Viscosity and the Shapes of Macromolecules A Physical Chemistry Experiment Using Molecular-Level Models in the Interpretation of Macroscopic Data Obtained from Simple Measurements John L. Richards Lehman College of the City University of New York, Bronx, NY 10468 Introduction Among the variety of techniques (1)that provide information about the molar mass and shape of macromolecules in solution, the measurement of viscosity stands out because it requires only simple equipment and the measurements are relatively easy to make. When viscosity is used to determine molar mass, an emuirical relationshi~is first established for a articular kind bf solute using several samples of known i o l a r mass; the relationshi~is then used to determine values for additional saml;les ( 2 , 3 ) . Alternatively, if approximate molar mass and specific volume data are available, viscosity can be used to identify and characterize the shapes of macromolecules1 as eompad globular or rodlike particles, or
flexible random coils And, if molecular models are used, viscosity coupled with specificvolume and molar mass data can yield information about the dimensions of macromolecules in solution. I n addition, the sensitivity of viscosity to molecular structure makes it useful for monitoring. urocesses that result in changes in molecular size or shLie, including (44) the interconversion of biological macromolecules between native (active)and denatured (inactive)forms .the intercalation of small molecules within%acromoleeules intermolecular crass-linking Using Viscosity and Density Data To Characterize Macromolecular Structure In this experiment, the protein bovine serum albumin (BSA) is studied in both its native (globular) and denatured (random-coil)forms. The viscosities and densities of several aqueous solutions of the protein are measured and used to determine (7) t h e panial spemfie wlume of BSA t h e mtnnsic visensirica nnrive and denatured RSA These macroscopic results are then used with certain theoretical expressions and asumptions to calculate t h e shape hetor for native RSA 'th? dpgreeof hgdrntinn of denntuwd HS.1 .rnr approximate dimrnswns of the natlve and drnaturpd BJA hydmdynarnw panicle^ 'In some cases, !he molecular shape may oe ascenained even
w thout a mo ar mass val~e.1 tne macromo ecd e concenlratlon s very high while the intrinsic viscosity is very low, the molecule must be globular with a compact, nearly spherical shape. If the converse is true, the molecule is probably rodlike, or it is an extremely high molecular weight random coil. Inducing a conformational change (e.g., denaturation) will generally differentiate between these possibilities because loss of a rodlike configuration is accompanied by adramatic decrease in intrinsic viscosity, while transitions from globular to random-coil conformations significantly increase this quantity.
Theory Viswsity
Viscosity is a measure of the resistance of a fluid to flow. If a pure liquid is placed in a cylindrical tube and flows slowly enough to avoid turbulence, it can be thought of as flowing in concentric cylindrical 'layers", with the layers near the wall moving most slowly and those near the center of the tube moving most rapidly. If solute molecules that are large compared to the solvent molecules are introduced, they will interfere with the flow of the "layers" of solvent, causing the solution's resistance to flow (and its viscosity) to increase.
The Specific Viscosity or Viscosity Increment The change in viscosity that results from the addition of a solute to a liquid is usually expressed as the specific viscosity or viscosity increment qsp
where q is the viscosity of the solution and q. is the viscosity of the solvent (which may be a solution itself). The qSp depends on concentration and on the strength and nature of interactions between solute and solvent. The Intrinsic Viscosity A quantity related to q, is the intrinsic viscosity [ql, which is independent of concentration and characteristic of the solute alone. It is defined as the value of I@ in the dilute solution (c + 0) limit.
where the units of [qI and solute concentration c are usually given in cm3/g and g h 3 , respectively. (The second expression in eq 2 is used more often because it approaches the limiting value somewhat more rapidly (81.1 Relating Viscosity to Size and Shape
The Hydrodynamic Solute Particle Because some solvent is invariably bound to solute molecules in solution, the information gained from viscosity measurements does not pertain to the "dry" solute molecule alone, but rather to the hydrodynamic solute particle, which comprises the solute molecule and the solvent that, on the average, travels with it in a flowing solution.
The Shape Factor and the Hydrodynamic Particle Volume As first shown by Albert Einstein, the viscosity increment qapdepends on Volume 70 Number 8 August 1993
685
the volume fraction Q of the solution occupied by the hydrodynamic salute particles t h e shape of the hydrodynamic solute particles, expressed in terms of the shape factor v (9). The shape factor is interpreted in terms of models for molecular structure (seebe low^. The volume fraction of solute @ (a unitless may be expressed in terms of the volume u h (in cm3) of a n individual hydrodynamic particle and the number of such particles per em3 of solution (NclM):
where c is the solute concentration (in glcm3);N is Avogadm's number; and M is the solute molar mass (10,l). If eq 4 is substituted into eq 3 and the result divided by c, an expression is obtained for the viscosity increment in terms of the shape factor and the hydrodynamic volume of a solute particle.
