J. theor. Biol. (1978) 74, 361-375
The Diameters of Membrane Vesicles Fit in Geometric Series W. S. BONT Department of Biophysics, Antoni Van Leeuwenhoek-Huis, The Netherlands Cancer Institute, Amsterdam, The Netherlanh (Received 23 January 1978, and in revisedform
17 March 1978)
Vesicles formed in vitro by fragmentation of biomembranes are restricted to certain dimensions; the diameters are represented by two geometric series. The diameters of membrane vesicles found in intact cells, including viral envelopes, are terms of the same two series. A tentative mathematical model is proposed, to explain this phenomenon by fusion of equally sized vesicles. 1. Introduction
All membranes found in nature show a basic architectural design (Robertson, 1959); the structure common to all membranes is the lipid bilayer (Danielli d Davson, 1935). Membranes are an integral part of living cells; every cell is confined by a membrane. In eukaryotic cells in addition they form the boundary of the nucleus. A host of other cellular structures also requires the presence of membranes. Often membranes form vesicles, i.e. small spherical bodies confined by a lipid bilayer, that serve the purpose, e.g. of storageof materials produced by the cell. Thus, materials like neurotransmitters and hormones destined for export or proteins required in a later phase of the cell’s existence are temporarily enveloped in membranes (Palade, 1975). Phospholipids are the major constituents of virtually all membrane bilayers and accordingly they primarily exist in the form of closed spherical bodies (Bangham & Horne, 1964). Therefore vesicles are formed when large membraneous structures are fragmented in vitro. Thus far no systematic investigation has been made on the size distribution of vesicles both present in vivo and prepared in vitro. Recently we started such an investigation (Boom, Bont, Hofs & De Vries, 1976; Bont, Boom, Hofs & De Vries, 1977). The size distribution of vesicles obtained in vitro by fragmentation of large membrane structures was studied by analytical centrifugation. It was demonstrated that the vesicles obtained 361
0022~5193/78/190361+15
$02.00/O
Q 1978 Academic Press Inc. (London)
Ltd.
362
W.
S.
BONT
in vitro occur in only a few sizes with the exclusion of other sizes (Boom rr al., 1976). On the other hand the size distribution of vesicles found in viva in various cell types has been determined with the electron microscope and it was shown that the standard deviation of the mean size was small (Geffen & Ostberg, 1969; Bunge, Bartlett-Bunge & Peterson, 1965; Costoff & McShaw, 1969). Especially from the comparison between vesicles occurring in vitro and those occurring in z&o it appeared that the masses of vesicles are terms of the same geometric series with a common ratio of 2 (Bont et al., 1977). The following aspects related to these findings need to be elucidated. (a) How can the existence of vesicles, that do not fit in the geometric series, be explained? (b) What is the mechanism underlying the process that leads to the results cited above? (c) Is the finding that vesicle size is determined by purely physical factors generally applicable to all lipid bilayers and therefore also to the envelopes of, e.g. viruses and chlamydiae ? In this paper we present further data supporting the assumption that the size of all vesicles is determined by purely physical factors and also propose a tentative mathematical model that might explain the size discreteness of vesicles.
2. Size of Hormone Vesicles, Viruses and Chlamydiae A comparison between the radii of vesicles formed in vitro and those of granular vesicles found in vioo revealed that these radii irrespective of their source are terms of the same geometric series (Bont et al., 1977). Recently Gray (1977) published diameters of granular hormone vesicles from human pituitary, and in Table 1 we compare also these data with those of the geometric series presented previously (Bont et al., 1977). Since a great number of other particles also is confined by a phospholipid bilayer we extended the comparison in Table 1 to viruses and chlamydiae. Again a striking agreement is found between both quantities. As to the real significance of this comparison the following must be kept in mind. When we restrict ourselves to spherical vesicles, the real diameter of the membrane vesicle sometimes cannot be derived unequivocally from the electron micrographs. Artifacts introduced by our manipulations give rise to distortion of the real dimension due to (a) deviations from the spherical form (Jones & Kwanbunbumpen, 1970), (b) various techniques used for fixation and staining (Feller, Dougherty & Di Stefano, 1971), and (c) surface projections on viruses obscuring the real diameter of the membraneous envelope (Luftig,
1
450
442
1 688
2
1200t
1355
TSH
FSH 1415
TSH 870 ACTH 975
1206
1800
1816
LH 1475 LH
1720
2260
35403
ACTH ; MSH 3568
LTH 3395
STH 2410 FSH
3496
6
2454
Serial number of vesicles 4 5 5
948
3
4500$
4529
STH
4468
8
5599
LTH
6374
9
t This value was obtained by subtracting 400 (the spike height times 2) from the overall diameter of 1600 (cf. McIntosh et al., 1967). $ The particles for meningopneumonitis (3540) are significantly smaller than the mean (4500) of the remainder of the group, that consists of six other types of chluq&ae (cf. Kurotchkin et al., 1947).
Sindbis virus (Cornpans, 1971) I.B.V. (McIntosh, Dees, Becker, Kapikian & Chanock, 1967) Herpes virus (Wildy, Russell & Home, 1960) Chhaydiae (Kurotchkin et al., 1947)
Human pituitary (Gray, 1977)
Fragmented membanes (Bent et al., 1977) Rat pituitary (Costoff et al., 1969)
Source of particles
Diameters of granular hormone vesicles, viruses and chlamydiae (by electron microscopy) compared with diameters of membrane vesicles in preparations of fragmented biomembranes(by sedimentation analysis). In all Tables diameters are given in Angstrom
TABLE
364
W.
S. BONT
McMillan, Culbreth & Bolognesi, 1974). Therefore it is difficult to select from the wealth of data given for the diameters of viruses (cf. Luftig et al., 1974). We therefore only present those data given in the literature from which the diameters of the membrane vesicles can unequivocally be determined. Often, measurements from the electron micrographs presented in literature, using the magnification factors given by the authors, yielded striking correlations with the values presented in Table 1. However, these values were omitted. In general, it is obvious that large groups of viruses have the same size (Fenner, McAuslan, Mims, Sambrook & White, 1974) and that in addition most of the sizes given in literature certainly do not exclude a priori the possibility that the size of the membrane envelope associated with these viruses corresponds to one of the members of the series. In this respect it is of interest to cite the work of Sarker, Manthey & Sheffield (1975) with MuMTV and MuLV. These authors measured the diameters of the particles and constructed a histogram. MuMTV is apparently less stable than MuLV and therefore it is disrupted under various conditions. Instead of the expected continuous size distribution of the breakdown products, they found that these products had discrete sizes, giving rise to a peak in the histogram. Sarker et al. (1975) expressed their surprise by the following statement “It is puzzling that a peak should occur in the distribution of these small diameters of MuMTV”. Apparently, the breakdown products of the viruses, as long as they are confined by a membrane, are restricted to a certain size. The work with particles from the psittacosis-lymphogranular group (cldamydiae) is included in Table 1. Six of the seven members of this group given by Kurotchkin, Libby, Gagnon & Cox (1947) have an average diameter of 4500 A (ranging from 4220 A to 4970 A). Only the one for meningopneumonitis is significantly lower (3540 A). But both size classes fit with the values of the series for n = 8 and M = 7 respectively.
3. A Tentative Mathematical
Model
Brouwer (1909) has formulated the following theorem : “A vector direction varying continuously on a singly connected two-sided closed surface must be indeterminate in at least one point”. In terms more appropriate to our problem this theorem reads: “ ‘Hairs’ cannot be implanted in a sphere in an orderly fashion; at least one singular point (‘crown’) must be formed”. If the vesicles are assumed to be spherical, the phosphodiester groups of the diglycerides can be considered as “hairs”. Therefore these groups must form singular points on the surface of the vesicles.
