March, 1901
P a R T I A L S P E C I F I C V O L U M E S I N A~!~ULTICOMPONENT SYSTEMS
TABLE VI APPARENT L)IPOLE MOMENTS OF SOMEALCOHOLS, CHLORIDES A N D BROMIDES IN BENZENE SOLUTION X =
0H
CI
Br
2.1-2.3 Do 2.1-2.3 Do C6HiiX 1.7-1.9 Do 1.67’ 1 .82-1.85” 1.85-1.87“ CBHSCH~X 1,57-1,60“ 1.51-1.56“ Cd&X 1.45h L. G. Wesson, “Tables of Electric Dipole Moments,” The Technology Press, Massachusetts Institute of Technology, Cambridge, Mass., 1948. A. H. Boud, D. Cleverdon, G. B. Collins and J. W. Smith, J . Chem. Soc., 3793 (1955). (I
427
TABLE VI1 INTERNUCLEAR DISTANCES OF C-X BONDS (IN d.). X = S 0 F c1 Br Paraffinic 1.81 1.43 1.38 1.77 1.94 Aromatic .. 1.36 1.30 1.70 1.85 “Tables of Interatomic Distances and Configuration in Molecules and Ions,” Special Publication KO. 11, The Chemical Society, London, 1968.
mental results reported in the present work by applying methods and approximations afforded by the molecular orbital theory may lead to a more satisfactory approach. the order cyclohexanethiol, a-toluenethiol and Acknowledgment.-The authors gratefully acben~enethiol,’~ am attempt to interpret the experi- knowledge the financial assistance given to this Laboratory by the Rockefeller Foundation (New (17) See for instance G. I. Brown, ”An Introduction to Electronic York) and the Conselho Nacional de Pesquisas Theories of Oriranic Chemistry,” Longmsns, Green and Co., London, (Rio de Janeiro). 195s.
PARTIAL SPECIFIC VOLUMES AND REFRACTIVE INDEX INCREMENTS IN MULTICOMPONENT SYSTEMS1 BY EDWARD F. CASASSA AND HENRYK EISENBERG Mellon Instituk, Pittsburgh, Pennsylvania Received August 8S,1960
Partial specific volumes and specific refractive index increments of proteins and polyelectrolytes in aqueous solution with a simple salt are diiscussed in relation to possible definitions of independent thermodynamic components. As suggested elsewhere, it is sometimes useful to define components by equating inner and outer salt concentrations in a dialysis equilibrium experiment ,and assigning any excess or deficiency of salt ions in the three-component phase as part of the macromolecular component. Since by need or by custom, protein solutions may be dialyzed against the solvent, volume and refractive index changes corresponding to addition of the non-diffusible component are often measured in a way relating directly to such a definition of components: that is, as the protein concentration is changed the chemical potential of the salt component is held a t least approximately constant. A consideration of available data indicates that in some three-component systems thermodynamic interaction between salt and the macromolecular sDecies is great enough for volume and refractive index changes to be significantly affected. It is important, therefore, that the partial volumes and refractive index increments applied in thermodynamic analysis of ultracentrifuge and light scattering data be defined and interpreted in consistent fashion. An earlier discussion of sedimentation equilibrium is extended to the special case of sedimentation in a density gradient produced by the distribution of a heavy salt. It is shown how an unambiguous molecular weight for the polymeric species can be obtained from equilibrium measurements without explicitly evaluating the interaction with the supporting electrolyte.
Introduction The usefulness of ultracentrifugation and light scattering as techniques for the physicochemical investigation of niacromolecular solutes depends in great part on the independent measurement of the partial specific vlolume and the specific refractive index increment. Aside from their intrinsic interest as physical prope:rties, both enter into the interpretation of centrifuge measurements while the determination of the refractive increment makes possible absolute determinations of molecular weight by light scattering. In two-component solutions (one solute and one solvent) the definition of these quantities is subject .to no ambiguity. I n multicomponent systems, on the other hand, the definition of independent components to be used in thermodynamic formulations is largely a matter of convenience2p3;but whatever the conventions chosen, it is essential that all the experimental quantities involved and the relations used be consistent with (1) This study was supported by a grant (NSF-G7G08) of the National Science Foundation. (2) G . Scatchard, J . A n . Chem. Soc., 68, 2315 (1946). (3) E. F. Casitssa and H. Eisenberg, J. Phus. Chem., 64, 753 (ISGO).
