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of enzymes toward inhibitors. In the present instance, it would be possible toenvisage conditions in vivo in which carboxydismutase would be insensitive towardiodoacetamide and other alkylating agents.

The authors are indebted to Professor M. Calvin for stimulating advice and for the interest hehas shown throughout this investigation. One of the authors (BRR) is indebted to the WellcomeTrust for a travel grant.The following abbreviations are used: Tris, Tris(hydroxymethyl)aminomethane; RuDP,

ribulose-1,5-diphosphate.* The work described in this paper was sponsored by the U.S. Atomic Energy Commission.t On leave of absence from the Department of Biochemistry, University College, London.I Bassham, J. A., A. A. Benson, L. D. Kay, A. Z. Harris, A. T. Wilson, and M. Calvin, J. Am.

Chem. Soc., 76, 1760 (1954).2 Quayle, J. R., R. C. Fuller, A. A. Benson, and M. Calvin, J. Am. Chem. Soc., 76,3610 (1954).3 Mayaudon, J., A. A. Benson, and M. Calvin, Biochem. Biophys. Acta, 23, 342 (1957).4 Pon, N. G., B. R. Rabin, and M. Calvin, Biochem. Z., 358, 7 (1963).5Trown, P. W., manuscript in preparation.Hughes, T. A., and J. M. Klotx, in Methods of Biochemical Analysis, ed. D. Glick (1956),

vol. 3, p. 265.7 Thiers, R. C., in Methods of Biochemical Analysis, ed. D. Glick (1957), vol. 5, p. 273.8 Pon, N. G., Ph.D. thesis, University of California, UCRL 9373 (1960).9 Hollaway, M., private communication."Watts, D. C., and B. R. Rabin. Biochem. J., 85, 507 (1962)."Watts, D. C., B. R. Rabin, and E. M. Crook, Biochem. J., 82, 412 (1962).1Rabin, B. R., and E. P. Whitehead, Nature (London), 196, 658 (1962).13 Rabin, B. R., and D. C. Watts, Nature (London), 188, 1163 (1960).14 Calvin, M., Fed. Proc., 13, 697 (1954).15 Calvin, M., and N. G. Pon, J. Cell. Comp. Physiol., Suppl. 1, 54, 51 (1959).

DETERMINATION OF DIFFUSION COEFFICIENTS AND MOLECULARWEIGHTS OF RIBONUCLEIC ACIDS AND VIRUSES*

By WIM J. MLLER

DEPARTMENT OF BIOPHYSICS, SCHOOL OF MEDICINE, JOHNS HOPKINS UNIVERSITY

Communicated by J. L. Oncley, January 2, 1964

It is widely believed that only a small fraction of the total number of large ribo-nucleic acid (RNA) molecules present in the cell can serve as direct informationcarriers in protein synthesis.' The formation of specific coat protein under thedirection of RNA isolated from f2-bacteriophage2 in a cell-free system from E. coliplaces viral RNA in this category. No clear function has been found for the bulkof the high molecular weight RNA which is present in the ribosomes, the site ofprotein synthesis. Better physical characterization of the different types of RNAmolecules in the cell might help to clarify this situation and to establish what,if any, direct relationships exist between these different large RNA molecules.For this reason we were interested in finding whether existing techniques mightbe modified so as to give for small amounts of material, reliable molecular weightsof RNA, both free and attached to protein. Light scattering, sedimentation-viscosity, sedimentation-diffusion, density-gradient centrifugation, electron-micros-

