T H E ACTIVITY COEFFICIENT OF EGG ALBUMIN IN THE PRESENCE OF AMMONIUM SULFATE CRAWFORD F. FAILEY Department of Biological Chemistry, College of Physicians and Surgeons, Columbia University, New York C i t y . Received March 4, 1933
The usual method of determining the activity coefficient of a protein in the presence of salt by measuring its solubility is limited, except in the case of globulins, to comparatively concentrated solutions. Furthermore, necessary information concerning the composition of the solid phase is usually lacking, so that one may actually be dealing with a hydrate whose activity is equal to the product of the activity of the protein by some power of the water activity. It may be pointed out that in the inorganic field solutes are usually treated without regard to hydration even though the solid phase, should it exist, contain water of crystallization. One does not, for example, speak of the activity coefficient of hexahydrated calcium chloride. PROTEIN ACTIVITIES FROM OSMOTIC EQUILIBRIA
The osmotic pressure of protein, both in the presence and in the absence of other substances, has been recently investigated by a number of workers (1,2,3). In the following a method will be developed for determining from such measurements what changes are brought about in the activity coefficient of the unhydrated protein by addition of a third component to the solution. The method is applicable at, any temperature between the freezing point of water and that temperature at which the protein coagulates, and may be used throughout the entire solubility range of protein and added crystalloid,-organic or inorganic, Consider an isoelectric protein solution in equilibrium against water across a membrane impermeable to protein but permeable to all other substances. If successive small amounts of salt or urea be added to the system, there will be reached after each addition a new state of equilibrium. B y exerting an appropriate pressure the protein molality may be held constant, leaving as the measured variables the osmotic pressure and the molality of added substance within and outside of the membrane. Suppose the excess pressure above atmospheric on the protein solution 1075
1076
CRAWFORD F. FAILEY
reduced from P to zero. Within the membrane when P is reduced t o zero, .let us set: al = activity of water. u2 = activity of added substance. a3 = activity of protein. m2 = molality of added substance. m3 = molality of protein. y3 = activity coefficient of protein. VI - = partial molal volume of water. V z = partial molal volume of added substance. Let a:, a ! , m i represent the corresponding quantities in the protein-free outer solution. The symbols and general method of treatment are those of Lewis and Randall (4). We have the two Duhem equations m3 d In aa = - 55.51 d In
a1
-
m2 d In a2
(1)
55.51 d In a,” = -mz# d In a:
(2)
Also RT In al = RT In a,” - v l P
or VI
d In al = d In a,” - - dP RT
Similarly d In a2 = d In a,”
v2 -dP RT
Substituting equations 4 and 5 in equation 1, ma d In
a3 =
55’5171 + mzV2d P RT
-
55.51 d In a: - mzd In a,“
(6)
From equations 6 and 2 d In a3 =
55.5171
- m2 + msvz d P + m,“ ___ d In a,“
maRT
(7)
m3
Changing to common logarithms and introducing ys d log
78
+ mtT2 d P + + d log m3 = 55.51v3 2.3 mrRT
Integrating d log 7 3 =
s
+
55.511i1 mzv, 2.3 meRT
m,“
rn;
- mz
~
- m2
ma
d log a,”
d log a,”
-
S
(8)
d log ma (9)
1077
ACTIVITY COEFFICIENT O F EGG ALBUMIN
This equation enables us to calculate changes in y3 from measurements of osmotic pressure and the distribution of added substance between the protein-containing and protein-free compartments of an osmometer. The partial molal volumes may be measured or assumed to be the same as in the protein-free solution. Activities outside are determined by the usual methods. This calculation has been carried through using the careful data of Sorensen (5) and his coworkers on the osmotic pressure of egg albumin in the presence of ammonium sulfate. Since the activity coefficient of ammonium sulfate is not known, it has been assumed equal to that of sodium sulfate as given by Akerlof (6). In the calculation of m3 the molecular weight was taken as 34,500 and the nitrogen factor as 6.45. The value of 55.51 TI m2Tz is from table 32, p. 158 of reference 5. In the most dilute salt solution y3 is set equal to one.
