Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979 233
of these machines, and further work is desirable to refine the understanding. Nomenclature a = fractional chamber height D = diameter, in. H = chamber height, in. H' = dimensionless chamber height N = revolutions per unit time (or revolutions per second) p = pressure, lbf/in.2 r = radial distance, in. R = a point on the second screw land t = dimensionless time T = dimensionless time required for a fluid particle to traverse a streamline in the chamber V = local fluid velocity, in./s or dimensionless x , y, z = Cartesian coordinate distances across the channel width, up the channel depth, and down the channel, respectively Greek Letters y = shear strain, dimensionless i. = shear rate (or rate of strain), s-l A = difference operator 0 = helix angle, rad p = fluid viscosity, CP T = actual residence time, s t = dimensionless residence time Subscripts b = refers to barrel c = refers to a complete cycle in the chamber
e = refers to end zones f = refers to the final or end position m = refers to the middle of the chamber ml, m2 = refers to two complementary planes in middle of chamber 0 = refers to the initial position r = refers to the screw root R = refers to a point on the second screw land x = refers to the cross-channel direction y = refers to the channel depth direction z = refers to the down-channel direction
Literature Cited Bigg, D., Middleman, S., Ind. Eng. Chem. Fundam., 13,66 (1974). Bird, R. B., Stewart, W. E., Lightfoot, E. N., "Transport Phenomena", Chapter 3, Wiley, New York, N.Y., 1960. Janssen, L. P. B. M., Ph.D. Dissertation, Delft University, Delft, The Netherlands, 1976. Kim, W. S., Skatschkow, W. W., Jewmenow, S. D., Pkste Kautsch., 20,696 (1973). Lidor, G.,Tadmor, Z.,Polym. Eng. Sci., 16,450 (1976). Maheshri, J. C., Ph.D. Dissertation, University of New Hampshire, Durham, N.H., 1977. McKelvey, J. M., "Polymer Processing", Chapter 12, Wiley, New York, N.Y., 1962. Middleman, S., "Fundamentals of Polymer Processing", Chapter 12, McGraw-Hill, New York, N.Y., 1977. Prause, J. J., Plast. Techno/., 13, 41 (1967). Tadmw, Z.,Klein, I., "Engineering Principles of Plasticating Extrusion", Chapter 7, Van Nostrand-Reinhold, New York, N.Y., 1970. Todd, D. B., Polym. Eng. Sci., 15, 437 (1975). Tung, T. T., Laurence, R. L., Polym. Eng. Sci., 15,401 (1975). Wyman, C. E.. Polym. Eng. Sci., 15, 606 (1975).
Received for review March 20, 1978 Accepted April 25, 1979
Diffusion and Interaction of Sodium Caprylate in Bovine Serum Albumin Solutions Christie J. Geankoplls," Eric A. Grulke,' and Martin R. Okos2 Department of Chemical Engineering, The Ohio State University, Columbus, Ohio 432 10
The diffusivity of sodium caprylate in aqueous bovine serum albumin solutions was measured using a Teflon diffusion cell. The diffusivity was found to be independent of concentration in the range of 0.5 to 1.5 g/100 mL when no protein was present. However, in the presence of 3 g of albumin/100 mL t h e diffusivity decreased by 23% at 0.5 g/100 mL and by 13% at 1.5 g/lOO mL concentration. These reductions are due to binding of the sodium caprylate to the protein and blockage of diffusion by the protein molecules. At the lower sodium caprylate concentrations, a greater fraction is bound. Increasing the protein concentration from 0 to 4 g/100 mL substantially decreased the diffusivity of sodium caprylate because of increased blockage and increased number of binding sites available. These data appear to be the first for a system where substantial binding occurs and both solute and protein concentrations were varied over a range. Experimental results were predicted using existing and modified models within an average deviation of f5 YO.
