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The heterogeneity of antibodies with respect to equilibrium constants
lmmunochemistry, 1976, Vol. 13, pp. 873-883. Pergamon Press. Printed in Great Britain
THE HETEROGENEITY OF ANTIBODIES WITH RESPECT TO EQUILIBRIUM CONSTANTS C A L C U L A T I O N BY A N E W M E T H O D U S I N G D E L T A FUNCTIONS, AND ANALYSIS OF THE RESULTS* PHILLIP M. E R W l N t and FREDERICK ALADJEM~: Department of Microbiology, University of Southern California, School of Medicine, Los Angeles, CA, U.S.A. (First received 22 July 1975; in revisedform 10 December 1975)
Abstract--A method of data analysis has been developed which approximates the probability density of association constants of the reaction of hapten with antibody by a minimum number of delta functions, and alternatively, by a large number of delta functions. The method has been computerized and found able to approximate any physically possible affinity distribution. We have calculated the delta function distributions which describe known distributions of association constants and analyzed experimental data. From 2 to 4 delta functions were required to describe the probability density of Sips distributions. Not more than three delta functions were needed to describe the probability density functions of each of three antibody populations obtained from pooled antisera. These delta function distributions were found in one case to fit the data significantly more accurately than a Sips distribution. In the other two cases the fit was about the same as the Sips distribution. Affinity heterogeneity analysis using a large number of delta functions gave roughly equivalent results in terms of closeness of fit to the data to the results using a minimum number of delta functions. It is shown that binding measurements do not contain sufficient information to prove homogeneity of antibody. An analysis of factors affecting the resolution of delta functions is given.
INTRODUCTION
Calculation of the probability density function of equilibrium constants which describes the reaction of univalent hapten with antibody,, antibody heterogeneous with respect to equilibrium constants, has been mathematically achieved using Fourier transforms of the binding equation, Bowman & Aladjem (1963). The difficulties of the calculations have prevented the use of this method to evaluate experimental data. Other less accurate methods have been used to approximate the association constant distributions. Two of the more common methods are the Gaussian and the Sips distribution. The availability of high speed computers has made possible the use of numerical methods to generate approximations to distributions of antibody association constants. The method published by Werblin & Siskind (1972) is an example of such a method. Data have shown that monoclonal antibody producing cells synthesize antibody of homogeneous affinity. This suggests that an antibody population is composed of a number of subpopulations of homogeneous affinity. A useful feature of a method of analysis of affinity heterogeneity would be the capability to determine the number of subpopulations, but as is shown below we have found that this is not
possible due to resolution factors. It is possible to determine the minimum number of subpopulations which is adequate to describe the experimental data. A more nearly continuous approximation to the distribution of association constants can also be calculated by assuming a large number of homogeneous subpopulations. The limits of variability of the solution could then be inferred from the two different results. The method presented in this paper has these capabilities.
* An abstract of this work has been presented (P. Erwin & F. Aladjem (1972) Fedn Proc. 31, 781). $ Predoctoral trainee supported by NIH Training Grant ROI-AI 00157. ~;Supported, in part, by Research Grant CA14089 from the U.S.P.H.S. 873 IMM. 13/11 A
THEORY
OF
ANALYSIS
The purpose of this analysis of the reaction of univalent homogeneous hapten with antibody is to determine the probability density distribution of free energies of binding. An assumption made to simplify the calculations is that there are no co-operative binding effects for multivalent antibody-univalent hapten binding. No data have been found which indicate that there are such co-operative binding phenomena. Structural studies of antibodies have not shown a mechanism to bring about such an interaction for the reaction of a small hapten with antibody. Given this assumption, a probability density function (PDF) of the association constants can be represented as the sum of delta functions, the number of delta functions being equal to the number of different affinity subpopulations participating in the reaction. A delta function is simply a spike of probability density such that the integral of the probability density function has a finite value over the range x - dx to x + dx. This integral is equal to the fraction of antibody combining sites with association constant K = x.
874
P.M. ERWlN and F. ALADJEM
The accuracy of a PDF is determined by calculation of the chi-square
and
S~Kfl-I//(1 + K i l o = S~K/-II.
P
Z2 = ~ ( B i - A/)/2a 2,
(3)
(1)
i=1
where B / = the experimentally determined value of bound hapten concentration at free hapten concentration H~; A~ = the concentration of bound hapten at H~, calculated from the PDF; cq = the standard deviation of B~; p = the number of data points. The values of A~ are determined using the equation:
Equation (3) shows that the binding of low affinity antibody is proportional to the free hapten concentration. If there are several different low afftnity subpopulations of antibody, the net binding is
y~ S~K~H, = Hi Y SjKj. J
J
The sum on the right can be replaced by a constant, Szo, and is the low affinity uncertainty term.
SLo = Z SjKj.
(4)
J
A~ = ~ S~KjHd(1 + H/Kj),
(2)
j=l
Sj = the concentration of antibody binding sites with association constant K j, and m = the number of subpopulations of antibody which bind hapten. Since m, the Sj's, and the Kj's are all unknown, a general method of approximation of the PDF must be used until these parameters can be estimated. The number of antibody subpopulations may be large enough to generate an essentially continuous distribution. An approximation to the PDF using a large number of delta functions sufficient to approximate a continuous distribution serves a dual purpose: the heterogeneity of binding is shown, and the large m PDF can be used to generate an estimate of the minimum number of delta functions necessary to approximate the antibody affinity distribution and to estimate the affinities and concentrations of these few delta functions. Calculation of a large m P D F A PDF, whether continuous or discontinuous, can always be approximated to any desired degree of accuracy using a sufficiently large number of delta functions. For data of accuracy ___19/~ one delta function for every factor of 3 in association constant is sufficient to approximate the distribution (see Resolution below). However, the theoretical range of possible association constants is from 0 to ~ , and spacing delta functions every factor of three over this interval would require an infinite number of delta functions. The number of delta functions to be used can be reduced to a manageable number by the introduction of two uncertainty terms, Sag and Sin. SLo can be used in a binding equation to represent antibody of very low affinity, and Sm can be used to represent antibody of very high affinity. These terms are uncertainty terms because the association constants of the antibodies represented cannot be determined from binding measurements. The first uncertainty term, Szo, has the following justification. For antibody of low affinity, the denominator of the binding equation (2), simplifies. If K j/-//,~ 1 for all Hi in the experiment, then (1 + Kfl-//) ~ 1
It cannot be determined from binding measurements alone how many different affinities of antibodies are included in the binding represented by SLO. The second uncertainty term, Sin, is obtained for very high affinity antibody, such that H/K i ~ 1 for all H/ in the experiment. The binding equation (2) again simplifies, thus (1 + K~I-I/) ~ Kfl-It and
SjKfld(1 + Kill) - S t.
(5)
Equation (5) shows that all of the binding sites of very high affinity antibody are combined with hapten even at the lowest hapten concentration. The total binding of many high affinity subpopulations can be represented by the high affinity uncertainty term S~ = Sin.
(6)
J
The units of Sm and Sro are not identical. Since Sm is a sum of St values, equation (6), its units are moles per liter. SLO is the sum of the products of concentrations of binding sites and association constants. Thus, SLO is dimensionless. When multiplied by the free hapten concentration, the result is the concentration of hapten bound by low affinity antibody, and the units are moles per liter. These two terms, SLO and Sin, are sufficient to describe the binding with K values outside the range of the experiment. A general equation to approximate hapten binding is therefore:
A(H) = Sm + H'SLo +
SjKjH .i= 1 (1 + Kill)"
(7)
In equation (7) the summation index j is defined to index delta functions which have Kj values between those which can be included in the uncertainty terms. Our computer program for the calculation of a large m PDF uses a simple procedure which fits the data using equation (7). The six steps in the procedure are given in Appendix I. The values of K~ and m are determined according to a set procedure, after which values of the S/s and SLo and Sm are calculated by an iterative procedure.
Heterogeneity of Antibodies
A theoretical model of delta function resolution The following is an analysis of factors which must be considered in determination of the significance of a PDF. The significance of a PDF depends on the precision with which subpopulations are resolved, which depends on the potential resolution of the method and on the number and precision of the data points. There is no general formula relating resolution to these factors, but examination of a simple example can give some insight into the problem. Consider the distribution of association constants within an interval K1 to K2. Let there be two possible distributions of association constants within this interval: (1) One homogeneous subpopulation within the interval with association constant K and binding sites concentration S; or (2) two subpopulations, one with parameters K/a and S/2 and the other with Ka and S/2, where a is a real number greater than one.. The resolution of the method of analysis can be described as the smallest value of a at which a choice between these two alternatives can be made. There are other possible definitions of resolution, but t h i s one is useful in calculation of the minimum number of delta functions, as will be shown below. For data of perfect accuracy a sufficiently general method should achieve nearly perfect resolution. A method of such generality is unnecessary in analysis of real data. For the alternative cases 1 and 2 to be resolvable, there must be a significant difference in the A(H) functions generated by the distributions. This difference should be at least as large as the error of the relevant portion of the experimental binding curve. We define the error of the data binding curve to be
where summation is of data standard deviations for data points at H values within a factor of 2 of X, and L is the number of these data points. For this simple example it is possible to calculate the differences between the binding curves generated by the two alternative PDF's. Let the binding due to antibody with K values outside the interval (K1, K2) be designated by a residual function B,(H). Then the total binding for case 1 is:
SKH A~(H) = - + B,(H). 1 +KH
(8)
The equation for case 2 is:
,( osK.
sK.
A2(H) = ~ 1 + aK-------H+ a + KHI + B,(H).