them. These considerations give rise to some rather subtle problems that have not always been fully appreciated. In some circumstances the uncritical acceptance of simple relations established for two-component systems may lead to serious errors in interpretation of experimental data or sometimes to failure to discern small effects that can furnish information about thermodynamic interactions in complex systems. These remarks apply with particular force to studies of materials of biological interest since the systems encountered almost always contain at least three components. Aqueous protein solutions, for example, are frequently buffered or a t least may contain an added salt. A further complication is the fact that preparative procedures often involve dialysis against the solvent medium with consequent redistribution of diffusible species across the membrane. As we have pointed out elsewhere3 this circumstance, when its implications are understood correctly, actually becomes an advantage in simplifying the analysis of light scattering and ultracentrifugal measurements. Rather full discussions of the question of defining
EDWARD I?. CASASSA AND HENRYK EISENBERG
428
components in interpreting light scattering from dialyzed solutions have been given recently in a paper by Scatchard and Bregman4 and in reference 3. Here, after considering the partial volume and refractive increment of a macromolecular solute in relation to useful definitions of components, we apply our conclusions in carrying further the discussion of sedimentation equilibrium given in reference 3. As in the earlier paper, we are concerned with a system of three components, of which one is non-diffusible : specifically the solution comprises water (component 1), a protein or polyelectrolyte (component 2) , and a uni-univalent salt (component 3). Partial Specific Volume In measurements of partial specific volume of a solute one usually compares solution and solvent densities and thus obtains directly the apparent partial volume + of solute from the change produced by a finite Concentration increment. Conceptually one visualizes adding the solute component 2 in some completely definite, and possibly actually available, form (dry isoionic protein or the stoichiometric composition XZ P of an alkali metal salt of the polyacid HzP, for example) to the solvent mixture consisting of components 1 and 3. Upon addition of g2 grams of component 2 to gs grams of solvent of density psl the volume of the system changes by AV and the density by Ap. The apparent partial specific volume of component 2 in these terms is + = -AT’ =g2
1 ?
1-@%E
(
=
m.>
1 [l P
(1
+ mrM~/1wo)Ap]
(I)
WZPS
where w2 is the solute concentration in grams per gram of water, m3 is the molality of the salt, and M 3 is its formula weight. By substituting into equation 1 the weight fraction z = q2/(gs g2), one obtains an alternative form
+
9 = -1 ( l - $ ) = z ( l1- $ )
(2)
Pn
in which c is the concentration of component 2 in grams per milliliter of solution. If the common laboratory practice of dialyzing solutions against the solvent has been adopted, it must be recognized that the distribution of small ions between solution and dialysate is affected by the presence of the macromolecular species. I n the thermodynamic formalism for experiments carried out in this way, it is useful3-*to equate the molality of salt in the dialysate with that in the solution and to assign any excess (or deficiency) or diffusible ions to component 2. If component 2*, as we shall aesignate the component defined in this way, were added to the solution a t osmotic equilibrium there would occur no redistribution of diffusible species across the membrane. This convention is generally useful only if the numbers of diffusible ions included are independent of the concentration of the macromolecular species over a fairly wide concentration range and are also independent of pressure. Fortu(4) G . Scatchard and J. Bregman. J . Am. Ckem. SOC., 81, 6095 (1959). ’
T‘ol. 65
nately these requirements hold to a good approximation in the few system^^-^ adequately studied, and one may therefore discuss a partial specific volume for component 2* in a concentration series equilibrated by dialysis, or equivalently prepared by dilution with d i a l y ~ a t e . ~Thus, .~ conceptually one can add to the solution a t constant pressure, g2* grams of component 2*, the new mass being related to ~2 by g2*
gz(l f t ) = g p ( 1
+ Z V Z ~ * M , / M-~XvZiMi/Mz)
The numbers of moles of diffusible ions of molecular weight i l l i included by each definition in one mole of the macromolecular component are designated vzi, Vzi*; and g2/Mz is the number of moles, the same in both cases, added to the system. For component 2*, equations 1 and 2 are still valid in terms of the corresponding density change A*p and the weight concentrations wz*,c*
L(1 P0
-$ )!
