Quarterly Reviews of Biophysics 25, 4 (1992), pp. 477-510 Printed in Great Britain

477

Theoretical perspectives on ion-channel electrostatics: continuum and microscopic approaches

MICHAEL B.PARTENSKII AND PETER C.JORDAN Department of Chemistry, Brandeis University, Waltham, MA 02254-gno

1. INTRODUCTION

477

2. MICROSCOPIC STRUCTURE AND DIELECTRIC CONTINUA

478

2.1 Fundamentals 478 2.2 Non-locality of electrical response 480 2.3 Non-linearity of electrical response 482 3. CONTINUUM MODELLING OF CHANNEL SYSTEMS 483 4. CONTINUUM MODELLING RESULTS 487 4.1 Image energies and electric field profiles in gramicidin

487

4.2 Lipid-water potential differences (LWPD) and gramicidin conductance 488 4.3 Polarity altering mutations and gramicidin conductance 489 4.4 Vestibule charges and sodium channel conductance 491 5. ' H Y B R I D ' MODELS

491

5.1 Dipolar chain models for water in transmembrane ion channels (TMIC) 491 5.2 A chain of electrical sources in a structureless medium 494 5.3 The influence of a model protein charge distribution 6. CONCLUSIONS AND FUTURE PROSPECTS

497

502

6.1 Summary 502 6.2 New perspectives 503 7. ACKNOWLEDGEMENT 8. REFERENCES

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504

I. INTRODUCTION

Peter Lauger introduced me (P.C.J.) to the field of ion-channel electrostatics while I was a sabbatical visitor at Konstanz in 1978-79. Lauger pointed out that the relative conductance of hydrophobic ions through phosphatidyl choline (PC) and glyceryl monooleate (GMO) membranes differed by a factor of about 100 19-2

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M. B. Partenskii and P. C. Jordan

(Hladky & Haydon, 1973), quite consistent with the difference in the watermembrane potential differences in the two systems (Pickar & Benz, 1978). However, cation conductance through gramicidin channels spanning these membranes only differs by a factor of 2-3 (Bamberg et al. 1976). Why? It is the pursuit of an answer to this question which led me into my researches in this field. In this review we present some answers to Lauger's question and other related ones. The general area of ion interaction with membrane proteins has been reviewed recently (Jordan, 1992); here we concentrate on an issue of special importance to theorists. We focus on the aqueous pore and when it can be adequately described as a dielectric continuum. We begin by discussing the dielectric constants of channel forming proteins and of aqueous pores. Are they ever meaningful ? How can they be estimated from molecular considerations ? Then we describe continuum electrostatic models of ion channels emphasizing the problems in relating molecular structure to the continuum dielectric picture. We discuss how, within a strict continuum approach, the dielectric problem has been treated and indicate what can reliably be concluded from such models and where they should be used with caution. We finally describe 'hybrid' models of the channel-protein-water-membrane ensemble designed to circumvent the basic limitation of continuum electrostatics, how it describes the phospholipid and protein charge distributions and the interior water molecules. We will see that the continuum mesoscopic picture could stand substantial reconsideration and that we have only limited understanding of ion-water-protein interaction in the narrow channels of physiological importance. The field remains fertile, with many unresolved questions. 2. MICROSCOPIC STRUCTURE AND DIELECTRIC CONTINUA

2.1 Fundamentals By electrostatically modelling channel systems we avoid specifying the details of ion-atom interaction in the pore. Changes in lipid, protein and water structure due to electrical stress are treated in an average sense and described by a local dielectric function eiocal(r) or by a series of dielectric constants, eu one for each distinct dielectric region. The difficulty is providing an internally consistent way to assign an elocal(r) to each point r in the dielectric. To illustrate the problem, we discuss some basic issues in electrostatics, mainly limiting detailed discussion to a uniform, isotropic system in which elocal would be constant. A dielectric constant measures a solvent's ability to shield an electric field generated by an external potential or a polar source. Shielding arises from two factors: polarization of the charge distribution of the solvent (electronic, atomic or bond polarization); reorientation of the solvent molecules (orientational or structural polarization) in the field. Dielectric response at r depends on the frequency of the source and the distance from the source; a complete analysis would account for non-locality and establish e(o), \r — r'\), with r' the location of the source. At sufficiently high frequencies ( > io 1 0 s - 1 ) molecular dipoles can not reorient and only the electronic term contributes to e; if w is very much higher

Ion-channel electrostatics

479

(> io 15 s"1), electronic degrees of freedom are also frozen and e = 1. For uniform, isotropic systems, in regions far enough from the electrical source, the basic relations connecting macroscopic electrical variables to statistical mechanical averages are: D = eE, D = E+4nP,

