Diffusion coefficient of DNA in solution at “Zero” concentration as measured by electron microscopy

J. Mol. Biol. (1968) 36, 137-151

Diffusion Coefficient of DNA in Solution at ccZeroyy Concentration as measured by Electron Microscopy

D.LANQ AND PENELOPE CoATEsj

Division of Biology Southwest Center for Advanced f&dies

P.O. Box 30365, Dallas, Texas 75230, U.X.A.

(Received 17 February 1968)

A theoretical analysis is given, supported by a model experiment, of diffusion- controlled adsorption of DNA from solution onto a protein monolayer as a function of diffusion time. The diffusion coefficients of intact DNA molecules from bacteriophages T3 and x were calculated from adsorption kinetics involving electron microscope count of individual DNA molecules adsorbed from 0.20 M-

ammonium acetate onto a cytochrome c surface film. The diffusion coefficients were found to be independent of concentration between 3 x 10e9 and 2 x 10M7 g/cm3, concentrations where DNA molecules of this size do not interact signi- ficantly. Hence, an extrapolation to zero concentration is unnecessary. The type of kinetics observed is evidence that the adsorption is irreversible. It also provides an independent estimate of the average coil diameter of T3 DNA (3 p), in agreement with previous electron microscope measurements (2.8 ,LL). The diffusion coefficients found are, respectively, 19 and 38% smaller than values calculated from the Svedberg equation, combined with an empirical formula (Crothers & Zimm, 1965) for the sedimentation coefficient of DNA extrapolated to zero concentration. Assuming that this combination gives true diffusion coefficients, then deviations from it found by us are explained by a linear density of DNA, bound to a cytochrome c film, which is about 15% higher as compared to the B-configuration in solution.

1. Introduction At present there seems to be no theory which can predict quantitatively the hydrodynamic behavior of DNA in solution. Empirical relations, however, are well estab- lished, e.g. for sedimentation coefficients and intrinsic viscosities of DNA in aqueous solution within a wide range of molecular weights (Crothers & Zimm, 1965; Eigner & Doty, 1965; Studier, 1965). Because of the high ratio of molecular contour length to diameter, such measurements are markedly dependent upon DNA concentration and-owing to experimental limitations in sensitivity-have to be extrapolated to zero concentration. This extrapolation is not guided theoretically, since the conoentra- tion dependence is not completely understood (Schachman, 1959).

This paper reports an attempt to measure another hydrodynamic property of DNA, namely, the diffusion coefficient, at practically zero concentration. Electron microscopy provides the mandatory high sensitivity.

The method applied is diffusion-controlled adsorption of DNA from solution onto a cytochrome c surface -film; it has been used earlier for preparation of individual DNA

t Present address: Southwestern Medical School, University of Texas, Dallas, Tex., U.S.A.

137

138 D. LANG: AND P. COATES

molecules for electron microscopy (Lang, Kleinschmidt & Zahn, 1961; Lang, Bujard, Wolff & Russell, 1967). Less than 6 x 10V8 g DNA/ml. bulk solution are required, so that the DNA molecules are sufficiently far apart that they will not interact signi- ficantly. The solution is therefore ideal, in physical-chemical terms. With the DNA concentration known, the diffusion coefficient can be determined from electron microscope counts of DNA molecules adsorbed onto the protein surface film as a function of time.

It will be shown that the diffusion coefficient of T3 DNA measured in this way is independent of DNA concentration up to 2 x 10m7 g/ml. Its value is smaller than that, expected from extrapolated empirical data of the sedimentation coefficient combined with the Svedberg equation. The observed type of adsorption kinetics is evidence that DNA is irreversibly adsorbed at the cytochrome c film.

2. Theory This section presents the basis for calculation of the diffusion coefficient of DNA

from the measured kinetics of the molecules adsorbed from solution onto cytochrome c. We will treat a general, more complicated, system first and then deduce practical solutions.

(a) General system Let the origin of the x-co-ordinate (cm) be in the surface of a solution of unlimited

depth and lateral extension, the positive direction pointing into the solution per- pendicular to the surface. At time t = 0 (see), a monomolecular protein film is produced at x = 0 and, simultaneously, the solution is assumed to contain the following numbers, n(x, t), of DNA molecules per cm3, in an upper layer of thickness 6, and below it, respectively:

n(x,t) = 0; o<x<s, t=o; Iz(x,t) = no; x > 6, t = 0. >

(1)

We also permit the DNA to exhibit different diffusion coefficients: D’ (cm2/sec)

near the surface 0 < x < E, and D in bulk, x > 4 (Fig. 1). If external forces, which might affect the position of a DNA molecule, e.g. by convection or sedimentation, are excluded or sufficiently small, then DNA is transported only by random thermal

I / I , 1 I 1 1 1 -__ D

1 --+

FIG 1. Relative DNA concentration, ~z(x,t)/m,,, as a function of the distance, z, from the surface; for diffusion times t = 0 (initial state) and t = 4 sec. Calculated from equations (8), (9) and (10) with parameters given in section Z(b) (i). c < 6.

