Jan. 5 , 1953
CONSIDERATION OF THE HYDRODYNAMIC PROPERTIES OF PROTEINS [CONTRIBUTION FROM THE
179
DEPARTMENT OF CHEMISTRY, CORNELL UNIVERSITY]
Consideration of the Hydrodynamic Properties of Proteins1$2 BY
HAROLD A. SCHERAGA A N D LEOMANDELKERN~ RECEIVED FEBRUARY 13, 1952
A treatment of the hydrodynamic properties of proteins is presented wherein the molecule is assumed to possess some degree of flexibility and solvation. The configuration of the protein molecule is represented in terms of an effective hydrodynamic ellipsoid whose axial ratio and size may be determined from accurate measurements of sedimentation constant, intrinsic viscosity, molecular weight and flow birefringence, all made in the same solvent.
Introduction
the value of p in the correction term for the bound The configurations of protein molecules have water. Thus, erroneous interpretations of experibeen determined from hydrodynamic measurements mental results are possible if the molecular domain by use of the theories of Simha4 and Perrin506 for is identified with the effective hydrodynamic volrigid ellipsoids of revolution. In the p a ~ t 7 ~ ~ume ~ ~ ~and ” J if the definition of w is retained. More the problem has been treated by considering a property w should be given that value which allows boundary to be assigned to the particle and approx- the observations to be reconciled with rigid particle imating this molecular domain by an ellipsoid of hydrodynamics. Since, in general, the value of w revolution. The volume of this domain was re- would have no direct relation to the amount of water garded as the sum of m / N and Mw/Np where M in the domain of the molecule, the introduction of V is the anhydrous molecular weight, N is Avogadro’s in the expression for V, becomes superfluous and number, V is the partial specific volume of the also misleading. lZb This is most clearly seen in the graphical methanhydrous protein, p is the density of the solvent and w is the g. of solvent bound per g. of dry od7s9which requires that curves of axial ratio as a protein; w has been taken to be the amount function of w for observed values of intrinsic of water associated with the protein molecule in viscosity and frictional coefficient intersect a t the s o l ~ t i o n . ~ It ~ ’has ~ been further assumed that the correct value of w and axial ratio, thus giving conhydrodynamically effective volume, Vet is equal to sistency between the two hydrodynamic measuref i ( 1 w/Gp)/N, ;.e., the effective volume has ments. However, in all cases cited71gthese curves been identified with the molecular domain of the could be made to intersect only by imposing large protein molecule in solution. There is, however, experimental errors on the data. In one caseno a priori reason to assume this equality, for it pepsin-the curves do not cross ai! all. If one exneglects possible flow of solvent through the do- amines the data closely, it can be seen that these main, deviations of the shape of the domain from curves can be made to intersect a t negative values that of an ellipsoid of revolution, deformation of the of w without introducing large experimental errors domain by the hydrodynamic forces, selective ad- in the data. The negative values of w, which are sorption from mixed solvent, electrostriction, and incompatible with the original definition of w, similar effects. Besides these problems, it does arise because of the arbitrary assignment of a pornot, in general, appear reasonable to obtain the tion of the effective volume to the term M i / N . Besides the difficulty involved in the interpretaeffective volume in the manner indicated, i.e., by expressing hydration as an excess over MG/N. tion of w (as originally defined), the tendency seems The value of V may be positive, negative, or zero, to have been to avoid determining it altogether. as is the case, for example, in aqueous solutions of Instead, as indicated by numerous examples in the magnesium sulfate.“ While such a behavior has’ literature, arbitrary values such as zero hydranot been observed for proteins, nevertheless, the tion 13,15,17--21 or 20-30ojO hydrationl4,1 6 8 1 7 have been value of V has to reflect the interaction with the assumed, thereby confusing the relative contribusolvent, making it impossible to identify M i / N (12b) The degree of hydration of proteins, obtained from hydrowith the volume of the anhydrous proteinlZa; dynamic measurements, has been compared to the hydration of protein in addition, electrostriction would render unknown crystals. It has even been suggested “that the degree of hydration of a
+
(1) Part of this work was carried out in connection with project N6-onr-26414 supported by a contract between Cornell University and the O5ce of Naval Research. (2) Presented before the Division of Biological Chemistry at the 121st Meeting of the American Chemical Society, Milwaukee, Wisconsin, April, 1952. (3) Rubber Section, National Bureau of Standards, Washington 25, D. C. (4) R. Simha, J . Phys. Chem., 44, 25 (1940). (5) F. Perrin, J . 9hys. radium, [7] 6,497 (1934). ( 6 ) F. Perrin, ibid., 171 7, 1 (1936). (7) J. L. Oncley, A n n . N . Y . Acad. Sci., 41, 121 (1941). (8) T. Svedberg and K. 0. Pcdersen, “The Ultracentrifuge,” Oxford University Press, London, 1940. (9) J. W. Mehl, J. L. Oncley and R. Simha, Science, 81,132 (1940). (IO) E. J. Cohn and J. T. Edsall, “Proteins, Amino Acids and Peptides,” Reinhold Publishing Corp., New York, N. Y..1948. (11) G. N. Lewis and M. Randall, “Thermodynamics,” McGrawHill Book Co., Inc., New York, 1923, p. 37. (12a) Ref. 10, p. 516.
