J. Mol. Bid. (1977) 116, 593-606

Hydrodynamic

Properties and Structure of fd Virus

JAY NEWMAN~, HARRY L. SWZNNEY of Physics, City College of the City University of New York 138th Street and Convent Avenue, New York, N. Y. 10031, U.X.A.

Department

AND LOREN A. DAY

Department of Biochemistry, The Public Health Research Institute of the City of New York, Inc., 455 First Avenue New York, N.Y. 10016, U.X.A. (Received 3 May 1977) A length of 8950f200 A and a diameter of 90% 10 A have been obtained for fd virus from a simultaneous solution of the Broersma equations relating the length and diameter of a rod-like particle to its rotational, D,, and translational, D,, diffusion coefficients. Measurements of D, were by transient electric birefringence, and of D, by low-angle intensity fluctuation spectroscopy. A mass of (16.410.6) x lo6 daltons was calculated from the Svedberg equation using our measured values of DT, the sedimentation coefficient and the density increment. These results, together with the molecular weight of fd DNA, give a total number of major coat protein subunits of 2710f 110 and a ratio of nucleotides to protein subunits which is definitely non-integral, 2.30f0.11. These measurements help delineate significant structural differences between fd and other filamentous viruses. Also included in this paper is an Appendix (by L. A. Day & S. A. Berkowitz) concerning the number of nucleotides, 6370&140, and the density and refractive index increments of fd DNA.

1. Introduction The hydrodynamic properties of the fd virus determined in the present study can be used to deduce information on the packaging of the protein and DNA components of the fd virus. Of particular interest are the values obtained for the numbers of protein subunits and nucleotides per unit length; these values are independent of analyses of X-ray diffraction patterns (Marvin et al., 197&J; Wachtel et al., 1976) and other physical measurements (Berkowitz & Day, 1976). In this study the rotational diffusion coefficient of fd virus was determined by a sensitive laser signal averaging transient electric birefringence technique, the translational diffusion coefficient by intensity fluctuation spectroscopy, the sedimentation coefficient by boundary sedimentation, and the density increment by resonant oscillator densimetry. The measured values are combined with the hydrodynamic t Present address: Department U.S.A.

of Biophysics,

Johns Hopkins

693

University,

Baltimore, Md 21218,

594

J. NEWMAN,

H.

L.

SWINNEY

AND

L.

A. DAY

equations of Broersma and of Svedberg to give the length, diameter and mass of the virus. These parameters, together with the number of nucleotides in fd DNA (see Appendix), allow us to deduce the average axial distances between neighboring protein subunits and between neighboring nucleotides for fd virus in solution.

2. Experimental Procedures (a) Preparation

and characterization.

of samples

Samples of fd virus were grown and isolated following procedures described by Berkowitz & Day (1976). The birefringence measurements caused no apparent decrease in the plating efficiencies of virus solutions. Length distributions were obtained by electron microscopy for the virus preparations using a microdroplet technique (Lang & Mitani, 1970) and negative staining. A typical example of the contour length distributions obtained for the fd virus samples is shown in Figure 1. Under the spreading conditions employed, the individual particles showed slight curvature but no apparent uniformity in curvature, similar to the observations of others (Marvin & Hoffmann-Berling, 1963; Frank & Day, 1970).

Fro. 1. The distribution of contour lengths, obtained by electron microscopy, of a sample of purified fd virus. Note the absence of any particles longer than 10,000 A or shorter than 7000 A. The lengths determined from the electron micrographs are relative rather than absolute values since the magnification was not calibrated.

The solvents used were 10 mM-KCl, 1 mm-Na phosphate (pH 7.5) and 1 mM-Na phosphate (pH 7-5) for the birefringence studies; 150 mm-HCl, 15 mM-Na phosphate (pH 7.0) for the other studies. Concentrations were determined by U.V. absorption measurements, using an extinction coefficient of 3.84 mg-1 cm2 at 269 nm (Berkowitz & Day, 1976) for the higher concentrations, and by volumetric dilution for the low concentrations used in the birefringence studies. (b) Densimetry Densities were determined in a resonant oscillator den&meter at 25.0 f 0.2”C (controlled to f O.Ol”C). The techniques are described in the Appendix. Solvent viscosities and densities necessary for the normalization of results to one or another solvent condition were obtained from the Handbook of Chemistry and Physics (1976).

