Proc. Nat. Acad. Sci. USA Vol. 69, No. 3, pp. 702-705, March 1972

On a Theory for the Passive Transport of Solute through Semipermeable Membranes (/cellophane membranes/peptides/activation energy) LYMAN C. CRAIG AND HAO-CHIA CHEN The Rockefeller University, New York, N.Y. 10021 Contributed by Lyman C. Craig, January 10, 1972

ABSTRACT It has been shown that thin-film dialysis the peptide or the membrane. This theory seemed reasonable can be performed in such a way that the limiting rate is the rate of entry of the solute into the membrane from the because of the high discrimination of the dialysis method, high-concentration side. The rate of diffusion, therefore, that appeared to be in the range of 2-3% of the Stokes reflects the probability of a molecule entering the pores on radius (3). the surface and does not depend on the resistance to Recently it has been discovered (5) that with peptides of diffusion offered by the internal structure of the mem- brane. The good correlation, so generally found, of the high charge density and with charges of the same sign, a order of escape times of given solutes with the order of free higher order of freedom from fixed charge on the membrane is diffusion rates is thus explained. The data from stretching required in order for dialysis rates to be independent of experiments with wet cellophane, in which the pore struc- charge. A method of covering any residual fixed charge has ture is distorted, are also explained. been developed; the charge is coupled with glycine amide with Soon after the method of thin-film dialysis was proposed as a the help of a water-soluble carbodiimide coupling agent. simple experimental way to study relative rates of diffusions Membranes so treated have reduced absorptive properties and, thereby, to permit estimations of Stokes radii, solute- and behave more ideally with highly charged solutes. It solvent behavior, and conformational change, preliminary seemed of interest to again investigate temperature, as well as studies (1) on the effect of temperature were also made. The salt, effects with these modified membranes. behavior of approximated that expected on the basis RESULTS AND DISCUSSION of the theory that their relative rates of diffusion through membranes would occur in the same order as their free diffu- The effect of change of temperature on free diffusion coeffi- sion coefficients, but that differences would be greatly en- cients in various solvents has been thoroughly investigated by hanced due to the resistance of the membrane. This finding different workers, and was reviewed by Longsworth (6). implied that the membrane would carry no fixed charge, and Ideal spherical solutes follow the Stokes-Einstein equation indeed evidence was supplied that Visking dialysis casing rather closely: available at the time (2) did have a low, and apparently D KT/6wrrqr, (1) negligible, order of fixed charge. With increasing temperature, up to 40 50°, it was somewhat surprising to find that the in- where D is the diffusion coefficient, K is a constant, T is the creased rate of dialysis of certain test solutes approximated absolute temperature, tq is the viscosity, and r is the particle that shown by free diffusion (the Stokes-Einstein relation- radius. The uncertainty of hydration always must be consid- ship), i.e. directly proportional to change of absolute temper- ered with polar molecules, particularly with amino acids. For ature, but inversely proportional to the viscosity change of instance, glycine amide (7) diffuses considerably faster than the solvent. Above 50°, proteins known to undergo heat de- the dipolar ion. This is observed also with the dialysis method: naturation showed a much slower rate of dialysis, as would be the simple amides of the amino acids always dialyze more expected from an expanded molecule. rapidly than the free amino acids. This may be due to a hy- A preliminary study with carbohydrates of known confor- dration effect. mation and rigidity (3) strongly supported the theory of in- For particles that are not spherical, formulas analogous to creased selectivity, as compared to free diffusion, when com- Stokes' law for prolate and oblate elipsoids of revolution were parisons were at a given temperature, but the temperature co- derived by Perrin (8). These have been used for derivation of efficients were often too high for particular carbohydrates. frictional coefficients for the larger molecules, such as pro- These higher temperature coefficients were thought to be due to teins. Again there is the uncertainty of hydration effects, and changes in the hydration of either the solute or the membrane. it is not even certain that these formulas apply to the smaller When polypeptides of both the linear and the rigid cyclic molecules. In fact, Longsworth's data (6) are in quite good antibiotic types were studied, individual temperature co- agreement with the simpler Stokes-Einstein equation. efficients again appeared to be unpredictable (4), but were re- Although it is obvious that the restricted type of diffusion producible for each peptide. It was thought that the linear that results from thin-film dialysis is more complicated than peptides could easily undergo conformational change, but the free diffusion, it is helpful perhaps to think of it along the lines too-high temperature coefficients of the cyclic peptides were set forth in free diffusion studies. Thus, from the Stokes- again thought perhaps to reflect changes of hydration either of Einstein equation, the reciprocal plot shown in Fig. 1 can be 702 Downloaded by guest on October 1, 2021 Proc. Nat. Acad. Sci. USA 69 (1972) Passive Transport of Solute 703

