Determination of Vesicle Size Distributions by Freeze- Fracture Electron Microscopy F.R. HALLETT, B. NICKEL, C. SAMUELS, AND P.H. KRYGSMAN Biophysics Interdepartmental Group, Department ofPhysics {F.R.H., B.N., C.S.1 and Department of Chemistry and Biochemistry (P.H.K.), University of Guelph, Guelph, Ontario, Canada N l G 2Wl


Vesicle, Freeze-fracture, Electron microscopy, Size distributions

ABSTRACT The most common electron microscopic technique for obtaining information on size distributions of uncollapsed membrane vesicles is based on the method of van Venetie (1980). This technique involves the sizing of only those vesicles that were freeze fractured a t their equatorial planes. As a result, only a small number of images can be used to generate size distributions. Further, the technique is susceptible to systematic error. An alternate approach is to consider the complete distribution of image sizes and use this distribution to determine the average size and distribution of the vesicles. It is shown that the mean vesicle size is 4 i ~ times r the mean image size. As well, a parameter, m, which can be determined from the image distribution, can be used to characterize the vesicle distribution. The advantage of this new approach is that images of all vesicles are used, leading to a statistically better determination of vesicle sizes.

INTRODUCTION The determination of vesicle sizes and size distributions is often important in studies of model membranes, in membrane permeability and transport studies, and in the encapsulation and delivery of drugs. A variety of methods have been used to characterize vesicle sizes, including dynamic light scattering (Carey e t al., 1985; Chen et al., 1977; Selser et al., 1976; Day et al., 1977; Go11 and Stock, 1977; Kremer et al., 1977; Ostrowsky and Hesse-Bezot, 1977); small angle neutron scattering (Muddle and Higgins, 1984), sedimentation (Johnson, 1973), and freeze-fracture electron microscopy (Mayer e t al., 1986; Olson et al., 1979; van Venetie, 1980). The initial objective of our investigation was to obtain number distributions of vesicle diameters using the dynamic light scattering method. To achieve this objective, it was necessary to compare the distributions with those obtained from a n alternate technique. The most direct alternate method, which was capable of providing size distribution information, was electron microscopy. The most common approach involves the analysis of electron micrographs from freeze-fractured specimens of vesicles that have been prepared by spray freezing (van Venetie, 1980). After fracturing, platinum shadow casting of carbon replicas (PtiC) is performed with the angle between the shadowing direction and the cleavage plane set a t approximately 45”. Under these conditions, those vesicles that have been cleaved a t the equatorial plane have 50% of their surface shadowed. Only those images that have this “half-moon” appearance are counted for the purpose of obtaining size distribution information. The requirement of a 50% shadow greatly reduces the number of images that can usefully be included in the analysis. Further, a subjective determination has to be made on each vesicle image. Since there is inevitably a gradual intensity change across the shadow boundary, there can be considerable error in the selec-



tion of equatorially cleaved vesicles. Further, the uncertainty in the shadow is not randomly distributed about the 50% value. Most vesicles that are cleaved below the equatorial plane appear as craters that are shadowed greater than 50%. Those vesicles that are cleaved above the equatorial plane usually appear as domes that are also shadowed greater than 50%. Only rarely are images shadowed less than 50%. As a consequence, a systematic error can arise, leading to a skewing of distributions thus obtained. Our objective in this investigation was to find a more rigorous method for analyzing electron micrographs of freeze-fractured and shadowed specimens, one that was free from subjective, systematic biases and one that might lend itself to automation.

THEORETICAL CONSIDERATIONS When a group of spherical vesicles, all of radius R, are cleaved randomly along the Y axes (see Fig. 11, the result is a n array of circular slices whose radii are distributed by the probability density function G(r). This function, G(r), can be determined as follows. The number of slices, N(y), between y and y + d y is the fraction dyiR of the total number of slices, N. Thus N(y) - dy N R This fraction corresponds to the probability G(y)dy of finding slices in the same interval, where, from equation (l),

Received November 17, 1988; accepted in revised form May 2, 1990




7 ,








r/R Fig. 2 . The probability density distribution (G(r))obtained by randomly slicing spheres of radius R. Fig. 1. Diagram showing how slices of radius, r, can be produced by randomly slicing a sphere of radius R. 1.5

1 G(y) = - = constant R


is the uniform probability density corresponding to the fact that cleavage is equally probable a t all points along y. The probability of obtaining a slice of radius r is given by G(r)dr and this can be obtained using the relationship G(r)dr = G(y)dy


& 0.5



0.0 0.0










Fig. 3 . Scaled probability density distributions of image slices, rI(r), for values of m of 3 (broadest distribution), 5 , 7 , 9 , 11, 15, 19, 29 (narrowest distribution), calculated using Equation ( 2 2 ) .