HYDRODYNAMICS

AND

STRUCTURE

(c) Analytical

OF fd VIRUS

595

ultracen&fugation

The sedimentation coefficient of fd virus was determined by boundary sedimentation in a Beckman model E centrifuge equipped with a photoelectric scanner. The calibration of the instrument has been previously described (Berkowitz & Day, 1974). For each run the sedimentation coefficient was obtained from the linear lea&-squares slope of a log (radial position) versus time graph of 7 to 12 points. The sedimentation coefficients were assigned to the average concentration during the course of a run. (d)

Transient

electric

birefringence

The rotational diffusion coefficient D, of the fd virus was determined by the transient electric birefringence (TEB) technique. The sensitivity and accuracy of the traditional TEB method was enhanced by several orders of magnitude through the use of a lowpowered laser and crystal polarizing optics, signal averaging, and digital data processing, as described by Newman & Swinney (1976). For each sample condition, typically 10 to 15 sets of signal-averaged data (each set consisting of about 500 separate measurements) were recorded. Electric field pulses ranged from 9 to 360 V cm-l in strength and from 1 to 10 ms in duration, and the birefringence decay times deduced from the data were independent of these variations. Values of D, were deduced from measurements of the field-free decay of the light intensity transmitted through the sample, which for our geometry (see Newman & Swinney, 1976) is given by I=B+A exp(-6D,t), (1) where A and B are constants, and the applied field pulse has been assumed to be small. The intensity data were, at all sample concentrations studied, accurately represented by a single exponential function, as indicated by the average values of the normalized second cumulant quality of fit parameter (Koppel, 1972), 0.014 for fd in 1 mM-Na phosphate and 0.024 in 10 mM-KCl, 1 mM-Na phosphate. (e) Intentiity

jhctuation

spectroscopy

The translational diffusion coefficient, D,, of fd virus was determined from measurements of light-scattering intensity autocorrelation functions. Measurements were made at City College with scattering vector Q = 3.55 x IO4 cm-l (where the scattering angle 8 w&8 11.86” and the laser wavelength h, was 488.0 nm), and at Johns Hopkins University with q = 2.08~ lo4 cm-l and 2.57 x lo4 cm-l (corresponding to 6 = 8.97’ and 11*19”, respectively, and h,, = 632.8 nm). At these small angles the contribution to the correlation function from possible configurational fluctuations is entirely negligible (see, for example, Berne & Pecora, 1976), and the contribution from rotational diffiusion is small and known (see below). Thus the deduction of D, from the data is direct and model-independent. The problem of quasi-elastic scattering from dust in the sample solution was reduced considerably by centrifuging the samples in the 1 cm x 1 cm optical cells used for the scattering experiments for 24 h or more at 1500 g before an experiment. The effect of dust or aggregation on each correlation function measured was determined from the average count rate, the size of the autocorrelation function at large time delays (see Newman et al., 1974), and the size of the cumulant (Koppel, 1972; the average normalized second cumulant was O-019). The data affected by dust were rejected. The intensity autocorrelation function data were analyzed using the theory for rigid rods in solution (Pecora, 1964) which gives for the normalized autocorrelation function: gca)(k) = 1 + A [l + (2B,/B,)exp(

-6D,t)

+ . . . ]exp (-2DTq2t),

(2)

where B2/B, vanishes aa qL +O (L = rod length), and the higher-order terms are negligible for the angles of our measurements. Using the value of D, deduced from the birefringence measurements (corrected for the temperature difference between the measurements) and the value of B,/B, calculated for a 8950 A rod (see Discussion), we analyzed the data in terms of a single exponential, after dividing g c2)- 1 by the term in square brackets in equation (2), which never differed from unity by more than 3 o!o in our experiments,

596

J. NEWMAN,

H.

L.

SWINNEY

AND

L.

