A numerical solver of 3D Poisson Nernst Planck equations for functional studies of ion channels

Transcription

1 Modellng n Medcne and Bology VI 111 A numercal solver of 3D Posson Nernst Planck equatons for functonal studes of on channels S. Furn 1, F. Zerbetto 2 & S. Cavalcant 1 1 Department of Electroncs Computer Scence and Systems, Unversty of Bologna, Italy 2 Department of Chemstry G. Camcan, Unversty of Bologna, Italy Abstract Recent results of X-Ray crystallography have provded mportant nformaton for functonal studes of membrane on channels based on computer smulatons. Because of the large number of atoms that consttute the channel protens, t s prohbtve to approach functonal studes usng molecular dynamc methods. To overcome the current computatonal lmt we propose a novel approach based on the Posson, Nernst, Planck electrodffuson theory. The proposed numercal method allows the quck computaton of on flux through the channel, startng from ts 3D structure. We appled the method to the KcsA potassum channel obtanng a good accordance wth the expermental data. Keywords: membrane proten, computer smulaton, on fluxes. 1 Introducton Ion channels are proten molecules embedded n the lpd blayer of the cell membranes. They control the on fluxes through the cellular membrane playng a central role n several cell functons,.e. the cellular exctablty [1]. In the last few years, thanks to the structural data provded by X-ray crystallography, t has become possble to analyze the channels at atomc level. In partcular, the atomc structures of several bacteral on channels selectvely permeable to potassum ons - KcsA [2], MthK [3], and KvAP [4] - were revealed. The pecular characterstc of potassum channels s the ablty to conduct at a rate close to the dffuson lmt (10 8 ons/s) keepng a hgh selectvty (K s 10 4 tmes more permeant than Na ). Snce hgh fluxes need conducton mechansms wthout

2 112 Modellng n Medcne and Bology VI energetc barrers, whle selectvty needs a close nteracton between ons and channel, these characterstcs seemed ncompatble. The knowledge of the KcsA atomc structure, the frst potassum channel crystallzed, has permtted to solve ths apparent contradcton by molecular dynamc methods [5]. KcsA s made by four dentcal subunts, symmetrcally placed around the channel axs. Each subunt conssts of 160 amno acds and s characterzed by three α helx structures - the outer helx, the pore helx and the nner helx placed lke n fgure 1. On the extracellular sde the conducton pathway s lned by the carbonyl oxygens of the amno acd sequence TVGYG, one from each subunt. Ths regon, named selectvty flter, s 12 Å long wth a mean radus of 1.4 Å and t s wdely conserved among dfferent potassum channels. Below the selectvty flter the channel opens n a wde chamber wth a mean radus of 6Å connected to the ntracellular space by a 18 Å long hydrophobc pore. Both the chamber and the hydrophobc pore are lned by the four nner helxes. Close to the ntracellular mouth a bundle between these helxes reduces the hydrophobc pore radus near to 0.5 Å, preventng on fluxes through the channel. Electron paramagnetc resonance experments [6] and structural data from other potassum channels [3], suggest that KcsA openng s due to an outward movement of the nner helxes, realzed by a bendng of these helxes at a hnge glycne (GLY 99). The conservaton of ths amno acd among many potassum channels suggests a common mechansm for channel gatng. Fgure 1: Two subunts of the KcsA channel. S1, S2, S3 and S4 are the four on bndng stes n the selectvty flter.

