difference and partial volume data in terms of a common molality of component 3. ... ally a minor component in terms of the mole fraction. The quantitatively.
[4]
MAGNETIC
SUSPENSION
29
proteins, human and bovine serum albumin,"-' transferrin,:' caeruloplasmin, '~ and ferredoxin r2 have been determined by this method. The density measurements can equally well be performed in the presence of various small molecules, such as KC1 (Fig. 3), provided adequate solute t e m p e r a t u r e ~ l e n s i t y - c o n c e n t r a t i o n correlation data are available. The volume of solution required for these measurements could be greatly reduced if a small thermister were used to measure temperature and if the divers were slightly shorter. The 3 Figures are reproduced from J. Phys. Chem. 70, 3285, (1966): 71, 3717 (1967) by permission from the American Chemical Society. ' A. G. Morell, C. J. A. Van Den Hamer. and 1. H. Scheinberg,,l. Biol. ('hem. 244, 3494
(1969). D. H. Petering, Ph.D. Thesis. University of Michigan, Ann Arbor, Michigan, 1969.
[4] M a g n e t i c
Suspension: Density-Volume, Osmotic Pressure
Viscosity, and
By D. W. KUPKE and T. H. CROUCH Improvements and additional applications in magnetic balancing for the study of proteins have appeared since this physical principle was last treated in this series.' In addition to density applications, the magnetic approach is now used also for viscosity and osmotic pressure. Viscosities of high accuracy can be obtained on a routine basis concurrently with the determination of the density on the same sample (:--(I.2 ml). Moreover, the viscosity method is enlarged by this approach because the change in viscosity with time is conveniently obtained and unusually small shearing stresses (:- 10 :; dyne, cm -') are applied without sacrifice of speed. The former attribute may be useful for the study of elongating proteins and of conformational changes, and the latter is applicable to systems thai are sensitive to shearing. A newel- application of magnetic balancing, while less developed, is in the measurement of very small osmotic pressures (to -10 ,2 cm H~O). Osmotic pressure is theoretically the simplest and least ambiguous means for getting out the chemical potentials of macromolecular solutions: however, the sensitivity of the measurements has not been improved appreciably for routine use with the proteins since the inception of the method. This limitation in sensitivity still stands at :-10 ' M in macromolecule concentration when results of high precision are required. Hence, conventional osmometry misses many of the systems of interest to enzymologists today. By the magnetic suspension approach, results at the 10 '~M range have been obtained with satisfactory precision. As currently practiced, this development is restricted to low-pressure studies (< 10 cm H,_,Oor, of each solution are added together on the analytical balance to give m~a~: the density of the mixture. p d V m ) V~>'-'
rtlm~ (V + V°) '' dV~T~,
(20)
For a fractional change in the pressure, P/P", where P is another pressure which is close to P", the density of the buoy b e c o m e s mm~ Pmml Pro,- V + ( P ° / P ) V " - P V + P°V'~
(21)
assuming that the ideal gas law applies for these small pressure differences (as has been shown by calibration). Differentiating Eq. (21) with respect to pressure and rearranging gives mmMP mm~dP d p ~ = pov,[1 + (pv/PoVO)]._, ~- poVO(l + V/VO)._,
(22)
where p/po is essentially unity. In the idealized case where V - + 0, or V(B) ~ V °, then dpm) ~-dP since p,> is nearly 1 g/ml and p 0 ~ 1 arm. Hence, the m a x i m u m useful sensitivity at AT ~ 0.0003 ° when Apm) = -+ 1 × 10-'; g/ml is - + 1 × 10 ~; atm or about _+ 1 × 10 :~ cm H.,O. The volume V of the buoy material, however, is finite, ranging from 0.1 V" to - 3 V" in the various buoy designs employed. For the most unfavorable case (V ~ 3 V"), the sensitivity reduces to dp(f~ ~ O. 169 dP (23) via Eq. (22) where m(B~ is typically --0.17 g (i.e., a change in density of 1.7 × 10 4 g/ml per cm H.~O). The latter type of buoy, having thicker walls, can be fashioned symmetrically for better alignment with the magnetic axis£" Although the sensitivity drops about 6-fold from the ideal by the use of such " h e a v y " buoys, the density difference between buoy and solution can be maintained by magnetic suspension to better than 10 7 g/ml if the temperature is stable to within 0.0003 ° at constant pressure (atmospheric pressure fluctuations limit the precision in ordinary magnetic densimetry to ~ 10 '~ g/ml, regardless of temperature control to better than -0.005°). In practice, the time taken for the final reading of the density difference and a calibration of the pressure upon releasing the developed pressure difference ( ~ 1 rain) is such that temperature fluctuations of more than 0.0003 ° are not observed. Thus, the density difference has been found to remain constant to +0.2/xg/ml. :"' T. H. Crouch, Dissertation, University of Virginia, Charlottesville, 1977.