protein is essentially the same in solution as in the crystal.”l@sd This conclusion is based on the interpretation of w cited above. See 71, 536 also H. R. Schachman and M. A. Lauffer, THIS JOURNAL, (1949). (124 T. L. McMeekin, M. L. Groves and N . J. Hipp, ibid., 71, 3662 (1950). (12d) A. D. McLaren and J. W. Rowen, J . Polymer Sci., 7 , 314 (1951). See for example: (13) H. Neurath and A. M. Saum, J . Biol. Chem., 1111, 347 (1939). (14) G. R. Cooper and H. Neurath, J . Phys. Chem., 47,383 (1943). (15) S. Brohult, J . Phys. Colloid Chem., 61,206 (1947). (16) J. L. Oncley, G. Scatchard and A. Brown, ibid., 61, 184 (1947). (17) J. T. Edsall, J. F. Foster and H. Scheinberg, THISJOURNAL, 68, 2731 (1947). (18) H. Kahler, J . Phys. Colloid Chcm., 61,676 (1948). (19) J. F. Foster, ibid., 68, 175 (1949). (20) E. H. Mercer and B. Olofsson, J . Polymer Sci., 6, 671 (1951). (21) C. S. Hocking, M. Ladtowski, Jr., and H. A. Schcraga, THIS JOURNAL, 74,775 (1952).
180
HAROLD A. SCHERAGA AND LEO MANDELKERN
VOl. 75
tions of both the solvation and the asymmetry to nized that only the dimensions of the effective ellipthe frictional behavior of the protein molecule. I n soid (for whatever degree of flexibility or permeafact this procedure is equivalent to assuming a vol- tion by the solvent the molecule may possess) may ume and then explaining the observations on the be obtained from the hydrodynamic measurements in dilute solution and not those of the molecular basis of asymmetry. A different procedure is considered here in which domain. In addition, relationships between axial the configurations of globular type protein mole- ratio, sedimentation constant, intrinsic viscosity cules in solution are represented in terms of an ej- and molecular weight are developed in a more confective hydrodynamic ellipsoid. The size and shape cise and systematic manner than heretofore. The of this ellipsoid are defined as those which allow the basis for determining the size of the effective ellipexperimental hydrodynamic observations to be soid and for distinguishing between a prolate and treated by the hydrodynamic equations developed oblate one is explicitly given in terms of two indeby Simha and Perrin for rigid ellipsoids of revolu- pendent hydrodynamic measurements. It is also tion even though the actual molecular configuration shown that there is a similarity in hydrodynamic may not be a rigid one; Le., there is a rigid ellip- behavior between globular type proteins and flexible soid (the effective or equivalent hydrodynamic chain molecules. ellipsoid) which exhibits the same hydrodynamic beTheory havior as the solvated protein molecule in solution. Intrinsic viscosity-translational coefficient: The size and shape of this rigid ellipsoid are such as Whatever the actual configuration of africtional protein molecule may will take into account possible flexibility of the mol- be in a particular solvent a t a given pH and ionic strength, ecule, permeation of the molecule by the solvent, there is associated with it an effective hydrodynamic ellipetc. This procedure is analogous to the use of an soid of volume V. and axial ratio 9, which will account for the frictional effects arising from the presence of the moleeffective hydrodynamic spherez2for flexible, solv- cule in the solvent. Ve and #, of course, depend on the ated polymers in consideration of the Einstein- solvent, temperature, flH and ionic strength. Stokes relations for spheres. In general, the relaIn the absence of soluttsolute interaction, i.e., a t infinite tionship between the effective ellipsoid and the dilution, the specific viscosity, vsp, can be written as a prodof three factors, (1) the number of particles per cc., actual protein configuration in solution is not uct (2) the effective volume of an individual particle, and (3) known, nor need it be specified. In the very spe- a shape factor. The number of particles per cc. is equal to cial case, where the molecules are completely rigid Nc/100M where N is the Avogadro number, c is the dryand impermeable to solvent, the effective ellipsoid weight concentration in g./lOO cc., and M is the unhydrated molecular weight. The intrinsic viscosity, [ v ] , may then be and the actual Configuration could coincide. written as It is worth noting that neither the configuration [VI = (Nv/100)(Ir,/M) (1) nor the effective hydrodynamic ellipsoid of protein where v is a shape factor which depends on the axial ratio, P , molecules in the anhydrous state can be determined of the effective hydrodynamic ellipsoid which is assumed to be by measurements of intrinsic viscosity, sedimenta- an ellipsoid of revolution. The quantity Y has been caltion and diffusion, flow birefringence, etc., in pro- culated by Simha' for the condition of prevalent Brownian tein solutions because it has not yet been possible motion. Values of Y as a function of p