BACTERIOLOGICAL REVIEWS, Mar., 1966Copyright © 1966 American Society for Microbiology

Symposium on Bioelectrochemistry of Microorganisms'I. Membrane Potentials and Permeability

VINCENT P. CIRILLODepartment of Biological Sciences, State University ofNew York at Stony Brook, Stony Brook, New York

HISTORICAL REVIEW...........................................................Resting Membrane Potential.................................................Donnan Equilibrium .......................................................Ion "Pumps"..............................................................

IONIC DIsTRIBUTION IN MICROORGANISMS........................................Ionic Distribution in Escherichia coli, Interpreted by the Membrane Theory..........Concept ofMembrane Carriers...............................................Ionic Accumulation and Exclusion by Intracellular Binding Sites....................

Entropy ofion associations in a fixed-charge system.............................Selectivity ofionaccumulation.......................................

Surface potentials .........................................................CONCLUSIONS.................................................................LITERATURE CITED............................................................

HISTORICAL REVIEW

Resting Membrane PotentialIt has been known since Galvani that it is possi-

ble to measure electrical potential differencesbetween tissues and across multicellular mem-branes of plants and animals. More recently,especially since the development of microelec-trodes with tips less than 1 1A in diameter, elec-trical potential differences have been found acrosscell membranes of individual cells wherever suchmeasurements are possible. This resting mem-brane potential in many cases corresponds to theelectrical potential expected from the asymmetricK+ ion distribution across the cell membrane ac-cording to the Nernst equation:

Em = (2.3RT)/F log (foK+o)/(fiK+i) (1)

where Em is the resting membrane potential (involts); R, the gas constant (1.98 cal/mole degree);T, the absolute temperature; F, the Faraday(23,062 cal/v); K4i , K4+, fi, andf. are the K4 ionconcentrations and their respective activity co-efficients inside and outside the cell; and 2.3 is thenumerical constant for conversion from loge tologeo. If it is assumed that the activity coefficientis the same inside and outside the cell (e.g., thereis not significant binding of Ki ions on either sideof the membrane), then equation 1 becomes:

Em = (2.3RT)/F log K+o/K+i (2)

'This symposium was held at the Annual Meetingof the American Society for Microbiology, AtlanticCity, N.J., 26 April 1965, with R. L. Starkey as con-vener and consultant editor.

The possibility that the activity coefficients forthe Kit ions on the two sides of the membrane are

not the same, because of intracellular Ki bind-ing, and the alternative interpretations of themembrane potential which such binding requires,will be discussed in a later section.A qualitative statement of the Nernst equation

is simply that an electrical potential differenceexists across a membrane if there is an asymmetricionic distribution. Quantitatively, the membranepotential is equal to the product of (2.3RT)/F,which is a constant at a given temperature, andthe log of the K4 ion concentration ratio. In thebiological range, the constant (2.3RT)IF is ap-proximately equal to 60 mv, e.g., 59.15 mv at 25 Cand 61.73 mv at 38 C (11, 28).

In most cells, the intracellular K4 ion concen-

tration is far greater than the extracellular con-centration (19): a K4./K i ratio of 1 :10 wouldrepresent an Em = -60 mv, whereas a ratio of1 :100 represents an Em, = -120 mv. The notionthat the membrane potential can be interpretedas a potassium concentration or diffusion poten-tial is supported by the observation that, withinlimits, in certain muscles and algae (where theconcept applies best), the membrane potentialchanges in conformance with the Nernst equa-tion when the Ki ion distribution ratio is experi-mentally altered (2, 4, 19).The thermodynamic basis of the correspond-

ence between the Em and the potassium ion dis-tribution expressed by the Nernst equation maybe reviewed profitably at this point, since it willplace this observation in the proper perspectivefor the discussion to follow.

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The asymmetry of the K4 ion across the cellmembrane represents a chemical potential differ-ence. If Kit were a nonionized solute, the magni-tude of the chemical potential difference wouldbe a measure of the work required to bring aboutthe observed asymmetry. The chemical potentialdifference Achem associated with the concen-tration difference across the cell membrane, is(18):

AAchem = 2.3RT log K+i/K+o (3)

However, because Kit is a charged particle,any conclusions about the work required to bringabout a given concentration difference across thecell membrane must take into account the elec-trical potential difference across the membrane(29). The latter represents the work done in trans-ferring a unit charge across the membrane. Theelectrical potential difference, A4elec, is given bythe product EmF:

Ajlelec = EmF (4)

The electrochemical potential difference, AM,of a charged particle is, therefore, the sum of itschemical potential difference and its electrical po-tential difference, as given in equation 5. Theelectrochemical potential difference for K4 ion isgiven in equation 6.

AP = AP-ehem + Apeleo (5)

AP = 2.3RT log K+i/K+o + EmF (6)

If the Ki ion distribution across the membraneis the result of a passive process, there will be noelectrochemical difference across the membraneat equilibrium (i.e., AM = 0 for a passive process).In such a case, equation 6 becomes:

EmF = -2.3RT log K+i/K+0 (7)

Equation 7 can be recognized as a form' of theNernst equation (equation 2).To state that the distribution of an ion con-

forms to the Nernst equation implies, therefore,that its distribution is the result of a passiveprocess.