Introduction The interaction of biological compounds is often an important parameter in physiological fluids such as milk and blood. Serum albumin has the ability to bind a wide variety of organic and inorganic ligands. Because of this property, albumin is believed to function as a transport protein and possibly serve in regulating the blood level of Chemical Engineering Department,Michigan State University, East Lansing, Mich. 'Agricultural Engineering Department, Purdue University, W. Lafayette, Ind. 0019-7874/79/1018-0233$01.00/0
certain metabolites (King and Spencer, 1970). Albumin binds almost all of the free fatty acids that are released into the blood stream from the adipose tissue allowing the long-chain fatty acids to be several hundred times more soluble in albumin solutions than in salt solutions (Spector et al., 1969). Since the diffusivity of albumin is an order of magnitude lower than fatty acids the effective diffusivity of the acid is substantially decreased. The phenomenon may be of importance in the biological uptake of many other small molecules. Since the possible function of albumin is in transport, the measurement and prediction of diffusivity can be of great importance in the interaction and diffusion of al0 1979 American Chemical Society
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Ind. Eng. Chem. Fundam., Vol. 18, No.
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bumin and fatty acids in biological solutions. Theories of the diffusivity and interaction of biological substances cannot be based solely on theories formulated for simple binary systems. Binding of the small solute to the protein often occurs and diffusion is partially blocked by the protein macromolecules. The purpose of this work was to determine experimentally the diffusivity of sodium caprylate (which is the salt of the common fatty acid often present in biological solutions) in bovine serum albumin solutions and to compare these results with theoretical predictions. Literature Review The systematic study of binding of fatty acids (ligands) to BSA has been going on for many years. Boyer et al. (1947) were among the first to measure quantitatively the binding of the anions of the sodium salts of fatty acids, C4through Clo. They found that binding increased with hydrocarbon chain length, that temperature (2-47 “C) had no effect on caprylate binding, and that the amount bound decreased with increase in pH. However, in the pH range of 7 to 7.8, binding decreased by less than 2%. Others (Teresi and Luck, 1952; Spector et al., 1969; Reynolds et al., 1968) also studied the binding of various fatty acid anions. The theories and correlations for diffusion in binary systems (Jost, 1960; Geankoplis, 1972) seldom also apply directly to diffusion in polymer solutions and protein solutions. Large macromolecules interfere with diffusion of small solute molecules in polymer and protein solutions. This interference with diffusion is by the so-called “blocking effect”, solvent-polymer interactions, and solutepolymer interactions. Theories attempting to describe diffusion in dilute polymer or protein solutions are usually based on the hydrodynamic, kinetic, or thermodynamic models for diffusion in binary systems. Navari et al. (1971) have proposed an expression based on Eyring’s absolute rate theory to predict the diffusivity of solutes in polymer and protein solutions. However, this theory is applicable only to noninteracting solutes and does not consider binding. In a quantitative treatment of the self-diffusion of water in protein solutions, Wang (1954) took into account the obstruction effect of proteins and the reversible “hydration” or “binding” of the water molecules to the protein molecule which must diffuse along longer paths in order to get around the protein. Wang neglected the diffusion of bound water and assumed that the exchange between “bound” and “free” water is instantaneously fast. It was found that 0.2 g of water was bound as an outside sheath of water of hydration to 1.0 g of protein. Binding of other solutes was not considered. Colton et al. (1970) developed a model for diffusion of solutes in plasma solution based on existing binding and diffusion theories. The model assumed that the rate of reversible binding is instantaneous, that the bound and free solutes are a t equilibrium, that the solute binding isotherm is linear for the solute concentration range, and that the Wang blockage term (1- a+,) is valid. The Colton diffusion equation is DAP= [DAB(1- 4,)/(1
-
4p) + Dpkp/(l - 4p)l/[1 + k p / ( l - 4p)l (1)
where D A B is the diffusivity of unbound solute A in protein-free solution, DAp is the diffusivity of A in the protein solution, and D, is the diffusivity of the protein-solute complex (assumed to be that of the protein). Geankoplis et al. (1978) recently reported a diffusionbinding equation similar to the Colton equation with a
t
50rnrn
a
Figure 1. Teflon diaphragm cell for diffusion studies.
modified blockage factor to account more accurately for nonbinding solutes. DAP = [DAB(1 - 1.2&~,)
+ Dpkp/(l - 4p)l/[1 + k p / ( l - 4p)l (2)
This is equivalent to approximately 1.7 layers of water molecules bound to one protein molecule compared to the 1.0 layer found by Wang and close to 2 layers found by Stroeve (1975). Using existing binding data this equation predicted the diffusivity of urea and uric acid which bind in protein solutions within about *5% of experimental data and the diffusivity of oxygen and sucrose in protein solutions where only blockage effects occur within about fl%.