3 Modellng n Medcne and Bology VI 113 Expermental data [2] and molecular dynamc smulatons [5] have revealed the presence of four on bndng stes nsde the selectvty flter, named respectvely S1, S2, S3, and S4 (see Fg. 1). The energetcally favoured confguratons have two potassum ons n the flter, separated by a water molecule. The rapd swtchng between the confguraton wth ons n S1 and S3, and the one wth ons n S2 and S4, s possble thanks to the presence of a low energetc barrer. Ion transport takes place when a thrd on enters the pore from one sde drvng out an on on the opposte sde. Molecular dynamcs smulatons have showed that no energy barrer s present between the states nvolved n the conducton mechansm, gvng an explanaton for the hgh conducton rate [5]. The lack of energetc barrers s due to the presence of the carbonyl oxygens of the selectvty flter. These atoms mmc the presence of water hydraton oxygens, cuttng out the energetc cost of dehydraton. The conducton mechansm works perfectly wth potassum ons, but fals wth smaller partcles, lke sodum ons. Free energy perturbaton smulatons have revealed that, nsde the flter, the substtuton of potassum by sodum s energetcally expensve, wth an energetc cost n lne wth the expermental selectvty of potassum channels [5]. Molecular dynamc smulaton has ponted out the atomc detals of conducton and selectvty, but s not able to reproduce the on fluxes through the channels. Snce electrcal current s the man functonal characterstc of on channels, and the only one expermentally determnable, smulaton of on fluxes s the only way to lnk structural and functonal data. The complexty of the system, due to the huge number of atoms, lmts the dynamc smulatons to the nanosecond scale whle bologcal process, lke on conducton, are far slower (mllsecond scale). A smplfed mathematcal model of on conducton s so needed. Snce most of the complexty n on channel dynamc smulatons les n the hgh number of water molecule, a contnuum descrpton of the solvent s the frst step to reduce the computatonal resources. Brownan dynamc smulatons are a possble approach based on ths dea. In a Brownan dynamc smulaton only ons preserved ther dscrete nature, the solvent s descrbed by dffuson coeffcents and stochastc collsons wth ons and the effect of the proten by a potental energy functon. The potental energy functon and the dffuson coeffcents are the basc elements of ths method, and to avod arbtrarness n the mathematcal model these functons are computed startng from the atomc structure of the channel. A Brownan dynamc smulaton based on ths approach was used by Bernechè [7] on KcsA, gettng a good accordance wth the expermental data. In ths study we present a further smplfed approach based on the Posson, Nernst, Planck (PNP) theory of electrodffuson, whch use a contnuum descrpton of the whole system. The next secton descrbes the theory and the numercal algorthm developed to solve the PNP equatons. The applcaton of the method to the KcsA channel s then descrbed. The paper concludes wth a bref dscusson of the results.

4 114 Modellng n Medcne and Bology VI 2 Theory and method 2.1 PNP theory The PNP electrodffuson theory descrbes a steady state condton for a system of moble charges. In membrane channels, the moble charges are the dfferent on speces n soluton, whch space dstrbutons are descrbed by the concentratons C (r) (subscrpt marks the -th on spece; r s the poston n the space). Assumng the electrc feld as the only drvng force actng on ons, the steady state flux of the -th on spece has the form: J = D C µ C z e ψ (1) where D (r), µ (r), z are respectvely dffuson coeffcent, moblty and valence of the -th on spece, e s the elementary charge and ψ the electrostatc potental. The frst term, whch s proportonal to the concentraton gradent, s due to dffuson processes, whle the last s produced by the electrc feld. Snce the PNP theory descrbes a steady state condton, fluxes are tme ndependent, and n vrtue of the mass conservaton law, the dvergence of J (r) s zero. Ths gves the set of dfferental equatons: ez dv D C C ψ = 0 = 1,..., N (2) kbt where N s the number of on speces, k B the Boltzmann s constant and T the absolute temperature. In (2) the Ensten s relaton between dffuson coeffcents and on moblty, µ/d=k B T, was ntroduced. To complete the mathematcal model t s necessary to defne how electrostatc potental and on concentratons are connected. Ths relaton may be defned by the Posson s equaton: [ ψ ] = ρ( r) dv ε ez C (3) where ε(r) s the delectrc constant and ρ(r) the charge dstrbuton of the proten atoms that dfferently from on charge dstrbuton s assumed fxed n the space. In the present study we ncluded n the water soluton only two monovalent on speces: one postve (=) and one negatve (=-). Then, the PNP dfferental equaton set (2) and (3) reduced to: dv D dv D C C e k B e C k T B N = 1 C ψ = 0 T ψ = 0 (4)