DISCRETENESS
OF
VESICLE
SIZE
365
If these singular points are reactive sites on the surface, one could postulate that they change in nature and perhaps also in number with the size of the vesicles. Now it seems possible that the vesicles fuse at these reactive singular points and that only fusion of identical points is allowed. This would explain the occurrence of the geometric series and the findings mentioned above. Furthermore it is in agreement with the assumption that for vesicle-cell fusion there is an interaction between the phospholipid bilayers (Pagan0 & Huang, 1975) and that cell-cell fusion takes place at sides where the bilayers, denuded of intramembraneous particles, are exposed (Ahkong, Fisher, Tampion & Lucy, 1975). Implicit in all our considerations then is the assumption that the peculiar properties of the biomembranes (especially in respect of the discreteness of vesicle size) reside in the phospholipid moiety. This assumption is strengthened by the fact that liposomes also occur in discrete sizes (Bont et al., 1977). The problem of the discreteness of vesicle size has, however, two aspects. First: why is the smallest vesicle discrete and secondly : why are the other vesicles multiples of the smallest one? Robertson (1964) has already stated that the smallest vesicles cannot exist below a certain critical value of about 300 A. But this statement does not exclude the possibility that above 300 A all diameters are allowed. Application of the proposed mathematical model would then result in a continuous size distribution of the fusion products. What we need is a theoretical justification for the discreteness of the smallest vesicles and the derivation of the equations with which the absolute value of these vesicles can be calculated. In other words, for the applicability of the mathematical model, a proof of the discreteness of the smallest vesicles and the determination of their diameters is a prerequisite. Relative simple systems can best be used both experimentally and theoretically, for this kind of studies. Huang (1969) demonstrated that vesicles formed with phosphatidylcholine from chicken eggs showed a very narrow size distribution with a diameter of 228 + 5 A. Johnson, Bangham, Hill & Korn (1971) confirmed these results and Helfrich (1974) could demonstrate on purely theoretical grounds that a diameter of about 400 A can be expected for these smallest vesicles. Later, Israelachvili, Mitchell & Ninham (1976, 1977) developed a unified theory that accounts quantitatively for many of the physical properties of membrane vesicles and that links thermodynamics, interaction free energy, and molecular geometry. These authors not only confirmed by the application of their theory the experimental findings of Huang (1969) and Johnson ef al. (1971) for egg lecithin but also were able to derive equations from which it followed (a) that the smallest vesicles are homogeneous in size, (b) that a change in the average length of the fatty acid chains causes a shift in size. In addition these authors were able to explain the findings of Gent &
366
W.
S. BONT
Prestegard (1974) that (c) in the phosphatidylcholine-phosphatidylethanolamine system the vesicle size increases by increasing the percentage of phosphatidylethanolamine. A mole fraction, F = 0.52, for phosphatidylethanolamine results in a diameter of 350 A for the smallest liposomes. All these studes were restricted to phospholipid vesicles. When we want to extrapolate all these data to real membrane vesicles found in situ we have to keep in mind that a substantial part of biomembranes consists of proteins. Phospholipids contribute about 45 A to the thickness of the bilayer and therefore the contribution of the other components, e.g. proteins, carbohydrates, present in biomembranes, is about 558. When we take hypothetical biomembrane vesicles with a mole fraction, F = 0.52, for phosphatidylethanolamine, we could then expect the diameter for the smallest vesicles to be 350+55 = 405A. This value differs less than 10% from the diameter of 442 A observed for the smallest vesicles in preparation of biomembranes. Furthermore Wickner (1977) demonstrated that for a given phospholipid composition the presence of protein per se might influence the radius of the smallest vesicles. In summary, we can say that the size of the smallest vesicles is influenced (a) by the length of the fatty acid chains, (b) by the chemical composition of the phospholipids, and (c) by the presence of non-lipid constituents such as proteins. Various subcellular membrane structures differ in their lipid composition (Van Hoeven & Emmelot, 1972). Furthermore, they also probably differ in other membrane constituents such as protein and carbohydrates. The only constant in all our calculations seems to be the average length of 18 C-atoms for the fatty acid chains. Therefore it is difficult to make an exact calculation of the diameters of the smallest vesicles in a preparation of biomembranes. Whatever the actual diameter of these vesicles, however, the discreteness of size seems well established. A few equations required for a further development of the model will now be derived.
4. The Concept of the Unit Vesicle
The smallest vesicle () 442 A) formed by biomembranes will be designated as unit vesicle. The mass of a vesicle is proportional with [D(v)- lOO]” (cf. Bont et al., 1977), where D(v) = diameter of a vesicle with V, the number of unit vesicles of which it is composed. The diameter of the unit vesicle is represented by D(1). Since the mass ratio of two vesicles is given by the ratio of the number of unit vesicles, Y, of which they are composed the following
DISCRETENESS
OF VESICLE
367
SIZE
eq,uation can be derived: (1) When D(V) is known, D(1) can be calculated or vice versa. The following equations can easily be derived: D(V) - loo D(1) = +loo JV and D(v) = [O(l)-- 1oo]Jv+ loo
(2)
(3) With the aid of the mass ratios for the geometric series presented in the first row of Table 1 (cf. Bont et al., 1977) this series can be written: D(l), D(3), D(6), D(12) . . . D(3.2“-‘) This series will be designated as series I. Using equation (2), D(1) can be calculated for each term D(V) of this series. In Table 2 the results of these calculations are presented and the mean D(1) and its standard deviation are given. When the fusion hypothesis is applied starting with the unit vesicle, we should obtain the following series. D(l), O(2), O(4), D(8) . . . 0(2”- ‘) This series obtained by fusion of equally sized vesicles is designated as series II. The real values for all terms D(v) of series II have to be calculated with the aid of equation (3), and with the value, D(1), calculated in Table 2. TABLE 2 The diameters, D(l), of the unit vesicle, calculated from the diameters D(V) in Series I. Equation (2) was used for the calculations. D(1) f s.d. = 433 f 12 V = number of unit vesicles
D( v> = diameter of vesicle -.
I 3 6 12 24 48 96 192 384
442 688 948
442 439
1206
419 431
1720 2454 3496
4468 6374
446 440 447 415 420
150 425 778 1016 1539 3653
Sedimentation coefficientt 442 939 1566 1988 2912 6644
Diameter2
the sedimentation
-~ 1 6 16 32 64 384
V-values 442 443 467 434 452 434
D(1) II 11.7 70.4 214.9 356.5 790.7 4282.4
6.02 3.05 1.66 2.22 5.42
Observed mass ratio: $3
6.00 2.67 2.00 2G-l 6.00 where
v1+1 Vf
[(R-5O)2/R],
2
Calculated mass ratio: P1:
coeJicients of membranes of the endoplasmic reticulum from rat liver
t The sedimentation coefficients were published previously (Boom et al., 1976) $ The diameter was obtained from the sedimentation coefficient S, with the aid of the equation S = [3.4/3] R = radius in Angstrom (see Bont et al., 1977) 8 The V-values were either from series I or from series II (see Table 4). /I For the determination of D(1) see Table 2; D(1) = 445113. tt The mass of a vesicle is proportional with [D(V)-lOO]a (cf. Bont et al., 1977). I$ The mass ratio was obtained both from the diameters D(V) (observed) and from the V values (calculated). On average the value (observed mass ratio)/(calculated mass ratio) = 1@010~13.