Experimental Methods To measure the diffusion of solutes, a diaphragm cell as described by Geankoplis et al. (1978) was used as shown in Figure 1. Two types of membranes were used for the diaphragm: MF-Millipore RA and Polyvic BS. The MF-Millipore membranes had a porosity of 82%, a mean pore size of 1.2 pm, and were made of mixed esters of cellulose. The Polyvic membranes had a porosity of 79%, a mean pore size of 2 pm, and were made of polyvinyl chloride. All Millipore membranes averaged 150 f 10 pm in thickness. In order to establish a linear concentration gradient, an unsteady-state diffusion period was allowed for each run. The approximate time required to establish this gradient was calculated using Gordon’s rule (Gordon, 1937). After the gradient was established, the upper solvent was replaced with fresh solvent and the run was started. The total run time for each compound was approximately 2 h for potassium chloride which was used to calibrate the cell and 31/2h for sodium caprylate. All runs were a t p H 7 using a Tris buffer and 25 “C. All mass balances for the sodium caprylate diffusion runs were generally within 1 to 2%. Diffusion data for solutes acetic acid, urea, and sodium caprylate obtained in the cell compared closely to
Ind. Eng. Chem. Fundam., Vol. 18, No.
3, 1979
235
Table I. Calibration of Cell with Diffusion of Potassium Chloride in Water Usine Different Membranes ( 2 5 C) concn of KC1, g-mol/L CLO
CL
Cu
time, s p x
0.8 NaCap-BSAIThis workl Urea-BSA IGeankoplisl 0 KCI-BSA IGeankoplisl Urea-Plasma IColton i Q 02-BSA IGoldstick I 02-Plasma IHersheyl 'r, Sucrose-Plasma icoltonl
DA P
0.09730 0.09730 0.09740 0.09550 0.09680 0.09600 0.09610 0.09590 0.09590 0.09610 0.09560 0.09590
MF-Millipore Membranesa 0.06310 0.03450 7204 0.06550 0.03200 7205 0.06540 0.03220 7205 0.06450 0.03120 7212 0.06280 0.03420 7202 0.06500 0.04120 7204 0.06220 0.03420 7203 0.06200 0.03410 7203 0.06340 0.03280 7203 0.06210 0.03420 7205 0.06440 0.03140 7202 0.06530 0.03080 7203
9.093 7.903 7.985 7.810 9.049 7.746 9.160 9.162 8.485 9.176 7.896 7.587
0.09419 0.09289 0.09348 0.09738
Polyvic Membranesb 0.06314 0.03127 7202 0.06194 0.03117 7204 0.06264 0.03107 7203 0.06693 0.03067 7209
8.045 8.202 8.059 7.327
0
DA B
x
0.6
0.4 0
I
I
2
1
4
6
8
10
Figure 2. Diffusivity ratio vs. protein concentration. Comparison of data from various investigators.
A t least three extractions were performed. Two milliliters of sample and 10 mL of petroleum ether were used. Once extracted, the acid was titrated with sodium hydroxide making the analysis a case of titrating a weak acid with a strong base. Once the concentration of the sodium caprylate was found in the upper and lower chambers a t the end of the run, the diffusion coefficient was determined from an integrated form of Fick's equation
a Average p for Millipore membranes = 8.43 x l o 4 m-*; average stirring speed for MF = 1 4 2 rpm. Average p for Polyvic membranes = 7.90 x l o 4 m-'; average stirring speed for Polyvic = 1 4 4 rpm.