5 [ ψ ] = ρ( r) e[ C C ] dv ε (5) Once solved these equatons, as descrbed n the next secton, on concentratons and electrostatc potental n the space are obtaned. Then, the on fluxes n the channel can be computed by equaton (1). 2.2 Computatonal method Modellng n Medcne and Bology VI 115 The dfferental equatons (4) and (5) were numercally solved on a cubc volume formed by cubc grd elements (the sde of the grd element was 0.5 Å). The cubc volume was dvded n three dstnct sub-volumes: the on channel, the membrane and the water soluton (see Fg. 2). The poston, the radus and the partal charge of all the atoms made up the on channel sub-volume. Channel was dscretzed on the grd by usng the dscretzaton algorthm mplemented n DELPHI v.4, a well-known Posson-Boltzmann equaton solver [8]. The channel was placed wth ts geometrc centre n the centre of the cube and wth the pore axs orthogonal to the upper and lower faces. The extracellular sde of the channel ponted to the upper face. The heght of cube was chosen double of the channel length along the pore axs (z axs). To separate extra- and ntra-cellular spaces outsde the on channel, a sub-volume surroundng the channel and extendng between two planes orthogonal to the pore axs was consdered. Ths volume smulated the lpd blayer of the cell membrane. The water soluton spread all over the volume not occuped by the channel and by the membrane. It was characterzed by two space-dependent dffuson coeffcents, one for each onc spece. A specfc delectrc constant was assgned to water soluton, membrane and channel sub-volumes. The Posson s equaton (5) was solved n the whole volume, whle the mass conservaton equatons (4) were solved n the water soluton only. As boundary condton for the Posson s equaton we assgned the electrostatc potental on the sx faces of cube. The potental was set to 0 on the upper face, whereas the membrane potental value (V m ) to be smulated was appled on the lower face. On the sde faces a lnear nterpolaton between 0 and V m was used. Two dfferent boundary condtons were assgned for the mass conservaton equatons. On the upper and lower faces the boundary condtons were the on concentratons to be smulated. On both faces anon and caton concentratons were set equal, to have electrcally neutral boundares. Boundary condtons on the sde faces and at the separaton surface wth channel and membrane were: J n = 0 J n = 0 (6) where n s the surface normal vector. In ths way no on flux was allowed through these surfaces. Once defned the boundary condtons, the dfferental equaton set was solved by an teratve scheme. The electrostatc potental was frst computed solvng the Posson s equaton wth on concentratons set to zero. Ths potental was used to compute on concentratons by the soluton of mass conservaton equatons, then the new concentratons were used to update the electrostatc

6 116 Modellng n Medcne and Bology VI potental. Ths procedure was repeated untl a self-consstent soluton was found. The convergence was tested by the root mean square devaton between two successve teratons. All the dfferental equatons were solved as descrbed n the Appendx. Fgure 2: A 2D slce of the cubc grd. The channel s n black, the membrane n grey and the water soluton n whte. 3 Applcaton to the KcsA channel 3.1 KcsA structure The three dmensonal atomc coordnates of the KcsA channel were taken from the crystallographc structure determned at 2 Å resoluton by Y. Zhou et al. [9] (fle 1K4C.pdb of the Proten Data Bank [10]). The atomc coordnates of the frst 22 amno acds at the N termnal and of the last 45 at the C termnal were not expermentally determned. Snce both C and N termnal are located n the cytoplasm, far from the conducton pathway, these amno acds are not crucal for the present study and therefore, they were not ncluded n the channel model. Sde chans wth mssng atoms were completed usng deal nternal coordnate from the AMBER force feld [11]. AMBER force feld was used to defne atomc rad and partal charges too. The expermentally determned structure of KcsA corresponds to a closed state of the channel. To compute correctly on fluxes t was necessary to use an open channel structure. Closng of KcsA s due to a bundle between the four nner helxes at the ntracellular channel mouth [3, 6]. A glycne hnge allows the nner helx bendng and the consequent channel openng. Ths mechansm was suggested by the expermental structure of MthK (fle 1LNQ.pdb of the Proten Data Bank). MthK s potassum channel from the methanothermbacter thermautotrophc that, dfferently from KcsA, was crystallzed n an open state. Indeed, MthK nner helxes are bent and the selectvty flter s connected to the

