On the mechanism of immunodiffusion

Immunodunnistr~. Pergamon Press 1968. VoL 5, pp. 217-252. Printed in Great Britain

ON THE MECHANISM OF I M M U N O D I F F U S I O N * F~sDsmcE ALADJEMt, RrrA L.

PALDINO,RUPERT

PERRIN and Fu-Wu CHANO~

Department of Microbiology, University of Southern California School of Medicine

(R~eived 17 August 1967) Almtract--The present experiments were designed to yield quantitative immunodiffusion data to provide an experimental framework upon which mathematical descriptions of immunodiffusion can be tested. Experiments were carried out over wide ranges of initial amounts of antigen (MA) and antibody (MAe) employing a new concentric arrangement which allows very precise measurement of the location (x,) and time of appearance (t,) of the initial zone of precipitation. From experimental data, and using two alternative assumptions about the boundary conditions of the diffusion equation, the concentrations or amounts of antigen and antibody and the antibody-antigen combining ratios in the initial zone of precipitation were computed. It was found that for wide ranges of MA and MAc a linear relationship exists between log MA and x,, and log MAs and xp. A similar relationship was observed between log MA and tp, and log M ~ and tp. Within these ranges, use of both types of boundary conditions leadS to reasonably consistent values for concentrations or amounts of antibody and antigen in the initial zones o f precipitation and yields experimentally reasonable values for antibodyantigen ratios. However, in plates in which MA was very large, unreasonably high antibodyantigen ratios for the initial zone of precipitation were computed. This finding is interpreted as being due to soluble antigen-antibody complex formation which occurs before the initial zone ofprecipitation has formed and gives rise to a shift in the location of the initial zone of precipitation toward the antibody well over where it would have occurred had there been no complex formation. INTRODUCTION IN PRgVIOUS communications we have shown that the location and time of appearance of the initial zone of precipitation between antigen and antibody are two of the important experimental parameters which can be used to obtain information about the mechanism of immunodiffusion reactions [1-3]. These analyses, however, were based upon measurements of only a few zones of precipitation. T h e purpose of the present study was (i) to measure the location and time of appearance of the initial zones of precipitation over as wide a range of initial amounts of antigen and antibody as was experimentally feasible, and at high point density, and (ii) to use the experimental results and previously developed mathematical methods to compute the concentrations of antigen and antibody which should constitute the initial zones of precipitation if the mathematical models completely describe the reactions. * This work was supported by a Public Health Service Research Grant (AI 03222) from the National Institute of Allergy and Infectious Diseases. I" Recipient of a Research Career Development Award (5KO3-GM04817-07) from the National Institute of General Medical Sciences. 2~Predoctoral trainee, supported by Training Grant (A1 00157-08) from the National Institute of Allergy and Infectious Diseases. 217

218

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T h e experiments were carried out over wide ranges of initial amounts of antigen and antibody since it appeared likely that the assumptions in the mathematical derivation might be applicable for certain ranges of initial amounts of reactants but not for others. A new geometric arrangement was used to obtain the most precise measurements o f the location and time of appearance of the initial zone of precipitation. T h i s geometric arrangement consists of a central cylindrical well and, concentrically, an annular well. Either well may contain antigen, the other antibody. The zone of precipitation is observed as a ring between reservoirs. The principal advantage of this concentric arrangement is that as many independent measurements of the location of the zone of precipitation can be made as are desired, and that the time of appearance of the zone of precipitation can be observed much more accurately than with any other arrangement since not just a small segment but the entire ring of the zone of precipitation appears at one time. This geometric arrangement also permits detection of very small amounts of antigen or antibody when they are placed in the center weU. While computational results must be interpreted cautiously, they do provide insight concerning the mechanism of the reaction which could not be obtained in any other way. This is discussed in the appropriate section below. T h e experimental data provide a necessary though not yet sufficient framework for future development of a better description of immunodiffusion processes and, hopefully, will lead to the non-empirical use of immunodiffusion reactions in immunochemical investigation. EXPERIMENTAL

