Theory of antigen-antibody induced particulate aggregation—II Theoretical analysis and comparison with experimental results

Bulletin of Mathematical Biology, Vol. 42, pp. 37-56 PergamonPress Ltd. 1980. Printed in Great Britain © Societyfor Mathematical Biology

0007-4985/80/0101-0037$02.00/0

T H E O R Y OF A N T I G E N - A N T I B O D Y INDUCED PARTICULATE AGGREGATION--II T H E O R E T I C A L ANALYSIS A N D C O M P A R I S O N WITH E X P E R I M E N T A L RESULTS •

KI-CHUEN

CHAK

and

HIRAM

HART

Department of Physics, City College of New York, NY and Departments of Radiotherapy and Nuclear Medicine, Montefiore Hospital and Medical Center, and Albert Einstein College of Medicine, Bronx, NY The roles of the concentrations of the three interacting constituents in the aggregation process (antibodies, antigens and particulates) are analyzed in detail. It is shown that the basic equation derived in Part I is consistent over a broad range of conditions with experimental findings previously reported.

1. Introduction. The basic equation derived in the preceding article, equation (10) can be rewritten in expanded form as g =h~tyx+hlh2 2fCo(1 _ g)2

+ ... +hlhah t

j.

'

(i)

which now exhibits explicitly the relationship between the fractional agglutination g and three commonly studied experimental parameters; the antibody concentration, the antigen site density and the interacting particulate concentration (i.e., hi, tg and co). The ratio of the rates of binding to dissociation x = s/r, will be treated in the main as a parameter. In Section 2 the relationship between the antibody concentration and the percentage of agglutination is investigated. Section 3 deals with the relationship between the concentration of antigen and antibody for fixed fractional agglutination. In Section 4, the effects of particulate concentration on the percentage of agglutination are examined. In Section 5, the effect of temperature is considered. Various extensions of the dimer model are considered in Section 6. 37

38

KI-CHUEN

CHAK AND HIRAM HART

2. The Effect of Variation of Antibody Concentration on Fractional Agglutination. The effect of antibody concentrations on the percentage of agglutination is predictable from equation (1). Theoretical curves are graphed in Figure 1 for t,x=3.2, 9.4 and 20. In comparing theory and experiments (Dybkjaer, 1966; Wilkie and Becker, 1955; Greenwalt and Steane, 1970; Solomon et al., 1965), it is necessary to relate the experimental titers of anti-sera to the n u m b e r of surface bound antibodies per particle No (see Appendix A). It is found that theory and experiment both give rise to a sigmoidal dependence. However, while theoretically only an

._~ +-

70

o

X=2

X=9, 4 50

X=3.

g 30 13-

I 05

I

2

3

4

5

Number of ontibodies

Figure 1. Percentage of agglutination vs number of antibodies inside the effective area Ae plotted for different values of x. In calculating the curves in this fig_ and in the following figs, the antibody length 1 is assumed to be 250 A, the particle surface area 100#mz and the average interparticulate separation q=l/2. approximate four fold increase in antibody concentration is required for the range of 10 to 90 ~o agglutination, experimentally a four to eight fold increase in antibody concentration is required. Agreement between theory and experiment can still be obtained, however, if the initial assumption that all orders of bonding are equally possible is relaxed and the physical and statistical effects of sequential bonding are considered. The formation of the first bond does not significantly affect the average number of available antigen sites within the effective sub-area a e of the remaining singly bound antibodies. This is because the two spherical particles comprising the singly bound dimer are still relatively free to translate and rotate with respect to each other and therefore the effective sub-areas comprise new and continuously changing portions of the particle surfaces.

ANTIGEN ANTIBODY INDUCED PARTICULATE A G G R E G A T I O N - - I I

39

The formation of subsequent interparticulate bridges will, however, tend to severely limit the relative motion of the two particles, so that the effective sub-areas available for still higher order bond formation will tend to be uniquely prescribed, involving specific areas and orientations. Steric and statistical conditions will then apply which tend to reduce the average value of t o for all of the singly bound antibodies remaining in the effective area A e. Although t o is therefore a monotonic decreasing function of the bond order, it is convenient to assume that to=(ae/A)(Go-No) is relatively unchanged for the first two bonds and that t o is given by t~o=a~(ae/A) (Go-No)=a~t o for all subsequent bonds where the e, are parameters of magnitude less than one. Equation (1) becomes (tox) 2 (tox) 3 g" _ h l t o x + +hlh2h3a3 2fco(1 - g)2 hlh2 ~ - . -~(~. + "-. + h 1 • ..hjaj (

Y.