However, to obtain values of v and u h that are independent of solute-solute interactions (and variations in solutesolvent interactions with changingconccntration,, eq 5 must be written for the dilute solution limit. This is accomplished by combining eq 5 with eq 2 to yield
Shape factor vs. axial ratio for ellipsoids, for relatively low values of alb. (After Tanford; see ref 2.)
cific volume of hound solvent ul (both in cm31g); and a solvation parameter S1 expressed as g solventlg solute. M N
U,, = -(uz
+ S1ul)
where M I N is the mass of a single solute molecule. Equation 6 may be used with experimentally derived values of [q] and uh to draw conclusions about the shape a n d molecular dimensions of macromolecular solute particles. The specific methods for evaluating v and uh in terms of model structures for rigid and for flexible polymers are described below Model Structures and the Shape Factor To interpret viscosity data in terms of molecular parameters, relatively rigid (globular or rodlike) hydrodynamic particles are approximated by regular geometrical solids: spheres, ellipsoids, or cylindrical rods. Macromolecules with relativelv flexible structures are described as "random coils." As shown bv Einstein. the shape factor for spherical particles is 2.5, while for nonsph&ical particles ;> 2.5. For nonspherical particles, equations relating the shape factor to the axial ratio of prolate (cigar-shaped) and oblate (disk-shaped) ellipsoids have been developed by Simha (111.~ a axial ratio = b
where a and b are the lengths of the axes of the ellipsoids and a > b. (For convenience, a graph of v vs. a / b for low values of a l b derived from Simha's relationships is given in the figure.) Rigid Macromolecules Globular and Rodlike Particles. In general, the volume of the hydrodynamic particle can be expressed in terms of the specific volume of the "dry" solute vz; the average spe'Rodlike molecules, which are best described as long cylinders, are approximated in this treatment as extended prolate ellipsoids of the same length and volume as the corresponding cylinder. 3Thi~treatment assumes that the solution flow rate is low enough that the solutes particles remain randomly oriented.
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Journal of Chemical Education
Assuming a Dilute Solution. The difficulty with using eq 7 is that uz, ul, and 61 are not known. However, if we assume that the solution is sufficiently dilute that 61is independent of concentration, then thermodynamic theory may be used to "replace" ul and uz with quantities that can be experimentally determined. To do this, the total volume of solution Vis expressed as the sum of the volumes of solute, hound solvent, and free solvent.
where g, and gz are the masses of solvent and solute, respectively; and u? is the specific volume of free solvent. The partial specific volume i& of the solute is then
assuming the solution is sufficiently dilute that S1 is indeoendent of concentration. Solving eq 9 for uz and substituting into eq 7, we find
Finally, substitutingeq 10 into eq 6, we get an expression for the intrinsic viscosity in terms of the shape factor and the solvation parameter, which is useful for rigid (globular or rodlike) particles.
where the value of i& used corresponds to the dilute solution limit because [qI is defined in that limit. (The symbol
-
vz,c,o will be used below to denote this limiting value. See the discussion of partial specific volume belowJ4
Flexible Macmmolecules In contrast to macromolecules with relatively rigid structures, molecules with flexible random coil conformations effectively "trap", on average, large amounts of solvent within their structures. This results in a comparatively large hydrodynamic volume for a given molar mass. In such cases, eq 11is not useful because the second term is overwhelmingly dominant. Because random-coil polymers have relatively flexible structures which constantly change with time, they are best characterized by an equivalent hydrodynamic sphere with a n effective radius R..5 TOrelate Re to the intrinsic viscosity for these particles, we substitute and
into eq 6 to yield
temperature-controlledtransparent water bath (stability:M.l 'C) .analytical balance (M.1mg) .stopwatch (fO.l s) pH meter assorted volumetric glassware Solutions KC1 Soluent. Prepare 500 mL of 0.010 M KC1 solution using freshly boiled (but cool) deionized water. This solution is the "solvent" for the experiment. Native BSA. Prepare 100.0 mL of 0.03000 g/mL native BSA stock solution that is 0.010 M in KC1 using the KC1 solvent and BSA (e.g., A4503 "fraction V" gradeifrom Sigma). The BSA should be added to the solvent, not vice versa. Gentle magnetic stirring will speed dissolution, but caution should be used because protein solutions foam easily. To help ensure reproducible viscometer flow rates, the stock solution should be vacuum-filtered using a membrane filter (ex.. - . 0.8-um . .ore-size Tvpe -. AAMillipore filter) before dilution. Three dilutions of the native BSAstock solution are ore---~~-~ pared using 15.00,10.00, and 7.00 mL of the stock solution and sufficient KC1 solvent to produce a total volume of 25.00 mL of each solution. (The KC1 solvent is used to keep the ionic strength of the solution roughly constant.) This yields a set of four native BSA solutions with concentrations 0.03000, 0.01800, 0.01200, and 0.008400 g/mL. ~