DIFFUSION COEFFICIENT OF DNA 139

collisions with other molecules. DNA, arriving at the surface by such a difksion, is assumed to be irreversibly adsorbed and mechanically fixed by the protein film such that the number of dissolved molecules per cm3 at the surface is always:

n(x,t) = 0; x = 0, t > 0. (2) Furthermore, we assume the DNA to be highly dilute. D is consequently independent of concentration and the protein film is far from saturated with adsorbed DNA.

This system is described by one-dimensional diffusion equations:

Pn, 1 an, - _-- 8x2 D’ at = 0; o<x<5, t>o;

a%, i an, --------= ax2 D at 0; 56x<& t>o;

ak, 1 an, ~_-~ ax2 D at = 0; x b 8, t >o;

with the initial conditions (1) and the boundary conditions (2) and

Dt?$D!!t$ x=5, t>o; (6)

n, = n2

an, an, -=- ax ax

1 x = 6, t > 0. (7)

n2 = n3 ) Guided by the monograph of Carslaw & Jaeger (1959), equations (1) through (7) have been solved by Laplace transformations, resulting in:

m n,(x,t) u =--

a+1 CC

am erfc Pm + 1bE + (6 - 63 - ax - “0 2dDt

m=o - erfc Pm + IbE + (6 - I) + a

2dDt >

) @)

m

n,(x,t) 1

Lx (

CCm erfc 2md + (6 - 4 erfc 2(” + 114 + (6 - 25) + 5 Z-Z-

n0 2 22/Dt - 2dDt >

+

m=o

+a erfc 2f7wf + (6 - 2%) + x

2dDt - - erfc 2(m + 1bE + (6 - 4

~ 2dDt >I 3 (9) m=o

+ erfc 2(m + 1bE + W3 - E) - (6 - 4 --

22/Dt > m

--cc

CC

M.m erfc 2~d + 2(S - E) - (6 - 4 +

2dDt WI=0

+ edc 2(77% + lb5 - (8 - x)

22/Dt >I ) (1o)

140 D. LANG AND P. COATES

with the notations u = ~D/~D’, (11)

x = (u - 1)/b + l), (- 1 < CL < l), (12) m

erfC (z) = 1 - erf (z) = A d”r s

exp (- ~2) du.

The total number of DNA molecules adsorbed per cm2 of surface area, n,(t), is evidently (see Fig. 1) given by:

Tdgp$?] ix+&ax-j+x. (14)

8 0 c Introducing (S), (9) and (10) into (14) and performing the integrations, one obtains the kinetic equation for the adsorption of DNA on t,he protein film:

where

ierfc (z) = s

erfc (u) du = L exp (- 2”) - x erfc (z). dn

(16) z

(b) Discussion of equation (15)

(i) Application to experimental conditions. Intact viral DNA has a molecular weight in the range of 2 to 200 million daltons and diffusion coefficients in water in the range of 4 x10m8 to 2 ~10~~ cm2/sec. As an example, let D = 6.4 x 10eQ cmz/sec. The possibility that the protein film influences the average water structure immediately below it, or that an energy barrier has to be overcome just prior to irreversible adsorption, both resulting in a changed diffusion coefficient near the surface, has already been taken care of by introducing D’ for 0 < x < E. Assuming e.g. D’ = 4.3 x 10mQ cm2/sec, we obtain u = 1.22 and a = 0.1. We might guess the thickness of such a layer to be E = lOA4 cm.

Consideration of a top layer of thickness 8, free of DNA at zero time, allows for a “skimming” effect. The front part of the protein film, during spreading at t = 0, skims away DNA which happens to be at the surface and leaves behind an empty solvent layer with an average thickness equal to the average effective diameter of a randomly coiled DNA molecule. For T3 DNA, 6 w 2-S x 10m4 cm (Lang et al., 1967).