This simple equation applies to the situation where all vesicles have identical radii. In real systems, however, R is distributed, and this, in turn, will affect the distribution of slice or image sizes actually observed. If H(R) is the probability density function that describes the distribution of vesicle radii, the function G(r) must be averaged over H(R). If the resulting distribution of slice radii is denoted as I(r), then

1 r G(r) = R (R2-r2)1'2


is the probability density (see Fig. 2) that describes the distribution of slice radii. The mean value of the radius, r can be obtained through the first moment,





or (8)


Complete evaluation of I(r) requires a knowledge of the functional form of H(R). Such a functional form will be introduced shortly, but even without any knowledge of H(R), it is still possible to evaluate the first two moments of I(r). Since



the parameters. One function for H(R) is the gamma density

(11) The integral in Equction (11) is the definition of the mean vesicle radius R, so that a simple relation 7-r-

?=-R 4


is obtained. Remarkably, this simple relationship holds, regardless of the functional form of the distribution of vesicle radii. The result is that, by determining the average radius, f, of all of the vesicle images on the electron micrograph, the average vesicle size, R, is simply 411r times this value. The second moment of I(r) yields the mean square value, r2. That is,


r3 - r2)ll2dr dR

which has the advantage that a single parameter, m, determines the form of the distribution. Small values of m lead to broad distributions which skew more heavily to the high end, whereas high m values lead to sharper, more symmetric distributions. The variance of this distribution is (19) Since u2 and R2 are available from the electron micrograph data (Equations 12 and 171, then it is possible to determine the value of m. With m known, it is then possible to compute a distribution I(r) and compare i t to the corresponding distribution determined experimentally. The computed form of I(r) can be obtained by replacing H(R) in Equation (9) with its functional form, Equation (18).This yields






r2 = -



R2 H(R) dR


Now the integral in Equation (14) defines the mean square value of the vesicle radius, Rz.Thus (15) and, again, this result is independent of the functional form of H(R). If we define S2 to be the variance associated with the distribution of image radii (slices) from the micrograph and u2 to be the variance associated with the distribution of vesicle radii then,


These results are quite powerful since measurement of F and F2 from the micrograph allows unambiguous determination of R and a ' . A somewhat more complex, but perhaps more rewarding, approach is to propose a parametized form for H(R) and then use the distribution, I(r), to determine

Note that the integration runs from r to since the vesicle radius R is never smaller than the slice or image size, r. By setting R = r cosh A, and R = $ ?,the Equation (20) can be placed in the form r"(coshh)"-

which, for odd values of m, has the solution (Abramowitz and Stegun (1965) 9.6.241,



Fig. 4. An electron-micrograph of freeze-fractured and Ptic shadowed vesicles designated as sample 2. Magnification is 69,120 x .


900 7-




50 1








0.1 5

Radius (pM) Fig. 5. Histograms of all vesicle image radii taken from electron micrographs similar to that of Figure 4.Solid bars correspond to sample 1,hashed bars correspond to sample 2.

n(m + l ) r



and K,(P) are the modified Bessel functions of order m. The scaled distribution rI(r) is shown in Figure 3 as a function of the ratio rh for several values of m. This scaling is simply a matter of convenience since generalized distributions are obtained. The treatment above is based on the two assumptions: that the vesicle images are spherical and that the vesicles are randomly cleaved. The first condition was easy to satisfy in the 5 2% of the images on any of the micrographs were measurably nonspherical. These images were ignored on the grounds that they were atypical. The second assumption was more difficult to justify. However, by demonstrating that the image distributions are independent of vesicle concentration,













Fig. 6. Comparison of normalized and scaled data from Figure 5 to computed distributions of image sizes rI(r) from Equation (22) for (a)sample 1 and m = 29, and (b)sample 2 and m = 3. The dots are the data points and the lines are the computed distributions.


