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Spatial distribution of surface immunoglobulin on B lymphocytes
. .....__ I .___,, Copyright @ 1978 by Academic Press, Inc. All rights of reproduction in any form reserved 0014-4827/78/l 122-0309$02.00/0
Experimental
SPATIAL
DISTRIBUTION
Cell Research 1I2 (1978) 309-321
OF SURFACE
IMMUNOGLOBULIN
ON B LYMPHOCYTES Local Ordering ALAN S. PERELSON Theoretical Division, University of California, Los Alamos Scientifc Laboratory Los Alamos, NM 87545, USA
SUMMARY Maps of the spatial distribution of surface immunoglobulin on B lymphocytes obtained by freezeetch electron microscopy were analysed mathematically. The pictures were automatically scanned by a digital microdensitometer and the number and coordinates of the immunoglobulin molecules determined. A statistical measure, the radial distribution function, g(r), commonly used to study the structure of liquids, was computed for each map. The radial distribution function of surface immunoglobulin was found to closely resemble g(r) for fluids, indicating the presence of shortrange (but not long-range) order. It was determined that a given surface immunoglobulin molecule has a high probability of being surrounded by other immunoglobulins at distances approximately one-half the mean interparticle spacing. From this one can conclude that the distribution of surface immunoglobulin is non-random and is characterized by local clustering. The mean number of first nearest neighbors surrounding each surface immunoglobulin, computed from g(r), was found to be near two. This value is consistent with a degree of linear order in the topographic distribution of immunoglobulin. The radial distribution function provides a method of quantitating the amount of clustering in a spatial distribution. This function may prove useful in accessing the amount of receptor cross-linking necessary to trigger cellular responses, and in elucidating mechanisms for aggregation and movement of membrane macromolecules.
The fluid mosaic model of the cell membrane suggests that proteins are free to diffuse within the plane of the membrane Cl]. Consequently, one might naively expect to find that proteins are randomly distributed with respect to one another, especially at large separation distances. Clearly, however, the existence of either short or long range forces acting on membrane proteins can lead to non-random spatial distributions . One type of protein found on the surface of bone marrow derived (B) lymphocytes is immunoglobulin (Ig), which serves as a receptor for antigen [2]. The distribution and
behavior of surface Ig has been extensively studied in attempts to understand the mechanisms by which lymphocytes recognize foreign materials and begin their immune functioning. One important finding has been that in the presence of a multivalent ligand which can cross-link receptors, the surface Ig rapidly redistributes into microaggregates or patches. At temperatures above 20°C at which the membrane lipids remain fluid, the patches undergo a further redistribution and aggregate at one pole of the cell in a process known as capping [3-51. The ligand induced redistribution of surface Ig is taken as further evidence in support of Exp Cell Res I12 (1978)
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A. S. Perelson
the fluid mosaic hypothesis [6-71. Capping can be interfered with by numerous agents, such as sodium azide [3, 8-91, dinitrophenol [3, 7-91, cyanide [7-91, cytochalasin B (CB) [3, lo], vinblastine [ 10, 121,and colchicine used in combination with CB [lo], or by lowering the temperature from 20” to near 0°C [34] suggesting that it is an energy requiring process in which microtubules and microtilaments may play a role [lo]. Edelman & Yahara [ 11, 121 have, in fact, hypothesized that there may be a direct link between surface molecules and an underlying cytoplasmic fibrillar network [ 13-151. As an alternative explanation of capping, Bretscher [16] has suggested that directed lipid flow may move surface macromolecules. If either of these hypotheses are correct one might not expect to find a random distribution of surface Ig molecules. The precise native distribution of surface proteins such as Ig before ligand binding has been difftcult to measure because of technical problems in cell labeling and ultrastructural examination [17]. Studies of the spatial organization of surface Ig on murine B lymphocytes labeled with a monovalent ligand and examined by immunofluorescence which has low resolution, or by electron microscopy of thin sections, which only reveals the pattern in the plane of the section, have disclosed patterns described as random [34]. Examination of freezeetched replicas of labeled murine [18] and human cells [19] which provide more accurate two-dimensional information, revealed that the surface Ig was distributed in micro-clusters and patches of varying sizes. These latter studies utilizing bivalent antiIg coupled to ferritin as a label, have been criticized because of the possible crosslinking of surface molecules that the label may have produced [20-211. A later study, conducted by Abbas et al. [17], of freezeExp Ceil Res 112 (1978)
etched replicas of murine B lymphocytes utilizing an indirect labeling technique in which cells were first labeled with fluorescein conjugates of monovalent Fab anti-Ig antibody and then treated with monovalent Fab anti-fluorescein antibody coupled to ferritin has provided more evidence for a non-random distribution. In these studies the surface Ig appeared in small clusters and networks with intervening areas of bare membrane. This pattern was the same on cells labeled at low temperatures or at 20°C and on cells previously fixed in paraformaldehyde. The possibility of the clustering being an artifact caused by several fluorescein molecules being coupled to one Fab anti-Ig was reduced by using a fluorochrome : protein ratio of one. Also changing this ratio had no effect on the overall distribution. The observed pattern was analysed statistically by reticulating the membrane into small areas and scoring the number of molecules in each area. For a purely random (Poisson) distribution the mean and variance should be equal. In all of the twenty freeze-etch replicas studied by Abbas et al. [17] the variance was found to be greater than the mean. This substantiated the subjective perception of nonrandomness in the pattern and further implied that the pattern derived from random by being clustered [22-231. If cell surface proteins are arranged nonrandomly how can one quantitatively describe their distribution? One measure of spatial pattern commonly used in statistical physics to analyse the structure of liquids is the radial distribution function, g(r) [24251. It is related to the crystallographer’s Patterson function [26]. Quantitative information about the clustering of receptors and the possible forces among surface Ig molecules responsible for their non%andom distribution, in principle, can be obtained
Surface immunoglobulin
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micrograph of a labeled cell surface, the radial distribution function is defined as the 1.4 ratio of the average local number density of grains at a distance r from an arbitrary grain to the overall density of grains .Q[27]. (It will be convenient to plot radial distribution functions in terms of a dimensionless distance x=r/r,,, where r,,=e-’ is the mean 0.6separation distance between grains in a random pattern.) In computing g(r) the local 0.6 density at r is determined in an annulus of width Ar, thus r is only specified +Ar/2. For a random distribution of an infinite 0.4 number of points the local density should be the same as the overall density, and one ex0.2 pects g(r)= 1 for all r. However, for a single l I I I I I I I pattern with a finite number of points fluc1.0 2.0 3.0 4.0 OO tuations will occur, especially at small Fig. 1. Abscissa: x; ordinate: g(x). The radial distribution function for a set of N=653 values of r where ni(r) is small, which propoints randomly placed in a rectangle the size of the duce deviations from unity. Fig. 1 shows a one illustrated in fig. 4a (rO=227 A, Ar=0.2 I-,,). typical computed radial distribution function for a random pattern. Any systematic from g(r). This function has previously deviations from this form of g(r) indicates a been used for similar purposes in other bio- non-random distribution. For example, for logical systems: the analysis of pore pat- molecules distributed on a finite two-dimenterns on nuclear membranes [27] and in the sional square lattice (e.g., a two-dimenanalysis of the spatial organization of con- sional crystal) with lattice spacing a, g(r) nective tissue [28]. Other quantitative meas- would be zero everywhere except within a ures of spatial structure can be obtained via small region of width Ar around r=a, 6, correlation and power spectral analysis. 2a, I&, 2m, 3a, . . . (fig. 2b). These Statisticians and ecologists interested in values of r correspond to distances between describing the clustered distributions of pairs of lattice points (fig. 2a). If the posiplants and animals found in various eco- tions of the molecules are allowed to vary, systems have devised a number of scalar in- as in a fluid, the discrete form ofg(r) is condices for measuring the degree of aggrega- verted into a continuous function. Although tion in a pattern [23]. Although these in- the short range order between molecules dices can be applied to surface Ig maps may be preserved, at large distances the net [29], under many circumstances a single effect of random (thermal) motion is to denumber may not suffice in describing all the stroy any long-range correlations in the relevant features in a pattern. relative positions of the molecules, hence g(r) will approach one as r becomes large. The radial distribution function The radial distribution functions for a typiFor a pattern such as that generated by the cal gas and liquid are illustrated in fig. 3. ferritin grains on a freeze-etch electron Notice that g(r) for a dense liquid contains Exp Cell Res 112 (1978)
312
A. S. Perelson
.