___~ 1 2 3 4 5 6
i
Data obtainedfrom
TABLE 3
DISCRETENESS
OF
VESICLE
SIZE
369
Although terms of series II are sometimes detected in preparations of biomembranes, by centrifugal analysis (see Table 3) it is striking that a vesicle corresponding to D(2) has never been observed. Therefore, in addition to the hypothesis that only fusion of equally sized vesicles is allowed, it is further postulated that the dimer, D(2), is so reactive that, as an exceptionto the general rule, it can also react with another unit vesicle. In this way the existence of D(3) and thus the formation of series I can be explained. 5. Diameters
Fit in Two Series
In Table 4, D-values taken from literature are compared with the values of both series I (obtained by centrifugal analysis) and series II [calculated with the aid of equation (3)]. All D-values, belonging to the same group of vesicles, are placed in a separate column. Next to this column, the calculated D(l)-values (cf. Table 2) are placed. The following conclusions can be drawn from the data in Table 4. First of all it is evident that those values that do not fit in series I do fit in series II. The value of 787 (V = 4) of synaptic vesicles of cat spleen (Geffen & Ostberg, 1969) deviates less than 3% from the calculated value [D(4) = 766)]. Using the equation [D(4)- lOO]’ = [D(3)- 100]2+ [D(l)lOO]“, D(4) can also be considered as resulting from a fusion between vesicles with D(1) (= 443) and D(3) (= 688) respectively, as given by Geffen & Ostberg (1969). D(4) obtained in this way (= 781) is still better in agreement with the actually measured diameter. The close agreement of the three values for synaptic vesicles given by Geffen & Ostberg (1969) with the first three D-values given in Table 4 supports the evidence that the common ratio for the first two terms of series I is indeed = 3. This value is therefore not due to uncertainties in the measurement of sedimentation coefficients (see Bont et al., 1977). The calculated values for D(1) are remarkably constant in each group; even the differences in D(1) are not large. For the calculation of D(1) with equation (2) the value of I’ must be known. Therefore the relatively small value ofm) = 417 for human hormone vesicles could be attributed to the fact that we did not compare them with the correct V-value. However, the best fit with terms of series I and series II was not the only criterion for the construction of Table 4. In addition the results summarized in Table 5 were of importance. In Table 5 the observed mass ratio between two adjacent members in a group is compared with the calculated mass ratio. Since both the D(l)-values and the calculated mass ratios depend on the V-values ascribed to the components (cf. Table 3) it is evident that Tables 4 and 5 cannot be constructed independently. Why is the calculated D(1) = 417 for the vesicles of human hormones relatively low in comparison
Mean
415
4468
of D(1)fs.d.
433+12
420
447
34%
6374
440
2454
ii? 384
431
446
948
1720
439
688
24 32 48 64 96 128
442
442
1977)
419
er d..
D(1)
12 16
of unit vesicles, V
NlUllbW
5428
3867
2764
1984
1432
1042
766
(571)
Calculated diameter D(V)= [D(l)-lOO]~/y -tloo (series II)
688 787
443
442f3
439 444
443
1250
950
4w!4
‘“%? (Bunge er al., 1965)
of vesicles Synaptic vesicles
D(1)
Source
432
447
D(1)
3395
2410
1415 1475
870 975
et al., 1969)
Hormone vesicles .craf> P;~I.$
42a+i3
436
433
429 444
414 409
D(l)
3568 4529 5599
1816 2260
1355
(Gray,
Hormone vesicles (hum= pituitary)
407 420 444
403 412
414
417*15
1977)
D(1)
Bunge et al., 1965) and of hormone vesicles (Costoff et al., membranes. In contrast to Table 1, the values of this Table of which they are composed. The calculated value for D(2) wasfound neither in vivo nor in vitro
Synaptic vesicles (cat spleen) (Geffen et ul., 1969)
Diameters of synaptic vesicles (Geflen & Ostberg, 1969; 1969; Gray, 1977) compared with diameters of fragmented are arranged according to the number of unit vesicles, V, is put in parentheses since this value
TABLE 4
3
L
vesicles cat et al., 1969) 1
Synaptic vesicles rat (Fh.mge et al., 1965) 1 2 Hormone vesicles rat (Costoff ef al., 1969) TSH 1 ACTH 2 FSH 3 LH 4 sTH5 LTH6 Hormone vesicles human (Gray, 1977) TSH 1 LH 2 FSH3 ACTH; MSH 4 STH 5 LTH6
Synaptic (Geffen
Source of vesicle and its number, j, in the group
533.61 1062.11
48 96
16 32 48 128 192 256
1355 1816 2260 3568 4529 5599
157.50 294.47 466.56 1202.70 196140 302390
59.29 76.56 18090
6 8 16
870 975 1415 1475 2410 3359
12.35 132.25
11.76 34.57 47.20
6 12
1 3 4
V-value
950 1250
443 688 787
Diameter D(Y) of vesicles
1.87 1.58 2.58 l-63 1.54
2.95 199
1.29 2.36
1.83
2.94 1.37
Observed mass ratios DO’,+&--100 D(V,)-100 :
1 a
2.00 1.50 2.67 1.50 1.33
3.00 2.00
1.33 2xIO
2.00
3.00 1.33
Vj
:
vj+1
Calculated mass ratios
A comparison of the mass ratios of vesicles of the same group, determined both from their diameters (observed) and their V-values (calculated). On average, the value (observed mass ratio/calculated mass ratio) = l-02 + 0308
372
W.
S.
BONT
-
with the other values for D(l)? There are two possibilities: either all measurements are - correct and small but significant differences exist between the various D(1) values, or only one value for D(1) exists and all deviations from this value are due to uncertainties in the measurements. At the moment we cannot give a solution for this problem. Since for large values of V, [D(V) 1001 can be approximated by D(V), it is evident [cf. equation (I)] that
where Vp and Vq are two randomly chosen V-values. This relation holds for most of the values in Table 5 especially within the same group. For example, for human hormone vesicle the ratios of measured diameters (Gray, 1977) D(LTH) 2 = 4 DW’H) = 2 and DWTW D(TSH) ’ D(FSH) ’ D(LH) = have to be compared with the square root of the ratios of the corresponding V-values :
respectively. 6. Chemical Physics and Vesicle Size Helfrich (1974) derived a formula to explain the formation of small lipid vesicles based on the edge energy of a hypothetical, circular, flat bilayer and the curvature elastic energy of the corresponding spherical vesicle. Arguments based on thermodynamics, energy of interaction and molecular geometry (Israelachvili et al., 1976, 1977) have been proposed to explain the discreteness of the smallest vesicles (see section 3). The only theoretical attempt to explain the occurrence of the size of large vesicles at preferred values was to point out from topology that packing of hairs on a sphere causes at least one “crown” and that these “crowns” are possible points for fusion. The discreteness of all membrane vesicles is then well understood in a mechanistic way. But we have still to prove, as was done for the smallest vesicles, why also the larger vesicles formed by the aggregation of smaller ones are thermodynamically stable. With the aid of statistical mechanics a better understandingof theco-operative phenomena in biomembranes is possible (McCammon & Deutch, 1975). These phenomena were based on the existence of kinks in the fatty acid chains (Lagaly 62 Weiss, 1971). The discreteness of vesicle size also suggests
DISCRETENESS
OF
VESICLE
SIZE
373
the existence of co-operative effects. How could, otherwise, these giant structures, composed of thousands of molecules, form such well-defined unities? The equations derived by using statistical mechanics must therefore ultimately create the firm basis for our understanding of the real nature of the discreteness of membrane vesicles. 7. Are Breakdown and Fusion Related? Implicit in all our considerations is the assumption that starting with the unit vesicle the existence of discrete sizes has to be explained on the basis of fusion of two equally sized vesicles with only one exception, i.e. the fusion of 3 unit vesicles via an unstable dimer. Difficulties arise when we look at the fragmentation of large membranes. In preparations of plasma membranes the smaller well-defined vesicles are certainly formed in vitro by fragmentation of very large structures. However, it cannot be ruled out that all these structures are ultimately degraded into unit vesicles, which in turn can form larger vesicles by fusion, as described above. The exact processes that in vitro lead to the discontinuous size distribution need still to be clarified. In vivo, however, we can also find a breakdown of large membrane structures into smaller vesicles, e.g. the granular vesicles that are formed from the Golgi membrane. There is no indication that these well-defined vesicles are formed by fusion of smaller ones. Another example is given by the breakdown of granular vesicles in the neurosecretory cell. After secretion by exocytosis the plasma membrane corrects its excess of membrane material via the formation of small vesicles by endocytosis (Norman, 1969). Unless these vesicles are unit vesicles (the size of which is tlhermodynamically determined) it is not easily understood, in the terms of flusion discussed above, why these vesicles are equally sized. These facts lead us to the conclusion that the fusion hypothesis needs to be amended. Instead of giving us the real mechanism for the formation of discrete vesicles it rather seems to enable us in a formal way to determine the diameters that are allowed. 8. Discussion Especially the agreement between calculated and observed diameters for the larger terms in the series, makes it highly improbable that the agreement between the diameters given in various Tables was a matter of chance. Since the difference in diameters in Table 4 is relatively small for the first six members of the combined series (about 500, 700, 800, 1000, 1100, 1200 A) and since most of the viruses fall within this range, only precise measurements
374
W.