Donnan EquilibriwnThe fact that the potassium ion distribution,

in those cases in which it conforms with theNernst equation, is the result of a passive processleaves unexplained the origin of the asymmetry.An early explanation for the K4 asymmetry wasproposed by Bernstein (19), who attributed theK+ ion asymmetry to a Donnan equilibrium re-sulting from the presence of intracellular non-diffusible anions (such as proteins). The way thatan asymmetry of diffusible ions results from an

asymmetry of nondiffusible ions is illustrated inFig. 1.A Donnan equilibrium for a nonrigid cell, how-

ever, will not actually reach equilibrium becauseat all times the internal osmotic pressure will begreater than that of the medium. As a result,water tends to move into the cell constantly,which alters the ionic concentrations and causescontinuous ion and water movements until thecell bursts (unless it is opposed by a physical orhydrostatic force). A nonpenetrating ion with acharge opposite that of the nondiffusible ion in-side the cell would set up an opposing Donnanequilibrium, thus stabilizing such a system. It waslong believed that extracellular Na4 served thisfunction, since the Na4 is usually present at ahigher concentration outside the cell than insideand, unlike the response of the resting membranepotential in certain cells to changes in K4t ionconcentration ratio, the resting membrane poten-tial of these cells is not affected by changes in theNa+ ion concentration ratio over the same range.

Ion "Pumps"This view of Na4 ion impermeability prevailed

unmodified until the availability of the radioiso-tope Na24, when it was found that cells postulatedto be impermeable to Na+ ion were in fact freelypermeable to Na24. To account for the apparentimpermeability of the cell membrane to Na4 ions,and for the correspondence between the mem-brane potential and the K4t diffusion potential,

Ka X

cl- X

[K+],[C1l]-J K1;[Cl1;xg2 *.y4,y+z)X2 atY( +sZ

X -Y.y +Z)7

Membrane

xex>y m [cl-,i>[cl]i;(y+zhX - [KI]J > [K ].

FIG. 1. Donnan equilibrium. It is proposed that themembrane surrounding a cell is impermeable to thelarge anions, P-, but freely permeable to K+ and Cl1ions. Electroneutrality must exist on each side of themembrane at all times. At equilibrium the productK+0Clh0 = K+iCI-i. If the external concentrations ofK+ and Cl- are represented by X, and the internalconcentrations by Y and Z as shown, at equilibriumthere will be an asymmetry ofK+ and Cl- ions acrossthe membrane.

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Dean (19) in 1941 proposed that the Donnan dis-tribution of K+ ion and, therefore, the K+ diffu-sion potential would not be altered if the sodiumion were expelled from the cell in a nonionic formas rapidly as it entered by diffusion from the ex-ternal medium. Although sodium efflux in asupposedly complexed nonionic form would notaffect the membrane potential determined by thepotassium ion distribution ratio, sodium effluxwould be an energy-dependent process, sinceNa+ ion would be distributed against its electro-chemical potential gradient (4, 10, 12, 18, 19, 30).

This proposal that metabolic energy is coupledto a process for the maintenance of Na+ ionagainst its electrochemical potential gradient isnow familiar as the "sodium pump theory." (Thequantitative aspects of this energy requirementwill be explored for a specific case below.)The fact that cell membranes are permeable to

Na4 ions, as revealed by the use of Na24, requiresthat the Nernst equation relating the membranepotential to ionic distribution must take into ac-count not only the Kit ion but the Na4 and Ch7ions as well. The requisite equation withoutmodification would be:

Em = (2.3RT)/F log (8)* (K+i + Na+i + C1-0)/(K+o + Na+0 + Cl-i)

However, Hodgkin and Katz (12) proposedthat the equation, to be applied appropriately to abiological system, must take into account for eachion its mobility in the membrane, v, and its parti-tion coefficient, b, between the membrane and theaqueous solutions, which determines the ion con-centration in the membrane. The appropriateequation becomes:Em = (2.3RT)/F log

(PKK+i + PNNa+i + PciCl0)/ (9)(PKK+o + PNaK+o + PciChi)

where PK , PNa , and PCL are permeability co-efficients for each ion defined by:

P = (RT)/(aF)vb (10)

where R, T, and F have their usual meaning, ais the thickness of the membrane, and v and bare the mobility and partition coefficients,respectively, for any given ion. (For derivation ofthis equation, see reference 12, appendix.) Someof the values for the permeability coefficients(PK, PNa, and PCL) used in the literature formuscle and nerve are: in frog sartorius, 1, 0.027,and 0.23, respectively; in several muscles of SouthAmerican frogs, 1, 0.015, and 0.17; and in thegiant axon of the squid, 1, 0.04, and 0.45 (12, 19).When PNa. is small relative to PK, which is

true in all of the cases mentioned above, equation10 reduces to equation 11, which is another form

of the Nernst equation describing the diffusionpotential which would be observed if only Kitand C1- ions penetrated the membrane.