.
.
.
.
.
Fig. 2. (a) Square lattice with lattice spacing a=1.05. (b)Abscissa:
x; ordinate:
g(x).
Radial distribution function (N=653, rO=a, Ar=
many peaks indicating the successive shells of nearest neighbors, whereas in a dilute gas there is only one peak. The attractive force between molecules is such that in a dense system many molecules will be ordered forming shells of first nearest neighbors, second nearest neighbors, and so forth. In a sparse system, such as a dilute gas, the distance over which the attractive intramolecular force acts is insufficient to provide long-range order and only the first nearest neighbor peak is seen in g(r). In both gases and liquids strong repulsive forces prevent molecules from occupying the same space. Thus g(r) is zero for small values of r. The average number of first nearest neighbors, the coordination number, of a ferritin grain can be computed from g(r). If n(r) is defined as the average number of grains at a distance r from a given grain, then n(r)=2mrgg(r)
respond to the successive shells of nearest neighbors. In a liquid the nearest neighbor shells overlap making it difficult to identify the positions of all but the first nearest neighbor shell. By integrating n(r) from r=O to a position R at the end of the first neighbor peak the coordination number of a ferritin grain can be determined. (There is no unique method of computing coordination numbers since the nearest neighbor peaks overlap. Our procedure corresponds to 2-
,
(1)
A plot of n(r) versus r starts at zero, increases, usually through a maximum, and then continues to increase with possible subsequent maxima. The peaks in n(r) corExp Cell RPS 112 (1978)
0.07 r,,). The positions of the peaks correspond to separation distances between lattice points.
01
/
Fig. 3. Abscissa: x; ordinate:
g(x).
The radial distribution function for a monotomic fluid: -, dense liquid showing some short-range order; ---, dilute gas showing a single peak. Adapted from Reed & Grubbins [25].
Surface immunoglobulin
Fig. 4. (a) The ferritin grain pattern traced from a freeze-etched replica of a splenic murine B lymphocyte labeled at 4°C first with a fluorescein conjugate of monovalent (papain-digested) anti-Ig antibody followed by monovalent anti-fluorescein antibody coupled to ferritin. The rectangle indicates the area analysed. (b) Abscissa: x; ordinate: g(x). The radial distribution function found for the designated region (N=653, r0=227 1$, Ar=0.2 rO). The error bars indicate the 95 % confidence limit, + l.% S.E. (c) Abscissa: x; ordinate: x&x). The average number of fenitin grains in an annulus lying at a distance x from a given grain is proportional to xg(x). The first nearest neighbor peak was assumed to terminate at x=0.7.
method D of Mikolaj & Pings [30] in which R is chosen at the first minimum of n(r).) Analysis of the surface immunoglobulin distribution
In the study of Abbas et al. [17] maps of the distribution of surface Ig on murine splenic B lymphocytes were obtained by freeze-etching. By using monovalent reagents (fluorescein conjugates of monovalent anti-Ig antibody and monovalent anti-fluorescein antibody coupled to fer?I-781810
.L/02
0.4
on B lymphocytes
0.6
08
1.0
3 13
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ritin) and low temperatures cross-linking and aggregation of the surface Ig was minimized. From the twenty freeze-etch replicas analyzed by Abbas et al. [ 171,five, picked at random, were kindly given to me for analysis. Tracings of the ferritin-grain patterns on these electron micrographs were automatically scanned by a digital microdensitometer and the coordinates of the grains determined by a computer analysis of the digitized pictures as described in the Appendix. The radial distribution funcExp Cd Res 112 (1978)
314
A. S. Perelson
Fig. 5. (a) A second ferritin grain pattern obtained under the same conditions as in fig. 4a. (b) Abscissa: x; ordinate: g(x). The radial distribu-
tion function found for the designated region (N= 1028, r0=340 8, Ar=0.2 r,J. The error bars indicate the 95 % confidence limit, f I.96 SE.
tions of these surface Ig maps were then calculated on a CDC 7600 computer according to eqs (l)-(3) of ref. [27]. The computed radial distribution functions for two representative examples are shown in figs 46 and 56. The error bars indicate the 95% confidence limits. The graphs of all six radial distribution functions share the common features of rising from zero near the origin, going through a maximum near x=0.5, and approximating one for large values of x. Some of the graphs, such as the ones in figs 4b and 5b, show what could be minor second and third peaks at distances that are roughly multiples of the position of the first peak. However, the heights of these peaks are not significantly greater than the random fluctuations in g(r). Thus the major discernible feature of the distributions is the strong first maximum well below the mean interparticle distance. Table 1 lists the height and position of this peak for all six pictures. At small values of x, g(x) is much smaller
than one, presumably because Ig molecules cannot get arbitrarily close to one another. However, because the ferritin grains are farily large (-120 A diameter [ 17]), and two overlapping grains may not easily be resolved, one cannot obtain reliable information at small distances. Furthermore, the number of grains that appear close together in the analysed pictures is small, leading to large statistical fluctuations in g(x) for values of x near zero. For these reasons we have not plotted g(x) for ~~0.2, although one can assume that the curves do fall to zero somewhere near the origin. Two independent measures of the statistical fluctuations in g(r) due to the finite sample size have been computed. (1) For a picture containing N grains in a rectangle with specified dimensions, I have computed g(r) for N points placed at random in the same rectangle. Fig. 1 shows the radial distribution function computed for a randomized version of fig. 4~. (2) I have used a method derived by Markovics, Glass &
Exp Cell Res 1 I2 (1978)
Surface immunoglobulin
Table 1. Summary of radial distribution
function
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calculations
Picture
Max. value of g(r)fS.E.”
Position of maximum (A)*
Mean se aration r0 (R )
Position of maximum/r,
Coordination no. (z.)
No. of grains (N)
1’ 2d 3 4’ 5’ 6
1.56f0.13 2.lliO.13 1.49f0.12 1.S8IO.16 1.60+0.20 2.10+0.15
125 153 184 171 163 155
221 340 334 343 327 344
0.55 0.45 0.55 0.50 0.50 0.45
2.2 3.0 2.6 2.0 2.0 1.5
653 1 028 620 438 263 803
a The standard error (S.E.) was estimated by the method of Markovics et al. [27]. ’ For peaks that were flat, such as that shown in fig. 5b (i.e., g(x)=g(x+O.l) where x was evaluated at intervals of 0.1) the position of the maxima was chosen in the center of the flat portion of the peak (i.e., if g(x) = g(x+O.l) thenx+0.05 was chosen as the position of the maximum). c Shown in fig. 4. d Shown in fig. 5. e Pictures 4 and 5 correspond to nearby regions on the same freeze-etch micrograph.