(9)
The difference between the two alternatives, 0(H), is:
o(I-I)
1( aSKH
\ l + aK~
+ __SKH
a + Kn
2SKH \ (1 O) f-;~)"
We define the fractional difference as D(H)/A(H). Since the error of the binding curve is often proportional to the concentration of hapten bound, the fractional difference is really the quantity of greatest interest. At very large or small values of KH and at KH = 1, D ( H ) = 0. The maximum fractional differ-
875
Table 1. D(H)/S as a function of a. The precision of the data necessary to achieve a given resolution factor a is shown in columns 3-5 for 3 cases: when the binding due to other subpopulations, Br(H), is S, 2S, and 4S. These three cases correspond to a high affinity subpopulation, a moderate affinity subpopulation, and a lower affinity subpopulation, respectively
D(H)/A(H) D(H) a
S
1.2 1.414 1.732 2.0 3.0 5.0
0.0016 0.0057 0.014 0.022 0.053 0.10
Br = S 0.0013 0.005 0.012 0.018 0.04 0.08
B r = 2S 0.0007 0.0025 0.0064 0.01 0.024 0.045
Br = 4S 0.0004 0.0014 0.0033 0.005 0.013 0.024
ences occur somewhere near the H values of 4/K and (1/4K), except in the special case B,(H) = 0. Consideration of the magnitude of D(H) at these values of H will allow the estimation of the resolution possible. At 1 n=~-,
A--~=2
4--~a + 4a ~ 1
•
(11)
At H = 4/K, D(H) has the same magnitude but opposite sign, and A(4/K) is greater. Therefore, the fractional difference is greatest at H values near (1/4K). Using equation (11) it is possible to determine the dependence of D(H) on a. Table 1 has values of D(H)/S for six values of a, and also shows the effect of various levels of residual binding, Br(H). The fractional precision of the data required to achieve resolution of a factor of 3, i.e. when a = 3, is 0.053 S/A(H), where A(H) is evaluated at H = (1/4K) (Table 1, column 1). The quantity S/A(H) is a key ratio in evaluation of resolution. A relatively high affinity subpopulation is one for which B,(H) is relatively small, and a low affinity subpopulation is one for which B,(H) is relatively large. The residual binding function for any particular subpopulation is dominated by the binding of other higher affinity subpopulations. For a relatively low affinity subpopulation, there will be many higher affinity subpopulations contributing to B,(H), and for a high affinity subpopulation, there will be few of higher affinity, and B,(H) will be relatively small. Since the fractional precision of the data required to achieve a given resolution is inversely proportional to A(H), it follows that a smaller fractional data error is required to achieve a given resolution factor for subpopulations of relatively low affinity than is required for high affinity subpopulations. This relationship is demonstrated by the recent observations of Kiln et al. (1974). It was found that more low affinity antibody could be detected in a globulin preparation after adsorption of high affinity antibody. This was because the high affinity antibody produced a large residual binding level which not only prevented accurate analysis of the heterogeneity of the low affinity antibody, but also prevented an accurate estimate of the
876
P.M. ERWIN and F. ALADJEM
quantity of low affinity antibody. The corollary relationship is that for any given size of A(H), a subpopuiation with a greater concentration of binding sites, i.e. a larger S, will be resolved much better than a subpopulation of lesser S. This is because of the direct proportionality of S and D(H). There is an upper limit on the magnitude of D(H). D(H) can never be greater than S/2. The affinity of a delta function cannot be resolved if S/2A(1/K) is less than the fractional data error. Such delta functions can be replaced by other delta functions at other K values without significantly increasing chi-square. A way to reduce the number of delta functions being used in a PDF is. suggested by this model. If the resolution factor a of a delta function is such that there is a higher affinity delta function of affinity less than Ka, and a lower affinity delta function of affinity greater than K/a, then the elimination from the PDF of the delta function with affinity K and resolution factor a does not prevent successful analysis of the binding data. Increases in the heights of the two bracketing delta functions can compensate for the eliminated delta function. This fact is used in our procedure to determine the PDF with minimum m.
tration S of the remaining delta functions. If the minimum chi-square is too much larger than that obtained with the large m approximation, then at least one additional delta function is needed to approximate some part of the PDF. Because of resolution factors discussed above, the additional delta function will usually be required in the high affinity region of the PDF.
aj = the resolution factor of the fl' delta function; Ej = the average of the fractional errors of the binding data at H = 4/Kj and H = 1/4 K~; C~ = the coefficient of M2; Mj = 4A(1/Kj)/Sj; A(I/Kj)= the bound hapten concentration at H = 1/Kj calculated from the PDF. The resolution factor obtained from equations (12) and (13) is only approximate, but the accuracy is sufficient for the use outlined below to determine the PDF with minimum m and reasonably small chizsquare.
Method of elimination of delta functions The method of elimination of delta functions is to selectively absorb some delta functions by other delta functions. Certain delta functions are selected to remain, i.e. not be absorbed by other delta functions. The selection of delta functions to remain depends on the resolution factors a~ which have been calculated for the particular delta functions. Since we have spaced the delta functions at factors of x//~, a delta function with an aj less than ~ cannot be eliminated and is selected to remain. Next, delta functions with a t between x ~ and 10 are examined. If they are adjacent to delta functions with aj < x / ~ , then they are eliminated. If there are no delta functions which have already been selected to remain which are adjacent in K, then the delta function being evaluated is selected to remain. The procedure is performed for increasing a intervals until all delta functions have been selected either to remain or to be absorbed. A delta function which can be absorbed by two bracketing delta functions is split between them. A detailed example of the delta function elimination procedure is given in Appendix II. After the process of absorption is completed, the chi-square of the PDF is reduced by refining both the association constants and concentrations of the remaining delta functions and the two error terms. The refinement procedure is similar to that used when only the Si values are being refined, except that the St and Kj values are varied simultaneously. The minimum chi-square is obtained after about 30 iterations of the refinement procedure. At this point, operator intervention is necessary. If the chi-square obtained after refinement is much larger than the chi-square obtained by the large m approximation, an additional delta function may have to be added and refinement restarted. If the chi-square obtained by the minimum m model is close to that obtained by the large m approximation, the minimum m solution can then be examined regarding the possibility of further reduc,tion of the number of delta functions.
Method of determination of the minimum m P D F The results of the large m analysis of the PDF will in general show a number of maxima and minima giving the appearance of a continuous and complex distribution of affinities. We have found that there is usually a less complex PDF that will fit the data very well if not better than the large m distribution. This distribution is produced by eliminating delta functions obtained by the large m approximation above, where possible, and retaining the two uncertainty terms. The method by which delta functions are eliminated is outlined below. The validity of the method is tested by determination of the minimum chi-square fit that can be obtained by refinement of the values of association constant K and concen-
Large m and minimum m distributions calculated for three Sips distributions Of course, any new method of data analysis must have a capability to correctly analyze known data. Werblin and Siskind validated their method of analysis using delta functions by showing that their method would correctly analyze data calculated theoretically from five postulated PDF's. The PDF's from which they calculated data were composed of 1-6 homogeneous subpopulations. Our methods correctly analyze data calculated from PDF's composed of several homogeneous subpopulations. Rather than showing examples of this type of analysis, we wish to demonstrate an additional capability of methods of analysis using delta functions.
A practical method of resolution evaluation The model above does not allow calculation of the resolution factor a of a delta function in a PDF. It is necessary, therefore, to determine a relationship empirically between the resolution factor of a delta function and other parameters of the PDF and the binding curve. Calculation of a large number of examples led to the following relationship: aj = CiM ] + 0.137Mj + 1.66
(12)
C~ = 74E~ + 0.17Ej,
(13)
where
Heterogeneity of Antibodies
877
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H.qE-
I.H~ILII .~"
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g
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~
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fl
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ff
2. .B-
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.
H.HE
I q
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i
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LnB Ig K Fig. 1. Three Sips distributions with mean affinities and heterogeneity indices corresponding to the experimental data. The parameters of these distributions are given in Table 3. The highest distribution, middle and lowest distributions have heterogeneity indices 0.64, 0.49 and 0.36, respectively. O u r large m a n d m i n i m u m m methods can analyze data generated by continuous PDF's. Figure 1 shows three Sips distributions which correspond in m e a n association constant and heterogeneity index to the experimental data analyzed in this paper. Figure 2 shows large m distributions obtained for data gener-
I1.11,
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q
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~ ~ 7 a 9F,,i
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B
91111
Jl----
L| tl
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Cilia I I
LI'I~I B K
LOr~ IE K
Fig. 3. The delta functions determined by the minimum m method for data calculated from Sips distributions. The data analyzed are as in Fig. 2, and the ordinates and abscissas are the same parameters. ated by these Sips distributions over free h a p t e n ranges corresponding to our experimental data. The agreement is quite good between the shapes of the large m distributions a n d the corresponding parts of the Sips distributions. The results of the m i n i m u m m analysis shown in Fig. 3 c a n n o t be easily evaluated by inspection. Table 2 contains parameters of the analyses. The chi-square values show that the minim u m m distributions are adequate for analysis of Sips distributions. It can also be seen that reduction of the n u m b e r of delta functions below the m i n i m u m
II.ll. q
E E 7
B
S IE I I
q
~
6
7
H q
Igll
11.6"
mll.:~ .J EL
_L_ q
K
E 7
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tl
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7
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q
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B
qlllll
q
E 6
7
e
E.E,
11.'4,
II.I!,
LI2B igl(
~t11III
LnB It~ K
Fig. 2. The delta functions determined by the large m method for data calculated from Sips distributions. The parameters of the Sips distributions are, top to bottom, Ko = 5 x 105M -1 a n d a = 0.36, Ko = 1.8 x 10SM -1 and a = 0 . 4 9 , and K o = 8 x 10SM -1 and a = 0 . 6 4 . The ordinate on the left set of graphs is the probability density of log~oK. The ordinate of the right set of graphs is PD(IogloK)/2A(I~),a quantity used in calculation of resolution factors. The abscissa is in all cases log ~oK. The lines drawn in are the actual probability densities of the Sips distributions. The delta function heights are all increased by a factor of two because they represent binding sites within intervals of x ~ , rather than probability densities of log xoK.
