracton energy should be the same as the self energy of the one old charge before: q k b k " q b q j b j q q j R solv j.4) The effectve Born rad b, b j, and b k can be assumed to be equal snce the effectve Born radus manly accounts for the geometry of the solute and all three charges are only nfntesmal separated.

9 =.. THE ANALYTICAL CONTINUUM SOLENT ACS) 9 If we set R solv j b b j b k, eq.4 s fulflled snce q k q q j. Thus, R solv j approaches the effectve Born radus for small dstances between q and q j. In the next step of the Gedankenexperment, we separate the two charges by a large dstance compared to the sze of the solute. G nt j wll then be manly determned by the smple Coulomb energy. Ths means that R solv j approaches the real dstance r j. A sutable functon to connect these two extreme cases was proposed by Stll et al. 990): R solv j :; r j b b j exp< r j 6 4b b j.4) Ths functon approaches r j f the dstance r j s large compared to the effectve Born rad, and approaches the geometrc mean of the two Born rad b and b j for a small dstance r j. The functon s not only n agreement wth the results of our Gedankenexperment, but s also based on some other consderatons Stll et al., 990), whch I have not mentoned here. The expresson for can now be wrtten wth only one sum over all possble pars j ncludng pars where j). To account for the double countng of nteracton energes, a factor of 6 has to be ncluded for the nteracton energes. " j r j q q j b b j exp< r j 6 4b b j However, to really calculate, a way to determne the self energy G sel f.43) has stll to be found...4 Parwse self energy potental To evaluate the ntegral n eq.35, a molecular densty functon P S r s ntroduced: P S r?> : r s nsde solute volume 0 : otherwse Eq.35 can now be wrtten wth the ntegral over all space:.44) G sel f q R 8π D r P S r dr.45) The molecular densty functon P S r s now spltted nto a sum of atomc densty functons. For each atom k, there s one atomc densty functon P k r descrbng the volume dstrbuton of the atom. P S r The contrbuton of atom k to the self energy of atom, G sel f k, s gven by G sel f k 8π P k r.46) k D r P k r dr.47) The sum of G sel f k for all atoms k leads to the second term n eq.45, so that ths equaton can be rewrtten as G sel f q " R G sel f k.48) k@ The restrcton k A s n prncple not necessary snce t does not change the exact result. However, due to approxmatons appled later, the explct excluson mproves the approxmated result. For detals see Schaefer and Karplus 996).

10 0 CHAPTER. AN INTRODUCTION TO CONTINUUM ELECTROSTATICS The contrbuton G sel f k to the self energy s always postve. Snce the delectrc dsplacement D generally decreases wth ncreasng dstance from the charge generatng the dsplacement, G sel f k represents an effectve repulson between atoms and k. Ths repulson s caused by the contrbuton of atom k to the solute volume, preventng atom from nteractng wth the solvent drectly...5 Atomc charge and volume representaton In practce, t s not easy to fnd a sutable set of atomc densty functons P k r. Smply representng atomc volumes by spheres, leads to overlaps of bonded atom pars, but at the same tme there are stll cavtes between closely neghbored non-bonded atoms. Dong t ths way, the solute delectrc medum s very dscontnuous. To flatten these dscontnutes, both the charge dstrbuton of atom, ρ r and the atomc densty functon of atom k, P k r, are represented by three-dmensonal Gaussans: ρ rb q π# 3C 3 ˆR# exp r Dr ˆR ; ˆR R = π6.49) P k rb 4 3E πα 3 exp r Fr α R k.50) R s, as before, the van-der-waals radus of atom. The wdth parameter ˆR of the charge dstrbuton s chosen such that the Born self energy of the Gaussan-dstrbuted charge s the same as f the charge was equally dstrbuted on a sphere of radus R. The wdth parameter R k of the volume dstrbuton s the effectve atom radus derved from the average solvent-naccessble volume contrbuton Ṽ k of the dfferent atom types n proten structures from the PDB. The Gaussans are normalzed such that q ρ r dr.5) and Ṽ k 4π R 3 k 3 P k r dr.5) The volume Ṽ k does not depend on the smoothng parameter α, whch s ntroduced to control the wdth of the ndvdual atomc volume functons P k r. Wth a value of α G 0, the fluctuatons of the volume densty are stll too strong to yeld reasonable results. A value for α n the range from. to.8 leads to a suffcently smooth volume densty but does not yet greatly alter the contour of the proten as a whole...6 Evaluaton of the self energy To solve the ntegral n eq.47, agan the delectrc dsplacement D s approxmated by the Coulomb feld, whch means neglectng the reacton feld contrbuton to the energy densty of the electrc feld. For detals and dscusson of the ntroduced error, see Schaefer and Karplus 996). In a homogeneous delectrc wth delectrc constant, the electrostatc potental φ of a Gaussan charge dstrbuton ρ r, see eq.49, s gven by where erf denotes the error functon: φ r q erf HI r Dr *6 ˆR J r Dr.53)

