94 DIFFERENTIAL RATES OF PERMEATION THE PHYSICAL CHEMISTRY OF THE DIFFERENTIAL RATES OF PERMEATION OF IONS ACROSS POROUS MEMBRANES BY R. NEIHOF” AND K. SOLLNER Laboratory of Physical Biology National Institute of Arthritis and Metabolic Diseases National Institutes of Health Public Health Service U.S. Department of Health, Education, and Welfare Bethesda 14, Maryland, U.S.A. Received 12th January, 1956 A possible explanation of the differential uptake by living cells of various species of ions of the same charge by an essentially aqueous process is developed on the basis of the con- siderations which previously have led to the quantitative treatment of “ poIyionic” potentials across membranes of extreme ionic selectivity. Relative rates of exchange across such membranes of any two species of “ critical ” ions coexisting in solution 1 for any third species of ions of the same charge in solution 2 can be calculated from their relative activities in solution 1 and the “ bi-ionic ” potential arising with the same two species of ions across the same membrane.The predictions of the theory are confirmed with a variety of combinations of critical ions in systems with cation selective as well as in systems with anion selective permselective membranes of extreme ionic selectivity and low resistance. With these membranes the ratios of the fluxes of, for instance, Kf and Li+ (being at the same concentration in solution 1) are of the order of 6 : 1 ; of I- and C1-, 2.5 : 1 ; and of SCN- and Ac-, 33 : 1 . The mechanism of the differential uptake by living cells of various species of ions of the same charge from the surrounding milieu is one of the major open problems of physical biochemistry.In this paper we try to show that recent * present address : Physiological Institute, University of Uppsala, Uppsala, Sweden.R . NEIHOF AND K . SOLLNER 95 studies on the electromotive action of membranes of extreme ionic selectivity 1 * 2 lead to the prediction of the existence and the magnitude of an heretofore over- looked, easily demonstrable, physicochemical effect arising with porous membranes, which, if it should occur across the membranes of living cells and tissues, could readily account for the selective uptake by the latter of different ions of the same charge. Andr6 and Demoussy,3 according to Brooks,4 first realized that the preferential uptake of certain ions (for instance of K+ over Na+, or of I- over Cl-) by meta- bolizing cells must be connected with a state of non-equilibrium, now commonly believed to be a steady drift toward a Gibbs-Donnan membrane equilibrium.In this state of non-equilibrium, maintained by the continuous production of some electrolytic metabolites such as carbonic acid, the faster-permeating ions are present in the cell at a relatively higher concentration than the more slowly penetrating ones. Brooks has stressed the fact that numerous mechanisms could explain differential rates of permeation of ions of the same charge.4 The classical papers of Osterhout and his school consider the formation of some undissociated or weakly dissociated compounds in a protoplasmic membrane of an essentially nonaqueous character.Brooks proposed an essentially aqueous mechanism involving highly cation- or highly anion-selective membranes of porous character.4 On the basis of the infor- mation then available he suggested that digerences in the diffusion velocities of the various ions in water might be the determining factor. These differences, however, are much too small to account for the observed erTects. The lack of a demonstration in vifro of greatly differing rates of permeation of the various common univalent cations and anions across porous membranes of high ionic selectivity has remained one of the most serious difficulties of the various theories which assume essentially aqueous processes for the uptake of ions by living cells.The problem is one particular aspect of the dynamics of the exchange of ions across porous membranes in aqueous systems with three or more species of