(2) For tg considerably less than one, the cq will drop rapidly with increasing bond order. Curves calculated from (2) for x = 3 0 , t0=0.3 and various ai sequences are displayed in Figure 2. The sequence "cd', e~ = 6.4/2 ~ for i > 3 give results in quite good agreement with the experimental result on anti-D by Dybjkaer (1966). Although this particular sequence is, of course, not uniquely necessary for good agreement it does exhibit the general type of bond order dependence which would be expected on physical and statistical grounds. For t o greater than one, statistical conditions are likely to be less important and the monotonic decrease in ei with bond order more gradual. It follows from (2) that higher order bonding would then play a more significant role in the agglutination process and therefore that the sigmoidal dependence would be steeper. Experimentally, a slight increase in steepness with an elevated t o may have been observed, but this was only within the range of experimental error and less than would be expected (Dybjkaer, 1966). It is unclear at this time why the principal slope of the sigmoidal curves appears to be relatively independent of the antigen site concentration t o. If this independence is valid, it suggests that factors such as steric effects and single surface double bonding of antibody may in some way suppress higher order bonding at large antigen concentrations. It is, of course, also possible that in order to obtain agreement in such detail, a generally more sophisticated theory may be necessary. 3. The Relation Between Antigen and Antibody Concentrations for Fixed Fractional Agglutination. The effect of antigen concentration on aggluti-

40

KI-CHUEN CHAK AND HIRAM HART

9o 80

o~

7o 60 I

40 ~, 30 u

g_ 20 ro

I

I

I

I

I

I

I

08

I

2

4

6

8

I0

Number

of antibodies

Figure 2. Percentage of agglutination vs number of antibodies within A t for particles with a low surface density of antigen sites. Curves are plotted from equation (2) with tox=9 and five different sequences of ~ (i>3). (a) e l = l , (b) ~i=2.4/i, (d) ~i =6A/21, (e) czi=4/2 i, (f) c£ =2/2 i. The small squares are experimental results taken from Dybkjaer (1966) on anti-D antibody where the comparison is made by assuming that the undiluted serum contains ~ 1 mg/ml of pure anti-D antibodies and ~ 1 0 % of the antibody is surface bound and sterically capable of participating in interparticulate binding

nation has been examined by relatively few investigators (Hoyer and Trabalt, 1970, Leikola and Pasanen, 1970). No agglutination is observed if the particulate antigen site concentration on the particulate surface falls below a characteristic value for each system. As the number of antigen sites is increased, the antibody concentration necessary to produce a given percentage of agglutination should be correspondingly reduced. A simple approximate relation between the antibody and antigen concentrations for fixed fractional agglutination g, and fixed particulate concentration co can be derived by considering only the first term on the right hand side of (2) and noting that the h~t o then completely determine the fractional agglutination g. From the definitions;

AEae

h~to -=R1 = ~ T - N o ( G o - N o )

(3)

Co Go/1_4c

N°=5--+ 2 x/

c0'

where the parameter Ca ~A2R1/Aeae, can be viewed, for fixed f and Co as being a measure of g alone. In Figure 3 the general dependence of (No)-1

ANTIGEN

ANTIBODY

INDUCED

PARTICULATE

AGGREGATION--II

41

upon Go is shown. The two branches of each iso-agglutination curve converge at a minimum value of Go, below which, the required value of g cannot be achieved. There is a discontinuity in the slopes of the two branches. It is interesting to note that the junction occurs at the minimum value of Go=2No. This means that at low antigen concentrations, the optimal agglutination occurs when half of the antigens are occupied. This result is in agreement with the very analogous findings of Singer et al. (1973) that, in an antibody free system, m a x i m u m agglutination of latex particles occurs when the particle surface is half coated with protein, and with the theoretical analysis of Schulthess et al. (1976).

3

2 Ct=6

I

C~ = 8

2

3

4

5

6

7

8

9

Go

Figure 3. Relationship between the number of antibodies and antigen sites for fixed fractional agglutination. Iso-agglutination curves are shown for integral values of C1

The upper branches of the curves correspond of course to the lesser of the two required antibody concentrations for each fractional agglutination specified, while the lower branches correspond to the higher antibody concentration for the same fractional value. It has been reported by Hoyer and Trabalt (1970) that near the minimum values of G o for 50~o agglutination, the slope of the experimental curves tends to be horizontal. A possible explanation for this apparent lack of dependence on Go is that in the neighborhood of the apex of the curves at which Go=2No is a minimum, the upper and lower branches are rather close together and the experimental observations may represent average values of the titer for this region of agglutination, with a resulting apparent horizontal slope. For larger values of Go, the upper branch reflects the condition that No ~ Go. Therefore t o is then essentially directly proportional to Go. The

42

KI-CHUEN CHAK AND HIRAM HART

requirement for fixed fractional agglutination (that toN o be roughly constant) implies that GoNo is nearly constant and therefore that G0~(Constant) x(N0) -1 with the graph of NoX versus Go very nearly a straight line. However, experimentally, (Hoyer and Trabalt, 1970) further increase in the antigen site concentration, Go, does not result in significantly less antibody being required. At still higher values of Go, the experimental curves flatten out with a vanishing slope. An examination of the Go values indicates that this begins to occur when the antigen sites are sufficiently close for an antibody molecule to bind with both ends on the same surface. The formation of single surface doubly bound antibody (ssdba) is evident at high antigen densities (Greenburg et al., 1963, 1964, 1965 and Economidou et al., 1967a). This double bonding reduces the number of antibody molecules otherwise available for cross linkage, and the total gross antibody concentrations required for a given fractional agglutination is correspondingly increased. One would therefore expect that some decrease in the slope of No 1 versus Go would be observed at concentrations of Go for which ssdba becomes more probable. An analysis of this decrease appears in Appendix B. The calculated effect of ssdba is shown in Table 1. It is seen that N1/No depends upon both the effective TABLE 1 Relationship Between Antigen Sites and the Ratio of Singly Bound to Total Bound Surface Antibodies as a Result of ssdba