Em = (2.3RT)/F log (PKK+i + PC1Ch-0)/(PKK+O + Pc1Cl-h) (11)

This equation is formally identical with equa-tion 1. However, there are numerous data fromseveral cell types that place some restrictions onthe applicability of equation 10 or 11 to mem-brane potentials. Among the most important ob-served inconsistencies are the following: therange of external Ki concentrations over whichthe Em of muscle cells obeys the Nernst equationis at concentrations greater than the normal invivo plasma level; whereas alterations of the ex-ternal potassium concentration (within the limitsjust listed) has the predicted effect on the mem-brane potential, corresponding alterations of theinternal concentrations do not give the predictedresponse; replacement of the axoplasm of thegiant axon of the squid with seawater has rela-tively little effect on the membrane potential.These and other inconsistencies have led to modi-fications of the membrane theory, reviewed byRidge and Walker (19).

Finally, although the measured resting mem-

a1.000.

z0a

zo >z

w

Z0-i4-i

s -i< L

<o E

iwi

[K];

'% % d- ._ [Na]1'x_ _ _ _ _ _ _ _ x4 0

[K]oI I

O 2 3 4 5 ' 6AGE OF CULTURE (HOURS)

FIG. 2. Ion distributions between the cell and mediumin Escherichia coli during growth. The cellular con-centrations of K+ and Na+ are shown. The externalconcentrations of these ions are indicated by the arrows.The optical density changes shown in the upper panelare a measure of cell growth (24).

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MEMBRANE POTENTIALS AND PERMEABILITY

TABLE 1. Intracellular Cl, Na, and K concentrations in Escherichia coli*

Phase of growth Cl0 Cl Clo/Cli Nao Nai NNai/ K0 Ki | /K

mm mm mm mMmm

Stationary 25 21.2 4 0.6 (8)t 1.18 145 177 +4 (4) 1.22 5 11.2 i 1.0 (4) 2.250 44.5 ± 2.0 (8) 1.12 170 188 i 3 (4) 1.11 5 9.6 ± 1.1 (4) 1.975 68.1 i 3.6 (15) 1.10 145 165 4 (8) 1.14 5 13.0 1.1 (8) 2.6100 69.9 i 1.0 (7) 1.43 170 148 i 1 (4) 0.85 5 28.3 i 0.5 (4) 5.7Avg 1.13 1.16 2.2

Logarithmic 25 8.3 4 1.5 (14) 3.0 145 81 + 2 (7) 0.56 5 217 ± 5 (7) 43.450 17.1 i 0.9 (27) 2.9 120 70 i 2 (7) 0.58 5 224 ± 8 (6) 44.875 23.7 ± 3.0 (19) 3.2 145 80 4 4 (6) 0.55 5 227 i 5 (7) 45.4100 33.5 i 1.7 (8) 3.0 170 89 :i 8 (4) 0.52 5 257 ± 1 (4) 51.5Avg 3.0 0.55 46.3

* The errors shown are the standard errors of the mean; the number of determinations is given inparentheses.

t Cli was determined by use of C136.

TABLE 2. Electrochemical potential differencesassociated with ion distributions in

Escherichia coliAp = 2.3RT log Cj/C0 + FE.,A!z = 2.3RT/F log Cj/C0 + E.

Em = 2.3RT/F log Cl-i/Cl-0 = -26 mv

Ion Ci/C,* ~Diffusion Em AIIon Ci/C0* potential Em A/F

Na+ 0.55 -20 -26 -46K+ 46.0 +100 -26 +74

* Cation concentration ratio.

brane potentials correspond to K+ diffusion po-tentials suggesting that the K+ ion distribution isthe result of a passive process, in other cases boththe K+ and the Na+ ions are distributed againsttheir electrochemical gradients. In such instances,both K+ and Na+ pumps are postulated to ac-count for the electrochemical potential differencesrepresented by their distributions (29, 30).

IONIC DISTRIBUTION IN MICROORGANISMS

Ionic Distribution in Escherichia coli, Interpretedby the Membrane Theory

In a recent series of studies, Schultz et al. (23-25) measured the cell-to-medium distribution ofK+, Na+, and Cl- ions in E. coli during growth(Fig. 2, Table 1). The ionic distributions observedduring the log phase are similar to those observedin higher cells, namely, a high cell-to-mediumratio of K+ ion and a low cell-to-medium ratiofor Na+ and C1- ions. To decide whether any, or

all, of these ionic distributions are the result of apassive process or depend upon a "pump," itwould be necessary to know the resting mem-brane potential. In E. coli it is not possible to

measure this by internal electrodes. However, anindirect approach is possible, based on the con-cept, reviewed in the previous sections, that themembrane potential represents a diffusion po-tential of an ion whose asymmetry results from aDonnan equilibrium. This indirect approach in-volves the calculation of the diffusion potentialrepresented in turn by the K+, Na+, and Cl- ion(Table 2). This calculation gives three differentvalues: -26 mv for Cl-, -100 mv for K+, and+20 mv for Na+. If it is assumed that only oneof these diffusion potentials corresponds to theresting membrane potential, and that the relatedion is, therefore, distributed by a passive process,then the other two ions are distributed againsttheir electrochemical gradients and must be trans-ported by an "active" process. Schultz et al. pre-sented evidence which indicated that, in the logphase of E. coli, the chloride ion is distributed bya passive process. If this is so, then both K+ andNa+ ions are transported against electrochemicalgradients by an energy-dependent process. Themagnitude of the electrochemical potential repre-sented by the K+ and the Na+ ion distributionsand, hence, a measure of the energy requirementsto maintain this potential, may be calculated fromthe definition of the electrochemical potentialwhich, for K+ ion, is indicated by equation 6.