(3)
Since the volumes #I and +* have been defined in the usual fashion for apparent extensive properties, the true partial volume D~ is given by 02 = w2 (d+/ dwz),and the analogous relation holds for +*. In the two procedures just described densities and concentrations are treated consistently. Sometimes however, though the solution is dialyzed and its density compared with that of dialysate, the weight concentration arbitrarily used in calculations does not correspond to component 2.* For example, in protein studies, concentrations are often based on nitrogen content multiplied by a factor to give a protein concentration corresponding presumably to the salt-free isoionic species, which we can regard as the component 2 to which equation 1 applied. Hence, the apparent partial volume calculated is
++
9’= 1 (1
+)
- A*
(4)
PS
which corresponds neither to 4 nor to I$*. The relation between c$~ and c$* is obtained by substituting c* = c(l E ) in equation 4 and eliminating A*p/c2* by equation 3
+
9‘ = 9* -
W
P
8
- 9*)
(5)
The expression of 4* or 4’ in terms of 4 involves the volume corresponding to the fractional mass increment 5 ; and therefore the partial volume of the ions (taken in neutral combinations) contributing to t is required. To use an example cited above, we consider component 2 as the completely ionized Component 3 is the salt XzP of the anion salt XY; v2x = 2; v 2 y = 0; and vzx* - v2x = $M2/M3. Then, adding g,* grams of component 2” to the solvent is equivalent to adding g2 grams of XzI’ and Eg2 grams (or Eg2/M3moles) of XY. If the partial specific volume of XY in the three-component system is &, the corresponding volume change is AV Ef13g2if 83 is independent of the change in salt concentration. It follows that
+
(5) G. Scatchard, A. C. Batchelder and A. Brown, ibid., 68, 2320 (1946). (6) U. P. Strauss and P. Ander, ibid., 80, 494 (1958). (7) €1. Eisenherg and E. F. Casaasa. J . Polymer Sn’., in press.
March, 1961
PARTIAL SPECIFIC VOLUMESIN MULTICOMPONENT SYSTEMS
429
TABLEI PARTIAL SPECIFICV O L U M EAT S ~25’ System
Concn.
A*p X 10:
€ X 10’
6
4*
4‘
0.1036 equiv.c/l. 5.67 -87.6 0.534 0.518 0 560 SHrPVS in 0 . 5 M b NHaCl KPVS in 0 . I M b KC1 0.1124 equiv.O/l. 8.77 -76.5 .411 ,413 ,457 BSA in 0.1 M b NaC1 58.3 g./kg. HzO 2.67 .7343 .7331 .7324 a The meaning of symbols and procedures for calculation are given in the text. The solvent composition rcfers t o dinlysate. c Equivalents of sulfonate groups per liter.
+
by Scatchard, et al., this amount of binding corresponds to 3.2 moles of sodium chloride per mole of protein. Partial specific volumes of salts and denBy combining equations 5 and 6, we obtain sities of salt solutions required in calculations for Table I were obtained from equations and pa.$f = - e,) (7) rameters given by Harned and Owen.g For a protein system we might designate the The tabulated apparent volumes exemplify a isoionic protein as component 2; but in defining variety of circumstances that may be encountered components by clsmotic equilibrium with a solution in practice. In the case of serum albumin t is of the salt XY, we would not generally have equal positive, and the solution density is close to unity numbers of moles of X+ and Y - included in the while 4 is smaller and of the order of twice g3. It electrically neutral components 2* because the pro- follows that the partial volumes should fall in the tein may “bind” protons or hydroxyl ions as well as order > 4* > 4’. If f is negative, so that salt is the salt ions. Tlo be in equilibrium with a protein rejected by the macromolecular species, the order is component of specified composition, the dialysate reversed as in the case of the KPVS solution in must in general contain the acid HY or base XOH potassium chloride. For NHdPVS in ammonium in addition to the salt; and the system then in- chloride, 4 is also negative; but the order of partial cludes four components if one wishes to consider volumes is 4* < 4 < d’, the interchange of @* and varying independently the concentration of acid or 4’ being a reflection of the fact that the partial specific volume of ammonium chloride (0.701) is Ordinarily, the concentration variable most easily greater than that of the polymer salt. measured is c; aiod though c* is for many solutions It is clear that in protein solutions in rather dilute determinable in principle by comparing weights of salt, the differences among 4, +* and 4’ are likely dried residues of solution and dialysate, practical to be of the order of only a few tenths of a per cent ., difficulties might sometimes make the procedure indistinguishable or nearly so as compared with unsatisfactory. For instance, in a protein system absolute uncertainties of concentration determinathe loss of the volatile acid HY from the protein tions. It is possible though that comparative component might have to be considered even measurements attain a level of precision a t which though no acid has been added. The apparent these differences become significant. Since ion volumes 4 or 4’ are therefore the quantities ob- binding to proteins is a mass-action phenomenon, tained directly from most density measurements, increasing the salt concentration increases the but 4* may be a primary result if concentrations effect; and in 1 M sodium chloride, the values of are based on dry weights. the partial volumes for serum albumin would exI n order to convey some idea of the magnitude of hibit differences of the order of a per cent. The the differences which may exist among the three par- tabulated values provide