U v


Radius (pM) Fig. 7. Vesicle density distributions H(R) obtained from the comparisons of experimental data to computed distributions (Figure 5). These distributions should correspond to the true density distributions of sample 1 (dashed line) and sample 2 (solid) as determined using m = 29 and 3 respectively in Equation (18).

we show that the second assumption is, a t least, reasonable. This is discussed in greater detail in a later section. In summary, most investigators would be satisfied to obtain R and a2 from their freeze-fracture electron micrographs and we have shown simple ways of obtaining these quantities. We have also shown that, by assuming a functional form for H(R), one can estimate the complete size distribution of the vesicles. The following sections show the application of both approaches.

EXPERIMENTAL METHODS Large unilamellar vesicles composed of the phospholipid dimyrisoylphosphatidylcholine (DMPC) were prepared by employing a n extrusion device similar to that available from Lipex Biomembranes, Inc. (Vancouver, B.C.). The method was similar to that of Hope et al.

(1985) and Olson et al. (1979). Briefly, dry > 99% pure DMPC (Sigma) was vortexed in 0.05 M phosphate buffer, pH 7.4. The resulting multilamellar vesicles were then extruded at least 10 times through two stacked polycarbonate filters that possess well-defined pore sizes (Nuclepore Corp., Pleasanton, CA) by employing pressures of up to 5.6 x lo5 Pa from a nitrogen gas cylinder. Extrusion was conducted at 30 to 35"C, above the DMPC phase transition temperature of 2223°C. For sample #1, a 145 mM DMPC preparation was extruded through 0.1 pm diameter pore size filters (lower concentrations were also used) and for sample #2, a 55 mM DMPC preparation was extruded through 0.2 pm diameter pore size filters. Aliquots of the vesicle preparations were mixed with glycerol (25% by volume) as cryoprotectant prior to freezing. Sample #1, which was mixed at 22"C, was frozen in liquid propane that had been cooled by liquid nitrogen. In the case of sample #2, a n attempt was made to freeze the vesicleiglycerol mixture from 35°C to avoid gel state phospholipid. Samples were fractured and platinum shadowed carbon replicas were prepared with a Balzers BA-360 M freeze-etching apparatus (Technical Marketing Assoc., Mississauga, Ont.) using techniques that have been described elsewhere (Hope et al. 1985). Micrographs of the replicas were obtained on a Philips 300 electron microscope (Philips Electronics Corp., Eindhoven, The Netherlands) operating a t 60kV under standard conditions with the specimen cold trap in place. The electron micrographs from both samples showed vesicles possessing ruffled surfaces, a n indication of gel-state phospholipids. The image distributions were determined by hand measuring over 3,000 electron micrograph images. The study of the effect of image density on the size analysis procedures was performed on a series of micrographs kindly supplied by M. Hope (Department of Biochemistry, Univ. of British Columbia). These micrographs had also been prepared using his previously published techniques (Hope et al., 1985).



TABLE 1. Size distribution parameters

Samole .~ ~~~

1 2




Filter size (Radius)

# Images



.05pm 0.lO~rn

2,271 1,404

A~~~~~~ image radius

F (urn)

3.55 x 4.70 x


Average vesicle radius R (urn) 4.52 x 5.99 x




6.8 x 10-z 8.97 x 10-



'The variance was obtained by using Equation (19) with m obtained from the best comparison of data and calculated function (Equation 22) .

The computation of the functions H(R), Equation

(18),and rI(r), Equation (22), were performed on a n IBM-PC microcomputer (IBM Corp. Boca Raton, FL) using the WATFOR FORTRAN language (Watcom Ltd., Waterloo, Ont.) and using subroutines for the computation of the modified Bessel functions from Numerical Recipes (Cambridge Univ. Press, Cambridge, England). The parameters in these subroutines had to be converted to double precision to calculate rI(r) for m-values greater than 19. Although we have computed rI(r) for odd values of m, it is also straightforward to compute this function for even values. The quality of the hand-measured electron micrograph data was such that the uncertainty on m was greater than k 2 so that calculation for even values was unnecessary. If automated image analysis were to be used, however, i t is likely that the counting statistics would be greatly improved and the resulting data quality might necessitate the ability to compute rI(r) for both even and odd values of m. The accuracy of the computer programs was checked using a simulated test vesicle size distribution and then recovering the same distribution through the analysis. As m becomes very large, the form of I(r) converges to that of G(r) as given by Equation (6). This is expected since a n infinite m would correspond to a monodisperse vesicle suspension.

