A computer program for determination of equilibrium constants and binding site concentrations in radioimmunoassay systems

Computer Programsin Biomedicine 12 (1980) 19-26 © Elsevier]North-HollandBiomedicalPress

A COMPUTER PROGRAM FOR DETERMINATION OF EQUILIBRIUM CONSTANTS AND B I N D I N G SITE C O N C E N T R A T I O N S IN R A D I O I M M U N O A S S A Y SYSTEMS A. FAURE *, C. NEMOZ **, B. CLAUSTRAT and J. SITE * D~partement Pharmaceutiquede Biophysique, Math~matiqueet lnformatique, Universit~ 'ClaudeBernard', 8, Avenue Rockfeller, 69008 Lyon and ** Laboratoire d'Informatique Mddicale (Pr. SITE], H6tel Dieu, 1, Placede l'H6pital, 69288 Lyon Cedex 1, France

We propose a method allowingthe automatic determination of equilibrium constants and binding site concentrations for an immune serum composed of a series of homogeneousantibody subpopulations. It is then possibleto decomposefile 'Scatchard plot' in its straight components correspondingto each antibody subpopulation. Bindings i t e

Computerprograra

Equilibriumconstant

1. Introduction

subpopulations. Given N experimental points (F, B/F), this computer program fits/~/F versus F accord. ing to eq. (1) (forp = 1,2, 3). The parameters are estimated in each case and the Scatchard plot [8] is decomposed in its straight components corresponding to each antibody subpopulation. Each of the calculated binding curves is compared with the experimentally obtained binding data, by calculation of the sum of the squares of the distances from the real experimental points to the calculated points: we select the better of the three models proposed.

It is well documented that the antibody produced on injection of an antigen is heterogeneous in its affinity for the antigen [1,2]. If one assumed the antibodies contained in an immune serum to be composed of a series of'subpopulations" with differing affinities with no inter- or ir,tra-molecular interactions, then for an antibody sample containing p subpopulations, we have under equilibrium conditions [3.4]: B

P

Seatchardplot

K,Ai

where B = the equilibrium concentration of bound antigen (M) F = the equilibrium concentration of free antigen (30 Ai - t~e total ~:oncentration of binding 'sites' for the ith anti'3o4y swJbpopulation (21/) Ki = equilibrium constant for the ith antibody subpopulation (M-1) p = the number of antibody subpopulations The determination of the parameters Ki, Ai, is an objective of great practical importance: several methods have been proposed for such studies [5-11 ]. We have developed a computa~tonal method to determine the parameters K~, Ai, for an immune serum composed of 1,2 or 3 homogeneous antibody

2. Mathematical framework and computational method 2.1. Data collection A fLxed quantity of a determined immune serum reacts with N different con cf~*:ations of an antigen. Under equilibrium conditions, we note B/and F/the concentration of bound and free antigen, respectively. We have then N experimental points (X/, Y/) with: (~= l , N )

19

20

2.2. Determination o f equilibrium constants and binch'ng site concentrations According to eq. (1), we can write: ,P

by an iterative least squares method: the N experimental points are used now. Formulation

If

KiAi

x'

2.Z1. C a s e p = i In the case involving an univalent antigen reacting with a single order of antibody combining sites, the Scatchard plot is linear. The parameters A ~, K~ are easily calculated by weighted linear regression.

2.2.2. Case p = 2 We suppose now that the immune serum is composed of two types of mutually non-interacting binding sites. We have then to estimate four parameters A b K:, A2, K2. The method involves two steps: Step 1: Initial estimation K~°), At °), K~°), A~e) of the parameters. We choose at random 4 points from the N experimental points and we write eq. (1) for these 4 points (XI, Yi), ~ - 1,4). We have to solve the four simultaneous linear equations: {. +

-

-

rj = rj

/=1,4

a = K (°)a (o) + K(O)A (o) 1 "11 2 2 = K(O)~.(O)(A (o) + "A2 (o)~I ! s~2 ~" 1

v = lqo) + it(,°'

I

The existence and uniqueness of the somtions of this linear system have been proved [ 10]. ~t follows that ~, ~, 7 and 6 are uniquely determined. Then we note that K~°) and K~°) are the roots of the second order equation: Z2 -7Z+6