the known published values of these solutes indicating boundary layer effects were essentially eliminated. The experimental data for diffusion of sodium caprylate in solutions of albumin (Okos, 1972) were obtained a t sufficiently high concentrations of the solute so that the ionic strength of the solutions was above 0.05. This ensured that no conformational changes of the protein occurred (Creeth, 1952) and that the diffusivity of the albumin was constant and essentially independent of protein concentration. The concentration of chloride ion was determined by using Mohr's method (Skoog and West, 1963). Crystalline bovine serum albumin was obtained from Sigma Chemical, Inc., and used without further purification. Solutions of bovine serum albumin were analyzed for concentration by the Beckman DU spectrophotometer. The concentration of protein was found proportional to the absorption at 280 nm. The concentration of sodium caprylate was determined by acidification of the sample with hydrogen chloride, subsequent extraction with petroleum ether, and titration with sodium hydroxide (AOCS, 1970; Boyer et al., 1947).
Since 6 and E cannot be measured directly, P can be found indirectly by diffusing 0.1 M potassium chloride whose diffusivity is 1.87 X mz/s a t 25 "C (Bidstrup and Geankoplis, 1963). The average cell constants p for Polyvic and MF-Millipore membranes were found to be 7.90 X lo4 m-2 and 8.43 X lo4 m-2, respectively. The data obtained for determining the average cell constants using KC1 are given in Table I and the data for determining the diffusion coefficient of sodium caprylate are given in Table 11. Experimental Results a n d Discussion Comparison of Diffusivity of Various Solutes i n P r o t e i n Solutions. The experimental values of the diffusivity DAp of sodium caprylate in the BSA solutions are given in Table 11. The values of the diffusivity ratio DAP/DAB were obtained by using a mean value of 8.78 X m2/s for the value of DABfor sodium caprylate in solutions containing no BSA. A comparison with data from other workers is given in Figure 2. The data presented represent conditions similar to those in this work.
Table 11. Experimental Diffusion Coefficients of Sodium Caprylate in Aqueous Bovine Serum Albumin Solutions ( 2 5 ' C, pH 7 ) init. concn of NaCap, g/100mL
albumin concn, g/lOO mL
diffusivity concn of NaCap, g-mol/L time, s
CL
12
P r o t e i n , g / l O O mL
CL
CU
0.84 0.92 1.012 0.982 1.016 0.685 1.476 0.490 0.655
0.00 0.00 3.00 2.00 1.00 3.00 3.00 3.00 3.00
11603 11604 11608 13059 12004 12303 12305 12802 13002
MF-Millipore Membranes 0.05064 0.03605 0.01469 0.05558 0.03970 0.01599 0.06095 0.04533 0.01574 0.05917 0.04309 0.01620 0.06120 0.04428 0.01704 0.04123 0.03032 0.01099 0.08891 0.06460 0.02449 0.02953 0.02193 0.00766 0.03947 0.02907 0.01047
1.03 0.51 1.44 0.88 0.999
0.00 0.00 0.00 0.00 4.00
11604 11623 11607 11608 11604
0.06219 0.03061 0.08767 0.05584 0.06018
Polyvic Membranes 0.04496 0.01735 0.02218 0.008492 0.06356 0.02428 0.04032 0.01563 0.04585 0.01443
DAB X m'/S
lolo,
DAP X
lo1', m'/S
DAp/DAB
7.387 7.166 8.000 7.306 7.676 6.741 6.866
0.8413 0.8162 0.9112 0.8321 0.8743 0.7678 0.7820
7.091
0.8076
8.828 8.712
8.860 8.766 8.758 8.902
236 Ind. Eng. Chem. Fundam., Vol. 18,No. 3, 1979