some indication of the tial volume quantities defined here, experimental and possibility of determining from the deiisitics of calculated values for several systems are collected dialyzed and undialyzed solutions. in Table I. The ammonium and potassium salts of In the comparisons made here, we have assumed poly-(vinyl sulfonic) acid (denoted as NHJ‘VS that 4 is the same in water and in salt solution. and KPVS) have been studied in this L a b ~ r a t o r y . ~In the work of Charlwoodlo on proteins no sigValues of 4’ listed in the table for these salts were nificant difference is evident, but accurate measuredetermined from density measurements in pycnom- ments have been made only in rather dilute salt. eters. Then with these results and f from data on For proteins, 4 does generally increase markedly as the distribution of chloride ion a t dialysis equilib- the pH is varied away from the isoelectric point.1° rium, equations 5 and 7 were used to calculate 4 Refractive Index Increment and 4*. For b o h e serum albumin (BSA) at ca. 5% concentration in 0.1 M sodium chloride, 4 was Specific and molar refractive index incremeiits in taken as 0.7343, the value found in water by Day- a multicomponent system may be defined by conhoff, Perlmann and MacInnes*; and cp’ and cp* were siderations analogous to those utilized above : for calculated by equations 6 and 7. The value of example, by adding a mass of component 2, defined was obtained from membrane equilibrium measure- in any convenient way, to a system a t constant ments of Scatchard, et aL16a t the pH corresponding temperature and pressure, we obtain the derivative to a protein species of zero valence; that is, under $2 E ( b n / b ~ 2 ) ~ , = ~ , ‘mk 2 / M z or more precisely in such conditions that equal numbers of moles of Na+ practice, the apparent value An/w2,where An is the and C1- are included in component 2*. For iso(9) G. S. Harned and B. B. Owen, “The Physical Cherniitry 01 ionic albumin of the molecular weight 69,000, found Electrolytic Solutions,” Reinhold Publ. Corp., New York, N. Y.,Third
- (;
(8) M. 0, Dayhoff. G. E Perlmann and D. A. hIacInncs, J . A m . Chem. Soc.. 74, 2.515 (1532).
edition, 1958, Chapter 8. (10) P. A. Charlwood, J. A m . Chem. Soc.. 7 9 , 776 ( 1 9 5 7 ) .
EDWARD F. C A S A S S A AND HENRYK EISENBERG
430
increment in refractive index relative to the solvent system. Alternatively we could add component 2 a t constant chemical potential p of the other components, considered as diffusible, to obtain #2f = (bn/bwz)T , = ~ \k2*/M2. These refractive increments are easily related by expressing the refractive index as a function of pressure, temperature and masses of all the other components in a fixed mass of the principal solvent, conveniently one kilogram of water
We then calculate the derivative with respect to m2 a t constant temperature and potential of the other components. The first term of the result (bn/bP)T,,,,(dPlbm2)T,,, depends on the isothermal compressibility of the system and the osmotic pressure, but we shall assume it to be negligible. Accordingly for the three-component system we write *2*
=
*2
+
h a
*3
where (bm3/bm2),,,~is the number of moles of component 3 which we include in the formulation of component 2*3. Putting this expression in terms of the specific increments 9 and recalling that M2* = M2 (1 E ) , we obtain the relation
+
This will be recognized as analogous to equation 6 for the partial specific volume.11 In studies of proteins and synthetic polymers it is usually convenient to use the specific increment for the macromolecular component but the molar quantity for component 3. In these terms equation 8 gives * - $2 w3 $2 1
and $2'
=
$2
+
+ IM3 + P
3
(9)
where [ sz (bm3/dw2),= t/M3. As we showed in reference 3, the light scattering equation for a three-component system reverts formally to the simpler relation for two components when expressed in terms of M2*, w2*, #2*, that is, in terms of component 2* (cf. also StigterI2and VrijI3). Since these quantities appear in the combination M ~ * ( I , ~ * ) ~= w zMz * ($z')2w2,knowledge of { is not required for determining Mz. In practice it is usual to employ specific refractive index increments An/c in terms of volume concentrations. One reason for doing this, aside from custom and the ease of measurement, is that empirically the specific increment in this form appears to be independent of c to quite high concentrations in (11) Although we chose to use a physical argument in discussing the partial specsc volume in order to arrive at equations 1 to 4 for the apparent values, relations among &, bt' and &* of the form of equations 5, 6 and 7, could equally well have been obtained by reasoning like that leading to equation 8. I t must be noted, however, that &' = &*(l f E ) E / p , a8 determined by density changes and equation 4, is not the same as ( b V / ' / b w z ) p= ~ &*(l E). (12) D Stigter, J. Phys. Chern., 64, 842 (1960). (13) A. Vrij, Thesis, Utrecht, 1959, as quoted by Stigter.1'
-
+
Vol. 65
solutions of proteins and organic polymers.*4 This observation implies that to a good approximation the refractive index of a mixture is a linear function of the composition by volume n =
nitjici
i = 1,2,3
Since we consider only solutions rather dilute in components 2 and 3, the constants n2, n3, need not correspond to the refractive indices of pure components. If the are independent of concentrations and if variations of cz = c are made without varying the relative proportions of components 1 and 3; that is, at constant molality of salt, it can be that