constructed. The slope indicates the relative diffusabilities at TABLE 1. Comparative half-escape times (T/2) with highly the different temperatures and permits calculation of the acetylated membranes activation energy of diffusion. In thin-film dialysis, a straight-line escape plot (9) indi- T/2 (min) T/2 at cates ideal behavior, i.e., the rate of dialysis for the particular 20 00: T/2 membrane and assembly is proportional only to the concen- Membrane Solute at 200C at 400C at 400C tration gradient across the membrane. The half-escape time A 3H20 15 8.7 1.75 ±0.22 is, therefore, a measure of the rate of dialysis of the solute for 14CH30H 32 18 1.83 i 0.19 that membrane and cell that can be used for comparison with CH314COOH 52 29 1.80 ± 0.10 other solutes in the same membrane and cell. It is thus of B 3H20 11 7 1.51±0.16 interest to substitute the reciprocal of the half-escape time for CH314COOH 23 14 1.67 ± 0.06 the diffusion coefficient in the reciprocal plot for the study of [14CIUrea 40 26 1.56 ± 0.04 temperature effects. The reciprocal is required since a rate CH314COO-K+ 16 10 1.60 ± 0.12 constant is proportional to the reciprocal of the half-escape [14C]Glycine 23 13.5 1.71 ± 0.24 time. Such a plot of the logarithm of the reciprocal of the half- escape time against the reciprocal of the absolute temperature (Fig. 1) then provides a curve whose slope conceptually could membranes (4), the ratios of the half-escape time at 250 to give an estimation of the activation energy required for di- that at 400 of the relatively large solutes ribonuclease, cyto- alysis through the membrane. Hydration effects, as well as chrome c, and ovalbumin were 1.44, 1.42, and 1.48, respec- changes in conformation, would be included in this estimate. tively, in 0.01 N acetic acid. The expected ratio (Fig. 1) is Interaction with the membrane by adsorption would also be 1.44. On the other hand, very highly acetylated membranes, included if there were appreciable interaction. Whether or not which allow only very slow diffusion of small molecules such there is adsorption, however, can be easily determined by as urea, also show surprising adherence to the Stokes-Ein- a simple recovery calculation. stein equation (10). In Table 1, concentrations were deter- In spite of these complications, it seemed of interest to per- mined by scintillation counting of the labeled solutes (Tri-Carb form the experiments this way. We were surprised to find that Scintillation Counter in Bray's scintillation medium). The numerous solutes with the completely neutral membranes temperatures were 200 and 400, which, from Fig. 1, should have practically straight-line reciprocal plots over the tem- give a ratio of half-escape times of 1.73 if adherence to the perature range 4-40', with a slope in close agreement with Stokes-Einstein equation were followed. The agreement is that calculated from the Stokes-Einstein equation for free surprisingly good. These highly acetylated membranes are diffusion. The agreement of this slope with those for the di- more hydrophobic than the unacetylated ones and are con- peptide L-Leu-I-Tyr and the cyclic antibiotic gramicidin-SA diserably less flexible. in 0.1 M NaCl is shown in'Fig. 1. The data for the dipeptide From the data given for thin-film dialysis, it would appear were obtained from a membrane treated with glycine amide to that adherence to the Stokes-Einstein equation for rigid remove charge, and then acetylated to reduce its porosity. The solutes does not depend on the porosity, flexibility, or hydro-- data for gramicidin-SA were obtained from a more-porous phobicity of the membrane, provided it is free of fixed charge membrane, not acetylated but treated only with glycine and does not adsorb the solute. It also does not depend on the amide. A similar agreement has been obtained for tri- and size of the solute molecule, an observation coinciding with that tetraglycine in water. Numerous other examples can be cited made by Longsworth (11), who found that the activation en- from our published work. Thus, with porous unacetylated ergy of free diffusion for HDO (partially deuterated water) and albumin differed by less than 10%, in spite of the fact 0 E 30 that HDO diffuses 36-times faster than albumin. It seems of theoretical and practical interest to ask why it is ci 20 0 that the overall activation energies of dialysis derived from (A temperature studies in a thin-film dialysis cell coincide so O 10 closely with the activation energy of free diffusion. This ques-