A. DAY

3. Results The results are summarized in Table 1. TABLET

Summ4zry of the properties of fd virus in solution& Value

Property Measured Density increment, (+/a~): Rotational diffusion coefficient, Do,,,, (s-r) Translational diffusion coefficient, D!& (10-s cma s-r) Sedimentation coefficient, & (lo- i3 8)

0.2761 O.OOV’ 20.9-&0.3c 2.68&0.04b 47.0f0.3b

Derived Hydrodynamic length, L (A) Hydrodynamic diameter, d (A) Molecular weight of virus (10s g mol- ‘) Number of nucleotides Number of major coat protein subunits Nucleotides/protein subunit Nucleotide sxial spacing (A) Protein subunit axial spacing (A)

8950 f: 200d 90& 1oa 16.4f0.68 6370& 140’ 2710&-llOK 2.30&0.11h 2.81 f0.091 3.24+0.16’

LLErrors quoted are 2 standard deviations from the mean. b Conditions 160 mn-KCl, 16 mM-Na phosphate (pH ‘7.0) at 26°C. The solvent density is 1.0068 g cm-s which yields an apparent specific volume $’ of 0.720f0.008 g-’ cm3. OThe low concentration measurements yielded the same value, within experimental error, in 1 mM-Na phosphate and in 10 rnbr-Na phosphate. d See Table 2 and Discussion. at the same temperature B From the Svedberg equation and the values of D r, a, and (+/a,); and aolvent conditions. The value is that for the dry, salt-free virus. r From Table Al (Appendix). An independent value of 6400&260 is calculated from the virus molecular weight reported here, the DNA content of the virus, 12.0f0.2o/0 (Berkowitz & Day, 1976), and the average nucleotide anion molecular weight of 308. g Calculated from the virus molecular weight composed of 12% DNA, 2% minor coat proteins, and 86% major coat proteins with subunit molecular weight 6240. See Discussion. h Independent of chemical composition data except for the assumption of 2% minor coat proteins. See Discussion. 1 Average spacing along each up or down strand of the circular DNA in the virus. ’ Calculated with the assumption that minor coat proteins are distributed over 2% of the length, presumably at the ends (see footnote g). The same assumption and the length by electron microscopy (Frank & Day, 1970; Wall, 1971) give a separation of 3.19f0.16 A.

(a) Density increment The results of differential densimetry measurements for three virus preparations are given in Figure 2 as a plot of (Ap/c), versus c, where Ap is the density difference between the solution of virus and the buffer against which it. had been dialyzed to equilibrium, and c is concentration. There was no statistically significant dependence of Ap/c upon c so the average value of Ap/c, 0~276~0608 is taken as (a~/&)$ the density increment at the zero concentration limit. This value is 3% lower than an earlier value in a similar solvent (see Appendix and Berkowitz & Day, 1976).

HYDRODYNAMICS

AND

I 0

I

4

I

STRUCTURE

I

I

I

12

16

/

8 Concentrotlon

OF fd VIRUS

697

(mg cmm3)

FIQ. 2. Density increment of different samples of fd virus in 160 mM-KCl, 16 m&x-N8 phosphate (pH 7) : ( n ) mixture of 3 preparations; (0) preparation number 63; (A) preparation number 64. The broken line indicates the average value. Sample handling techniques are described in the Appendix. The density of the buffer was 1.0068 g cm- 3. Density differences were measured between the individual virus solutions and samples of buffer which had been dialyzed against the same outer buffer solutions.

