Numerical method for equilibrium fluorescence polarization

ANALYTICAL

61, 528-537 (1974)

BIOCHEMISTRY

Numerical

Method

for

Fluorescence

Polarization

DARIUS .%X96 Raymell

Equilibrium

MAVIS

Drive, San Diego, California

HARRIETTE Lije Sciences Department,

6WB

C. SCHAPIRO

California

State University

at San Diego

AND

WALTER Biochemistry

Department,

B. DANDLIKER

Scripps Clinic and Research Foundation, La Jolla, California

Received January 4, 1974; accepted April

29, 1974

A computer program for making first approximations of equation constants, and for fitting the equilibrium fluorescence polarization equation to data by successively improving the constant values is described. The techniques used in making the first approximations represent a substantial improvement over methods previously available.

The polarization of fluorescence has been used for determining thermodynamic parameters of the antigen-antibody reaction under equilibrium conditions (l-6). The mathematical derivation applies equally to systems with uniform and with nonuniform binding affinities (2-4). However, when the antibodies have nonuniform binding affinities, the descriptive equations cannot be transformed into a closed form suitable for least squares analysis. This makes direct calculation of the equation constants impossible in the case of nonuniform binding, but other indirect methods can be used. Most, frequently, the equation constants are estimated graphically and then refined by successive approximations. Stepping functions which perturb each constant to slightly higher and lower values are used to improve the values of the constants. The root-mean-square of the deviations of the observed of the goodness-of-fit.

values

of p, about

the fitt.ed

The method of stepwise convergence is frequently 528 Copyright @ 1974 by Academic Press, Inc. All rights of reproduction in any form reserved.

curve

is the measure

unsuccessful because

MAKING

FIRST

APPROXIMATIONS

529

first approximations of the equation constants oft’en may not be obtained with sufficient accuracy. Furthermore, the calculation of Ffi from Eq. (2) below, is usually difficult because the function is ill conditioned in the vicinity of its real root. The present paper describes methods for making first approximations of the equation constants, and for fitting the general equations to the data. Two principal equations describe the model. In Eq. (1) the molar concentration of fluorescence-labeled material added to the solution being Gtrated is the independent variable Mi. Polarization pi, and fluorescence quenching Qi are dependent variables and are measured after each addition of the labeled material. Equation constants pr and Q, are determined by direct measurement, and constants pb and Qb are approximated using methods described below. Variable Ffi, the molar concent,ration of unbound fluorescence-labeled material, is calculated from Eq. (1) after the constants have been obtained. Pairs of Mi and F,i values are then used to make accurate first approximations for the three constants in Eq. (2)) the average binding const.ant K,, the molar concentration of binding sites on the unlabeled molecule Fb ,I,ax, and the index of binding affinity uniformity a. After all the const’ants for Eqs. (1 and 2) have been estimated, the values are progressively refined to fit calculated polarization values to the observed pi. Numerical values for a, K,, and Fa m,ax are the end result of the computation. The derivation of Eqs. (1 and 2) is based on two assumptions, namely, that the molecules in a solution fluoresce independently and the observed intensity is simply the sum of the intensities of the individual molecules, and second, t,hat molecules bearing a fluorescent label can be grouped into two distinct categories: those that are bound and those that are free. Both assumptions may be argued, but considering the extremely low concent8rations of reactants, they constitute the simplest and most realistic approach to the model. The derivation is covered in detail b\ Dandliker et al., in an early paper (2). EQUilTIONS

The general equation

has the form

pi = CP~Q~ - PbQbIFfi + PbQdMi, C&f - &b)F,i + Q&i

(1)

where, pi is the polarization of fluorescence (dependent variable), Mi is the molar concentration of labeled material (independent variable), and the constants are pf and Qr, the polarization and quenching, respectively, of molar fluorescence of the unbound labeled molecule ; and pb and Qb,

530

MAVIS,

SCHAPIRO

AND

DANDLIKER

the fluorescence and quenching of the ‘bound labeled molecule. Quenching (molar fluorescence), Q-subscripted, is the ratio of the fluorescence intensity divided by the molar concentration of the fluorescence labeled molecule. i is an indexing variable. Ff; is the molar concentration of the unbound fluorescence-labeled material and can be expressed in terms of Mi and three additional constants; a, the binding affinity uniformity index; K,, the average molecular binding constant; and Fa mx, the molar concentration of binding sites of the unlabeled component (assumed to be antibody in the ensuing discussion) : l/a 1 Mi - Ffi Ffj E (2) illj + Ffi K,’ F bmsx Ffi is also found in the identity Mi SEFfi + Fbi,

(3)

where, Fbi is the molar fluorescence of bound labeled material. FIRST Group

1:

Qf, Qb,

pf,

and

APPROXIMATIONS

pb

Both Q and p must be determined once for completely free and for completely bound labeled material. &f and pf are obtained by directly measuring the labeled component alone. However, &b and p. cannot be measured directly, since at least a small amount of free labeled material would exist at equilibrium with bound labeled material under any experimental conditions. To approximate pb Eqs. (2 and 3) are combined and rearranged into the form

Flli

F&-i = KoVtmax

- FtJ.