+
+
where ns = (nlfilcl n3fi3C3)/(fllCI Q3c3)is the refractive index of the solvent. It follows that if the refractive index increment in one solvent a is known, it can be calculated for another solvent b from equation 10 and a measurement of n s b - n,. In particular if the specific increment (dn/dc),, in water is available, that in salt solution is For proteins in water # is positive; hence increasing the salt concentration, and thus ns,should decrease $20. l6
The most careful determinations of refractive increments for proteins are perhaps those of Perlmann and Long~worth'~ who employed a differential prism method for direct measurements of small differences of refractive index. It is therefore of interest to examine some of their data in light of the discussion just given. I n two series of experiments these workers added successive small amounts of sodium hydroxide to solutions of bovine serum albumin and ovalbumin in water and followed the change in An with respect to pure water at a wave length of 5780 A. and temperature of 0.5'. They found An/c, c being the concentration in weight of isoionic protein, to increase linearly with the amount of added base in identical fashion for both proteins according to the relation An
(An
C = T ) , =(1~+ O.O45u/c) where u is the molarity of the base. They interpreted this as indicating a variation of the refractive increment with the charge of the protein ion. I n advancing a different interpretation we can ignore the fact of chemical reaction between protein and base and regard the system as a mixture of three components-water, protein, sodium hydrox(14) In the case of solutions of simple salts, An/c exhibits a linear dependence on c ' h to much higher concentration than doea An/w with respect to w'/% Here, where we are concerned with dilute solutions of component 2, any variation in salt concentration attending variation in CI is much too small for the square-root term to contribute to the variation in the refraction due to the salt. (15) E. F. Casassa, J . Phya. Chern., 60, 926 (1966). (16) Equation 11 cannot be regarded as universally valid. In polyphosphates [TJ.P. Strausa and P. L. Wineman; J . Am. Chern. Soc., 80, 2366 (1958)l $zo decreaaes with increasing sslt concentration qualitatively as predicted, but the observed effect seems much too great. (17) G.E. Perlmann and L. G. Longsworth, ibid., 70, 2719 (1948).
PARTIAL SPECIFIC VOLUMESIN MULTICOMPONENT SYSTEMS
March, 1961
ide. If we know the total refractive increment An with reference to water and that for a solution of sodium hydroxide of the same molality, we can calculate the specific increment for protein as $p0
= (An
- A~N~oH)/c
This expression implies no theory; it is simply a definition of #Zc for the process of adding isoionic protein to sodium hydroxide solution. Then for comparison we c m calculate #zc by substituting the measured value in water into equation 11. The results are exhibited in Table 11. The experimental values of #zc in the fourth column are now independent of sodium hydroxide concentration, except possibly rtt the highest concentrations; and the calculated values in the fifth column confirm that the concentration of sodium hydroxide is never great enough to cause a sensible decrease in +zc through narrowing of the refractive index difference between protein and the solvent mixture. Alternatively we can regard these systems as containing two components, water and a sodium protein salt to which, denoting the isoionic protein by I+, we could assign formulas such as Pr.(NaOH), or Pr.(?SaOH), - yHzO with the composition given by the amount of base added. Using the experimental values of A n and multiplying c for Pr by 1 0.040 u / c to obtain the weight concentration of Pr.(XaOH),, we have calculated for this, component as given in the sixth column of Table 11. It so happens, as is evident from the empirical result of Perlmann and Longsworth, that with the component defined in this way, #2 appears to be independent of the composition y. This calculation is like that of Perlmann and Longsworth except that while they included the contribution of NaOH to An, they did not include its corresponding contribution to the weight concentration. Had we chosen, on the other hand, to regard the solute as Pr.(NaOH), - yH20 and increased c by the factor 1 0.22 u/c, we would have obtained a +2c increasing linearly with the NaOH content a t about half the rate reported by Perlmann and Longsmorth . I n carrying through the calculations on the systems containing base we estimated the molar refractive increment of sodium hydroxide in water as 10.1 X loM31. mole-l. This figure was obtained from the limiting value a t infinite dilution, 9.50 X 10-3 for the NaD line a t 15°.18 The GladstoneDale rule and partial molar volumes calculated from data given by Harned and Owengwere used to convert to 0.5'; but the wave length dispersion correction was omitted as unimportant. Perlmann and Longsworth also measured refractive increments against dialysate for solutions of ovalbumin, bovine serum albumin and human serum albumin, in sodium chloride, and compared the results with solutions dialyzed against water. Averaged values of #2c' (bn/bc),, that they obtained are reproduced in the second column of Table 111. From equation 11 and the experimental $2 in water, we have calculated gZc in salt and thence from equation 9, (neglecting the distinction between t,b2 and #zc) the inoles of salt { bound per gram of
+
+
(18) C. C h h e v e a u . Compl. rend., 138, 1483 (1904).