ex- changing ions of the same charge. The simplest systems of this type are those with membranes of extreme ionic selectivity, which (for exclusively cation-perme- able anion-impermeable membranes [ +@+ I ) are represented by system I : solution 1 1 membrane I solution 2 (system I) where a1 and a, are activities, and A+, B+, and L+- the exchangeable, " critical " cations ; X- are the nonexchanging anions. Anionic systems, with exclusively anion permeable membranes, I +-a+ I, are analogous. The systems sought would be characterized by a ratio in the rates of simul- taneous exchange of A+ for L+ and of B+ for L+, which is large compared with the ratio that might result from the differences in the diffusion velocities in water (which are proportional to the ionic mobilities) of the A+ and Bt ions.With the common ions these latter differences are relatively small, except for the H30f and OH- ions. For instance, the ratios of the diffusion velocities of K+, Na+ and Lii- are 1-00 : 0-68 : 0.53 ; of I-, C1-, SCN-, IO3-, and Ac- 1.01 : 1.00 : 0.86 : 0.54 : 0.54. There are no data in the literature concerning the kinetics of the exchange of ions between solutions across membranes in systems such as system I. However, the closely related problem of the origin of the electrical potentials which arise in systems of type I, and in numerous other more complex types of " polyionic " systems with membranes of high ionic selectivity, has recently been treated theoretically and experimentally in some detail.These potentials were interpreted in terms of relative intrinsic permeabilities of the different ions when present simultaneously in the same system. The concepts developed in this connection 19296 DIFFERENTIAL RATES OF PERMEATION are applicable to the problem on hand and lead, in combination with the empirical information on the electromotive behaviour of polyionic systems,S~ 6 to the pre- diction of the effect looked for and to the selection of appropriate experimental systems, as will be shown presently. The simplest polyionic systems are the " bi-ionic " systems in which a membrane of either extreme anionic or extreme cationic selectivity separates the solutions of two electrolytes at the same activities, al, having different " critical " ions, which are able to exchange across the membrane, and the same " noncritical " ion species for which the membrane is impermeable.l.2 Such bi-ionic membrane systems may be represented as shown here for a cationic system, system 11, (system 11) in which the univalent critical ions A+ and B+ exchange in equivalent quantities.The " bi-ionic " potentials arising in such systems, according to the nature of the membrane and the combinations of critical ions, may be 150 mV and more. The bi-ionic potential (b.i.p.) results from the different tendencies of the two species of critical ions to penetrate across the membrane and from the restriction that they can exchange only on a one to one basis.The more readily permeable species of critical ions impresses its charge on the other solution. The absolute magnitude of the b.i.p., Ebip, is a function of the relative contributions of the two species of critical ions toward the (virtual) transportation of electricity across the membrane, according to the equation where by definition T i + + r;+ = 1. TL+ + T;+ represent the transference numbers of critical ions A+ and B+ across the membrane. The sign of the potential refers to the charge of solution 2. The relative abundances of the two species of critical ions in the pores of a membrane multiplied by their relative diffusion velocities determine the ratio T ~ + / T % + . These ratios vary considerably according to the nature of the membranes.Typical figures (as given below for low resistance permselective membranes) are : rL1-/rYo3- = 7.1 ; r ~ l - / T ~ c - = 7.3 ; riCN-/TiC- = 39.0, etc. All such ratios are far in excess of, and in some instances in an order inverse to the ratios of the diffusion velocities of the various ions in water. With other membranes