P=(M)E,

(I)

where D is the electric displacement, E the local (Maxwell) electric field, P the molar polarization and (M}E the thermally averaged total solvent dipole moment (Bottcher, 1973; Davidson, 1962) and e the dielectric constant. Statistical mechanics then provides the link between (M)£ and the electronic and molecular structure of the solvent. For an isotropic, uniform, pure polar solvent in the limit of a small electric field, its application, while subtle in practice, is straightforward in principle. The energy of the individual dipole in an applied field depends on the local electric field E; this differs from the applied field because of interactions with the surrounding solvent (Bottcher, 1973; Frohlich, 1958). Taking this reaction field into consideration, e is uniquely determined by 0, the mean square dipole moment at zero field (Kirkwood, 1939; Frohlich, 1958; Neumann & Steinhauser, 1980; Neumann, 1983). This approach, employing plausible intermolecular potentials for liquid water, leads to reasonable values for e(w) and its temperature dependence in the low frequency limit (Neumann, 1986; Alper & Levy, 1989)There is no simple generalization which describes the dielectric properties of water in a channel. Not only are aqueous pores anisotropic, they are non-uniform and they are rarely well approximated as continua. Even thermally averaged, each point along the permeation pathway is unique; the local environment changes substantially as the ion moves through a channel and e(\r — r'\;w) must be generalized to e(r, r';w). To assign a local dielectric constant to a region r of the dielectric requires in invoking an averaging procedure, a consideration recognized in assigning elocal for proteins (Warshel & Aqvist, 1991 ; Sharp & Honig, 1990; Harvey, 1989). Not only does eiocal vary in different regions of the protein, but, for reasons to be discussed, different measures of electrical response suggest distinctly different identifications; elocal varies depending on whether it is determined by the Born energy of a buried charged residue or by the interaction energy between a pair of buried residues (Warshel & Aqvist, 1991; Warshel & Russell, 1984; Simonson et al. 1991). The two measures average over different spatial domains. Frequency does not appear explicitly; however, the w dependence of e and its dependence on r — r' are related through a modified fluctuation-dissipation theorem (see e.g. Dolgov et al. 1981). Analogous issues arise in characterizing the dielectric properties of water in or near the aqueous channel (Green & Lewis, 1991; Partenskii & Jordan, 1992). In polar solvents dielectric response depends on the solvent molecules' capacity to reorient in an electric field and on the reaction field, created by interactions with surrounding solvent, which shields the applied field (Bottcher, 1973; Frohlich, 1958). In a protein, structural considerations substantially limit both the reorientational ability of any polar groups and the formation of a large reaction

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field, thus reducing elocal (King et al. 1992). Water filled channels represent an intermediate situation. The water molecules in the channel can rotate freely, but the surrounding solvent (protein) has only limited capacity to shield electrical interactions and create substantial reaction fields. Why is it so hard to determine the local dielectric constant in a solvent P1 There are two basic reasons: dielectric response to electrical stress is both non-local and non-linear. In any domain, polarization depends both on the distance from the source and on the strength of the field. 2.2 Non-locality of electrical response Equation 1 expresses Maxwell's equation for macroscopic electrostatics: the local constitutive relation between the displacement D and the macroscopically averaged electric field E, D = eE. More generally the relation between the field and the displacement is non-local (Harrison, 1970; Kornyshev, 1985), D(r)=fdr'e(r',r)E(r').

(2)

Restricting ourselves to uniform, isotropic media and focusing attention on nonlocal effects, it is useful to introduce a non-local definition for polarization, AnP{r)

= D(r)-E(r) = jdr'[e(r',r)-S(r- r'

(3)

which, when contrasted with the local relation 477P = (e— i)E, shows that nonlocality prescribes that the value of the polarization at a point r is determined not only by the field E at that point, but also by the field at other points r' in the vicinity of r (Kornyshev, 1985). This arises from spatial correlation of polarization, which is extremely important over microscopic distances. In uniform, isotropic media such as polar liquids, the electric response (redistribution of charges and polarization of the medium) is independent of the location of the electric source and e(r, r') — e(\r — r'\); this is most easily described in Fourier representation2 [ e(k) = exp ( - ik • r) e{r) dr,

(4)

where e(k) describes the screening of the electric field component with wave length A = 2n/k. The potential due to a point charge is {r) = (2/n) T [dk/e(k)] • [sin (kr)/kr] = i/[eeff(r) r], Jo 1

Here the term ' solvent' is generalized to mean the ion's surroundings, which for channels includes pore water, protein, membrane and bulk water. 2 As already discussed, non-uniform domains, such as transition regions near interfaces or aqueous pores, are often described by a local dielectric constant eioc^(r)- This is different from the issue addressed here, non-locality in a uniform medium; a local e treatment describes non-uniform media in a local approximation, with the local constitutive relation D(r) = elocal(r)2s'(r).