7 Modellng n Medcne and Bology VI 117 ntracellulare space by a wde pore (mnmum radus 12 Å). To compute a structure of KcsA n an open state we mnmzed the dstance between the nner helx of KcsA and MthK by a rgd rotaton of the ntracellular sde of the KcsA nner helxes around the glycne hnges. KcsA channel comes nto contact wth the membrane lpd blayer by the four outer helxes. These helxes have a hydrophobc segment between the amno acds TRP 113 (z=14.5 Å) and TRP 87 (z=-15.5 Å). We used the poston of these amno acds to defne the thckness of the membrane n our model. 3.2 Results The relatve delectrc constant was set to 80 n the water soluton and to 2 n the channel and the membrane. Small changes around these values were tested but they dd not affect the results sgnfcantly. The dffuson coeffcents were set to the expermental value, respectvely of potassum and chlorde (DK =1.96*10-9 m 2 /s, DCl - =2.03*10-9 m 2 /s). Settng the membrane potental to 25 mv and both the extracellular and ntracellular on concentratons to 100 mm, the computed current s 4 tmes the expermental data [12]. The approxmaton gets worst at hgher on concentraton and membrane potental (Fg. 3). A possble approach to reproduce the expermental data s to use the dffuson coeffcents as fttng parameters. Computed currents reproduce qute well the expermental data wth a reducton of both the dffuson coeffcents to 25%. However, a smlar reducton of the dffuson coeffcents n the whole water soluton s not ustfable. Otherwse, nsde the channel, dffuson processes take place dfferently and t s reasonable to use dfferent values for the dffuson coeffcents. We dvded the channel n two regons: the hydrophobc pore (0<z<-24) and the selectvty flter (14<z<0). The dffuson coeffcents were reduced only n these two regons accordng to the value computed by molecular dynamc smulatons (10% of the expermental value n the selectvty flter, 50% n the hydrophobc pore) [5]. Usng these dffuson coeffcents the accordance wth the expermental data mproves (see Fg. 3). Fgure 3: Current-voltage (A) and current-concentraton (B) characterstcs. Expermental data ( ), dffuson coeffcents set to the expermental values ( ), reduced dffuson coeffcents ( ).

8 118 Modellng n Medcne and Bology VI 4 Dscusson In ths paper we have attempted to relate structure and functon of on channels, makng use of the KcsA expermental data. To ths end we developed a numercal solver of the PNP equatons. In on channel smulaton studes, an approach based on the PNP electrodffuson theory s complementary to an approach based on molecular dynamcs. The last can reveal the atomc detals about on channel functonng, whle the former, gvng up the atomc detal, provde a way to compute on currents. To smplfy the mathematcal model of on conducton, the PNP theory descrbes the on dstrbuton wth a contnuum functon. Despte ths smplfyng assumpton the computed currents are n good accordance wth the expermental data. It s mportant to hghlght that the accordance between smulated and expermental data was obtaned wth dffuson coeffcent values matchng the ones computed by molecular dynamcs smulatons. Therefore these coeffcents must not to be nterpreted as fttng parameters. Instead, they connect the mcroscopc descrpton provded by molecular dynamcs to the macroscopc of the PNP theory. The accordance wth the expermental data s good especally wth on concentratons and membrane potental n the physologcal range. At hgher values an overestmaton of the current was obtaned, probably due to saturaton effects not ntroduced n the model. Even f an mprovement of the mathematcal model n ths drecton s desrable, the present approxmaton provdes a good tool to connect structure and functon of on channels. Appendx: numercal method The numercal procedure used to solve the mass conservaton equatons (4) s presented here. For sake of smplcty we wll refer to a bdmensonal case, generalzaton to the three dmensonal case s mmedate. The same numercal procedure was used to solve the Posson s equaton (5). The on flux n the x drecton between the grd elements (-1,) and (,) was expressed as: J x 1 D =, D 2 1, C, C h 1, ze k T B C, C 2 1, ψ, ψ h 1, (A1) that s the lattce verson of equaton (1) (h s the grd step). The on fluxes y J 1 y J 1 x J 1 - n the x drecton between the elements (,) and (1,) -, and were expressed consstently. Accordng to the mass conservaton law, net steady state flux through any grd element s zero, that s: x x y y J J J J 0 (A2) =

