J. theor. Biol. (1975) 52, 419-440

The Kiuetics of Hemolytic Plaque Formation IV. IgM Plaque Inhilbitiont $ CHARLES

DELISI

University of California Los Alamos ScientiJic Laboratory, Los Alamos, New Mexico 87544, U.S.A. (Received 20 May 1974) An analysisof the inhibition of hemolytic plaquesformed againstIgM antibodiesis presented.The starting point is the equationsof DeLisi & Bell (1974) which describe the kinetics of plaque growth, and DeLisi & Goldstein (1975) which describeinhibition of IgG plaques. However, the physical chemical models which were used previously to describe IgG inhibition data are shown to be inadequate for describing the characteristicsof IgM inhibition curves. Moreover, it is shown that the experimental results place severerestrictions on the possiblechoicesof physical chemicalmodelsfor IgM upon which to basethe calculations. It is argued that in order to account even qualitatively for all the data, one mustassume(1) a very restricted motion of IgMs about the Fab hinge region and (2) a very narrow secretionrate distribution of IgM by antibody secretingcells.

1. Introduction In a previous paper we formulated

a mathematical model for the growth of hemolytic plaques (DeLisi & Bell, 1974). The model allows antibodies which are secreted isotropically by a point source to diffuse in an infinite medium in which they may interact specifically and reversibly with RBC epitope. This picture leads to a nonlinear diffusion equation in the free antibody concentration, coupled to a first order differential equation in the number of bound red blood cell (RBC) sites. Solutions to these equations can be found under various conditions (DeLisi & Bell, 1974; DeLisi 1975b) and the plaque size predicted by requiring that some fixed minimum number of antibodies be bound per RBC in order to have lysis (DeLisi & Bell, 1974; Jerne, 1974). The model was later extended and applied to the analysis of experiments in which free epitope (inhibitor) as well as RBC bound epitope is present in t Work performedunderthe auspices of the U.S. Atomic EnergyCommission. $ Resent address:Laboratory of Theoretical Biology, National Cancer Institute, NationalInstitutesof Health,Bethesda,Md 20014,U.S.A. 419 21 T.B.

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the medium (DeLisi & Goldstein, 1975; DeLisi, 197%). The inhibitor competes with RBC bound epitope for antibody combining sites and thus retards the growth of plaques. Experiments are generally carried out at several inhibitor concentrations and the number of plaques observed after a fixed period of incubation is recorded as a percentage of the number counted when no inhibitor is present. A plot of the percentage of plaques remaining against the total inhibitor concentration is referred to as an inhibition curve. The derivative (slope) of this curve plotted against the total inhibitor concentration is referred to as a differential plot. The analysis of inhibition experiments leads to the prediction that the type of information which can be obtained from differential plots may be critically dependent upon the way in which the RBCs are prepared. If they are densely haptenated and bimolecular interaction with antibody is assumed to be followed rapidly by an irreversible intramolecular reaction, the differential plot will be a reasonable reflection of the affinity distribution of antibody combining sites for inhibitor (DeLisi & Goldstein, 1975). This prediction is a direct consequence of treating the interaction between antibody and RBC as irreversible so that only the interaction with inhibitor is controlled by affinity. If, on the other hand, the RBCs are sparsely haptenated so that there is a strong bias against intramolecular reaction, then the interaction between antibody and RBC epitope as well as between antibody and inhibitor is controlled by afhnity. In this case the slope of the inhibition curve cannot be affinity dependent and the breadth will probably reflect the spread in secretion rates (DeLisi & Goldstein, 1974, 1975). Neither of these predictions takes account of the details of antibody structure, but they may nevertheless be reliable. It is clear that, in general, the antibody hinge region as well as the RBC epitope density must play a role in determining the equilibrium constant governing intra-molecular reaction (Crothers & Metzger, 1972; DeLisi, 1974). However, under certain conditions the details of the hinge structure need not be important. For example, if bivalent attachment of an antibody to a RBC is irreversible and rate limited by the formation of the first bond, the intra-molecular nucleation step will not enter the description of the process. As mentioned above, this possibility is most likely when the RBCs are highly haptenated. One of the models developed previously pursues the mathematical consequences of this description. The predictions of that model appear to support intuitive expectations and available experimental evidence. In particular, the free hapten concentration at which a particular degree of inhibition is achieved was found to reflect the antibody affinity, and the differential plot was found to reflect the affinity distribution, The description, however, implicitly treats IgG.