S. BONT
can give us the ultimate answer whether or not the size of the viral envelope corresponds to one of the members of the series. Especially the discontinuity in size for larger viruses, however, makes it highly probable that also the size of the viral envelope is determined by purely physical factors and that this size is always one predicted by the series. For instance, a morphological study of eight strains of Newcastle disease virus (Waterson & Cruickshank, 1963) showed a considerable variation in size among the strains. Some strains had relatively small sized particles (1200 A), another strain had particles of about 4000 A, and still another strain particles of even 8000 A. As already pointed out deviations from the spherical form in electron micrographs can be artificial (Jones et al., 1970). Therefore especially for large particles it is difficult to give the exact size when the particles deviate from the spherical form. For instance, the cross section of the immature vaccinia virus is often described as an ellipse with considerable variation in length and breadth. However, in the actual electron micrographs shown in literature often almost circular diameters are observed and using the magnification factor given by the authors Dales & Mosbach (1968) a size of about 3300 A (cf. IZ = 7 in Table 1) is obtained. It is evident that accurate measurements of the real size of vesicles found in vivo are required. In this respect it is also of importance to repeat and extend the sedimentation studies with membrane vesicles, obtained by fragmentation of biomembranes in vitro (Boom et al., 1976; Bont et al., 1977). In addition, since artificial membranes (liposomes), assembled in vitro from phospholipids, also occur in discrete sizes (Bont et al., 1977) they have to be included in future work. Although the fusion model proposed here appears to fit the available data, future work will have to establish the possible significance of this model. In addition we are fully aware of the fact that the theorem of Brouwer (1909) can be applied rigorously only to a continuously varying vector direction and not to a discontinuous distribution of phosphodiester groups. In order to explain in a formal way how singularities can occur on membrane vesicles we adapted the theorem to our problem by making the following assumptions. First: a phosphodiester group can be represented by a vector. Secondly: the theorem also holds for the discontinuous distribution of phosphodiester groups, because the distance between two nearest neighbours is small. In conclusion we can say that each of the examples given is by itself no proof for the existence of the discreteness of vesicle size in nature. However, the finding that so many vesicles have diameters that fit into the geometric series is highly suggestive that the dimensions of vesicles are determined by mainly physical factors.
DISCRETENESS
OF
VESICLE
375
SIZE
I thank Professor S. A. Wouthuyzen (Department of Theoretical Physics. University of Amsterdam) for drawing my attention to the theorem of Brouwer, and Professor J. Th. G. Overbeek (Department of Physical Chemistry, University of Utrecht) and Dr R. P. H. Rettschnick (Department of Physical Chemistry, LJniversity of Amsterdam) for fruitful discussions. REFERENCES Q. F., FISHER, D., T-ION, W. & LUCY, J. A. (1975). Nature, Lond. 253, 194. BANGHAM. A. D. & HORNE. R. W. (1964). J. mol. Biol. 8.660. BONT,W.‘S., BOOM,J., HO&, H. P.‘& Di VRIES, M. (1977)..7.mem. Biol. 36, 213. BOOM, J., BUNT, W. S., HOFS, H. P. & DE VRIES, M. (1976). Mol. biol. Rep. 3, 81. BROUWER, L. E. J. (1909). Proceedings Koninklijke Nederlandse Academic Van WetenAHKONG,
schappen 11,850. BUNGE, R. P., BARTLETT-BUNGE, M. & PETERSON, G. R. (1965). J. cell Biol. 24, 163. COMPANS, R. W. (1971). Nat., new Biol. 229, 114. CUFF, A. & McSn~w, W. H. (1969).J. ceI1. Biol. 43, 564. DALES, S. & M~~BACH, E. H. (1968). Virology 35, 564. DANIELLI, J. F. & DAVSON, H. (1935). J. cell. camp. Physiol. 5,495. FELLER, U., DOUGHERTY, R. M. & Dr STEFANO, H. S. (1971). J. mtn. Cancer Inst. 47,1289. FENNER, F., MCAUSLAN, B. R., MIMS, C. A., SAMBROOK, J. & WHITE, D. 0. (1974). In
The Biology of Animal Viruses, 2nd edn,p. 14.London: AcademicPress. GEFFEN, L. B. & OSTBERG, A. (1969).J. Physiol. 204, 583. GENT, M. P. N. & PRESTEGARD, J. H. (1974). Biochemistry
A. B. (1977).Acta Em&r. 85,249. HELFRICH. W. (1974‘). Phv. Lett. SOA.115-116.
13, 4027.
GRAY,
HUANG, d. H. i196!$. ISRAELACHVILI, J. N., Trans. ZZ,72, 1525. ISRAELACHVILI, J. N.,
Bhxhem. 8, 344. MITCHELL,
D.
MITCHELL,
D. J.
470,185. JOHNSON,
S. M.,
BANGHAM,
A. D.,
J. & NINHAM,
HILL,
& NINHAM, M. W.
B. W.
(1976). J. them. Sot., Farad.
B. W. (1977).
Biochim. biophys. Acta
& KORN, E. D. (1971).
Biochem. biophys.
Acta 233, 820. JONES, S. F. & KWANBUNBUMPEN, S. (1970).J. Physiol. 207, 31, KUROTCHKIN, T. J., LIBBY, R. L., CAGNON, E. & Cox, H. R. (1947). J. Zmmunol. 55,283. LAGALY, G. & WEISS, A. (1971).Angew. Chem. Znt. ed. Engl. 10, 558. LUUFIIG, B., MCMILLAN, P. N., CULBRETH, K. & BOLOGNESI, D. P. (1974). Cant. Res. 34,
1694. MCCAMMON, MCINTOSH,
J. A. & DEUTCH, J. M. (1975). J. Am. them. Sot. 97, 6675. K., DEES, J. H., BECKER, W. B., KAPIKIAN, A. Z. & CHANOCK,
Proc. natn. Acad. Sci. U.S.A., 57,933. NORMANN. T. C. (1969). EXD. cell Res. 55.285. PAGANO, R. E. &‘HuA;G, L. (1975). J. ceB. Biol. 67, 49. PALADE, G. (1975).Science 189,347. .ROBERTSON, J. D. (1969). Biochem. sot. Symp. 16, 3. ROBERTSON. J. D. (1964). In Cellular Membranes in Develooment CM. Locke. Yolk: Acahemk Press. ;SARKAR, N. H., MANTHEY, W. J. & SHEFFIELD, J. B. (1975). Cant. Res. 35, VAN HOE~EN, R. P. & EMMEL~T, P. (1972). J. mem. Biol. 9, 105. WATERSON, A. P. & CRUIKSHANK, J. G. (1963). Z. Naturforsch. 8B, 114. WICKNER, T. (1977). Biochem. 16,254. WILDY, P., RUSSELL, W. C. & HORNE, R. W. (1960). Virology 12, 204.
R. M.
(1967).
ed.). ,,_.DD. 1-81.