9 Modellng n Medcne and Bology VI 119 Replacng n ths equaton the fluxes expressed as n (A1) t s possble to wrte: C, = k D, Dk ez Ck 1 2 2kBT D, Dk ez 1 2 2kBT k ( ψ ψ ) ( ψ ψ ),, k k (A3) where k s an ndex that rounds over the four surroundng grd elements, k=(- 1,), (1,), (,-1), (,1). Equaton (A3) was used n an teratve scheme based on the successve over relaxaton technques [14] to fnd the soluton of the mass conservaton equaton. At the frst step on concentraton n each grd element s set randomly, these concentratons are then updated, accordng to: n n1 C, = ( 1 w) C, wc, (A4) n n1 where C, and C, are respectvely the concentraton computed at the step n and n-1, and C, s obtaned by (A3). Settng correctly the weght w, the number of teraton to reach the solutons drops apprecably. To test the convergence to soluton the root mean square dstance (RMSD) between two successve teratons s used, the procedure s stopped when RMSD falls below References [1] Hlle, B., Ionc channels of exctable membranes, 2nd Ed. Snauer Assocates, Sunderland, MA, [2] Doyle, D.A., The structure of the potassum channel: molecular bass of K conducton and selectvty, Scence, 1998, vol. 280, no. 5360, pp [3] Jang, Y., Crystal structure and mechansm of a calcum-gated potassum channel, Nature, 2002, vol. 417, no. 6888, pp [4] Jang, Y., X-ray structure of a voltage-dependent K channel, Nature, 2003, vol.4030, no. 7001,pp [5] Allen, T.W., The potassum channel: structure, selectvty and dffuson, Bophys J., 1999, vol. 77, no. 5, pp [6] Perozo, E., Structural rearrangements underlyng K-channel actvaton gatng, Scence, 1999, vol. 285, no. 5424, pp [7] Bernechè, S., A mcroscopc vew of on conducton through the K channel, PNAS, 2003, vol. 100, no. 15, pp

10 120 Modellng n Medcne and Bology VI [8] Roccha, W., Rapd grd-based constructon of the molecular surface for both molecules and geometrc obects: applcaton to the fnte dfference Posson-Boltzmann method, J. Comp. Chem., 2002, vol. 23, pp [9] Zhou, Y., Chemstry of on coordnaton and hydraton revealed by a K channel-fab complex at 2.0 Å resoluton, Nature, 2001, vol. 414, no.6859, pp [10] Proten Data Bank, [11] Pearlman, D.A., AMBER, a package of computer programs for applyng molecular mechancs, normal mode analyss, molecular dynamcs and free energy calculatons to smulate the structural and energetc propertes of molecules. Comp. Phys. Commun., 1995, vol. 91, pp [12] LeMasurer, M., KcsA: t's a potassum channel, J Gen Physol, 2001, vol. 118, no. 3, p [13] Young, D.M., Iteratve soluton of large lnear systems, 1971, Academc press: New York and London.