Maul [27] to estimate the standard error (SE.) for g(r) at each point. The 95 % confidence limits specified in table 1 and figs 4b and 5b are given by +1.96 S.E. Both computations of statistical fluctuations show that the first peak seen in g(r) for the surface Ig maps is statistically significant, whereas subsequent peaks may not be. Estimates of the total number of surface Ig molecules can be made from the density of molecules found in the radial distribution function analysis. Using the values of r,, listed in table 1, one finds that Q=ro2 is on the order of 10” molecules/cm2. Hence for a typical B cell with a 7 pm diameter the total number of receptors is on the order of lo5 which agrees with other estimates [3 l-321. The average number, z, of first nearest neighbors of a receptor was calculated from eq. (1). From table 1 one can see that z varies from 1.5 to 3.0, with mean 2.2. This computation was not uniformly accurate since it was difficult to identify the end of the first nearest neighbor peak of the n(r) vs r plots for pictures l-3 (see fig. 4c), whereas the remaining pictures had distinct minima which marked the termination of the first peak (see fig. 6).
DISCUSSION The predominant feature of the radial distribution functions computed for the six surface Ig maps is the presence of a major peak at distances near one-half the mean interparticle separation. A comparison with fig. 3 shows that these radial distribution functions closely resemble those of a dilute gas or liquid. The presence of the major peak, which corresponds to a shell of nearest neighbor atoms in a fluid, indicates that the ferritin grains are clustered into small groups. The number of grains per group is related to the area under the first peak of the function 2rrg(r). The distance along the x-axis between the peak in g(x) and x= 1, and the sharpness of the peak, determine the amount of cluster character in the distribution [27]. Thus on the basis of this g(r) calculation one can say that the points in fig. 5a are somewhat more clustered than those in fig. 4a. Identifying a cluster as a ferritin grain and its shell of first nearest neighbors, the average cluster size in fig. 5a is 4.0, whereas for fig. 4a it is 3.2. The five freeze-etch pictures were all marred by the presence of a small number of ice crystals. The position of one is Exp Ceil Rrs I I2 (1978)
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A. S. Perelson
Fig. 6. Abscissa: x; or&a&: q(x). The radial distribution function for picture 5 (not shown) was used to calculate xg(x). The minimum at x=0.7 was taken as the end of the first nearest neighbor peak.
denoted by the kidney-shaped object near the left border of the rectangle in fig. 4a. These crystals which presumably hid some receptors, could slightly distort the distance relations in the nearest neighbor and g(r) calculations. The picture in fig. 5a was the worst offender, having 6 ice crystals (not shown) in the analysed rectangle. The additional clear area created by these crystals may be the reason that the nearest neighbor distance for this picture was the greatest of all the pictures analysed. The mean number of first nearest neighbors surrounding a ferritin grain was found to be 2.2. For a pattern in which ferritin grains were organized along well separated lines, possibly running in different directions, each grain would have two first nearest neighbors. This would also be true for points lying on well separated equilateral triangles. At distances of 2-5 mean interparticle spacings the radial distribution functions of all six patterns approached one. This strikExp Cell Res 112 (1978)
ingly demonstrates that at large separation distances there is no correlation in the positions of surface Ig molecules. The absence of long-range order in the topographic distribution of membrane proteins is an important prediction of the fluid mosaic model of membrane structure [6]. (Strictly speaking, the existence of a fluid membrane in which proteins are free to move does not imply the lack of long-range order. Clearly, if there were long-range forces acting among proteins or even short-range interactions which resulted in large aggregates then long-range order could exist. However, Singer 8z Nicolson [6] hypothesized that long-range random arrangements of proteins in membranes are the norm and whenever non-random distributions are found, special mechanisms, such as cross-linking agents extrinsic to the membrane, must exist which are responsible for the observed order.) Radial distribution function analysis provides a very precise quantitative confirmation of this prediction for surface immunoglobulin. The mechanism which generated the clustering of surface Ig molecules cannot be determined from radial distribution function analyses alone. It is possible that the clustering reflects the manner in which the Ig is inserted into the membrane. However, since receptors are believed to diffuse within the plane of the membrane, memory of the initial distribution of receptors should quickly be lost. If the movement of Ig is subject to some intermolecular interaction then a degree of short-range order is not inconsistent with the fluid mosaic model of membrane structure [6]. In fact, if one assumes that the interaction of surface Ig molecules is analogous to the interactions of molecules in an equilibrium fluid which can be described by an intermolecular potential, u(r), the techniques developed to
Surface immunoglobulin
study the structure of materials can be applied. For a dilute gas g(r)=exp [-u(r)/kTl while for a denser system a correction term, r(r), is needed
g(r)=7(r) expC-u(rYkTl where k is Boltzman’s constant and T is the absolute temperature [33]. In the theory of liquids experimental measurements of g(r) are used to infer information about the potential u (r). Similar techniques might be of use in the study of membrane proteins; however, a great degree of caution is necessary in applying such techniques to the surface Ig data analysed here. (1) One has no assurance that the Ig molecules are completely free to move within the plane of the membrane. There may be weak or transitory associations with some anchoring structure within the membrane or components of the cell’s cytoskeleton. In fact, the observed clustering could simply be explained on the basis of such associations. (2) The positions of the ferritin grains are not solely determined by membrane Ig. Since immunoglobulin is not the only protein on the cell surface, and in fact, has been estimated to occupy less than 10% of the membrane surface [34], protein-protein, protein-lipid, and lipid-lipid interactions within the membrane are certainly present and will effect the positions of the surface Ig. Further, the portions of the Ig molecules protruding from the membrane can interact with the liquid medium surrounding the cell, and interactions between the ferritin, fluorescein, and Fab fragments, used in labeling the surface immunoglobulin, and other
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molecular species in the system probably occur. Additionally, because a double labeling procedure was used there is no assurance that the ferritin grain is directly over the surface Ig; a statistical distribution of ferritin positions surrounding the Ig must be allowed. From a statistical mechanical viewpoint such a system is enormously more complicated than the simple liquids usually studied by radial distribution functions. Electrically charged globular proteins in concentrated aqueous solutions may have radial distribution functions similar to those found here for surface Ig [35-361. In such systems locally ordered clusters, characteristic of liquid structure, are established through the action of the Coulomb force screened by a solvent containing electrolytes of low molecular weight and the shortrange intermolecular repulsion determining the molecular diameter [35]. Since macromolecular solutions, which in some respects are analogous to the current fluid-mosaic view of membrane structure, give rise to liquid-like radial distribution functions, one has some basis for believing that the surface Ig distribution can be attributed to the net intermolecular force between two surface macromolecules. Even if one assumes that the computed g(r) is a reflection of the net intermolecular potential between surface Ig molecules, determining u(r) from the available data will not be an easy task. Many types of potentials give rise to radial distribution functions similar to those shown in figs 4 b and 5 b. A two-dimensional system of hard disks which interacts only by repulsion through contact collision gives rise to liquid-like radial distribution functions [37]. This fact implies that one cannot rule out the possibility that the observed clustering of Ig may result solely from repulsive interacExp CdRrs