Co( × 104)

(a) N1/No

(b) N1/No

1 2 3 4 5 6 7 8 9 10 20 40

0.961 0.904 0.847 0_793 0.742 0.694 0.648 0.606 0.567 0.530 0.273 0.077

0.988 0.966 0.941 0.915 0.890 0.865 0.839 0.815 0.791 0.768 0.568 0.312

Go: number of antigen sites on a particle surface of 100 #2. (a) and (b): calculations based upon effective average b!nding length of an antibody for ssdba assumed to be 150A and 100 ,~ respectively; e~X, assumed = 20 in equation (B.6).

ANTIGEN-ANTIBODY

I N D U C E D PARTICULATE A G G R E G A T I O N

II

43

antibody binding length and the antigen site surface density. At a density of Go> 10 s both theoretically and experimentally (see also Hoyer and Trabalt, 1970) the effects of ssdba become evident. 4. The Role of Particle Concentration in the Agglutination Process. The relationship between the fractional agglutination and the particle concentration is also given by (1), which can be written in the form

(1 g g ) 2 -

(la)

----YCo,

where

For a given antibody-antigen system with all physical and chemical factors fixed, y is then a function of h~, which in turn is specified by the number of singly bound antibodies per particle. If only the total concentration of antibody Ab (bound and unbound) is known, then for low concentrations of Ab and co, y is a single valued function of the quotient Ab/co. Consider a given value of y arising from two different pairs of antibody, and particle concentrations: (Abh (C0)I, and (Ab)2 (Co)2. From (la):

(1 -

gl

----Y(Co) 1 ;

(1 -

g2

=

y(coh.

Therefore gl (l-g1)

2

g2 (Co)l (l--g2) 2 (Co) 2 "

(4)

Equation (4) exhibits the theoretical dependence of the fractional agglutination upon the particle concentration in dilute systems when the ratio of Ab/co is kept constant. Relatively little experimental data exists on the effects of particle concentration on agglutination. Solomon et al. (1965); Wilkie and Becker (1955) and Greenwalt et al. (1970) studied the dependence of agglutination upon the amount of antibody for a series of different particle concentrations. In Table 2, some of their results are retabulated (Data 1 from Solomon et al. ; Data 2 from Wilkie and Becker).

44

KI-CHUEN CHAK AND HIRAM HART

Each column of experimental data n o w refers to data points taken from the graphs which correspond to one or another arbitrarily selected value of (Ab/co). The theoretical values are obtained by equating the theoretical and experimental values of g at the most dilute concentrations of co and then using (4) to calculate the agglutination to be expected at higher values of Co. It is seen that for over a ten fold range of dilute particle concentrations, the calculated values are reasonably close to the experimental results. At very high concentrations (the last values of each column), however, deviations exist between the experimental and theoretical results. Whether this is due to the possible increased concentration of inhibitors in the system or to other experimental or theoretical limitations, remains to be seen.

TABLE 2 Effect of Particle Concentration on Percentage of Agglutination

Co

Percentage of Agglutination Data 1 Calculated (a) (b) (a) (b)

5_5 11.3 14_0 22.6 81.0

16 25 27 36 40

103/rnm3

50 58 64 74 74

16 25 29 37 58

50 61 64 71 83

Percentage of Agglutination Data 2 Calculated (a) (b) (a) (b)

Co 103/ram3 11155 50.52 119.3 384.3 873.65

23 58 58 60 46

50 80 82 78 61

23 46 61 76 83

50 72 87 88 92

5. The Effect o_f Temperature. The association rate has the temperature dependence x/kTexp(-Ea/kT). F o r a temperature increase from T1 to T2,, the change of ratio of the association rate isEATz

-

For T2-T1 =10°C, the QA factor is ~ 1 . 4 to 2 for E A between 5 k c a l / M t o 14 k cal/M (the usual range of antibody-antigen energies)--in agreement with results of Hughes-Jones (1963). A similar result for the temperature

ANTIGEN-ANTIBODY INDUCED PARTICULATE AGGREGATION--II

45

dependence of dissociation is: fEB(T2 -

Therefore the temperature dependence of the parameter x is given by

Q =( T,~2/T~)exp{-(En-Ea)(T2- TO} k T2T~

(5)

For exothermic reactions, E ~ > E A and an increase in temperature results in a decrease in the parameter x and so in the percentage agglutination g - - a n effect observed qualitatively by Greendyke and Swisher (1968) and Solomon et al. (1965).