and combinations of ions the ratios may be much higher, of the order of 100 : 1 and more. The ratios T O in bi-ionic systems can be interpreted on the basis of the fixed charge theory of membrane behaviour, according to which the membranes are ion exchangers.1 If an ion exchanger is equilibrated with a solution containing two species of exchangeable ions, these ions compete as counter ions for positions of the fixed &sociable groups.In general, they are taken up to a different extent. The sequences of the relative adsorbabilities of the various ions are the two Hofmeister series, unless steric hindrance-the preferential screening out of some species of exchanging ions because of their size-comes into play. In explaining the mechanism of the origin of the bi-ionic potential two assumptions were made : (i) that the two species of critical ions are present within the membrane in the same ratio as if the membrane were equilibrated with a solu- tion prepared by mixing equal volumes of the two solutions of the bi-ionic system ; and (ii), that the ionic mobility of any species of ions within the membrane is independent of the presence or absence of other ions.In extending this approach to " polyionic " potentials across membranes of ideal ionic selectivity, in systems with more than two species of " critical " ions, rR+/rR,+ = 2.5 ; 7k+/Tli+ = 6.3 ; Tk,+/Te+ 2.6 ; 7' I- /TO c1- = 2.9 ; T&..N-/r&- = 6.2 ;R. NEIHOF AND K . SOLLNER 97 two additional assumptions were made ; (iii) the ratio of the adsorbed quantities of two species of ions is not changed by the presence of other competing species of ions; and (iv) the ratio of the adsorbed quantities of two competing species of ions is linearly proportional to their relative activities in the solutions.2 On the basis of these four assumptions it was possible to correlate quantitatively the poly- ionic potentials which arise in various polyionic systems with the same membrane and the same species of critical ions, The usefulness of this approach was confirmed by extensive experimental tests.6 Applying these assumptions to the exchange kinetics of polyionic membrane systems-the simplest of which is system I above-one concludes that the ratio of the rates of exchange across the membranes of two species of critical ions, A+ and B+ from solution 1 into solution 2 must be determined by the ratio of their relative abundances within the membrane times their respective mobilities.Thus, according to assumptions (i) to (iii), the ratio of the rates of exchange for A+ for L+ ions and of B f for L+ ions, $A+/~B+ in system I should be the same as the ratio T ~ + / T ; + in the corresponding bi-ionic system 11, independent of the nature or concentration of the L+ ions : Similarly, on the basis of assumptions (i) to (iv), the ratio of the rates of simul- taneous exchange of A+ and Bf across the same membrane can be predicted when these two ionic species are present in solution 1 at the different activities, a:; and a$>.We may write the general expression independent of the nature and activity of the L+ ions. Thus, our basic assumptions lead directly to the prediction of ratios of the rates of exchange, that is of the fluxes of two coexisting species of critical ions across porous membranes which are far in excess, in some instances in an inverse direction, of any flux ratios explainable by differences in the diffusion \,elocities of these ions in water. These considerations may be extended to any number of critical ions, A+, B+, Cf, etc., in solution 1 which exchange against one or several different species of ions L+, Mt, N+, etc., in solution 2.The ratios of +A+/$B+, #B+/$c+, #c+/#;, etc., for any pair of critical ions in solution 1 are defined by equations corres- ponding to eqn. (4). The experimental test of eqn. (4) is obvious: the bi-ionic potentials across a given membrane with several pairs of ions are determined and the corres- ponding T~+/T;+ , T ~ + / T E + , etc., ratios are computed. The initial rates of exchange of two or more ionic