for prolate and obellipsoids have been tabulated by Mehl, Oncley and to calculate the solute-solvent and intramolecular late Simha .'interactions, as has been done for randomly coiled Accordinn to Perrin.6 the frictional coefficient of the efchain polymers.22 As a consequence of these in- fective hyd;odynamic ellipsoid is given by the equation teractions, it is reasonable to expect that a protein f/fQ= 1 / F (2) molecule should not be completely rigid but flexi- where ble enough to swell to expanded configurations in p=/t 1 +d m for prolate various solventsz3and exhibit a hydrodynamic beF-In ellipsoids P 0 (P < 1) havior similar to that of randomly coiled chain molP'/l for oblate ecules. Thus if an unhydrated protein were placed P arc tan , / F j ellipsoids in an aqueous solution a swelling effect should oc1( P > 1) cur. Evidence for such a phenomenon has been f is the mean frictional coefficient a t infinite dilution, /O is reported on the basis of low angle X-ray scattering frictional coefficient of a sphere of radius a. having the from protein s o l ~ t i o n s . ~ ~For ~ ~the b case of south- the same volume as the equivalent hydrodynamic ellipsoid and ern bean mosaic virus almost a two-fold increase in obeying Stokes' law, and p = b/a where a and b are the volume has been found for the virus in solution com- semi-axes of the equivalent ellipsoid, a being the semi-axis revolution and b the equatorial radius. It should be pared to that in the dry ~ t a t e . * ~Similarly, a there of noted that our use of fo for a hydrated sphere differs from is an indication that serum albumin swells in solu- the usual interpretation of f~ in terms of an unhydrated sphere.' ti~n.~~b Since f e = 67nla0, where 9 is the viscosity of the solvent In the present treatment it is, first of all, recogf = Guvao/F (3) (22) P. J. Florp and T. G. Fox, Jr., THIS JOURNAL, 73, 1904 (1951). (23) The type of flexibility envisaged for proteins is that which allows them to imbibe solvent and deform from the configuration in which they exist in the anhydrous state. In general, the globular type urotein molecule is not necessarily to be thought of in terms of the random dight configuration of a flexible chain polymer. However, as in the case of polymerr, the swelling can involve immobilization of solvent within the molecule, in excess of stoichiometric quantities. (24a) B. R. Leonard,Jr., J. W. Anderegg, P. Raesberg, S. Shulman, and W. W. Bee", J . Chum. Phys., 19, 793 (1951). (24b) J. W.Anderegg, W. W. Beeman, and S Shulman, Phys. Rcu., 87, 186 (1952).
The value of a0 is (3/4s)%( V.)'/:. Therefore f = ( 162xt)'/:( V,)'/r?/F (4) In other words [VI and f both depend on p and V.. A solution of the simultaneous equations (1) and (4)would then give both p and Ve. The Svedberg equation for the sedimentation constant a t infinite dilution is8 where
p
s Mh(1 - hP)/Nf is the density of the solution and
(5) 6h
is the partial
Jan. 5 , 1953
CONSIDERATION OF THE HYDRODYNAMIC PROPERTIES OF PROTEINS
specific volume of the hydrated particle. Since eqn. (5) is being considered at infinite dilution Mh(1 VhP) may be replaced by M(1 V p ) , where 5 refers to the unhydrated particle. This relation is true for a binary system and holds approximately for three component system^.^*^^*^ Therefore eq. (5) becomes s = M ( l - 5p)/Nf (5') Combination of eqs. (1). (4) and (5') gives @ G N s [ ~ ] ' /7/Mz/a(l : fp) yFv'/: (6) where y = N%/( 16200 +)I/: and fl corresponds to the function W / : P - l used in a similar discussion of flexible chain molecules."*2* @ is determinable from the sedimentation constant a t infinite dilution, the intrinsic viscosity, the molecular weight, the partial specific volume, the solution density, and the solvent viscosity. It should be emphasized that all measurements must be made in the same solvent in order that p and V, shall not vary. For flexible chain polymers +l/:P-l should be a universal constant independent of the nature of the polymer or of the solvent and temperature. This conclusion is reached on the assumption that, as far as frictional effects are concerned, the polymer molecule may be replaced by an effective hydrodynamic sphere. The constancy of 9'/aP-l for flexible chain polymers has been confirmed by experiment .nJ9 For proteins, where an effective hydrodynamic ellipsoid is used, @ should depend only on p according to eq. (6). This dependence can be calculated from the functions of Perrine and Simha' and the results are shown in Table I. For oblate ellipsoids p is essentially independent of axial ratio whereas for prolate ellipsoids it varies considerably with axial ratio, providing a criterion to distinguish between prolate and oblate ellipsoids. If accurate data are available to calculate p it is thus possible to determine the axial ratio and dimensions of the effective hydrodynamic ellipsoid as discussed below.