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A mathematical formulation of the physical chemistry of plaques formed against IgM will, in general, be more difficult for several reasons. First, the additional combining sites increase the complexity of the problem. However, this presents no difficulty in principle. Much more important is the spatial disposition of the sites relative to one another. When five Y-shaped IgG-like structures polymerize, the relation of the thermodynamic and kinetic properties of the resulting IgM to those of the Y-shaped subunits (IgMs) may be relatively complicated. For example, there can be, and probably is, a change in the ease with which the Fab arms on the same subunit can move relative to one another (Feinstein, Munn & Richardson, 1971). In addition, the morphology of IgM is expected to have an important bearing on its physical behavior. Unfortunately, geometrical information detailed enough to evaluate these effects is not available. Although various morphologies have been visualized (e.g. planar and staplelike conformations-see Fig. I), neither their relative free energies nor the extent to which some may be artifacts imposed by preparation is known (Feinstein et al., 1971). In this paper a general formulation of the theory of IgM plaque inhibition will be presented. In some cases the theory will: contain unknown parameters related to IgM structural characteristics. I will show how it may be possible, by comparing theory to experiment, to gain some insight into these. In addition the results of certain inhibition experiments are qualitatively different for IgG and IgM and these differences can be traced to possible differences

.

330-35oi-------

1. A schematic representation of IgM in a planar conformation (taken from Crothers & Metzger, 1972). FIG.

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in structural features (section 3). Finally, as indicated above, it may be possible to set experimental conditions such that many structural features play a relatively minor role in determining the results of inhibition experiments, and in such cases the results should reflect the thermodynamic properties of the combining sites. 2. Theory (A)

INDEPENDENT

UNITS

The simplest model is one in which its five IgG-like subunits are treated as completely independent of one another but are nevertheless required to diffuse with the diffusion coefficient of the entire IgM. In this case expressions describing IgG and IgM inhibition will be formally identical. I will first review a model which was used previously to describe IgG plaque inhibition (DeLisi & Goldstein, 1975) and then develop additional and more general models. Example 1: hapten inhibition

When free hapten is present in the plaque assay, diffusing IgMs can be in one of three states: both, one or neither of its sites may be bound by inhibitor. Let cz, cr and cO denote the concentrations of these species respectively (for convenience these symbols will also be used as names) and let c be the total concentration of diffusing IgMs at time t, and distance r from the lymphocyte. Then c = c,+c,+c,. (1) If the distribution of diffusing antibodies among the three species is controlled by equilibrium conditions, then C

c&J= z cl=-c

2KH z

c2 = (KH)’

-;

@cl

where Z = 1+2KH +(KH)“. (3) H is the total inhibitor concentration and K the affinity of the antibody combining site for inhibitor. To develop the model further, the preparation of the RBC must be considered. When epitopes are sparsely distributed on the RBC membrane

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only univalent attachment is likely, whereas when the epitope coating is dense, multivalent attachment may be favored. We showed previously that these two possibilities lead to inhibition curves containing different types of information. In this example I will suppose that conditions favor irreversible bivalent attachment and that the intramolecular step occurs very rapidly. These assumptions correspond to those of example 3 in DeLisi & Goldstein (1975). Let p1 be the concentration of IgMs subunits which are bivalently attached to RBCs; p2, the concentration of univalently attached subunits having the other site free, and pa the concentration of univalently attached subunits having the other site bound by inhibitor. Because of the assumption of rapid intramolecular reaction, p2 is negligible. For the same reason, the rate-limiting step in forming pi by collision between cO and a RBC is the bimolecular reaction. Consequently, 3Pi -=-

at

%w,

z

+ b,

ap3 2krCKHp, -= - 2krp,at Z

(4a) (W

k, is the forward rate constant for interaction between an antibody combining site and a RBC epitope, k, is the reverse rate constant and p,, the total concentration of RBC epitopes. The first term on the right in equation (4a) reflects the formation of antibody bivalently bound to a RBC as the result of collision between a combining site and a RBC. The second term is the contribution resulting from dissociation of hapten from p3, followed by an instantaneous intramolecular reaction. The first term on the right in equation (4b) represents the formation of p3 by bimolecular interaction between ci and RBC epitope; the second term represents its disappearance either as the result of the antibody dissociating from the RBC, or the inhibitor dissociating from the antibody. In addition the concentration of diffusing IgMs satisfy equation (5). (5) where D is the IgM diffusion coefficient. Since S is the number of antibodies secreted per second it must be multiplied by five to obtain the rate at which IgMs are produced. A five multiplies the second term on the right because every time an antibody reacts with a RBC, five IgMs disappear. If the concentration changes resulting from chemical reaction occur rapidly compared to those due to diffusion, local equilibrium can be assumed. Such an assumption is expected to be most applicable for those interactions

424

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that can occur only univalently. KHKp,

With this assumption c

P3 =

DELIS1

(64

Z

ap3 KHKP, ac dt=-z-ai ap, 2k,p,c -Yg-=z+

(DeLisi & Goldstein, equation (5)

(W

kfHKp,c Z

1975). Substituting

- Kp,KH ~ 22

equations

ac ---

at

(64

(6b) and (SC) into

where

and 1 c-

W

- -(2+KH) ZD

the solution to equation (7) in three dimensions is

+ e’lLl

l-erf 9 + iitimr)l} (9) 1 1 [ ( and NA the number of antibodies bound per RBC is, from equations (6a) and (64, N A

= ktW+KW Z

’ ,5 c dt’ -j- K+;E

c

(10)

where c is given by equation (9), and E is the number of epitopes per RBC. Example 2: hapten inhibition, reversible intramolecular

binding to RBC

This example differs from those previously considered in that intramolecular binding of an antibody to a RBC is allowed to be reversible. However, the assumption of completely independent subunits is retained. If k, and kd are the forward and reverse rate constants for the intramolecular

IBM

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step, then -

at

= k,p,

ap2=

_

at ai3 ---Z at

- 2kdp,

2kf 7

+ 2k,p,

k 2KHc~o ~

f

+ k,p, - k,p, - kfHp,

+ k,Hp,

2

- k,p,

- 2k,p,.