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tions due to contact collisions or electrostatic forces between similarly charged membrane macromolecules. However, potentials which have both attractive and repulsive parts, such as the Lennard Jones 6-12 potential, will also generate liquid-like radial distribution functions. So far we have implicitly assumed that the lipids of a membrane form a homogeneous fluid. However, the lipid composition of a membrane may vary from one location to another. Phase separations in mammalian cell plasma membranes have been suggested to occur as cells are cooled from 37°C to below 15°C [38]. There is also evidence for an asymmetrical division of the major phospholipids between the two halves of the erythrocyte membrane bilayer [39]. If the lipids of the B cell membrane were segregated into domains, some with high affinity and some with low affinity for surface Ig, then spatial maps would tend to show regions of bare membrane interposed between regions of high surface Ig density. It is anticipated that the radial distribution function for such a system would indicate the presence of clusters and hence the existence of lipid domains with different partition coefficients for sIg is another possible explanation for the observed g(r). One limitation of radial distribution function analysis is that angular anisotropies are averaged out. (Such anisotropies can be analysed using the pair distribution function which depends on both the distance and angle separating particles.) Thus if surface Ig molecules tend to align themselves with an underlying oriented cytoplasmic tibrillar network the resulting orientation of the pattern would not be readily discernible from g(r). However, we have presented nearest neighbor evidence which is consistent with linear organization. If surface Ig molecules are arranged in anisotropic linear patterns Exp Cd Res 112 (1978)
then it seems doubtful that assigning a simple intermolecular potential acting between pairs of surface molecules can lead to accurate predictions of the topographic distribution of surface Ig. One further limitation of g(r) analysis is that its ability to reflect the true distribution of surface Ig is necessarily limited by the accuracy of the labeling and fixation technique. Abbas et al. [17] critically discuss their labeling procedure and seem assured that artifacts did not significantly affect their conclusion of a non-random distribution for surface immunoglobulin. However, by increasing the precision of the analysis as I have done, one must be concerned with artifacts that manifest themselves at small distances. At present one simply does not know how much distortion of surface architecture occurs during the freeze-etch procedure. It is doubtful that the distortion is great since the cells are fixed, but small positional disturbances can affect the short-range order seen in g(r). In the calculation of g(r) an average is done over all ferritin grains which may cancel the effects of small non-systematic distortions. In the indirect labeling procedure Abbas et al. used, one does not know with certainty how many ferritin grains label each surface Ig molecule. The large size of ferritin makes it unlikely that more than three or four ferritin grains can occupy the area represented by a single Ig [17], but this is the size of the clusters detected by nearest neighbor analysis. More sensitive labeling controls coupled with g(r) analysis may help alleviate such concerns. The distance between ferritin grains used in the calculations of g(r) were those obtained from freeze-etch micrographs. Since a micrograph is in effect a two-dimensional projection of the actual surface some distance distortion may be present. First, for a
Surface immunoglobulin
fragment of the cell surface 1 pm in length (a typical size for the samples analysed in this study), the distortion due to the curvature of the fragment, assuming it came from a sphere with a 4 pm radius, is negligible (a measured distance of 1 pm would correspond to a distance on the sphere of 1.0026 pm). However, if the surface were crenated then the distance distortion would be greater. In this regard it is important to point out that Lipscomb et al. [40], using scanning immunoelectronmicroscopy with tobacco mosaic virus as a marker, have described the surface immunoglobulin on murine splenic lymphocytes as being randomly distributed, appearing both on microvilli and the smooth surface of the cell. Thus the presence of microvilli would not be expected to cause a gross redistribution of surface immunoglobulin, but could cause some distance distortion. This effect could be minimized by restricting the analysis to cells prepared at 0”-4’C, since there appears to be a lack of microvilli expression at such temperatures [41, 421. In our present state of knowledge, the usefulness of radial distribution functions in the analysis of membrane structure comes from ignoring the detailed molecular interactions that produce a given spatial distribution of surface molecules, and using g(r) as a quantitative indicator of overall features of spatial pattern. Markovics et al. [27] show very dramatically how pore patterns on nuclear membranes can be distinguished by radial distribution function analysis. I have shown how the calculation of g(r) can be used to quantitatively confirm the absence of long-range order in the distribution of membrane proteins as predicted by the fluid mosaic hypothesis. However, there are other important potential applications of radial distribution functions in the study of surface macromolecules. Using the
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methods presented in this paper, one might be able to address questions about the amount of receptor cross-linking necessary to trigger specific cellular reactions. For example, if as is commonly believed capping is triggered by patch formation, is the effect of many small patches the same as the effect of a few large patches? Is there a relationship between the amount of receptor cross-linking and the induction of tolerance in B lymphocytes? The release of histamine by mast cells or basophils has been shown to depend upon receptor cross-linking [43, 441. In these systems radial distribution function calculations may provide a way of quantitatively relating the amount of crosslinking with the measured dose response curves for histamine release and thus relating cell surface events to stimulatory and tolerizing signals. Further, given maps of surface molecules at different times during the processes of patch and cap formation, analyses such as the one described herein, may be useful in distinguishing mechanisms involved in receptor aggregation and the dynamics of receptor movement.
CONCLUSIONS The spatial distribution of surface immunoglobulin on murine splenic B lymphocytes has been mathematically analysed by computing a statistical measure, the radial distribution function. The analysis confirms the conclusions of Abbas et al. [ 171:(1) surface immunoglobulin molecules are nonrandomly distributed; and (2) immunoglobulin receptors in the absence of crosslinking agents form small clusters. The analysis also demonstrated the absence of long-range order as predicted by the fluidmosaic hypothesis [6]. Computation of the average number of first nearest neighbors Exp Cell Res 112 (1978)
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indicates that there may be a linear order to the surface Ig patterns. The radial distribution function allows one to quantitate the clustering of surface molecules and can be used to provide information about the intermolecular forces between surface macromolecules. Its application to other experimental models is highly recommended.
analysis is complicated by the size of P, which may contain on the order of lo6 elements. In order to handle large matrices, P was analysed one row at a time. The sum of all the elements of Si is the total density of a ferritin grain. The central elements of S, having the highest densities. Those central areas, say of sizer’ XC’ (r’
Surface immunoglobulin 10. de Penis, S, Nature 250 (1974) 54. 11. Edelman, G M, Yahara, I & Wang, J L, Proc natl acad sci US 70 (1973) 1442. 12. Yahara, I & Edelman, G M, Nature 246 (1973) 152. 13. Wessells, N K, Spooner, B S, Ash, J F, Bradley, M 0, Luduena, M A, Taylor, E L, Wrenn, J T & Yamada, KM, Science 171(1971) 135. 14. Albertini, D F & Clark, J I, Proc natl acad sci US 72 (1975) 4976. 15. Brinkley, B R, Fuller, G M & Highfield, D P, Proc natl acad sci US 72 (1975) 4981. 16. Bretscher, M S, Nature 260 (1976) 21. 17. Abbas, A K, Ault, K A, Karnovsky, M J & Unanue, E R, J immunol 114 (1975) 1197. 18. Karnovsky, M J, Unanue, E R & Leventhal, M, J exp med 136(1972) 907. 19. Ault, K A, Kamovsky, M J & Unanue, E R, J clin invest 52 (1973) 2507. 20. Davis, W C, Alspaugh, M A, Stimpfling, J H & Walford, R L, Tissue antigens 1 (1971) 89. 21. Davis, W C, Science 175(1972) 1006. 22. Zar, J H, Biostatistical anslysis. Prentice-Hall, Englewood Cliffs, NJ (1974). 23. Pielou, E C, An introduction to mathematical ecology. Wiley-Interscience, New York (1969). 24. Hill. T. Statistical mechanics. McGraw-Hill. New York (i956). 25. Reed. T M & Grubbins. K E. Aoolied statistical mechanics. McGraw-Hih, New Y&k (1973). 26. Finney, J L, Nature 266 (1977) 309. 27. Markovics, J, Glass, L & Maul, G G, Exp cell res 85 (1974) 443. 28. Farrell, R A & Hart, R W, Bull math biophys 31 (1969) 727. 29. Perelson, A S, Theoretical immunology (ed G Bell, A Perelson & G Pimbley). Marcel Dekker, New York (1978).