-
-
-
TABLE I DEPENDENCE OF p ON AXIAL RATIO FOR PROLATE AND OBLATEELLIPSOIDS' I/P = a/b
Prolate @
x
10-1
fi
-
Oblate b/a
t9
x
2.12 1 2.12 2 2.13 2 2.12 3 2.16 3 2.13 4 2.20 4 2.13 5 2.23 5 2.13'. 6 2.28 6 2.14 8 2.35 8 2.14 10 2.41 10 2.14 12 2.47 12 2.14 15 2.54 15 2.14 20 2.64 20 2.15 25 2.72 25 2.15 30 2.78 30 2.15 40 2.89 40 2.15 50 2.97 50 2.15 60 3.04 60 2.15 80 3.14 80 2.15 100 3.22 100 2.15 200 3.48 200 2.15 300 3.60 300 2.15 a v was obtained from Mehl, Oncley and Simha,' and Perrin's function F from Svedberg and Pedersen,O p. 41. 1
As can be seen in eq. (1) and eq. (4), [7]andf depend on both p and V.. Therefore, p cannot be determined from (25) W. D. Lansing and E. 0. Kraemer, THISJOURNAL, 68, 1471 (1938); see also reference 8, pp. 62-66. (28) H.K.Schachman and M. A. Lauffer, ibid., 72, 4266 (1950). (27) L. Maadelkem and P. J. Flory, J . Chcm. Phys., BO, 212 (1952). (28) A relation analogous t o eq. (6) is obtainable from the translational diffusion constant at infinitedilution, D,using D kT/fand eq. (1) and (4) giving 6 G D[7]'/:M'/+/RT yFv'/:. (29) L. Mandelkun, W,R. Krigbaum, H. A. Scheraga and P. J. Flory, J . Chem. Phys., 20, 1392 (1952).
-
-
[7] or f alone but only by a combined measurement of [ q ] andf. The pair measurement, [ ~ l - f ,then gives both p and V. for the solvated protein. In other words, high viscosity and high frictional coefficient have usually been attributed to high asymmetry, whereas, as can be seen from eq. (J) and (4), increased effective volume, rather than (or in addition to) increased asymmetry, could be just as important a factor. Intrinsic viscosity-rotary frictional coefficient: The rotary diffusion constant of protein molecules can be treated in a similar manner. The rotary frictional coefficient, -?,for ellipsoidal particles a t infinite dilution has been considered by Perrin6 and is given by the equation $0
where PY2
JZ2 3 =3 -
- P2)
4p""
4-1
(7) J
+
1 vT=-p In .(1 - P')
- 2, arc tan 4 (P' - 1)
F i
Pz
for prolate ellipsoids
$2
for oblate ellipsoids
+
2 Again, lo refers t o a sphere of the same volume as the equivalent hydrodynamic el1ipsoid.m Values of f/ro are given in Table I1 for various values of p . Since lo = 8 ~ 9 7 5=~ 6qV. (8) the rotary frictional coefficient may be written
S 67Ve/J The rotary diffusion constant,a18, is related t o f . 8 = kT/f = kTJ/6qVe
(9) (10)
TABLE I1 DEPENDENCE OF
r/co AND 8 ON AXIALRATIOFOR PROLATE AND
OBLATEELLIPSOIDS
Prolate
10-
181
1/P = a/b
1 2 3 4 5 6 8 10 12 15 20 25 30 40 50 60 80 100
200 300
S/h 1 1.505 2.340 3.395 4.638 6.061 9.401 13.37 17.94 25.86 41.80 61.05 83.45 137.3 202.9 279.9 465.9 694.5 2428 5085
Oblate 8
2.50 1.93 1.57 1.37 1.25 1.17 1.07 1.02 0.990 .959 .923 ,904 .893 .880 .870
.865 .859 .854 .845 ,841
fi = b/a
1
2 3 4 5 6 8 10
12 15 20 25 30 40 50 60 80 100 200 300
t/b 1 1.132 1.464 1.843 2.240 2.645 3.471 4.305 5.143 6.407 8.519 10.64 12.75 16.99 21.24 25.48 33.96 42.44 84.86 127.3
a 2.50 2.52 2.34 2.20 2.10 2.03 1.93 1.87 1.83 1.78 1.74 1.71 1.69 1.67 1.65 1.64 1.62 1.62 1.60 1.60
Combination of eq. (1) and ( I O ) gives (11) (30) For rotational diffusion of ellipsoids of revolution there are two rotary frictional coefficients corresponding to rotation about the aand b-axes, respectively. However, rotation about the a-axisdoes not affect the orientation of the particle. Therefore, since only rotation about the b-axis is observable in a flow birefringence experiment, only the rotary frictional coe5cient for this case is considered here. (31) The relaxation time, I, determined from dielectric dispersion measurements" is related to 8 by the equation 7 = l/2e. (32) J. L. Oncley, Chcm. Reus., 80, 433 (1942).