New
740.
The Diameters of Membrane Vesicles Fit in Geometric Series W. S. BONT Department of Biophysics, Antoni Van Leeuwenhoek-Huis, The Netherlands Cancer Institute, Amsterdam, The Netherlanh (Received 23 January 1978, and in revisedform
17 March 1978)
Vesicles formed in vitro by fragmentation of biomembranes are restricted to certain dimensions; the diameters are represented by two geometric series. The diameters of membrane vesicles found in intact cells, including viral envelopes, are terms of the same two series. A tentative mathematical model is proposed, to explain this phenomenon by fusion of equally sized vesicles. 1. Introduction
All membranes found in nature show a basic architectural design (Robertson, 1959); the structure common to all membranes is the lipid bilayer (Danielli d Davson, 1935). Membranes are an integral part of living cells; every cell is confined by a membrane. In eukaryotic cells in addition they form the boundary of the nucleus. A host of other cellular structures also requires the presence of membranes. Often membranes form vesicles, i.e. small spherical bodies confined by a lipid bilayer, that serve the purpose, e.g. of storageof materials produced by the cell. Thus, materials like neurotransmitters and hormones destined for export or proteins required in a later phase of the cell’s existence are temporarily enveloped in membranes (Palade, 1975). Phospholipids are the major constituents of virtually all membrane bilayers and accordingly they primarily exist in the form of closed spherical bodies (Bangham & Horne, 1964). Therefore vesicles are formed when large membraneous structures are fragmented in vitro. Thus far no systematic investigation has been made on the size distribution of vesicles both present in vivo and prepared in vitro. Recently we started such an investigation (Boom, Bont, Hofs & De Vries, 1976; Bont, Boom, Hofs & De Vries, 1977). The size distribution of vesicles obtained in vitro by fragmentation of large membrane structures was studied by analytical centrifugation. It was demonstrated that the vesicles obtained 361
0022~5193/78/190361+15
$02.00/O
Q 1978 Academic Press Inc. (London)
Ltd.
362
W.
S.
BONT
in vitro occur in only a few sizes with the exclusion of other sizes (Boom rr al., 1976). On the other hand the size distribution of vesicles found in viva in various cell types has been determined with the electron microscope and it was shown that the standard deviation of the mean size was small (Geffen & Ostberg, 1969; Bunge, Bartlett-Bunge & Peterson, 1965; Costoff & McShaw, 1969). Especially from the comparison between vesicles occurring in vitro and those occurring in z&o it appeared that the masses of vesicles are terms of the same geometric series with a common ratio of 2 (Bont et al., 1977). The following aspects related to these findings need to be elucidated. (a) How can the existence of vesicles, that do not fit in the geometric series, be explained? (b) What is the mechanism underlying the process that leads to the results cited above? (c) Is the finding that vesicle size is determined by purely physical factors generally applicable to all lipid bilayers and therefore also to the envelopes of, e.g. viruses and chlamydiae ? In this paper we present further data supporting the assumption that the size of all vesicles is determined by purely physical factors and also propose a tentative mathematical model that might explain the size discreteness of vesicles.
2. Size of Hormone Vesicles, Viruses and Chlamydiae A comparison between the radii of vesicles formed in vitro and those of granular vesicles found in vioo revealed that these radii irrespective of their source are terms of the same geometric series (Bont et al., 1977). Recently Gray (1977) published diameters of granular hormone vesicles from human pituitary, and in Table 1 we compare also these data with those of the geometric series presented previously (Bont et al., 1977). Since a great number of other particles also is confined by a phospholipid bilayer we extended the comparison in Table 1 to viruses and chlamydiae. Again a striking agreement is found between both quantities. As to the real significance of this comparison the following must be kept in mind. When we restrict ourselves to spherical vesicles, the real diameter of the membrane vesicle sometimes cannot be derived unequivocally from the electron micrographs. Artifacts introduced by our manipulations give rise to distortion of the real dimension due to (a) deviations from the spherical form (Jones & Kwanbunbumpen, 1970), (b) various techniques used for fixation and staining (Feller, Dougherty & Di Stefano, 1971), and (c) surface projections on viruses obscuring the real diameter of the membraneous envelope (Luftig,
1
450
442
1 688
2
1200t
1355
TSH
FSH 1415
TSH 870 ACTH 975
1206
1800
1816
LH 1475 LH
1720
2260
35403
ACTH ; MSH 3568
LTH 3395
STH 2410 FSH
3496
6
2454
Serial number of vesicles 4 5 5
948
3
4500$
4529
STH
4468
8
5599
LTH
6374
9
t This value was obtained by subtracting 400 (the spike height times 2) from the overall diameter of 1600 (cf. McIntosh et al., 1967). $ The particles for meningopneumonitis (3540) are significantly smaller than the mean (4500) of the remainder of the group, that consists of six other types of chluq&ae (cf. Kurotchkin et al., 1947).
Sindbis virus (Cornpans, 1971) I.B.V. (McIntosh, Dees, Becker, Kapikian & Chanock, 1967) Herpes virus (Wildy, Russell & Home, 1960) Chhaydiae (Kurotchkin et al., 1947)
Human pituitary (Gray, 1977)
Fragmented membanes (Bent et al., 1977) Rat pituitary (Costoff et al., 1969)
Source of particles
Diameters of granular hormone vesicles, viruses and chlamydiae (by electron microscopy) compared with diameters of membrane vesicles in preparations of fragmented biomembranes(by sedimentation analysis). In all Tables diameters are given in Angstrom
TABLE
364
W.
S. BONT
McMillan, Culbreth & Bolognesi, 1974). Therefore it is difficult to select from the wealth of data given for the diameters of viruses (cf. Luftig et al., 1974). We therefore only present those data given in the literature from which the diameters of the membrane vesicles can unequivocally be determined. Often, measurements from the electron micrographs presented in literature, using the magnification factors given by the authors, yielded striking correlations with the values presented in Table 1. However, these values were omitted. In general, it is obvious that large groups of viruses have the same size (Fenner, McAuslan, Mims, Sambrook & White, 1974) and that in addition most of the sizes given in literature certainly do not exclude a priori the possibility that the size of the membrane envelope associated with these viruses corresponds to one of the members of the series. In this respect it is of interest to cite the work of Sarker, Manthey & Sheffield (1975) with MuMTV and MuLV. These authors measured the diameters of the particles and constructed a histogram. MuMTV is apparently less stable than MuLV and therefore it is disrupted under various conditions. Instead of the expected continuous size distribution of the breakdown products, they found that these products had discrete sizes, giving rise to a peak in the histogram. Sarker et al. (1975) expressed their surprise by the following statement “It is puzzling that a peak should occur in the distribution of these small diameters of MuMTV”. Apparently, the breakdown products of the viruses, as long as they are confined by a membrane, are restricted to a certain size. The work with particles from the psittacosis-lymphogranular group (cldamydiae) is included in Table 1. Six of the seven members of this group given by Kurotchkin, Libby, Gagnon & Cox (1947) have an average diameter of 4500 A (ranging from 4220 A to 4970 A). Only the one for meningopneumonitis is significantly lower (3540 A). But both size classes fit with the values of the series for n = 8 and M = 7 respectively.
3. A Tentative Mathematical
Model
Brouwer (1909) has formulated the following theorem : “A vector direction varying continuously on a singly connected two-sided closed surface must be indeterminate in at least one point”. In terms more appropriate to our problem this theorem reads: “ ‘Hairs’ cannot be implanted in a sphere in an orderly fashion; at least one singular point (‘crown’) must be formed”. If the vesicles are assumed to be spherical, the phosphodiester groups of the diglycerides can be considered as “hairs”. Therefore these groups must form singular points on the surface of the vesicles.