The first term on the right in equation (lla) represents the formation of bivalently bound antibody as the result of intramolecular reaction. The second term represents the disappearance of bivalently bound antibody as the result of intramolecular dissociation. The first term on the right in equation (1 lb) represents the association of co with RBC epitope; the second, intramolecular dissociation and the third dissociation of inhibitor from p3. The last three terms on the right represent the disappearance of pz by dissociation from RBC, association with inhibitor (transformation to p3) and intramolecular nucleation (transformation to pl). The only difference between equations (1 lc) and (4b) is the second term on the right in (1 lc) which represents formation of p3 by collision between pz and inhibitor. These equations are coupled to equation (5) which governs the concentration of diffusing antibody combining sites. The system must, in general, be solved numerically. However, if chemical reactions are rapid enough for local equilibrium to apply, equations (1 la)+ lc) become apI K&PO at=---z

ac

Wa)

at ap2 2Kp, ac dt=-- z at ap3 at=

where K,, E k,/k,. Equation into equation (5).

ZKp,KH z

VW ac at

(1 w

(13) follows upon substituting

g=v2c+pDSSS(r)

equations (12) (13)

2

where D2 E

D 1 + ~OKPO z (1 +K,+KH)

.

(14)

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The solution to equation (13) is 5s c = --- erfc (r/2viDz t).

(15)

4zDr

From equations (12), the number of antibodies bound per RBC, NA, is

Example 3 : inhibition by antigen

If the antigen is haptenated with E’ epitopes [e.g. dinitrophenylated bovine serum albumin (DNP,,--BSA), where E’ is the number of DNP groups per BSA molecule] then there is a possibility for intramolecular reaction between antibody and antigen. Let c3 be the concentration of antibody subunits which are bivalently bound to the antigen. Then c = c,+c,+c,+c, c, c,,, cr and c2 retain their previous definitions. for a diffusing antibody molecule is Z = 1 + 2KHE’ + (KHE’)’

(17)

The partition

+2KHE’K:,

function, Z, (18)

where H is the number of antigens per ml; KA the equilibrium constant governing intramolecular nucleation of an antibody which complexed with an antigenic determinant and 2KHE’K; c3 = z--

*

If the chemical equilibrium assumption is retained, the form of equations (5) and (12) are unchanged so that equation (I 6) is the solution for the number of antibodies bound per RBC. Numerically, of course, D, and Z are both different since for this model Z is given by equation (18). Equations (14) and (18) involve both bimolecuIar and intramolecular equilibrium constants. Their numerical values will reflect the reactivity of the chemical groups that are interacting and the probability that the groups are correctly positioned. One generally assumes that the reactivity of the chemical groups is the same whether the reaction is intramolecular or bimolecular, and that the difference in equilibrium constants arises from restrictions on the freedom of reacting units which are present in the former reaction but not in the latter (DeLisi & Crothers, 1971; Crothers & Metzger, 1972). It is therefore evident that there is a relation between the two equilibrium constants which can be written as K, = Kf(S’,

6, N/V)

UW

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where f is a function which depends upon antibody structure, S’; epitope density, o and some reference concentration N/V. For example, if IgG is assumed to have a freely jointed hinge region (Crothers & Metzger, 1972) then f= 1.9 x 10-2’a 1*9x 10-21E = (lgb) b bA for intramolecular reaction with groups on BSA. A is the area of the BSA molecule; b, the length of the Fab fragment and 1 mole/l has been used as the standard state concentration. With b = 6.5 x lo-’ cm, A = 1.7 x lo-‘* cm2 and E’ = 5; then, for this particular example K, = O-0086 K. (Bj

A STAPLE

MODEL

In order to obtain an estimate of the extent to which different IgM conformations can lead to different predictions, I will develop a model in which IgM has a staple-like morphology. The salient feature of this model is that intramolecular reaction with a RBC will be easier than in previous cases since it will only be necessary for any two of the ten combining sites to be free. As a result there will be more possibilities for intramolecular reaction than there would be if all subunits were separate. In addition two free combining sites on different subunits can span relatively large distances and intramolecular reaction should therefore be possible at lower hapten densities than in previous cases. In order to make a specific comparison with the independent subunit model for inhibition by hapten, a model will be developed in which intramolecular reaction with a RBC is instantaneous and irreversible. Because of this assumption, the result will not depend upon the details of the intramolecular reaction (for example, it will not depend upon whether one, two, three, etc., sites remain free, nor will it depend upon their spatial disposition after bimolecular interaction). Let xi be the probability that a diffusing IgM has exactly i free sites. Then in analogy to equations (4)t

h ---=

10iXicpo+9k,p, kf C

ap,=

krxl CP, -

at

at

i=Z

1(&p,.