Table 2. Analysis of Sips distributions by delta functions. The closeness of fit to data generated by Sips distributions of different heterogeneity are shown by chi-squares in column 4. All chi-squares are due to systematic differences between the models and the data because these data are theoretically calculated. The data ranges (in orders of magnitude of free hapten concentration) in the right column correspond to the experimental data presented in this paper. SLo and Sx~ are given as fractions of total binding at the highest free hapten concentration Chisquare
Snt
a = 0.64 Sips -Large m 14 Min m 4 Min m - 1 3 Min m - 2 2
0 0.1 1.8 7.0 22.3
-0.0002 0.003 0.004 0.004
-0.004 0.002 0.07 0.07
4.5
a = 0.49 Sips -Large m 10 Min m 2 Min m - 1 1
0 0.1 1.0 9.3
-0.016 0.024 0.039
-0.086 0.22 0.73
3.0
0 0.26 3.1 17.2
-0.02 0.08 0.097
-0.10 0.28 0.48
3.0
Data
Model
m
a = 0.36 Sips -Large m 11 Min m 2 Min m - 1 1 '
Data
SLo range
878
P.M. ERWIN and F. ALADJEM W K
l.m-
d'
B.I{-
¢.
rl." --.I B.B
HFIPTEN CnNC. X I B '~ (MDLFIR)
I.¢I
X
Z
It
V Z
i
l.g"
I""IZ
m.~"
,r
[] . m =,it
m.B
-7
-:
LDISt,, HRPTEN CDNC. (MDLFIR)
Fig. 4. Binding curves of p-tSliodophenylarsonate with rabbit antiphenylarsonyl-BgG. The points at high total hapten concentration are actually the averages of 4-6 measurements.
cedure. Therefore, all delta function heights are divided by the greatest measured value of hapten bound. In the right column of Fig. 5 we plot S/2A(H) for each delta function vs the log K, where A(H) is evaluated at H equal to the reciprocal of the association constant of the particular delta function. As pointed out in the discussion of resolution, the quantity S/2A(H) is important in the determination of the resolution of the delta functions, i.e. the resolution factors are a quadratic function of the inverse of S/2A(H). Therefore, the largest delta functions in the graphs on the right can be expected to have the smallest resolution factors. These graphs show which delta functions can be expected to have the smallest resolution factors and also show the importance of some of the high affinity subpopulations which appear to be insignificant in the graphs on the left. Figure 6 is a presentation of the results of minimum m analysis in the same format as Fig. 5. Table 3 contains relevant parameters for comparison of the results of large m, minimum m, and Sips heterogeneity analysis. The low affinity uncertainty term is presented as its fraction of A(H) at H equal to the largest experimentally measured value. Since the binding due to this term is SLoH, the term is presented as SLoH/A(H). The values of association constant of the delta functions obtained for fluorescence polarization data by minimum m analysis are very similar. The data in Table 3 show that a considerable fraction of the R-FI binding is described by the low affinity uncertainty term. For the equilibrium dialysis
number determined by our method introduces substantial systematic errors. Conversely, increasing the number of delta functions does not substantially decrease chi-square. The correlation coefficient R was not found to be sufficiently sensitive for detection of differences between the data and the curves generated by the distributions. Rxr for all of the above distributions varied between 0.99 and 1.0.
Experimental data used for analysis We analyzed the reaction between phenylarsonate (PA) and rabbit antiphenylarsonate antibody. Two sets of binding data were analyzed: (1) The reaction of fluorescein-coupled phenylarsonate, namely (R-F1) with the water-soluble and water-insoluble immunoglobulin, fractions as described in the preceding paper and graphed in Figs. 4a and b of that paper (Medof & Aladjem, 1976); and (2) para -is1 iodophenylarsonate reacting with the IgG fraction of antiphenylarsonate antiserum. The latter reaction was studied by equilibrium dialysis. The equilibrium dialysis data are given in Fig. 4. This immunoglobulin fraction was obtained from a different antiserum pool than that of the R-F1 data. The details of these experiments will be published elsewhere (Fukahara K. & Aladjem F., to be published). RESULTS
The PDF's obtained by the large m method are presented in Fig. 5. In the left column we plot delta function height vs the logarithm of the delta function association constant. Since the total antibody binding site concentration is unknown for these data, it was necessary to arbitrarily pick a normalization pro-
I l
x rf-
E
7
B
[t I l l l l
LI ~
E
,I,nPll
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'7
B
HIIlll
4
6
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glmll
q
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7
B
~llmll
• : il.LI-
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+till I!. H
,ll.
~
q
~
E
7
LD~
B
l<
[i
7
B 911111
,+lllllll [;
E
7
LDG
g
BIBII
K
Fig. 5. Delta functions obtained by the large m method. The delta functions obtained by the large m approximation method are plotted from top to bottom for water-soluble and water-insoluble immunogtobulin fractions reacting with R-FI, and the reaction of immunoglobulin with lalI-iodophenylarsonate. The delta functions are plotted vs the loglo of the K value. The right column graphs show the concentration of binding sites Sj divided by twice the bound hapten concentration evaluated at H = 1/Kj. The left column graphs show the binding sites concentration which for normalization is divided by the maximum concentration of bound hapten.
Heterogeneity of Antibodies
879
I.IIILB-
ILB
IZ I z < rv. ~ il.S-
In 1,-1 ...I
B.EI
-~!.ll-
-3.~-t-E.g.
n.el
q [
fi 7 fl 911111
i;
1
,
I -H
,
~ tr fi 7 B Cllt~ll
1 '-'B
I -7
1 -G
I
LDE I~ H
; , e.,+
la,ta,
L+_
Fig. 7. Sips heterogeneity plot of the reaction of IgG with p.1 a,iodophenylarsonate.
e.,~
H.B
tt ~ 6 7 O U I H I I Lira m I~
. . . . . . . tt K 6 7 B g l l ~ l l t.nr. m K
Fig. 6. The delta functions obtained by minimum m analysis are plotted from top to bottom, water-soluble and water-insoluble immunoglobulin fractions reacting with R-F1, and the reaction of immunoglobulin with para 13q-iodophenylarsonate. As in Fig. 5, the left column is of graphs of S/(max Bi) vs logloK~, and the right column is of graphs of SJ2A(1/Kj) vs logl0K j. data, the low affinity and high affinity uncertainty terms, are of little significance, i.e. they can be neglected without significantly increasing chi-square. It is clear from the chi-square values that the large m and m i n i m u m m distributions fit the equilibrium dialysis data significantly better t h a n a Sips distribution. Figure 7 is a Sips plot of these data. The data appear to deviate little qualitatively from the Sips
model, but the precision of the data is adequate to show the difference by calculation of chi-squares. Figures 8(a and b) show Sips plots of the R-FI water-soluble and water-insoluble IgG binding data binding sites. D e t e r m i n a t i o n of the total concentration of binding sites is necessary if one wishes to calculate the m e a n affinity of the antibody population or if one wishes to make a Sips plot. W h e n there is a substantial low affinity uncertainty term, as in the cases of these data, accurate determination of the total binding site concentration is impossible. Table 4 gives the m e a n affinities determined for three possible binding site concentrations. The chief point to be noted is that as the concentration of binding sites is raised, the data more nearly fit the Sips model, b u t the m e a n affinity then more poorly relates the information in the binding data. Instead, the mean a t f n i t y describes an extrapolated distribution of low mean affinity.
Table 3. The relevant parameters for comparison of the closeness of fit of the distributions. The mean association constants in column five are obtained in the normal way for the Sips distribution. For the delta function distributions, the Ko values are determined by ignoring the high and low affinity error terms. SLO and Sat are given as fractions of the total binding at the largest experimental free hapten concentration. For the large m and minimum m distributions, K0 = antilog
Si log Ki i
Data
Model
m
Si i
Chisquare
Watersoluble globulin fraction
Sips -Large m 10 Min m 2 Min m - 1 1
19.1 15.8 17.7 25.5
Waterinsoluble globulin fraction
Sips Large m Min m Min m - 1
-10 2 1
Equilibrium dialysis data
Sips -Large m 14 Min m 3 Min m - 1 2
Ko(M- 1) 5.3 1.1 2.1 3.9
x x x x
a
SLo
Snt
105 108 108 10a
0.36 ----
--0.32 0.04 0.44 0.069 0.55 0.084
16.3 13.7 15.3 38.4
1.75 x 105 2.7 x 107 9.4 x 1 0 7 1.8 x 108
0.49 --
--0.48 0.015 0.65 0.024 0.79 0.034
29.1 13.0 12.5 16.0
8.3 5.6 1.54 2.14
0.64 ---0.024 0.001 -0.09 0.0033 -0.20 0.0035
x x x x
105 105 106 106
--
--
880
P.M. ERWIN and F. ALADJEM Table 4. The concentrations of antibody combining sites per mg protein per ml as determined in three ways: (A) by extrapolation of a graph of the reciprocal of the bound sites concentration (I/B) vs the reciprocal of the free hapten concentration (l/H) to 1./H = 0; (B) by varying the antibody sites concentration to obtain linearity of a Sips heterogeneity graph [log R/(N-R) vs logH]; and (C) by treating the low affinity error term obtained for the large m distribution as if it were of K = (1/4 H m J , where Hm,~ is the largest experimental free hapten concentration. Using these binding site concentrations, K0's were calculated as
1/n at R = N/2 Data
Parameter
1/B vs 1/H
S Ko
3.47 x 1 0 - 7 M 2.9 x 107M - I
8.8 x 10-TM 5.3 x 105M -1
Water-insoluble globulin fraction
S Ko
9.91 x 1 0 - 7 M 1.4 x 107M - 1
4.65 × 1 0 - 6 M 1.75 x 105M - 1
1.96 x 1 0 - 6 M 1.7 × 106M - 1
Equilibrium dialysis data
S K0
1.18 x 1 0 - 6 M 1.25 x 106M -1
1.36 x 10-6M 8.3 x 105M -1
1.47 x 10-6M 6.1 x 10SM -1
Ilu I
rv
---I .1~ rn ._1
-2.B
I
I
I
-g
-B
-7
LDG IB H
(b)
I
trl r-1
5.33 × 1 0 - 7 M 4.3 x 106M -1
DISCUSSION
?..
Z
Delta functions
Water-soluble globulin fraction
(a)
Z
Linear Sips plot
-I~
!
-g
.-B
I
-7
LDE 1121 H
Fig. 8. Sips heterogeneity plots of the reaction of antibody with R-FI using the three possible concentrations of binding sites (5) from Table 4. S determined by: (1) D,I/B vs 1/H; (2) I , delta functions; a n d (3) X, Sips. (a) the reaction of water-soluble immunoglobulin with R-F1; (b) the reaction of water-insoluble immunoglobulin with R-FI.