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copy, and autoradiography have all been used with varying success3-" to determinethe molecular weights of nucleic acids, bacteriophages, and viruses. The lasttwo methods'0 are useful for nucleic acids but can only be applied if some informa-tion about the molecular structure is already at hand. The results of densitygradient ultracentrifugation are affected by interaction between salt and largepolyions.'2 Light scattering has the great advantage of providing a measure ofthe molecular dimensions as well as the weight-average molecular weight.4' 6. 8Howeverf for large asymmetric particles the results may be inaccurate owing to intra-and intermolecular interference and the difficulty of making accurate extrapolationsto zero angle and concentration., Sedimentation-viscosity measurements alsorequire extrapolations to zero-concentration and a theoretical assumption aboutthe hydrodynamic behavior of the molecule.4' 5. 7. 8 The interpretation of sedi-mentation-diffusion experiments does not require such an assumption, but particleinteraction3' I again sets a limit to the accuracy of the final result when schlierenor interference optics are used. Moreover, at the concentrations required for theschlieren optical system a sedimenting boundary often shows a marked self-sharpening and, before a meaningful diffusion coefficient can be derived from theboundary spreading, a large correction based on a knowledge of the dependenceof s upon c has to be made."' 14 Schachman,16 in an effort to circumvent the effectof self-sharpening, studied the boundary spreading of TMV under low-speed ultra-centrifugation. With the schlieren optical system then in use, boundary sharpen-ing still occurred, and no conclusion was drawn about the real diffusion coefficientof TMV. Therefore, we have recently returned to the light absorption method asoriginally developed by Svedberg and his collaborators for protein systems.'6These authors showed that in sedimentation-velocity experiments the boundaryspreading may be analyzed in terms of a true diffusion-coefficient provided thatthe material is homogeneous and that a plateau value between the boundary andthe top and bottom of the cell is maintained. Several reports7-20 have appearedon the analysis of the boundary spreading of nucleic acids by means of UV ab-sorption optics. Owing to the polydispersity of the systems used, the informationobtained was confined to sedimentation distributions rather than molecular weights.However, Schumaker and Schachman,'8 using a synthetic boundary cell, measuredthe diffusion of cytochrome with absorption optics at 50 ey/ml.

In the present work, the light absorption method has been adapted for themeasurement of sedimentation and diffusion coefficients of mono- and paucidis-perse systems containing ribonucleic acids. The diffusion experiments wereperformed in two phases of the same operation: a relatively short, high-speedultracentrifugation followed by a prolonged, low-speed centrifugation suitable forthe analysis of the boundary spreading (see also Pickels2'). The purpose of thehigh-speed centrifugation is twofold: first, to establish a permanent plateau regionbetween the boundary and the meniscus; secondly, in the case of paucidispersesystems, to separate the individual components and to prevent any overlap ofthe different concentration gradients during the second phase. Since, under mostconditions, diffusion was limited to a rather narrow fraction of the total cell length,an analysis of a homogeneous boundary, unaffected by the presence of a neighbor-ing one, appeared possible. Low-speed centrifugation was performed for 10-40hr at a speed low enough to permit an analysis of the diffusion process uninfluenced

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by back diffusion. During this time the spreading boundary did not travel beyondone half to two thirds of the length of the cell. In this way molecular weightsof ribonucleic acids and viruses have been measured over the range of 30,000 to40 million and, with one exception, good agreement was obtained with previouslyreported values.Among the advantages of this method are its simplicity, the absence of particle

and salt interactions,22 the small amount of material needed (e.g., 4 X 10-5 gmof RNA for a determination of the sedimentation and diffusion coefficients), andthe possibility of obtaining information about the diffusion coefficient and themolecular weight of individual components in paucidisperse systems.

D-.

FIG. 1.-Heights of analytrol tracings for different concentrations of E. colitransfer RNA measured in a 12-mm Kel F cell. Concentration range corre-

X sponds with an absorbancy interval between zero and one OD2W0 units. Filmswere exposed for 3 sec and developed for 3 min in X-ray developer (Kodak)at 220C.

Micwgroms/ml. 50

Materials and Methods.-Dr. R. L. Steere generously supplied a highly uniform preparation of300 mu rods of virus (TMV) of established homogeneity.23MS2 bacteriophage and its host, E. coli strain C3000, were obtained from Dr. R. Sinsheimer.

