proteins, channels and crowded ions

Biophysical Chemistry 100(2003) 507–517

0301-4622/03/$ - see front matter� 2002 Elsevier Science B.V. All rights reserved.PII: S0301-4622Ž02.00302-2

Proteins, channels and crowded ions

Bob Eisenberg*

Department of Molecular Biophysics and Physiology, Rush Medical College, 1750 West Harrison, Chicago, IL 60612, USA

Received 22 April 2002; accepted 31 May 2002

Abstract

Ion channels are proteins with a hole down their middle that control a vast range of biological function in healthand disease. Selectivity is an important biological function determined by the open channel, which does not changeconformation on the biological time scale. The challenge is to predict the function—the current of ions of differenttypes and concentrations through a variety of channels—from structure, given fundamental physical laws. Walls ofion channels, like active sites of enzymes, often contain several fixed charges. Those fixed charges demand counterions nearby, and the density of those counter ions is very high, greater than 5 molar, because of the tiny volumes ofthe channel’s pore. Physical chemists can now calculate the free energy per mole of salt solutions(e.g. the activitycoefficient) from infinite dilution to saturation, even in ionic melts. Such calculations of a model of the L-typecalcium channel show that the large energies needed to crowd charges into the channel can account for the substantialselectivity and complex properties found experimentally. The properties of such crowded charge are likely to be animportant determinant of the properties of proteins in general because channels are nearly enzymes.� 2002 Elsevier Science B.V. All rights reserved.

Keywords: Physical; Selectivity; Saturation; Density

1. Introduction

Ion channels are proteins with holes down theirmiddle of enormous biological importancew1–3xthat perform many of their biological functionswithout changing structure. The physics of ionmovement through channels, once they are open,is simple: only electrodiffusion, convection andheat flow occur in condensed phases under biolog-ical conditions, and electrodiffusion clearly domi-nates the properties of permeation through openchannels in many cases. The challenge is to predict

*Corresponding author. Tel.:q1-312-942-6467; fax:q1-312-942-8711.

E-mail address: [email protected](B. Eisenberg).

the current of ions of different types and concen-trations through the many different types of chan-nels using only the language of physics ofcondensed phases. The opportunity is the substan-tial experimental knowledge of channels:thousands of scientists(literally) measure the ionsgoing through channels, one molecule at a time,every day.Ions and channels are inseparable. Without ions,

channels do not function. The fixed charge of thechannel protein demands counter ions in a nearbyionic atmosphere: otherwise, impossible electricalforces are created, as shown in the first paragraphsof Feynman’s textw4x. Cohn and Edsall recognizedlong ago that proteins bristle with charge(cf. w5x

508 B. Eisenberg / Biophysical Chemistry 100 (2003) 507–517

as cited on p. 73 ofw6x) and thus with counterions.We argue that many of the properties of channels

are produced by the crowding of counter ions nearactive sites, allowing their different sizes andcharges to determine and modulate the free energyavailable for ion permeation. Similar energeticsare likely to be important in protein function ingeneral. After all, the phrase ‘channels areenzymes’ is a tautology as well as oxymoronw7x.Of course, some proteins(particularly enzymes)use additional sources of energy beyond those inthe simplest physical models of ion channelselectivity.The fixed charge that crowds ions is permanent

in the sense that it arises from the nature of thechemical bonds of the protein, i.e. from the solu-tion of Schrodinger’s equation. Those bonds are¨built into proteins following the blueprint of theirDNA. The permanent charge of proteins and thesurrounding ionic atmosphere is under genetic, andthus evolutionary control. The modulation of theenergy of crowded charge is a mechanism bywhich the genome and protein might control bio-logical function.The ionic atmosphere surrounding an active site

can be viewed as a compressible plasma. The punon ‘plasma’ is precise. The ions of the ionicatmosphere are non-cellular components of a bio-logical fluid. The ionic atmosphere contains ionsmoving in a dielectric background quite like adense gaseous plasma, whether the plasma is thatof a real gasw8–10x or that of a gas of quasi-particles, like the holes and ‘electrons’ of a semi-conductorw11–18x.The ionic atmosphere is compressible because

its number and charge density is variable, althoughits conventional (gravitational) density is not.Large amounts of energy are involved in changingthe number and charge density of the ionic atmos-phere near active sites, even if the water andsolution itself are(nearly) incompressible. A majorresult of modern physical chemistry is the fact thatmany properties of salt solutions arise from thevariation in volume and charge density of solutesw19–36x.