Randall (18) suggested that, as a simple com-putational method to arrive at a measure of theelectrochemical potential represented by a givenionic distribution, both sides of the equation bedivided by F, to give an equation in the form:

(A)/F = (2.3RT)/F log K+i/K+o + Em (12)

For the biological range:

(AIA)/F = 60 mv log K+i/K+o + Em (13)

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Chemical Electrical "Active"

I t-100 -26

l l

Na+Chemical Electrical "Active"

CoChemical Electrical "Active"

-46

I+20 -26l -26 E-26

l I

+74

FIG. 3. Potential differences associated with K+, Na+, and Cl- ion distributions in log-phase cells ofEscherichiacoli. The data in Table 2 are shown in graphic form. The electrical potential difference, -26 mv, is calculatedfrom the Cl- ion distribution. The chemical potentials represent the sign and magnitude, in millivolts, expectedfrom the distribution ofK+, Na+, and Cl- ions, respectively, ifeach ion were distributed passively across the mem-brane in accordance with the Nernst equation. The dotted lines represent the magnitude of the difference betweenthe calculated potential and the chloride potential. Assuming that the chloride potential is the actual membranepotential, the magnitude of the difference and its direction indicate the work required to establish the observedelectrochemical potential differences. The work requirement (Ap/F of equation 12) represents "active" transport.For K+, the active transport is directed into the cell; for Na+, it is directed outward. This representation is adaptedfrom Randall (18).

The term, (A MA)/F, is a measure in millivolts ofthe total electrochemical potential; the middleterm is a measure of the chemical potential andis simply the negative of the diffusion potential;the last term is a measure of the electrical poten-tial, and is the membrane potential. This calcu-lation for K+, Na+, and C1- ions for log-phaseE. coli is shown in Table 2 and Fig. 3.These calculations are predicated on two as-

sumptions: that the membrane potential is a

diffusion potential, and that the chloride ion ispassively distributed across the membrane ac-

cording to the Nernst equation. The second as-

sumption may be wrong, because the chloride ionmay also be distributed against its electrochemicalgradient. Without knowledge of the actual mem-brane potential, this cannot be ascertained. Morewill be said about the first assumption later.

Support for the idea that both K+ and Na+ ionsare transported by active processes comes from a

study of the characteristics of the distribution of

these ions when stationary-phase cells of E. coliare inoculated into fresh medium (Fig. 2). Thecells taken from the stationary phase at a low pHshow a cell-to-medium distribution ratio for allthese ions of nearly 1 (Table 1). On the basis ofthe C1- ion distribution, the Em has fallen from-26 mv of the log phase to -6 mv. When suchcells are placed in fresh medium at pH 7.0, thereis an immediate accumulation of intracellularK+ and an extrusion of the Na+ ion. Within 1hr, the log-phase distributions are reached (Fig.2). The effect of various conditions and inhibitorson the ionic movements is shown in Fig. 4. Themetabolic dependence of these net movements, as

shown by these data, is expected. The differentialinhibition by NaF of the Na and K net fluxes,however, is particularly noteworthy, since inmany cells the movements of these two ions istightly coupled. Inhibition of the flux of one has a

marked effect on the flux of the other (10). In E.

Inside

Membrane

Outside

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MEMBRANE POTENTIALS AND PERMEABILITY

-J0cr

z0

FIG. 4. Effect of glucose deprivation and metabolicinhibitors on net Na and K movement (24).

coli, these fluxes appear to be completely un-coupled.

Concept ofMembrane Carriers

It is not unexpected that processes associatedwith ionic movements show characteristics whichresemble enzyme reactions. Most transportprocesses show saturation kinetics, ion discrimi-nation, competitive inhibition, and counterflow,all of which indicate that the transported ionsduring transport are somehow associated with amembrane component. These components, forthe most part, have only conceptual existence atthe present time. However, one component prob-ably involved in the active transport of Na+ ionshas been purified from the membrane fractions ofhigher cells (17, 26, 27). It shows "adenosinetriphosphatase" activity, and the unusual features,distinguishing this activity from other "adenosinetriphosphatase" activity, of requiring both Natand K+ ions for maximal activity as well as a re-markable sensitivity to the glycoside, ouabain, aspecific inhibitor of the process of active extrusionof sodium ions (10). A Na-K dependent "adeno-sine triphosphatase" has been sought, but notfound, in microorganisms (8). Ion uptake be-lieved to represent carrier-mediated transport hasbeen studied in a number of microorganisms.However, the most extensive studies have beencarried out with bakers' yeast and E. coli. Inbakers' yeast, as in the majority of cells studied,