Preparation of reagents. Antigen used was bovine plasma albumin (BPA), crystalline, obtained from Armour and Company; dimer and a negligible amount of polymer were removed by column chromatography on Sephadcx. Antisera were prepared in rabbits by repeated intradermal injections. No adjuvant was used. Antibody content was determined by quantitative precipitln analysis. All experiments were carried out with one preparation of BPA and one antiserum pool. Preparation and reading of agar plates. Agar plates were prepared as previously reported [1], except that 0.75~o Ionagar No. 2 was used. Plastic molds formed a central cylindrical well of radius 0.4509 cm and an annular well of inner radius 4.1275 cm and outer radius 4.4275 era. After preliminary experiments, immunodiffusion plates were set up again and examined at 15 rain intervals at about the time when the zone of precipitation was expected to appear. The distance of the initial zone of precipitation from the center of the center well was determined from at least 4 replicate measurements in different radial directions. The temperature of the incubator and reading room was 26 -4- 2°C. COMPUTATIONS The concentrations of antigen and antibody which should constitute the initial zone of precipitation wcrc calculated on the basis of two alternative assumptions about the boundary conditions to the diffusion equation: free diffusion and time invariant sink [3] ; for the latterthe approximate solution was used for diffusionfrom

Mechanism of Immunodiffu.sion

219

both reservoirs. T h e exact solution was used only for diffusion from the center well because it was again found that the ratio of concentrations of exact solution/approximate solution was 1.9 for all experiments; it seemed unlikely that further computation using the exact solution would give new information. All the equations which were used have been described in detail previously [3] except equations (2) and (4) which can readily be derived. The equations are given here primarily for ease of reference. T h e concentration distribution due to free diffusion from the center well is given by sl C (r, t) _ 4 M ,r .O-----~e_,./,o,

e -''/4a' Io 2--Dt

s ds

(1)

0

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Sa

where s, and s, are now the radii to the inner and outer edges, respectively, of the annular reservoir. T h e approximate solution for the time invariant sink boundary condition, i.e., it is assumed that antigen and antibody accumulate at the location where the initial zone of precipitation will form but do not trespass that location, is, for diffusion from the center well oO

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xp $2 Computer programs were written as previously described [3]. RESULTS The results of experiments in which antigen was in the center well and antibody in the annular well are given in Tables 1-8 and Figs. 5-8. The results of experiments in which antibody was in the center well and antigen in the annular well are given in Tables 9-20 and in Figs. 5-8.

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Mecban;sm of Immunodiffusion

249

The first part of each table gives the experimental data averaged for experiments which were done in replicate or only once (see Table It). The second part of each table gives the computed concentrations or amounts of antibody a n d antigen and the antibody-antigen ratios which should constitute the initial zone of precipitation if the assumed boundary conditions completely describe lmmunodiffusion. The experimental and computational results are, in addition, presented graphically in the figures for ready visualization.* The relationship between MAre MA, and xp. The location of the initial zone of precipitation as a function of initial amounts of antigen and antibody in the wells is given in Figs. la, b for antigen in the center well and in Figs. 5a, b for antibody in the center well. Figures l a and 5a show that for wide ranges of MA and MA~ an almost linear relationship exists between log MA and x~ (MA~ constant) and log MAB and xp (MA constant). The restrictions on the range for which this relationship holds are clearly seen in Figs. lb and 5b; these are the same data as in Figs. la and 5a but are plotted on enlarged scales. Thus, in Figure lb, for values MAB2 I 1 " 5 2 m g and MA:> 40 mg there is a change in slope. This suggests that at these values of MA and MAS one or more new processes become important to the development of the initial zone of precipitation, processes which are not important at the other concentrations. Also, in Fig. 5b, at MAm~ 2.88 mg and MA<~ 1 mg there is a change in slope suggesting a similar phenomenon. The relationship between MA, MAa and b. The data in Figs. 2 and 6 strongly suggest that the same kind of relationship exists between MA, MAR and to as exists for MA, MAB and x,. In Figure 2, at very low values of MAre, to is higher than would be expected from extrapolations of the other data points; in Fig. 6 it is seen that for very low values of MA, to is similarly higher.