The first term on the right in equation (20a) represents the formation of a multivalently bound antibody as the result of a bivalent interaction between In section (A) the p, (i = 1, 2, 3) weredefined as concentrations of RBC bound IgMs. However implicit in those examples wasthe condition that each time a subunit complexed with RBC epitope, a total of five diffusing subunits was lost. Therefore, the p, are also bound antibody concentrations.

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a combining site and an epitope, followed immediately by an irreversible intramolecular reaction. Since an antibody with i free sites can form the first bond in i different ways, this factor is included within the sum. The other terms in equations (20) have meanings analogous to those introduced m example 1. If p is the probability of a free site then I P = r+-iH

(214

and an explicit impression for xi is

Xi= ‘0 pi(l-p)lO-i. 0 Equations (20) are coupled to equation (22) (22)

where c is the concentration of diffusing antibody molecules. in example 1, equations (20) and (22) become

aPI = at

_

krx,wo

lOk,pcp, - T

-

Proceeding as

W-4 f2W (24)

Wb) Since equations (23) and (24) have the same form as (6) and (7), the concentration of free antibodies and the number of antibodies bound per RBC is given by equations (9) and (lo), respectively, with L2 replacing L,, D, replacing D,. 3. Results It is clear that when IgM is modeled as five independently acting IgG molecules, only minor quantitive differences between IgG and IgM inhibition curves can exist. Figure 2 compares these inhibition curves using the hypothetical affinity distribution in Table 1 under the conditions of example 1

IgM PLAQUE

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429

FIG. 2. I is the fraction of the aflinity distributionin Table 1 that producesobservable plaquesat inhibitor concentrationH. Thecurveswerecalculatedusingequation(10)with s = 5OOantibodies/sec; kr = 6 x 106/M-set;E = 6 x 1V epitopes/RBC; p. = 2.4 x lOl4 epitopes/cm3;r=2000 set, -xIgM, D = 1.7 x lo-” cma/sec;-@IgG, D = 3.6 x lop3 cma/sec. His in molecules/ml.

TABLE

1

Hypothetical afiraity distribution K (M-I)

1.02x 10’0 5*07x109 2.50x 109 1.25x 100 6.23x 108 3.12x 108 1.56x 108 840 x 107 390x 10’ 1.95x 10’ 9.72x 106 4.80x lo6 2.43x 10”

n(K)t

0.050 0.057 0.066 0.082 0.093 0~100 0*104 O*lOO 0.093 0.082 0+066 0.057 0.050

t n(K) isthe fractionof antibodycombiningsiteshavingaffinity K for hapten.The table is takenfrom DeLisi& Goldstein(1975).

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(bivalent attachment is irreversible and rate limited by the bimolecular reaction step). The details of the procedure used to construct an inhibition curve were described previously (DeLisi & Goldstein, 1975). Briefly the plaque radius is defined as a distance beyond which the average number of antibodies bound per RBC is insufficient for lysis [approximately five for IgM plaques (Jerne, 1974)] and the smallest observable plaque is assumed to have a radius of 0@)20 cm. Then as NA is calculated at fixed H for various values of K, one need only consult Table 1 in order to determine what fraction of the plaques remain. The result shown in Fig. 2 is that the IgM inhibition curve is displaced from the IgG curve towards higher concentrations. A shift of this sort is generally attributed to affinity variations. However, in this case IgG and IgM have the same affinity distribution. The reason for the shift resides partly in the diffusion coefficient difference, and partly in the difference in the number of subunits per immunoglobulin. Both lead to higher concentrations of IgM subunits within distances typical of plaque radii, and thus increase the difficulty in achieving inhibition. It is immediately possible to dismiss, on the basis of experimental evidence, independent subunits as a model for describing the physical behavior of IgM in plaque experiments. In particular, Clafin & Merchant (1972) report that total inhibition of IgM plaques is produced by about a threefold increase in the concentration of trinitrophenylamino caproic acid (TNP-EACA). The curves in Fig. 2, on the other hand, are very broad and even at fixed secretion rate, span four decades in TNP concentration. Inhibition curves of comparable breadth are, however, found for IgG plaques (Clafin, Merchant & Inman, 1973 ; Davie & Paul, 1973 ; Segre & Segre, 1973). The model, therefore, leads to predictions which are in accord with observations on IgG plaque inhibition, but in striking contrast to observations on IgM inhibition. It is evident that treating IgM as five independent IgG molecules omits characteristics which are very important in determining the thermodynamic and/or kinetic properties of IgM. It is also evident that merely allowing a staple conformation for IgM while retaining the conditions of example 1 will not bring the predictions into closer agreement with experiment. As indicated in section 2, allowing IgM access to a staple conformation will increase the number of ways in which intramolecular reaction can be achieved. Figure 3 shows that the effect of this decreased activation entropy on the inhibition curve is a rigid translation to higher inhibitor concentrations. In order to understand the inadequacies of this model and to interpret the experimental results, it is necessary to understand the factors making the main contribution to the slope of an inhibition curve. In a previous paper

IBM

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FIG. 3. Inhibition curves predicted by the staple (-•-) model. Parameters are the same as in Fig. 2.