Introducing into equation (15) the values estimated in the preceding two paragraphs, together with practical diffusion times up to 7 x lo3 see, the result is Figure 2, which shows that the number of molecules at first increases more than proportional to dt, then becomes linear in dt for Z/t > 10 seeliz, with a slope of (2/2/r) d/o. The upper curve in Figure 2 corresponds to E = 6 = a = 0, giving nF/no = (2/2/n) 4Dt. An approximation of equation (15) suitable for the experiments presented is:

DIFFUSION COEFFICIENT OF DNA

0 50 100 (Diffusion time)l’2 (sK)“~

141

I?IG. 2. Lower curve: adsorption kinetics according to equation (15) calculated for a typical ease (see 2(b)(i)). Upper curve: same, but for 4 = 6 = CC = 0. For z/r > 10sec112 both curves have identical slope, (2/l/57) z/D.

It is the pertinent conclusion from equation (17) that D, the diffusion coefficient in bulk, can be obtained from the slope,ifthe measured numbers of molecules adsorbed, n,(t), fall on a straight line when plotted against the square root of the diffusion time, dt, and if the initial concentration, n,, is known. The occurrence of a skimming effect and of a diffusion coefficient near the surface differing from that in bulk, is of no consequence for this conclusion. On the other hand, with reasonable assumptions for ,$ and 0, the skimming depth, 6, which is probably identical with the molecular coil diameter, can be estimated (or vice versa) from the intercept of the straight-line por- tion with the dt-axis at nF = 0, according to equation (17).

(ii) The injluence of a different d#usion coeficielzt near the surface as predicted by equation (15), was tested by a model experiment. A glass tube of 15 cm diameter, filled with agar containing the water-soluble dye lissamine-green, was partly immersed in a l-liter vessel containing stirred water (Fig. 3(a)). The concentration of dye having diffused through the agar into water was measured photometrically with red light as a function of time. Then the tube was refilled and .its lower opening closed by a Millipore alter (0.3 p pore size) previously soaked in O*25o/o B’ormvar (polyvinyl- formal dissolved in dichloroethylene). Subsequently the dye concentration as a function of time was measured again. In this model, the dye, Formvar and large water vessel, represent respectively DNA, a surface layer with reduced diffusion coefficients and the adsorbing cytochrome G film. The upper and lower curves in Figure 3(b) (without and with diffusion through Formvar-Millipore fnter) have identical slopes after 50 se@, yielding identical diffusion coefficients of the dye in agar, independent of the presence of a Formvar layer, as predicted by equations (15) or (17).

(c) Discussion of other diflu.sion conditions

(i) Convection currents. If the DNA is transported by solution convection, caused, for example, by temperature gradients, n,(t) will be proportional to t, corresponding to adsorption from a well-stirred liquid onto a solid (e.g. Trurmt, 1954).

(ii) Reversible adsorption of DNA. The appropriate solution (Carslaw $ Jaeger, 1959; Hellwig & Thomssen, 1962) of the diffusion equation allowing for a significant probability of desorption of DNA from the protein film predicts for n,(t)lno < 1: that n,(t) will be proportional to t.


2

0 50 100 150

(Diffusion timel”2 ke~)“~

FIG. 3. (a) Model experiment for DNA (represented by dye) diffusing through bulk solution (agar) and becoming irreversibly adsorbed at the cytochrome c film (highly diluted after passing the surface at the lower end of the tube).

(b) Optical density of dye in the vessel as function of (diffusion time)lla. Lower curve: bottom of tube closed with a thin Formvar layer (representing a layer with reduced diffusion coefficient). Upper curve: without Formvar. For l/t > 50 se&l2 both curves have identical slopes.

(d) Assumptions and restriction used in the theoretical development

The diffusion coefhcient of DNA in solution can be obtained from equation (17) under the following conditions, (i) to (vi).

(i) The depth, x0, of the solution is mficiently large that the DNA concentration at the bottom does not change within the maximum &&&on time. This is easily fulfilled if the depth is, e.g. x0 = 0.8 cm, since for T3 DNA approximately 53 days have to pass before the concentration at the bottom is reduced by 0.1% [from equation (lo), with E = 8 = cc = 0, nFlno = 0.999 = erf(x,/2y/Dt)].

(ii) The DNA is transported by &;f&on only. In 5 hours T3 DNA sediments about 0.6 p owing to gravity; this is less than its coil diameter and can be neglected compared to the average displacement by diffusion. As discussed above, thermal convection of DNA would result in a time dependence of adsorption measurably different from equation (17).

