J. theor. Biol. (1975) 51, 337-345

The Kinetics of Hemolytic Plaque Formation III. Inhibition of Plaques by AntigenQ CHARLES DELISI Theoretical Division, University of California, Los Alamos ScientiJc Laboratory, Los Alamos, New Mexico 87544, U.S.A. (Received 19 April 1974) Equations are presented which can be used to describe the inhibition of plaques by multifunctional antigen which binds yG antibody bivalently. The interaction is treated as a bimolecular reaction which is irreversible within the time of the experiment. It is shown that under these conditions the characteristics of the inhibition curve, and their relationship to kinetic and thermodynamic parameters are strongly dependent upon how antibody interacts with RBCs. When the epitope coating is dense, multivalent attachment of antibody is likely and the interaction is considered irreversible. When the epitope coating is sparse, only rapid, reversible univalent attachment is considered and local equilibrium is assumed to hold. The first case leads to an abrupt inhibition curve whose position is determined by the forward rate constant and RBC density. The second case leads to broad asymmetric curves. For this situation the relation between the extent of inhibition and the affinity of the population is generally complicated and reliable affinity information is difficult to obtain. This is contrasted to results obtained previously for unifunctional inhibitors from which reliable affinity information can, in principle, be obtained. The results emphasize the need for carefully designed experiments if affinity information is to be obtained from inhibition studies.

1. Introduction In a previous paper a methematical model was developed for the growth of a hemolytic plaque (DeLisi & Bell, 1974). The model describes the diffusion of antibodies which are emitted at a constant rate by a point source (lymphocyte) embedded in an infinite isotropic three-dimensional medium. Red blood cells (RBCs) with hapten covalently coupled to their surfaces are t Work performed under the auspices of the U.S. Atomic Energy Commission. $ The previous paper is by DeLisi & Goldstein (1975). 331

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uniformly dispersed throughout the medium. Antibody reacts specifically, and in the most general situation reversibly with the hapten. The model leads to a nonlinear diffusion equation in the free antibody concentration, coupled to a first order partial differential equation in the free hapten concentration. In general these equations must be solved numerically. However, limiting closed form solutions for the number of bound antibodies, valid at short and long times, can be easily obtained, the short time solution being particularly useful when antibody binds bivalently to the RBC as would be expected when the hapten coating is dense. The plaque radius is predicted by requiring that at the edge of the plaque, the number of antibodies bound per RBC is just sufficient for lysis. Models have also been developed to describe the inhibition by hapten of plaques formed by yG antibody (DeLisi & Goldstein, 1975). For this situation the medium contains free hapten (hereafter referred to as inhibitor) in addition to the RBC bound hapten (hereafter referred to as epitope). The inhibitor binds antibody combining sites thus reducing their availability for association with RBC epitope. When the reaction between inhibitor and antibody site is rapid compared to the rate of change of antibody concentration resulting from diffusion, local equilibrium between hapten and antibody site is assumed. Under this condition a differential plot of the inhibition curve will reflect the affinity distribution if the RBC is densely haptenated so that reaction with it is irreversible. Alternatively, if the RBCs are sparsely haptenated or the binding is weak so that local antibody-epitope equilibrium prevails, a differential plot reflects the secretion rate distribution. In this paper we analyze the inhibition of yG plaques by haptenated antigen which binds antibodies bivalently and irreversibly within the time of the experiment. We will be particularly interested in the type of information that can be obtained from such experiments. Two models will be developed. The first treats antibody binding to a RBC as an irreversible bimolecular reaction as might be expected if the hapten coating is dense. In the second model RBC epitopes are considered to be sparsely distributed so that only rapid reversible univalent attachment is allowed. 2. Theory CASE I. DENSELY

HAPTENATED

RBCS

Let c be the concentration of free antibody; D, the antibody diffusion coefficient; k,, the forward rate constant for antibody site hapten association ; p,,, the total epitope concentration; pl, the concentration of antibodies bivalently bound to RBCs; H, the free inhibitor concentration, and S, the number of antibodies secreted per second by the lymphocyte under consid-

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eration. Then the equations of the model are II

;

e

= DV’c - !&

SW) + 9,

-2k,Hc

di

(2)

The first term on the right in equation (I) is the rate at which the free antibody concentration changes due to diffusion, the second and third terms the rate at which it changes as the result of complexation with epitope and inhibitor, respectively, and the fourth term the rate of antibody production by the lymphocyte. 6(r) is the delta function. Equation (2) describes the rate at which RBC epitopes disappear. Since the epitope and inhibitor concentrations are each many orders of magnitude larger than the total antibody concentration, the total concentrations of epitope and inhibitor can be used (DeLisi & Bell, 1974; DeLisi & Goldstein, 1975). Then the solution to equation (1) is ic = -~S ,-rlL $!! - -_1L8nDl”

2JDt >I +

where D L2 = 2k,(p, + H)’

(4)

The number of antibodies bound per RBC, NA, is obtained by integrating equation (2) -2k,jcdt’

(5)

0

N 2Ek, i c dt’

for

p N p.