43 1
TABLEI1 EFFECTOF ADDEDSODIUM HYDROXIDE O N THE REFRACTIVE INCREMENTS OF PROTEINS IN AQUEOCSSOLUTION" Protein conon., m1.-I 102
NaOH added, mole L-1 X 108
An X 10a (m.Hz0)
6.451 5.951 5.739 5.545 5.334 5.128 5.025
0 7.754 11.065 14.073 17.426 21.69 22.41
Ovalbumin 12.106 0.18766 11.234 .1875 0.1876 0.1878 10.868 .1875 ,1876 ,1879 10.523 ,1873 ,1876 ,1879 10.187 ,1878 ,1875 ,1885 9.77 .1864 .1875 ,1874 9.593 .1865 ,1875 .18iG
g.
x
$20
protein (calcd., eq. 11)
$20
protein
+L~C
NE proteinate
BSA 4.740 0 9.015 0.19019 4.340 8.502 8.239 ,1905 0.1901 0.1904 4.077 10.041 7.885 ,1909 .1901 ,1908 3.782 20.302 7.35 .1904 ,1900 ,1903 3.606 23.349 7.036 .1905 ,1900 ,1909 3.525 25.722 6.978 ,1927 ,1900 ,1923 a From data of Perlmann and Longsworth."
protein. Multiplication by the molecular weight of the protein-we used 67,000 for the serum albumins and 45,000 for ovalbumin-then gives the number of moles of salt bound per mole of protein as shown in the last column of Table 111. In calculating the refractive increment of NaCl for the experiment'al conditions, 5780 A. and 0.5", we employed the extensive refractive index data a t 25" of I < r u i ~inter'~ polating to the desired wave length. Then with the aid of a value for the temperature dependence given by Perlmann and Longsworth, we obtained finally ~3
=
11.36
- 1.17mi/2
a t 0.5". I n converting molarities a t 0.5"to molalities we again used partial volumes calculated from the parameters given by Harned and Owen. TABLE I11 SPECIFIC REFRACTIVE INDEX INCREMENTS FOR PROTEINS IN AQUEOUSSOLUTION" System
icd,
ml.
g.
iLtc
-1
Ovalbumin : in HzO in 0 . 1 M NaCl
0.1871 .1874
0.1871 .1863
BSA: in HzO in 0 . 1 M NaCl in 0 . 5 M NaC1
,1921 ,1938 .1948
.1921 ,1913 .1881
Human serum albumin: .1887 .1887 in HzO in 0 . 5 M XaC1 ,1918 ,1847 From data of Perlmann and Longsworth.''
rM2
4.5
5 26
26
0
At best, the binding calculated from these refraction determinations can be regarded as revealing no more than an order of magnitude. The results depend on very small differences and there are some unexplained variations in the measurements. Also, we assumed the protein species to be uncharged so that the binding involves only sodium (19) A. Kruis. Z. physik. Chem., 34B,13 (1936).