+
Table 1 of this paper gives values of m:, m2, m3,
mf - m a mo
1
- 1% Y L
and log a! obtained from tables 55, 56, and 57, pp. 334, 335, 336, and 337 of reference 5, and values of yz from reference 6. These are plotted in figure 1with
m! - m 2 m3
as ordinate and log
a! as abscissa. A smooth curve
through these points permits a graphical integration of the second term of the right-hand member of equation 9. The first term of the same member contributes little to log y3 and is easy to evaluate. The quantity
is found to be approximately constant', so that one may use an average value over any interval and set 1 pzv - dP = 5.3 X dP = 2 . 3 R T 1 1 ma
( P z - PI) (10)
where P is expressed in centimeters of water and T = 291. In table 2 are given the steps in a graphical integration of equation 9. The solutions are selected from table 55, pp. 334 and 335 of reference 5 in such a manner as to cover the concentration range of ammonium sulfate in approximately equal steps. Values for m' -
m3
arereadfromfigure 1. It
may be observed that the osmotic pressure term is negligible. Figure 2 shows that the logarithm of the activity coefficient of egg albumin i s a linear function of the ammonium sulfate molality within the membrane. The decrease in the activity coefficient of egg albumin with increasing salt
1078
CRAWFORD F. FAILEY
TABLE 1 EXPERIMENT
ml$
ma
ma X 10s
1.763 1.693 1.433 1.358 1.335 1.186 1.160 1.137 1.140 1,137 1.131 1.123 1.123 1.118 1.010 1.004 0.8214 0.6362 0.3200 0,3133 0.1930 0.1305 0.0686 0,0633 0.0349 0.0347 1.518 1.463 1,355 1.161 0.9960 0.7979 0.6605 0,3216 0.1907 0,1279 0.0679 0.0635 0.0332 0.0318 1.478 1.352 1.154 1.046 1.009 0.7591 0.3249
3.657 3.700 4.074 3.374 3.615 3.550 3.932 3.951 3.850 3.921 3.977 4,038 3.977 3.580 3.338 3.295 3.375 3.461 3.193 3.336 3.381 3,349 3.772 3.186 3.772 3,350 3.445 3.396 3.500 3,315 3,274 3.282 3.194 3.145 3.233 3.140 3.851 3.086 3.802 3.087 3.452 3.587 3.333 3.515 3.435 3.693 3.092
NO.
98 104 62 112 97 103 61 90 87 89 88 163 164 137 110 111 105 96 146 95 94 93 82 92 81 91 117 114 108 116 113 115 107 129 122 121 84 120 83 119 167 166 127 165 106 168 133
1.836 1.761 1.502 1.407 1.388 1.229 1.207 1.183 1.182 1.181 1.178 1.170 1.168 1.167 1.046 1.042 0.8470 0.6537 0.3283 0.3191 0.1945 0.1299 0.0677 0.0642 0.0336 0.0331 1.577 1.519 1.412 1.209 1.030 0.8209 0.6812 0,3291 0.1923 0.1298 0.0680 0.0637 0.0331 0.0317 1.531 1.407 1.193 1.081 1.039 0.7625 0.3312
19.96 18.38 16.94 14.52 14.66 12.11 11.95 11.64 10.91 11.22 11.82 11.64 11.32 13.69 10.78 11.53 7.59 5.06 2.60 1.74 0.44 -0.18 -0.24 $0.28 -0.34 -0.48 17.12 16,49 16.29 14.48 10.38 7.01 6.48 2.39 0.50 0.61 0.03 . 0.06 -0.03 -0.03 15.35 15.33 11.70 9.96 8.73 9.20 2.04
- logY*xI
- log .,N
0.803 0.794 0.761 0.748 0.745 0.722 0.719 0.715 0.714 0.714 0.714 0.713 0.712 0.712 0.692 0,691 0.657 0.619 0,521 0.517 0.446 0.396 0.320 0,315 0.250 0.246 0.772 0.764 0.749 0.719 0,690 0.652 0.624 0.522 0.445 0.396 0.322 0,315 0.246 0.242 0,766 0.748 0.716 0.698 0.691 0.641 0.523