0

where E is the number of epitopes per RBC. CASE

In this case the appropriate

ac at =

T.B.

equations are

DV2c - g

aP2 at =

II

2kh

- 2k,cH

-

~21~

-

+ $$;

(6)

(7)

kp2 22

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where pz is the concentration of antibodies singly bound to RBCs. The second term on the right in equation (6) represents the rate at which the free antibody concentration changes as the result of reversible chemical reaction with RBC epitope. The explicit expression for this is given by equation (7) in which the first term represents the loss of free epitope as the result of collision with free antibody, and the second term, the appearance of free epitope as the result of antibody dissociation. k, is the reverse rate constant. When the number of epitopes per RBC is small and they are uniformly distributed, multivalent attachment will not be possible. In this case the reaction between antibody site and epitope occurs rapidly and reversibly and local equilibrium can be assumed. Then, if K is the antibody site-epitope equilibrium constant (8) Since about lo3 antibodies/set are secreted (DeLisi & Bell, 1974) the total number of antibodies contained within the plaque boundary during the time of a typical experiment is ,5 3.6 x 106. Therefore and

Equation

(6) then becomes

where D* E I) 1+2Kp,

(11)

and 1 FE-1

2k,H D .

w

From equation (8) the number of antibodies per RBC is given by NA = 2KEc

where c is given by equation (3) with dD*t

(13)

replacing JDt and L1 replacing L.

3. Discussion

Equations have been presented which can be used to describe the inhibition of yG plaques by antigen when binding of antibody to the antigen can be considered irreversible within the time of the experiment.

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I

I

16.0

IT.0

I8.c I

log (4

FIG. 1. Inhibition curves for the affinity distribution in Table 1. The dashed line is the result for hapten inhibition obtained using equation (14) as explained in the previous paper. S = 100; kr = 6 x lo5 (M-set)-I; D = 3.68 x 1O-7 cmz/sec; pO= 1.2 x 1Ol4 sites/cm3, E = 6 x lo5 epitopes/RBC, t = 2000 sec. The other two curves were obtained using equation (5) with (100 < S < 1300) antibodies/set. kr = 6 x lo6 (M-WC)-l (high concentration curve); kf = 6 x lo5 (M-~ec-~ (low concentration curve).

The inhibition curve to which the first model leads is shown in Fig. 1. Also shown for comparison is the predicted result when hapten is the inhibitor (DeLisi & Goldstein, 1975). Both curves were generated for the affinity distribution shown in Table 1. The antigen inhibition curve was obtained by TABLE 1

Hypothetical

afinity distribution

K(M-I) 1.02 5.07 2.52 1.26 6.30 3.15 156 7.80 3.90 1.95 9.72 4.86 2.43

x x x x x x x x x x x x x

n(K) 1010 109 lo9 lo9 lo8 108 lo8 lo7 10’ 10’ 106 lo6 lo6

n(K) is the fraction of antibodies having equilibrium

0.0044 0.0700 0.1210 0.1590 0.1400 0.1100 0.0700 0.0150 0.0460 0.0840 0.1100 0.0540 0.0166 constant K for hapten.

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using equation (4); the hapten inhibition

curve by using (14)

where c is given by equation (3) with L replaced by

The only difference between the models which lead to the two results shown in Fig. 1 is that hapten antibody binding is assumed to be rapid and reversible, whereas antigen antibody binding is assumed irreversible. The consequence of this difference together with the assumption that kf is the same for all antibodies, is that the antigen inhibition curve occurs over a range of inhibitor concentration which is several orders of magnitude smaller than that for hapten inhibition. It is evident from equation (14) that plaque inhibition by hapten begins when l/(1 +KH) becomes small enough to reduce the free antibody concentration below a threshold required for plaque formation. This will happen at relatively low hapten concentrations for high affinity antibodies. As the hapten concentration i-s increased the lower affinity plaques will be gradually inhibited. Since the inhibition plot shows a linear relation between log M and log K, this type of inhibition is referred to as linear inhibition. By contrast, in the first model for antigen inhibition, affinity plays no role since binding to both BBC and antigen is assumed irreversible. Under these circumstances the only source of broadening is the secretion rate distribution. Although the secretion rate distribution is probably not as broad as the affinity distribution, this difference is not the main reason for the sharp curve. The abrupt inhibition arises primarily because H affects the diffusion length, E, which enters equation (5) exponentially. The diffusion length also appears in the error function. Its occurrence there, however, was found numerically to have little effect on inhibition. The reason is that the error function reaches 99 % of its asymptotic value in a fraction of a second when L is in the range required for inhibition. Even if L is varied by a factor of 10 (100 in H)? there is an effect on the integrand only at very short times so that the value of the integral is essentially unaffected. On the other hand inhibition occurs at values of exp (-r/L) which are particularly sensitive to changes in L. Inhibition by antigen is therefore mediated by an exponential function in contrast to inhibition by hapten which is mediated by a linear function. In an actual experiment, it is unlikely that the reverse rate constants of all antibodies will meet the conditions of the theory. In particular low