Partial Specific Volume If Vzistaken to be the volume increase that results when gz grams of solute are added tog, grams of solvent that occupy Vl mL, then a mean value of the partial specific volume Ez,, over the range c = 0 to c may be defined as
The extrapolation to c = 0 of a plot of the values of Fz,, vs. c for a set of solutions with a range of concentrations then yields the dilute-solution limiting value of the partial specifie volume Cz,, ,as the c = 0 intercept. The calculation of &,, values is facilitated by writing "conservation equations" for the mass G and volume V of a particular sample of solution. G=g~+gz V=Vl+Vz
(14) (15)
G can be calculated from Given a value for V(e.g., 1ma), the measured solution density, andgz from the concentration of the solution. Theng, can be calculated using eq 14. If we assume that the mean partial specific volume of the solvent in the solution may be approximated by the specific volume of pure solvent vy, then Vl can be calculated from gl and v?. The value of Vz can then be obtained from eq 15 and used withg, to calculate G,, for a particular solution (12,13). Experimental Equipment Ostwald-Fenske viscometer (size #50) Weld (capillarystopper-type)pycnometer with cover (25mL) 4Whiie it is tempting to say that Vz is the contribution of the solute to the hydrodynamic palticie volume and 6,@ is the solvent contribution (and thus that eqs 7 and 10 are equivalent term for term),this is not the case in general; V9 and 6 , depend on many factors, and V2 is even negat ve n some cases However, I does seem to oe a reasonab e approxmat on for prole ns ( I ) "or arge ranoom cots Re :0 88 Re, where RG 1s the radfusof gyration oithe molecule (10):
~
~
~
~
Denatured BSA. Prepare 100.0 mL of 0.01200 M, pH 3, denatured BSAstock solution from 40.00 mL of the native BSA stock solution, using HCI (1 M and 0.1 M) to adjust the pH, and the KC1 solvent solution. Three dilutions of the denatured BSA stock solution are prepared using 30.00,20.00, and 15.00 mL of the stock solution and sufficient KC1 solvent to produce a total volume of 50.00 mL of each solution, yielding a set of four denat u r e d BSA solutions with concentrations 0.01200, 0.007200,0.004800, and 0.003600 g1mL. Measurements All measurements are made at 25.0 f 0.1 Z. Solutions should be degassed just before use by applying reduced pressure (using an aspirator) until the evolution of dissolved gas ceases. (Vacuum membrane filtration also yields degassed solutions.) Density Measurements The pycnometer is calibrated with degassed, deionized water (three trials) and used to determine densities of the KC1 solvent and the four denatured BSA solutions (one trial each). Viscosity Measurements For each sample, the viscometer should be charged once with 10.00 mL of solution, then clamped vertically in the water bath so that the uppermost bulb is a t least partially submerged and allowed to equilibrate 5 mins before beginning timed runs. Three trials are made for each of the following solutions: deionized water (for calibration) KC1 solvent four native BSA solutions four denatured BSA solutions For most samples, the range for the flow times should be less than 2 s, though for the most concentrated solutions they may be somewhat greater. Volume 70 Number 8 August 1993
687
Cleanup and Maintenance
When changing solutions, the viscometer should be cleaned and dried using water, detergent, water, deionized water, and acetone in that order. Use an aspirator with a short piece of tubing to speed the addition and removal of liquids. To check the viscometer for cleanliness, a water run should be made just before the acetone drying step by adding 10.0 mL of water to the drained but wet viscometer. A 5-6% NaOCl (household bleach) solution will remove any protein residue from the viscometer. Stock solutions typically may be stored in the refrigerator for a week or two without noticeable precipitation; they should be warmed to room temperature and vacuum membrane filtered before use. Dilutions should be prepared on the day they are used.6 Calculations and Results Macroscopic Properties
Solution Density. A graph of the density of the denatured BSA solutions vs. concentrationis made and used to determine the densities of the native BSA solutions. (Density measurements on native protein solutions are generally unreliable due to the presence of microscopic bubbles.) Solution Viscosity. Solution viscosities are calculated. using an equation derived from Poiseuille's law. q =Bpt
(16)
where q is the viscosity; B is the viscometer constant; t is the flow time; and p is the density of the solution. (B is determined by calibrating the viscometer with water for which q = 0.8904 CPat 25 'C.) Partial Specific Volume. The values of i&m for denatured BSA in the four solutions studied are determined using eqs 13-15. Then &,,o is found by plotting the Cz,, values vs. c and extrapolating to c = 0. It is assumed that the resulting value for C2,,,o applies to BSA in both its native and denatured forms. Intrinsic Viscosity. The intrinsic viscosities of native and denatured BSA are determined from graphs of (In (qlq.))/c vs. c for the two respective sets of solutions. MolecularParameters