431

and independent units (- x -)

we indicated that when the conditions of example 1 are satisfied the inhibition curve for antibodies produced by lymphoid cells having identical secretion rates will be as broad as the affinity distribution (DeLisi & Goldstein, 1975). The reason involves the assumption that the antibody-inhibitor interaction is affinity controlled, whereas the antibody-RBC interaction is not; the latter process being treated as irreversible within the time of a typical experiment. More generally it is reasonable to suppose that the breadth of an inhibition curve may reflect both the spread in a%nities and secretion rates. An inhibition curve which drops precipitously implies that either inhibition conditions are not sensitive to the breadth of these distributions or that the distributions are very narrow. There are conditions under which the inhibition curve will be sensitive to neither of these distributions (DeLisi, 197%). If the reaction with both RBC and inhibitor is irreversible, very abrupt inhibition will occur and this prediction has in fact been verified by experiments with IgG inhibition by antigen (G. Siskind, personal communication; DeLisi & Petersen, unpublished results). However, it appears unlikely that the interaction of TNP specific antibody sites with free hapten can reasonably be approximated by an irreversible chemical reaction during an experiment that typically lasts about 1 hr. Reactions of this sort probably occur rapidly and are more nearly approximated as equilibrium controlled (DeLisi & Goldstein, 1975). We have, in fact, shown that such an equilibrium assumption leads to a number of results which are intuitively reasonable and which are in accord with numerous experimental observations on IgG plaque inhibition.

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With this in mind it is necessary to ask what changes in the model might lead to inhibition curves in agreement with those observed experimentally for IgM. The most obvious choice is to relax the conditions on antibody RBC interaction. Two conditions were used in example 1. The interaction between antibody and RBC can occur multivalently (and therefore irreversibly) and multivalent attachment is rate limited by the bimolecular reaction step. The most efficient way to relax these restrictions is to allow slow (rather than instantaneous) intramolecular reaction. A model which prohibits multivalent attachment then arises naturally as a special case. However, since the conditions of example 1 seem to lead to reasonable predictions for IgG, one must ask what structural differences between the two molecules might necessitate the different choice of models. It may appear that multivalent attachment of IgM to a RBC will be easier than multivalent IgG attachment because of the increased number of reaction possibilities. However, it is easy to imagine situations in which this may not play an important role during the time of the experiment. For example, if motion about the Fab hinge in IgMs is more restricted than it is in IgG, multisite adherence as the result of attachment of both sites in the same sub-unit may not be likely under the usual haptenation conditions. That would imply that IgM can interact multivalently only by assuming a staple conformation. Such attachment probably requires a conformational change from a planar to a staple-like morphology (Metzger, 1974). Rate constants associated with isomeric transitions in proteins generally range from 102-104/sec (Hammer+ 1968) so that such changes are generally faster than antibody site epitope dissociation (Pecht, Givol 8~ Sela, 1972). This may mean that the conformational change in the antibody occurs much more slowly than most of the measured times for isomeric changes in proteins. Alternatively, it may reflect the fact that not only must the antibody change conformation, but the antibody site which is free (most will be bound by inhibitor) must find a free epitope before the antibody dissociates or converts to the planar conformation. In any case, slow intramolecular reaction is the only way in which the antibody RBC reaction can be equilibrium controlled, and such an equilibrium controlled reaction appears to be the only way to account for the experimental observations. Figure 4 shows that this model (example 3 with E’ = 1) predicts abrupt inhibition with secretion rate held constant; i.e. inhibition is insensitive to affinity. The reason is that during the time of the experiment the IgM RBC interaction is controlled by the equilibrium constant for the antibody siteepitope interaction. Since the antibody site-inhibitor interaction is also affinity controlled, selective affinity inhibition cannot be achieved by varying inhibitor concentration. This can also be understood algebraically. A term

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log (HI FIG. 4. Inhibition curves predicted by equation (16) using equation (18) for Z [i.e., using example (311. -.-, K; = 4.35 x lO-5 K. (---) K; = 1.45 x 1O-5K. S = 1000 antibodiesjsec. Other parameters the same as in previous examples. &’ = 0.

of the form KE(1 +KH+K,) (1 +KHE’)2+2KHE’K, enters linearly on the right side of equation (16) for NA and also enters the expression for D,. When intramolecular reaction is impossible and E’= I this term becomes KE/( 1 + KH). Increasing H has two opposing effects on plaque formation. It increases the likelihood that a diffusing antibody can bind to RBC epitope. The latter effect is the dominant one (DeLisi k Goldstein, 1975). As H is increased low affinity antibodies are inhibited first. Plaques corresponding to the lowest affinity subpopulation will disappear when KH is sufficiently greater than 1 to bring NA below a critical value at a distance corresponding to the smallest observable plaque radius. It is, therefore, clear that for most affinity groups inhibition occurs when KH 9 1. Then KE

E

As H increases, the decrease in E/H causes entropic (and therefore almost simultaneous) inhibition of all antibody subpopulations. These results are applicable to a single secretion rate. More generally a distribution of secretion rates must be considered. If the distribution spans two decades, then the inhibition curve will span two decades (DeLisi & Goldstein, 1975). Consequently if the model discussed above is accepted, the conclusion must be that the secretion rate distribution is very narrow.