Computedconcentrations or amounts of antigen and antibody in the initial zones ofpreci~tation. The results of these computations are given in the Tables and in Figs. 3, 4, 7, and 8. For antigen in the center weU it can be seen in Fig. 3a that CAB ranges between 5 × l0 -4 and 45 × 10 -6 g/cm a. The most startling result in Fig..3a is that for any given initial amotlnt of antibody higher values for CA~ are computed to be in the initial zone of precipitation at high values of MA than at lower values of MA. Thus, for MAB = 2 3 . 0 4 m g and MA = 160rag, CAa = 44"7 × 10-* g/cm 2, while for the same value of MAa and MA = 2 "5 rag, CAB = 18 "3 × 10 -6 g/cm 6. A similar relationship is obtained for all other values of MA~, i.e., as MA increases, so does the computed value of CAm. Another interesting relationship is observed in Fig. 3b where the relationship between MA, MAS, and CA is given. Here, for any given value of MAB, higher values of CA are computed for plates with low MA than for plates with high MA. For example, when MAre = 1 "44 mg and MA ----- 160 rag, CA = 1.54 × 10 -6 g/cm*, while for the same value of MAa but MA ---- l0 rag, CA -~ 2.8 × 10 -6 g/cm*. * The complete data on individual immun.ocliffusion plates have been submitted to the American Documentation Institute, ADI Auxiliary Puhfication Project, Photodupllcation Service, Library of Congress, Washington 25, D.C. Copies of Tables may be obtained from ADI.

250

FP,~DERICKALADJ~M,RaTA L. PALDINO,RUPERTPERRXNand Fu-Wu CI-IANo

Similarly, when MAB = 23-04 mg and MA -= 160 rag, CA ~ 0"65 × 10 -e g/era 2, but when MA = 10 mg, CA = 1 "93 × 10 -6 g]cm I. Furthermore, for a given value of M^, as MAB is increased, the computed values of CABalSO increase. Figure 3c gives the antibody-antigen ratios. There is a marked increase of CAB[CAas MA is increased. For constant values of MA there is a slight increase in the values of CAB[CAas MAB is increased. For antibody in the center well it can be seen in Fig. 7a that CAB ranges between 2 × 10 -e to 26 × 10 -e g/era z, and that CAB tends to increase as MA and/or MAB iS increased. Figure 7b shows that CA ranges between 0.4 × 10 -e to 2.0 × 10 -6 g/era ~ and that there is a general tendency for high values of CA at low values of M,B. Figure 7c gives the antibody-antigen ratios. There is here also a marked increase of CAB/CAas MA is increased. For some values of MA there is an increase in the values of CAB/CA as MAB is increased. Qualitatively similar results are obtained when calculations were made using equations (3) and (4). These results are given in Figs. 4a, b, e for antigen in the center well and in Figs. 8a, b, c for antibody in the center well. As can be seen from the tables and the figures, the values ofSAA, SAABare generally proportional to the values of CA and CA~, respectively. The values of CAB/CAand SAAB/SAA as a function of MAB and MA are given in the Tables and in Figs. 3c, 4c, 7c, and 8c. It can be seen that reasonable values of antibody-antigen ratios for the zones of precipitation are computed for plates with low MA and low M~,B. At low MA and high MAB high but still reasonable ratios are obtained; for example, MA~ = 23"04 mg and MA ---- 2"5 rag, CAB/CA --~ 10. Most interesting, however, are again the computed ratios at high values of MA. CAB/CAmole ratios of 35-70 were found, and the SAAs/SAA mole ratios were as high as 95. These implausible ratios provide an important clue to the mechanism of immunodiffusion processes as described below. The exact solution of the "sink" boundary condition [3] equation (8) was also used for the computation of amounts of antigen and antibody in the initial zone of precipitation, for diffusion from the center well. It was again found that the ratios SA[SAA and SAB/SAA,, (SA and S,~ are the amounts of antigen or antibody, respectively, per unit length of the zone of precipitation computed by equation (8) [3] were for all data points in the Tables 1.9 -4- 0.1. It did not appear interesting, therefore, to carry out similar computations for diffusion emanating from peripheral wells. DISCUSSION The simplest hypothesis concerning the conditions which give rise to the initial zone of precipitation and one that is intuitively plausible is that at the location where the initial zone of precipitation will form, at the time it forms, two conditions must prevail: (i) nC,, = CoAn and (ii) CoA q- Co,~ -~ Q where Co^ and Co,B are the concentrations or amounts of antigen and antibody respectively, which constitute the initial zone of precipitation, n is the antibody-antigen ratio in the initial zone of precipitation, and Q is the minimal concentration of precipitate which can be detected. There is evidence that this hypothesis leads to an approximately correct description of immunodiffusion, at least for the few zones of precipitation which have been analyzed to date [3].