Approximation of antibody association constants by a sum of delta functions has several advantages relative to approximation by a Sips or Gaussian distribution. First, a delta function distribution can locate local maxima in a distribution of association constants, whereas the maximum of the best fit Sips or Gaussian distribution may not correspond to any feature of the actual distribution. Second, a distribution obtained with delta functions is not restricted to any type of symmetry. Third, an estimate of the amount of antibody with very high and very low association constants can be obtained from the values of $~.o and S~I. Delta functions are useful for approximation of discontinuous distributions (a small number of delta functions) and continuous distributions (large number of delta functions). An additional advantage of this type of analysis is that once the antibody population has been divided into subpopulations using delta functions, other properties of the antibody can be investigated. For example, Medof & Aladjem (1971) were able to determine values of specific molar quenchence for delta functions which would reproduce hapten fluorescence quenching data for the reaction of fluorescein with antibody. Previous attempts to calculate the heterogeneity of specific molar quenchence using a variety of graphical procedures but without knowledge of the delta function distribution, had not been as successful• Our large m and small m methods are programmed to be run completely on a desk top calculator• There are no restrictions on data input, as far as computations are concerned. Of course, more accurate data will allow a more accurate calculation of the distribution of association constants, and a greater range of free hapten concentrations reduces the amount of the antibody population in the uncertainty terms SL~z.gnd Snl. The use of a delta function spacing of x/10 in our large m method has three advantages: (1) this close spacing is sufficient to provide what is essentially a best fit of the data; (2) the delta functions are close enough together to excellently approximate
Heterogeneity of Antibodies continuous distributions; and (3) this spacing gives a pleasing appearance on graphs. This spacing of the delta functions is narrower than experimenters not interested in testing the method may wish to use. We have found that use of a spacing factor of 10 is almost as accurate as ~/]-0 for our data and affords a significant reduction in computer time (almost a factor of two). Another method of approximation of antibody association constant distributions is in use which is similar to our large m approximation. This method was developed by Werblin & Siskind (1972). It approximates the distribution of association constants by a number of delta functions spaced by a constant factor. Their method refines the S~ values of the delta functions by calculating the sum of the squares of the deviations (SSD) between the binding data and calculated values of bound hapten for a large number of variations of an approximation to the PDF. The method then chooses the variation with the least error as the basis for another sequence of similar calculations. The number of variations of the PDF tested in each iteration of this refinement procedure is 2m, where m is the number of delta functions. Our method tests 3 m + 6 variations per iteration of our refinement procedure. A second difference between the methods is that we minimize chi-square, whereas Werblin and Siskind's method minimizes the SSD. It is our understanding that minimizing the SSD is equivalent to minimizing chi-square when all standard deviations are equal to unity. We have tried this approach and found that significant differences in the PDF's occur when the SSD is minimized instead of chi-square. In particular, the data at low hapten concentrations contribute little to the SSD. A third difference between the methods is that we introduce and calculate the high affinity and low affinity uncertainty terms. These terms give insight into what fractions of the antibody population are of indeterminate affinity. In addition to these differences between our large m approximation and that of Werblin and Siskind, we have the additional capability to calculate the minimum m PDF. Using the minimum m PDF, we can model other aspects of the reaction, such as quenchence, much more easily than could be done using a large number of delta functions. The minimum m PDF also gives us an indication of the minimum possible heterogeneity of the real distribution. The results of the analysis of experimental data show that there are many possible PDF's which can adequately describe the association constant distribution of an antibody population• For each of the two sets of data for the reaction of the hapten R-F1 with antibody, three PDF's of approximately equal chisquare were obtained: a Sips distribution (Fig. 1), a large m delta function distribution (Fig. 4), and a minimum m distribution (Fig. 5). An infinite number of different solutions of equal quality fit can be obtained by combinations of the three solutions for the same data. Another result of this analysis is that for these antibody preparations, a significant fraction of the antibody is contained in the high and low affinity uncertainty terms. It is impossible therefore, to determine the actual mean association constant Ko for either of these antibody populations without making an ad hoc
881
assumption about the concentration of binding sites. For the equilibrium dialysis data, a significant 10w atfmity error term was not obtained, and the high affinity error term was quite small. Because there is a measureable high affinity uncertainty term, it would be impossible to predict accurately the results of an experiment carried out in far antibody excess in which very high affinity antibody would play a dominant role. For example, phage inhibition by antibody at very dilute antibody and phage concentrations would be dependent on very high affinity antibody. Although the equilibrium dialysis data appear linear (Rxy = 0.998) on a Sips plot, Fig. 7, there is a substantial contribution to the chi-square due to systematic deviation between the data and the Sips line. The chi-square for the large m and minimum m distributions are significantly smaller because of the absence of this systematic deviation. Examples such as this confirm that chi-square is significantly more sensitive than Rxr as an indicator of closeness of fit. The results of the analysis of our data by the minimum m method were in one sense quite surprising. Examination of the results for R-F1 binding data analyzed by the large m method did not show any striking similarities between the two antibody fractions. Examination of the results of the minimum m analysis showed for the first time the close agreement in association constant values of these two different antibody preparations. Apparently, too many delta functions can mask a similarity of this type. More data will have to be analyzed before conclusions can be drawn concerning the generality of similarities of water-soluble immunoglobulin to water-insoluble immunoglobulin affinities. In view of the fact that we obtained excellent fit to our experimental data with only two delta functions for two cases and three delta functions in one case and that the antibodies were obtained from serum pools with 30 or more rabbits, it seems very unlikely that each delta function represents an exactly homogeneous antibody subpopulation. The analysis of resolution presented in the theoretical section shows that these delta functions may represent a large number of different subpopulations that are not resolvable by this method. The fact that an association constant distribution using two or three delta functions fits the data is an indication that the heterogeneity may be very limited relative to the Sips or Gaussian distributions of affinities. It is believed that monoclonal antibody producing cells produce antibody of homogeneous affinity. A limited number of different affinity antibody subpopulations are likely to be produced in response to any single antigenic determinant. Calculation of the minimum m distribution shows the minimum possible number of subpopulations of differing affinity. No method of analysis of binding data can determine the maximum possible number of subpopulations. The method of analysis which we have developed seems to extract the useful information of hapten-antibody binding data and is computationally efficient. REFERENCES
Bowman J. D. & Aladjem F. (1963) J. theor. Biol. 4, 242. Erwin P. & Aladjem F. (1972) Fedn. Proc. 31, 781.
882
P . M . ERWIN and F. ALADJEM
Fujio H. & Karush F. (1966) Biochemistry 5, 1856. Kim Y. T., Werblin T. P. & Siskind G. W. (1974) Immunochemistry 11, 685. Medof M. E. & Aladjem F. (1971) Fedn. Proc. 30, 657. Werblin T. P. & Siskind G. W. (1972) Immunochemistry 9, 987.
Table 5. Elimination of delta functions Cycle i 1 2
APPENDIX I The procedure for calculation of the large m association constant distribution is outlined below. (1) Data input. (2) The initial values of the K / s and S/s are calculated: Starting a factor of 10 greater than the reciprocal of the smallest Hi and extending a factor of 10 less than the reciprocal of the largest H value, delta functions are initialized in the computer program and spaced by factors of ~/]O. The initial values of the S/s and Sm are set equal to the largest Bi divided by m + 2. m + 2 is the number of delta functions plus the two uncertainty terms. SLo is initially set equal to S1K1/IO. (3) The initial values of Sj, Sm and SLo are refined. The chi-square depends parabolically on the value of one of these parameters when other quantities are held constant. A short calculation determines the minimum of the parabc/la for S~ and changes the value of S~ to reach this minimum chi-square. S~ is then held constant while the procedure is repeated for $2. As this process is performed sequentially on all of the S/s and the uncertainty parameters, the chi-square decreases monotonically. If the procedure is performed many times, the result is very close to the minimum chi-square, an extremely accurate fit. In general, about 50 iterations are required to achieve this fit, although the number of iterations does depend on the number of delta functions and the data to be approximated. (4) Parameters of interest, such as total antibody combining sites concentration, Sips ratio (r/n-r), mean association constant, best fit Sips parameters K0 and a, etc, are calculated. (5) Results are printed. The only step that requires a significant amount of computer time is step (3). The calculation time depends linearly on three parameters: the number of iterations, the number of delta functions, and the number of data points. With 10 delta functions and 30 data points, about 15 min of HP9830 calculator time is required.
APPENDIX I1
An example of elimination of delta functions The following example of the delta function elimination procedure is taken from the analysis of the reaction of IgG with p.la~ iodophenylarsonate. The calculated resolution factors of the delta functions are listed in column 2 of Table 5. Delta functions 1 and 2 are automatically eliminated because S1/2A(H) and S2/2A(H) are less than the data error. The computerized elimination procedure evaluates the delta functions in cycles. In each cycle, the delta functions are examined in order from the first delta function to the last delta function. During the first cycle, delta functions with 0 < logloa~ < 0.5 are marked to remain. In this example, none of the delta functions had resolution factors in this range. In the second cycle, delta functions with 0.5 _< logtoa~ < 1.0 are evaluated. The first delta function with a~ within this range is delta function 8. The computer then checks for delta functions previously selected to remain which are within a factor of 10 in association constant. There are none, and delta function 8 is selected to remain. In Table 5, a plus sign ( + ) is placed in row
3 4 5 6 7 8 9 10 11 12 13 14
a~
2
3
4
5
10
Net
1.1 x 107
7.0 x 106 58.0 6.3 x 104 19.7 11.0 202.0 9.5 8.8 18.3 197.0 13.6 58.5 15.4
+ + + -
+ -+ + -
8 under the column of cycle two. The second delta function to be evaluated in cycle 2 is delta function 9. The computer checks for delta functions already selected to remain which are within a factor of 10 in K, and finds delta function 8. Delta function 9 is then marked for elimination, as noted by the minus sign ( - ) in row 9 in cycle 2. No more delta functions are evaluated in cycle 2. The first delta function evaluated in cycle three is number 5. The resolution factor is within the range 1.0 < logloa~ < 1.5. The computer checks for delta functions which have already been selected to remain and which are within a factor of 10 x / ~ in association constant. Delta function number 8 is exactly 10 x / ~ higher than delta function number 5 in association constant, but it is not within this distance. Therefore, delta function 5 is selected to remain. The next delta function evaluated, delta function number 6, is less than a factor of 10 x/T0 from both delta function 5 and delta function 8, so delta function 6 is eliminated. Delta function 10 is less than 10 from delta function 8, so delta function 10 is eliminated. Delta function number 12 is marked to remain because it is more than 10 x/qO from delta function 8, and delta function 14 is marked for elimination because it is less than 10 x/l-0- from delta function 12. In succeeding cycles of evaluation, the remaining delta functions are marked for elimination because of their proximity to delta functions 5, 8 or 12. The net result of the procedure to this point is shown in the last column of Table 5. The second part of the process is the addition of the concentrations of the delta functions marked for elimination to the concentrations of the remaining delta functions. S L through S 4 are added to $5, and SL3 and $14 are added to $12, the nearest remaining delta functions. $6 and $7 must be split between delta functions 5 and 8 in proportion to proximity. $6 is split in a way which preserves the mean binding properties of delta function 6 after its elimination. The two equations below show the concentrations added to delta functions 5 and 8. To
Ss = S~'log(Ks/K6)/Iog(Ka/Ks)
(A-I)
to
$8 = S6"Iog(K6/Ks)/Iog(Ks/Ks).
(A-2)
Similarly, delta function 7 is split in fractions given in equations A-3 and A-4. To
$5 = S7"log(Ka/kv)/log(Ka/ks)
(A-3)
to
Sa = S7"log(K7/Ks)/log(Ks/ks).
(A-4)
In mum delta delta
this way, a good initial approximation to the minim distribution is generated. The concentrations of functions 9, 10 and 11 are similarly split between functions 8 and 12.