DISCRETENESS
OF
VESICLE
SIZE
365
If these singular points are reactive sites on the surface, one could postulate that they change in nature and perhaps also in number with the size of the vesicles. Now it seems possible that the vesicles fuse at these reactive singular points and that only fusion of identical points is allowed. This would explain the occurrence of the geometric series and the findings mentioned above. Furthermore it is in agreement with the assumption that for vesicle-cell fusion there is an interaction between the phospholipid bilayers (Pagan0 & Huang, 1975) and that cell-cell fusion takes place at sides where the bilayers, denuded of intramembraneous particles, are exposed (Ahkong, Fisher, Tampion & Lucy, 1975). Implicit in all our considerations then is the assumption that the peculiar properties of the biomembranes (especially in respect of the discreteness of vesicle size) reside in the phospholipid moiety. This assumption is strengthened by the fact that liposomes also occur in discrete sizes (Bont et al., 1977). The problem of the discreteness of vesicle size has, however, two aspects. First: why is the smallest vesicle discrete and secondly : why are the other vesicles multiples of the smallest one? Robertson (1964) has already stated that the smallest vesicles cannot exist below a certain critical value of about 300 A. But this statement does not exclude the possibility that above 300 A all diameters are allowed. Application of the proposed mathematical model would then result in a continuous size distribution of the fusion products. What we need is a theoretical justification for the discreteness of the smallest vesicles and the derivation of the equations with which the absolute value of these vesicles can be calculated. In other words, for the applicability of the mathematical model, a proof of the discreteness of the smallest vesicles and the determination of their diameters is a prerequisite. Relative simple systems can best be used both experimentally and theoretically, for this kind of studies. Huang (1969) demonstrated that vesicles formed with phosphatidylcholine from chicken eggs showed a very narrow size distribution with a diameter of 228 + 5 A. Johnson, Bangham, Hill & Korn (1971) confirmed these results and Helfrich (1974) could demonstrate on purely theoretical grounds that a diameter of about 400 A can be expected for these smallest vesicles. Later, Israelachvili, Mitchell & Ninham (1976, 1977) developed a unified theory that accounts quantitatively for many of the physical properties of membrane vesicles and that links thermodynamics, interaction free energy, and molecular geometry. These authors not only confirmed by the application of their theory the experimental findings of Huang (1969) and Johnson ef al. (1971) for egg lecithin but also were able to derive equations from which it followed (a) that the smallest vesicles are homogeneous in size, (b) that a change in the average length of the fatty acid chains causes a shift in size. In addition these authors were able to explain the findings of Gent &
366
W.
S. BONT
Prestegard (1974) that (c) in the phosphatidylcholine-phosphatidylethanolamine system the vesicle size increases by increasing the percentage of phosphatidylethanolamine. A mole fraction, F = 0.52, for phosphatidylethanolamine results in a diameter of 350 A for the smallest liposomes. All these studes were restricted to phospholipid vesicles. When we want to extrapolate all these data to real membrane vesicles found in situ we have to keep in mind that a substantial part of biomembranes consists of proteins. Phospholipids contribute about 45 A to the thickness of the bilayer and therefore the contribution of the other components, e.g. proteins, carbohydrates, present in biomembranes, is about 558. When we take hypothetical biomembrane vesicles with a mole fraction, F = 0.52, for phosphatidylethanolamine, we could then expect the diameter for the smallest vesicles to be 350+55 = 405A. This value differs less than 10% from the diameter of 442 A observed for the smallest vesicles in preparation of biomembranes. Furthermore Wickner (1977) demonstrated that for a given phospholipid composition the presence of protein per se might influence the radius of the smallest vesicles. In summary, we can say that the size of the smallest vesicles is influenced (a) by the length of the fatty acid chains, (b) by the chemical composition of the phospholipids, and (c) by the presence of non-lipid constituents such as proteins. Various subcellular membrane structures differ in their lipid composition (Van Hoeven & Emmelot, 1972). Furthermore, they also probably differ in other membrane constituents such as protein and carbohydrates. The only constant in all our calculations seems to be the average length of 18 C-atoms for the fatty acid chains. Therefore it is difficult to make an exact calculation of the diameters of the smallest vesicles in a preparation of biomembranes. Whatever the actual diameter of these vesicles, however, the discreteness of size seems well established. A few equations required for a further development of the model will now be derived.
4. The Concept of the Unit Vesicle
The smallest vesicle () 442 A) formed by biomembranes will be designated as unit vesicle. The mass of a vesicle is proportional with [D(v)- lOO]” (cf. Bont et al., 1977), where D(v) = diameter of a vesicle with V, the number of unit vesicles of which it is composed. The diameter of the unit vesicle is represented by D(1). Since the mass ratio of two vesicles is given by the ratio of the number of unit vesicles, Y, of which they are composed the following
DISCRETENESS
OF VESICLE
367
SIZE
eq,uation can be derived: (1) When D(V) is known, D(1) can be calculated or vice versa. The following equations can easily be derived: D(V) - loo D(1) = +loo JV and D(v) = [O(l)-- 1oo]Jv+ loo
(2)
(3) With the aid of the mass ratios for the geometric series presented in the first row of Table 1 (cf. Bont et al., 1977) this series can be written: D(l), D(3), D(6), D(12) . . . D(3.2“-‘) This series will be designated as series I. Using equation (2), D(1) can be calculated for each term D(V) of this series. In Table 2 the results of these calculations are presented and the mean D(1) and its standard deviation are given. When the fusion hypothesis is applied starting with the unit vesicle, we should obtain the following series. D(l), O(2), O(4), D(8) . . . 0(2”- ‘) This series obtained by fusion of equally sized vesicles is designated as series II. The real values for all terms D(v) of series II have to be calculated with the aid of equation (3), and with the value, D(1), calculated in Table 2. TABLE 2 The diameters, D(l), of the unit vesicle, calculated from the diameters D(V) in Series I. Equation (2) was used for the calculations. D(1) f s.d. = 433 f 12 V = number of unit vesicles
D( v> = diameter of vesicle -.
I 3 6 12 24 48 96 192 384
442 688 948
442 439
1206
419 431
1720 2454 3496
4468 6374
446 440 447 415 420
150 425 778 1016 1539 3653
Sedimentation coefficientt 442 939 1566 1988 2912 6644
Diameter2
the sedimentation
-~ 1 6 16 32 64 384
V-values 442 443 467 434 452 434
D(1) II 11.7 70.4 214.9 356.5 790.7 4282.4
6.02 3.05 1.66 2.22 5.42
Observed mass ratio: $3
6.00 2.67 2.00 2G-l 6.00 where
v1+1 Vf
[(R-5O)2/R],
2
Calculated mass ratio: P1:
coeJicients of membranes of the endoplasmic reticulum from rat liver
t The sedimentation coefficients were published previously (Boom et al., 1976) $ The diameter was obtained from the sedimentation coefficient S, with the aid of the equation S = [3.4/3] R = radius in Angstrom (see Bont et al., 1977) 8 The V-values were either from series I or from series II (see Table 4). /I For the determination of D(1) see Table 2; D(1) = 445113. tt The mass of a vesicle is proportional with [D(V)-lOO]a (cf. Bont et al., 1977). I$ The mass ratio was obtained both from the diameters D(V) (observed) and from the V values (calculated). On average the value (observed mass ratio)/(calculated mass ratio) = 1@010~13.