2. Physical chemistry of concentrated solutions

Many specific chemical properties of concen-trated salt solutions(having nothing to do withproteins) are known now to arise from changes inthe number density, and thus density of charge andexcluded volume. According to modern physicalchemistry w37–45x, the diameter and charge ofions alone is enough to explain many complexproperties of concentrated salt solutions withoutconsidering other specific chemical properties ofthe ionic species, e.g. its orbital structure. Specificchemical interactions with the solvent(producedby electron delocalization, for example) are notincluded in these modern successful treatments ofconcentrated salt solutions. Models that analyzethe free energy of crowded spheres, treating thewater as a uniform dielectric, with dielectric con-stant of the ionic solution(not of pure water), doremarkably well because the dominant effect inconcentrated salt solutions comes from the electro-static energy and entropy of crowded spheres. Thecenters of crowded spheres can only approachwithin 1 diameter of each other(more precisely,the sum of the radii of two perhaps unequalspheres) w46,47x. Debye–Huckel theory treats its¨central ion as a sphere but it treats other ions aspoints—even though they are identical to thecentral ion—so their centers of charge canapproach within a radius, not diameter of eachother. The electric field is strongest close to an ionso the errors in Debye–Huckel theory are large,¨particularly at high concentrations, where it is nowcustomarily replaced by the Mean SphericalApproximation MSAw37–43,46–54x.The MSA (or its analogs) represents solvation

energies only in a primitive way as the interactionof finite charged spheres(ions) with each otherand with a uniform dielectric(water). Nonetheless,the theories quite successfully predict the activitycoefficients of ions over a range of concentrations.w37–42x The finite (i.e. excluded) volume of theions distorts the electric field(compared to that ofpoint particles) and accounts for much of theexcess free energy; the finite volume of unchargedspheres accounts for much of the entropic com-ponent of the excess free energy of the solution.Specific quantum chemical effects exist, but are

509B. Eisenberg / Biophysical Chemistry 100 (2003) 507–517

small in comparison, fundamentally becausechanges in density of the excluded volume anddisturbances in the electric field swamp the ener-getic effects of orbital delocalization. It seemslikely then that crowding effects might be impor-tant in the concentrated ionic atmosphere nearactive sites of proteins, particularly because theconcentration of charge density there is so large,accounting for a substantial fraction of all theatoms, and that charge density is variable.

3. Selectivity in channels

Selectivity can be explained by a simple modelof the crowded charge in and near their selectivityfilter, using the same principles that are used tounderstand concentrated salt solutions without pro-teins w55–58x.In this treatment, selectivity arises from the

properties of the concentrated ion plasma near theactive site, more than anything else. The complexproperties of selectivity arise from changes in thedensity of the concentrated ion plasma near theactive site. Changes in the density of permanentcharge at the active site change the density ofcounter ions, whether the active site changes involume, conformation, or a number of fixed charg-es. That change in density has large effects becausethe energy stored in the ionic atmosphere is solarge. The permanent charge of the active site andstrength of the electric field enforce severe crowd-ing of mobile ions, allowing their specific chemicalproperties(that arise chiefly from their excludedvolume and charge) to become important deter-minants of protein function. In this view, selectivityand other properties of proteins arise from thebalance of electrical and excluded volume in placescrowded with charge, e.g. the selectivity filter ofchannels and active sites of proteins. In this view,the protein modulates and uses the energies of ionsin the dense plasma near the active site much asan automobile engine modulates and uses gasolineand air.Indeed, the energetics of a compressible plasma

is enough to explain rather complex chemical andbiological properties of the channel without invok-ing specific chemical properties or a specific geo-metrical arrangement of atoms of the channel

protein. In the MSA version of this idea, theenergy of the protein is(nearly) ignored; the MSAmodel leaves out both specific chemical bondenergies and energy stored in the protein awayfrom the active site. The theory, nonetheless, isquite successful in dealing quantitatively with thephenomena of selectivity, using only the knownproperties of concentrated salt solutions and twophysical parameters to describe the role of theprotein, although these models will undoubtedlyneed successive extension.This physical explanation of selectivity is quan-

titative and physically specific in contrast to verbaldescriptions of selectivity. Physical models arespecific and testable and thus can be systematicallyimproved by the usual scientific process.I think utilitarian engineering analysis is likely

to be more productive than narratives of trajecto-ries or traditional literary discussions of bindingsites. What is needed in my view is engineeringas usual, but now engineering on the(sub) nanos-cale of proteins, relying mostly on the techniquesof ‘reverse engineering’(i.e. the solution of inverseproblems w59x), trying to design systems ratherthan just describe them.