RESULTS AND DISCUSSION The results from hand measuring vesicle image radii from electron micrographs, such a s the ones shown in Figure 4,are presented a s histograms in Figure 5. In all, 2,271 images were measured from sample #1 and 1,404 images from sample #2. The statistical information from these measurements is listed in Table l, together with values for R, and u2. Our main interest in this study was to find the best means for analyzing the electron micrograph data, and as a result, the extrusion and filtering techniques were not optimised or checked for reproducibility. With this in mind, we discuss the results given in Table 1. As expected, the com uted mean vesicle radius of sample #1 (4.52 x 10- pm) was smaller than that obtained for sample #2 (5.99 x l o p 2 pm). This is consistent with the fact that the pore radii of the filters used in their preparation was 5 x l o p 2 pm and 10 x lop2 pm, respectively. These mean vesicle radii are lower than those obtained by Mayer et al. (1986), who obtained mean radii of 5.15 x pm and 7.55 x lop2 pm, respectively, but these authors used a preparation of egg phosphatidyl choline (PC) rather than DMPC .

The most striking differences we observe between samples #1 and #2 is in the variance of their respective distributions. The smaller variance for sample #1 indicates a much narrower distribution. This observation is even more striking if the complete distributions are estimated, using the m-value described earlier. One could evaluate this after obtaining m by the use of Equation (19). Such a n evaluation does give approximate estimates of m, but these estimates are subject to considerable uncertainty because of the relative uncertainty in u2. A narrow distribution, for example, leads to a small u2. If the uncertainty in u2 for a narrow distribution is ?50%, which is quite likely given the number of measurements made, and the uncertainty in R is - lo%, then the error in m can be considerable, > 200%. Accurate determination of m in this manner would require the measurement of many more image radii than we performed here. The best approach is to fit, using least-squares methods, the calculated distributions using Equation (22) to the experimentally determined distributions, with m being the adjustable parameter. For the purpose of this study, we chose a n intermediate approach that was simply to calculate the distributions for a range of m values and determine which corresponded most closely to the measured function. We found it a simple matter to determine m = 29 for sample #1 and m = 3 for sample #2 (see Fig. 6a,b). The vesicle distributions, H(R), which correspond to these m values, are shown in Figure 7. These results indicate that, under the preparative conditions used, the main difference between sample #1 and sample #2 is the relative breadth of the vesicle size distributions. The difference in the average size is less pronounced between the two cases. If the analysis technique we suggest here is to be generally useful, then the underlying assumption, namely that the local fracture plane randomly cleaves the vesicles, requires validation. We presume that a single vesicle would have great difficulty in redirecting the fracture plane, but as the concentration increases, so does the possibility that the plane will distort, taking the path of least resistance, and tend toward cleaving neighboring vesicles a t their equatorial planes (see Fig. 8). We have, therefore, examined all micrographs closely for evidence of immediate distortion of the fracture plane near the vesicle image and especially in regions where images happen to be clustered. Although we did, from time to time, find such distortions, they were not common, and affected less than 1%of the images. As a further check, however, we analyzed a series of micrographs of similar vesicle preparations but with varying concentrations and, a s a result, image density. All vesicles used in this comparison were pre-





Fig. 8. At high vesicle concentrations, the fracture plane, which ideally would cleave the vesicles randomly (dashed lines) could become distorted (solid lines).

= F



00 0


0 5


0 6


Image Density (pm)-* Fig. 9. The average image radius (solid circles) and the standard deviation of the image radius distribution (open circles) plotted as a function of image density.

pared using 0.20 pm diameter pore sizes similar to that for sample #2. If the fracture plane is undergoing distortion to produce more equatorial cleavages, the average image radius should increase with the concentration. Second, the standard deviation of the image distribution should decrease with increasing concentration since more equatorial as opposed to random cleavages should lead to narrower distributions. The characteristics of image distributions obtained as a function of image density is shown in Figure 9. Both the average image radius and the standard deviation of the image radius distribution are found to be independent of image density and suggest that the cleavage is indeed random. In separate studies, using smaller vesicles prepared using 0.1 pm diameter filters, no evidence of plane distortion was noted even when image densities were as high as 10 pmp2.Nevertheless, we do caution that serious distortions may occur a t higher concentrations and suggest that lipid concentrations less than 75 mM be used. It is also possible that the concentration limit may change with different lipid and solvent conditions. As a result, image density effects on image distributions should be checked before too much faith is placed in the result.