1.015 1.042 1.150 1.198 1.207 1.294 1.309 1.324 1.321 1.324 1.327 1.333 1.333 1.333 1.417 1.417 1.585 1.810 2.410 2.437 2.869 3.244 3.865 3.877 4.570 4.576 1.120 1.144 1.195 1.309 1.429 1.612 1.771 2,413 2.881 3.247 3.868 3.931 4.576 4.621 1.141 1.198 1.315 1.390 1.420 1.675 2.407
1079
ACTIVITY COEFFICIENT OF EGG ALBUMIN
I
I
I
I
I
I
I
I
1.1
1.6
2.1
2.6
3.1
3.6
41
4.6
-109
1 Solutions of pH 4.6;
a;
FIQ. 1. solutions of pH 4.9; -. solutions of pH 5.3
I
I
!
I
0.5
1.0
1.5
20
2.5
Ammonium sulfate moldity
. 1
FIG.2. Log activity coefficient egg albumin from osmotic equilibria. Log activity coefficient egg albumin from solubilities, assuming log y = Constant - log ma. The broken line shows log activity coefficient egg albumin from solubilities corrected for changing water activity, assuming the solid phase to contain 0.22 g. water per gram of protein. In the case of the two curves from solubilities only the slopes are significant.
1080
CRAWFORD F. FAILEY
concentration a t low salt concentrations may be real, as in the case of globulins, or may possibly be an error due to a small amount of ammonia in the egg albumin, which would reverse the sign of
m$ -
m3
m2.
The author
is inclined to accept the first interpretation. __
___
__
~
TABLE 2 ~
i r 2,
;
[
8
s"
X
91 82 93 95 96 105 111 137 90 61 97 112 62 104 98
E
5
B
0.03 0.07 0.13 0.31 0.64 0.82 1.00 1.12 1.14 1.16 1.34 1.36 1.43 1.69 1.76
E
b
3.35 3.77 3.35 3.37 3.46 3.38 3.30 3.58 3.95 3.93 3.62 3.37 4.07 3.70 3.66
1OC3 1005 1008 1019 1039 1052 1064 1072 1073 1075 1087 1088 1095 1112 1118
__
73.1 0 79.5 t 0 . 0 3 75.7 t O . O 1 0 73.7 0 73.1 67.9 -0.03 62.1 -0.06 61.7 -0.07 72.3 0 66.8 -0.03 58.2 -0.07 53.0 -0.10 59.3 -0.07 43.9 -0.15 39.6 -0.17
EXPERIMENT N O . m: ____________
1 2 3 4 5 6
1.81 1.88 1.95 2.03 2.22 2.29
m3 X 105
-0.3 -0.2
+o. 1 1.8 5.5 7.9 10.3 11.7 12.0 12.2 14.6 14.8 16.0 19.2 20.0
__ -
59.42 40.17 26,18 14.98 3 487 2.080
F M
I
a,
4.576 3.865 3.244 2.437 1.810 1.585 1.417 1.333 1.324 1.309 1.207 1.198 1.150 1,042 1.015
log ms
9
0
-0.18 -0.18 t-0.59 2.88 4.38 5.91 6.84 6.94 7.13 8.49 8.62 9.36 11.26 11.79
+5
1.77 1.60 1.42 1.18 0.54 0.32
0 0.06 0 0.01 0.02 0.01 0 0.03 0.08 0.07 0.03 0.01 0.09 0.05 0.04
1.57
0
-0.21 -0.17 -0.58 2.86 4.34 5.85 6.74 6.86 7.03 8.39 8.51 9.20 11.06 11.58
- log nk8
4.80 4.97 5.15 5.39 6.03 6.25
PROTEIN ACTIVITIES FROM MEASUREMENTS O F SOLUBILITY
That the solubilities of proteins are affected by salts has long been known. Cohn (7) pointed out that the logarithm of protein solubility is, in the case of egg albumin and of pseudoglobulin, a linear function of salt concentration. This relation has since been found by Florkin (8) to hold for fibrinogen, and by Green (9) for hemoglobin. It is sometimes assumed that
ACTIVITY COEFFICIENT O F EGG ALBUMIN