Heterogeneity of Antibodies This process is by no means sufficiently accurate for this initial approximation to the minimum m P D F to be used without further refinement. The chief benefits of this delta function elimination process are that only a relatively small
883
number of additional refinement cycles are required and usually the number of delta functions found is the minimum sufficient number.
THE HETEROGENEITY OF ANTIBODIES WITH RESPECT TO EQUILIBRIUM CONSTANTS C A L C U L A T I O N BY A N E W M E T H O D U S I N G D E L T A FUNCTIONS, AND ANALYSIS OF THE RESULTS* PHILLIP M. E R W l N t and FREDERICK ALADJEM~: Department of Microbiology, University of Southern California, School of Medicine, Los Angeles, CA, U.S.A. (First received 22 July 1975; in revisedform 10 December 1975)
Abstract--A method of data analysis has been developed which approximates the probability density of association constants of the reaction of hapten with antibody by a minimum number of delta functions, and alternatively, by a large number of delta functions. The method has been computerized and found able to approximate any physically possible affinity distribution. We have calculated the delta function distributions which describe known distributions of association constants and analyzed experimental data. From 2 to 4 delta functions were required to describe the probability density of Sips distributions. Not more than three delta functions were needed to describe the probability density functions of each of three antibody populations obtained from pooled antisera. These delta function distributions were found in one case to fit the data significantly more accurately than a Sips distribution. In the other two cases the fit was about the same as the Sips distribution. Affinity heterogeneity analysis using a large number of delta functions gave roughly equivalent results in terms of closeness of fit to the data to the results using a minimum number of delta functions. It is shown that binding measurements do not contain sufficient information to prove homogeneity of antibody. An analysis of factors affecting the resolution of delta functions is given.
INTRODUCTION
Calculation of the probability density function of equilibrium constants which describes the reaction of univalent hapten with antibody,, antibody heterogeneous with respect to equilibrium constants, has been mathematically achieved using Fourier transforms of the binding equation, Bowman & Aladjem (1963). The difficulties of the calculations have prevented the use of this method to evaluate experimental data. Other less accurate methods have been used to approximate the association constant distributions. Two of the more common methods are the Gaussian and the Sips distribution. The availability of high speed computers has made possible the use of numerical methods to generate approximations to distributions of antibody association constants. The method published by Werblin & Siskind (1972) is an example of such a method. Data have shown that monoclonal antibody producing cells synthesize antibody of homogeneous affinity. This suggests that an antibody population is composed of a number of subpopulations of homogeneous affinity. A useful feature of a method of analysis of affinity heterogeneity would be the capability to determine the number of subpopulations, but as is shown below we have found that this is not
possible due to resolution factors. It is possible to determine the minimum number of subpopulations which is adequate to describe the experimental data. A more nearly continuous approximation to the distribution of association constants can also be calculated by assuming a large number of homogeneous subpopulations. The limits of variability of the solution could then be inferred from the two different results. The method presented in this paper has these capabilities.
* An abstract of this work has been presented (P. Erwin & F. Aladjem (1972) Fedn Proc. 31, 781). $ Predoctoral trainee supported by NIH Training Grant ROI-AI 00157. ~;Supported, in part, by Research Grant CA14089 from the U.S.P.H.S. 873 IMM. 13/11 A
THEORY
OF
ANALYSIS
The purpose of this analysis of the reaction of univalent homogeneous hapten with antibody is to determine the probability density distribution of free energies of binding. An assumption made to simplify the calculations is that there are no co-operative binding effects for multivalent antibody-univalent hapten binding. No data have been found which indicate that there are such co-operative binding phenomena. Structural studies of antibodies have not shown a mechanism to bring about such an interaction for the reaction of a small hapten with antibody. Given this assumption, a probability density function (PDF) of the association constants can be represented as the sum of delta functions, the number of delta functions being equal to the number of different affinity subpopulations participating in the reaction. A delta function is simply a spike of probability density such that the integral of the probability density function has a finite value over the range x - dx to x + dx. This integral is equal to the fraction of antibody combining sites with association constant K = x.
874
P.M. ERWlN and F. ALADJEM
The accuracy of a PDF is determined by calculation of the chi-square
and
S~Kfl-I//(1 + K i l o = S~K/-II.
P
Z2 = ~ ( B i - A/)/2a 2,
(3)
(1)
i=1
where B / = the experimentally determined value of bound hapten concentration at free hapten concentration H~; A~ = the concentration of bound hapten at H~, calculated from the PDF; cq = the standard deviation of B~; p = the number of data points. The values of A~ are determined using the equation:
Equation (3) shows that the binding of low affinity antibody is proportional to the free hapten concentration. If there are several different low afftnity subpopulations of antibody, the net binding is
y~ S~K~H, = Hi Y SjKj. J
J
The sum on the right can be replaced by a constant, Szo, and is the low affinity uncertainty term.
SLo = Z SjKj.
(4)
J
A~ = ~ S~KjHd(1 + H/Kj),
(2)
j=l
Sj = the concentration of antibody binding sites with association constant K j, and m = the number of subpopulations of antibody which bind hapten. Since m, the Sj's, and the Kj's are all unknown, a general method of approximation of the PDF must be used until these parameters can be estimated. The number of antibody subpopulations may be large enough to generate an essentially continuous distribution. An approximation to the PDF using a large number of delta functions sufficient to approximate a continuous distribution serves a dual purpose: the heterogeneity of binding is shown, and the large m PDF can be used to generate an estimate of the minimum number of delta functions necessary to approximate the antibody affinity distribution and to estimate the affinities and concentrations of these few delta functions. Calculation of a large m P D F A PDF, whether continuous or discontinuous, can always be approximated to any desired degree of accuracy using a sufficiently large number of delta functions. For data of accuracy ___19/~ one delta function for every factor of 3 in association constant is sufficient to approximate the distribution (see Resolution below). However, the theoretical range of possible association constants is from 0 to ~ , and spacing delta functions every factor of three over this interval would require an infinite number of delta functions. The number of delta functions to be used can be reduced to a manageable number by the introduction of two uncertainty terms, Sag and Sin. SLo can be used in a binding equation to represent antibody of very low affinity, and Sm can be used to represent antibody of very high affinity. These terms are uncertainty terms because the association constants of the antibodies represented cannot be determined from binding measurements. The first uncertainty term, Szo, has the following justification. For antibody of low affinity, the denominator of the binding equation (2), simplifies. If K j/-//,~ 1 for all Hi in the experiment, then (1 + Kfl-//) ~ 1
It cannot be determined from binding measurements alone how many different affinities of antibodies are included in the binding represented by SLO. The second uncertainty term, Sin, is obtained for very high affinity antibody, such that H/K i ~ 1 for all H/ in the experiment. The binding equation (2) again simplifies, thus (1 + K~I-I/) ~ Kfl-It and
SjKfld(1 + Kill) - S t.
(5)
Equation (5) shows that all of the binding sites of very high affinity antibody are combined with hapten even at the lowest hapten concentration. The total binding of many high affinity subpopulations can be represented by the high affinity uncertainty term S~ = Sin.
(6)
J
The units of Sm and Sro are not identical. Since Sm is a sum of St values, equation (6), its units are moles per liter. SLO is the sum of the products of concentrations of binding sites and association constants. Thus, SLO is dimensionless. When multiplied by the free hapten concentration, the result is the concentration of hapten bound by low affinity antibody, and the units are moles per liter. These two terms, SLO and Sin, are sufficient to describe the binding with K values outside the range of the experiment. A general equation to approximate hapten binding is therefore:
A(H) = Sm + H'SLo +
SjKjH .i= 1 (1 + Kill)"
(7)
In equation (7) the summation index j is defined to index delta functions which have Kj values between those which can be included in the uncertainty terms. Our computer program for the calculation of a large m PDF uses a simple procedure which fits the data using equation (7). The six steps in the procedure are given in Appendix I. The values of K~ and m are determined according to a set procedure, after which values of the S/s and SLo and Sm are calculated by an iterative procedure.
Heterogeneity of Antibodies
A theoretical model of delta function resolution The following is an analysis of factors which must be considered in determination of the significance of a PDF. The significance of a PDF depends on the precision with which subpopulations are resolved, which depends on the potential resolution of the method and on the number and precision of the data points. There is no general formula relating resolution to these factors, but examination of a simple example can give some insight into the problem. Consider the distribution of association constants within an interval K1 to K2. Let there be two possible distributions of association constants within this interval: (1) One homogeneous subpopulation within the interval with association constant K and binding sites concentration S; or (2) two subpopulations, one with parameters K/a and S/2 and the other with Ka and S/2, where a is a real number greater than one.. The resolution of the method of analysis can be described as the smallest value of a at which a choice between these two alternatives can be made. There are other possible definitions of resolution, but t h i s one is useful in calculation of the minimum number of delta functions, as will be shown below. For data of perfect accuracy a sufficiently general method should achieve nearly perfect resolution. A method of such generality is unnecessary in analysis of real data. For the alternative cases 1 and 2 to be resolvable, there must be a significant difference in the A(H) functions generated by the distributions. This difference should be at least as large as the error of the relevant portion of the experimental binding curve. We define the error of the data binding curve to be
where summation is of data standard deviations for data points at H values within a factor of 2 of X, and L is the number of these data points. For this simple example it is possible to calculate the differences between the binding curves generated by the two alternative PDF's. Let the binding due to antibody with K values outside the interval (K1, K2) be designated by a residual function B,(H). Then the total binding for case 1 is:
SKH A~(H) = - + B,(H). 1 +KH
(8)
The equation for case 2 is:
,( osK.
sK.
A2(H) = ~ 1 + aK-------H+ a + KHI + B,(H).