The diffusivity ratios found by Geankoplis et al. (1978) for urea fall near those found in this work for sodium caprylate. However, this similarity does not indicate that the mechanism of interaction of the two compounds to BSA is similar. Sodium caprylate binds by hydrophobic and ion interaction. The interaction of urea to protein is very poorly understood and several theories have been proposed (Steinhardt and Reynolds, 1969; Kauzman, 1959; Jenck, 1969; Tanford, 1970). The diffusivity ratio for KC1 indicates an even greater BSA interaction than existed for urea or sodium caprylate. Binding studies of KC1 to BSA indicate that under physiological conditions (pH 7.4 and chloride near 0.15 M) 6 to 7 mol of chloride ion are bound per mole of albumin (Edsall and Wyman, 1958). Scatchard et al. (1950) have reported that 40 sites are available for chloride binding. The number of moles of chloride binding increases with a decrease in pH (Scatchard and Yap, 1964), which is consistent with the observation that the diffusivity ratio decreases with decrease in pH (Geankoplis et al., 1978). Also shown in Figure 2 are reported diffusivity ratios for oxygen and sucrose. Giles and McKay (1962) reported that no disaccharide-bovine serum albumin complexes were formed in their work. Oxygen is generally also assumed not to bind significantly to albumin (Colton et al., 1970). These data represent the physical blockage effect of the protein molecules on the diffusion of small solute molecules when binding is not present. Modified Wang Equation to Predict Diffusion. The equation of Wang (Wang, 1954) was originally developed for the self-diffusion of water in protein solutions. This equation, although developed for bound water, can be modified to include other solutes such as sodium caprylate interacting with proteins. Since approximately 800 mol of water are bound to one mole of protein and the water of hydration is one of the major factors determining the protein obstruction factor, (1- ad,), it may be assumed that the protein obstruction factor remains independent of the solute. The diffusion of the bound solute may also be taken into account by assuming that the bound solute-protein complex will diffuse the same as the protein diffuses when present alone in solution. Therefore, the Wang equation is modified to
Diffusivity and Binding of Sodium Caprylate in BSA Solutions. The average diffusivity obtained for sodium caprylate at 25 "C and pH 7 was 8.78 X m2/s when no BSA was present. No effect on the diffusivity was found in the concentration range of 0.5 to 1.5 g/100 mL as shown in Figure 3. The diffusivity of sodium caprylate for the same concentration range used in this work is given by Lamm and Hogberg (1940) as 7.5 x lo-'' m2/s a t 20 "C. To correct for the temperature effect on diffusivity, the Stokes-Einstein equation (Einstein, 1906) was used. The ratio of the viscosity of water a t the two temperatures is assumed to be the same as the ratio of viscosities for sodium caprylate a t 20 and 25 "C. The corrected diffusivity of Lamm and Hogberg was calculated to be 8.58 x m2/s, which is within the experimental error of that found in this work. The effect of sodium caprylate concentration on its diffusivity in albumin solutions is also given in Figure 3. Increasing the sodium caprylate concentration from 0.5 to 1.5 g/100 mL in a 3 g/100 mL albumin solution increased the diffusivity from 6.741 X to 7.676 X m'/s. Compared to the diffusivity of the solute sodium caprylate
"r
o
Millipore membrane
t 41
0
"
0.4
'
I
0.8
"
1.2
'
'
1.6
'
2
Total sodium caprylate, g/lOOrnL
Figure 3. Comparison of experimental diffusivity of sodium caprylate in BSA solutions with predictions using the Geankoplis ( G ) ,modified Wang (W), and Colton (C) models.