EDWARD F. CASASSA AND HENRYK EISENBE~ZG
432
Vol. 65
Mh(1 - 6hP) = Mz(1 - 6zp) chloride. Considering these uncertainties, the agreement with membrane distribution data and the subscript h denoting the hydrated species, are other measurements of Scatchard and ~ o - w o r k e r s ~not ~ ~pertinent ~ to the present problem. Whether it is on serum albumin is gratifying. Perlmann and based on a physical modelz1 or a thermodynamic Longsworth remarked correctly that the binding of argument,22the simple reasoning which leads to the the salt by serum albumin is reflected in an increase conclusion that sedimentation is unaffected by in &’ when sodium chloride is added to the protein hydration involves the assumption that the density in water. Independence of &‘ of the concentration of the adsorbed hydration layer is the same as that of salt (as in the case of ovalbumin) does not, on the of the solvent in bulk. Obviously this assumption other hand, indicate that there is no binding since in cannot apply in a multicomponent system in which the absence of interaction +2c‘ is equal to $2c, and the macromolecular solute interacts selectively with decreases with increasing amounts of salt, at least other species. if the assumptions implied by equation 11 are valid. Equilibrium Sedimentation in a Density Gradient Application to Sedimentation Equilibrium Ordinarily, equilibrium centrifugation experiI n reference 3, u-e pointed out that the entity designated here as component 2*, to a good approxi- ments are carried out at speeds of rotation so low mation sediments independently of component 3 in that ordinary salts do not sediment appreciably; an ultracentrifuge experiment; hence the equilib- hence the concentration of Component 3 remains rium distribution in the centrifugal field depends uniform in spite of the redistribution of component on the molecular weight M z * . Formulated in 2*. Interpretation is then simplified because M2* terms of this component (homogeneous in molecular and e2* are constant throughout the solution weight and density) the expression for sedimenta- column if our thermodynamic approximations8 are tion equilibrium reverts to the simple form appli- valid and if hydrostatic pressure effects can be neglected. Recently, however, Meselson, Stahl cable in a two-component system: i.e. and VinogradZ3tz4have studied sedimentation of nucleic acids in the presence of concentrated cesium chloride. At a fairly high speed of rotation this 1 r n z ( Z v z i * 2 / ~ ~ i P 2 2 * ) (12) heavy salt forms a density gradient, a t some point where mi,m2 are molalities of species and com- of which the centrifugal and bouyant forces acting ponents, Mz* is the molecular weight of component on the macromolecular species are in balance. 2*, DZ2* is the derivative of the logarithm of its The studies of the equilibrium distribution in the activity coefficient with respect to m2, x is the gradient have provided important information distance from the center of rotation, and w is the about these substances which, owing to their extremely high molecular weights, cannot be investiangular speed of the rotor. The correct measurement of M2* in the centrifuge gated by the ordinary equilibrium method. requires an accurate value of 82* (or of d*) the For the three-component system, the ideas predetermination of which in turn requires, in addition sented here concerning partial volumes and refracto density measurements, data on distribution of tive index increments are readily applied in interdiffusible species across an osmotic membrane or preting data obtained by this important new techequivalent information. It is easy to show, how- nique. With components defined by osmotic equiever, that a meaningful molecular weight for the librium, the distribution of component 2* does not macromolecular component can be obtained, even affect the distribution of the diffusible salt comwithout knowlcdge of the u*. Let us suppose ponent; but the uZi* depend on the salt concentrathat a protein is dialyzed against the supporting tion and therefore vary with position in the cell. electrolyte and the density of solution and dialysate Although equation 12 is still valid at any position 5, is measured. Then by some means of analysis this means that Mz*, i&* and p are also functions of (e.g., optical density, nitrogen content) the weight 2. It is still possible to use equation 13, however, in concentration corresponding to salt-free protein is determining the molecular weight M2, which of determined and used to calculate the apparent par- course does not vary with 2. Three independent tial volume. In other words, +’ is calculated. The sets of observations are required for a complete substitution of this value in equation 12 to deter- analysis of results: (a) the sedimentation experimine an unknown molecular weight leads in fact to ment on the polymeric material to measure its disthe molecular weight of the isoionic protein. To see tribution in the dense electrolyte a t equilibrium, (b) this, we need merely use equation 5 and the relation the sedimentation of the salt alone to determine its distribution in the centrifuge cell, and (c) measureAI2* = df2(1 f ) to find that ment of density difference between solutions of the llf2*(l = polymer and dialysate as a function of salt concentration in the dialysate in order to determine the = df2[1 k (1 - P/P.) - +’PI apparent partial volume 4‘ as a function of 2. In the limit as m2 approaches zero, p / p s approaches If we can presume that it is possible to use such a unity and
+
+
+
+ +
Mz*(l -
D**p)
rl L
=
dT(+!+ 91
Mz(l
- +‘p)
(13)
It may be mentioned that the many discussions of “hydration” in two-component systems, in which it is shown that (20) G. Scatchard and E. S. Black, J . Phys. CLm., LIS, 88 (1949).
(21) ,I. T. Edsall in “The Proteins,” Academia Press, Inc., New York. N . Y.,1953, Vol. IB, Chapter 7. (22) H. K. Schachmann, “Ultracentrifugation in Biochemistry,” Academic Preas, New York, N. Y., 1959. (23) M. Meaelson, F. W. Stahl and J. Vinograd, Proc. NaU. Acod. Sei., U. S.,CS, 581 (1957). (24) M. Meaelson and F. W. Stahl, ibid., 44, 671 (1968).