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affinity antibodies are expected to dissociate within the time of the experiment. Since most of these dissociations will be from free hapten, inhibition will be less efficient and the high concentration part of the curve will broaden. An asymmetric inhibition curve is therefore expected, with a large low concentration slope, that gradually decreases as concentration increases. If the incubation period is varied, the effect should become more pronounced for longer periods of incubation. The effect of pO, the total epitope concentration, on the inhibition curve is also easy to predict. Since inhibition depends upon the sum p,+ H, a smaller value of p,; e.g., a smaller red blood cell density will result in a I.0

I A

!

0,80.6 I

A 4

0.4 -

/

o-2 -

./A,a/' A----

00

A4

Il.6

I 12.0

./L 14.0

I30

15.0

log WI FIG. 2. Inhibition curves for the affinity distribution in Table 1, predicted according to equation (13). pa = 1.0 x 1Ol3 sites/en?; E = 5.0 x lo4 epitopes/RRC, k, = 6 x 106 (M-see)-I. (A), S = 100; (--g-), S = 1000.

longer diffusion length. This means that, provided optical contrast is still good, a larger H will be needed to achieve inhibition. The curve will therefore be shifted to higher inhibitor concentrations. Figure 2 shows the prediction of the second model. For this case broad inhibition curves do occur. The reason for the breadth involves the appearance of affinity in equation (13). It is evident that because of the way K enters this equation, the low affinity antibodies will be inhibited at the lowest free hapten concentrations. In the present example inhibition begins at r/L = 0+002/0.00084 = 2.4. Because the exponential function is not changing rapidly at this point, a small change in H will not be amplified and the initial portion of the curve will be broad. As His increased in order to inhibit the high affinity plaques the amplification region is approached and the inhibition curve begins to sharpen as shown in Fig. 2.

344

C. DELIS1

Exponential inhibition has several consequences which can lead to erroneous interpretation of experimental data. To begin with, a differential plot will not be an accurate reflection of the affinity distribution. This is a direct consequence of the asymmetric nature of the inhibition curve which reflects primarily the mode of inhibition rather than the shape of the distribution. Consequently a differential plot will exhibit skewing toward high concentration, even if the affinity distribution is symmetric. This spurious asymmetry will be most pronounced for the high alIinity antibody populations since low affinity populations can be inhibited at lower values of exp (- r/L). If the concentration of antigen at which 50 % inhibition is achieved is used as a measure of the average afhnity of the population, the effect of this differential skewing will be to underestimate the affinity difference between two populations. A potentially more important difficulty can arise as the result of differences in the forward rate constants for two populations. As an antibody population “matures”, most of the affinity change will of course reflect changes in the reverse rate constants. However, there is some evidence that the forward rate constant also increases (Davie & Paul, 1972). If this is so, the populations with higher forward rate constants will reach the amplification region at lower values of H. The result of the increased affinity on the other hand, will be to shift the inhibition curve toward higher H. These two effects will tend to cancel, with the result that the affinity difference between two populations can be severely underestimated. This effect is illustrated in Fig. 3 which shows two inhibition curves. One was generated using the aflinity distribution in Table 1 and the other using a distribution with the same form, but with the

log(H)

FIG. 3. (-A-)

refers to the affinity distribution in Table 1 with kr = 1.2 x 30’ -) refers to a distribution of the same form as that in Table 1, but with the mean affinity reduced by l/l0 and k, = 6 x lo6 (M-see)-l S = 1000 antibodies/set. The other parameters are the same as in Fig. 2.

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mean decreased by a factor of l/10, The high affinity population has a forward rate constant twice that of the low affinity population. If the antigen concentration at which 50 % inhibition is achieved was used as a measure of affinity, the conclusion would be that there is no d$jkence in the afkity of the populations. This result emphasizes the need to interpret plaque inhibition experiments cautiously, and with haptenation conditions clearly in mind. The author is grateful to Dr Gregory Siskind for stimulating discussions which led to the initiation of theseinhibition studies.Drs G. I. Bell and L. M. Simmons are alsothanked for their careful reading of the manuscript. REFERENCES DAVIE, DELIS, DELIS,

J. M. & PAUL, W. E. (1972). J. exp. C. & BELL, G. I. (1974). PVOC. mm. C. & GOLDSTEIN, B. (1975). .K U~ear.

Med. 135, 643. Acnd. Sci. U.S.A. 71, 16. Biol. 51, 313.