The phage was grown on a medium, which he has described,'4 or on a synthetic medium of thefollowing composition per liter: 1.33 gm KH2PO4, 13.33 gm K2HPO4, 2 gm NH4Cl, 3 mg FeSO4'7H20, 400 mg MgSO4.7H20, 10 mg CaCl,22H20, 6.7 gm monosodium glutamate, 10 mg thiamineHCl, glucose 20 gm. The isolation and purification were done according to the procedure givenby Loeb and Zinder,u with the exception that the purification was carried out using 0.01 MMgCl2, and the high-speed centrifugation following the CsCl banding was omitted.

70S E. coli ribosomes were isolated using the method described by Tissibres.'6 The RNA wasextracted with phenol, precipitated with ethanol, and dialyzed against 0.01 M Tris (HC1), pH7.6, 0.01 M MgCl2.

Dr. G. von Ehrenstein kindly provided a sample of E. coli transfer RNA freed from boundamino acids.27The sedimentation-diffusion experiments were performed with a Spinco Model E ultracentrifuge

equipped with standard ultraviolet light optics. All measurements were performed using theAn-D analytical rotor with a normal 4°, Kel F 12-mm cell. After the boundary had movedbetween 0.1 and 0.5 mm from the meniscus, the rotor was decelerated until a preset low-speed valuebetween 2,500 and 15,000 rpm was reached. Spreading measurements made before, during, andafter deceleration showed that, in the absence of external braking, no serious deterioration of theboundary takes place (see Fig. 3b). Convective disturbances, due to adiabatic temperaturechanges during the acceleration and deceleration of the rotor and other sources of temperaturefluctuations, were eliminated by performing the experiments at a temperature of 50C, close tothat at which water has its maximum density. Photographic exposures were made at knowntime intervals using Kodak Blue Brand Medical X-ray film. The blackening of the film wasmeasured with a modified Spinco analytrol photodensitometer. A series of exposures repre-senting known concentrations of RNA was made, and conditions were found under which theheight of the tracing was inversely proportional to the concentrations in the cell (see Fig. 1).All absorption films were exposed, developed, and traced in the same way. Partial specificvolume measurements were made with a 5-ml pycnometer if no data were available in the litera-ture. The measurements were performed at 250C in a concentration range of 0.5-5% using thesame solvent as in the sedimentation-diffusion measurements. Concentrations of MS2 phageand E. coli transfer RNA were determined by drying aliquots at 105'C over P20 to constantweight. The dry weight of the solvent was subtracted from the total particle weight. Theamount of RNA in the phage was estimated by the orcinol method'8 using as a standard, E. coliribosomal RNA. In this procedure the base compositions of E. coli ribosomal RNA and MS2RNA

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were taken to be identical. Concentrations ofA the standard solutions of E. coli ribosomal

RNA were determined by phosphorus analysis.-2The digestion was performed in H2SO4 and1H202. Inorganic phosphate was used as astandard. Protein in MS2 phage was estimatedwith the Folin phenol reagent3 using crystal-line insulin as a standard. The calculated per-centages of RNA in the phage are based either

____________________:.on dry weight or on the sum of the amountsof protein and RNA, determined separately.

B Analysis of the boundary spreading and cri-teria for homogeneity: Figure 2 shows bound-ary patterns for TMV made at different timesduring the low-speed centrifugation. Diffusioncoefficients were calculated from the spreadingof the boundaries and the equations for a homo-geneous boundary diffusing in a centrifugalfield16

C D 2(1 - SW2t)(1I ~~~~~~~~~~~~~~~~4y2t

0ODJ and

ODEX A A ~~~~~~~~~~CO2it Vt y2 (2)