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4. Discussion The reliability of conclusions drawn from analyses of plaque inhibition experiments is based largely upon assumptions regarding the physical characteristics and behavior of antibodies. For example, it is often stated that a differential inhibition plot will reflect the distribution of antibody affinities (equilibrium constants) for inhibitor. The reasoning is that at the lowest free hapten concentration at which inhibition is observed, primarily the highest affinity antibody subpopulation will be inhibited. As the free hapten concentration is incrementally increased lower affinity subpopulations will be incrementally inhibited. Consequently, a differential plot should reflect the affinity distribution. Underlying this argument, although not explicitly stated, are the assumptions that (1) the interaction between free hapten and antibody is equilibrium controlled and (2) the interaction between antibody and RBC bound hapten is not. Assumption (1) may be valid under the usual conditions which prevail in plaque experiments, but the extent of its validity is not obvious since there will, in general, be a spatial dependence in the antibody concentration gradient. Assumption (2) may be reasonable if the epitope density and antibody hinge geometry are such that bivalent interaction is favorable and, to a first approximation, irreversible during times of experimental interest. It is clear that, starting from a set of physical premises, one can draw conclusions by intuitive reasoning (as above) or by formulating a set of mathematical equations which embody the premises and which can be used to predict their consequences. Often the equations are complicated and additional assumptions are made in solving them. However, it is important to realize that the significance of this level of approximation is trivial in the sense that (1) the errors involved can be estimated and (2) the errors can generally be made as small as desired. Consequently, it is not the details of the mathematical procedure, but the reasonableness of the physical or biological assumptions (i.e. the model) which must generally be called into question. In this paper different models have been presented which might be used to describe the inhibition of IgM plaques. Each arises from the more general model developed previously to describe plaque growth and inhibition (DeLisi & Bell, 1974). One most therefore ask (a) how reasonable is the general model and (b) how reasonable are the specific assumptions which lead to the model presented here. The former question has been considered (Goldstein, DeLisi & Abate, 1975; DeLisi & Goldstein, 1975) and excellent agreement of the predictions with a wide variety of data found. Consequently the main questions which arise here are those specifically involving the

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relation between IgM geometry and its thermodynamic properties. These relations enter the theory as assumptions and if the assumptions are reasonable the predictions of the theory should be in satisfactory agreement with experiment. Fortunately there is an excellent set of experiments to which the predictions of the theory can be compared. The results can be summarized as follows: (1) IgM plaque inhibition is very sensitive to small variations in the free hapten concentration, but relatively insensitive to small variations in hapten concentration when the hapten is presented as a protein conjugate such as TNPE-BSA. (2) Precisely the opposite is true for IgG inhibition. When free hapten is used the curves are broad (Davie & Paul, 1972; Segre & Segre, 1973) whereas when the hapten is aggregated on the surfaces of proteins inhibition occurs abruptly (DeLisi & Petersen, unpublished results; G. Siskind, personal communication). (3) IgM inhibition curves are displaced toward low inhibitor concentrations as the number of haptens per antigen molecule is increased. (4) A gradual broadening of IgM inhibition curves is observed as the extent of haptenation (E) increases. (5) Both IgG and IgM inhibition curves are displaced toward low inhibitor concentrations for high affinity populations. Items (l), (3), (4) and (5) are from the Clafin 8z Merchant (1972) experiments with anti-TNP antibodies. Statements (1) and (2) provide a striking example of how changes in antibody morphology can lead to dramatic quantitative and qualitative changes in the results of inhibition experiments. The results for IgG have been discussed previously and a plausible explanation is readily advanced in terms of a model which requires that (1) multivalent attachment of antibody to aggregated hapten (i.e. hapten conjugated to either a RBC or an antigen) is irreversible during the time of the experiment and (2) such attachment is rate limited by the bimolecular reaction step. These assumptions lead to the prediction that IgG plaque inhibition by haptenated antigen is abrupt even if both the secretion rate and affinity distributions are broad (DeLisi, 1975a). A correspondingly simple model for IgM plaque inhibition cannot be found as readily. It was argued in section 3 that the very sharp inhibition which occurs when free hapten is used as inhibitor can be accounted for provided that within the time of the experiment, IgM interacts only univalently with RBC epitope under typical haptenation conditions and provided that the secretion rate distribution is narrow. Associated with the first assumption are definite implications related to structural differences between IgG and IgMs; associated with the second are biological implications. The latter will not be pursued in this paper. However, the consequences of the structural assumptions lead to definite predictions and the compatability of these with experimental information can be discussed. Items (3) and (4) T.P. 2.8