(9)
The difference between the two alternatives, 0(H), is:
o(I-I)
1( aSKH
\ l + aK~
+ __SKH
a + Kn
2SKH \ (1 O) f-;~)"
We define the fractional difference as D(H)/A(H). Since the error of the binding curve is often proportional to the concentration of hapten bound, the fractional difference is really the quantity of greatest interest. At very large or small values of KH and at KH = 1, D ( H ) = 0. The maximum fractional differ-
875
Table 1. D(H)/S as a function of a. The precision of the data necessary to achieve a given resolution factor a is shown in columns 3-5 for 3 cases: when the binding due to other subpopulations, Br(H), is S, 2S, and 4S. These three cases correspond to a high affinity subpopulation, a moderate affinity subpopulation, and a lower affinity subpopulation, respectively
D(H)/A(H) D(H) a
S
1.2 1.414 1.732 2.0 3.0 5.0
0.0016 0.0057 0.014 0.022 0.053 0.10
Br = S 0.0013 0.005 0.012 0.018 0.04 0.08
B r = 2S 0.0007 0.0025 0.0064 0.01 0.024 0.045
Br = 4S 0.0004 0.0014 0.0033 0.005 0.013 0.024
ences occur somewhere near the H values of 4/K and (1/4K), except in the special case B,(H) = 0. Consideration of the magnitude of D(H) at these values of H will allow the estimation of the resolution possible. At 1 n=~-,
A--~=2
4--~a + 4a ~ 1
•
(11)
At H = 4/K, D(H) has the same magnitude but opposite sign, and A(4/K) is greater. Therefore, the fractional difference is greatest at H values near (1/4K). Using equation (11) it is possible to determine the dependence of D(H) on a. Table 1 has values of D(H)/S for six values of a, and also shows the effect of various levels of residual binding, Br(H). The fractional precision of the data required to achieve resolution of a factor of 3, i.e. when a = 3, is 0.053 S/A(H), where A(H) is evaluated at H = (1/4K) (Table 1, column 1). The quantity S/A(H) is a key ratio in evaluation of resolution. A relatively high affinity subpopulation is one for which B,(H) is relatively small, and a low affinity subpopulation is one for which B,(H) is relatively large. The residual binding function for any particular subpopulation is dominated by the binding of other higher affinity subpopulations. For a relatively low affinity subpopulation, there will be many higher affinity subpopulations contributing to B,(H), and for a high affinity subpopulation, there will be few of higher affinity, and B,(H) will be relatively small. Since the fractional precision of the data required to achieve a given resolution is inversely proportional to A(H), it follows that a smaller fractional data error is required to achieve a given resolution factor for subpopulations of relatively low affinity than is required for high affinity subpopulations. This relationship is demonstrated by the recent observations of Kiln et al. (1974). It was found that more low affinity antibody could be detected in a globulin preparation after adsorption of high affinity antibody. This was because the high affinity antibody produced a large residual binding level which not only prevented accurate analysis of the heterogeneity of the low affinity antibody, but also prevented an accurate estimate of the
876
P.M. ERWIN and F. ALADJEM
quantity of low affinity antibody. The corollary relationship is that for any given size of A(H), a subpopuiation with a greater concentration of binding sites, i.e. a larger S, will be resolved much better than a subpopulation of lesser S. This is because of the direct proportionality of S and D(H). There is an upper limit on the magnitude of D(H). D(H) can never be greater than S/2. The affinity of a delta function cannot be resolved if S/2A(1/K) is less than the fractional data error. Such delta functions can be replaced by other delta functions at other K values without significantly increasing chi-square. A way to reduce the number of delta functions being used in a PDF is. suggested by this model. If the resolution factor a of a delta function is such that there is a higher affinity delta function of affinity less than Ka, and a lower affinity delta function of affinity greater than K/a, then the elimination from the PDF of the delta function with affinity K and resolution factor a does not prevent successful analysis of the binding data. Increases in the heights of the two bracketing delta functions can compensate for the eliminated delta function. This fact is used in our procedure to determine the PDF with minimum m.
tration S of the remaining delta functions. If the minimum chi-square is too much larger than that obtained with the large m approximation, then at least one additional delta function is needed to approximate some part of the PDF. Because of resolution factors discussed above, the additional delta function will usually be required in the high affinity region of the PDF.
aj = the resolution factor of the fl' delta function; Ej = the average of the fractional errors of the binding data at H = 4/Kj and H = 1/4 K~; C~ = the coefficient of M2; Mj = 4A(1/Kj)/Sj; A(I/Kj)= the bound hapten concentration at H = 1/Kj calculated from the PDF. The resolution factor obtained from equations (12) and (13) is only approximate, but the accuracy is sufficient for the use outlined below to determine the PDF with minimum m and reasonably small chizsquare.
Method of elimination of delta functions The method of elimination of delta functions is to selectively absorb some delta functions by other delta functions. Certain delta functions are selected to remain, i.e. not be absorbed by other delta functions. The selection of delta functions to remain depends on the resolution factors a~ which have been calculated for the particular delta functions. Since we have spaced the delta functions at factors of x//~, a delta function with an aj less than ~ cannot be eliminated and is selected to remain. Next, delta functions with a t between x ~ and 10 are examined. If they are adjacent to delta functions with aj < x / ~ , then they are eliminated. If there are no delta functions which have already been selected to remain which are adjacent in K, then the delta function being evaluated is selected to remain. The procedure is performed for increasing a intervals until all delta functions have been selected either to remain or to be absorbed. A delta function which can be absorbed by two bracketing delta functions is split between them. A detailed example of the delta function elimination procedure is given in Appendix II. After the process of absorption is completed, the chi-square of the PDF is reduced by refining both the association constants and concentrations of the remaining delta functions and the two error terms. The refinement procedure is similar to that used when only the Si values are being refined, except that the St and Kj values are varied simultaneously. The minimum chi-square is obtained after about 30 iterations of the refinement procedure. At this point, operator intervention is necessary. If the chi-square obtained after refinement is much larger than the chi-square obtained by the large m approximation, an additional delta function may have to be added and refinement restarted. If the chi-square obtained by the minimum m model is close to that obtained by the large m approximation, the minimum m solution can then be examined regarding the possibility of further reduc,tion of the number of delta functions.
Method of determination of the minimum m P D F The results of the large m analysis of the PDF will in general show a number of maxima and minima giving the appearance of a continuous and complex distribution of affinities. We have found that there is usually a less complex PDF that will fit the data very well if not better than the large m distribution. This distribution is produced by eliminating delta functions obtained by the large m approximation above, where possible, and retaining the two uncertainty terms. The method by which delta functions are eliminated is outlined below. The validity of the method is tested by determination of the minimum chi-square fit that can be obtained by refinement of the values of association constant K and concen-
Large m and minimum m distributions calculated for three Sips distributions Of course, any new method of data analysis must have a capability to correctly analyze known data. Werblin and Siskind validated their method of analysis using delta functions by showing that their method would correctly analyze data calculated theoretically from five postulated PDF's. The PDF's from which they calculated data were composed of 1-6 homogeneous subpopulations. Our methods correctly analyze data calculated from PDF's composed of several homogeneous subpopulations. Rather than showing examples of this type of analysis, we wish to demonstrate an additional capability of methods of analysis using delta functions.
A practical method of resolution evaluation The model above does not allow calculation of the resolution factor a of a delta function in a PDF. It is necessary, therefore, to determine a relationship empirically between the resolution factor of a delta function and other parameters of the PDF and the binding curve. Calculation of a large number of examples led to the following relationship: aj = CiM ] + 0.137Mj + 1.66
(12)
C~ = 74E~ + 0.17Ej,
(13)
where
Heterogeneity of Antibodies
877
H.fiH II.D-
H.qE-
I.H~ILII .~"
H ~
B 7
g
~I IH I I
q
~
B 7
fl
9 IEII
N II.Bm
ff
2. .B-
~ B.IE-
.
H.HE
I q
I E
I B
I 7
I B
i
.~-
I 9 1
LnB Ig K Fig. 1. Three Sips distributions with mean affinities and heterogeneity indices corresponding to the experimental data. The parameters of these distributions are given in Table 3. The highest distribution, middle and lowest distributions have heterogeneity indices 0.64, 0.49 and 0.36, respectively. O u r large m a n d m i n i m u m m methods can analyze data generated by continuous PDF's. Figure 1 shows three Sips distributions which correspond in m e a n association constant and heterogeneity index to the experimental data analyzed in this paper. Figure 2 shows large m distributions obtained for data gener-
I1.11,
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~
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.
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Jl----
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l=
7
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Cilia I I
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Fig. 3. The delta functions determined by the minimum m method for data calculated from Sips distributions. The data analyzed are as in Fig. 2, and the ordinates and abscissas are the same parameters. ated by these Sips distributions over free h a p t e n ranges corresponding to our experimental data. The agreement is quite good between the shapes of the large m distributions a n d the corresponding parts of the Sips distributions. The results of the m i n i m u m m analysis shown in Fig. 3 c a n n o t be easily evaluated by inspection. Table 2 contains parameters of the analyses. The chi-square values show that the minim u m m distributions are adequate for analysis of Sips distributions. It can also be seen that reduction of the n u m b e r of delta functions below the m i n i m u m
II.ll. q
E E 7
B
S IE I I
q
~
6
7
H q
Igll
11.6"
mll.:~ .J EL
_L_ q
K
E 7
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BIIIII
tl
E 6
7
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B
qlllll
q
E 6
7
e
E.E,
11.'4,
II.I!,
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LnB It~ K
Fig. 2. The delta functions determined by the large m method for data calculated from Sips distributions. The parameters of the Sips distributions are, top to bottom, Ko = 5 x 105M -1 a n d a = 0.36, Ko = 1.8 x 10SM -1 and a = 0 . 4 9 , and K o = 8 x 10SM -1 and a = 0 . 6 4 . The ordinate on the left set of graphs is the probability density of log~oK. The ordinate of the right set of graphs is PD(IogloK)/2A(I~),a quantity used in calculation of resolution factors. The abscissa is in all cases log ~oK. The lines drawn in are the actual probability densities of the Sips distributions. The delta function heights are all increased by a factor of two because they represent binding sites within intervals of x ~ , rather than probability densities of log xoK.