In equation (1), - is the mean experimental_________________________________,distance in centimeters at a time t from a level

in the boundary where the concentration ratioFIG. 2.-UV absorption patterns of TMV c/co is0.5 tothe twoequidistant levels withcon-

during a diffusion run of 30 hr at 2,994 rpm.Tracings correspond to the boundary spreading centration ratios determined by equation (2).at different time intervals. Temperature: 50C. For a definite value of c/co the factor y, the so-The arrow indicates the direction of travel of lution of the probability integral Yfo e-2dy, canthe boundary. The dotted line in C cor-eresponds to the inner reference distance of 1.60 be found easily (cf. ref. 16). The influence of acm measured from the center of the rotor. A, centrifugal field on the diffusion process is giventaken after 19.0 min; B, taken after 595 min.; by the factor 1 - sC02t in which s = sedimen-C, taken after 1,235 min. tation coefficient underthe conditions of the low-

speed centrifugation. (These sedimentation co-efficients are lower than those measured at high speed with negligible diffusion. The differencein s was found larger, the lower the molecular weight and the speed at which the diffusion was meas-ured.31) co = angular velocity and t = time from the start of the experiment. The exact timediffusion begins is uncertain; in all our experiments zero-time was taken from the moment at whichwe began to take the rotor up to high speed. Corresponding values of u lying on opposite sidesof the centroid were determined from the photodensitometer tracings either directly or afterenlargement. The enlargement factors were derived from a reference distance as given by themanufacturer. The distance between the top and bottom reference points on different densitom-eter tracings was found to be constant with 1%. Representative plots of f&2(1 - sco2t) vs tare given in Figures 3a, b, and c, representing the spreading of TMV, E. coli transfer RNA, andMS2 coli bacteriophage for two different sets of pairs of u in the boundary. The procedure usedto measure a from the tracings was the following: two horizontal lines were drawn through theplateau areas adjacent to the boundary with 0.2 cm. From the measured, vertical distance be-tween those lines the 20(80)%, 30(70)%, and 50% points were located. The variation in cewas of the order of 1-3% giving rise to a probable error of 2-10% in any individual measurementof f2. The accuracy of the measurement of ii itself was 2-5%. If both errors are simply additive,

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VOL. 51, 1964 BIOCHEMISTRY: W. J. MULLER 505

40 200- 100-

30_CM oEE E,V L0

WN20- CY100 N350-

10 A

5 10 2 3 5 10tXIO4sec. tXl04sec. tX O14sec.

a b cFIG. 3. a2(1 -Sw2t) vs t for two different C/CO ratios in the boundary. Temperature 50C.; for

conditions, see text and Table 1. O(C/C0) = 0.2 (0.8), A(C/Co) = 0.3 (0.7). (a) Diffusion ofTMV for 30 hr. From 20(80%) C/Co point DJ° = 0.267 X 10-7 cm2 sec-', from 30(70%) C/C0point Deo = 0.257 X 10-7 cm2 sec.-'. (b) Diffusion of E. coli transfer RNA for 9 hr. From20(80%) C/CO point Deo = 4.08 X 10- cm2 sec-, from 30(70%) C/Co point Do = 4.00 X 10-cm2 sec-'. First 5 points were taken during the deceleration of the rotor from 59,780 rev/min to7,928 rev/min. (c) Diffusion of MS2 coli bacteriophage for 20 hr. From 20(80%) C/CO pointDeo = 0.724 X 10- cm2 sec-', from 30(70)% C/CO point Dbo = 0.731 X 10- cm2 sec.