0 tion can be studied in the following way. Fig. 2a is a schematic

4-0 6 drawing based on the dimensions of a thin-film dialysis cell. At time 0, a thin-film, about 100,um in depth, of a solution of 0._ Ba, 4 known concentration is spread over a 50- cm2 surface of a cel- 3 lophane membrane wet with the solvent only, and about 0 40 urn thick. A concentration gradient of unknown shape or 2.9 (70) 3.1 (50") 3.3(300) 3.5(10) slope is quickly established across the membrane, as repre- Reciprocal of temperature (K) x10-3 sented by the diagonal line. The concentration on the diffusate side is kept low, less than 1/100 that on the side containing the FIG. 1. The reciprocal of the absolute temperature plotted against diffusion coefficient (D) for molecules that follow the retained material, by frequent changes of the diffusate solvent. Stokes-Einstein relationship, or against the reciprocal of the half- Accordingly there is negligible back-diffusion. Analysis of the escape times of various solutes. The curves of the latter are nor- diffusate solutions then permits escape plots, such as those malized to the diffusion data, except for ,81-24-ACTH. Free diffu- shown in Fig. 2b and c, to be constructed. A straight line indi- sion, Stokes-Einstein relationship, 0. Half-escape time: grami- cates ideal behavior. The half-escape time is a convenient in- cidin-SA, 0; iLeu-iTyr, 0; ,62-4-ACTH, X. dication of the rate of dialysis [k = 0.693/(T/2) ], which is a Downloaded by guest on October 1, 2021 704 Chemistry: Craig and Chen Proc. Nat. Acad. Sci. USA 69 (1972)

100 . 80 . a b 60 Ret 50 I) 40

O 30 D l- ._- ._c IEi 20 A=Ao(l-r/R)2 FIG. 4. Schematic representation of the theory of restricted 10 20 30 40 50 60 70 80 10 diffusion. A, effective cross-sectional area of pore; A, total cross- 8 Minutes sectional area of pore; r, radius of particle; R, radius of cross- C sectional area of the pore. 6 110 CRet 5- I 0cDif0 4 lowed as in a standard experiment, except that the solution within the dialysis tubing is omitted, causing the membrane to FIG. 2. (a) A schematic representation of thin-film dialysis. collapse against the center expander tube. The solute re- Membrane thickness = 40 Ajm; retentate film thickness = 100 maining in the membrane can now diffuse only into the out- ,um. See text for details. (b and c) Escape plots at 400 and 110, respectively, determined with #I -24-ACTH. side diffusate solution. 5% of the original charge was recov- ered from the membrane, but at a rate 4- to 5-fold faster (Fig. 3, curve b) than that of the overall dialysis rate. (Curve b combination of three rates: the rate of entry into the mem- does not extrapolate through the origin because of the time brane, the rate of diffusion through the membrane, and the necessarily lost in the washing.) rate of entry into the diffusate solution. In a third experiment, the bacitracin solution was inserted Since the actual overall rate of dialysis is quite slow per cm2 as the film, but the diffusate solution was omitted. After of dialyzing surface (0.5 mg of ,B1-24-ACTH passing through a period of time, the membrane should have been fully 50 cm2 of membrane area in 18 min at 400; Fig. 2b), the third charged, since there was no diffusate solution to remove it on rate would be so much faster than the other two that it can be the diffusate side. The bacitracin solution on the inside of the neglected. If the controlling rate should be the rate of diffusion tubing was then removed, both sides were quickly washed through the membrane,-it seems entirely improbable that the twice with the solvent, and the elution was started again in activation energy could parallel so closely the energy of free the absence of solvent within the tubing. In this way more diffusion. Nonetheless, a direct measure of the rate of diffusion solute, about 8% of the original charge, was recovered. The within the mertbrane is desirable. The rate of diffusion can be escape rate was slightly faster than that in the previous ex- estimated as follows. periment, as would be expected. By choice of a solute with a moderately slow rate of dialysis It would, therefore, appear that the controlling rate (such as bacitracin A, molecular weight 1422, which behaves in thin-film dialysis is the probability of the solute molecule ideally and does not adsorb to the membrane), a standard finding its way into a pore at the membrane boundary on the escape curve (Fig. 3, curve a) at 25° can be obtained. This side of the membrane containing the high concentration of the same membrane and dialysis cell can then be used for a solute. Since this rate is controlled by free diffusion, the second experiment with a higher concentration of bacitracin overall energy of dialysis should parallel rather closely that of A. After 2 min, at which time the gradient across the mem- free diffusion for ideal solutes whose conformation, state of brane (Fig. 3) is established, the solutions on both sides of the aggregation, or hydration does not change over the tempera- membrane are quickly removed. Both sides are quickly ture interval involved. washed twice with the solvent, and the elution is then fol- Some years ago, stretching experiments with wet cello- phane casing showed that if the molecular dimensions of a large solute molecule in solution were known, it would be 00o possible to estimate how much a calibrated membrane should to a desired rate. 70 be two-dimensionally stretched give escape \ b These experiments are entirely consistent with a two-dimen- 50 sional barrier, but not with a three-dimensional one. It is not 0 \ ** possible to stretch the membrane three-dimensionally, and if o0 the barrier were three-dimensional, a certain collapse of the 0 0. \ c 30 \ pores with two-dimensional stretching would be expected. This would lead to a reduced rate of diffusion, as was experi- 201 mentally found for one-dimensional stretching. Finally there is the extensive experience we now have had with many different types of solutes, where the order of the relative dialysis rates coincides with that to be expected from 6 18 30 42 free diffusion (3). Minutes Many polypeptides, which are not covalently crosslinked