K+ and Na+ ions are distributed against theirelectrochemical gradients, and K+ ion accumula-tion (as well as Na+ ion extrusion) is an energy-dependent process (20). In a study of K+ ionaccumulation by bakers' yeast, Armstrong andRothstein (1) presented evidence that the alkalimetal ions as well as H+ ions are transported by acommon carrier system. From kinetic studies andthe pattern of competitive inhibition among thealkali metal ions and H+, they were able to assigna "relative affinity" of these ions for the mem-brane carrier (Table 3).The inhibition of K+ transport by H+ ion is

seen in Fig. 5. From a comparison of the slopeand intercepts of the curves for pH 2.0(10 M [H+]) andpH 3.5 (10-3- 6M [H+]), it can beseen that, over this range ofH+ ion concentration,H+ ion is a competitive inhibitor. However, afurther analysis of the effect of H+ ion concen-tration on K+ transport shows that, between 10-and 104 M, the H+ ion acts as a noncompetitiveinhibitor (compare the curves for pH 3.5, 4.5,and 6.7 and 8 in Fig. 5). From these data, theauthors inferred that there are present on thecarrier molecule two sites with which H+ ionsmay combine.Only one of these sites is the "transport" site,

namely, the site with which the transported cationmust combine for transport to be effected. H+ ionsmay compete with the alkali metal ions for thissite. However, there is an additional site on thecarrier with which H+ ions (as well as other ca-tions) may combine. When this second site isoccupied by an H+ ion (which it is at an H+ ionconcentration greater than 10o M or at a pH be-low 4.0), then the maximal rate of carrier trans-port is reduced. These authors refer to this secondsite as a "modifier" site. The "affinities" of othercations for this site are also given in Table 3. Notethat the "affinities" of each ion for the transport

TABLE 3. Affinity constants of the transport siteand the modifier site for cations

Transport ModifierIon

Concn Relative Conca Relative

H+Li+Na+K+Rb+Cs+Mg+Ca+

mm

0.320150.71.3

12300

>1,000

0.4302012

17400

>1,000

mm

0.0520203

483

0.0277

1131

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and modifier site may differ markedly. This isespecially so for Mg++ and Ca++, which are nottransported to any significant extent by the alkalimetal carrier (6). The most important conse-quence of the existence of two distinct sites withdifferent affinities is the fact that, depending uponthe concentrations of metal ions or the H+ ionconcentrations, there will be a different order ofcation discrimination. The manner in which theorder of cation preference may be changed isillustrated in Fig. 6 and 7, which show how theparameters of ion discrimination, namely, theKm and the Vmiaz, are altered by changes in thehydrogen ion concentration.These data show how the concept of membrane

carriers can provide an understanding of themechanism of selective ion accumulation in bio-logical systems. The mechanism of coupling ofmetabolic energy to carrier transport to accom-plish ion movements against electrochemicaldifferences has been reviewed recently (27, 30);the subject is beyond the scope of this review.

Ionic Accumulation and Exclusion byIntracellular Binding Sites

Up to this point it has been assumed that, be-cause of the correlations between ion distributionand the electrical potential, the activity coefficientsof the ions inside and outside the cell are identical,or nearly so. If, however, the ion activities are

0.5

0.4 to

.. 0.3

.3-

~0.2-

0.1 61*'

1/K conc. in mM/liter

FIG. 5. Double reciprocal plots of the effect of theH+ ion concentration on K+ uptake by bakers' yeast(1).

16-

14-

12-

.C

0

x

HE

10-

8-

6-

4.

3 4 5 6 7 8pH

FIG. 6. Effect ofpH on the Vma. of alkali metaltransport in bakers' yeast (1).

significantly different on the two sides of the cellmembrane, then the conclusions based on thisassumption are invalid. The correspondence be-tween ion distributions and electrical potentialspredicted from the Nernst equation would thenhave to be reinterpreted. Others have emphasizedthis in their efforts to interpret ion accumulationas the result of intracellular binding of ions tometabolically controlled sites rather than as aresult of the activity of membrane carriers. Theseinterpretations are sometimes referred to as"sorption" theories, the concept of which is indi-cated by Cowie and Roberts (7) as follows, basedon their early work on E. coli and Torula utilis:"The protoplasm may be likened to a sponge, thecell membrane to a surrounding hair net unableto exclude the entrance or emergence of smallmolecules. Into this system water may diffusefreely and carry with it many of the dissolved sub-stances which it contains. Thus the nutrients ofthe environment diffuse into the reactive sites ofthe protoplasm and become nondiffusible."

Entropy of ion associations in a fixed-chargesystem. For understanding the role of ion bindingin accumulation of elements in cells, there isneed for consideration of the thermodynamicfactors that determine anion-cation associations(13, 14, 16).The extent of anion-cation association for the

reaction:

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MEMBRANE POTENTIALS AND PERMEABILITY

120

100

E

By

80-

60-

40-

20- LINo

* i Cs

10- A

8-

4-

2-Rb

0 K

3 4 5 6 7 8pH

FIG. 7. Effect ofpH on the Km ofalkali metal tport in bakers' yeast (1).

n Anionm- + m Cation-+AnionnCationm complex

depends upon the association constant, K:

K = [Anion.Cationml/[Anionfl]n[Cationn+]mFrom classical thermodynamics we know thaequilibrium constant depends upon the chantfree energy, AF, associated with reaction 14cording to:

AF = -2.3RT log K

From the second law of thermodynamics weknow that the change in free energy associwith a reaction such as shown above (occujin solution and at constant temperature) ispressed by:

AF = AE - TAS

The second law states that only reactions w]lead to a loss of free energy occur spontaneotHow far the reaction goes depends upondifference in internal energy, AE, and the dirence in the entropy, AS, between the productthe reactants. If the product of the reaction,association complex (nAmC), has less energy tthe reactants, the dissociated ions (Am- + Cthen AE is negative and the system goes tomthe associated state (i.e., Eassoc - = - ZBesides loss of internal energy (-AE), a reacis also favored if it goes from a less randomstatistically less probable) state to a more ranc

(or statistically more probable) state; AS is anindex of the difference between the randomness(or statistical probability) of the final state andthat of the initial state. [Usually F refers to theGibbs function, which would be given by: AF =AH - TAS. The right side of equation 17 refersto the Helmholtz free energy usually designatedby A, which would give: AA = AE - TAS. How-ever, the difference betweenAH andAE is negligi-ble for ion associations occurring in solution atconstant pressure and temperature, and equation17 is, therefore, valid.]An increase in entropy (Sfinal - Sinitial) =

+AS) also favors a spontaneous change. SinceAS is multiplied by - Tin equation 17, a positivechange in entropy will make AF negative. Sincethe two conditions which favor a spontaneouschemical reaction, namely, a decrease in internalenergy (-AE) and an increase in entropy(- TAS), both result in a negative AF; the sinequa non of a spontaneous reaction is that thereresults a decrease in the free energy. (This is arestatement of the second law of thermody-

rans- namics.)With the meaning of AF defined, we may pro-

ceed with an inquiry into the conditions which(14) affect anion-cation associations in biological sys-

K tems.We may rewrite equation 16 in a more useful

form:(15)

It thelog K = -AF/2.3RT (18)

&e mn This equation indicates that, if the association re-; ac- action is spontaneous, that is, AF is negative, then

the right-hand term of equation 17 is positive,since - (-AF) = +. Thus, K must be greater

(16) than 1, which is what would be expected for aalso spontaneous reaction. However, how muchated greater than 1.0 would depend upon how largeTin -AF would be. Equation 15 can be changed toexg equation 19 by replacing AF by the definition of

AF given by equation 17.

(17) log K = -AE/2.3RT + AS/2.3R (19)

hich As has been stated before, the entropy termLisly. of equation 19 is a measure of the probabilitythe of occurrence of the final state compared with

Bfer- the probability of occurrence of the initial state.and This relationship is expressed by:

than

vard5E.

tion(ordom

AS/2.3R = log P.asoc/Pdiss (20)

where P..5,, is the number of equally probableconfigurations that can be assumed by the associa-tion complex (nAmC), and Pdi88 is the number ofequally probable configurations that can be as-sumed by the ions when they are dissociated.Substituting this definition into equation 19 gives:

9

4

hrs

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log K = -AE/2.3RT + log Pa8soo/Pdies (21)

This equation may be converted to the more use-ful, exponential form:

K = PaSOC/Pd,38al-ARI2 8RT (22)

The association constant is, therefore, determinedby the ratio Pas8Oc/Pdjss (or the partition func-tion), which represents the entropy factors of thereaction, and the exponential expression, whichrepresents the energy factors of the reaction.

It will be useful to analyze the influence of theentropy and energy factors separately. Since theprobability function relates to the relative prob-ability of occurrence of the associated state andthe dissociated state, association is usually smallin dilute solutions. This is due in part to the factthat the fraction Passoc/Pdiss is very small, be-cause by becoming associated both ions losetranslational freedom. One might state that thecomplex represents a less random state. This de-crease in randomness (or entropy) will, therefore,not favor association. However, Ling (13-15) em-phasized that the entropy factor is different inionic associations when one of the ions is fixed ina three-dimensional lattice. In such a case, onemust consider the volume which is available tothe dissociated ions as compared with the volumeavailable to the complex. As long as the volumefor the dissociated state exceeds the volume occu-pied by the associated state, the ratio Passoc/Pdisswill be very small and the association constant(see equation 15) will also be small. If, however,the fixed charges are very close together but uni-

Space occupied byassociated countercations

formly distributed so that their association sitesdo not overlap, as shown in Fig. 8, then the rela-tive volume occupied by the associated ions maybe equal to, or even larger than, the volume avail-able to the dissociated counter cation plus fixedanion. In the latter case, there may actually bean entropy gain upon association. Ling indi-cated that the extent of association may also befavored by decrease in the dielectric constant ofwater between ions separated by only one or twowater molecules. The dielectric constant of bulkwater is very high, namely, 80, which decreasesattraction between charged groups to V0th thatwhich would exist in air at the same distance.However the first water molecule, and to someextent the second water molecule, around a cationhas a reduced dielectric constant, possibly as lowas 1. However, the importance of the dielectricsaturation effect on ion association in biologicalsystems has been challenged by Conway (4, 5).

Selectivity of ion accumulation. A combinationof the entropy factors which Ling suggested forthe fixed-charge system, plus the possible effectof dielectric saturation on ion associations in atightly ordered lattice, can account for the ac-cumulation of ions by biological structures. How-ever, the selectivity of this accumulation requiresfurther analysis of the energy factors associatedwith anion-cation combinations. Ling (13, 14) andEisenman (9) calculated the association energiesfor the combination between oxyanions of variousstrength and alkali metal cations and H+ ions.These data are represented in Fig. 9, in which theordinate represents the association energy, AE.

associated countercations

FIG. 8. Effect ofconfigurational entropy on ion association in a fixed-charge system and in free solution (14).