the maximal predicted scattering of the f&2(1 - sct) points of Figure 3 could be as high as 20%.However, as the diffusion proceeds, the accuracy of the measurement of a increases, whereasthe accuracy determined by the correct baseline location decreases. In practice, the scatteringwas found to be of the order of 10%. The most reliable results were obtained for f40.2, 0.8 valuesbetween 0.03 and 0.15 cm. The standard error of estimate for MS2 coli bacteriophage (Fig. 3c)was found to be 3 cm2 by the method of the least squares. In three separate experiments ondifferent preparations of MS2 coli bacteriophage the diffusion coefficients were found to be re-producible within 5%. The results ior E. coli transfer RNA were reproducible within 2%.That the points of Figure 3 fit a linear curve and that the diffusion coefficients derived from the30 (70%) and 20 (80%) point agree within a few per cent, allows us to assume both boundarystability and homogeneity. That the lines for different values of c/co cross the time axis at thesame point is another indication of boundary stability and uniformity. A symmetrical boundarywith minimum spreading under normal, high-speed centrifugation could be regarded as furtherevidence of homogeneity. In the absence of particle interaction, high-speed centrifugation is asensitive criterion for homogeneity provided the molecular weight is at least of the order of halfa million.16 This criterion cannot be applied to E. coli transfer RNA since its relatively lowmolecular weight and high diffusion coefficient may lead to masking of polydispersity by diffusion.With one exception, these criteria for homogeneity were met in all cases. Equations (1) and (2)are derived by Lamm3" starting from Faxen's32 solution of the differential equation of the centri-fuge, through a comparison of the normal diffusion process with that in a centrifugal field. Fujitallhas shown that Faxen's solution is only valid in the absence of a concentration dependence of s.Since at the dilutions used the concentration dependence of s is small or zero, the error made byusing the modified Faxen's solution must be negligible. Our use of equations (1) and (2) is there-fore justified by: (a) the homogeneity of the material in the boundary region as judged fromthe criteria described above; (b) the negligible concentration dependence of s at the dilutionsused. The factor 8co't was smaller than 0.15 in the experiments described.The slope of the line, tan 0 = (uP(1 - 8,w2t)/t) corresponding to the 20(80%) point was

used for the calculation of the diffusion coefficients (Fig. 3). The sedimentation coefficientsused in equation (1) were calculated from the distance traveled by the diffusing boundary duringthe slow-speed centrifugation at 50C. Normal sedimentation coefficients were determined bysubjecting the solution used for the diffusion experiments to high-speed centrifugation, using

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UV absorption optics. After standardizing thenormal sedimentation constants and diffusion

C) o ° tcoefficients to the same solvent (water) and<aJ°XXo ° X temperature (,200C),16 molecular weights were

;p;Xe'?XX= X calculated from the Svedberg equation:B" 1-0 1cc RTse° ° ¢ M= R(1 (3)r_4 ~~~~~~~~D(1- VP)

@eX X00X,, X X in which R = gas constant, 8.314 X 107 ergIs ,_c.*<o c mole-' degree-'; T = 293°K; 8 = S°2o sec;1 0-H+° o D = D02o cm' sec-1; V = partial specific vol-4 e:, ec H ume cm' gm'1; p = density of water at 2930K.t . e e _Results and Discussion.-Table 1

summarizes the results of our studies

~ -riC -H -H 4 for a number of substances containingribonucleic acid.

5os - -The sedimentation coefficient of S'20X<n -H. tH -H = 189 for TMV agrees with the value

0s - , -.,:,° ° of S020 = 188, which Boetker and Sim-0E0 o - mons3 obtained by extrapolation to in-

finite dilution. Using V = 0.738's and1 41 i Ic the S value found above, the molecular

'>4 "_4o>¢,0 0 weight of TMV was calculated to beo5 55o0 41.3 X 101 in good agreement with the

X4 d °° <,, oaccepted average value of 40 X 106.soa=F cs Weber et al.4 using the identical TMVB > > dq U: sample employed in this study have

.z .~ .~ found a value of 41.5 X 106 by meansS ,@0 g * E of low-speed equilibrium centrifugation.