6. Extensions of the Dimer Model. A. Linear model. Consider an extended linear model in which monomers are now theoretically free to interact not only among themselves, but also with dimers, trimers, tetramers, etc. so that larger aggregates can be formed. Cyclic bound aggregates will still be excluded. For simplicity, the interactions between aggregates themselves are prohibited and only the formation of linear chains by monomer accretion will be permitted initially. From the principle of detailed balancing, the relationship between the concentration of monomer and dimer at equilibrium remains: n~ =KD. Similarly,

2noD =KT,

(6)

where D and T stand for the total number of dimers and trimers (singly, doubly, and multiply bound, etc.) and the same value of K applies because the intrinsic association and dissociation properties of the particles are assumed t o be unaltered. The factor 2 reflects the two elements of the dimer with which the monomer can combine. The reduction in the collision cross section due to steric effects as well as to the relatively smaller dimer velocity are considered subsequently. Since only the two end particles of each polymer combine with no the corresponding relationships for the formation of higher aggregates are simply:

2noT =KQ

(7) 2noQ =KP,

46

K I - C H U E N CHAK AND HIRAM HART

where Q , P , . . . represent the concentrations of quatrimers, pentamers, etc. The conservation of particles n o w requires that: no + 2D + 3 T + 4Q, + ... =Co

no + 2 n~) 2n3 + 4 4n4° K +3~K3- + ' " =Co.

(8)

Multiplying (8) by 4/K, and defining z = 2 n o / K , I1 =co/K (8) b e c o m e s J

z + ~ qz q = 4t/.

(9)

q=l

If ]z] > [ t h e n no < 2 D = < 3 T = < - . - a n d this implies complete clumping of the extended system. If [z[ <] the s u m m a t i o n converges and for large j, z

z + ~( 1 - z )

=4q,

or

z 3 - 2(1 + 2rl)Z 2 q-2(1 + 4 q ) z - 41/=0.

(lo)

Since 1/is greater than zero, (10) has one and only one positive real root corresponding to the physical solution. F r o m the definitions of z and q it follows that: y =-no/Co = Zo/2~l can be determined. B. Steric factors. Consider n o w the case in qhich steric or other factors reduce the formation probabilities of higher order aggregates. Let H be a general parameter representing such a limiting effect. The value H = I indicates that no hindrance at all exists, while H = 0 represents a hindrance so large that trimers and higher aggregates cannot be formed. (This actually is the dimer model.) The equilibrium equations for the aggregates are:

no = K D 2HnoD = K T 2HnoT=KQ.

Since side chains or branches continue to be excluded, only the terminal elements are assumed to interact with the monomer, and the same value of H applies throughout. The condition of the conservation of particles n o w

ANTIGEN ANTIBODY INDUCED PARTICULATE A G G R E G A T I O N - - I I

47

leads to : J

( 2 H - 1)z+ ~

qz q

=4.jH 2,

(11)

q=i

where

z=2Hno/K,

and for I z l < l , (11)reduces to:

(2H - 1)z 3 - 2(2Hg~/+ 2H - 1)z 2

-t-

2(4H2t/-I- H ) z

-

4H2t/= 0.

(12)

A physical solution for z can again be obtained provided H and t/ are known. C. Numerical comparison. A numerical comparison between the dimer model and the above extended model is now in order. The a m o u n t of aggregation for various values of t/ and H appears in Table 3. F r o m Table

TABLE 3

0.04 0.06 0.08 0.10 0.20 0.40 0_60 0.80 1.0 2.0 4.0 10.0

Dimer model (H=0)

H=0.5

Extended model H=0.7

H=I

1 - no~co

1 - no~Co

1 - no/Co

1 - no~Co

0.069 0.098 0.123 0.146 0.234 0.344 0_413 0.462 0.500 0_610 0.704 0.800

0.072 0.105 0.134 0.161 0.271 0_414 0.505 0.570 0.618 0.750 0.847 0.927

0.073 0.111 0.142 0_167 0.290 0.450 0_549 0.162 0.667 0.795 0.875 0.946

0.075 0.117 0.148 0.180 0.312 0.485 0.590 0.659 0.707 0.827 0.902 0.957

3, it can be seen that the agglutination predicted by the dimer model and the extended model are in close agreement for small values of r/(--~0 0.2). This is to be expected since at low values of t/, the formation of higher aggregates is very much suppressed. For intermediate values of t/(,-~0.2--2) the extended model again as expected predicts more agglutination than the dimer model with successively more aggregation as the value of H increases. Note that the a m o u n t of agglutination predicted by the dimer model is always least, supporting the original hypothesis that the dimer model can provide a basis (i.e. sufficient condition) for predicting the onset

48

K I - C H U E N CHAK AND HIRAM HART

of agglutination. For t / > 4 both the dimer and extended model predict almost complete aggregation. D. A spherical model. Another analytically tractable model assumes that all binding and dissociation occur only on the surface of purely spherical aggregates. Then, for a monomer of radius Ro, the spherical model gives the surface area: A1 =47cR~, with radius R1--Ro. For a dimer, the model gives A2=4rcR 2 with radius R2=,~/2Ro. For a trimer, the radius R3 = , ~ R o and so on. Although this model is only a rough approximation when considering mini-aggregations, it should tend to be somewhat more realistic when applied to the formation of larger aggregates. Assuming the association and the dissociation are proportional to the surface area: k1

kl n2 = k2D, or n 2 ----~2 D = K D ~/4klnoD = ½~ 4 k 2 T , or noD = K T

(7a)