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-5.0 -4.0 -3.0 -2.0

c-value

(angstrom units)

FIG. 9. Relation between the calculated association energy, AE, of various cations and the strength of the aniongroup (14).

The more negative the association energy, themore stable the association will be. This is shownin equation 22, in which the equilibrium constantfor association is shown to be proportional toexpo - AE/2.3RT. When AE is negative, the ex-ponent is positive and K becomes greater than1.0. When AE is positive, the exponent is nega-

tive and K is less than 1.0. The abscissa representsthe oxyanion strength, which increases from leftto right. A c value of -3.0 corresponds to an

oxyanion of the strength of trichloroacetic acid;a c value of -1.0 corresponds to an oxyanion ofthe strength of acetic acid. Therefore, for anycation, the strength with which it is held by an

-30

-25

-20

_, -

To

uE

-10

= -5'o - 5

- _

-10A

- -5

- 1.0

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BACTERIOL. REV.

oxyanion and the extent to which association isfavored increases from left to right. Thus, AEchanges from near zero values to relatively highnegative values in going from left to right. Theselectivity of an oxyanion of any given strengthfor the cations included in the figure decreasesfrom top to bottom. Thus, for an oxyanion witha c value of -3, the order of selectivity is NH4+,K+, Rb+, Cs+, Na+, Li+, H+; for an oxyanionwith a c value of -1.0, the order of selectivity isH+, Li+, Na+, K+, NH4+, Rb+, Cs+. The inver-sion of the order of selectivity with change in theoxyanion strength is the most significant resultof this analysis. There is a remarkable similaritybetween the inversions of selectivity shown byFig. 9 and those described by Armstrong andRothstein (Fig. 6) for the effect of pH on alkalimetal uptake by bakers' yeast.

Surface potentials. A central question in thecontext of this symposium is whether the fixed-charge system can account for the observed"membrane potentials" (and their reversal whenthis can be observed, such as in muscle and nerve).Ling derived equations to predict the electricalpotentials, and he reported that surface potentialswhich were predicted by the equations were con-sistent with experimental observation. His gen-eral equation:= 2.3RT/F constant (23)

- log (KK [K+lo + KNB [Nall,)

may take the form of the classical Nernst equa-tion, and predicts that the surface potential, 4t, isproportional to the absolute temperature and tothe log of both the K+ and Na+ ion concentra-tions. However, under appropriate conditions-that is, when changes of oxyanionic strength re-sult in alterations of the affinity constant for Kbinding (KK) or Na binding (KNa)-this equationreduces to a Nernst-type equation for a K+ or aNa+ diffusion potential (13-15).

CONCLUSIONSThe controversy between the membrane and

sorption theories will undoubtedly be resolved bysome "vigorous hybridization." The virtues ofLing's and Eisenman's concepts are self-evident.The concepts indicate what must happen at cellsurfaces or in any other region of the cell wherethe kind of lattice structures they describe occur.However, it is doubtful that this is the exclusivestructure throughout the protoplast. In fact,there is evidence against this concept (5). None-theless, Ling and Eisenman have provided whatwill probably prove to be an atomistic basis forcarrier selectivity upon which most transmem-branal transport depends. In this respect, the dis-

crimination of the alkali metal carrier in bakers'yeast, described by Armstrong and Rothstein,shows remarkable similarity to the alterations ofion selectivity described by Ling and Eisenman.(Reviewing the arguments over whether thereare or are not membrane carriers for transportacross the cell membrane is reminiscent of thestory of the theistic and atheistic goldfish swim-ming around in the fishbowl discussing whetherthere is or is not a God. When the discussionreached the inevitable impasse, the theist goldfishthrew up his fins and asked: "If there is no God,who changes the water?")

AcKNowLEDGMENIs

I express appreciation to the following authors andpublishers who have given permission to reproducefigures and tables from their published works: to S. G.Schultz, A. K. Solomon, W. Epstein, and N. L. Wil-son and The Rockefeller Institute Press for Table 1and Fig. 2 and 4; to W. McD. Armstrong and A.Rothstein and The Rockefeller Institute Press forFig. 5, 6, and 7; to A. Rothstein and Prentice Hall,Inc., for Table 3; and to G. N. Ling and The BlaisdellPublishing Co. for Fig. 8 and 9.

LITERATURE CITED

1. ARMSTRONG, W. MCD., AND A. ROTHSTEIN.1964. Discrimination between alkali metalcations by yeast. I. Effect of pH on uptake. J.Gen. Physiol. 48:61-71.

2. BLINKS, L. R. 1955. Some electrical properties oflarge cells, p. 187-212. In T. Shedlovsky [ed.],Electrochemistry in biology and medicine.John Wiley & Sons, Inc., New York.

3. CHRISTENSEN, H. N. 1962. Biological transport.W. A. Benjamin, Inc., New York.

4. CONWAY, E. J. 1957. Membrane equilibrium inskeletal muscle and the active transport ofsodium, p. 73-114. In Q. R. Murphy [ed.],Metabolic aspects of transport across cellmembranes. Univ. Wisconsin Press, Madison.