-~°se> The equilibrium sedimentation was per-06 to° ° o oformed at a virus concentration of 0.1zOD-gd 3 per cent, where particle interaction wasoldCtg*f:, ono longer detectable by their criteria.'4

The sample of E. coli transfer RNAfreed from bound amino acids, was found

- C4Z to have a sedimentation coefficient ofo ioim oive R ni sst¢i S°2o = 4.13, a diffusion coefficient D%0

Z><i1lb1l b = 6.49 X 10-, a partial specific volume°Ep S-4 > V = 0.51, which lead to a molecular

°. O CD "40-.0 weight M = 31,600. Estimates of the5 physical molecular weight of E. coli

transfer RNA range from 24,000 tozZ; ;0 ;: ;, 35,000."-31 Probably the most reliablee estimate, obtained with chemical meth-

qds, is that given by Ofengand, Dieck-.S m>; ,>, tann, and Berg9 who report an average

°,A, chain length in E. coli transfer RNA of91 nicleotides, leading to a molecular

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weight of 31,000 E 1,500.In addition to 40 per cent of a 21S compo-

O.D. nent and 30 per cent of a 31S component, oursample of E. coli ribosomal RNA contained 10

1 T l per cent of 8-12S material and 20 per cent ofa faster 40S component sharing with the 70Sribosomes a sensitivity for magnesium andprobably providing an RNA component for

,_______'_____ each of the two dissociation products of 70SFIG. 4.-Sedimentation-diffusion pat- ribosomes40 (see Fig. 4). E. coli ribosomal

tern of E. coli ribosomal RNA after 60 RNA has a very compact structure in themin. Speed 15,220 rpm. Direction of presence of magnesium which causes a 30 pertravel indicated by arrow. For condi-tions, see Table 1. cent increase in the sedimentation coeffi-

cient.40 The values given for the components21S and 31S refer to those usually designated at "16S" and "23S.' The 21Scomponent showed a drift in the apparent diffusion-coefficient with time. The31S component of E. coli ribosomal RNA was homogeneous and resolved from the21S and 40S components. The molecular weight of the 31S component was foundto be 1.1 X 106, in agreement with the value reported by Kurland.41The sedimentation coefficient of MS2 bacteriophage was found to be 81S and its

diffusion coefficient D'20 = 1.16 X 10-v. Substituting these values in the Sved-berg equation one obtains a molecular weight of 5.3 X 106 (V = 0.683). In theelectron microscope, MS2 phage appears as a spherical object about 25 mu indiameter (personal communication, Dr. J. W. Greenawalt) in agreement with thevalue of 26 mu found by Strauss and Sinsheimer with the same technique.42 How-ever, the diameter calculated from the experimental diffusion coefficient and therelation D RT/6rnrN is 37 mju which is consistent with a highly hydrated sphere,having 2.3 gm of water per gram dry virus. The RNA content of the phagewas found to be 21.5 per cent on the basis of dry weight and ribose content. Onthe assumption that MS2 phage is composed solely of protein and RNA, combinedprotein and ribose estimates gave an RNA percentage of 23.5 per cent. Theabsorbancy of MS2 phage was 7.20/mg/mi when measured in a Zeiss spectro-photometer at 260 mju in a solution of 0.1 M NaCl, 0.01 M tris (HCl) pH 7.6.Since the RNA content was found to be 22.5 1 1 per cent and the molecular weightof the phage 5.3 X 106, the expected total molecular weight of the RNA com-ponent in the phage is (1.2 ± 1) X 106. Our molecular weight estimate of MS2RNA is in agreement with the value of 1.05 X 106, reported by Strauss and Sin-sheimer.42 However, these authors report a molecular weight of the phage of3.6 X 106 and an RNA percentage of 31.5 per cent on the basis of phosphorusanalysis.42 Assuming that the two viral strains are identical, the difference inprotein content and thereby molecular weight would seem to lie in the isolationand purification procedure of this phage.Summary.-Ultraviolet absorption techniques have been adapted to the measure-

ment of the diffusion coefficients of ribonucleic acids, both free and complexed toprotein at low centrifugal force. Details of the method are given, and the validityof the calculation procedure is discussed. With the aid of the diffusion coefficientsfound, the molecular weights of particles which contain ribonucleic acids were

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determined in the explored range of 30,000 to 40 million. The method is ap-plicable to mono- and paucidisperse systems; in the latter case estimates can bemade of the molecular weights of individual components. Other advantages ofthe method are the absence of particle or salt interactions and the small amount(4 X 10-5 gm) of material needed for a separate determination of the sedimenta-tion and the diffusion coefficients.