___~ 1 2 3 4 5 6
i
Data obtainedfrom
TABLE 3
DISCRETENESS
OF
VESICLE
SIZE
369
Although terms of series II are sometimes detected in preparations of biomembranes, by centrifugal analysis (see Table 3) it is striking that a vesicle corresponding to D(2) has never been observed. Therefore, in addition to the hypothesis that only fusion of equally sized vesicles is allowed, it is further postulated that the dimer, D(2), is so reactive that, as an exceptionto the general rule, it can also react with another unit vesicle. In this way the existence of D(3) and thus the formation of series I can be explained. 5. Diameters
Fit in Two Series
In Table 4, D-values taken from literature are compared with the values of both series I (obtained by centrifugal analysis) and series II [calculated with the aid of equation (3)]. All D-values, belonging to the same group of vesicles, are placed in a separate column. Next to this column, the calculated D(l)-values (cf. Table 2) are placed. The following conclusions can be drawn from the data in Table 4. First of all it is evident that those values that do not fit in series I do fit in series II. The value of 787 (V = 4) of synaptic vesicles of cat spleen (Geffen & Ostberg, 1969) deviates less than 3% from the calculated value [D(4) = 766)]. Using the equation [D(4)- lOO]’ = [D(3)- 100]2+ [D(l)lOO]“, D(4) can also be considered as resulting from a fusion between vesicles with D(1) (= 443) and D(3) (= 688) respectively, as given by Geffen & Ostberg (1969). D(4) obtained in this way (= 781) is still better in agreement with the actually measured diameter. The close agreement of the three values for synaptic vesicles given by Geffen & Ostberg (1969) with the first three D-values given in Table 4 supports the evidence that the common ratio for the first two terms of series I is indeed = 3. This value is therefore not due to uncertainties in the measurement of sedimentation coefficients (see Bont et al., 1977). The calculated values for D(1) are remarkably constant in each group; even the differences in D(1) are not large. For the calculation of D(1) with equation (2) the value of I’ must be known. Therefore the relatively small value ofm) = 417 for human hormone vesicles could be attributed to the fact that we did not compare them with the correct V-value. However, the best fit with terms of series I and series II was not the only criterion for the construction of Table 4. In addition the results summarized in Table 5 were of importance. In Table 5 the observed mass ratio between two adjacent members in a group is compared with the calculated mass ratio. Since both the D(l)-values and the calculated mass ratios depend on the V-values ascribed to the components (cf. Table 3) it is evident that Tables 4 and 5 cannot be constructed independently. Why is the calculated D(1) = 417 for the vesicles of human hormones relatively low in comparison
Mean
415
4468
of D(1)fs.d.
433+12
420
447
34%
6374
440
2454
ii? 384
431
446
948
1720
439
688
24 32 48 64 96 128
442
442
1977)
419
er d..
D(1)
12 16
of unit vesicles, V
NlUllbW
5428
3867
2764
1984
1432
1042
766
(571)
Calculated diameter D(V)= [D(l)-lOO]~/y -tloo (series II)
688 787
443
442f3
439 444
443
1250
950
4w!4
‘“%? (Bunge er al., 1965)
of vesicles Synaptic vesicles
D(1)
Source
432
447
D(1)
3395
2410
1415 1475
870 975
et al., 1969)
Hormone vesicles .craf> P;~I.$
42a+i3
436
433
429 444
414 409
D(l)
3568 4529 5599
1816 2260
1355
(Gray,
Hormone vesicles (hum= pituitary)
407 420 444
403 412
414
417*15
1977)
D(1)
Bunge et al., 1965) and of hormone vesicles (Costoff et al., membranes. In contrast to Table 1, the values of this Table of which they are composed. The calculated value for D(2) wasfound neither in vivo nor in vitro
Synaptic vesicles (cat spleen) (Geffen et ul., 1969)
Diameters of synaptic vesicles (Geflen & Ostberg, 1969; 1969; Gray, 1977) compared with diameters of fragmented are arranged according to the number of unit vesicles, V, is put in parentheses since this value
TABLE 4
3
L
vesicles cat et al., 1969) 1
Synaptic vesicles rat (Fh.mge et al., 1965) 1 2 Hormone vesicles rat (Costoff ef al., 1969) TSH 1 ACTH 2 FSH 3 LH 4 sTH5 LTH6 Hormone vesicles human (Gray, 1977) TSH 1 LH 2 FSH3 ACTH; MSH 4 STH 5 LTH6
Synaptic (Geffen
Source of vesicle and its number, j, in the group
533.61 1062.11
48 96
16 32 48 128 192 256
1355 1816 2260 3568 4529 5599
157.50 294.47 466.56 1202.70 196140 302390
59.29 76.56 18090
6 8 16
870 975 1415 1475 2410 3359
12.35 132.25
11.76 34.57 47.20
6 12
1 3 4
V-value
950 1250
443 688 787
Diameter D(Y) of vesicles
1.87 1.58 2.58 l-63 1.54
2.95 199
1.29 2.36
1.83
2.94 1.37
Observed mass ratios DO’,+&--100 D(V,)-100 :
1 a
2.00 1.50 2.67 1.50 1.33
3.00 2.00
1.33 2xIO
2.00
3.00 1.33
Vj
:
vj+1
Calculated mass ratios
A comparison of the mass ratios of vesicles of the same group, determined both from their diameters (observed) and their V-values (calculated). On average, the value (observed mass ratio/calculated mass ratio) = l-02 + 0308
372
W.
S.
BONT
-
with the other values for D(l)? There are two possibilities: either all measurements are - correct and small but significant differences exist between the various D(1) values, or only one value for D(1) exists and all deviations from this value are due to uncertainties in the measurements. At the moment we cannot give a solution for this problem. Since for large values of V, [D(V) 1001 can be approximated by D(V), it is evident [cf. equation (I)] that
where Vp and Vq are two randomly chosen V-values. This relation holds for most of the values in Table 5 especially within the same group. For example, for human hormone vesicle the ratios of measured diameters (Gray, 1977) D(LTH) 2 = 4 DW’H) = 2 and DWTW D(TSH) ’ D(FSH) ’ D(LH) = have to be compared with the square root of the ratios of the corresponding V-values :
respectively. 6. Chemical Physics and Vesicle Size Helfrich (1974) derived a formula to explain the formation of small lipid vesicles based on the edge energy of a hypothetical, circular, flat bilayer and the curvature elastic energy of the corresponding spherical vesicle. Arguments based on thermodynamics, energy of interaction and molecular geometry (Israelachvili et al., 1976, 1977) have been proposed to explain the discreteness of the smallest vesicles (see section 3). The only theoretical attempt to explain the occurrence of the size of large vesicles at preferred values was to point out from topology that packing of hairs on a sphere causes at least one “crown” and that these “crowns” are possible points for fusion. The discreteness of all membrane vesicles is then well understood in a mechanistic way. But we have still to prove, as was done for the smallest vesicles, why also the larger vesicles formed by the aggregation of smaller ones are thermodynamically stable. With the aid of statistical mechanics a better understandingof theco-operative phenomena in biomembranes is possible (McCammon & Deutch, 1975). These phenomena were based on the existence of kinks in the fatty acid chains (Lagaly 62 Weiss, 1971). The discreteness of vesicle size also suggests
DISCRETENESS
OF
VESICLE
SIZE
373
the existence of co-operative effects. How could, otherwise, these giant structures, composed of thousands of molecules, form such well-defined unities? The equations derived by using statistical mechanics must therefore ultimately create the firm basis for our understanding of the real nature of the discreteness of membrane vesicles. 7. Are Breakdown and Fusion Related? Implicit in all our considerations is the assumption that starting with the unit vesicle the existence of discrete sizes has to be explained on the basis of fusion of two equally sized vesicles with only one exception, i.e. the fusion of 3 unit vesicles via an unstable dimer. Difficulties arise when we look at the fragmentation of large membranes. In preparations of plasma membranes the smaller well-defined vesicles are certainly formed in vitro by fragmentation of very large structures. However, it cannot be ruled out that all these structures are ultimately degraded into unit vesicles, which in turn can form larger vesicles by fusion, as described above. The exact processes that in vitro lead to the discontinuous size distribution need still to be clarified. In vivo, however, we can also find a breakdown of large membrane structures into smaller vesicles, e.g. the granular vesicles that are formed from the Golgi membrane. There is no indication that these well-defined vesicles are formed by fusion of smaller ones. Another example is given by the breakdown of granular vesicles in the neurosecretory cell. After secretion by exocytosis the plasma membrane corrects its excess of membrane material via the formation of small vesicles by endocytosis (Norman, 1969). Unless these vesicles are unit vesicles (the size of which is tlhermodynamically determined) it is not easily understood, in the terms of flusion discussed above, why these vesicles are equally sized. These facts lead us to the conclusion that the fusion hypothesis needs to be amended. Instead of giving us the real mechanism for the formation of discrete vesicles it rather seems to enable us in a formal way to determine the diameters that are allowed. 8. Discussion Especially the agreement between calculated and observed diameters for the larger terms in the series, makes it highly improbable that the agreement between the diameters given in various Tables was a matter of chance. Since the difference in diameters in Table 4 is relatively small for the first six members of the combined series (about 500, 700, 800, 1000, 1100, 1200 A) and since most of the viruses fall within this range, only precise measurements
374
W.