(iii) The DNA is irreversibly adsorbed on the surface Jilm. Otherwise the adsorption kinetics would show a proportionality to a power of t higher than l/2.

(iv) The DNA is highly diluted. Otherwise D is dependent on DNA concentration and equations (3) to (5) would not apply. Low concentrations also ensure that the protein film does not become saturated with DNA.

(v) The time required for spreading of protein is small compared to the d#ueion time. Otherwise equation (15) and its derivation would not apply, and nr, at a given time, would not be constant for all points of the surface.

(vi) Ial < 1; (~5 + 6 - [)/(ZdDt) @ 1. These restrictions of equation (17) must be met in order for the experiment to show linearity between the number of adsorbed molecules and the square root of the diffusion time. Typical values for the quotient in the second restriction are between 0.1 and 0.02 (see section 2(b)).


3. Materials and Methods (a) DNA

A l-liter culture of Escherichia coli B/lgw with 2 x lo7 cells/cm3 in nutrient broth was infected with T3 phage at a multiplicity of infection of 10w3. The lysate (2 to 5 x lOlo phage/cms) was filtered through a hydroxyapatite layer and the filter then &ted by 0.5 M-NaH,P04 buffer (pH 75). The eluate, containing about 90% of total phage, was centrifuged twice at high and low speed. The final supernatant fraction (about 1013 phage/cm3) was further purified by CsCl density-gradient centrifugation.

DNA was extracted by freshly distilled buffered phenol and stored at 4°C in sterile 0.01 M-phosphate buffer (pH 7.5) containing O*OOl M-sodium EDTA. Phenol-extracted DNA from bacteriophage x was a gift from Michiko Mitani of this laboratory.

(b) DNA concentration

The optical densities of 3 samples from DNA stocks, diluted 1 : 10 with buffer, containing 0.01 M-sodium phosphate, 0.001 M-sodium EDTA and 0.182 M-NaC1 (pH 7.0), have been measured (without scattering correction) in l*OO-cm quartz cuvettes with ultraviolet light at 2600 A against buffer in a Beckman DU spectrophotometer. Taking a molar extinction coefficient for DNA with respect to phosphorus (Felsenfeld & Hirschman, 1965; without scattering correction) of 6540 l/(mole x cm) for T3 DNA with 50% GC content and 6710 l/(mole x cm) for x DNA with 57% GC content (Schade & Adler, 1967) and a molecular weight per Naz-dinucleotide of 661.8 daltons, stock concentrations were determined to be 7.18 x 10 -4 g/cm3 for T3 DNA and 8.50 x 10m5 g/cm3 for x DNA. All dilutions were done by weighing.

T3 DNA of 7.4 x low5 g/cm3 in the same buffer as used for concentration measurements was sedimented 14 hr, 20°C, at 42,500 rev./min (Beckman L2 ultracentrifuge, SW65 rotor). The supernatant fraction, 91% of the total solution, retained 1.8% of the initial concentration of material absorbing ultraviolet light at 2600 A. Under these conditions, DNA having a sedimentation coefficient less than 5.6 x lo-l3 set (no longer visible on electron micrographs) would remain in that supernatant fraction.

(c) Diffusion experiments

Solutions with DNA concentrations from 2.84~ 10Ms to 2.01 x IO-’ g/cm3 in 0.20 M-

ammonium acetate (pH 6.5) were slowly poured into Teflon-coated aluminum dishes (18 cm diameter, 0.8 cm depth, flat rim). After temperature equilibration (about 23”C), cytochrome c (Nutritional Biochemicals Corp.) was spread from a needle within a few seconds. The film thus produced covered about 90% of the surface and was slightly compressed (2%) by a clean Teflon-coated barrier. Portions of the surface monolayer containing adsorbed DNA were transferred to carbon-coated electron microscope specimen grids, of the Siemens type, at different times ranging from 9 to 200 mm after spreading. The adhering droplets of bulk solution were removed by contact with ethanol, and the grids were shadowed with platinum. Adsorption of DNA, transported by convection within the adhering droplet, apparently does not occur. Otherwise such molecules should be preferentially oriented; this has not been found. Local disturbances of the surface film, during pick-up of portions of it by grids, seem to be negligible, since the total surface area is large compared to the sum of the areas disturbed. For further details, see Lang et al. (1967).