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30. Mikolaj, P G & Pings, C J, Phys them liquids 1 (1%8) 93. 31. Rabellino. E, Colon, S. Grev, H M & Unanue, E R, J exp med 133 (1971) 156: 32. Ault. K A & Unanue. E R. J exe. med 139 (1974) 1110: 33. EgelstatI, P A, An introduction to the liquid state. Academic Press, New York (1%7). 34. Unanue, E R, Am j path01 77 (1974) 2. 35. Kirkwood, J G & Mazur, J, J polymer sci 9 (1952) 519. Compt rend reunion ann avec comm 36. thermodynam, p. 143, Union intern phys, Paris (1952). Reprinted in John Gamble Kirkwood collected works, Macromolecules (ed P L Auer) p. 187. Gordon & Breach, New York (1%7). 37. Chae, D G, Ree, F H & Ree, T, J them phys 50 (1969) 1581. 38. Petit, V A & Edidin, M, Science 184(1974) 1183. 39. Singer, S J, Ann rev biochem (1974) 805. 40. Lipscomb, M F, Holmes, K V. Vitetta, E S, Himmerling, U & Uhr, J W, Eur j immunol 5 (1975) 255. 41. Loor, F & H&g, LB, Eur j immunol5 (1975) 854. 42. Lin, P S, Wallach, D F H & Tsai, S, Proc natl acad sci US 70 (1973) 2492. 43. Siraganian, R P, Hook, W A & Levine, B B, Immunochemistry 12 (1975) 144. 44. Segal, D M, Taurog, J D & Metzger, H, Proc natl acad sci US 74 (1977) 2993.
Received July 20, 1977 Revised version received October 14, 1977 Accepted November 2, 1977
Exp CellRes 112 (1978)
Experimental
SPATIAL
DISTRIBUTION
Cell Research 1I2 (1978) 309-321
OF SURFACE
IMMUNOGLOBULIN
ON B LYMPHOCYTES Local Ordering ALAN S. PERELSON Theoretical Division, University of California, Los Alamos Scientifc Laboratory Los Alamos, NM 87545, USA
SUMMARY Maps of the spatial distribution of surface immunoglobulin on B lymphocytes obtained by freezeetch electron microscopy were analysed mathematically. The pictures were automatically scanned by a digital microdensitometer and the number and coordinates of the immunoglobulin molecules determined. A statistical measure, the radial distribution function, g(r), commonly used to study the structure of liquids, was computed for each map. The radial distribution function of surface immunoglobulin was found to closely resemble g(r) for fluids, indicating the presence of shortrange (but not long-range) order. It was determined that a given surface immunoglobulin molecule has a high probability of being surrounded by other immunoglobulins at distances approximately one-half the mean interparticle spacing. From this one can conclude that the distribution of surface immunoglobulin is non-random and is characterized by local clustering. The mean number of first nearest neighbors surrounding each surface immunoglobulin, computed from g(r), was found to be near two. This value is consistent with a degree of linear order in the topographic distribution of immunoglobulin. The radial distribution function provides a method of quantitating the amount of clustering in a spatial distribution. This function may prove useful in accessing the amount of receptor cross-linking necessary to trigger cellular responses, and in elucidating mechanisms for aggregation and movement of membrane macromolecules.
The fluid mosaic model of the cell membrane suggests that proteins are free to diffuse within the plane of the membrane Cl]. Consequently, one might naively expect to find that proteins are randomly distributed with respect to one another, especially at large separation distances. Clearly, however, the existence of either short or long range forces acting on membrane proteins can lead to non-random spatial distributions . One type of protein found on the surface of bone marrow derived (B) lymphocytes is immunoglobulin (Ig), which serves as a receptor for antigen [2]. The distribution and
behavior of surface Ig has been extensively studied in attempts to understand the mechanisms by which lymphocytes recognize foreign materials and begin their immune functioning. One important finding has been that in the presence of a multivalent ligand which can cross-link receptors, the surface Ig rapidly redistributes into microaggregates or patches. At temperatures above 20°C at which the membrane lipids remain fluid, the patches undergo a further redistribution and aggregate at one pole of the cell in a process known as capping [3-51. The ligand induced redistribution of surface Ig is taken as further evidence in support of Exp Cell Res I12 (1978)
3 10
A. S. Perelson
the fluid mosaic hypothesis [6-71. Capping can be interfered with by numerous agents, such as sodium azide [3, 8-91, dinitrophenol [3, 7-91, cyanide [7-91, cytochalasin B (CB) [3, lo], vinblastine [ 10, 121,and colchicine used in combination with CB [lo], or by lowering the temperature from 20” to near 0°C [34] suggesting that it is an energy requiring process in which microtubules and microtilaments may play a role [lo]. Edelman & Yahara [ 11, 121 have, in fact, hypothesized that there may be a direct link between surface molecules and an underlying cytoplasmic fibrillar network [ 13-151. As an alternative explanation of capping, Bretscher [16] has suggested that directed lipid flow may move surface macromolecules. If either of these hypotheses are correct one might not expect to find a random distribution of surface Ig molecules. The precise native distribution of surface proteins such as Ig before ligand binding has been difftcult to measure because of technical problems in cell labeling and ultrastructural examination [17]. Studies of the spatial organization of surface Ig on murine B lymphocytes labeled with a monovalent ligand and examined by immunofluorescence which has low resolution, or by electron microscopy of thin sections, which only reveals the pattern in the plane of the section, have disclosed patterns described as random [34]. Examination of freezeetched replicas of labeled murine [18] and human cells [19] which provide more accurate two-dimensional information, revealed that the surface Ig was distributed in micro-clusters and patches of varying sizes. These latter studies utilizing bivalent antiIg coupled to ferritin as a label, have been criticized because of the possible crosslinking of surface molecules that the label may have produced [20-211. A later study, conducted by Abbas et al. [17], of freezeExp Ceil Res 112 (1978)
etched replicas of murine B lymphocytes utilizing an indirect labeling technique in which cells were first labeled with fluorescein conjugates of monovalent Fab anti-Ig antibody and then treated with monovalent Fab anti-fluorescein antibody coupled to ferritin has provided more evidence for a non-random distribution. In these studies the surface Ig appeared in small clusters and networks with intervening areas of bare membrane. This pattern was the same on cells labeled at low temperatures or at 20°C and on cells previously fixed in paraformaldehyde. The possibility of the clustering being an artifact caused by several fluorescein molecules being coupled to one Fab anti-Ig was reduced by using a fluorochrome : protein ratio of one. Also changing this ratio had no effect on the overall distribution. The observed pattern was analysed statistically by reticulating the membrane into small areas and scoring the number of molecules in each area. For a purely random (Poisson) distribution the mean and variance should be equal. In all of the twenty freeze-etch replicas studied by Abbas et al. [17] the variance was found to be greater than the mean. This substantiated the subjective perception of nonrandomness in the pattern and further implied that the pattern derived from random by being clustered [22-231. If cell surface proteins are arranged nonrandomly how can one quantitatively describe their distribution? One measure of spatial pattern commonly used in statistical physics to analyse the structure of liquids is the radial distribution function, g(r) [24251. It is related to the crystallographer’s Patterson function [26]. Quantitative information about the clustering of receptors and the possible forces among surface Ig molecules responsible for their non%andom distribution, in principle, can be obtained
Surface immunoglobulin
on B lymphocytes
311