(b) Sedimentation velocity The sedimentation coet%icient for fd virus at 25.OfO.2 deg. C in 150 mM-KCI, 15 mM-Ns phosphate was determined at 13 concentrations in the range between 0.04 and 0.25 mg cme3. A linear least-squares fit of the data to l/s = (l/so) (l+ B ‘c) yielded B’ = 225f50 g-l cm3 and s&,, = (47+0f0*3) x lo-l3 s in this buffer. (c) Rotational diffusion Birefringence measurements were performed at 20.0fO.l deg. C on two different preparations of fd virus, in two different ionic strength solvents, over a large concentration range, OWO55 to I.1 mg cm- 3. Figure 3 is an intensity decay record for the lowest concentration studied, together with the best fit to equation (1) obtained in a three-parameter least-squares analysis. The values of D, deduced from the intensity decay data at different concentrations are graphed in Figure 4. In 1 mM-Na phosphate, D, is independent of concentration below about O-002 mg cm-3. At higher concentrations, the average interparticle spacing becomes comparable to the particle length (they are equal at 0.03 mg cmV3), and the measured concentration dependence presumably reflects the interparticle interactions. At an ionic strength of 12 mM, D, remained constant up to 0.01 mg cme3 and, within experimental error, was equal to the value at the lower ionic strength. At the higher ionic strength the concentration dependence was significantly reduced, as would be expected if the interactions were electrostatic. The weighted average of the low concentration values for the rotational diffusion coefficient is D&, = 20.9*0.3 s-l. (d) Translational

diffdon

The translational diffusion coefficients determined from spectroscopic measurements at the three scattering vectors were, in order of increasing scattering vector, 2.58~ 1O-8, 2.58 x 10-s and 2.60 x 10e8 cm2 s-l at c = O-068 mg cme3 at 25.0fO.l deg. C. The three sets of data, representing 80 independent measurements, yield D,.,, = (2*59&0*03)x lo-* cm2 s-l at c = O-068 mg cm- 3. Some spectroscopic measurements

In -

-0 .

-1

0

HYDRODYNAMICS

I 0~0001

AND

I O*OOl

I

I

STRUCTURE

I 0.01

1

OF fd VIRUS

I

I 0-I

I

,

I I-0

1

Concentmtmnimq/cm3f FIQ. 4. The concentration dependence of the rotational diffusion coefficient of fd virus in 1 mmNa phosphate (pH 7.6; (0) and ( A) indicate 2 different preparations of the virus) and in 10 mMKCl, 1 miwNa phosphate (pH 7.5; ( n )). Th e smooth curves are drawn to guide the eye.

were made at c = l-38 mg cmW3 and these yielded DT,25 = (2.74hO.08) x lo-* cm2 s-l. Extrapolating the data at the two concentrations to zero concentration, we obtain Dg,25 = (2.58kO.04) + 10-O cm2 s-l.

4. Discussion (a) Dimensions of fd virus in solution The birefiingence decay data are accurately described by a single exponential function and the D, values are independent of the applied field strength; thus we have no evidence of flexibility for the fd particle in solution. This may imply that either fd is more rigid than electron micrographs seem to indicate, or perhaps that the stiffness of the molecule at the low ionic strengths used in these experiments results in a flexing mode with a characteristic time which is short compared to the rotational time, so that all the molecules have the same effective hydrodynamic length. The values of D, and D,, although obtained for fd in solutions of different ionic strength, are presumed to characterize the same hydrodynamic structure for fd since no ionic strength dependence was found for D, in the zero concentration limit (see Fig, 4). The independently measured rotational and translational diffusion coefficients determine uniquely, through the Broersma (1960a,b) theory for diffusion of cylindrical rod-shaped particles, the length L and diameter d of fd virus in solution. The Broersma equations are (see Note Added Proof) Da = (3kT/7nlL3) (S-0,

(3)

600

J. NEWMAN,

H.

L.

SWINNEY TABLE

AND

L. A. DAY

2

Values for the hydrodynamic dimension of the fd virus calculated from the value for D, and the value for D,

Dkmeter

(A)

60 70 80 90 100

Length From D, (eqn (3))

(103A) From D, (eqn (4))

9.29 9.16 9-06 8.96 8.86

10.0 9.6 9.3 9.0 8.7

Uncertainties in the tabulated values we 1.8% for lengths obteined from D, and 3.2% for length.3 from DT. They arise from the uncertainties in the measured diffusion coefficients (1.6% for DT and 1.4% for DR) and from uncertaintiea in the equations for &‘(1.7 %) and y,, and yL (2.8 %).

where 6 = l.n(ZL/d) and 4 = 1*45--7*5(1/S - 0*27)2, and DT = +“/37~L)