Mecha.~ism of Immunodiffusion

251

Based upon certain assumptions, the present experimental data allow computation of CoAaand CoA over wide ranges of initial amounts of antigen and antibody. Computations were made on the basis of two alternative assumptions concerning the boundary conditions of the diffusion equation: free diffusion (equations 1 and 2) and the time invariant sink at xp (equations 3-5). I f the location and time of appearance of the initial zone of precipitation were dependent upon diffusion only or at least were insensitive to other processes, then the CAB and CA should equal CoABand CoA,respectively. These values should be relatively constant for all plates, Q should be relatively constant, and n should vary over the restricted range of combining ratios which are observed in quantitative precipitin analysis; for the present system n might vary between about two and thirteen. It can be seen from the tables and figures that constancy of CAa and CA and reasonable combining ratios prevail for wide ranges of MAB and MA. There are a number of data points, however, for which constancy of CA, and CA does not hold and for which CAp~CAlies outside any acceptable range. A better approximation than free diffusion should be obtained if SAAp and SAA are used for CoApand CoA respectively, since SA allows for antigen-antibody complex formation at the location where the initial zone of precipitation will form. 8-4ABand S.4A have been calculated for all data points. It can be seen from the tables and figures that the SA values indeed tend to be more self-consistent over a wider range of MA and MAp than the corresponding values of CAp and CA, but that antibody-antigen ratios show the same type of deviation but even more pronounced than that obtained with the assumptions of free diffusion. We now come to the analysis of the most interesting part of the computational results: the relations between computed antibody-antigen ratios and the initial amounts of antigen and antibody in the system. The Tables as well as Figs. 3c, 4c, 7c, and 8c show that for plates with high initial amounts of antigen, unrealistically high antibody-antigen ratios are computed. What is the reason for these computational results and what does it tell us about immunodiffusion processes ? We suggest that these artificially high antibody-antigen ratios may be due to a shift of the location of the initial zone of precipitation as a consequence of soluble antigen-antibody complex formation throughout the plate before the initial zone of precipitation becomes visible, as follows: even after a very short period of time of diffusion there will be some antigen molecules near the antibody well and some antibody molecules near the antigen well. Some of these molecules will meet and combine to form soluble complexes. Near the antibody well complexes will consist mostly of one antigen molecule surrounded b y f (jr = 5 or 6 -- valence of antigen) antibody molecules. Near the antigen well complexes will consist of two antigen molecules and one antibody molecule (antibody bivalent). Formation of these complexes will cause a corresponding decrease in the concentration of free antigen and antibody. Complexes will diffuse according to their concentration gradients but their contribution to the initial zone of precipitation can almost certainly be neglected. However, on the antibody side of the location where the zone will eventually be observed, every antigen molecule that has entered will decrease the concentration of free antibody by 5 or 6 molecules; on the antigen side every antibody molecule that enters will decrease the free antigen concentration only by two. Antigen-antibody complex formation at locations other than xp will thus cause a shift of the location of

252

FREDERICK ALADJZM, Rrrh L. PAx.nxNo,RuPzRT PZRRm and Fu-Wu CHANO

the initialzone of precipitation from the location that would have been observed had there bccn no antigen-antibody complex formation, i.e.,frcc diffusion, or complex formation only at xp, i.e.,the timc-invariant sink. The shiftwill bc toward the antibody well if M A is high relative to MAC and toward the antigen wall if MAB is high relative to M'A. Because of the valence dlffcrcnccs between antigen and antibody, high initialamounts of antigen will be 5/2 to 3 times more effective in causing the shift than high initialamounts of antibody. The differences in diffusion coefficients between antigen and antibody will enhance this effect in the system studied. This is our present view why in plates with high MA excessively high antibody-antigen ratios are computed. At high values of MAB, the computed antibody-antigen ratios are low as would be expected from the above reasoning. Experiments were apparently not done at sufficiently high MAreto observe unreasonably low antibody-antigen ratios. The present results, then suggest that soluble antigen-antibody complex formation can occur throughout the immunodiffusion plate at locations other than that at which the initial zone of precipitation will form, before the zone has formed, but that this process does not significantly affect the location and time of appearance of the initial zone of precipitation unless either MA or M ~ are very large for the particular system (geometry, diffusion coefficients, etc.). Concentric immunodiffusion may be a practically useful procedure for the quantitative determination of antigen or antibody. REFERENCES 1. AJ.AVJSMF., KLOS'rEaOAZDH. and TAYLORR. W. Jnl t]'~0f.Biol. 3, 134"(1962). 2. ALADJEMF.,JARoSSR. W., PALDXNOR. L. and LACK~ERJ.A., J. Immun.83, 221 (1959). 3. ALADJEMF., J. Immun. 93, 682 (1964).