micrograph of a labeled cell surface, the radial distribution function is defined as the 1.4 ratio of the average local number density of grains at a distance r from an arbitrary grain to the overall density of grains .Q[27]. (It will be convenient to plot radial distribution functions in terms of a dimensionless distance x=r/r,,, where r,,=e-’ is the mean 0.6separation distance between grains in a random pattern.) In computing g(r) the local 0.6 density at r is determined in an annulus of width Ar, thus r is only specified +Ar/2. For a random distribution of an infinite 0.4 number of points the local density should be the same as the overall density, and one ex0.2 pects g(r)= 1 for all r. However, for a single l I I I I I I I pattern with a finite number of points fluc1.0 2.0 3.0 4.0 OO tuations will occur, especially at small Fig. 1. Abscissa: x; ordinate: g(x). The radial distribution function for a set of N=653 values of r where ni(r) is small, which propoints randomly placed in a rectangle the size of the duce deviations from unity. Fig. 1 shows a one illustrated in fig. 4a (rO=227 A, Ar=0.2 I-,,). typical computed radial distribution function for a random pattern. Any systematic from g(r). This function has previously deviations from this form of g(r) indicates a been used for similar purposes in other bio- non-random distribution. For example, for logical systems: the analysis of pore pat- molecules distributed on a finite two-dimenterns on nuclear membranes [27] and in the sional square lattice (e.g., a two-dimenanalysis of the spatial organization of con- sional crystal) with lattice spacing a, g(r) nective tissue [28]. Other quantitative meas- would be zero everywhere except within a ures of spatial structure can be obtained via small region of width Ar around r=a, 6, correlation and power spectral analysis. 2a, I&, 2m, 3a, . . . (fig. 2b). These Statisticians and ecologists interested in values of r correspond to distances between describing the clustered distributions of pairs of lattice points (fig. 2a). If the posiplants and animals found in various eco- tions of the molecules are allowed to vary, systems have devised a number of scalar in- as in a fluid, the discrete form ofg(r) is condices for measuring the degree of aggrega- verted into a continuous function. Although tion in a pattern [23]. Although these in- the short range order between molecules dices can be applied to surface Ig maps may be preserved, at large distances the net [29], under many circumstances a single effect of random (thermal) motion is to denumber may not suffice in describing all the stroy any long-range correlations in the relevant features in a pattern. relative positions of the molecules, hence g(r) will approach one as r becomes large. The radial distribution function The radial distribution functions for a typiFor a pattern such as that generated by the cal gas and liquid are illustrated in fig. 3. ferritin grains on a freeze-etch electron Notice that g(r) for a dense liquid contains Exp Cell Res 112 (1978)
312
A. S. Perelson
.
.
.
.
.
.
Fig. 2. (a) Square lattice with lattice spacing a=1.05. (b)Abscissa:
x; ordinate:
g(x).
Radial distribution function (N=653, rO=a, Ar=
many peaks indicating the successive shells of nearest neighbors, whereas in a dilute gas there is only one peak. The attractive force between molecules is such that in a dense system many molecules will be ordered forming shells of first nearest neighbors, second nearest neighbors, and so forth. In a sparse system, such as a dilute gas, the distance over which the attractive intramolecular force acts is insufficient to provide long-range order and only the first nearest neighbor peak is seen in g(r). In both gases and liquids strong repulsive forces prevent molecules from occupying the same space. Thus g(r) is zero for small values of r. The average number of first nearest neighbors, the coordination number, of a ferritin grain can be computed from g(r). If n(r) is defined as the average number of grains at a distance r from a given grain, then n(r)=2mrgg(r)
respond to the successive shells of nearest neighbors. In a liquid the nearest neighbor shells overlap making it difficult to identify the positions of all but the first nearest neighbor shell. By integrating n(r) from r=O to a position R at the end of the first neighbor peak the coordination number of a ferritin grain can be determined. (There is no unique method of computing coordination numbers since the nearest neighbor peaks overlap. Our procedure corresponds to 2-
,
(1)
A plot of n(r) versus r starts at zero, increases, usually through a maximum, and then continues to increase with possible subsequent maxima. The peaks in n(r) corExp Cell RPS 112 (1978)
0.07 r,,). The positions of the peaks correspond to separation distances between lattice points.
01
/
Fig. 3. Abscissa: x; ordinate:
g(x).
The radial distribution function for a monotomic fluid: -, dense liquid showing some short-range order; ---, dilute gas showing a single peak. Adapted from Reed & Grubbins [25].
Surface immunoglobulin
Fig. 4. (a) The ferritin grain pattern traced from a freeze-etched replica of a splenic murine B lymphocyte labeled at 4°C first with a fluorescein conjugate of monovalent (papain-digested) anti-Ig antibody followed by monovalent anti-fluorescein antibody coupled to ferritin. The rectangle indicates the area analysed. (b) Abscissa: x; ordinate: g(x). The radial distribution function found for the designated region (N=653, r0=227 1$, Ar=0.2 rO). The error bars indicate the 95 % confidence limit, + l.% S.E. (c) Abscissa: x; ordinate: x&x). The average number of fenitin grains in an annulus lying at a distance x from a given grain is proportional to xg(x). The first nearest neighbor peak was assumed to terminate at x=0.7.
method D of Mikolaj & Pings [30] in which R is chosen at the first minimum of n(r).) Analysis of the surface immunoglobulin distribution
In the study of Abbas et al. [17] maps of the distribution of surface Ig on murine splenic B lymphocytes were obtained by freeze-etching. By using monovalent reagents (fluorescein conjugates of monovalent anti-Ig antibody and monovalent anti-fluorescein antibody coupled to fer?I-781810
.L/02
0.4
on B lymphocytes
0.6
08
1.0
3 13
12
ritin) and low temperatures cross-linking and aggregation of the surface Ig was minimized. From the twenty freeze-etch replicas analyzed by Abbas et al. [ 171,five, picked at random, were kindly given to me for analysis. Tracings of the ferritin-grain patterns on these electron micrographs were automatically scanned by a digital microdensitometer and the coordinates of the grains determined by a computer analysis of the digitized pictures as described in the Appendix. The radial distribution funcExp Cd Res 112 (1978)
314
A. S. Perelson
Fig. 5. (a) A second ferritin grain pattern obtained under the same conditions as in fig. 4a. (b) Abscissa: x; ordinate: g(x). The radial distribu-
tion function found for the designated region (N= 1028, r0=340 8, Ar=0.2 r,J. The error bars indicate the 95 % confidence limit, f I.96 SE.
tions of these surface Ig maps were then calculated on a CDC 7600 computer according to eqs (l)-(3) of ref. [27]. The computed radial distribution functions for two representative examples are shown in figs 46 and 56. The error bars indicate the 95% confidence limits. The graphs of all six radial distribution functions share the common features of rising from zero near the origin, going through a maximum near x=0.5, and approximating one for large values of x. Some of the graphs, such as the ones in figs 4b and 5b, show what could be minor second and third peaks at distances that are roughly multiples of the position of the first peak. However, the heights of these peaks are not significantly greater than the random fluctuations in g(r). Thus the major discernible feature of the distributions is the strong first maximum well below the mean interparticle distance. Table 1 lists the height and position of this peak for all six pictures. At small values of x, g(x) is much smaller
than one, presumably because Ig molecules cannot get arbitrarily close to one another. However, because the ferritin grains are farily large (-120 A diameter [ 17]), and two overlapping grains may not easily be resolved, one cannot obtain reliable information at small distances. Furthermore, the number of grains that appear close together in the analysed pictures is small, leading to large statistical fluctuations in g(x) for values of x near zero. For these reasons we have not plotted g(x) for ~~0.2, although one can assume that the curves do fall to zero somewhere near the origin. Two independent measures of the statistical fluctuations in g(r) due to the finite sample size have been computed. (1) For a picture containing N grains in a rectangle with specified dimensions, I have computed g(r) for N points placed at random in the same rectangle. Fig. 1 shows the radial distribution function computed for a randomized version of fig. 4~. (2) I have used a method derived by Markovics, Glass &
Exp Cell Res 1 I2 (1978)
Surface immunoglobulin
Table 1. Summary of radial distribution
function
on B lymphocytes
3 15
calculations
Picture
Max. value of g(r)fS.E.”