P - i (r,, + rJ1,

(4)

where y,, = 1.27 - 7.4(1/S - 0*34)2 and yI = 0.19 - 4*2(1/S - O*39)2. From the measured values for the diffusion coeEcients we calculate a hydrodynamic length of 8950&2OOA and a hydrodynamic diameter of 9OflO A, where the uncertainties are from the propagation of experimental errors and, uncertainties of &O-l in [,y,, and yL (Broersma, 1960a,b; personal communication). To indicate how length and diameter are related through the two Broersma relations we list in Table 2 lengths of the virus calculated for different diameters using the measured values of DT and D,. The hydrodynamic dimensions of fd can be compared with those obtained from X-ray analysis and electron microscopy. Dunker et al. (1974) found from X-ray diffraction studies that the spacing between the axes of neighboring virus particles varied from 55 A at 0% relative humidity to about 85 il as the relative humidity approached lOOo/o. If the 85 A spacing from X-ray analysis does indeed re%ect the diameter of fd at 100% relative humidity then our diameter of 9OflO A indicates that the hydrodynamic boundary layer is quite small. Dunker et al. (1974) also found that distances along the axis of a given particle increase by 4 to 5% as the relative humidity is increased from 0 to 100%. Measurements of the length by Frank & Day (1970), who used a conventional electron microscope, gave a value of @@Of150 d (S.D.) and measurements by Wall (1971), with a scanning transmission electron microscope, gave a length of 883O&llOA (S.D.), and, in addition, a diameter of SO&4 A (s.D.); calibrations in each study were with carbon replicas of ruled gratings. The length of the virus from birefiingence decay is 1.5% greater than that determined from electron microscopy; this difference is slightly less than the 4 to 5% expected from the difference in the relative humidity of the two measurements. However, the length from birefringence decay was deduced assuming that the virus in solution is a cylindrical rod although there may be slight curvature that would lead to a hydrodynamic length shorter than the actual contour length in solution. Moreover, the exact meaning of the fd length determined by electron microscopy is uncertain due to uncertainty, albeit small, in the microscope calibrations and due to uncertainty m

HYDRODYNAMICS

AND

STRUCTURE

OF fd VIRUS

602

the exact state of dryness and condition of the particles. The conclusion from the comparison of the hydrodynamic length from birefringence and the contour length from electron microscopy is that the hydrodynamic particle is accurately modeled as a rod, and that any apparent shortening clue to curvature is at most about 5%. A volume for the hydrodynamic species of (5.7f1.2) x lo7 A3 is readily calculated from its dimensions with the assumption that it is cylindrical. A volume of 2-Ox lo7 A” is occupied by the dry virus as calculated from the apparent specific volume, I$‘, given in (L$/&)z = 1-#‘p and the molecular weight. The volume difference of (3.7 51.2) x lo7 A3 comprises solvent associated with the hydrodynamic species which corresponds to 1.4AO.5 cm3 solvent per g virus. Since we concluded above that the hydrodynamic boundary layer is small, this indicates that considerable water may be located within the DNA core region of the virus, or within the interstices between subunits. Similar conclusions have been reached by Camerini-Otero et al. (1974) for a number of spherical viruses on the basis of hydrodynamic measurements. (b) Molecular

weight and the non-integral

ratio of nucleotides to protein subunits

The sedimentation and translational diffusion coefficients can be combined with the density increment in the Svedberg equation M = s”RT/Do(ap/ac);,

(5)