2

t=l 1 + KjX

=g(a,, a2, a3, a4, X)

We want to determine al, a2, an, a4 by minimizing the weighted sum of residual squares. N

S =~

w/fi2(a)

minimum

(2)

i=1

with = rj - g(al, a v a3, a4, x j )

wi

0" = l , N )

= weight affected to each point (it is possible to weight each point by a quantity which is in inverse ratio to its variance)

Under some conditions, we have then to resolve the system:

with

v

Then

=0

The parameters A ~o) and A (20) occur linearly in the expression of(~ and ~, and are easily calculated. Step 2: The second s[ep improves on the provisional ~stimates of parameters K~°), A~°), K(2°), A(2°)

(3)

(J(a))tF(a) -- 0 where

F(a) = ((wi~(a)k t)

: aai !,.jI aai I~,i'

matrix (N × 1) Jacoby matrix (N X 4)

weighted Jacoby matrix (N X 4)

Itemtive method o f resolution o f eq. (3) The approximative values obtained in the first step are improved by iteration:

a(r+l) = a(,) + ~(r)

(4)

I

Data

Caiiect

I

21

c

Calculation

of linear

regiession

IChoice of 4 (or 6) experimental to calculate

L

estimation

Ki*’

,

points

(initial

Ai”

of Ki Ai)

I

I

Subroutine ________Calculation

Iterative

Fig. 1. Flowchsr!,

of Kc” i

u ,

Ai”

least-squares I

for

F = 3

method /

22 a (P÷t) and a (') are values of the parameters obtained at the (r + l)th and rth iteration. The matr;.x ,5(r) is calculated by:

6(') : -[M(a('))i-' (J(a(r)))tF(a) with

K 1(°) and A t(°) ~,'6 - 1,3) for the model with three subpopulations; then it comes back to the F2 subroutine PLOT - it plots expelhnental and calculated points, and model curve with the decomposition in its two or three straight components

M(a(r)): (J(a(r)))rJw (a(r)) 4. Sample run r

4.1. Input We use the Taylor development of system (eq. (3)) to prove the iterative fommla (eq. (4)). Number of iterations is fixed before calculations, and we select the step where the weighted sum of residual squares is minimum (eq. (2)). Calculations are available in MIS [ 11].

2.2.3. Case p 3 :

Under the hypothesis of three antibody subpopulations, we have to determine six parameters At, Ki (i = 1,3). We calculate a first estimation by solving a set of six simultaneous linear equations and by the resolution of a 3th order polynomial by the Newton method (the Newton method is useless in the case p : 2). In a second step we improve on the provisional estimates of the parameters by the same iterative least squares method.

The coordinates of each experimental pgint are recorded in the calculator "by hand" or with magnetic tape. We have three options for weight: ! - No weight - Weights are recorded with coordinates of each point - Weights are calculated by the program as the inverse ratio of the variance of each point 4.2. Output

We present a first example with 1 ! points (F, B/F) from anti-aldosterone antibody (fig. 2). We obtain a model with 2 antibody subpopulations. We note no significant difference between nnweighted (fig. 3) and we+ghted (fig. 4, 5) method [ 12].

3. The program

Determination of and b t [ f i d I n g s i t e

Main program reads and stores experimental data, and calls subroutL-tds which calculate each model (fig. i).

- Essay 1

im ~lm~am

.uw~

1 2 3 4 5 6 7 8 9

~ o

Im am . m l ~ a m

o o

o

am ~ul

I

,mlr.lm

Experimental

Subroutines F 1 - it calculates weighted linear regression of the model with one antibody subpopulation F2 - it calculates first estimates of parameters K(O) i .~,.d . . . .~. i(o) (i- = 1,2) for the model with two antibody subpopulations and h-nproves on the provisional estimates by the iterative least squares method for the model with two or three subpopulations F3 - it calculates first estimates of parameters

o ~ . m

mamlmm

eauilibgiua constants concentrations

P 0.0270 0.0540 0.0858 0.1254 0.1629 0.2440 0.3243 0.4899 0.6993

10

1.0140

11

1.3646

Fig. 2. Essay 1: data.