, ~ k , n o r = ½ , ~ / g k 2 Q , or n o T = K Q

where ka and k 2 a r e the association and dissociation rate constants and the factor 1/2 arises from the fact that a dimer dissociates into two monomers whereas all higher polymers upon dissociation give rise to only one monomer. The resulting equations paradoxically correspond exactly to (7) of the extended dimer model. E. The effect of side chains. The actual interactions between different mini-aggregates in the agglutinating system are of course more complicated than the simple extended models assumed above. There are interactions between mini and higher order aggregates and also side chains are formed. While the association and dissociation of higher order aggregates do not significantly change the concentration of no (the monomer), the extended dimer model tends to underestimate the extent of monomer binding in neglecting the process of side chain formation. In formulating the simple extended linear model, it is assumed that both association and dissociation of the aggregates are independent of the number of particles in the aggregates. This is because in the reactions, only the two end particles of the chain are assumed to participate. If side chain formation is considered, the association between monomers and aggregates is then more nearly proportional to the total available surface areas. This available surface area is rather difficult to evaluate in general. It

A N T I G E N - A N T I B O D Y I N D U C E D PARTICULATE A G G R E G A T I O N - - I I

49

seems reasonable, however, that as aggregate size increases the "interior" particles will become increasingly inaccessible. In the limit of large aggregates, the surface will of course be proportional to n z/3 as given already by the spherical model. The difficult intermediate case in which side chain formation occurs can at present be treated only very approximately. While the rate of association should be essentially proportional to the available surface area, the rate of dissociation of monomer from the surface of a given aggregate depends upon the number of exterior particles which instantaneously are not firmly bound (i.e. held in place by only one antibody bridge). The number of such exterior particles for a given n-fold aggregate will depend not only upon the concentration of the three interacting constituents (antibody, antigen, particles) but upon the specific gross structure of the aggregate. Clearly even if they are at the surface, individual particles bound to more than one neighbor in the aggregate cannot directly enter the no (monomer) pool. To estimate association and dissociation characteristics, it is first appropriate to identify the various non-cyclic mini-aggregates possible and to estimate their relative probabilities of formation. It will be assumed that monomer is equally likely to bind with any constituent particle of a mini-aggregate (i.e. steric effects are not considered in the formation process) and bonds once formed are equally likely to break-~. T h e topologically distinct structures associated with the first few mini-aggregates appear in Figure 4. The average number of singly linked particles Na within a miniaggregate can now be calculated for each aggregate number i and the result is tabulated below together with the approximate exterior surface area Ai for each aggregate species. It is of interest to note that the ratio of successive dissociable particles (nd)~/(nd)~_ 1 and the ratio of the successive surface areas of the spherical model A i / A i _ l = ( i / i - 1 ) El3 correspond closely. The equilibrium equations for this side chain model are similarly in close agreement with the equations of (7a) for the previous models. nZo= K D

(1.31)

2noD = K T

(1.211)

2noT=(1.165)KQ

(1.165)

2noQ=(1.180)KP

?Note that in general, the relative probability of different structures may depend, in addition, upon dissociation occurring in larger aggregates. However, a truly complete analysis involving details of the dissociation process as well as the effects of cyclic bonding becomes extremely complicated and is not considered here.

50

KI-CHUEN CHAK AND HIRAM HART (I) n=3

2 (2)

(I)

(4)

(7)

3

(r)

@

3 (8)

4 (15)

2 (18)

(7)

n=6

Figure 4. The different structures of mini-aggregates from n = 3 to n = 6. The superscripts indicate the relative probability of each structure being formed by monomer accretion compared to others with the same value of n. The subscripts indmate the number of singly linked particles

TABLE 4

Particles in aggregate (i)

Average number of singly bound particles (nn)

2

2

3 4 5 6 7 8

2 2.33 2.75 3.20 3.68 4.07

Successive ratios (neh Ai (n.),_l A, L1 __

1.000 1.165 1.180 1_164 1.150 1.106

L

1.310 1.211 1.160 1.129 1.108 1.093

ANTIGEN ANTIBODY INDUCED PARTICULATE AGGREGATION--II

51

Thus when association, which should be closely proportional to the total available surface area of the aggregates, and dissociation, proportional to the average number of accessible particles ha, are both examined separately and in somewhat more detail, the results seem to leave essentially unaltered the predictions of the simple extended dimer model and the spherical model.

Concluding Remarks.

The dimer theory proposed relates the three interacting constituents (antigen, antibody and antigen coated particulates) to the percentage of agglutination under a variety of conditions. Predictions of the theory appear to be in reasonable agreement with the results of a rather broad range of experiments in antigen-antibody agglutination. Based upon rather incomplete experimental data, there are still uncertainties and in some areas an apparent lack of quantitative agreement between theory and experiment. In order to resolve many of these questions, there is a need for further carefully controlled quantitative experiments on the average number of cross-linkage antibodies as a function of fractional agglutination; on the dependence of the principal slope of the sigmoidal ~ agglutination vs antibody concentration curve on the antigen site surface density; on the ratio of dimers to higher order aggregates as a function of the extent of agglutinatiofi and on such basic questions as the effects of the initial monomer concentrations. It is to be hoped that the rather simple theory here proposed will serve to illuminate marly of the unresolved problems in this field and so ultimately motivate a more complete theory and a better understanding of antigen-antibody induced particulate aggregation. This work was supported in part under Contract ERDA (11-1)-3253 with the United States Energy Research and Development Adrn~nistration. Portions of this work are to be submitted by K. C. Chak in partial fulfillment of the requirements of the Ph.D. degree of the City University of New York. We wish to thank Dr. J. Michael Schurr for his careful review of the theory and for helpful suggestions relating to the requirements of microscopic reversibility and a more exact approach to some of the probability calculations. Thanks are also due to Dr. Micah Dembo for helpful comments motivating a more explicit treatment of the theory and the comparison with experiment. APPENDIX A