log
yprotein=
Constant,- log Solubility,,,tein
1081 (11)
and a calculation made in accordance with the above from table 39, p. 221 of reference 5 is shown in table 3 and plotted in figure 2. One may see that a great dis.crepancy exists between the slope of the straight line through these points and the values of log y3 obtained by the previous method. If the solid phase contains water as water of crystallization and we choose unhydrated protein as one component, however, equation 11 cannot hold. The reaction is not
* Protein (solid)
(12)
+ nH2O + Protein hydrate (solid)
(13)
Protein (dissolved)
but Protein (dissohed)
so that if al = activity of water, m2 = molalily of ammonium sulfate, a3 = activity of protein, m3 = solubility of protein, y3 = activity coefficient of protein, a4 = activity of protein hydrate, and n = moles of water per mole of prot'ein in the hydrate, n
a, ' 1 x 3 = a4 = Constant
therefore TI.
d logal
+ dlogya + d logm3 = 0
and b log ma b m2
b log y3 bmz
a log a1
n-
b rnz
but bloga1 3 m2
-=-_
bloga, de be bm?
where 0 is the freezing point depression. b log a,
be
= -0.00421
(See reference 4, p. 284.) Also for ammonium sulfate (reference lo), when m2 is about 2, e
- = 3.21 m2
(19)
1082
CRAWFORD F. FAILEY
We shall set
be equal t o this, and neglect the effect of protein on al. dm2
Solid
egg albumin hydrate contains 0.22 g. of water per gram of protein (reference 5, p. 210). 34,500 X 0.22 = 422 18
?&a
From table 3
Thus
*'amz
= 3.0 -I-422 X 0.00421 X 3.21 = 8.7
(22)
which agrees satisfactorily with the slope 7.6 obtainable from table 2 when one considers all the assumptions made in getting constants necessary for the computation, and the fact that in one case the protein concentration is held constant while in the other it varies greatly. SUMMARY
The logarithm of the activity coefficient of egg albumin in the presence of ammonium sulfate has been calculated from osmotic data and found to be a linear function of the salt concentration, These activity coefficients are in satisfactory agreement with solubility measurements when account is taken of the composition of the solid phase. REFERENCES (1) ADAIR,G. S.: Proc. Roy. SOC.London A120,573 (1928);Al26, 16 (1929);J. Am. Chem. SOC.61, 696 (1929). (2) BURK,N.F.,AND GREENBERG, D. M.: J. Biol. Chem. 87,197 (1930). (3) BURK,N. F.: J. Biol. Chem. 98, 353 (1932). M.: Thermodynamics and the Free Energy of (4) LEWIS, G. N.,AND RANDALL, Chemical Substances. McGraw-Hill Book Co., New York and London
(1923). ( 5 ) SORENSEN, S. P. L.: Compt. rend. trav. Lab. Carlsberg 12 (1917). (6) WKERLOF, G.:J. Am. Chem. SOC. 48,1160 (1926). (7) COHN,E.J.: Physiol. Rev. 6, 349 (1925). (8) FLORKIN, M.:J. Biol. Chem. 87, 629 (1930). (9) GREEN,A. A.: J. Biol. Chem. 93, 495 (1931). (10)International Critical Tables, Vol. IV, p. 255. McGraw-Hill Book York (1928).
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