For simplicity, the constant a in this expression is assumed equal to unity. With this assumption, Eqs. (1 and 3) can be combined with Eq. (4) t.0 give

If M approaches zero, then Fb necessarily does, too, and the polarization p’. Thus, for the limit M--jr 0, Eq. (5) can be rearranged into linear form: pi, approaches some limit

p’

=

pb

-

Qge,

(6)

MAKING

FIRST

APPROXIMATIONS

531

In practice, several titrations with different concentrations of the unlabeled component are performed. To estimate pb values of p’ would be plotted against (p’ - pf)/Z, where 2 is the concentration of the unlabeled component expressed in any convenient units, and pb fomld at the intercept as l/Z* 0. The extrapolation method for pb just described can be computer simulated by expressing pi as a power expansion in terms of Mi. Where there are n data pairs, pi and Mi, p’ is the constant term in the following set of power expansions. p, = p’ + c&f, + . . . + c,-$fp-’ pi = p’ +

+ . . . + c,-lh4ine1

~lh4i

(7)

p, = p’ + clhln + . . . + cn - 1114n n-1.

The Gauss-Seidel method described by McCracken (7) was used to compute the coefficients, In performing the linear extrapolation to find pb, p’ and (p’ - p,)/.Z are treated as dependent and independent variable, respectively, and pb is approximated by computing the constant when the independent variable equals zero. The method for approximating Qb is analogous to that for pb. Changes in fluorescence intensity, Q, due to molecular combination are assumed to follow Eq. (8) (2).

Q’ for each titration is obtained by expressing Qi as a power expansion in Mi. The constant term in the expansion is Q’. Equations (4 and 8) are combined to give Eq. (9), again assuming the binding affinity uniformity index a equals unity, and the linear extrapolation is performed. Q’

=

Qb

+ !k/. b max

(9)

Q’ and (Qf - Q’)/Z are treated as dependent and independent variables, respectively, and Qb is approximated by computing the intercept. Group N: a, K,, and Fb ma+ Previously investigators have approximated values for a, K,, and Fbmax, using graphical methods. The techniques have varied depending upon the extent of the binding affinity nonuniformity, but when

532

MAVIS,

SCHAPIRO

AND

DANDLIKER

>>I a > 0.5, errors in the first approximations can be large, causing stepwise least squares fitting efforts to fail. An improved method for approximating a, K,, and Fbmax with greater accuracy than was previously possible has been developed based on techniques that are practical by computer. With this method, Eq. (2) is solved for Fb maXin terms of a, K,, F,,i, and Ffi. Since the expression is equally valid in terms of a, K,,, Fbi+l and Ffi+l, Fa mazmay be eliminated from expressions in terms of any two consecutive values of Ff, Fb, and the following relationship obtained.

Inherent in the Sips’ distribution model is the notion that 1 5 a > 0. A set of values between 0 and 1 is chosen for a, and for each a, a set of K, values is calculated for each consecutive pair of Fb and Ff. The normalized relative deviation, 8 is computed for the set of K. values at each a, and the a for which 6 is minimum, is the first approximation for that constant. This is shown in Fig. 1. With a known, K, and Fb max can be calculated.

N = the number of Fb, F, data pairs. Least Square Fitting Fluorescence polarization data are fit to Eq. (1) in the least squares sense, with Ff given by Eq. (2). To find the best values for the seven equation constants, each of the first approximat,ions is perturbed to a slightly larger, and to a slightly smaller value. Polarization values are computed from all possible combinations of the perturbed equation constants, and the quantity S given by s _ 100 p exp - p talc ’ 1’2 (12) R (( >> P exp is calculated for each combination. Values for the equation constants giving the smallest value for S are perturbed with an increment half the size of the previous one, and the 5’ values for all combinations are again computed. The process is repeated until the uncertainty in the equation constants is less than some preselected value. Equation constants have been determined with precision of less than l%, using this method on simulated data.

MAKING

FIRST

.5

533

APPROXIMATIONS

.6

.7

.6

.S

a FIG. 1. Graphical depiction of the method for approximating a. The normalized relative deviation in KO,denoted 6 in Eq. (ll), is plotted against a to determine the a for which 6 is minimum. This estimate of a, and the corresponding Ko and Fbrnlx are the first approximations used in the least square analysis of fluorescence polarization data.