Relation Between Anti-serum Dilution and Particle Surface Concentration of Antibodies. The relationship between free and bound antibodies in the system depends

52

KI-CHUEN CHAK AND HIRAM HART

upon the antibody-antigen species as well as the number of antigen sites. At the first stage of (prior to particulate aggregation) equilibrium, the condition can be specified by the law of mass action:

N B = k(G T - NB)(N T - Nn),

(AI)

where N B is bound antibodies per unit volume in the system; N r is the total antibodies per unit volume in the system; G r is the total antigens per unit volume in the system; K is the equilibrium constant. Solving for NB: N n = 1/2 {NT + Gr + 1/k - V / ~ T ~ G r + l/k) a - 4NTGT }.

(A2)

For GT>>NB,

KGTNr NB--

1 +'KG T "

The number of singly bound antibodies in the effective area A~ is given by v0 = (AeNJAeo), or v0 = kBNT, where

kB = A~K GT/(Aco(1 + K Gv)).

(A3)

Since experimental reports are usually expressed in terms of dilution of anti-sera (titers), N~ is generally only specified to within an unknown proportionahty constant kt, N r = k r (titers)- 1. The constant k T depends, of course, upon the specific anti-serum used and the method of producing it. In commercially prepared antisera, the concentration of antibody is of the order of 1 mg/ml. F o r antibody of molecular weight 1.6 x 105, k r is approximately around 101S/ml. The value of kB can then be evaluated by (A3), using the values of the equilibrium constant k, the estimated antigen sites G r and the particle concentration Co.

APPENDIX B

Effect of ssdba on the Number of Singly Bound Surface Antibodies. Consider a given antigen site at the origin. Let Po be the probability that about the origin there is no other site inside an area of b 2, where b is the maximum binding length of an antibody which can bind both its ends on the same surface. Let Pl be the probability that at least one or more sites are inside ~b 2. The parameters Po and Pa can be calculated by use of the Poisson Distribution. The probability of having k randomly distributed antigen sites within the radius b from a chosen site is:

mke-m P,.(k) =

k~-'

(B1)

where m=b2Go/47rR 2 is the number of antigen sites within the neighbourhood nb 2 on the spherical surface of radius R and G0 is the average number of antigen sites on the surface of a particle. (The length b should be less than l, the maximum length of an antibody, since physically, binding on the same surface an antibody cannot extend to 180 ° . Throughout the calculation, it is assumed that l ~ 2 5 0 A and b ~ 150 A.)

ANTIGEN-ANTIBODY INDUCED PARTICULATE AGGREGATION--II

53

Therefore p,.(0) = Po = e - "~ p l = l - - e -m. Suppose there are No antibodies bound to the surface of a single particle. Let N1 of them be singly bound and N z doubly bound (No = N i +Nz). Also let N1 = N i l +N12, where N i l are those singly bound antibodies whose nearest free antigen site is further than b from its binding position and N2 are the remaining antibodies of N 1. It follows that: N i l =p0Ni =e-raN1

(B2)

N12=plN1 =(1 - e-")N i.

(B3)

21N12 = ~2N2,

(B4)

By detailed balancing:

where 21 and 22 are the rate constants for formation and dissociation respectively. Therefore: N1 - 21 e-m+22 "~1+22 N o

(B5)

It is reasonable to assume that the quantity 21, for the formation of doubly bound antibody is directly proportional to the available antigen site vg within the area rob2 about a singly bound antibody or vg=Gm where G is a proportionality factor 0 < c ~ < 1 arising from the existence of steric hindrance effects. Since the dissociation constant 22 is independent of the antigen density vg, 21/12 =x,vg where xu represents the ratio of the association rate to the unstretched dissociation rate of an antibody attached to a single surface equation (B_5) can be written as: N1 -

1 -t-O~sXume-'n 1 + O~sXum

No.

(B6)

Equation (I3.6) indicates the dependence of the number of antibodies instantaneously available for interparticulate cross-linkages upon the number of bound antibodies and the surface density of antigen sites. The steric factor G would depend upon the roughness of the surfaces and the relative location of the antigens on the surface. A simple approximation is to assume cq is a constant in (B.6). The value of x u can be approximated as ~100 (Appendix B of Part I). However, the product can be treated as a parameter and the value of N1/No can then be calculated as a function of m by (B.6) if the antibody concentration is expressed in terms of N. The effect is listed in Table 1.

APPENDIX C

Estimation of the Parameter x.

A detailed expression for x--s/r, using equation (14), (15) and also equation (A10) in Part I is given by: 96,/2 / M 7i72did2e~(o~t fE~-EA" ~ x = -1.67 - ~ 7t ~ / m (1 +e)(l1-2e--e2)/2 e x p ~ ) .