6, like 0, should depend only on p for proteins whose configuration is represented in terms of a.n effective hydrodynamic ellipsoid. 6 is determinable from the rotary diffusion constant at infinite dilution (by means of flow birefringence or dielectric dispersion measurements), the intrinsic viscosity, the molecular weight, and the solvent viscosity. Values of 6, calculated from J and Y are given in Table I1 for various values of p . The identical form (except for numerical coefficients) of eq. (11) for several models for macromolecules has been discussed by several authors.a*-s' For effective ellipsoids the explicit dependence of 6 on axial ratio has not been given, hrretofore, as it is hy the factor Ju tnbulated in Table 11.
Discussion From the foregoing it is apparent that it is possible to calculate the dimensions of the effective hydrodynamic ellipsoid from the types of experiments considered here. Equation (6) provides a basis for considering the determination of the axial ratio. For /3 > 2.15 X lo6 it is possible to rule out an oblate ellipsoid from consideration and find the axial ratio of the prolate ellipsoid from the data of Table I. This value of p determines Y which together with the experimental values of intrinsic viscosity and molecular weight determines Ve according to eq. (1). Alternatively, V, can be obtained from F and the sedimentation constant according to eq. (4)and (5'). a and b are then determinable from p and Ve. An independent determination of a can be made by means of flow birefringence measurements which give the rotary diffusion constant 0. For the known value of $J (and, therefore, J) Ve is determinable from eq. 1.57 would (10). It may be noted that a 6 value also rule out an oblate ellipsoid from consideration. For 0 I2.15 X lo6 the shape may be either a prolate ellipsoid of axial ratio 24 or an oblate ellipsoid of any axial ratio. In such a situation flow birefringence measurements can be helpful in determining the dimensions especially since p , for an oblate ellipsoid, is not determinable with accuracy from p. Even though /3 is independent of p for oblate ellipsoids, 6 does vary sufficiently with p so that the axial ratio is determinable from 6. Alternatively, use may be made of the fact that the rotary diffusion constant of an oblate ellipsoid5 depends only on b for b >> a. b is thus determinable from 0. Once b is determined a can be calculated from the intrinsic viscosity according to eq. (1) or from the sedimentation constant according to eq. (4) and (5'). The experimental value of 6 can also be examined from the point of view of a prolate ellipsoid of axial ratio 2s to decide which type of ellipsoid is consistent with the observed 0 and also the various experimental data used in the original calculation of p. For molecules which are not large enough it may be dificult to determine 8. ,R is more amenable to accurate determination, especially since 6 is not a very sensitive function of p for l i p > 15. Thus eq. (6) and the auxiliary eq. (11) provide a (33) R. Simha. "High Polymer Physics," Chemical Publ. Co., Xew York, N. Y., 1948, p. 398. (34) J. Riseman and J. G. Kirkwood, J . Chcra. Phys., 17,442 (1949); 18,512 (1950). Also see footnote 6 in the 1949 paper for quotation of a personal communication from Simha about a similar conclusion. (35) J. G.Kirkwood and P. L. Auer, ;bid., 19. 281 (1951). (36; N. Saito, J . Phys. .?or. J a p a n , 6 , 302 (1951).