with no protein present, however, the diffusivity of the solute in a 3 g of BSA/100 mL solution is reduced by 23% at 0.5 g of solute/100 mL and by 13% at 1.5 g of solute/loO mL. These substantial reductions in diffusivity can be attributed to the blockage effect of the large protein molecules and to the reversible adsorption or binding of the caprylate ion to the surface of the relatively slow moving protein. The increase in sodium caprylate diffusivity with increase in solute concentration a t a given protein concentration can be accounted for by the decrease in binding coefficient. Increasing the solute concentration decreases the ratio of bound solute to free solute, thereby providing a higher effective diffusivity. Using the experimental Boyer binding data to predict the effect of sodium caprylate concentration on its diffusivity DApat a constant protein concentration of 3 g/100 mL (Figure 3), the Colton equation appears to have the best fit with an average deviation of f 2 % , with the modified Wang and Geankoplis equations having an average deviation of f 3 % and f 5 % , respectively. In using these equations values for a and 4, used were those also employed by Colton et al. (1970) and Geankoplis et al. (1978). For D , a value of 6.81 X lo-" m2/s (Charlwood, 1953) was used and for sodium caprylate the average value of Dm = 8.78 X mz/s was used as found in this work. The effect of protein concentration on the diffusivity of sodium caprylate is shown in Figure 4. Increasing the protein concentration while holding the sodium caprylate solute concentration constant a t 1.0 g of solute/100 mL decreases the diffusivity of the solute due to increased blockage and increased number of available binding sites. For purposes of showing the effect of blockage only with no binding, the data for the diffusion of the nonbinding solutes oxygen and sucrose are also plotted in Figure 4. These data give considerably higher values of the diffusivity ratio DAp/DAB because of the absence of binding. Using eq 1 , 2 , and 4 again but with no binding, the three predicted lines are plotted in Figure 4 for only blockage effects being present. The Geankoplis equation predicts the data best having an average deviation of f l % , with the modified Wang and Colton equations having an average deviation of f370 and f12%, respectively. The Colton equation greatly underpredicts the blockage effect with a maximum deviation of +17%. Predictions for the effect of protein concentration on the diffusivity ratio of sodium caprylate with binding and blockage both present are also shown in Figure 4. All three equations predict the data with an average deviation of
Ind. Eng. Chem. Fundam., Vol. 18, No.
-P
0.6
A'
be used to predict binding coefficients from these three equations. The data in this work appear to be the first known results for a system where binding coefficients are known and both solute and protein concentrations are varied over a range. Even though results have been predicted reasonably well, more work is necessary on widely different systems of binding solutes and different proteins to more accurately relate the parameters affecting diffusion in these systems.
-
DAB
0.4
# NaCap-ESAIThis 002-BSA
0
work1
Nomenclature
fl 02-Plasma
0.2
il,
1
0
1
2
Sucrose-Plasma
1
1
4
1
6
i
1
I
l
l
10
8
12
Protein, g / 1 0 0 mL
Figure 4. E f f e c t o f concentration o f BSA o n d i f f u s i v i t y r a t i o for s o d i u m caprylate a n d comparison w i t h predictions using t h e Geankoplis (G), m o d i f i e d W a n g (W), a n d Colton (C) models. Also, comparisons f o r n o n b i n d i n g solutes oxygen a n d sucrose.
I \
3 g BSA/100mL
0 Geankoplis
,d C o l t o n 0
3, 1979 237
Wang
m2 A = mass transfer area, 3.98 X Cb = concentration of protein-bound solute A, g/m3 of solution Co = concentration of total solute A, g/m3 of solution CLo = original concentration of solute in lower chamber, g-mol/L CL = final concentration of solute in lower chamber, g-mol/L Cuo = original concentration of solute in upper chamber, g-mol/L Cu = final concentration of solute in upper chamber, g-mol/L DAB= diffusivity of solute A in solution with no protein present, m2/s DAp= diffusivity of solute A in protein solution, m2/s D, = diffusivity of protein in solution, m2/s k, = protein binding coefficient, [(g of bound solute)/(mL of solution)]/ [ (g of free solute)/(mL of protein-free solution)] t = time, s VL = volume of lower chamber, 21.04 X lo4 m3 Vu = volume of upper chamber, 20.89 X lo4 m3 Greek L e t t e r s CY = diffusivity reduction shape factor for protein p = cell constant, m-2 d = effective pore length, m e = fraction of area open for diffusion 4, = volume fraction proteins in solution
a t a 01 Boyer
.E 0.15 m
Literature Cited 0
0
0.4
0.8
1.2
1.6
2
Total sodium caprylate, W 1 0 0 mL
Figure 5. Prediction o f t h e b i n d i n g coefficient o f sodium caprylate using the diffusion models o f Geankoplis, modified Wang, and Colton.