(c1)

54

KI-CHUEN CHAK AND HIRAM HART

The numerical values of the factors appearing in (C.1) can be estimated in turn: (i) The effective mass m of the antibody molecule (Ig) is taken to be 0.26×10-~Sgm (mol.wt 1.6× 105), while the mass M of the particle or red cell is taken to be 10-1°gin. Therefore (M/m) ~ ~ 104. (ii) The value of the parameter e=q/l for the average separation of particle pairs can be evaluated by considering the relative probability of all possible positions when the particle centers are within the reacting distance (L=2R+I). However, as the particle radius R is much larger than the length of the bond, the value of q is approximately ½(l+qo) where q0 is the distance of closest approach. A more detailed evaluation yields q = ~(l + qo) + { (1- q0)2 R +qo The parameter qo may change according to the system, e.g. it may be a function of the surface charge or the form of the surface of the particles. When q0 changes from 0 to 0.9l, the value of 100/((1 + e ) ( l 1 - 2 e - e ) 2) is only minorly changed, ranging from 6.9 to 6.3_ (iii) The value of 1 can be taken to be the maximum extension of the two Fab sections of the antibody. From electron microscopy, this value is close to 250 A. (iv) The fractional values Yl and 72 refer to the maximum allowable offsets of the antibody and antigen binding sites upon impact that can still result in a bond. There is no general prescription for an exact estimation, since the geometrical dependence is probably a very complicated function of both the relative translational and angular orientations_ As shown in Part I, if 6nly a small portion of the binding areas between the two sites overlap, it seems unlikely that bond will be formed. Let d, and d2 be the two dimensional rectangular binding sides of site, then the overlapping area is: aov = (1 --71)(1-72)did2, and the ratio of ao~ to the whole cross-sectional area of the site is : R 0 . = (1 - 7 1 ) ( 1 - e 2 ) -

To estimate the minimum value of Roy likely to be required for bond formation to be favored, it is useful to note that there are three hypervariable regions in the hght chain and four in the heavy chain, which are believed to be responsible for the highly specific bonds involved (Capra and Edmundson, 1977). Assuming for simplicity that these seven regions are "uniformly" distributed on a planar rectangular surface, then fractional offset of ¼ for each of the corresponding sides results in an overall Ro, of 9/16 with all the hypervariable regions dislocated from their matching configurations. It seems unhkely, therefore, that bonding upon impact will occur if the values of Yl and 72 are greater than ¼ or R0~ is less than 9/16. The value of the binding site cross-section of the antibody and that of the antigen may be assumed roughly the same and to be approximately: dl

×

d2 ~36 × 12/~

(Barrett, 1974)_

(v) The maximum angular offsets which still permit antibody-antigen binding to occur are difficult to evaluate. Even without any translational dislocation of the two binding sites, a rotational shift of one site can still cause off matching positions for the binding elements or regions situated near the edge of the sites while there is only a purely orientational difference at the center. For antibody-antigen binding consistent with the lock and key hypothesis it is reasonable to expect that too large a value for either of the rotational angles au and ~ u would prohibit binding. If the same approach as in (iv) is taken, with the seven

ANTIGEN-ANTIBODY INDUCED PARTICULATE AGGREGATION--II

55

hypervariable regions altogether constituting the binding surface, then no matter how they are arranged (in the shape of a ring with one region at center and the rest near the rim or in a double array or other forms) a rotation of about n/6 at the center axis is close to causing the regions to be completely dislocated from corresponding binding elements on the opposite site and even the one matching pair of sites will be orientationally stressed. While the choice of the maximum possible relative orientation angles is necessarily arbitrary, it seems probable that neither a M nor q5u are likely to exceed 1t/6 and in fact are probably somewhat less. It will therefore be assumed that: 2c%tq5M n2

1 20

(vi) Substituting the above estimates: x = 0.16 exp(AE), where A E = E n - E A. The values of x for T = 3 0 0 K and a range of values of AE are tabulated below:

AE (K cal. M - 1)

1

2

3

x

0.86

4.64

25.44

4 138.40

Although malay of the factors (i)-(vi) as estimated above will vary for different antigen~ antibody pairs, the overall order of magnitude of the multiplicative coefficient is not likely to be very much altered. As the table indicates, however, the value of AE substantially determines the order of magnitude of x and so the relative binding properties of different antigen-antibody systems, If AE > 4 Kcal.M- ~, then the value of x would be > 100 and in the absence of ssdba, relatively complete agglutination should occur at very low antibody concentrations. For AE around 3 K cal. M-1, however, x assumes values of around 30 or less and the formation of multiple bonds probably plays a significant role in the agglutination process. In applying (1), it was found in general that values of x ranging from 3 to 30 yielded results in reasonable agreement with experiments.