to give a mass of (16.4f0.6) x lo6 daltons for the anhydrous salt-free fcl virus. This value agrees with the value calculated from the chemical composition and the DNA molecular weight but is significantly higher than the value previously reported by Berkowitz 8c Day (1976; see Appendix). The molecular weight determined from the Sveclberg equation is a weight-average molecular weight (Newman et al., 1974); differently averaged molecular weights would be in close agreement with this value because of the high degree of monoclispersity of the samples used in this study. If we subtract from the molecular weight of the virus the molecular weight of the DNA anion calculated from the number of nucleoticles (Table Al, Appendix) and also (2.4kO.6) x lo8 for the molecular weight of three or four subunits of gene III protein (Beaudoin, 1970; Goldsmith, 1976; Goldsmith & Konigsberg, 1977; Woolford et al., 1977), and divide the result by 5240 (Asbeck et al., 1969; Nakashima & Konigsberg, 1974), we obtain 2710&110 for the number of major coat protein subunits in the virus. The number of nucleotides interacting, on the average, with each subunit of major coat protein is 2.30&0.11, where it is assumed that the fraction of nucleotides interacting with minor protein components corresponds to the weight fraction of the minor components, 2%. This ratio is independent of any determinations of the DNA content of fd virus. It is in excellent agreement with the value of 2.32f0.07 obtained by Berkowitz & Day (1976) from the DNA content based on chemical data in the literature and two independent physical methods. Thus, the ratio of nucleoticles to subunits in fd is clearly not an integer quantity. (c) Structure

A distinguishing feature of X-ray diffraction patterns of fcl virus is the crystalline meridional reflection corresponding to a distance of 16.1 d in the axial direction (Marvin, 1966). The complex diffraction patterns have been interpreted by Marvin et al. (19743) on the basis of a model proposed for Pfl virus which calls for 22 protein

602

J. NEWMAN,

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L.

SWINNEY

AND

L. A. DAY

subunits in five turns (4.4 subunits per turn) of a helix with a pitch of 15 A. A perturbation was introduced into the model for Pfl to give a model with 4.5 subunits per turn for fd; the perturbation, which might derive from a non-integral nucleotide/ subunit ratio, would repeat every 16.1 A. The pitch of the Pfl helix is about 15.4 A in wet fibers and about 14.4 A in dry fibers (Marvin et al., 1974a). Over these distances our data for fd (Table 1, see footnote j) give 4.75f0.22 and 4*51*0*20 subunits, respectively. Therefore, our data are consistent with the model of Marvin et al. (1974b) for fd calling for 4.5 subunits per turn based on the model for Pfl virus. However, recent data on Pfl (Wiseman et al., 1976; Wiseman BEDay, 1977) rule out a model for Pfl with 4.4 subunits per helical turn (15 b) and support a model with 5.4 subunits per turn. Thus, as concluded by Wiseman et al. (1976), fd and Pfl are not related by a small shift in subunit positions. A number of models for fd structure could probably be proposed which fit both the X-ray data and other physical data such as those gathered in this study. One possibility we would like to see tested would have exactly five subunits per helical turn of 16.1 a pitch. This suggestion is based on the observation that our value for the subunit separation for fd in solution corresponds to 5.0 protein subunits per 16.1 8. The total protein structure might repeat every 32.2 A (Marvin, 1966), which would correspond to 10 subunits in two helical turns. Ten subunits would correspond to 23&l nucleotides, but the repeats of the DNA structure need not coincide with those of the protein. We thank Professor F. D. Carlson for his interest and for permitting one of us (J. N.) to perform some of the intensity fluctuation spectroscopic measurements in his laboratory. We thank Dr F. C. Chen for helpful discussions. This work was supported by National Science Foundation grant DMR76-11033, and grants AI-09049 and K04-GM 70363 from the U.S. Public Health Service. Note added in proof: S. Broersma, in work to be published, has refined the calculations of 1960 a$). Expressions for these quantities following our equa6 , y,, , and yl (Broersma, tions (3) and (4) and the hydrodynamic dimensions derived from them reflect these refinements. We thank Professor Broersma for suggestions and for sending us his unpublished results. REFERENCES Asbeck, F., Beyreuther, K., Kahler, H., von Wettstein, G. & Braunitzer, G. (1969). Hoppe-Seyler’s 2. Physiol. Chem. 350, 1047-1066. Beaudoin, J. (1970). Ph.D. thesis, University of Wisconsin. Berkowitz, S. A. & Day, L. A. (1974). BiochemistmJ, 13, 4825-4831. Berkowitz, S. A. & Day, L. A. (1976). J. Mol. Biol. 102, 531-547. Berne, B. J. & Pecora, R. (1976). Dynamic Light Scattering, chapter 8, John Wiley, New York. Broersma, S. (1960a) J. Chem. Phys. 32, 1626-1631. Broersma, S. (19606). J. Chem. Phys. 32, 1632-1635. Camerini-Otero, R. D., Pusey, P. N., Koppel, D. E., Schaefer, D. W. & Franklin, R. M. (1974). Biochemidy, 13,960-970. Dunker, A. K., Klausner, R. D., Marvin, D. A. & Wiseman, R. L. (1974). J. Mol. Biol. 81,115-117. Frank, H. Q Day, L. A. (1970). Virology, 42, 144-154. Goldsmith, M. E. (1976). Ph.D. thesis, Yale University. Goldsmith, M. E. & Konigsberg, W. H. (1977). Biochemistry, 16, 2686-2694. Handbook of Chemistry and Physics, (1974), 56th edit., C.R.C. Press, Cleveland. Koppel, D. (1972). J. Chem. Phys. 57, 4814-4820.