FIG. 3. Escape rate. * - , Overall rate. O -O, Rate from but are of the so-called "random coil" type, give straight-line membrane alone. escape plots in a favorable solvent environment, but their be- Downloaded by guest on October 1, 2021 Proc. Nat. Acad. &ci. USA 69 (1972) Passive Transport of Solute 705

havior at different temperatures does not follow the Stokes- This study was supported in part by NIH Grant AM 02493 Einstemnrelationship (as shown by in Fig. 1). and Contract N IH-71-2250 from the Artificial Kidney-Chronic #1-24-ACTH Uremia Program of the National Institute of Arthritis and The diffusional size seems to decrease in the interval 10-40', Metabolic Diseases. We thank Helen Kac for the determinations but then to increase in the interval 40-55°. of the rates of dialysis. Solutes that associate, such as the tyrocidines (13), give too low a temperature coefficient in the temperature range 10-250, but much too large a coefficient above 250. The exact shape of the curve is concentration dependent. 1. Craig, L. C., Konigsberg, W., Stracher, A. & King, T. P. It is now well established that cellulose acetate membranes, (1957) in "Symposium on Structure," ed. Neuberger, so widely used in reverse for recovery of sea water, A., 1UPAC Symposium, July 1967 (John Wiley and Sons, Inc., N.Y.), pp. 104-115. owe their effectiveness to a very thin, dense layer of the 2. Craig, L. C. & Ansevin, A. (1963) 2, 1268-1271. polymer on the surface (14); they are called skin membranes. 3. Craig, L. C. & Pulley, A. 0. (1962) Biochemistry 1, 89-94. Perhaps such a barrier plays more of a role with other mem- 4. Craig, L. C. (1962) Arch. Biochem. Biophys., Suppl. 1, branes than has been suspected. i12-118. The results presented in this paper support the view ex- 5. Craig, L. C., Kac, H., Chen, H. C. & Printz, M. P. (1971) Proc. of 2nd International Symposium on Protein and Poly- pressed earlier (3) that the selective data from thin-film peptide Hormones, Sept. 27 (Excerpta Medica, Arristerdam, dialysis can be used for determination of Stokes radii, pro- Holland), Vol. 1, pp. 176-179. vided comparisons are properly made with suitable solute 6. Longsworth, L. G. (1955) in "Electrochemistry in Biology models of known size and shape. The reasoning is similar to and Medicine," ed. Shedlovsky, T. (John Wiley and Sons, of with N.Y.), pp. 2262-247. that used in the determination size Sephadex gels (15). 7. Dunlop, P. J. & Gosting, L. J. (1953) J. Amer. Chem. Soc. 75, The mechanism for the membranes is uncomplicated, and the 5073-5075. method can be more versatile and selective than with 8. Perrin, F. (1936) J. Phys. Radium 7, 1-11. Sephadex. 9. Craig, L. C. & King, T. P. (1956) J. Amer. Chem. Soc. 78, 4171-4172. The demonstration that thin-film dialysis can be performed 10. Stewart, K. K. & Craig, L. C. (1970) Anal. Chem. 42, 1257- in such a way that it clearly reflects the probability of a solute 1260. 11. Longsworth, L. G. (1960) J. Phys. Chem. 64, 1914-1917. molecule entering a pore on the surface of the membrane is 12. Craig, L. C. & Konigsberg, W. (1961) J. Phys. Chem.. 65, consistent with an explanation of potentially high discrim- 166-172. ination offered by a theory of restricted diffusion suggested 13. Burachik, M., Craig, L. C. & Chang, J. (1970) Biochemistry many years ago (16, 17) in studies. This is 9, 3293-3300. shown schematically in Fig. 4. As the size of the particle ap- 14. Loeb, S., Sourirajan, S. & Uuster, S. T. (1960) Chem. Eng. News 38, 64. proaches the cross-sectional area of the pore, the probability 15. Andrews, P. (1965) Biochem. J. 96, 595-606. of its entering the pore is controlled by the equation given in 16. Elford, W. J. (1937) Trans. Faraday Soc. 33, 1094-1106. Fig.4. 17. Ferry, J. D. (1936) J. Gen. Physiol. 20, 95-104. Downloaded by guest on October 1, 2021