(d) Number of DNA molecules adsorbed per cm2 of surface area

Electron micrographs were taken randomly from non-overlapping fields at 4.95 x lo3 magnification on Kodak projection plates, contrast, 6.5 x 9 cm, and optically projected 23.7 : 1 onto a white board on which a rectangle had been drawn corresponding to an area of 31.84 cm2 on the micrograph. All DNA molecules with more than half of their contour length within the rectangle were counted, or traced and the contour length measured, including an average correction of 1.7% for pincushion distortion of the projector lens of the electron microscope.


(e) Sedimentation of T3 DNA

Three band sedimentations (Vinograd, Bruner, Kent & Weigle, 1963) of 1.7 to 3.3 x 10m5 g/cm3 DNA, total amounts 1 to 3 x 10M2 cm3, in 0.20 M-NaCl, 10e2 M-phosphate buffer (pH 8) and one boundary sedimentation with 2 x 10e5 g/cm3, in the same solvent, were analyzed with a Beckman model E analytical ultracentrifuge equipped with ultraviolet optics and a monochromator.

(f) Intrinsic viscosity of T3 DNA

This was determined in 0.20 M-Nacl, 10e2 M-phosphate buffer, 10M3 M-EDTA (pH 7.5) with a rotating aylinder visoometer, similar to that described by Zimm & Crothers, 1962 (W. Krannich, Giittingen), applying 4 concentrations between 4 and 30 x 10e6 g/cm3 at a shearing stress of 1.9 x 10d3 dyne/cm 2. This viscometer was also used to measure the viscosities of 0.20 M-ammonium acetate and water at different temperatures.

4. Results (a) Characterization. of DNA samples used

The length distributions of T3 DNA as measured by electron microscopy before, between and after diffusion experiments are shown in Figure 4, normalized and corrected for differential diffusion rates. Since DNA diffuses to the surface at a rate proportional to 40 (see section 2(b, ii)), the true length population was approximated by dividing each length fraction by the corresponding value of l/D, as given by equation (24). The data on molecular shape (Lang et al., 1967), sedimentation and intrinsic viscosity are given in Table 1. In terms of fragments present (14.3% by

TABLE 1

Data on the T3 DNA sample used

%Iean length of intact (L > 10.0 CL) molecules in 0.20 M-

ammonium acetate, e, & sample standard deviation of 3 determinations over 1 year: (11.93 f 0.47)p

Number of fragments (L < 10.0 p): 14.3% Weight of fragments (L < 10.0 p): 2.9%

Shape in solution, as obtained from electron microscopy (Lang, et al., 1967): Random coil

Root-mean-square end-to-end distance: 2.8 p Distribution of end-to-end distances, h: 1.9 x lOI h2 exp (- 1.92 x 10’ h2)

Sedimentation coefficient, +, , w: 31.8 s Intrinsic viscosity, [ 71: 108 dl./g

number or 2.9% by weight), this DNA preparation is the best one so far studied by us with the electron microscope; 2.9% fragments by weight are hardly detectable by sedimentation analysis or viscometry. The sedimentation coeEcient and the intrinsic viscosity in Table 1 give molecular weights of 23.9 x lo6 and 24.7 x 106, respectively, according to the empirical formulas of Crothers & Zimm (1965). Most of the fragments are very short and cannot be explained by random scission of intact viral DNA. Two possibilities remain: non-random scission, preferentially at the ends, or bacterial DNA fragments adhering to the outside of the isolated phages or packed into the phage head “by mistake”.

10


Remarkably, the proportion of fragments did not change within one year of storage of the DNA stock solution of 7.18 x lo-* g/cm” in sterile 0.01 M-phosphate buffer containing O-001 M-EDTA, adjusted to pH 75 by NaOH, 4°C. Thus it was justified to count large molecules on micrographs as intact instead of measuring all lengths (except a few doubtful large fragments). As required for equations (15) or (17), the number of intact DNA molecules per cm3, n,, was then given by:

NC0 NM,E no =.- =--

M 103M@) (18)

where c, is the total concentration of DNA (g/cm”), determined from optical density at 2600 A, minus 2.9% fragments by weight; N = 6.025 x 10z3 (Avogadro’s number); M, molecular weight of intact DNA; MN = 330.9 daltons, molecular weight of the average sodium dinucleotide as calculated from its atomic constituents; E, optical density of DNA solution per cm path at 2600 A wavelength; c(P), molar extinction coefficient with respect to phosphorus, in liter/(mole x cm). M was calculated from M = M’ x .I?, where M’ = l-913 x lOlo daltonslcm for Na-DNA assuming B-configuration (34.6 A per 20 nucleotides, Langridge, Wilson, Hooper, Wilkins & Hamilton, 1960) and z the average contour length of intact DNA as measured by electron microscopy after adsorption to cytochrome c from O-20 M-

ammonium acetate. The x DNA sample was less homogeneous (61.0% fragments by number, 14.0%

by weight, including a few dimers). The average contour length of intact molecules (17.75 < L < 22.25 p) in O-20 M-ammonium acetate was L = 20-O & O-6 p (sample standard deviation), giving a molecular weight of 38.3 x lo6 dalton.