basis for correlating the hydrodynamic behavior of proteins with an effective hydrodynamic ellipsoid whose dimensions are calculable. Of course, if the protein is anhydrous in solution then the effective ellipsoid applies to the anhydrous parti~le.~'In general, though, the protein will be considerably hydrated. In such cases only the effective ellipsoid for the hydrated and not the anhydrous particle can be determined. Table I11 gives the values of /3 for several proteins to show that the numerical values of 0fall in the range indicated in Table I. VALUESOF p
FOR
Substance
TABLE I11 SEVERAL PROTEINS AND POLYMERS~ [VI
PM
x
10"
fO5
M
Egg albumin 0.043 3.55 44,000 2.40 Horse serum albumin ,049 4.46 70,000 2.33 Hemoglobin .040 4.48 63,000 2.34 $mandin ,052 1 2 . 5 330,000 2.30 Octopus hemocyanin ,067 49.3 2,800,000 2.36 Gliadin ,105 2 . 1 27 ;500 2.39 Homarus hemocyanin ,047 22.6 760,000 2.28 Helix pomatia hemocyanin .047 98.9 6,600,000 2.36 .067 7.1 167,000 2.27 Serum globulin 630,000 2.35 Thyroglobulin ,071 19.2 Lactoglobulin .045 3.12 41,500 2.28 Pepsin ,039 3.3 35,500 2.52 Tobacco mosaic virus ,285 185 33,200,000 2.63 2 ,3 Polystyrene in toluene 2.G Polystyrene in methyl ethyl ketone 2.7 Cellulose acetate in acetone Polyisobutylene in cyclohexane 2.5 Polysarcosine in water 2.3 a For the first 12 proteins, intrinsic viscosities are from Polson,a* and sedimentation constants and sedimentation velocity-diffusion molecular weights are from Svedberg and Pedersen.8 As pointed out by Cohn and EdsalllO other values have been reported for these proteins. Data for tobacco mosaic virus are from Lauffer.38 See reference 29 for the polymer data.
It should be kept in mind that all the quantities appearing in must be determined in the same solvent. This may not be true for all the data listed and, therefore, no further calculations of TABLE IV CALCULATION OF DIMENSIONS OF EFFECTIVE ELLIPSOIDS IX UREASOLUTIONS OF HORSESERUMALBUMIN USIXGDATA OF KE-CRATH AND S A T J M AT~25" ~ Urea concn., M
[VI
D X lo7 9 , X 10-6
r,a
A.
0.049 6.85 2.23 a = 82, b = 1 6 . 4 .050 6.20 2.04 38 .056 6.08 2.07 39 1.5 3 .0 ,065 5.69 2.04 41 4.45 1.98 51 4.5 .123 6.0 ,147 4.27 2.01 54 6.66 ,170 4.15 2.05 56 a For the native albumin the axes of the effective ellipsoid are listed. All other values in this column are the radii of the respective effective hydrodynamic spheres. 0 0.5
(37) For example, tobacco mosaic virus has negligible hydration.38 (38) M. A. Lauffer, THIS JOURNAL, 66,1188 (1944). (39) -4.Polson. Kolloid Z . , 88,51 (1939).
Jan. .?, 1053
CONSIDERATION OF THE HYDRODYNAMIC PROPERTIES OF PROTEINS
183
dimensions have been made with these values.40 Ooty and kat^'^ concluded from light scattering Data for several chain type polymers are also studies of urea-water mixtures of bovine serum included41 in Table 111. The similar values of albumin that the “principal change undergone by p for proteins and polymers seem to indicate that the serum albumin molecule in concentrated urea the same kind of hydrodynamic treatment is ap- solutions is that of approximately isotropic swellplicable to both kinds of molecules and that the ing” with preferential adsorption of either urea or point of view presented here appears reasonable. water depending upon the pH. As another example we may cite the recent flow Several examples, where the effects discussed a ~the ~ here are large and rather striking, will serve to birefringence studies of Foster and S a m ~ on emphasize the need for considering the effective denaturation of ovalbumin in the presence of urea. Here again, interpretations were based on rotary hydrodynamic ellipsoid. Neurath and SaumIa have investigated the diffusion constants alone instead of a pair measuredenaturation of horse serum albumin by performing ment of [ q ] 4 to obtain 6. It was concluded that, parallel diffusion and viscosity measurements in in those cases where no aggregation occurred, the solvents containing various amounts of urea. The denaturation involved essentially an intramolecular intrinsic viscosity increased and diffusion coefficient unfolding. Foster and Samsa reported their data decreased with increasing urea concentration. ’ in terms of lengths which are presumed to have This was interpreted in terms of increasing asym- been calculated from observed 0-values. As can metry (from about 5 : 1 to 20: 1) arising from the be seen in eq. (lo), decreased values of 0 upon uncoiling of polypeptide chains, the axial ratios denaturation could arise either from increased being obtained from the intrinsic viscosity or diffu- asymmetry or increased effective volume or both. sion measurements alone. If they are combined, Thus the interpretation that decreased 0 means and /3 computed as indicated in footnote 28, signifi- chain unfolding is not necessarily correct. A pair cantly different results are obtained as shown in measurement of [q]-0in the same solvent would be required to answer this question. As an illustraTable IV. Within the experimental error, all values of 6, tion, particles whose effective ellipsoids have except for the native albumin, are 2.12 indicating dimensions of a = 300A., b = lOA.and a = 228& that p 2 1/2. Thus, an asymmetrical prolate b = 57&, respectively, would both have the same ellipsoid is completely ruled out. With these data rotary diffusion constants. Therefore, since