Table 2. Analysis of Sips distributions by delta functions. The closeness of fit to data generated by Sips distributions of different heterogeneity are shown by chi-squares in column 4. All chi-squares are due to systematic differences between the models and the data because these data are theoretically calculated. The data ranges (in orders of magnitude of free hapten concentration) in the right column correspond to the experimental data presented in this paper. SLo and Sx~ are given as fractions of total binding at the highest free hapten concentration Chisquare
Snt
a = 0.64 Sips -Large m 14 Min m 4 Min m - 1 3 Min m - 2 2
0 0.1 1.8 7.0 22.3
-0.0002 0.003 0.004 0.004
-0.004 0.002 0.07 0.07
4.5
a = 0.49 Sips -Large m 10 Min m 2 Min m - 1 1
0 0.1 1.0 9.3
-0.016 0.024 0.039
-0.086 0.22 0.73
3.0
0 0.26 3.1 17.2
-0.02 0.08 0.097
-0.10 0.28 0.48
3.0
Data
Model
m
a = 0.36 Sips -Large m 11 Min m 2 Min m - 1 1 '
Data
SLo range
878
P.M. ERWIN and F. ALADJEM W K
l.m-
d'
B.I{-
¢.
rl." --.I B.B
HFIPTEN CnNC. X I B '~ (MDLFIR)
I.¢I
X
Z
It
V Z
i
l.g"
I""IZ
m.~"
,r
[] . m =,it
m.B
-7
-:
LDISt,, HRPTEN CDNC. (MDLFIR)
Fig. 4. Binding curves of p-tSliodophenylarsonate with rabbit antiphenylarsonyl-BgG. The points at high total hapten concentration are actually the averages of 4-6 measurements.
cedure. Therefore, all delta function heights are divided by the greatest measured value of hapten bound. In the right column of Fig. 5 we plot S/2A(H) for each delta function vs the log K, where A(H) is evaluated at H equal to the reciprocal of the association constant of the particular delta function. As pointed out in the discussion of resolution, the quantity S/2A(H) is important in the determination of the resolution of the delta functions, i.e. the resolution factors are a quadratic function of the inverse of S/2A(H). Therefore, the largest delta functions in the graphs on the right can be expected to have the smallest resolution factors. These graphs show which delta functions can be expected to have the smallest resolution factors and also show the importance of some of the high affinity subpopulations which appear to be insignificant in the graphs on the left. Figure 6 is a presentation of the results of minimum m analysis in the same format as Fig. 5. Table 3 contains relevant parameters for comparison of the results of large m, minimum m, and Sips heterogeneity analysis. The low affinity uncertainty term is presented as its fraction of A(H) at H equal to the largest experimentally measured value. Since the binding due to this term is SLoH, the term is presented as SLoH/A(H). The values of association constant of the delta functions obtained for fluorescence polarization data by minimum m analysis are very similar. The data in Table 3 show that a considerable fraction of the R-FI binding is described by the low affinity uncertainty term. For the equilibrium dialysis
number determined by our method introduces substantial systematic errors. Conversely, increasing the number of delta functions does not substantially decrease chi-square. The correlation coefficient R was not found to be sufficiently sensitive for detection of differences between the data and the curves generated by the distributions. Rxr for all of the above distributions varied between 0.99 and 1.0.
Experimental data used for analysis We analyzed the reaction between phenylarsonate (PA) and rabbit antiphenylarsonate antibody. Two sets of binding data were analyzed: (1) The reaction of fluorescein-coupled phenylarsonate, namely (R-F1) with the water-soluble and water-insoluble immunoglobulin, fractions as described in the preceding paper and graphed in Figs. 4a and b of that paper (Medof & Aladjem, 1976); and (2) para -is1 iodophenylarsonate reacting with the IgG fraction of antiphenylarsonate antiserum. The latter reaction was studied by equilibrium dialysis. The equilibrium dialysis data are given in Fig. 4. This immunoglobulin fraction was obtained from a different antiserum pool than that of the R-F1 data. The details of these experiments will be published elsewhere (Fukahara K. & Aladjem F., to be published). RESULTS
The PDF's obtained by the large m method are presented in Fig. 5. In the left column we plot delta function height vs the logarithm of the delta function association constant. Since the total antibody binding site concentration is unknown for these data, it was necessary to arbitrarily pick a normalization pro-
I l
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E
7
B
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7
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~
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~
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7
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B
l<
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7
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,+lllllll [;
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g
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K
Fig. 5. Delta functions obtained by the large m method. The delta functions obtained by the large m approximation method are plotted from top to bottom for water-soluble and water-insoluble immunogtobulin fractions reacting with R-FI, and the reaction of immunoglobulin with lalI-iodophenylarsonate. The delta functions are plotted vs the loglo of the K value. The right column graphs show the concentration of binding sites Sj divided by twice the bound hapten concentration evaluated at H = 1/Kj. The left column graphs show the binding sites concentration which for normalization is divided by the maximum concentration of bound hapten.
Heterogeneity of Antibodies
879
I.IIILB-
ILB
IZ I z < rv. ~ il.S-
In 1,-1 ...I
B.EI
-~!.ll-
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,
~ tr fi 7 B Cllt~ll
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Fig. 7. Sips heterogeneity plot of the reaction of IgG with p.1 a,iodophenylarsonate.
e.,~
H.B
tt ~ 6 7 O U I H I I Lira m I~
. . . . . . . tt K 6 7 B g l l ~ l l t.nr. m K
Fig. 6. The delta functions obtained by minimum m analysis are plotted from top to bottom, water-soluble and water-insoluble immunoglobulin fractions reacting with R-F1, and the reaction of immunoglobulin with para 13q-iodophenylarsonate. As in Fig. 5, the left column is of graphs of S/(max Bi) vs logloK~, and the right column is of graphs of SJ2A(1/Kj) vs logl0K j. data, the low affinity and high affinity uncertainty terms, are of little significance, i.e. they can be neglected without significantly increasing chi-square. It is clear from the chi-square values that the large m and m i n i m u m m distributions fit the equilibrium dialysis data significantly better t h a n a Sips distribution. Figure 7 is a Sips plot of these data. The data appear to deviate little qualitatively from the Sips
model, but the precision of the data is adequate to show the difference by calculation of chi-squares. Figures 8(a and b) show Sips plots of the R-FI water-soluble and water-insoluble IgG binding data binding sites. D e t e r m i n a t i o n of the total concentration of binding sites is necessary if one wishes to calculate the m e a n affinity of the antibody population or if one wishes to make a Sips plot. W h e n there is a substantial low affinity uncertainty term, as in the cases of these data, accurate determination of the total binding site concentration is impossible. Table 4 gives the m e a n affinities determined for three possible binding site concentrations. The chief point to be noted is that as the concentration of binding sites is raised, the data more nearly fit the Sips model, b u t the m e a n affinity then more poorly relates the information in the binding data. Instead, the mean a t f n i t y describes an extrapolated distribution of low mean affinity.
Table 3. The relevant parameters for comparison of the closeness of fit of the distributions. The mean association constants in column five are obtained in the normal way for the Sips distribution. For the delta function distributions, the Ko values are determined by ignoring the high and low affinity error terms. SLO and Sat are given as fractions of the total binding at the largest experimental free hapten concentration. For the large m and minimum m distributions, K0 = antilog
Si log Ki i
Data
Model
m
Si i
Chisquare
Watersoluble globulin fraction
Sips -Large m 10 Min m 2 Min m - 1 1
19.1 15.8 17.7 25.5
Waterinsoluble globulin fraction
Sips Large m Min m Min m - 1
-10 2 1
Equilibrium dialysis data
Sips -Large m 14 Min m 3 Min m - 1 2
Ko(M- 1) 5.3 1.1 2.1 3.9
x x x x
a
SLo
Snt
105 108 108 10a
0.36 ----
--0.32 0.04 0.44 0.069 0.55 0.084
16.3 13.7 15.3 38.4
1.75 x 105 2.7 x 107 9.4 x 1 0 7 1.8 x 108
0.49 --
--0.48 0.015 0.65 0.024 0.79 0.034
29.1 13.0 12.5 16.0
8.3 5.6 1.54 2.14
0.64 ---0.024 0.001 -0.09 0.0033 -0.20 0.0035
x x x x
105 105 106 106
--
--
880
P.M. ERWIN and F. ALADJEM Table 4. The concentrations of antibody combining sites per mg protein per ml as determined in three ways: (A) by extrapolation of a graph of the reciprocal of the bound sites concentration (I/B) vs the reciprocal of the free hapten concentration (l/H) to 1./H = 0; (B) by varying the antibody sites concentration to obtain linearity of a Sips heterogeneity graph [log R/(N-R) vs logH]; and (C) by treating the low affinity error term obtained for the large m distribution as if it were of K = (1/4 H m J , where Hm,~ is the largest experimental free hapten concentration. Using these binding site concentrations, K0's were calculated as
1/n at R = N/2 Data
Parameter
1/B vs 1/H
S Ko
3.47 x 1 0 - 7 M 2.9 x 107M - I
8.8 x 10-TM 5.3 x 105M -1
Water-insoluble globulin fraction
S Ko
9.91 x 1 0 - 7 M 1.4 x 107M - 1
4.65 × 1 0 - 6 M 1.75 x 105M - 1
1.96 x 1 0 - 6 M 1.7 × 106M - 1
Equilibrium dialysis data
S K0
1.18 x 1 0 - 6 M 1.25 x 106M -1
1.36 x 10-6M 8.3 x 105M -1
1.47 x 10-6M 6.1 x 10SM -1
Ilu I
rv
---I .1~ rn ._1
-2.B
I
I
I
-g
-B
-7
LDG IB H
(b)
I
trl r-1
5.33 × 1 0 - 7 M 4.3 x 106M -1
DISCUSSION
?..
Z
Delta functions
Water-soluble globulin fraction
(a)
Z
Linear Sips plot
-I~
!
-g
.-B
I
-7
LDE 1121 H
Fig. 8. Sips heterogeneity plots of the reaction of antibody with R-FI using the three possible concentrations of binding sites (5) from Table 4. S determined by: (1) D,I/B vs 1/H; (2) I , delta functions; a n d (3) X, Sips. (a) the reaction of water-soluble immunoglobulin with R-F1; (b) the reaction of water-insoluble immunoglobulin with R-FI.