Note: The molecular weight of E. coli transfer RNA, using sedimentation-equilibrium cen-trifugation and UV absorption techniques was found to be 31,000 :i 1,000, in agreement with thevalue of 31,600 :1: 2,000 found by sedimentation-diffusion measurements on the same sample.

The author wishes to thank Drs. H. Dintzis, J. L. Oncley, H. Schachman, and P. Doty for theirinterest and their comments on the manuscript prior to publication. The author thanks Mr.J. Bendler for growing the phage. The excellent technical assistance of Mr. R. Anderson isgratefully acknowledged.

* This work was supported by grant (62-97-A-6006-(RI) AMPP) from the National Institutesof Health, U.S. Public Health Service.

1 Jacob, F., and J. Monod, J. Mol. Biol., 3, 318 (1961).2 Nathans, D., G. Notani, J. Schwartz, and N. Zinder, these PROCEEDINGS, 48, 1424 (1962).3 Schachman, H. K., Ultracentrifugation in Biochemistry (New York: Academic Press, 1959).4 Doty, P., in Biophysical Science, A Study Program (New York: John Wiley and Sons, 1959).6 Sadron, C. L., in The Nucleic Acids (New York: Academic Press, 1960), vol. 3.6 Gierer, A., in Progress in Biophysics (London: Pergamon Press, 1960), vol. 10.7 Schachman, H. K., in Methods in Enzymology (New York: Academic Press, 1957), vol. 4.8 Steiner, R. F., and R. F. Beers, Polynucleotides (Amsterdam: Elsevier Publishing Co., 1961).9 Berg, P., Ann. Rev. Biochem., 30, 293 (1961).10 Thomas, C. A., in Molecular Genetics (New York: Academic Press, 1963), part 1.11 Cavalieri, L. F., and B. H. Rosenberg, Ann. Rev. Biochem., 31, 79 (1962).12 Hearst, J. E., and J. Vinograd, these PROCEEDINGS, 47, 825 (1961).13 Fujita, H. J., Chem. Phys., 24, 1084 (1956).14 Baldwin, R. L., Biochem. J., 65, 503 (1957).16 Schachman, H. K., J. Am. Chem. Soc., 73, 4808 (1951).16 Svedberg, T., and K. 0. Pedersen, The Ultracentrifuge (Oxford University Press, 1940).17 Shooter, K. V., and J. A. V. Butler, Trans. Faraday Soc., 52, 734 (1956).18 Schumaker, V. N., and H. K. Schachman, Biochim. et Biophys. Acta, 23, 628 (1957).19 Butler, J. A. V., D. J. R. Laurence, A. B. Robins, and K. V. Shooter, Proc. Roy. Soc.

(London), A250, 1(1959).20 Schumaker, V. N., and B. Marano, J. Biol. Chem., 235, 2698 (1960).21 Pickels, E. G., Chem. Rev., 30, 340 (1942).22 Meselson, M., F. W. Stahl, and J. Vinograd, these PROCEEDINGS, 43, 581 (1957).23 Steere, R. L., Science, 140, 1089 (1963).24 Davis, J. E., and R. L. Sinsheimer, J. Mol. Biol., 6, 203 (1963).a' Loeb, T., and N. Z. Zinder, these PROCEEDINGS, 42, 282 (1961).26 Tissibres, A., J. D. Watson, D. Schlesinger, and B. R. Hollingworth, J. Mol. Biol., 1, 221