the alone, a further distinction cannot be made be- larger effective volume of the second particle could tween a sphere or an oblate ellipsoid. Flow bire- be the result of increased solvation, the flow birefringence measurements on the same systems, mak- fringence measurements on urea-denaturated ovaling use of 6, would help. However, interpreting bumin cannot be unambiguously interpreted in terms of chain unfolding. the data in terms of a sphere, for the present, the The use of an effective hydrodynamic ellipsoidal radii of these effective spheres are listed in the last model thus permits the protein molecule to be concolumn of Table IV. The denaturation process sidered as a partially flexible, solvated one with thus appears to involve an increased effective vol- its hydrodynamic properties related to the axial ume due to swelling instead of uncoiling of poly- ratio of the effective ellipsoid by eq. ( 6 ) or (11). peptide chains. It is of interest to point out that It is therefore.possible in conjunction with eq. (40) For bovine fibrinogen” a 6 value of 0.99 is obtained from [ q ] = (l), (4) and ( 5 ’ ) , or (lo), to decide between an 0 25, @/T = 1.34, and M = 407,000. oblate and prolate shape and determine the dimen(41) It should not be inferred from the values of p for polymers t h a t sions of the effective ellipsoid. It remains to be the chain molecule must be considered in terms of a n effective ellipsoid. I n the treatments previously given for the intrinsic viscosity*% seen whether suitable variation of the solvent medium can give rise to changes in the soluteand for the frictional coefficients’ the assumption was made in each case t h a t the radius of the effective hydrodynamic sphere is proporsolvent and intramolecular interactions which will tional to an average linear dimension of the chain molecule in solution, be reflected in variations in a/b and Ve. This would and t h a t the constants of proportionality applying t o t h e viscosity be analogous to the variation of the root-meanand to the frictional coefficient, respectively, are the same for all high square end-to-end distance of a flexible polymer polymer chains. These constants were not assumed to be identical, Ewever. On this basis, p should be a universal constant as is conchain caused by such interactions and would promed by experiment.21’29 If it is further assumed t h a t the equivalent vide a deeper insight into the problem of the conspheres for the intrinsic viscosity and for the frictional coefficient are figuration of protein molecules in solution. Also, identical in size then p should be equal to 2.12 X 108 in disagreement this approach will be helpful for denaturation with experiment. The hydrodynamic theory of Kirkwood and Riseman,‘Z on the other hand, gives B = 2.5 X 106. It would appear t h a t studies. For example, as can be seen from eq. the assumption of the identity of the equivalent spheres for chain (1) and as illustrated in the examples cited above, molecules is not valid. This is not an unexpected result since the averthe increased viscosity usually observed in deage distribution of segments for a chain polymer molecule is approxinaturation need not necessarily imply increased mately Gaussian and, presumably, the linear parameter of the Gaussian distribution required for the intrinsic viscosity is not identical axial ratio but possibly increased effective volume with t h a t required for the frictional coefficient. For proteins, on the of denaturated proteins. Thus, correct values of other hand, i t is reasonable t o assume t h a t the distribution of segments p and Ve in such cases can be obtained by the along the axes of the molecule is rectangular so t h a t one would expect method indicated here, and a decision made as to the equivalent ellipsoids for the intrinsic viscosity and for the frictional coefficient to be identical. However, for 00w birefringence studies in the relative importance of increased asymmetry, protein solutions, it is possible t h a t the effective ellipsoid for the rotary on the one hand, and increased effective volume or diffusion constant may not be identical with those for the intrinsic visswelling on the other, in protein denaturation. cosity and frictional coefficient. This arises because of the relatively high rates of shear used in 00w birefringence measurements. (42) See ref. (27) for the revisions introduced into the KirkwoodRiseman theory and for further discussion of this point.
( 4 3 ) P. Doty and S. Katz, Abstracts of A.C.S. Meeting, p. 14‘2, Chicago, Ill., September, 1950. (44) J. F. Foster and E. G. Samsa. TRISJOURNAL, 73, 5388 (1951).
184
C,.
FREDERICK SMITH
Vol. 75
The method developed here furnishes values of
This shows clearly that the interpretation of w
taneous equations. Such solutions, of course, should also be obtainable by graphical method^'^^ without the necessity of introducing large experimental errors in the data. The effective volume in many cases cited, calculated according to- the procedure developed here, is less than iWd/N. Since, in the previous MGlN has been interpreted as a part of the effective volume (;.e., using the assumption that Ve = Mv(1 w/Vp)/N), negative w values are, therefore, required to obtain consistency between intrinsic viscosity and frictional coefficient measurements in that procedure.