species, A+, B+, C+, etc., of known activities in solution 1 of a polyionic system (with the same membrane), against L+, M+, etc., ions of solution 2 are determined from the initial slopes, that is the slopes at zero time, of the curves in which the quantities of exchanged A+, B+, etc., ions are plotted against time.The ratios of each two of these initial flux rates ( ~ A + / $ B + ) ~ ~ ~ ~ . , ( $ B + / $ c + ) ~ ~ ~ ~ . , etc., are computed and compared with the ratios of the rates calculated from eqn. (4), ($~+/$~+)caic , ($~+/$~+)caic , etc. In the experiments reported below no attempt was made to select the experi- mental systems and conditions so that an optimum agreement between calculated and experimental 4 ratios would result. In all instances it must be considered that the idealized assumptions on which the calculated $ ratios are based are not neces- sarily strictly fulfilled, particularly over wide ranges of concentrations and con- centration ratios,51 6 and that a small error in the bi-ionic potentials from which a ro ratio is computed causes a relatively large error in this ratio since it appears in eqn.(1) as a logarithmic term. D98 DIFFERENTIAL RATES OF PERMEATION EXPERIMENTAL The experimental conditions for the study of dynamic membrane systems in general have recently been discussed and are not restated here.5 The membranes used were " perm- selective " collodion matrix membranes of almost ideal ionic selectivity.7. 8, 9 The permselective membranes are ion exchangers ; all ionic processes which occur across them are generally assumed to take place in the aqueous medium which fills their pores. Their absolute permeability (as measured by their electrolytic conductance) may be varied at will over a very wide range ; their water permeability is very low and can be disregarded in experiments of short duration. They were prepared according to methods previously described?, g Y 9 and tested by our standard methods (a) for their elec- trical resistance p * in 0.1 N KCI as an indication of the rates at which ions diffuse across them, and (b) for their electromotive properties in concentration cells 0.4 N KCl I membrane I 0-2 N KCl as a measure of their ionic selectivity.89 9 These potentials, corrected for the asymmetry of the liquid-junction potentials, were in the range of f 14.9 to rt 15.5 mV, the theoretical maximum potentials being & 15.95 mV, the plus sign referring to selec- tively cation-permeable, the minus sign to selectively anion-permeable membranes.The bi-ionic potentials, &p, with the various pairs of critical ions were measured at 25.0" by the Poggendorf compensation method, with an accuracy of f 2%, using saturated calomel half-cells with saturated KC1-agar bridges as reference electrodes.5 No correction was made for the asymmetry of the two liquid-junction potentials. Using eqn. (l), the corresponding ratios of the transference numbers T : + / T ~ + , 7;+/7e+, etc., and T ~ - / T & , %-IT%- , etc., were evaluated and used to compute the calculated flux ratios.* For the measurement of the rates of exchange a bag-shaped membrane was filled with 30 ml of solution 2 and immersed in 1 1. of solution 1 at 25.0" C. Both solutions were stirred at such rates that further increases in stirring did not increase the rates of the ex- change of ions.The effective membrane area in contact with the solutions was about 50 cm2 ; it was constant during a given experiment. To establish a steady-state condition across the membrane the inside solution was renewed repeatedly during the 1-2 h before starting the experiment proper at zero time when both solutions were renewed. At measured intervals small aliquots of solution 2 were removed and analyzed for the,ions A+, B+, C+, etc., or X-, Y-, Z-, etc., entering from solution 1. After the concentration of the critical ions initially present in solution 2 had decreased not more than 10 %, the experiment was interrupted, and a second run with fresh solutions started. (In a few systems rather large quantities of the " inside " solution were required for analysis ; in these instances several runs of different durations were made and the entire inside volume used for analysis at the end of each run.) These procedures were repeated until two succes- sive runs gave the same average rate of increase within k 5 % or less.All analyticaI determinations were carried out by standard (if necessary, suitably adapted) micro-methods,lo due regard being taken of the presence of other, potentially interfering electrolytes. The error in any individual analysis was never more than rt 5 %, in most cases less than rt 3 %. The rate of increase in concentration of a particular ionic species in the inside solution was determined by plotting concentration against time after correcting the measured concen- trations (after the first) for the withdrawal of previous samp1es.t The curves drawn through the plotted points were straight lines within experimental error up to at least 5 % depletion of the concentration of the ion in question in solution 2.Their slopes at zero time are the initial flux rates which were used in computing the experimental flux ratios ('#'A+/+B+)expt. Y (+B+/k+)expt. , . . . ($X-/+Y-)expt. . . . etc. * In view of the limitations of the assumptions on which eqn. (4) is based, and because of the similarity of the critical ions used, the calculated $ ratios were computed on the basis of concentrations without regard to the minor differences of the activity coefficients applic- able to the different ions in the various mixed solutions 1.t The formula used for computing the concentration Cn' corrected for withdrawal of previous samples was tn Cn cn' = t l + (v1/Y2) ( t 2 - t l ) + (YI/J'~) ( t 3 - 12) + - - (J'dVn) (tn - tn-1) ' where c, is the measured concentration at time tn when the total volume inside the membrane is V,. Vl, V2, V3, etc. and tl, t 2 , t 3 , etc. are the volumes and times, respectively, when the h t , second, third, etc. samples were withdrawn.R . NEIHOF AND K . SOLLNER 99 TABLE 1.-A COMPARISON OF THE CALCULATED AND EXPERIMENTAL RATIOS OF THE FLUXES OF TWO SPECIES OF COEXISTING CRITICAL IONS FROM SOLUTION 1 INTO SOLUTION 2 ACROSS PERMSELECTIVE MEMBRANES. ( t = 25.0" C) A.-SYSTEMS WITH CATION SELECTIVE MEMBRANES solution 1 (crit. ions A+ and B+) 0-1 N KCI 0.1 N LiCl 0.05 N KC1 0-05 N LiCl 0.3 N KCl 0-3 N LiCl 0.02 N KC1 0.2 N LiCl 0.2 N KCl 0-05 N LiCl 0.1 N KCI 0.1 N NaCl 0.1 N HCI 0.1 N NaCl solution 2 0.2 N NQCI 0.3 N NH4Cl 0.05 N NH4CI 0.2 N NHiCl 0.2 N NH4C1 0.2 N NH4Cl 0.2 N NH4C1 sulphonated polystyrene- oxidized collodion collodion membrane membrane 6.3a 6-3 6.8 7-lc 7.1 9-2 6.3a 6-3 6.2 7.lc 7.1 8.0 6 .3 ~ 25 28 7*lc 28 30 22b 22 24 a membrane with p* of 220 R cm2, b membrane with p* of 185 Q cm2, c membrane with p* of 295 L? cm2, dmembrane with p* of 325 L? cm2. K-SYSTEMS WITH ANION SELECTIVE MEMBRANES solution 1 (crit. ions solution 2 X- and Y-) KSCN 0.2N KN03 0.1 N KC1 0.05 N KCI KSCN 0.2N KN03 0.05 N KSCN 0.2 N KNo3 0-2 N KC1 0.05 N KSCN 0.3 N KN03 0.05 N KCI 0.1 N KSCN 0.2 KAc 0.1 N KCl 0.1 N KAc KBr 0.2 N KN03 KSCN 0.2 N KN03 0.1 N KAc KC' 0.2 N KN0-j 0.1 N KIO3 O.I KT 0 .2 ~ K N O ~ 0.1 N KCl 0.1 N KSCN 0.1 N KNO3 0.1 N KCI 0.1 N KAc poly-2-vinyl-N-methyl pyridinium protamine collodion collodion membrane membrane 8.2e 2.05 1.69 2.8g 0.70 0.70 8.2e 8.2 5.3 2.8g 2.8 2 7 12.9f 12.9 12-9 6.0h 6.0 6.8 39f 39 33 13.7h 13.7 15.3 7.1 7.1 8.6 4.3h 4.3 5.3 2.9f 2.9 2.4 1.62h 1.62 1.56 6.0f 6.0 4-5 3-1h 3.1 2.7 e membrane with p* of 170 R cm2, f membrane with p* of 11 5 R cm2 8 membrane with p* of 140 9 cm2, h membrane with p* of 150 52 cm2.100 DIFFERENTIAL RATES OF PERMEATION The probable error in these experimental flux ratios may be 3 % in the most favourable instances, and may reach 10 % under unfavourable conditions. ratios, an agreement of calculated and experimental q!