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which indicate that as E’ increases, the inhibition curves shift to lower total TNP concentrations and broaden, are of particular interest. The theory predicts that as the extent of haptenation increases with total TNP concentration kept constant, the probability of intramolecular reaction increases, but the probability of bimolecular reaction does not change. The assumption that the bimolecular reaction depends only upon total TNP concentration and is insensitive to clustering is, strictly speaking, not correct. Reducing the freedom of TNP groups by constraining them to the surfaces of BSA molecules is expected to decrease their collision frequency with antibody and thus result in a smaller effective forward rate constant. This effect should make inhibition less efficient; i.e. the inhibition curves should shift toward higher total TNP concentrations, whereas they are found to shift to lower concentrations. The effect therefore cannot play a major role in explaining the two results under consideration. The other effect of increasing E’ is to increase the probability of intramolecular reaction. Since the BSA molecule has a relatively small diameter, the intramolecular reaction probably involves combining sites on the same IgM subunit. Such intramolecular reaction with epitopes on a RBC was not permitted, but it may be possible in the case of BSA because of the very high density of TNP groups per molecule. For example, if five TNP groups are uniformly distributed on the BSA surface, they will be spaced at about 65 A from one another, using 3.7 x lo-’ cm for the radius of BSA (Tanford, 1961). To achieve this spacing on a RBC which has a radius of about 4 x 10s4 cm would require 5 x lo6 epitopes; well beyond the several hundred thousand usually obtained under optimum haptenation conditions (Inman, Merchant, Clafin & Tacey, 1972). It is evident from expression (26) that with K,, = 0, and Ki + 0 inhibition is much more efficient and can, in fact, occur when KH < 1. A typical curve is shown in Fig. 4. The two important features of this result are that (1) the curve is broad and (2) the curve is displaced toward lower inhibitor concentrations. Both these features can be understood by again referring to equation (26). Because of the presence of K,‘,, KH need not be large for the expression to be small enough to cause inhibition. It is also clear that when KH is small compared to 1, the expression becomes, using equation (19), KE

1+2K%f’

(27)

where E’ is being included in J: Since inhibition can only occur when 2K’Hf > 1, expression (27) is approximately EI2KHf within the range of H which brings about inhibition. Consequently as H increases, K must decrease by the same amount in order to

IBM

PLAQUE

INHIBITION

437

keep N,, constant at the minimum observable plaque radius. Inhibition, therefore, occurs gradually and reflects the affinity distribution. Although the shift and broadening of the inhibition curve are readily predicted by the model presented here, it appears from Fig. 4 that the breadth is very sensitive to changes in haptenation, whereas experimentally the breadth varies gradually as a function of E’. This difference can be understood by considering in more detail the dependence of intramolecular reaction on the extent of haptenation. As E’ increases, the intramolecular reaction should become more favorable. However, the free energy decrease associated with this reaction cannot continue indefinitely for two reasons, First, as the spacing between most of the determinants becomes very much smaller than the mean separation of the antibody combining sites, very little will be gained by further increase in the determinant density since most combining site separations will lead to intramolecular reaction (provided orientatian is favorable). Second, there is a physical limit on the local density of antigenic determinants. Consequently one expects KA to be a generally increasing function of E’ at small E’ and to eventually become independent of E’. As the E’ increases, Ki may become large enough so that it is the important factor in the denominator of expression (26) and as seen in Fig. 4, the slope of the inhibition curve changes abruptly. However, it is important to realize that at the “critical” value of E’ at which this large to small slope transition occurs, Ki can still be a sensitive function of E’. This hypothesis is illustrated in Fig. 5. Consequently, an increase in E’ beyond its critical value will have the effect of causing inhibition at smaller values of H, but the breadth of the curve will

Number

of hoptens/oti~genrn&de

FIG. 5. A hypothetical curve showing the intramolecular equilibrium constant on the abscissa (scaled by’ W/K) plotted as a function of the emtent of antigen haptenation, E’. The dashed line is the slope of the inhibition curve as a function of (Ki/K). The diagram illustrates that Ki can still be far from its asymptotic limit when the “small-to-large-slope” transition occurs.