APPENDIX

603

Lang, D. & Mitani, M. (1970). Biopolymers, 9, 373-379. Marvin, D. A. (1966). J. Mol. Biol. 15, 8-17. Marvin, D. A. & Hoffmann-Berling, H. (1963). 2. Naturforschg. B18, 886893. Marvin, D. A., Wiseman, R. L. & Wachtel, E. J. (1974~). J. Mol. BioZ. 82, 121-138. Marvin, D. A., Pigram, W. J., Wiseman, R. L., Wachtel, E. J. & Marvin, F. J. (19746). J. Mol. BioZ. 88, 581-600. Nakashima, Y. & Konigsberg, W. (1974). J. Mol. BioZ. 88, 598-600. Newman, J. & Swinney, H. L. (1976). Biopolymers, 15, 301-315. Newman, J., Swinney, H. L., Berkowitz, S. A. & Day, L. A. (1974). Biochemistry, 13, 4832-4838. Pecora, R. (1964). J. Chem. Phys. 40, 1604-1614. Wachtel, E. J., Marvin, F. J. & Marvin, D. A. (1976). J. Mol. BioZ. 107, 379-383. Wall, J. (1971). Ph.D. thesis, University of Chicago. Wiseman, R. L., Berkowitz, S. A. & Day, L. A. (1976). J. Mol. BioZ. 102, 549-561. Wiseman, R. L. & Day, 1,. A. (1977). J. Mol. BioZ. 116, 604-613. Woolford, J. L., Jr, Steinman, H. M. &Webster, R. E. (1977). Biochemistry, 16, 2694-2700.

APPENDIX

The Number of Nucleotides and the Density and Refractive Index Increments of fd Virus DNA LOREN A. DAY AND STEVEN A. BERKOWITZ~ Department of Biochemistry, The Public Health Research Institute of the City of New York, Inc., 455 .First Avenue New York, N.Y. 10016, U.S.A.

Sedimentation-diffusion (S.-D.) measurements (Newman et al., 1974) and sedimentation equilibrium (S.E.) and light-scattering (L.S.) measurements (Berkowitz & Day, 1974) gave three values for the molecular weight of fd DNA, the mean of which was about 12% lower than the value one calculates from 5375 nucleotides in $X174 DNA (Sanger et al., 1977) and electron microscopy data on the relative sizes of these DNA species (Ray et al., 1966 ; Frank & Day, 1970). In this Appendix, we report a new value and describe experiments which have led us to conclude that there were systematic errors in the density and refractive index increments used to obtain the earlier values. We then use the correct molecular weight to calculate new density and refractive index increments. With +X174 DNA as an absolute standard, the molecular weight of fd DNA can be obtained from relative sedimentation velocities. Results of two experiments are given in Table Al. The relative molecular weights should have little systematic error since: (a) the fractions of linear DNA were less than 10% for each DNA, (b) fd DNA and +X174 DNA are about the same size and have the same average nucleotide molecular weight (331, Na+ form), and (c) conditions were identical since both DNA species were in the rotor for each experiment. Combining the mass ratio with the number of nucleotides in $X174 DNA, one obtains 6370f140 nucleotides for t Present address : Department 39

of Chemistry,

Yale University,

New Haven,

Conn. 06620, U.S.A.