LITERATURE Barrett, J. T. 1974: Textbook of Immunology, pp. 30-47. C. V. Mosby. Capra, J_ D_ and A. E. Edmundson. 1977. "The Antibody Binding Site." Sci. Am., 236, 5 0 59. Collins, F. C. and G. E. Kimball. 1949_ "Diffusion-Controlled Reaction Rates." J. Colloid Sci., 4, 425-437. Dybkjaer, E. 1966. "A New Technique for the Quantitation of Haemagglutination?' Vox Sang., 11, 21-32. Economidou, J., N. C. Hughes-Jones and B. Gardner. 1967. "The Functional Activities of IgG and IgM Anti-A and Anti-B." Immunology, 13, 501-508. and , 1967a. "Quantitative Measurements Concerning A and B Antigen Sites", Vox Sang., 12, 231-238. Frommel, D., P. J. Grob, S. P. Masoriredis and H. C. Isliker. 1967. "Studies on the Mechanism of Immunoglobulin Binding to Red Cells". Immunology, 13, 501-508_

56

K1-CHUEN CHAK AND HIRAM HART

Goldberg, R. J., 1952_ "A Theory of Antibody-Antigen Reactions. I." Am. Chem. Soc., 74, 5715-5725_ Greenburg, C. L., D. H. Moore and A. C. Nunn. 1963. "Reaction of Rabbit Antisera with Human Red Cells." Immunology, 6, 421~433. and - 1964. "Combination with Red Cells of Immune Rabbit 7S Antibody and its Sub-units." Nature, 203, 1147-1148. and - - . 1965. "The Reaction with Red Cells of 7S Rabbit Antibody, its Sub-units and their Recombinants." Immunology, 8, 420-431. Greenwalt, T. J. and E. A. Steane. 1970. "Quantitative Haemagglutination. II." Br. J. Haematol., 19, 691-700. Greendyke, R. M_ and S. N. Swisher. 1968. "Quantitative Studies of Hemagglutination." Vox Sang., 15, 321-337. Hoyer, L. W . and N_ C. Trabalt. 1970. "The Significance of Erythrocyte Antigen Site Density." J. Clin. Invest., 49, 87-95. Hughes-Jones, N. C_ 1963. "Nature of the Reaction Between Antigen and Antibody." Br. Med. Bull., 19, 171-177. Leikola, J. and V. Pasanen. 1970. "The Influence of Antigen Receptor Density on Agglutination of Red Blood Cells." J. Int. Arch. Allergy, 39, 352-359. Ming, T. K., H. S. Goodman and B. Brown. 1965. "Mathematical Model for the Process of Aggregation in Immune Agglutination." Nature, 208, 84-85. Nezlin, R. S., Y. A. Zagyansky and L. A. Tumerman. 1970. "Strong Evidence for the Freedom of Rotation of Immunoglobulin G Sub-units." J. Molec. Biol., 50, 569-572_ and - - - . 1972. "The Flexibility of Antibody Molecule." Haematologia, 6, 313-316. Pollack, W., 1965. "Some Physiochemical Aspects of Hemagglutination." Ann. N.Y. Acad. Sci., 127, 892-900. , H. J. Hager, R. Reckel, D. A. Toren and H. O. Singher. 1965. "'A Study of the Forces Involved in the Second Stage of Hemagglutination." Transf., 5, 158-183. Reif, F. 1965. Fundamentals of Statistical and Thermal Physics. pp. 269-273. New York: McGraw-Hill_ Romano, E. L. and P. L. Mollison. 1973. "Mechanism of Red Cell Agglutination by IgG Antibodies." Vox Sang., 25, 28-31. Schulthess, G. K., R. J. Cohen, N. Sakato and G, B. Benedek. 1976. "Laser Light Scattering Spectroscopic Immunoassay for Mouse IgA." Immunochemistry, 13, 955-962_ Schurr, J. M, 1970. "The Role of Diffusion in Bimolecular Solution Kinetics." Biophys. J., 10, 700-716. Singer, J. M., F. C. A. Vekemans, J. W. Th. Lichtenbelt, F. Th. Hesselink and P_ H. Wiersema. 1973. "Kinetics of Floculation of Latex Particles by Human Gamma Globulin." J. Colloid Int. Sci., 45, 608-614. Smoluchowski, V. M. 1916. "Drei Vortage uber Diffusion Brownshe Bewegang und Koagulation yon Kolloidteilchen." Physik Z., 17, 557-571, 585-599. - - . 1917. "Versuch einer mathematischen Theorie der Koagulations-Kinetik Kolloider Losungen." Z. Physik Chem., 92, 129-168. Solomon, J_ M., M. B. Gibbs and A. J. Bowdler. •965. "Methods in Quantitative Hemagglutination, I and II." Vox Sang., 10, 54-72, 133-148_ Somers, W. R., R. A. Brown and T. Makinodan. 1966. "The Possibility of Determining the Absolute number of Hemagglutinating Antibody Molecules in Antisera." lmmuno. Chem., 3, 343-345. Sugerman, D. and H. E_ Hart. 1973. "Thermal Dissociation of Antigen-Antibody Antigen Like Systems." Bull. Math. Biol., 35, 219-235. Wilkie, M. H. and E. L_ Becker. 1955_ "Quantitative Studies in Hemagglutination I and II." J. lmmunol., 74, 192-204. Yguerabide, J., H. F. Epstein and L. Stryer. 1970_ "Segmental Flexibility in an Antibody Molecule." J. Molec. Biol., 51,573-590. RECEIVED 8-22-77 REVISED 11-14-78