11 THE ANALYTICAL CONTINUUM SOLENT ACS) erf z E π 0 z e# t dt.54) Based on the Coulomb feld approxmaton, the delectrc dsplacement of the Gaussan charge dstrbuton s D r" q erf H r Dr *6 ˆR J r Dr.55) Inserton of eq.55 and eq.50 nto eq.47 leads to the followng expresson for the energy G sel f k that atom k contrbutes to the self energy of atom : G sel f k q 6π 3C α 3 erf HI r Dr *6 ˆR J r Fr exp r Dr k 9 dr.56) H α R kj Ths ntegral s not analytcally solvable. However, a sutable approxmaton can be used f G sel f k s fnte and monotoncally decreasng wth the dstance r k. These assumptons fal n the lmt R K 0 and f the rato R 6! α R k s larger than a crtcal value that s close to unty. By numercally ntegratng eq.56, t turns out that at short range r k 5 α R k ), G sel f k decreases lke a Gaussan, whereas t approaches q Ṽk6! 8π r k 4 at long range. Ths behavor leads to the followng Ansatz: G sel f k q ω k exp L r k σ k M q Ṽk 8π L 3 r k r k 4 µ 4 k M 4.57) The parameters ω k and σ k determne the heght and wdth of the Gaussan that approxmates G sel f k n the short-range doman. The frst term n eq.57 becomes neglgble for large r k, and the second term vanshes at r k N 0 due to the ntroducton of the parameter µ k. The parameters n eq.57 are determned by calculatng analytcally the exact value for G sel f k and the second dervatve G sel f k 6 r k at r k ) 0 usng eq.56. Ths s possble snce r k O 0 means that r Pr k. At r k ) 0, the second term of eq.57 as well as ts second dervatve vanshes, so that the parameters ω k and σ k can be determned as follows: ω k 3πα 4 R k k 4 Q k arctanq k.58) wth σ k 3α R k k Q k arctanq k.59) 3 f k Q k 4arctanQ k Q k q k ; q k ; f k q k q k ; q k π L α k R k R M ; α k max L αq R R k M.60) The long-range parameter µ k s determned by makng sure that the value of the Born energy term of a charge q n a homogenous delectrc medum wth delectrc constant p s mantaned. Ths yelds µ k 5 R 77πE R π 3S σ 3 k R ω k Ṽ k T.6)

12 CHAPTER. AN INTRODUCTION TO CONTINUUM ELECTROSTATICS..7 Non-polar solvaton term To extend ACE to ACS, a term for the non-polar parts of the solvaton energy G np s added to the electrostatc solvaton energy: G np G np 4πσ R R s R b.6) R s s the radus of a water probe sphere. The emprcal solvaton parameter σ has the dmenson J6! mol Å, whch appears n conventonal solvent-accessble surface approxmatons of the hydrophobc effect. It s typcally set to values of about 0 to J6! mol Å. All other parameters n eq.6 are known from the electrostatcs part, so that the calculaton s straght forward. ACS was ntroduced by Schaefer et al. 998). In ther paper, they appled also some other mnor mprovements of the basc ACE potental, whch I wll not descrbe here. Unfortunately, the paper tself Schaefer et al., 998) does not gve a lot of detals, ether. The publcaton of a more extensve descrpton s n preparaton.