S. BONT
can give us the ultimate answer whether or not the size of the viral envelope corresponds to one of the members of the series. Especially the discontinuity in size for larger viruses, however, makes it highly probable that also the size of the viral envelope is determined by purely physical factors and that this size is always one predicted by the series. For instance, a morphological study of eight strains of Newcastle disease virus (Waterson & Cruickshank, 1963) showed a considerable variation in size among the strains. Some strains had relatively small sized particles (1200 A), another strain had particles of about 4000 A, and still another strain particles of even 8000 A. As already pointed out deviations from the spherical form in electron micrographs can be artificial (Jones et al., 1970). Therefore especially for large particles it is difficult to give the exact size when the particles deviate from the spherical form. For instance, the cross section of the immature vaccinia virus is often described as an ellipse with considerable variation in length and breadth. However, in the actual electron micrographs shown in literature often almost circular diameters are observed and using the magnification factor given by the authors Dales & Mosbach (1968) a size of about 3300 A (cf. IZ = 7 in Table 1) is obtained. It is evident that accurate measurements of the real size of vesicles found in vivo are required. In this respect it is also of importance to repeat and extend the sedimentation studies with membrane vesicles, obtained by fragmentation of biomembranes in vitro (Boom et al., 1976; Bont et al., 1977). In addition, since artificial membranes (liposomes), assembled in vitro from phospholipids, also occur in discrete sizes (Bont et al., 1977) they have to be included in future work. Although the fusion model proposed here appears to fit the available data, future work will have to establish the possible significance of this model. In addition we are fully aware of the fact that the theorem of Brouwer (1909) can be applied rigorously only to a continuously varying vector direction and not to a discontinuous distribution of phosphodiester groups. In order to explain in a formal way how singularities can occur on membrane vesicles we adapted the theorem to our problem by making the following assumptions. First: a phosphodiester group can be represented by a vector. Secondly: the theorem also holds for the discontinuous distribution of phosphodiester groups, because the distance between two nearest neighbours is small. In conclusion we can say that each of the examples given is by itself no proof for the existence of the discreteness of vesicle size in nature. However, the finding that so many vesicles have diameters that fit into the geometric series is highly suggestive that the dimensions of vesicles are determined by mainly physical factors.
DISCRETENESS
OF
VESICLE
375
SIZE
I thank Professor S. A. Wouthuyzen (Department of Theoretical Physics. University of Amsterdam) for drawing my attention to the theorem of Brouwer, and Professor J. Th. G. Overbeek (Department of Physical Chemistry, University of Utrecht) and Dr R. P. H. Rettschnick (Department of Physical Chemistry, LJniversity of Amsterdam) for fruitful discussions. REFERENCES Q. F., FISHER, D., T-ION, W. & LUCY, J. A. (1975). Nature, Lond. 253, 194. BANGHAM. A. D. & HORNE. R. W. (1964). J. mol. Biol. 8.660. BONT,W.‘S., BOOM,J., HO&, H. P.‘& Di VRIES, M. (1977)..7.mem. Biol. 36, 213. BOOM, J., BUNT, W. S., HOFS, H. P. & DE VRIES, M. (1976). Mol. biol. Rep. 3, 81. BROUWER, L. E. J. (1909). Proceedings Koninklijke Nederlandse Academic Van WetenAHKONG,
schappen 11,850. BUNGE, R. P., BARTLETT-BUNGE, M. & PETERSON, G. R. (1965). J. cell Biol. 24, 163. COMPANS, R. W. (1971). Nat., new Biol. 229, 114. CUFF, A. & McSn~w, W. H. (1969).J. ceI1. Biol. 43, 564. DALES, S. & M~~BACH, E. H. (1968). Virology 35, 564. DANIELLI, J. F. & DAVSON, H. (1935). J. cell. camp. Physiol. 5,495. FELLER, U., DOUGHERTY, R. M. & Dr STEFANO, H. S. (1971). J. mtn. Cancer Inst. 47,1289. FENNER, F., MCAUSLAN, B. R., MIMS, C. A., SAMBROOK, J. & WHITE, D. 0. (1974). In
The Biology of Animal Viruses, 2nd edn,p. 14.London: AcademicPress. GEFFEN, L. B. & OSTBERG, A. (1969).J. Physiol. 204, 583. GENT, M. P. N. & PRESTEGARD, J. H. (1974). Biochemistry
A. B. (1977).Acta Em&r. 85,249. HELFRICH. W. (1974‘). Phv. Lett. SOA.115-116.
13, 4027.
GRAY,
HUANG, d. H. i196!$. ISRAELACHVILI, J. N., Trans. ZZ,72, 1525. ISRAELACHVILI, J. N.,
Bhxhem. 8, 344. MITCHELL,
D.
MITCHELL,
D. J.
470,185. JOHNSON,
S. M.,
BANGHAM,
A. D.,
J. & NINHAM,
HILL,
& NINHAM, M. W.
B. W.
(1976). J. them. Sot., Farad.
B. W. (1977).
Biochim. biophys. Acta
& KORN, E. D. (1971).
Biochem. biophys.
Acta 233, 820. JONES, S. F. & KWANBUNBUMPEN, S. (1970).J. Physiol. 207, 31, KUROTCHKIN, T. J., LIBBY, R. L., CAGNON, E. & Cox, H. R. (1947). J. Zmmunol. 55,283. LAGALY, G. & WEISS, A. (1971).Angew. Chem. Znt. ed. Engl. 10, 558. LUUFIIG, B., MCMILLAN, P. N., CULBRETH, K. & BOLOGNESI, D. P. (1974). Cant. Res. 34,
1694. MCCAMMON, MCINTOSH,
J. A. & DEUTCH, J. M. (1975). J. Am. them. Sot. 97, 6675. K., DEES, J. H., BECKER, W. B., KAPIKIAN, A. Z. & CHANOCK,
Proc. natn. Acad. Sci. U.S.A., 57,933. NORMANN. T. C. (1969). EXD. cell Res. 55.285. PAGANO, R. E. &‘HuA;G, L. (1975). J. ceB. Biol. 67, 49. PALADE, G. (1975).Science 189,347. .ROBERTSON, J. D. (1969). Biochem. sot. Symp. 16, 3. ROBERTSON. J. D. (1964). In Cellular Membranes in Develooment CM. Locke. Yolk: Acahemk Press. ;SARKAR, N. H., MANTHEY, W. J. & SHEFFIELD, J. B. (1975). Cant. Res. 35, VAN HOE~EN, R. P. & EMMEL~T, P. (1972). J. mem. Biol. 9, 105. WATERSON, A. P. & CRUIKSHANK, J. G. (1963). Z. Naturforsch. 8B, 114. WICKNER, T. (1977). Biochem. 16,254. WILDY, P., RUSSELL, W. C. & HORNE, R. W. (1960). Virology 12, 204.
R. M.
(1967).
ed.). ,,_.DD. 1-81.
New
740.