4. Calcium channels

We illustrate these general ideas by consideringthe selectivity of calcium channelsw55–58,60x.The L-type calcium channel is one of the most

selective channels known and has complex prop-erties w61–63x. Nearly all the current through thechannel is carried by Ca ions under normal2q

conditions, when Ca is approximately 1% of the2q

external cations and 0.001% of the internal cations.The functional group of the L-type calcium chan-nel is formed(mostly) from four glutamates, madefrom 8 half charged carboxylate ions, the ‘selectiv-ity oxygens’, to emphasize their function.Here, we consider a model in which the selec-

tivity of the channel arises from physical effects,involving no quantum mechanical delocalizationw64–66x. This model does not depend on theprecise location of individual atoms of the proteinw67–72x but it does depend on some properties ofthe channel protein in an important way: thevolume of the channel’s pore determines selectiv-


ity, as does the ‘concentration’(i.e. number densityand excluded volume) of its glutamates and(to alesser extent) the mechanical properties of theprotein, and its effective dielectric ‘constant’.Indeed, physical models are specific and falsi-

fiable precisely because they depend on only asmall number of measures of specific propertiesof the protein. The remarkable fact is that thesefew measures are all that are needed to predict theselectivity of the L-type calcium channel under awide range of conditions. These measures are theonly traces of the channel structure found in thismodel; yet the model is quantitatively successfulin dealing with many of the properties of selectiv-ity w55–58x.In the physical approach, the crystallographic

structure of the active site is unimportant, exceptinsofar as it determines these specific measures,i.e. properties of the protein, e.g. volume, chargedensity, excluded volume, effective dielectric con-stant, and elasticity. These properties are veryimportant. Other properties of the protein areunimportant in the model and perhaps in reality.Of course, the selectivity of some types of channelsfor some ions undoubtedly will depend on orbitalinteractions and the specific location of ions andpeptides.A physical model of this sort can be written at

many levels of resolution. Here, I consider onlymean field models because of limitations of space,although physical models of ion channels at higherresolution are of considerable importancew73–76xand I spend a good fraction of my time trying todevelop them and learning to average their trajec-tories in a mathematically defined wayw77x.Molecular dynamics models are not considered

here because regrettably they cannot yet estimatethe macroscopic quantities and behaviors knownto be important in channel function, e.g. concen-tration, electrical potential, ionic current, Ohm’slaw, and Fick’s law. We adopt an engineeringapproach developed and used in computationalelectronics w12,14,18,78,79x. This engineeringanalysis of proteins uses only as much detail asneeded to explain biological phenomena. The goalis to use as little atomic detail as possible, althoughof course some atomic detail is essential.

5. L-type Ca channel

In our treatment of the L-type calcium channelthe protein is treated as 8 half charged oxygens ina pore of(to be determined) volume and effectivedielectric constant. The ions and oxygens aredescribed by the MSA using crystal radii(in thesimplest versionw56x) or radii that change insimple ways(in more complex versions).The calculation of the properties of the L-type

calcium channel can be done in different ways,even if one is concerned only with mean fieldtheories. Nonner et al.w56x show that the mainproperties of the Z-type calcium channel—its bind-ing and anomalous mole fraction effect—can bereproduced by MSA using only two adjustableparameters, the dielectric coefficient and volumeof the selectivity filter, which are set to reasonablevalues (dielectric coefficient 63.5 and volume0.375 nm) to fit one data point, and then not3

changed. Patterns of selectivity closely resemblethose reported experimentally, although the affinityof Mg is incorrectly predicted(by a factor of2q

10= or so).The physical origin of selectivity between two

ions is predicted without ambiguity in the model,as well as the dependence of that selectivity onthe properties of the channel protein, namely itscharge, size of its charges, volume of its selectivityfilter, and dielectric coefficient. But the physicalbasis involves a number of terms, most of whichdepend on the concentration of all other ions.Thus, the selectivity depends on the(1) electricalpotential;(2) ideal chemical potential(i.e. concen-tration); (3) entropy of charged and(4) unchargedhard spheres;(5) energy of charged hard spheres.Each of these terms is different for each type ofion and most depend on concentration. Thus, tounderstand the selectivity between two particularsituations of biological interest one must compareall the terms. This is not hard to do, since theequations are algebraic and explicit, and all termsare known without possibility of adjustment. Theterms are of nearly the same size, however, andso the results must be computed. Their relativesize cannot be evaluated by verbal discussion.Qualitative analysis is possible, on the other

hand, if a few terms dominate. Interestingly, the


most important biological property of L-type cal-cium channels—their selectivity for Ca over2q

Na —is dominated by two terms. This is not theq

only case in which evolution seems to have useda particular subset of all physical possibilities: seeany text of physiology or biophysics. It seems thatevolution often chooses the strategy of a sensibleengineer, KISS—Keep it Simple Stupid—when itchooses the physics to create a biological function.Complexity often arises in the structure and diver-sity, not the physics of biological systems. Oneimagines that when a few terms dominate thephysics of a system, genetic control is easier, andthe resulting phenotype is more likely to followsimple reproducible rules that are robust and sur-vive unforeseen changes in environment. Here theselectivity of the L-type calcium channel forCa vs. Na is dominated by only a few terms2q q

and so can be understood qualitatively.