(5)

Ion-channel electrostatics

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which provides a definition for a distance dependent effective e, ee!t(r), a concept used commonly in molecular modelling (see e.g. Brooks et al. 1988)3; r is the distance from the source. The analogous effective e for a charged sphere (Born model) is (Kornyshev, 1985) [dk/e(k)] • [sin (kr)/kr] • [sin (kRJ/kRJ, o where RT is the ionic radius. The non-local Born energy is W(RJ = qVizR,) [1 - i/e ef f(*i)], which reduces to the familiar Born result -iAsoivL

(6)

(7a) (7b)

In macroscopic electrostatics e(k) = esolv, a wave vector independent constant, i.e. all Fourier components create the same polarization and equation 5 reduces to (j)(r) = i/[re solv ]. This can not be true in real systems. Consider a simple illustration, where polarization is due to the deformation of molecular dipoles, having equilibrium dipole moment fi0 and bond length d0. Due to thermal motion the dipole senses a field averaged over d. If the force constant for the bond is KB, the induced dipole moment, A/i, yields !

;

(8)

Ek-cosk-r, the &th Fourier component of the electric field, is averaged over the bonding region. Thus, because of the oscillatory weighting factor, long wavelength (small k) components contribute heavily to the induced dipole moment, while the components with A 15-20 A. For ionic solvation it has been shown (Kornyshev, 1985; Dogonadze & Kornyshev, 1974) that eett in equations 5-70 is lower than the solvent value which accounts in part for the relatively small experimental values of the free energy of solvation as compared to the classic Born charging energy, equation 76, a consideration also applicable to channel systems (Partenskii & Jordan, 1992). Thus eeff determined from Born energy calculations may be much lower than values based on using the long wave length (k —* o) approximation to e(k). Consequently, elocal(r) must be different depending on the measure of electrical 3 The differences between eef((r) and elocal(r) can not be stressed too strongly. We must reemphasize that the former is associated with dielectric response a distance r from an electrical source, which is not constant even in an isotropic fluid. The latter defines the mean value of e to be assigned to a local region in an anisotropic dielectric; it is associated with a point r in space.

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M. B. Partenskii and P. C. Jordan

response being considered and the use of elocal(r) rather than e(r, r') in the Poisson equation could create serious problems in calculating the local field due to a point charge. 2.3 Non-linearity of electrical response Equally important is the non-linearity of the electrical response since electrical fields in the vicinity of a solvated ion can be > io 7 V/A. To gain insight into this phenomenon, consider a situation where the solvent's dielectric response is due to reorientation of molecular dipoles. In low fields the dipoles are free to rotate and readily reorient. At high fields there is a deep potential well, W(a) = —/lEcosa, so that the dipoles become tightly ordered and ever less reorientable as E increases. Hence the dielectric constant drops sharply in the vicinity of an ion and the system becomes 'dielectrically saturated' (Millen & Watts, 1967; Abe, 1986; Bucher & Porter, 1986; Eherson, 1987). The simplest example is the Debye formula for the polarization, explicitly justified for dilute gases, but often applied to the study of polar fluids (Warshel & Russell, 1984), P = /inL{/iE/k T),

where

L(u) = ctnh(u) -i/u;

(9)

n is the concentration of molecular dipoles. At large E,u$> I , L ( M ) - » I , and P approaches a constant value, /in, indicative of saturation. It is worth noting that the idea that solvent near an ion may be dielectrically saturated can be traced back to Debye (see Brown, 1956). Applications of the nonlinear approach to theories of solvation are also motivated by attempts to improve upon the Born formula for the energy of transfer from vacuum into solvent, which often overestimates the free energy of solvation, the errors being largest for smaller ions (see Liszi & Ruff, 1985). This is different from the distance dependence of eett, since it relates specifically to the size of the fields generated by the ions. As already noted, equation 76, with e = esolv, exaggerates the screening of the ion's field by the solvent; this has been treated operationally by assuming ionic radii in solution are larger than crystallographic radii (Glueckauf, 1964; Hush, 1948; Grahame, 1950, 1953). However it is the large electric fields in solvent near an ion which are the more probable causes for the discrepancies. This has been heuristically accommodated within the distance dependent e framework by assigning each solvation shell a different dielectric constant, with e being small in the inner shell (Schnuelle & Beveridge, 1975; Abraham & Liszi, 1978, 1980). Recent work (Jayaram et al. 1989 a) explicitly incorporated nonlinearity of response by describing e in the first solvation shell as a function of ionic charge eT(q). The Born charging process involves increasing the charge in the Born sphere from o to q and the charging energy, equation 7, becomes V(Rl,q')dq'-q2/2RI

W(Rl)=\

(10)

Jo where V(Rvq) is the electrical potential of a Born sphere incorporating charge q. This is equivalent to equation 7 only if e is a charge independent constant, e7(

Theoretical perspectives on ion-channel electrostatics: continuum and microscopic approaches.

Quarterly Reviews of Biophysics 25, 4 (1992), pp. 477-510 Printed in Great Britain 477 Theoretical perspectives on ion-channel electrostatics: conti...
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