Taking into account the results for the three cases, the student is asked to decide whether, based on the data, the native BSA hydrodynamic particle is more likely to be spherical or ellipsoidal. Ellipsoidal. If the results suggest the particle is ellipsoidal, it is assumed to be an ellipsoid of the prolate type, and Simha's relationships (see the figure) are used to determine the axial ratio corresponding to the case of a typical amount of solvation. Then uh is calculated from eq 10 and used with the axial ratio and the formula for the volume of a prolate ellipsoid (4mb2/3,a > b) to calculate a and b. Spherical. If the results indicate that the protein particle is spherical, then uh is calculated from eq 10 aqd used with the formula for the volume of a sphere (4a?/3) to calculate a radius for the particle. Denatured BSA Assuming denatured BSA may be described as a random coil (see footnote 11, the experimental [ql and the known molar mass of BSA are used to calculate the effectiveradius R, and volume vh of the denatured BSAhydrodynamic particle. Then uh and i&,,o are used to estimate the volume fraction of water "contained" in the denatured BSA hydrodynamic particle, assuming that G,, ,o can be used to approximate the specific volume of "dry" protein (see footnote 4). Finally, the student is asked to use the experimental results to compare the structures of the native and denatured BSA hydrodynamic particles with respect to size, shape, and extent of hydration. As part of this discussion, the two forms of the protein are sketched to scale, showing the probable location of the water of hydration in each case. Time Economy
The experiment requires approximately 6 to 9 h of laboratory time for students working in pairs. Eliminating the density determinations and providing the students with the value O ~ G ~ , ,should + ~ reduce the time needed by about 2 h. Another 2 h can be saved if the nmtein stock solutions are prepared for the students. (The Gme estimates assume two viscometers are used: one for the native and one for the denatured BSA solutions.)
The resulting values for [q] and Cz,, , are then used with the known the molar mass of BSA(66,267)to draw conclusions about the shapes and molecular dimensions of the native and denatured BSA hydrodynamic particles.
Assessing Student Results
Native BSA
and
For native BSA, the experimentally determined values e v or of in1 and i% -,- , ."are used to calculate the s h a ~ factor the &vation parameter S1 for the following three cases
To provide an indication of the quality of student results, plots were made of -
V2,,
vs. C
.
No Soluation. First, it is assumed that no solvation of the pmtein occurs, allowing the shape factor corresponding to the greatest possible value of the axial ratio to be calculated. Maximum Solvation. Next, the lowest possible of v, namely that for a spherical particle, is used to calculate the maximum possible value of the solvation parameter S1. Typical Soluation. Finally, a value of v is calculated assuming a typical amount of solvation (e. g., S1= 0.2 g HzO/g protein). (See the discussion in ref 7). 'A detailed procedure will be sent on request.
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Journal of Chemical Education
using density and viscosity data taken h m the reports of fourteen student pairs representing the full range of student performance. The plots (of 56 points each)were essentially linear and had small slopes, as expected. After rejecting the obviously discordant points in each plot, linear regression analysis yielded the following intercept values, which are reported with their respective standard errors and the number of points used in the fit in Table 1. The relatively larger fractional error for the value of the denatured BSAintrinsic viscosity reflects the considerably greater scatter in the (In (qlq.))/c vs. c plot for that species. The number of points rejected in the C2,, vs. c plot reflects the dificulty in obtaining good iz,, values. (This is probably due to problems with small bubbles in the solutions
Table 1. The Limiting Value of the Partial Specific Volume and the Intrinsic Viscosities for BSA from Extrapolation N -
Y2.c-0
39
[q] for native BSA
44 50
[q] for denatured BSA
Student Values Lit. Values (7) mug mug 0.71 &0.10 0.73 3.7 3.4f 0.6 11 k 4 H
and the inaccuracies associated with t h e small quantities derived from differences between large numbers.) Students who carry out the experiment with considerable care generally obtain results that agree reasonably well with the literature values. Nevertheless their values of Cz., for t h e more dilute solutions can deviate significantiy from expected values. Less careful work generally results i n considerable scatter i n t h e d o t s of (In (q/qJ)lc and i&.
Dealing with Less Careful Student Work I n some of these cases, the intercept values can still be related t o t h e literature values through their limits of error. I n the remaininp. cases. a reasonable a . p. ~ r o a c h(considering t h e small slope values) is to calculate means from the four m i n t s in each plot a n d use them i n place of the intercep