6. Selectivity of the L-type Ca channel

The selectivity between Ca and Na arises2q q

in a simple way. The four glutamates of thechannel demand the presence of four mobile pos-itive charges nearby. If only Ca is present, two2q

Ca provide the four charges. If only Na is2q q

present, four Na provide the four charges, butq

the four Na are twice as crowded as two Ca .q 2q

The four Na occupy twice the space of the twoq

Ca (because Ca has the same diameter as2q 2q

Na ). The free energy necessary to pack the extraq

two charged spheres into the channel accounts for;60% of selectivity. Approximately 35% of theselectivity comes from the different electricalpotential found in the channel when Ca is2q

present. The electrical potential is different becauseCa provides better screening(of the negative2q

charge of the glutamates). The double valence ofCa allows two charges to approach within one2q

ion diameter of the glutamate oxygen where Naq

allows only one. Five percent of the selectivitycomes from other effects, e.g. entropy resultingfrom specific arrangements of the ions andglutamates.It is important to emphasize how different this

view of selectivity is from the traditional view.Traditional models of binding more or less ignore

the electrical term altogether, as can be easilyverified by noting the absence of Coulomb’s lawin their derivation or the absence of a dielectricconstant(or permittivity parameter) in their outputequations. Physical models predict electrical poten-tials of hundreds of millivolts—i.e. of the order of4 k Tye—and those potentials vary more or lessB

linearly with the logarithm of Ca concentration2q

under standard conditions. Thus, a substantial frac-tion of selectivity in a physical model comes froman effect ignored in most traditional treatments ofbinding and selectivity, the electrical energy need-ed to bind an ion to a charged sitew80x.Traditional models of binding and selectivity

(and of enzyme kinetics, for that matterw81,82x)also assume rate constants and binding constantsindependent of concentration of the binding spe-cies. This seems unlikely on physical grounds,because the binding of a charged group to acharged site will inevitably change the electricalpotential, according to Gauss’ law of electrostatics(i.e. Maxwell’s equations) w80x. The change inpotential will be large because the binding sitesare so small and the ‘capacitance to infinity’ is sosmall (i.e. the self-energy is so large), as waspointed out sometime agow80,83x. The existenceof such effects is the basis of single charge devicesw84,85x studied experimentally in hundreds of lab-oratories every day, and the effects are likely tobe larger in proteins than in single charge devicesbecause the binding sites are smaller. One can useother language, to describe the same physics. Thechange in potential with concentration is an exam-ple of shielding. Shielding determines many prop-erties of channelsw60,77,80,86–104x as it doesmany properties of ionic solutionsw19,105–107x,plasmas of a real gasw8–10x, or plasmas of quasi-particles—the holes and ‘electrons’ in a semi-conductorw11–18x.When shielding is important, the binding con-

stant changes many fold as conditions vary. Thus,many of the qualitative features of binding areproduced by a variation of binding constant notpermitted in traditional theories of chemicalkinetics.Of course, the MSA treatment of binding has

significant limitations. Perhaps most importantly,it is a theory of a spatially homogeneous solution


but has been applied to a strikingly inhomogeneoussystem. Henderson and colleaguesw25,57,58x haveaddressed some of these issues, as has Hansen ina different contextw32,33,108x and all agree thatmore precise simulations and theories confirm theeffects found in the MSA model.Of course, there are other limitations in the