March, 1901
COUKTERCURRENT DISTRIBUTION OF CHEMICALLY REACTING SYSTEMS
low polymer concentration that the concentration dependent term on the right-hand side of equation 12 is negligible, the only problem remaining is that of evaluating the quantity (dm/dx2)mz.-1 = (dwz/dz2)w2-1 = (dc/dzz))c-l
in equation 12. The last equality becomes exact in the limit of small c. When light absorption by the polymer species can be measured, c can be determined directly :is a function of x. If the datum obtained is the difference in refractive index between the solution and the reference solvent a t the same level x (e.g., as by interferometry) the concentration is obtained from this difference An, by An = #‘2w2 = #20’c where t,hz’ and I,&‘ are now functions of x through their dependence on salt concentration. It is thus necessary to measure the refractive increment of polymer solution referred to dialysate as a function of salt concentration. If the refractive index gradients rather than indices are measured, as is the case for the schlieren optical system, the concentration profile in the centrifuge can be obtained if $2t or $2c’, together with \E3 or are known, but in general a numerical integration is required. We conclude, therefore, that even though the analysis of sedimentation equilibrium in a density gradient is complicated by the variation of salt concentration through the cell and therefore by the variation in interaction between salt and polymeric
433
component, it is still possible to determine a molecular weight unambiguously without explicit knowledge of the amount of salt bound. The above remarks are confined to the case of a single macromolecular component. For a macromolecular solute heterogeneous in molecular mass, the expressions for sedimentation equilibrium in terms of the starred components assume the form22-26applicable to a mixture of species of uniform partial volume in a single solvent, provided iP is the same for all solute components. The question of heterogeneity with respect to partial volume, which does arise in the case of nucleic acids, has been discussed recently by Baldwin. NOTEADDEDIN PRooF.-It is pertinent to make note here of two emendations to the previous paper, reference 3. First, the indicated derivation of the osmotic relation, equation 5, is faulty in not including the effect of pressure on the potential of component 2. The result given is correct, however, m a limiting law for dilute solutions (which is all we actually require). As well as yielding the proper limit for T[/m2 when ma is zero, it gives the eecond virial coefficient correctly except for a term smaller than errors of measurement in protein solutions. Finally, equation 8 of the same paper should read 2 H’V,M*c* e H’V,Mc-
(25) R. J. Goldberg, J . Phya. Chem., 67, 194 (1953). (26) R. L. Baldwin, Proc. N d l . A d . Sei., 46, 939 (1959).
COUXTERCURRENT DISTRIBUTION OF CHEMICALLY REACTING SYSTEMS. I. POLYMERIZATION BY J. L. BETHUNE~ AND GERSON KEGELES Department of Chemistry of Clark University, Worcester, Mass. Received August $4, lS60
Although the simple extraction behavior of systems undergoing chemical reactions has been widely studied, general quantitative predictions of the expected behavior of such systems during countercurrent distribution experiments have been lacking. The recent theories of Gilbert and Gilbert and Jenkins for the separation of schlieren peaks in ultracentrifugation and electrophoresis, assuming instantaneous chemical re-equilibration between species, have suggested the possibility of hitherto unexpected behavior in countercurrent distribution experiments as well. The present study, involving the use of a high speed digital computer to solve the material balance equations involved, serves to predict the expected behavior of certain polymerizing s,ystems over a wide range of the parameters governing the equilibria. One prediction is that a system undergoing trimeririation may show quite different behavior from that of a dimerizing system.
bution of penicillins, hm been reported briefly. Introduction Most interpretations of the concentration us. The results of this calculation, done for 24 transtube number patterns obtained from the process fers, indicated that the position of the individual of countercurrent distribution have been based penicillins in a countercurrent distribution train upon the assumption of constant partition ratios, could not be predicted by the use of a partition which should be expected to hold for dilute solu- ratio measured by simple extraction a t a single tions of solutes not undergoing chemical reac- concentration. Calculations have also been done,’ tions2+ One quantitative calculation16 showing with the aid of an analog computer, for cases of the possible effect of dimerization upon the distri- continuous countercurrent extraction, in which the effective partition ratio may be represented by a (1) Submitted by J. L. Bethune to the Faculty of Clark University power series in concentration. Note has also in partial fulfillment cif the requirements for the degree of Doctor of been made of the effects on countercurrent distriPhilosophy. June, 1961. bution behavior of non-linear partition isotherms5 (2) E. Stene. Arkis Kemi, Mineral. Geol., Ala, No. 18 (1944). (3) B. WillLmson and L. C. Craig, J . Bioi. Chem., 168, 687 (1947). as well as of such possible reactions as X a Y b $ (4) R. M.Bock, J . .Am. Chem. Soc., l a , 4269 (1950). aX For non-linear partition isotherms (5) L. C. Craig and D. Craig in A. Weissberger. “Technique of
+
Organic Chemistry,” Vol. 111, Interscience Publ., New York, N. Y., 1950. (6) W. R. Boon, Analyst. 73, 202 (1948).
(7) A . Acrivos and N. R. .4rnundson, Ind. Eng. Chem., 46, 467 (1953). (8) E. H.Ahrena and L. C. Craig, J . B i d . Chem., 196, 763 (1952).