[4]
MAGNETIC SUSPENSION
65
If the volume of the liquid phase surrounding the buoy is truly constant (i.e., no bubbles or m e m b r a n e distortions), the amount of fluid crossing the m e m b r a n e during the a p p r o a c h to equilibrium may be calculated from the ideal gas law. That is, (dV~/dP),,, = V"/P °
(24)
where V~ is the volume o f the gas in the buoy at any pressure P. Substituting the gas volume of 0.063 ml for a typical buoy at the reference conditions V~'w h e n P " = 1030 cm HeO (~1 atm), the derivative [Eq. (24)] is ~ 6 0 nl per cm H._,O or ~ 1.5 nl crosses the m e m b r a n e when the nondiffusible solute is at 10 " M (H ~ 0.025 cm H,O). Hence, the method holds the capability of being very rapid with respect to achieving equilibrium at small values of I1. Methodology O s m o m e t e r s can conform to almost any g e o m e t r y as long as the two liquid phases are in contact via a semipermeable m e m b r a n e and a suitably quantitative manifestation of the pressure difference is observable. At present, the magnetic o s m o m e t e r s do not include the technology for changing the protein concentration without disassembling the cell compartment. Such an i m p r o v e m e n t would permit better evaluation of the slopes, [ d ( I I / c i ) / d c ~ ] , , > for studying the various factors contributing to nonideal behavior :'':~ (p~ = constant chemical potential of the diffusible components, and i refers to the nondiffusible components). Although an improved model, with the buoy on the solvent side and with more convenient t e m p e r a t u r e control, has been effected, a" the overall precision with the simpler, original design '-'7 has not been surpassed. The o s m o m e t e r assembly of the latter, which amounts to a test tube containing the solvent, protein solution, m e m b r a n e , and buoy, is shown in Fig. 15. Several such tubes are assembled and allowed to come to equilibrium at the desired temperature, after which each is inserted, in turn, into the magnetic suspension system for the measurements, A given tube is positioned reproducibly in an insulated, thermostable water bath with optical ports so that optical sensing and buoy positioning with the aid of a telemicroscope are facilitated. Alternatively, a thermostable brass block with appropriate portholes has been used for containing the tubes. The detailed circuitry for the optical height-sensing s y s t e m of the buoy is to be found in the original paper"r: it differs in some respects from that for the basic densimeter of Fig. 5, but the changes are not critical. This is because the :" G. Scatchard,.l. A m . Chem. Soc. 68, 2315 (1946). :~ A. V. Giintelberg and K. U Linderstrcm-Lang, C. R. Tray. Lab. Car&berg. Set. Chim. 27, I (1949).
66
MOLECULAR WEIGHT
d
DETERMINATIONS
[4]
F
FIG. 15. Schematic diagram of the osmotic cell assembly. The a s s e m b l e d glass tube G ( - 15 x 1.8 cm) is inserted in a magnetic suspension system for measuring changes in density of the buoy B as a function of pressure. The cylindrical glass cell C (--~2 × 0.6 cm), which contains B and the protein solution ( - 0 . 5 ml), is cemented into a nylon plug Q, which has an axial channel for filling. Membrane N is made to fit tightly over the bottom of C by slipping a latex band, 4-5 m m wide, over N and around C (the O-ring and gasket assembly, as shown, and perforated plastic plates against N to minimize ballooning were found to be unnecessary). The upper end of the channel in C is sealed by a neoprene ring O and a short nylon screw D, which is rigidly e m b e d d e d into plastic rod J. J extends out of G through a tightly fitting stopper E. Q, which slides into G, contains a n u m b e r of vertical holes H to permit flee m o v e m e n t of the solvent. A small hole is drilled into G at about the desired height for Q, and a nylon screw (not shown) backed by an O ring protrudes through G and is aligned with a tapped hole in Q: this arrangement prevents any lateral m o v e m e n t of C when rod J is turned to open or close the liquid phase in C from the solvent phase. C is filled slowly by m e a n s of a gas-tight syringe into the channel until the solution runs out (at a temperature a little lower than that of the experiment): a hand magnet is used to prevent the buoy from floating and blocking offthe channel during the filling. Solvent is then added into G through H until at Q: excess solution from C is blotted off, and J is slowly turned to close offC. Solvent is added to the dashed line and the stopper E is slid down along J until tightly e m b e d d e d into G. For the m e a s u r e m e n t s , the a s s e m b l e d o s m o m e t e r is slipped into the densimeter cavity and is then attached to the insulated constant-atmosphere tank (see text) from glass tube F emerging from stopper E. A stopcock assembly between F and the tank (not shown) allows for opening the s y s t e m to a manometer, the outside atmosphere, and/or to syringes for adjusting the pressure of the constant-atmosphere tank. It is best to submerge the tank and all leads from F in a constant-temperature bath: however, the system performed adequately when all e x p o s e d items were insulated. The tube of highest concentration in a series of assembled tubes m a y be kept connected to the measuring system for following the approach to equilibrium: this approach is monitored by the change in current across the reference resistance (Fig. 5 or Fig. 14) as observed with a digital voltmeter. At equilibrium, J is turned gently to release the developed pressure difference (for other details, see text). The plessure differences in the remaining tubes are then m e a s u r e d , in turn, quite rapidly. The protein solutions are then resealed from the solvent atmospheres, and the procedure is repeated to test for reproducibility. Adapted from J. W. Beams, M. G. Hodgins. and D. W. Kupke, Proc. N~ltl. Acad. Sci. U.S.A. 70, 3785 (1973).
4.00-
(a)
~,
3.50-
,
3.00
~.
2 50
~'.00-
1.50
$
0
~ B
1'0 C2 [mg/ml]
1'5
20
1;o
!'.5
2.0
4.00, (b)
3.50
3.00' 0
e% m i... 2.5o.
2.00"
1.50
~..-..._...,.
o.o
o;s
C2 [ mg/m I ]
F[G. 16. Reduced osmotic p r e s s m e versus concentration of defatted and deionized bovine serum albumin (BSA) and of its peptic fragments A and B when separate and when mixed in equimolar proportions (see text) in 0.01 M bis-tris propane~). 10 M KCI, pH 8.6 at 20 ° . (a) Composite plot. (b) Expanded plot of the A + B complex at