Accurate measurements of sedimentation constants, intrinsic viscosities, molecular weights, partial specific volumes and rotary diffusion constants for monodispersed native and denatured proteins in various solvents are required to explore the implications of the point of view presented here. Acknowledgment.-We should like to thank Professor Paul J. Flory for helpful discussions in connection with this work.
p and V . by an analytical solution of two sinid- and the procedure used f ~ r m e r l yare ~ *incorrect. ~~~~
'ag
+
[CONTRIBUTIONFROM
THE XOYES
ITHACA, NEWYORK
LABORATORY OF CHEMISTRY, UNIVERSITY OF ILLINOIS]
The Preparation of Anhydrous Perchloric Acid BY G. FREDERICK SMITH RECEIVED AUGUST15, 1952 An improved method is described for the preparation of anhydrous perchloric acid. The procedure involves the use of 72% perchloric acid and 20% fuming sulfuric acid. Mixtures of these acids in the proportions of 1 t o 4, in the order given, serve as the reaction medium. The anhydrous perchloric acid is evolved from this mixture a t pressures of 1 mm. or less and at temperatures from 27-75' in 75% yield. The product is completely recovered by chilling t o Dry Ice temperatures. A discussion of the hazards involved is given. The process is favored also for the preparation of oxonium perchlorate OHaC104. The finished product is not contaminated by sulfuric acid.
Introduction For many operations in the study of perchloric acid and its salts the use of anhydrous perchloric acid may be required. Bemuse of the hazards involved, a procedure for the preparation of this product, to be suitable, should involve the use of starting materials which are readily available in pure form, an apparatus assembly of simple but effective design, and a procedure that involves the least hazardous manipulations. The objective in the present investigation was the study of operative procedures leading to this goal. Applicable Reactions.-The most appropriate reactions serving as a preparative scheme are KClO, + HzS04 vac. distln. (50 mm). -+
+
+ KHS04 xHC104.2H20 vac. distiln. (18mm., 120') --+ He104 t 4-HC104.H20t HC104.3H20 t HC104.HzO vac. distiln. + HClO4t HC104.2H20 ClzO, H20 +2HC104 HC104.2H~0 2SOa vac. distiln. -+HClOit 2HzSO4
+
+
+
+
+
+
+
+
(2)
(3) (4) (5)
Reaction 2 was employed by Goehler and Smith4 in their study of the improved Preparation of anhydrous Perchloric acid and in the study of the dissociation of the concentrated acid a t moderately low, (8-18 Pressures. Yield of anhydrous acid by this Procedure attained UP to 10% of the Stadkg material only. Reaction 3 is convenient and effective but a SUPPlY of the monohydrated Perchloric acid is dePendent upon the Preparation of anhydrous acid followed by dilution with water or the dihydrate of perchloric acid. Reaction 4 involves the Synthesis Of the anhydride of perchloric acid, (ClzO,), by the method of Michael and C ~ h n . Aqueous ~ perchloric acid is dehydrated by reaction with excess phosphoric anhydride, (PzOs),followed by distillation. Reaction 5 is utilized in the present study. The reaction ingredients taken are commercially available in pure form. The apparatus employed is of simple design. The reaction temperature covers the range 25-80'. The yield ranges from 50-8070 and the raw materials may be recovered. General Description of the Process.-Fuming
sulfuric
Reaction 1 has been employed frequently in studies acid (15-20%0), is added in various proportions to 72% perchloric acid. The heat of reaction is moderate and the involving the preparation of anhydrous perchloric reaction mixture is chilled to 25". This mixture is digested acid for example b y Roscoe' in one of the pioneer a t gradually increasing temperatures, 25-80', and at low studies in this field. It is not a convenient process pressure to volatilize anhydrous perchloric acid. The and was ori&ally employed for the preparation of finished product is condensed using Dry Ice as coolant and as a colorless liquid, freezing point -112'. Ananhydrous perchloric acid to be at Once diluted with collected hydrous perchloric acid may be stored without explosive water to 20 Or 60% acid composition. This method decomposition for 30-60 days a t liquid air temperatures and was employed by van Wyk2 and by van Emstera without the accumulation of the least coloration from dein important studies of the physical constants composition products. Pure'samples do not explode when stored a t ordinary temperatures for approximately 30 days. of anhydrous perchloric acid. (1) H. E. Roscoe, J . Chcm. Soc., 16, 82 (1863). (2) H.S. van W y k , 2 . unorg. Ckcm., 48, l ( 1 9 0 6 ) . (3) K. van Emster, ibtd., 62, 270 (1907).
( 4 ) 0. E. Goehler and G . F. Smith, Ind. Eng. Chem., Anal. Ed., 3, 55
(1931). ( 5 ) A. Michael and W. T. Cohn, Am. Chcm. J . , 23, 444 (1900).