f 4 % with the Colton equation having the largest maximum deviation of + l o % . It appears that the Geankoplis eq 2 has the best overall fit to experimental data since it predicts the overall results for nonbinding and binding solutes with an overall average deviation of less than rt5% and a maximum deviation of 7%, which is within engineering accuracy. The modified Wang equation has approximately the same overall accuracy with a maximum of 8%; however, it tends to underpredict the blockage effect. Predictions for four experimental points for diffusion of urea in protein and one point for uric acid in plasma (Geankoplis et al., 1978) using the Geankoplis equation also had an average deviation of f 5 % and a maximum of 7%. The Geankoplis, modified Wang, and Colton equations can also be used, conversely, to predict the binding coefficients from experimental diffusion data of D A p . When these equations were used to calculate the binding coefficient from the experimental diffusivity data of Table 11,a reasonable fit resulted as shown in Figure 5. The line represents the binding data of Boyer e t al. (1947). However, a small deviation in diffusivity causes a large deviation in the predicted binding data. A 1% decrease in diffusivity results in about a 5% increase in predicted binding coefficient when using any of the three equations. Because of this, it is not recommended that diffusivity data
American Oil Chemist's Society, "Official and Tentative Methods of the American Oil Chemist's Society", 3rd ed, Chicago, Ill., 1970. Bidstrup, D. E., Geankoplis, C. J., J . Chem. Eng. Data, 8 , 170 (1963). Boyer, P. D. Ballou, G. A,, Luck, J. M., J . Biol. Chem., 167, 407 (1947). Colton, C. K., Smlth, K. A,, Merrill, E., Reece, J. M., Chem. Eng. Prog. Symp. Ser., 66(99), 85 (1970). Creeth, J. M., Biochem. J . , 51, 10 (1952). Edsall, E., Wyman, "Physical Chemistry", Vol. 1, Academic Press, New York, N.Y., 1956. Einstein, A,, Ann. Phys., 19, 371 (1906). Geankoplb, C. J., "MassTransport p h e m n a " , Ohio State University Bookstores, Columbus, Ohio, 1972. Geankoplis, C. J., Okos. M. R., GsuUce, E. A,, J. Chem. Eng. Data, 23, 40 (1978). Giles, C. H., McKay, R. B., J . Biol. Chem., 237, 3388 (1962). Goldstick, T. K., Fan, I., Chem. Eng. Prog. Symp. Ser., 66(99), 101 (1970). Gordon, A. R., J . Chem. Phys.. 5. 522 (1937). Hershey, D.. Karhan, T., AIChE J . , 14, 969 (1968). Jenck, W. P., "Catalysis in Chemistry and Enzymology", McGraw-Hill, New York, N.Y., 1969. Jost, W., "Diffusion", Academic Press, New York, N.Y., 1960. Kauzmann, W., "Advances in Protein Chemistry", Vol. 14, Anson, Bailey, Edsall, Ed., Academic Press, New York, N.Y., 1959. King, T. P., Spencer, M., J . Biol. Chem., 245, 6134 (1970). Lamm, Von Ole, Hogberg, H., Kolloid Z . , 91, 10 (1940). Navari. R . M., Gainer, J. L., Hall, K. R., AIChE J., 17, 1028 (1971). Okos, M. R., M.S. Thesis, Ohio State University, 1972. Reynolds, J., Herbert, S..Steinhardt. J., Biochemistry, 7 , 1357 (1968). Scatchard, G.. Scheinberg, I. H., Armstrong. S. H., J . Am. Chem. SOC.,72, 540 (1950).
Scatchard, G., Yap, W. T., J . Am. Chem. Soc., 86, 3434 (1964). Skow, D. A., West, D., "Fundamentals of Analytical Chemistry", Hok, Rinehart a d Winston, New York, N.Y., 1963. Spector, A,, Kathryn, J., Fletcher, J., J . Lipid Res., I O , 56 (1969). Steinhardt, J., Reynolds, J. A,, "Muttiple Equilibria in Proteins", Academic Press, New York. N.Y.. 1969. Stroeve, P., ind. €ng. Chem. Fundam., 14, 140 (1975). Tanford, C., "Advances in Protein Chemistry", Vol. 24, Anson, Bailey, Edsall, Ed., Academic Press, New York, N.Y., 1970. Teresi, J. D., Luck, J. M., J . Biol. %hem., 194, 623 (1952). Wang, J. H., J . Am. Chem. Soc., 76, 4755 (1954). Received for r e v i e w April 27, 1978 A c c e p t e d February 8, 1979