Points B/P 2.6000 1. 7380 1. 3080 0.9770 0. 8270 0.6196 0.5288 0.4162 0.3329 0.2720 0.2345

B 0.0702 0.0939 0.1122 0.1225 0.1347 0,1512 0.1715 0.2039 0.2320 0.2758 0.3200

o

23

Case P-2

Case P - 2

Weight I Height 2 Height 3 : NoUght 4 Height 5 Weight 6 8 Height 7 : Height 8 Height 9 " W e i g h t 10 H e i g h t 11 -

1.0000 1.0000 1.0000 1.0000 1.0000 1. OO00 1.0000 1.0000 1.0000 1.0000 1.0000

Height 1 : Weight 2 : Weight 3 Height 4 : weJ.ght 5 = lleigbt 6 : Weight 7 = Weight 8 " Weight 9 e H e i g h t 10 : W e i g h t 11 =

index-n,mber of exp.pts choosed for initial estimation 1 1 2 2 3 10 4 11

la]5

gl AI

45.1473 0.1182

K2 A2

0. 2192 0.8913

Variation between (exp. -calc. ) points 1 2 3 4 5 6 7 8 9 10 11

0.00040 -0.00703 0.02087 -0.01446 -0.00057 -0.00998 O. 00519 0.00892 -0.00035 - 0 . 00193 -0.00112

W e i g h t e d sun o f r e s i d u a l Pop. i 1 2

~ndex-nusbec ¢hoosed for =--I~ 1 2 2 3 10 :~ 4 11

tal 3

:

Calculated B/F 2.5,9960 1, 74503 1, 28713 0, 99146 0.82757 0,62958 0. 52361 0.40728 0.33325 0 . 27393 0. 23562

points B 0.07019 0.09428 0.11041 0.12431 0.13479 0.15364 0. 16982 0. 19953 0.23305 0. 27775 0. 32153

squages : 0.0009050

0.0029 0.0114 0.0284 0.0649 0.1078 0.2432 0.4012 0.8312 1.5760 2.930? 4.71:8 of exp.pts initial estimation

gl Al

46.4714 0,1163

K2 A2

0,2629 0.7784

Va:iat ion between -=: (exp.-calc. I points 1 2

-0.00014

-0.00319

3 4

0.02400

-0.01279 ~ . 00006 - 0 . 01067 O. 0 0394 0.00749 -0.00132 -0.00190

S 6 7 8 9 10 U

-0.00002

~lculated BR

points S

2.60014 1. 74119

0. 07020 0. 09407

1.28400

0.11014

0.98979 0. 82694 0.63027 0. 52486 0. 40871 0.33422 0.27390 0.23452

0. 12410 O. 13469 0.15380 0. 17022 0. 20023 0.23372 0.27772 0.32003

W e i g h t e d sum o f ~ e s L d u a l s q u a r e s

: 0.0001210

KiAi 5. 3 3 7 5 4 O. 19534

1 2

5.40439 0.20467

Fig. 3. Essay I : case p : 2 (without weight),

Fig. 4. Essay l : case p = 2 (with calculated weights).

The second run presents a model with 3 antibody subpopulations. We had to calculate experimental points in this case (fig. 6-8).

kilobytes) with magnetic tape and plotter. Programs are written in specific language of the calculator: they request 2420 bytes max. Data and parameters request: 152 N + 680 bytes (N is number of experimental points). Computer programs with method diagrams, listing and sample runs are available at the 'Departement Pharmaceutique de Biophysique' at the above-mentioned address.

5. Hardware and software specification

We use a Hewlett Packard 9825-A calculator (16

24

ESSRY I

P :

2

E/F=H(E) + CRLCUL. PTS X EXP. PTS

| .B~ ~

I. B¢] I .qll 1.18 B.gq il.7n

R.I~BU B.B29

B.B[B

m EBB m.I

Fig. 5. Graph of essay 1.

~n~e~mLuation.of equLlibzium constant~ p z m z x n 9 szr~e c o n c e n t r a t x o n s Essay 2 •eqw

~ , ~ l m

qw~

~

Experimental

gmma,~

m ~ , m

Point;s

1 2 3 4

0.0184 0.0316 0.0450 0.05v5

6.402.0 s.3020 4.61v0 4.16e0

0.1].~8 0.1675 0.20'78 0.23s7

5

0.0693

3. 8 4 2 0

0.2663

6 7

o.090z o.1118

3.4060 3.0'700

0.3069 0.3432

8 ~_9= 10 1i

0.1541 0.1881 0.2893 0 . 3 577

2.6060 2.3400 1.8290 1. 6080

0.4016 0.4402 0.5291 O. 5752

Fig. 6. Essay 2: data.