CONCLUSION Vesicle size distributions may be obtained by analyzing all the images present on a n electron micrograph. The average vesicle radius is :times the average image radius for all distributions. Since all images are included in the analysis, the procedure could be used in automatic image analysis devices.

ACKNOWLEDGMENTS This study was supported by the National Science and Engineering Council of Canada. The authors are indebted to R. Harris of the NSERC Guelph Regional Stem Facility for performing the electron microscopy, and to T. Beveridge, R. Humphries, and D. Sullivan for valuable suggestions. The authors also thank Dr. Michael Hope and Dr. Pieter Cullis of the Department of Biochemistry, University of British Columbia, for kindly supplying many of the other micrographs used in this study.

REFERENCES Carey, M.C., Benedek, G.B., and Donovan, J.M. (1985) Micelles and vesicles of human apolypoproteins (APO) A-I and A-11, lecithins and bile salts: new insights from quasi-elastic light scattering (QLS).In:



Recent Advances in Bile Acid Research, L. Barbara et al., eds. Raven Press, New York, pp. 183-189. Chen, F.C., Chrzeszczyk, A,, and Chu, B. (1977) Quasi-elastic light scattering of monolayer vesicles. J. Chem. Phys. 66:2237-2238. Day, E.P., Ho, J.T., Kunze, R.K. Jr., and Sun, S.T. (1977) Dynamic light scattering study of calcium-induced fusion in phospholipid vesicles. Biochim. et Biophys. Acta 470503-508. Goll, J.H., and Stock, G.B. (1977) Determination by photon correlation spectroscopy of particle size distributions in lipid vesicle suspensions. Biophys. J. 19:265-273. Hope, M.J., Bally, M.B., Webb, G. and Cullis, P.R. (1985) Production of large unilamellar vesicles by a rapid extrusion procedure: Characterization of size, trapped volume and ability to maintain a membrane potential. Biochim. et Biophys. Acta 812:55-65. Johnson, S.M. (1973) The effect of charge and cholesterol on the size and thickness of sonicated phospholipid vesicles. Biochim. et Biophys. Acta 307:27-41. Kremer, J.M.H., van de Esker, M.W.J., Pathmamanoharan, C., and Wiersema, P.H. I 1977) Vesicles of variable diameter prepared by a modified injection method. Biochemistry 16:3932-3941.

Mayer, L.D., Hope, M.J. and Cullis, P.R. (1986) Vesicles of variable sizes produced by a rapid extrusion procedure. Biochim. et Biophys. Acta 858:161-168. Muddle, A.G., and Higgins, J.S. (1984) Light scattering and neutron scattering from concentrated dispersions. Faraday Discuss. Chem. 76:77-122. Olson, F., Hunt, C.A., Szoka, F.C., Vail, W.J., and Papahadjopoulos, D. (1979) Preparation of liposomes of defined sized distribution by extrusion through polycarbonate membranes. Biochim. et Biophys. Acta 557:9-23. Ostrowsky, N., and Hesse-Bezot, Ch. (1977) Dynamic light scattering study of the conformational change and fusion phenomenon of phospholipid vesicles. Chem. Phys. Letters 52:141-144. Selser, J.C., Yeh, Y. and Baskin, R.J. (1976) A light-scattering characterization of membrane vesicles. Biophys. J . 16:337-356. van Venetie, R., Leunissen-Bijvelt, J., Verkleij, A.J., and Ververgaert, P.H.J.Th. (1980) Size determination of sonicated vesicles by freezefracture electron microscopy, using the spray-freezing method. J. Microscopy 118:401-408.

Determination of vesicle size distributions by freeze-fracture electron microscopy.

The most common electron microscopic technique for obtaining information on size distributions of uncollapsed membrane vesicles is based on the method...
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