Schade $ Adler (1967) found an average contour length of 21.7 p for x DNA spread on water. The difference of 8.5% can be understood as the influence of ionic strength upon the measured lengths (Lang et al., 1967; Inman, 1967).

(b) Measurement of the diflusion coeficient

Figure 5 shows the number of molecules adsorbed per cm2 surface divided by the initial concentration in bulk as a function of the square root of diffusion time for intact T3 DNA (upper curve) and x DNA (lower curve), both dissolved in 0.20 M-

ammonium acetate. Within a diffusion time interval from 10 minutes to at least 2 hours, the curves are straight lines, as expected from equation (17). However, the indicated sample standard deviations could permit a slight curvature. Therefore, by applying logarithmic plots, the power oft was determined graphically as 0.50 f 0.02 maximum error. Within this error limit, it follows (see section 2(c) (i) and (ii)) that the DNA solution is free of convection and that the molecules are irreversibly adsorbed at the surface. At a given time, the counts of adsorbed molecules, from field to field within a grid and from grid to grid, varied within the expected limits of statistical fluctuation as given by plus or minus the square roots of these counts. The diffusion coefficients given by the slopes in Figure 5, (2/d/n)dD, have been reduced to standard conditions (water, 20°C) by means of:

D ?Is To 20,w =

D 1720, w To

8,

where subscript 6 indicates the temperature in deg.C at the time of measurement; 17, absolute temperature (To = 293.2 “K); Q, solvent viscosity and Q~,~, viscosity


of water at 0 = 20°C. The viscosity ratio for 0.20 M-ammonium acetate was found to be:

%?lrl2o.w = - 0.0228 + 1.501 for 20 < 6 < 26°C.

The results are:

Dfm,, = 5.73 X 10Vg cm2/sec (T3 DNA), (20)

D&,, = 3.42 x 10Wg cm2/sec (x DNA). (21)

Neglecting the alpha-term in equation (17), the intercepts with the abscissa in Figure 5 give 6 M 3 p as compared to 2.8 p, the independently measured (Table 1) root-mean-square end-to-end distance of T3 DNA in solution.

(Diffusion time)“’ k.ec)“2

FIG. 5. Experimental adsorption kinetics for T3 DNA (0) and x DNA (0) at (23.0 &- 0.5)‘C. Initial concentrations were no = 1.85~ lOa molecules/cm3 for T3 DNA and n0 = 8-2x 10s molecules/cm3 for x DNA. The numbers of molecules counted are indicated at the points. Error margins are & sample standard deviations of the average counts of molecules per specimen grid (3 to 8 grids per point). The diffusion coefficient obtained from the slopes and from equation (19) are D!& = 5.73 x 10-O cmz/sec for T3 DNA and 3.42 x 10m9 cm2/sec for x DNA. The interoepts with the abscissa lead to average DNA coil-diameters of about 3 11.

(c) Independence of D of concentra$ion

The superscript 0 in equations (20) and (21) indicates that D20,w is independent of T3 DNA concentration up to 2 x low7 g/cm3 as shown in Figure 6. The tiled circle is the value from equation (20). The open circles represent additional measurements (with less precision, 303 to 666 molecules per point, single diffusion time only, taking the intercept at nFlno = 0 (Fig. 5) as second point to determine the slopes). Because of extremely low concentration, the two points on the left were not obtained from micrographs but by directly counting molecules on the fluorescent screen at 10,000 magnification, where the molecules can be seen with the naked eye. Also shown in Figure 6 are T3 DNA random coils within their average available volume, l/n,, at the indicated concentrations, illustrating that the molecules are practically independent of one another. Consequently, the DNA solutions applied are ideal in the physical- chemical sense and the diffusion coefficients are independent of concentration.


Average volume per T3 DNA molecule

,:;

I 1 I I I 0 0.1 0.2

DNA concentration ()cg/ml.)

FIG. 6. Independence of the T3 DNA diffusion coefficient of concentration in bulk. The filled circle was obtained from Fig. 5. Open circles represent additional measurements. For each concentration applied the average available volume per DNA molecule is drawn to scale.