Approximation of antibody association constants by a sum of delta functions has several advantages relative to approximation by a Sips or Gaussian distribution. First, a delta function distribution can locate local maxima in a distribution of association constants, whereas the maximum of the best fit Sips or Gaussian distribution may not correspond to any feature of the actual distribution. Second, a distribution obtained with delta functions is not restricted to any type of symmetry. Third, an estimate of the amount of antibody with very high and very low association constants can be obtained from the values of $~.o and S~I. Delta functions are useful for approximation of discontinuous distributions (a small number of delta functions) and continuous distributions (large number of delta functions). An additional advantage of this type of analysis is that once the antibody population has been divided into subpopulations using delta functions, other properties of the antibody can be investigated. For example, Medof & Aladjem (1971) were able to determine values of specific molar quenchence for delta functions which would reproduce hapten fluorescence quenching data for the reaction of fluorescein with antibody. Previous attempts to calculate the heterogeneity of specific molar quenchence using a variety of graphical procedures but without knowledge of the delta function distribution, had not been as successful• Our large m and small m methods are programmed to be run completely on a desk top calculator• There are no restrictions on data input, as far as computations are concerned. Of course, more accurate data will allow a more accurate calculation of the distribution of association constants, and a greater range of free hapten concentrations reduces the amount of the antibody population in the uncertainty terms SL~z.gnd Snl. The use of a delta function spacing of x/10 in our large m method has three advantages: (1) this close spacing is sufficient to provide what is essentially a best fit of the data; (2) the delta functions are close enough together to excellently approximate
Heterogeneity of Antibodies continuous distributions; and (3) this spacing gives a pleasing appearance on graphs. This spacing of the delta functions is narrower than experimenters not interested in testing the method may wish to use. We have found that use of a spacing factor of 10 is almost as accurate as ~/]-0 for our data and affords a significant reduction in computer time (almost a factor of two). Another method of approximation of antibody association constant distributions is in use which is similar to our large m approximation. This method was developed by Werblin & Siskind (1972). It approximates the distribution of association constants by a number of delta functions spaced by a constant factor. Their method refines the S~ values of the delta functions by calculating the sum of the squares of the deviations (SSD) between the binding data and calculated values of bound hapten for a large number of variations of an approximation to the PDF. The method then chooses the variation with the least error as the basis for another sequence of similar calculations. The number of variations of the PDF tested in each iteration of this refinement procedure is 2m, where m is the number of delta functions. Our method tests 3 m + 6 variations per iteration of our refinement procedure. A second difference between the methods is that we minimize chi-square, whereas Werblin and Siskind's method minimizes the SSD. It is our understanding that minimizing the SSD is equivalent to minimizing chi-square when all standard deviations are equal to unity. We have tried this approach and found that significant differences in the PDF's occur when the SSD is minimized instead of chi-square. In particular, the data at low hapten concentrations contribute little to the SSD. A third difference between the methods is that we introduce and calculate the high affinity and low affinity uncertainty terms. These terms give insight into what fractions of the antibody population are of indeterminate affinity. In addition to these differences between our large m approximation and that of Werblin and Siskind, we have the additional capability to calculate the minimum m PDF. Using the minimum m PDF, we can model other aspects of the reaction, such as quenchence, much more easily than could be done using a large number of delta functions. The minimum m PDF also gives us an indication of the minimum possible heterogeneity of the real distribution. The results of the analysis of experimental data show that there are many possible PDF's which can adequately describe the association constant distribution of an antibody population• For each of the two sets of data for the reaction of the hapten R-F1 with antibody, three PDF's of approximately equal chisquare were obtained: a Sips distribution (Fig. 1), a large m delta function distribution (Fig. 4), and a minimum m distribution (Fig. 5). An infinite number of different solutions of equal quality fit can be obtained by combinations of the three solutions for the same data. Another result of this analysis is that for these antibody preparations, a significant fraction of the antibody is contained in the high and low affinity uncertainty terms. It is impossible therefore, to determine the actual mean association constant Ko for either of these antibody populations without making an ad hoc
881
assumption about the concentration of binding sites. For the equilibrium dialysis data, a significant 10w atfmity error term was not obtained, and the high affinity error term was quite small. Because there is a measureable high affinity uncertainty term, it would be impossible to predict accurately the results of an experiment carried out in far antibody excess in which very high affinity antibody would play a dominant role. For example, phage inhibition by antibody at very dilute antibody and phage concentrations would be dependent on very high affinity antibody. Although the equilibrium dialysis data appear linear (Rxy = 0.998) on a Sips plot, Fig. 7, there is a substantial contribution to the chi-square due to systematic deviation between the data and the Sips line. The chi-square for the large m and minimum m distributions are significantly smaller because of the absence of this systematic deviation. Examples such as this confirm that chi-square is significantly more sensitive than Rxr as an indicator of closeness of fit. The results of the analysis of our data by the minimum m method were in one sense quite surprising. Examination of the results for R-F1 binding data analyzed by the large m method did not show any striking similarities between the two antibody fractions. Examination of the results of the minimum m analysis showed for the first time the close agreement in association constant values of these two different antibody preparations. Apparently, too many delta functions can mask a similarity of this type. More data will have to be analyzed before conclusions can be drawn concerning the generality of similarities of water-soluble immunoglobulin to water-insoluble immunoglobulin affinities. In view of the fact that we obtained excellent fit to our experimental data with only two delta functions for two cases and three delta functions in one case and that the antibodies were obtained from serum pools with 30 or more rabbits, it seems very unlikely that each delta function represents an exactly homogeneous antibody subpopulation. The analysis of resolution presented in the theoretical section shows that these delta functions may represent a large number of different subpopulations that are not resolvable by this method. The fact that an association constant distribution using two or three delta functions fits the data is an indication that the heterogeneity may be very limited relative to the Sips or Gaussian distributions of affinities. It is believed that monoclonal antibody producing cells produce antibody of homogeneous affinity. A limited number of different affinity antibody subpopulations are likely to be produced in response to any single antigenic determinant. Calculation of the minimum m distribution shows the minimum possible number of subpopulations of differing affinity. No method of analysis of binding data can determine the maximum possible number of subpopulations. The method of analysis which we have developed seems to extract the useful information of hapten-antibody binding data and is computationally efficient. REFERENCES
Bowman J. D. & Aladjem F. (1963) J. theor. Biol. 4, 242. Erwin P. & Aladjem F. (1972) Fedn. Proc. 31, 781.
882
P . M . ERWIN and F. ALADJEM
Fujio H. & Karush F. (1966) Biochemistry 5, 1856. Kim Y. T., Werblin T. P. & Siskind G. W. (1974) Immunochemistry 11, 685. Medof M. E. & Aladjem F. (1971) Fedn. Proc. 30, 657. Werblin T. P. & Siskind G. W. (1972) Immunochemistry 9, 987.
Table 5. Elimination of delta functions Cycle i 1 2
APPENDIX I The procedure for calculation of the large m association constant distribution is outlined below. (1) Data input. (2) The initial values of the K / s and S/s are calculated: Starting a factor of 10 greater than the reciprocal of the smallest Hi and extending a factor of 10 less than the reciprocal of the largest H value, delta functions are initialized in the computer program and spaced by factors of ~/]O. The initial values of the S/s and Sm are set equal to the largest Bi divided by m + 2. m + 2 is the number of delta functions plus the two uncertainty terms. SLo is initially set equal to S1K1/IO. (3) The initial values of Sj, Sm and SLo are refined. The chi-square depends parabolically on the value of one of these parameters when other quantities are held constant. A short calculation determines the minimum of the parabc/la for S~ and changes the value of S~ to reach this minimum chi-square. S~ is then held constant while the procedure is repeated for $2. As this process is performed sequentially on all of the S/s and the uncertainty parameters, the chi-square decreases monotonically. If the procedure is performed many times, the result is very close to the minimum chi-square, an extremely accurate fit. In general, about 50 iterations are required to achieve this fit, although the number of iterations does depend on the number of delta functions and the data to be approximated. (4) Parameters of interest, such as total antibody combining sites concentration, Sips ratio (r/n-r), mean association constant, best fit Sips parameters K0 and a, etc, are calculated. (5) Results are printed. The only step that requires a significant amount of computer time is step (3). The calculation time depends linearly on three parameters: the number of iterations, the number of delta functions, and the number of data points. With 10 delta functions and 30 data points, about 15 min of HP9830 calculator time is required.
APPENDIX I1
An example of elimination of delta functions The following example of the delta function elimination procedure is taken from the analysis of the reaction of IgG with p.la~ iodophenylarsonate. The calculated resolution factors of the delta functions are listed in column 2 of Table 5. Delta functions 1 and 2 are automatically eliminated because S1/2A(H) and S2/2A(H) are less than the data error. The computerized elimination procedure evaluates the delta functions in cycles. In each cycle, the delta functions are examined in order from the first delta function to the last delta function. During the first cycle, delta functions with 0 < logloa~ < 0.5 are marked to remain. In this example, none of the delta functions had resolution factors in this range. In the second cycle, delta functions with 0.5 _< logtoa~ < 1.0 are evaluated. The first delta function with a~ within this range is delta function 8. The computer then checks for delta functions previously selected to remain which are within a factor of 10 in association constant. There are none, and delta function 8 is selected to remain. In Table 5, a plus sign ( + ) is placed in row
3 4 5 6 7 8 9 10 11 12 13 14
a~
2
3
4
5
10
Net
1.1 x 107
7.0 x 106 58.0 6.3 x 104 19.7 11.0 202.0 9.5 8.8 18.3 197.0 13.6 58.5 15.4
+ + + -
+ -+ + -
8 under the column of cycle two. The second delta function to be evaluated in cycle 2 is delta function 9. The computer checks for delta functions already selected to remain which are within a factor of 10 in K, and finds delta function 8. Delta function 9 is then marked for elimination, as noted by the minus sign ( - ) in row 9 in cycle 2. No more delta functions are evaluated in cycle 2. The first delta function evaluated in cycle three is number 5. The resolution factor is within the range 1.0 < logloa~ < 1.5. The computer checks for delta functions which have already been selected to remain and which are within a factor of 10 x / ~ in association constant. Delta function number 8 is exactly 10 x / ~ higher than delta function number 5 in association constant, but it is not within this distance. Therefore, delta function 5 is selected to remain. The next delta function evaluated, delta function number 6, is less than a factor of 10 x/T0 from both delta function 5 and delta function 8, so delta function 6 is eliminated. Delta function 10 is less than 10 from delta function 8, so delta function 10 is eliminated. Delta function number 12 is marked to remain because it is more than 10 x/qO from delta function 8, and delta function 14 is marked for elimination because it is less than 10 x/l-0- from delta function 12. In succeeding cycles of evaluation, the remaining delta functions are marked for elimination because of their proximity to delta functions 5, 8 or 12. The net result of the procedure to this point is shown in the last column of Table 5. The second part of the process is the addition of the concentrations of the delta functions marked for elimination to the concentrations of the remaining delta functions. S L through S 4 are added to $5, and SL3 and $14 are added to $12, the nearest remaining delta functions. $6 and $7 must be split between delta functions 5 and 8 in proportion to proximity. $6 is split in a way which preserves the mean binding properties of delta function 6 after its elimination. The two equations below show the concentrations added to delta functions 5 and 8. To
Ss = S~'log(Ks/K6)/Iog(Ka/Ks)
(A-I)
to
$8 = S6"Iog(K6/Ks)/Iog(Ks/Ks).
(A-2)
Similarly, delta function 7 is split in fractions given in equations A-3 and A-4. To
$5 = S7"log(Ka/kv)/log(Ka/ks)
(A-3)
to
Sa = S7"log(K7/Ks)/log(Ks/ks).
(A-4)
In mum delta delta
this way, a good initial approximation to the minim distribution is generated. The concentrations of functions 9, 10 and 11 are similarly split between functions 8 and 12.
Heterogeneity of Antibodies This process is by no means sufficiently accurate for this initial approximation to the minimum m P D F to be used without further refinement. The chief benefits of this delta function elimination process are that only a relatively small
883
number of additional refinement cycles are required and usually the number of delta functions found is the minimum sufficient number.