MSA theory.(1) The theory needs to be extendedto non-equilibrium by embedding in PNP or some-thing similar.(2) The theory needs to be extendedto a range of pH. The original theory was com-puted at basic pH because the physical parametersof un-ionized species are not available in thephysical chemistry literature.(When available, thevariable ionization can be easily incorporated intothe numerical schemew60x). (3) The treatment ofwater should be more realistic.(4) The treatmentof the protein should have higher resolution.(5) The macroscopic parameters, crucial to selec-tivity—volume of the pore, diffusion coefficient,and dielectric coefficient—need to be understoodin atomic detail.The SPM is a step towards improving the theory

of selectivity w55x. The properties of water andmechanical properties of the protein are includedby representing the solvent molecules as unchargedspheres in a uniform dielectric. That treatmentremoves most of the difficulty in dealing withMg although it still has too low resolution to2q

deal with all blocking effects of interest experi-mentally. The SPM shows how binding producesmechanical forces and how mechanical propertiesof the protein modify selectivity. These mechanicalforces are large and inescapable; they may be theforces that drive conformational changes. Interest-ingly, the theory shows that all the properties ofselectivity (in the SPM model) are functions ofjust one variable, the free energy per mole of theselectivity oxygens of the channel protein. It istantalizing that this is exactly the variable con-trolled by the protein and its genome. The ‘activ-ity’ of the selectivity oxygens is the actualphenotype of the genome, not the location of theatoms(in the SPM model and perhaps in reality).Despite the evident limitations, what is striking

is that simple physical models like the MSA andSPM can deal quantitatively with a large range ofselectivity phenomena. The physical effects in

these models are more or less confined to theconcentrated ionic plasma in the selectivity filter.The properties of that plasma are calculated usingknown results from the physical chemistry ofconcentrated salt solutions.In view of the success of these simple models,

I believe that understanding of binding and selec-tivity should start with known physical propertiesof concentrated ionic solutions and add in thecomplexities brought by the protein, one by one.This approach might lead to progress in otherareas, e.g. the study of drug docking and proteinfolding.

7. New work

Work is well underway extending this approach.Recently, Density Functional Theory(of statisticalnot quantum mechanicsw19,24,27x) has beenappliedw34x using the approach of Rosenfeldw20–23,28,30,31,35,51,109–119x to extend our work toexplicitly inhomogeneous systems. The theorycomputes easily enough in complex situations andreproduces the MSA quantitatively when appliedto systems where both are valid. DFT has beencombined with a coupled treatment of flux andelectrostatics using the Poisson and Nernst–Planckequations, called PNP in the biophysical literaturew60,77,80,86–104x and Poisson Drift Diffusion inthe(enormous) literature of computational physics,where the integrated method was bornw12,78,79,120,121x. These equations are solvedessentially as the semiconductor community learn-ed to do many years agow11–15,17,122–131x.The tricks they use hold much promise for bio-physics and physical chemistry in my view, wherecoupled solutions of Poisson and transport equa-tions are difficult to find.The selectivity of other types of channels has

been analyzed. A Monte Carlo simulation of theDEKA locus of Na channels has been performedq

with satisfactory results(Nonner and Henderson,personal communication) and the recent structureof the CP channelw132x has been used to calculatethe selectivity of anion channels(Nonner andGillespie, personal communication). A mean fieldtheory of single filing has been used to build aphysical theory of the K channel with some


success(Nonner and Gillespie, personal comm-unication).

8. Summary and historical note

Writing on this proud occasion of Dr Edsall’s100th birthday, I note that we are stumbling alongin the direction implied by Cohn and Edsallw5x, along time ago. They recognized that proteins bristlewith charge(cf. p. 73 of w6x). Here, we considerthe counter ions of that charge. We focus attentionon the permanent charges of the active site of theprotein, discovering that protein function can bedriven by the large energies stored in the(non-ideal) plasma of crowded charge surrounding theactive site. Protein function can be controlled bya few properties of the active site, its volume andpermanent charge, its local dielectric constant, andits mechanical strength. Specific arrangements ofatoms are not needed to account for the strikinglycomplex specificity of active sites of channels.This view of protein function is in the engineer-

ing tradition, engineering as usual, but here ofproteins. The engineering approach seeks controlof a system by building a specific physical model,a device equation so simple that it demands to befalsified and then extended. The engineeringapproach builds a succession of device models andequations of increasing accuracy, as new structuresand energy sources are discovered.The sequence of engineering models of protein

function will yield more understanding and control,with enormous consequences for science, technol-ogy, medicine, and life. All depend importantly onproteins, as John Edsall was the first to show me,thereby enriching my life, for which I am forevergrateful.

Acknowledgments

The stamp of the Edsall approach should beevident throughout this paper and the work itreports. That work has been done in a collaborationled by Wolfgang Nonner. It is a personal andprofessional pleasure to interact with Wolfgangevery day. Dirk Gillespie, Doug Henderson, andmany others have made important contributions.Funding provided generously by DARPA has been

indispensable and we thank the several programmanagers who have supported us, particularly,Shaun Jones, Mildred Donlon, Anantha Krishnan,and Sri Kumar(in chronological order), and theother program managers and their bosses whomade that support possible.

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proteins, channels and crowded ions

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