5. Discussion (a) The method

This procedure of measuring diffusion coefficients of DNA has two essential features. First, individual molecules are observed and measured. This includes simultaneous determination of distributions of contour lengths and end-to-end distances, knowledge of these being essential for most interpretations of hydrodynamic data. This is particularly true for polymers like DNA which are difficult to prepare without fragmentation. Second, the method is sensitive enough to apply it to low DNA concentrations (e.g. from 3 x 10dg to 2 x 10e7 g/cm3), at which the solutions behave ideally, showing no significant concentration effects in spite of the unusually large spatial extension of high-molecular-weight DNA.

A disadvantage of the method is that, since l/D is measured, the experimental errors are about doubled with respect to D.

The most serious error in D is probably introduced by equation (18). c0 was obtained from the optical density of the DNA stock solutions, using a molar extinction co- e%cient taken from the literature (Felsenfeld & Hirschman, 1965) and is likely to be accurate to within &3%, including the possible presence of DNA too short to be seen on electron micrographs. The upper limit of that DNA amount is 1+3o/o by weight (see section 3(b)). At the low final concentration of DNA, one could expect a significant loss of DNA by adsorption to glass of the final container prior to pouring the solution into the Teflon-coated, non-adsorbing diffusion vessel. This was checked with radioactively labeled DNA and, independently, by pseudoreplicas (Hall, 1953) made from glass previously in contact with DNA solutions. On glass cleaned in the usual way with detergent and water, no significant amount of adsorbed DNA was found (only glass cleaned by chromic acid and water adsorbed DNA). The molecular weight given by M = M’L might be a critical source of error. The expeoted error and the measured sample standard deviation of contour lengths of all intact DNA molecules visible on random micrographs are identical and about j, 4% (Lang et al.,


1967). Assuming B-configuration of DNA in ammonium acetate at ionic strengths greater than 0.1, where the electron microscope contour length becomes approximately constant (Lang et al., 1967), then the molecular weight M’L should be correct, with M’ = 1*913x1O1o daltons/cm Na-DNA. Any correction factor which might turn out to be necessary for the applied value of M’ or, equivalently, for 34.6 A per 20 nucleotides, would affect our values for d/D by an inverse proportion. For this reason (and others) a method of direct electron microscope determination of no, the number concentration of DNA, would be very desirable.

Other sources of error are the electron microscope magnification (i 3%), optical projector magnification (& 0*67$), the area on the micrograph in which the DNA was counted (& O*l%), the graphically determined slopes in Figure 5 (& 2%) and the temperature during diffusion (-& 0.2%). Since the above quantities are proportional or inversely proportional to 2/D, and since the signs of their errors are unknown, the combined error is the geometric sum and amounts to & 6.2%. In this figure the error of M’ is excluded because no data are available. The total relative error of 2/D is then:

i [(6.2J2 + e x 100~]l’z yo. (22)

For example, if the error of M’ is 2%, one obtains a total error of 13% for D. The method is confined to DNA concentrations of about the range shown in Figure

6. At very low concentrations counting becomes impractical; at very high concentrations the molecules are not well separated.

(b) Comparison of results with other data

At present there have been no other methods applied to high-molecular-weight DNA for unequivocal determination of diffusion coefficients, for the reasons mentioned in the Introduction. However, one can calculate the diffusion coefficient from the Svedberg equation:

kTN sio W 0

D;,,, = -; l-fip M

= 5.417 x lOlO SF cm2/sec (23)

where k = 1.38 x lo-l6 erg/“K (Boltzmann constant), 1 - Gp = 0.45, the buoyant term for DNA, and s$,, w, the sedimentation coefficient of DNA at zero concentration in water at 20°C. Taking our hydrodynamic data from Table 1 and M = 2.43 x lo7 daltons, calculated from the empirical formulas of Crothers & Zimm (1965), we find, for T3 DNA, Dz,,, = 7-l ~10~~ om2/sec. The experimental value is 19% below that. Based on a molecular weight of 3.83 x lo7 daltons, the corresponding difference for x DNA is 38%.