(1959).27 von Ehrenstein, G., and D. Dais, these PROCEEDINGS, 49, 669 (1963).?8 Schneider, W. C., in Methods in Enzymology (New York: Academic Press, 1957), vol. 3.29 Taussky, H. H., and E. Shorr, J. Biol. Chem., 202, 675 (1952).30 Lowry, 0. H., N. J. Rosebrough, A. L. Farr, and R. L. Randall, J. Biol. Chem., 193, 265

(1951).31 Lamm, O., Z. Physik. Chem. A., 143, 177 (1929).32 Faxen, 0. H., Arkiv. Math., Astron., Fysik., 21B (1929).33 Boetker, H., and N. S. Simmons, J. Am. Chem. Soc., 80, 2550 (1958).34 Weber, F. N., R. M. Elton, H. G. Kim, R. D. Rose, R. L. Steere, and D. W. Kupke,

Science, 140, 1090 (1963).

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35 Tissibres, A., J. Mol. Biol., 1, 365 (1959).36 Zillig, W., D. Schachtschnabel, and W. Krone, Z. Physiol. Chemie, 318, 100 (1960).37 Brown, G. L., Z. Kosinski, and C. Carr, in Acides Ribonucl~iques et Polyphosphates, Editions

du Centre National de la Recherche Scientifique, Paris, 1962.38 Brown, G. L., and G. Zubay, J. Mol. Biol., 2, 287 (1960).39 Ofengand, E. J., M. Dieckmann, and P. Berg, J. Biol. Chem., 236, 1741 (1961).40 Moller, W., and H. Boedtker, in Acides Ribonucteiques et Polyphosphates, Editions du Centre

National de la Recherche Scientifique, Paris, 1962.41 Kurland, C. B., J. Mol. Biol., 2, 83 (1960).42 Strauss, J. H., Jr., and R. L. Sinsheimer, J. Mol. Biol., 7, 43 (1963).

ON THE SIGNIFICANCE OF AVERAGE MOLECULAR WEIGHTS FROMSEDIMENTATION EQUILIBRIUM EXPERIMENTS*

BY E. T. ADAMS, JR.tCHEMISTRY DEPARTMENT, UNIVERSITY OF WISCONSIN, MADISON

Communicated by J. W. Williams, January 3, 1964

It is proposed here to call attention to certain precautions which require con-sideration in interpreting average molecular weights from sedimentation equi-librium experiments (Parts 1 and 2), and to demonstrate the existence of a newtype average which is derivable from equations which could lead to the deter-mination of the number average molecular weight (Part 3). The discussion isrestricted to ideal, incompressible solutions with a polydisperse macromolecularcomponent. No review of the theory basic to sedimentation equilibrium is pre-sented, since this information is readily available.'-3

1. Average Molecular Weights Over the Cell for Nonassociating Solutes.-Here isconsidered elaboration of statements of Lansing and Kraemer4 regarding theexistence of two different weight average (or any other average) molecular weightsover the cell, one of which represents the average molecular weight of the originalsolution, while the other does not.The usual Mw cell mass describes an average molecular weight over the mass of

solute in the solution column in the ultracentrifuge cell, thus

fab Mwrcd(r2) fab MwdmMto cell fb c~d(r2) ab dm

(Cb - ca)2RTco(l -_ p)W2(b2 - a2)

in which 0 = partial specific volume; p = density of the solution; Mwr = weightaverage molecular weight at any radial position, r, in the cell; c, = concentrationat position r; dm = Ohcd(r2)/2 = c dV; b = radial distance to cell bottom; aradial position of the meniscus; and w = angular velocity.

in arriving at this equation it was assumed that all the solute partial specificvolumes are equal and the quantity (1 - Vp) is constant.5 For ideal, nonassociat-ing solutes, it is readily shown that Mw cell represents the -original weight averagemolecular weight of the sample. Required to this end are the definitions of the

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