Position of maximum (A)*
Mean se aration r0 (R )
Position of maximum/r,
Coordination no. (z.)
No. of grains (N)
1’ 2d 3 4’ 5’ 6
1.56f0.13 2.lliO.13 1.49f0.12 1.S8IO.16 1.60+0.20 2.10+0.15
125 153 184 171 163 155
221 340 334 343 327 344
0.55 0.45 0.55 0.50 0.50 0.45
2.2 3.0 2.6 2.0 2.0 1.5
653 1 028 620 438 263 803
a The standard error (S.E.) was estimated by the method of Markovics et al. [27]. ’ For peaks that were flat, such as that shown in fig. 5b (i.e., g(x)=g(x+O.l) where x was evaluated at intervals of 0.1) the position of the maxima was chosen in the center of the flat portion of the peak (i.e., if g(x) = g(x+O.l) thenx+0.05 was chosen as the position of the maximum). c Shown in fig. 4. d Shown in fig. 5. e Pictures 4 and 5 correspond to nearby regions on the same freeze-etch micrograph.
Maul [27] to estimate the standard error (SE.) for g(r) at each point. The 95 % confidence limits specified in table 1 and figs 4b and 5b are given by +1.96 S.E. Both computations of statistical fluctuations show that the first peak seen in g(r) for the surface Ig maps is statistically significant, whereas subsequent peaks may not be. Estimates of the total number of surface Ig molecules can be made from the density of molecules found in the radial distribution function analysis. Using the values of r,, listed in table 1, one finds that Q=ro2 is on the order of 10” molecules/cm2. Hence for a typical B cell with a 7 pm diameter the total number of receptors is on the order of lo5 which agrees with other estimates [3 l-321. The average number, z, of first nearest neighbors of a receptor was calculated from eq. (1). From table 1 one can see that z varies from 1.5 to 3.0, with mean 2.2. This computation was not uniformly accurate since it was difficult to identify the end of the first nearest neighbor peak of the n(r) vs r plots for pictures l-3 (see fig. 4c), whereas the remaining pictures had distinct minima which marked the termination of the first peak (see fig. 6).
DISCUSSION The predominant feature of the radial distribution functions computed for the six surface Ig maps is the presence of a major peak at distances near one-half the mean interparticle separation. A comparison with fig. 3 shows that these radial distribution functions closely resemble those of a dilute gas or liquid. The presence of the major peak, which corresponds to a shell of nearest neighbor atoms in a fluid, indicates that the ferritin grains are clustered into small groups. The number of grains per group is related to the area under the first peak of the function 2rrg(r). The distance along the x-axis between the peak in g(x) and x= 1, and the sharpness of the peak, determine the amount of cluster character in the distribution [27]. Thus on the basis of this g(r) calculation one can say that the points in fig. 5a are somewhat more clustered than those in fig. 4a. Identifying a cluster as a ferritin grain and its shell of first nearest neighbors, the average cluster size in fig. 5a is 4.0, whereas for fig. 4a it is 3.2. The five freeze-etch pictures were all marred by the presence of a small number of ice crystals. The position of one is Exp Ceil Rrs I I2 (1978)
3 16
A. S. Perelson
Fig. 6. Abscissa: x; or&a&: q(x). The radial distribution function for picture 5 (not shown) was used to calculate xg(x). The minimum at x=0.7 was taken as the end of the first nearest neighbor peak.
denoted by the kidney-shaped object near the left border of the rectangle in fig. 4a. These crystals which presumably hid some receptors, could slightly distort the distance relations in the nearest neighbor and g(r) calculations. The picture in fig. 5a was the worst offender, having 6 ice crystals (not shown) in the analysed rectangle. The additional clear area created by these crystals may be the reason that the nearest neighbor distance for this picture was the greatest of all the pictures analysed. The mean number of first nearest neighbors surrounding a ferritin grain was found to be 2.2. For a pattern in which ferritin grains were organized along well separated lines, possibly running in different directions, each grain would have two first nearest neighbors. This would also be true for points lying on well separated equilateral triangles. At distances of 2-5 mean interparticle spacings the radial distribution functions of all six patterns approached one. This strikExp Cell Res 112 (1978)
ingly demonstrates that at large separation distances there is no correlation in the positions of surface Ig molecules. The absence of long-range order in the topographic distribution of membrane proteins is an important prediction of the fluid mosaic model of membrane structure [6]. (Strictly speaking, the existence of a fluid membrane in which proteins are free to move does not imply the lack of long-range order. Clearly, if there were long-range forces acting among proteins or even short-range interactions which resulted in large aggregates then long-range order could exist. However, Singer 8z Nicolson [6] hypothesized that long-range random arrangements of proteins in membranes are the norm and whenever non-random distributions are found, special mechanisms, such as cross-linking agents extrinsic to the membrane, must exist which are responsible for the observed order.) Radial distribution function analysis provides a very precise quantitative confirmation of this prediction for surface immunoglobulin. The mechanism which generated the clustering of surface Ig molecules cannot be determined from radial distribution function analyses alone. It is possible that the clustering reflects the manner in which the Ig is inserted into the membrane. However, since receptors are believed to diffuse within the plane of the membrane, memory of the initial distribution of receptors should quickly be lost. If the movement of Ig is subject to some intermolecular interaction then a degree of short-range order is not inconsistent with the fluid mosaic model of membrane structure [6]. In fact, if one assumes that the interaction of surface Ig molecules is analogous to the interactions of molecules in an equilibrium fluid which can be described by an intermolecular potential, u(r), the techniques developed to
Surface immunoglobulin
study the structure of materials can be applied. For a dilute gas g(r)=exp [-u(r)/kTl while for a denser system a correction term, r(r), is needed
g(r)=7(r) expC-u(rYkTl where k is Boltzman’s constant and T is the absolute temperature [33]. In the theory of liquids experimental measurements of g(r) are used to infer information about the potential u (r). Similar techniques might be of use in the study of membrane proteins; however, a great degree of caution is necessary in applying such techniques to the surface Ig data analysed here. (1) One has no assurance that the Ig molecules are completely free to move within the plane of the membrane. There may be weak or transitory associations with some anchoring structure within the membrane or components of the cell’s cytoskeleton. In fact, the observed clustering could simply be explained on the basis of such associations. (2) The positions of the ferritin grains are not solely determined by membrane Ig. Since immunoglobulin is not the only protein on the cell surface, and in fact, has been estimated to occupy less than 10% of the membrane surface [34], protein-protein, protein-lipid, and lipid-lipid interactions within the membrane are certainly present and will effect the positions of the surface Ig. Further, the portions of the Ig molecules protruding from the membrane can interact with the liquid medium surrounding the cell, and interactions between the ferritin, fluorescein, and Fab fragments, used in labeling the surface immunoglobulin, and other
on B lymphocytes
3 17
molecular species in the system probably occur. Additionally, because a double labeling procedure was used there is no assurance that the ferritin grain is directly over the surface Ig; a statistical distribution of ferritin positions surrounding the Ig must be allowed. From a statistical mechanical viewpoint such a system is enormously more complicated than the simple liquids usually studied by radial distribution functions. Electrically charged globular proteins in concentrated aqueous solutions may have radial distribution functions similar to those found here for surface Ig [35-361. In such systems locally ordered clusters, characteristic of liquid structure, are established through the action of the Coulomb force screened by a solvent containing electrolytes of low molecular weight and the shortrange intermolecular repulsion determining the molecular diameter [35]. Since macromolecular solutions, which in some respects are analogous to the current fluid-mosaic view of membrane structure, give rise to liquid-like radial distribution functions, one has some basis for believing that the surface Ig distribution can be attributed to the net intermolecular force between two surface macromolecules. Even if one assumes that the computed g(r) is a reflection of the net intermolecular potential between surface Ig molecules, determining u(r) from the available data will not be an easy task. Many types of potentials give rise to radial distribution functions similar to those shown in figs 4 b and 5 b. A two-dimensional system of hard disks which interacts only by repulsion through contact collision gives rise to liquid-like radial distribution functions [37]. This fact implies that one cannot rule out the possibility that the observed clustering of Ig may result solely from repulsive interacExp CdRrs