438

C. DELIS1

be unchanged. Since KL as a function of E’ is approaching an asymptotic value, the rate at which the inhibition curve is shifting should diminish as E’ increases and the midpoint of the curve should approach an asymptotic value of H in approximately the same way KA approaches its asymptotic value. This account, of course, tells what can happen; i.e. what is physically possible. It may not predict what actually does happen. In order to predict that, more information related to the geometry of IgM and the distribution of epitopes on antigens is needed. However, it does provide a coherent interpretation of observations (l)-(5) and since there may be a range of E’ for which the slope of the inhibition curve is broad, it is easy to understand how the breadth increases as the extent of haptenation increases. E’, it should be recalled, is the average value of some distribution. Therefore, for any given E’, the inhibition curve must be predicted by weighting and summing inhibition curves for particular degrees of haptenation over the entire distribution. Since the spread in the distribution increases as E’ increases, the width of the inhibition curve should also increase.? If this interpretation is correct, it means that the width of some IgM inhibition curves can, depending upon the extent of haptenation, reflect the dispersion in the degree of haptenation, as well as the dispersion in the affinity distribution. Finally some remarks must be made about the role of complement. Since the addition of complement is necessary before a plaque can be seen it may appear that the kinetics of RBC lysis by complement should enter the theory. The reason this effect does not appear explicitly in the equations can be understood by considering the effect which the rate of RBC lyses by complement will have on a plaque. If k; is the rate constant governing lysis by complement of a RBC having IZ bound antibodies, then to a good first approximation k; = nk, provided there are no co-operative effects between bound antibody molecules. In any case, k; will be a generally increasing function of )t and that is all that is necessary for the following remarks to be valid. If RBC lysis occurs rapidly compared to concentration changes due to diffusion, complement kinetics will not enter the theory. To be more specific, call N: the number of antibodies per RBC that defines the edge of the plaque under these limiting conditions, where Nz is a definite but unknown number, and suppose complement is added to the medium just prior to incubation (i.e. at t = 0). t As E’ becomes very large, the relative width (width/E’) is of course the important quantity, since the vast majority of any random sample of haptenated RBCs will have the mean number of haptens bound so the experimental characteristics will reflect primarily this mean number.

IgM

PLAQUE

INHIBITION

439

The rate at which the plaque grows will reflect exactly (i.e. with no lag) the rate at which N: moves outward from the source and complement kinetics does not enter the theory explicitly. For example, if the plaque radius is defined as the distance at which one out of every two RBCs are lysed, then in this limiting situation, NA* = l/2 and that is all that enters the equations as far as complement behavior is concerned. As k. is reduced below its limiting value, a larger number of bound antibodies per RBC will be necessary to lyse a RBC in a given period of time. Consequently, the value of Nz which defines the plaque radius will increase over what its value would be for the limiting situation. So complement kinetics will effect the value of Nz that defines the plaque size. Knowledge of complement kinetics may, therefore, be important if quantitative information is to be obtained from the theory, but it cannot effect the qualitative conclusions discussed above. There is also evidence that the efficiency with which complement binds IgM depends upon whether or not IgM has interacted with antigen. When IgM is free, complement binding efficiency is low; when it is bound by multivalent antigen, binding efficiency is high (Metzger, 1974). This difference has been interpreted by Metzger (1974) as being the result of a conformational change from a planar to a staple morphology when IgM interacts with antigen, possibly exposing sites for complement fixation. Since it was assumed in section 3 that such conformational changes are unimportant during times typical of plaque experiments, it must also be assumed, if Metzger’s hypothesis is to be accepted, that complement will not act with maximum efficiency. However, this need not be important under the saturating concentration conditions generally used in plaque experiments. Moreover, the effect of efficiency, if it exists, would be to alter definition of Nz, and therefore to alter the quantitative but not the qualitative predictions of the theory. REFERENCES CLAFIN, L. & MERCHANT, B. (1972). Cell. Immun. 5,209. CLAFIN, L., MERCHANT, B. & INMAN, J. (1973). J. Immun. 110,241. CROTHERS,D. M. & METZGER, H. (1972). Zmmlmochemistry 341. DAVIE, J. M. & PAUL, W. E. (1972). J. exp. Med. 135,660. DELISI, C. (1974). J. heor. Biof. 45, 555. DELIS. C. (1975a). J. heor. Biol. 51, 337. DEL& C. il975bj, J. math. Biol. (in press). DELISI. C. & BELL. G. I. (1974). Proc. natn. Acud. Sci. U.S.A. 71, 16. DELISI’& CROTHE& D. Ik (1971). Biopolymers 10, 1809. DELISI, C. & GOLDSTEIN, B. (1974). Immunochemistry 11, 661. DELISI, C. & GOLDSTEIN, B. (1975). J. theor. Biol. 51, 313. FEINSTEIN, A., MUNN, E. A. & RICHARDSON, N. E. (1971). Ann. N. Y. Acad. Sci. l!JO,lO4.

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GOLDSTEIN, B., DELISI, C. & ABATE, J., (1975) J. theor. Biol. 52, 317. HAMMES, 6. G. (1968); A&. Protein Chek. 23, 1. INMAN. J. K.. MERCHANT. B.. CLAFIN. L. & TACEY. S. E. (1972). Immunochemistrv. JERNE,‘N. K: (1974). In ‘Meihods insZmmwwlogy and Zknukchemistry (C. Wiiliams & M. Chase, eds.). Rockefeller Univ. Press: New York. METZGER, H. (1974). Adv. 1-n. 18, 169. F%CHT, I., GIVOL, D. & SELA, M. (1972). J. molec. Biol. 67, 421. SEGRE, D. & SEGRE, M. (1973). Science, N. Y. 181,852. TANFORD, C. (1961). Physical Chemistry of Macromoleudes. John Wiley & Sons: New York.