5. CONWAY, E. J. 1957. Discussion, p. 175-187. InQ. R. Murphy [ed.], Metabolic aspects oftransport across cell membranes. Univ. Wis-consin Press, Madison.

6. CONWAY, E. J., AND F. DUGAN. 1958. A cationcarrier in the yeast cell wall. Biochem. J. 69:265-274.

7. COwIE, D. B., AND R. B. ROBERTS. 1955. Per-meability of microorganisms to inorganic ions,amino acids and peptides, p. 1-34. In A. M.Shanes [ed.], Electrolytes in biological systems.American Physiological Society, Washington,D.C.

8. DRAPEAU, G., AND R. A. MACLEOD. 1962. Ionactivation of ATP-ase in membranes of marinebacterial cells; Bacterial. Proc., p. 122.

9. EISENMAN, G. 1961. On the elementary atomicorigin of equilibrium ionic specificity, p. 163-179. In A. Kleinzeller and A. Kotyk [ed.],

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Membrane transport and metabolism. Aca-demic Press, Inc., New York.

10. GLYNN, I. M. 1964. The action of cardiac glyco-sides on ion movements. Pharmacol. Rev. 16:381-407.

11. HITCHCOCK, D. I. 1945. Selected principles ofphysical chemistry, p. 59-74. In Physical chem-istry of cells and tissues. R. Hober, The Blakis-ton Co., Philadelphia.

12. HODGKIN, A. L., AND B. KATZ. 1949. The effectof sodium ions on the electrical activity of thegiant axon of the squid. J. Physiol. 108:37-77.

13. LING, G. N., 1960. The interpretation of selectiveionic permeability and cellular potentials interms of the fixed charge-induction hypothesis.J. Gen. Physiol. 43(suppl. 5) :149-174.

14. LING, G. N. 1962. A physical theory of the livingstate. Blaisdell Publishing Co., New York.

15. LING, G. N. 1965. Physiology and anatomy ofthe cell membrane: the physical state of waterin the living state. Federation Proc. 24(suppl.155) :103-112.

16. MAHAN, B. H. 1964. Elementary chemical thermo-dynamics. W. A. Benjamin, Inc., New York.

17. POST, R. L., C. R. MERRITT, C. R. KINsOLvING,AND C. D. ALBRIGHT. 1961. Membrane adeno-sine triphosphatase as a participant in the ac-tive transport of sodium and potassium in thehuman erythrocyte. J. Biol. Chem. 235:1796-1802.

18. RANDALL, J. E. 1962. Elements of biophysics.Year Book Medical Publishers, Inc., Chicago.

19. RIDGE, R. M. A. P., AND R. J. WALKER. 1963.Biolectric potentials, p. 92-206. In G. E.Kerkut [ed.], Problems in biology, vol. 1.MacMillan Co., New York.

20. ROTHsmN, A. 1959. Role of the cell membranein the metabolism of inorganic electrolytes bymicroorganisms. Bacteriol. Rev. 23:175-201.

21. ROTHSTEIN, A. 1961. Interrelationship betweenthe ion transporting systems of the yeast cell,p. 270-284. In A. Kleinzeller and A. Kotyk

[ed.], Membrane transport and metabolism.Academic Press, Inc., New York.

22. RoTHsTEIN, A. 1964. Membrane function andphysiological activity of microorganisms, p.23-39. In J. F. Hoffman [ed.], Cellular functionsof membrane transport. Prentice-Hall Inc.,Englewood Cliffs, N.J.

23. SCHULTZ, S. G., W. EPSTEIN, AND A. K. SoLo-MON. 1963. Cation transport in Escherichia coil.IV. Kinetics of net K uptake. J. Gen. Physiol.47:329-346.

24. SCHULTZ, S. G., AND A. K. SOLOMON. 1961.Cation transport in Escherichia coli. I. Intra-cellular Na and K concentrations and netcation movement. J. Gen. Physiol. 45:355-369.

25. SCHULTZ, S. G., N. L. WILSON, AND W. EPSTEIN.1962. Cation transport in Escherichia coli. II.Intracellular chloride concentration. J. Gen.Physiol. 46:159-166.

26. SEN, A. K., AND R. L. POsT. 1964. Stoichiometryand localization of adenosine triphosphate-dependent sodium and potassium transport inthe erythrocyte. J. Biol. Chem. 239:345-352.

27. SKOU, J. C. 1961. The relationship of a (Mg+' +Na+)-activated, K+-stimulated enzyme orenzyme system to the active linked transport ofNa+ and K+ across the cell membrane, p. 228-236. In A. Kleinzeller and A. Kotyk [ed.],Membrane transport and metabolism. Aca-demic Press, Inc., New York.

28. SOLLNER, K. 1954. Electrochemical studies withmodel membranes, p. 144-188. In H. T. Clarke[ed.], Ion transport across membranes. Aca-demic Press, Inc., New York.

29. UssING, H. H. 1954. Ion transport across bio-logical membranes, p. 3-22. In H. T. Clarke[ed.], Ion transport across membranes. Aca-demic Press, Inc., New York.

30. UssING, H. H. 1963. Transport of electrolytes andwater across epithelia. Harvey Lectures Ser. 59(1961-1962):1-30.

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