112 (1978)
318
A. S. Perelson
tions due to contact collisions or electrostatic forces between similarly charged membrane macromolecules. However, potentials which have both attractive and repulsive parts, such as the Lennard Jones 6-12 potential, will also generate liquid-like radial distribution functions. So far we have implicitly assumed that the lipids of a membrane form a homogeneous fluid. However, the lipid composition of a membrane may vary from one location to another. Phase separations in mammalian cell plasma membranes have been suggested to occur as cells are cooled from 37°C to below 15°C [38]. There is also evidence for an asymmetrical division of the major phospholipids between the two halves of the erythrocyte membrane bilayer [39]. If the lipids of the B cell membrane were segregated into domains, some with high affinity and some with low affinity for surface Ig, then spatial maps would tend to show regions of bare membrane interposed between regions of high surface Ig density. It is anticipated that the radial distribution function for such a system would indicate the presence of clusters and hence the existence of lipid domains with different partition coefficients for sIg is another possible explanation for the observed g(r). One limitation of radial distribution function analysis is that angular anisotropies are averaged out. (Such anisotropies can be analysed using the pair distribution function which depends on both the distance and angle separating particles.) Thus if surface Ig molecules tend to align themselves with an underlying oriented cytoplasmic tibrillar network the resulting orientation of the pattern would not be readily discernible from g(r). However, we have presented nearest neighbor evidence which is consistent with linear organization. If surface Ig molecules are arranged in anisotropic linear patterns Exp Cd Res 112 (1978)
then it seems doubtful that assigning a simple intermolecular potential acting between pairs of surface molecules can lead to accurate predictions of the topographic distribution of surface Ig. One further limitation of g(r) analysis is that its ability to reflect the true distribution of surface Ig is necessarily limited by the accuracy of the labeling and fixation technique. Abbas et al. [17] critically discuss their labeling procedure and seem assured that artifacts did not significantly affect their conclusion of a non-random distribution for surface immunoglobulin. However, by increasing the precision of the analysis as I have done, one must be concerned with artifacts that manifest themselves at small distances. At present one simply does not know how much distortion of surface architecture occurs during the freeze-etch procedure. It is doubtful that the distortion is great since the cells are fixed, but small positional disturbances can affect the short-range order seen in g(r). In the calculation of g(r) an average is done over all ferritin grains which may cancel the effects of small non-systematic distortions. In the indirect labeling procedure Abbas et al. used, one does not know with certainty how many ferritin grains label each surface Ig molecule. The large size of ferritin makes it unlikely that more than three or four ferritin grains can occupy the area represented by a single Ig [17], but this is the size of the clusters detected by nearest neighbor analysis. More sensitive labeling controls coupled with g(r) analysis may help alleviate such concerns. The distance between ferritin grains used in the calculations of g(r) were those obtained from freeze-etch micrographs. Since a micrograph is in effect a two-dimensional projection of the actual surface some distance distortion may be present. First, for a
Surface immunoglobulin
fragment of the cell surface 1 pm in length (a typical size for the samples analysed in this study), the distortion due to the curvature of the fragment, assuming it came from a sphere with a 4 pm radius, is negligible (a measured distance of 1 pm would correspond to a distance on the sphere of 1.0026 pm). However, if the surface were crenated then the distance distortion would be greater. In this regard it is important to point out that Lipscomb et al. [40], using scanning immunoelectronmicroscopy with tobacco mosaic virus as a marker, have described the surface immunoglobulin on murine splenic lymphocytes as being randomly distributed, appearing both on microvilli and the smooth surface of the cell. Thus the presence of microvilli would not be expected to cause a gross redistribution of surface immunoglobulin, but could cause some distance distortion. This effect could be minimized by restricting the analysis to cells prepared at 0”-4’C, since there appears to be a lack of microvilli expression at such temperatures [41, 421. In our present state of knowledge, the usefulness of radial distribution functions in the analysis of membrane structure comes from ignoring the detailed molecular interactions that produce a given spatial distribution of surface molecules, and using g(r) as a quantitative indicator of overall features of spatial pattern. Markovics et al. [27] show very dramatically how pore patterns on nuclear membranes can be distinguished by radial distribution function analysis. I have shown how the calculation of g(r) can be used to quantitatively confirm the absence of long-range order in the distribution of membrane proteins as predicted by the fluid mosaic hypothesis. However, there are other important potential applications of radial distribution functions in the study of surface macromolecules. Using the
on B lymphocytes
3 19
methods presented in this paper, one might be able to address questions about the amount of receptor cross-linking necessary to trigger specific cellular reactions. For example, if as is commonly believed capping is triggered by patch formation, is the effect of many small patches the same as the effect of a few large patches? Is there a relationship between the amount of receptor cross-linking and the induction of tolerance in B lymphocytes? The release of histamine by mast cells or basophils has been shown to depend upon receptor cross-linking [43, 441. In these systems radial distribution function calculations may provide a way of quantitatively relating the amount of crosslinking with the measured dose response curves for histamine release and thus relating cell surface events to stimulatory and tolerizing signals. Further, given maps of surface molecules at different times during the processes of patch and cap formation, analyses such as the one described herein, may be useful in distinguishing mechanisms involved in receptor aggregation and the dynamics of receptor movement.
CONCLUSIONS The spatial distribution of surface immunoglobulin on murine splenic B lymphocytes has been mathematically analysed by computing a statistical measure, the radial distribution function. The analysis confirms the conclusions of Abbas et al. [ 171:(1) surface immunoglobulin molecules are nonrandomly distributed; and (2) immunoglobulin receptors in the absence of crosslinking agents form small clusters. The analysis also demonstrated the absence of long-range order as predicted by the fluidmosaic hypothesis [6]. Computation of the average number of first nearest neighbors Exp Cell Res 112 (1978)
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A. S. Perelson
indicates that there may be a linear order to the surface Ig patterns. The radial distribution function allows one to quantitate the clustering of surface molecules and can be used to provide information about the intermolecular forces between surface macromolecules. Its application to other experimental models is highly recommended.
analysis is complicated by the size of P, which may contain on the order of lo6 elements. In order to handle large matrices, P was analysed one row at a time. The sum of all the elements of Si is the total density of a ferritin grain. The central elements of S, having the highest densities. Those central areas, say of sizer’ XC’ (r’
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Received July 20, 1977 Revised version received October 14, 1977 Accepted November 2, 1977
Exp CellRes 112 (1978)