B. lYE Z. 17;" E. 2iag m.23q

B.2E-3 B.PJ2

B ~22

B.

25 Case P-3

Weight 1 = .f$ggE ; c = Mistc; 4 = ilcijht 5 * .migbt 6 v&Gght 5 = j#&iaht 8 Weigbt 9 might LO Weight 11

* =

1.0000 l.OOO? I.f?l?(!!. l.OI;C’I

1.0000 1.0000 1.0000 LOO00 1.0000 1.0000 1.0000

inderaumber of exp.pts fat initial estimation

s8wsed

Variationbetween Calculated (exp.-calc.)pointe B/e 1 -0.00004 2 0.00029 3 -0.00037 .: -0.00017 0.00036 6

-0.00023 0~00039 B' 0*0001B -0.00046 0.00001 0.00013

k

3 .: 1'0 11

6' Cal4 z K2 A2 K3 A3

7.2969 0.4093 68.9332 O.lO;LB 0.9952 0.6909

6.482 S.8 s.238 Y.SsE 4.074 3.492 2.510 2.3a 1.746

I.164 : PI.

SEE’

0.0El0

Fig.

t

8.Graph of essay 2.

6.40204 5.30171 4.61737 4.16817 3.84164 3.40623 3.06961

0.11780 0.16753 0.20778 0.23967 0.26623 0.3069G

2.60582 2.34048 1.82899 1.60787

0.40156 0.44024

Heighted sum of residual squares 8

2’

p =3

points B

0.34318

x*. EE

26 References [ 1] A. Nisonoff and D. Pressman, Heterogeneity and average combining constants of antibodies from individual rabbits, J. lmmun. 80 0958) 417-428. [2] L. Pauling, D. Pressman a~td A.L. C~rossberg,The serological properties of sire .qe substances. VII - A quantitative theory of the inhibit-ion by haptens of the precipitation of heterogeJ~eous esti-sera with antigens, and comparison with experimental results for polyhaptenic simple substances and for azoprotcin, J. Am. Chem. Soc. 66 (1944) 784-792. [ 3 ] T.P. Werblin and G.W. Siskind, Distribution of antibody aft-mities: technics of measurement, lmmunochemistry 9 (1972) 987-101 !. [4] T.K.S. Mukkur, M.R. Szewczuk and D.E. Schmidt, Determination of total affinity constant for heterogeneous hapten-antibody interactions, lmmunochemistry 11 (1974)9-13. [5] O.A. Roholt, A.L. Grossberg, Y. Yagi and D. Pressman, Limited heterogeneity of antibodies. Resolution of hapten binding curves into linear components, Immunochemistry 9 (1972) 961-965.

[6] M. Klotz and D.L. Hunston, Properties ~ffgraphical representations of multiple classes of binding sites, Biochemistry, Vol. 10, No. 16 (1971) 3065. [ 7 ] S.l. Rubinow, A suggested method fog the resolution of Scatchard plots, lmmunochemistry 14 (1977) 573576.

[8] G. Scatchard, The attractions of proteins for small molecules and ions, Ann. New York, Acad. Sci. 5 I (1949) 660-672. [9] H. Linewcaver and D. Burk, Tl~.edetermination of enzyme dissociation constants, J. Am. Chem. Soc. 56 (1934) 658-665. [ 10] H.E. Hart, Determination of equilibrium constants and maximum binding capacities in complex in vitro systems, Bull. Math. Biophys. 27 (1965) 87-98. [ 1 ! ] C. Nemoz, A. Faure, B. Claustrat and J. Site, Calcul des constantes d'~quilibre et de la concentration en sites anticorp~ dans les syst~'mes antig~nes - anticorps, No. 17 MIS, Lyon, 1979. [ 12] J.A. Jacquez, F.J. Mater and C.R. Grawford, Linea~ regression with non constant, unknown error variances. Sampling experiments with least squares, weighted least squares and maximum fikelihood estimators, Biometrics 24 (1968) 607.