According to equation (22), only a systematic error in M’, the molecular weight per cm contour length of duplex Na-DNA, could account for such a deviation. The value M’ = 1.913 x lOlo daltons/cm was calculated from the B-configuration. If binding with cytochrome c in the surface film changes DNA structure (Kleinschmidt, Kass, Williams & Knight, 1965), then M’ must be changed accordingly in order to obtain a correct molecular weight from length measurements, M = ilf’ x .E, for application in equation (18). Earlier work has shown that the length of DNA adsorbed to cytochrome c films is influenced by the ionic strength of the bulk solution (Lang et al., 1967; Inman, 1967). In addition, we find that the average electron micrograph length of DNA, in O-15 M-ammonium acetate containing varying amounts of


cytochrome c and adsorbed to incomplete cytochrome c films which form spontaneously from the bulk solution, can be increased by 15%. When combined with mechanical stretching, molecules with lOOo/o length increase have occasionally been found.

We therefore believe that our value of M’ is too low, and that the Svedberg equation (23) combined with the empirical formula of Crothers & Zimm (1965) results in correct D values:

D:o,w = 8.218 x 10 - 5 .Mo’445 + 1,468 x 10 -2

M cm2/sec. (24)

We must then assign to DNA, adsorbed from 0.20 M-ammonium acetate onto a coherent cytochrome c film, a linear density of M’ = 2.2 x lOlo daltons/cm (with a possible error of -0.2 x lOlo daltonlcm). This gives a best fit of the experimental diffusion coeacients with equation (24):

Dio,w = 7.56 x 10mg cm2/sec (T3 DNA), (25)

Die,, = 4.51 X 10mg cm2/sec (x DNA). (26) The curve in Figure 7 represents equation (24). Our corrected values, equations (25) and (26), indicated as points in Figure 7, differ from that curve by about &lOo/o, which is now within the experimental error.

s I,, I,, I

-$ lo-'-

2 -s .9 .u- 2= 91 10-8- 8 6

.\. - 'si 3 tz '6 10-9, I 1 I I 1 I 1 1 -

105 106 IO' 108 Molecular weight

FIG. 7. Diffusion coefficient as function of DNA contour length. The curve is a combination of an empirical formula for sedimentation coefficients (Crothers & Zimm, 1965) and the Svedberg equation. The points are corrected experimental values, see equations (25) and (26).

We are very grateful to Dr H. Bujard for the T3 DNA preparation, sedimentation and viscosity measurements, to Michiko Mitani for the x DNA and to Brigitte Wolff for most of the length measurements. This work was supported by U.S. Public Health Service career development award K3-GM-34,964; divisional Public Health Service research grant GM-13234; National Science Foundation research grants GB-4388 and GB-683’7; Public Health Service predoctoral fellowship GM-34,772; and a Sigma Delta Epsilon grant.

REFERENCES

Carslaw, H. S. & Jaeger, J. C. (1959). Conduction of Heat in Solids, 2nd edition. Oxford: Clarendon Press. Reprinted 1962 by Hillman & Sons.

Crothers, D. M.. & Zimm, B. H. (1965). J. MoZ. Bid. 12, 525. Eigner, J. & Doty, P. (1965). J. Mol. Biol. 12, 549. Felsenfeld, G. & Hirschman, S. Z. (1965). J. Mol. Biol. 13, 407. Hall, C. B. (1953). Introduction to EZectrow Microscopy. New York: McGraw-Hill Book

co., Inc.


Hellwig, R. & Thomssen, R. (1962). 2. Naturforschg. 17b, 29. Inman, R. B. (1967). J. Mol. Biol. 25, 209. Kleinschmidt, A. K., Kass, S. J., Williams, R. C. & Knight, C. A. (1965). J. Mol. Biol. 13,

749. Lang, D., Bujard, H., Wolff, B. & Russell, D. (1967). J. Mol. Biol. 23, 163. Lang, D., Kleinschmidt, A. K. & Zahn, R. K. (1964). Biochim. biophys. Acta, 88, 142. Langridge, R., Wilson, H. R., Hooper, C. W., Wilkins, M. H. F. & Hamilton, L. D. (1960).

J. Mol. Biol. 2, 19. Schachman, H. K. (1959). Ultracentrtifugation in Biochemistry, Chapter IV(4), p. 90. New

York and London: Academic Press. Schade, 8. Z. & Adler, J. (1967). J. Virology, 1, 591. Studier, F. W. (1965). J. Mol. BioZ. 11, 373. Trurnit, H. J. (1954). Arch. Biochem. Biophys. 51, 176. Vinograd, J., Bruner, R., Kent, R. & Weigle, J. (1963). Proc. Nat. Acad. Sci., Wash. 49,

902. Zimm, B. H. & Crothers, D. M. (1962). Proc. Nat. Acad. Sci., Wash. 48, 905.

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Diffusion coefficient of DNA in solution at “Zero” concentration as measured by electron microscopy