~ ratios within 5 % is likely to be fortuitous or due to unusually favour- able conditions. In general, deviations of 10 to 15 % might be expected.Considering the uncertainties in the calculated RESULTS Tables 1 and 2 are self-explanatory. In the experiments of table 2 all species of critical ions in the solutions 1 were at the same concentration, so that the calculated 4 ratios are identical with the corresponding 7' ratios ; the latter are therefore omitted from the table. TABLE 2.-A COMPARISON OF THE CALCULATED AND EXPERIMENTAL RATIOS OF THE FLUXES OF THREE SPECIES OF COEXISTING CRITICAL IONS FROM SOLUTION 1 INTO SOLUTION 2 ACROSS solution 1 0.1 N KCl 0.1 N NaCl 0.1 N LiCl 0.1 N KCl 0.1 N NaCl 0.1 N LiCl solution 1 005 N KSCN 0.05 N KC1 0-05 N KAc 0.05 N KSCN 0.05 N KCl 0.05 N KAc solution 1 0-05 N KSCN 0.05 N KI 0.05 N KC1 0.05 N KSCN 0.05 N KI 0.05 N KCI PERMSELECTIVE MEMBRANES.(f = 25.0" C ) A.-SYSTEMS WITH CATION SELECTIVE MEMBRANES jk+- 4K+ +Na+ 4G membrane solution 2 calc. expt. calc. expt. sulphonated polystyrene 0.3 N NH4C1 2.5 2.8 6-4 7.1 collodion (p* = 185 52 cm2) oxidized (p* = 325 Qcm2) collodion 0.3 NNH4CI 3.0 3.2 7.1 9.5 B. SYSTEMS WITH ANION SELECTIVE MEMBRANES membrane solution 2 poly-2-vinyl-N- methyl pyridin- 0.15 N KN03 ium-collodion (p* = 115 L? cm2) protamine collodion 0-15 N KNO3 (p* = 150 s? cm2) membrane solution 2 poly-2-vinyl-N- methyl pyridin- 0.15 N KN03 ium-collodion (p* = 115 D cm2) protamine collodion 0-15 N KNO3 (p* = 150 12 cm2) 4SCN- 4SCN- dC1- $Ac- calc. expt. calc. expt. 6.0 4.8 39.0 42.0 3.1 2.7 13.5 14.9 4SCN- h C N - h- $CI- calc.expt. calc. expt. 1.98 1.80 6.0 4.6 1.86 1.78 3.1 2.7 DISCUSSION +Na+ G calc. expt. 2.6 2 4 2.4 3.0 d c l - calc. expt. 7.3 8.9 4.8 5.5 41- G- calc. expt. 2-9 2.6 1.63 1.52 The calculated and experimental 4 ratios in tables 1 and 2 demonstrate clearly the existence of the postulated effect and also show that its magnitude can be pre- dicted quantitatively on the basis of independent electrometric measurements in appropriate bi-ionic systems.R . NEIHOF AND K. SOLLNER 101 The agreement between the calculated and the experimental # ratios is in most instances within or near the estimated limits of the probable accuracy of the data. A discussion of the several larger deviations, of little importance with respect to the main purpose of this paper, may be left to a more comprehensive investigation of the general problem of ionic fluxes across permselective membranes.To what extent the effect demonstrated here plays a role in living cells and tissues is a problem outside the scope of this paper. One point, however, should be stressed here. The forces which regulate the chemical specificity of the various ions in exchange adsorption must be assumed to be of the same nature as those which determine ionic distribution equilibria between liquid phases. In micro- heterogeneous systems, in addition, steric hindrance may come into play. Thus it might be concluded that at least as great and varied degrees of ionic specificity can arise with micro-heterogeneous membranes as with homogeneous phase, " oil " membranes. The highly organized membranes of living cells and tissues might easily be much more effective in this respect than our synthetic membranes. 1 Sollner, J. Physic. Chem., 1949, 53, 1211, 1226. 2 Dray and Sollner, Biochim. Biophys. Acta, 1956, 21, 126. 3 Andr6 and Demoussy, Bull. SOC. Chim. biol., 1925, 8, 806. 4 Brooks, Protoplasma, 1929, 8, 389. 5 Dray and Sollner, Biochim. Biophys. Acta, 1955, 18, 341. 6 Dray and Sollner, Biochim. Biophys. Acta, 1956 (in press). 7 Sollner, J. Electrochem. SOC., 1950, 97, 139c ; Ann. N. Y. Acad. Sci., 1953, 57, 177 ; Electrochemistry in Biology and Medicine, ed. Shedlovsky (John Wiley and Sons, Inc., New York, 1955), Chap. 4, p. 33. 8 Neihof, J. Physic. Chem., 1954, 58, 916. 9 Gottlieb, Neihof and Sollner, J. Physic. Chem. (in press). 10 Kolthoff and Furman, Poterztiomefric Titrutions (John Wiley and Sons, Inc., New York), 2nd ed., 1931 ; Kolthoff and Stenger, Volumetric Analysis, vol. I1 (Inter- science Publishers, Inc., New York), 2nd ed., 1947 ; Snell and Snell, Colorimetric Methods of Analysis, vol. I1 (D. Van Nostrand Co., Toronto, New York, and London), 3rd ed., 1949.