F, Calculation Pf is computed frequently from Eq. (2) during the least square fitting routine. The function is ill-conditioned, having a singularity in the vicinity of the real root. The subroutine in Fig. 2 limits the root search to the real root and avoids the singularity by using progressively decreasing increments in successive trial values for Ff. Ff is refined until agreement between computed and trial values agrees within lo-*, or until the increment in the trial value is smaller than the truncation limit of the computer. Results and Use of the Program. A set of simulated data was constructed to test the operation of the program. Polarization “data” were computed from preselected values for molar&y, M, and the equation constants. Certain polarization values

534

MAWS,

SCHAPIRO

AND

DANDLIKER

START

FIG. 2. The F~s calculation logic diagram. F 16 is computed from M,, a, K,, and Fa mnT, using EIq. (2). When MC is large relative to Fa m.x, the solution lies in the neighborhood of a singularity. The solution is obtained by approaching the root stepwise from the direction opposite the singularity, while simultaneously reducing the step size.

were altered in order to simulate noise and experimental error. The input list consisted of molarity, quenching, and polarization values, the relative antibody concentration, quenching, and polarization “measurements” for the free fluorescent molecule,, and the maximum allowed relative deviations in the fitted equation constants. The program was written in FORTRAN 63 and required a total compile and execution time of less than 5 min on a commercial CDC 3600 computer. The program and data were read from cards. The printout was made on a 120-column off-line printer and listed equation constants; the root-mean-square deviation in p, 8; and the relative antibody concentration and sets of antigen molarity, observed polarization, and calculated polarization values for each titration. Table 1 compares ideal, input, and fitted values used to test the pro-

MAKING

FIRST

535

APPROXIMATIONS

TABLE 1 Simulated Fluorescence Polarization Data Including Ideal Data, Ideal Data wit,h Simulated Experimental Error, and Polarization Values Computed from the Optimized Equation Constants

Relative concentration of unlabeled mat,erial

Ideal Molarity of labeled material x 10’0

1.0

6.672 11.01 15.34 24.90 35.75 62.31 98.23

1.5

2.0

nolarization computed from const,ants shown below 0.156288 0.151655 0.148345 0.143104 0.138835 0.131704 0.125568

0.1599 0.1517 0.1483 0.1431 0.1388 0.1317 0.1257

0 15694 0.15225 0.14887 0.14347 0.13903 0.13156 -*

7.444 10.16 15.852 25.021 31.558 65.95 103.25

0.1675 0 164935 0.160898 0.156183 0.15351 0.143521 0.136303

0.1675 0.1649 0.1609 0.1562 0.1535 0. 1435 0.1363

0.16779 0.16527 0.16126 0.15653 0.15382 0.14357 0 .1360.5

5.756 11.97 25.342 39.90 55.75 92.31 113.81

0.176776 0.171712 0.165181 0.160241 0.155942 0.148142 0.144411

0.1768 0.1717 0.1652 0.1603 0.1559 0.1481 0.1441

0.1766 0.17173 0.16533 0.16041 0.15608 0.14814 0.144257

0.2000 0.1000 2.000 0.75 1 x 10s 1 x 10-a *

F, could

Simulated polarization (adiueted ideal Fitted values) used as polarization values comput.ed from program input data constants shown below

not be computed

(see Fig.

0.197442 0.099375 2.0833 0.743688 1.1138 X lo8 1.0095 x 10-s 0.101

2).

gram. Observed and calculated values for polarization plotted against molarity in Fig. 3 show there is good agreement between computed and measured polarization. In practice fluorescence polarization can be measured to within + .002, and sometimes to within f.001. If the simulated data had actually been

536

AMAVIS,

SCHAPIRO

AND

DANDLIKER

.I2

*I

0

B”

20

moloritY

(x10”)

Fm. 3. Results of computer program use on simulated fluorescence polarization data. Polarization values calculated from Eqs. (1 and 2) () agree well with simulated polarization data (A), to which the equations were fitted. With the equation constants at their optimum values, the polarization at one data point (A) could not be calculated because the solution to Eq. (2) lay too near the singularity.

obtained from a real experiment and the maximum error in the polarization measurement. were +.OOl, then for the experiment to be “valid,” the program must be able to fit the data so S z 0.145. A more stringent criterion would be imposed by requiring the program to fit the data to within the root-mean-square of the deviations in the simulated data relative to the “ideal” values; in this case S must not exceed 0.1109. The result of program use on the synthetic data was S = 0.101, which is within the limits of precision required by either of these criteria. When the program was executed with real data, limitations were identified which had not been discovered using the simulated data. For example, when the fluorescent intensity is unchanged with increasing concentrations of the labeled component, Qf,/Qb = 1, the linear extrapolation to find Qb fails, since there is no intercept. The problem is overcome by examining the slope of the linear equation from whose intercept &a is determined; if the slope approaches zero, then Qb is set equal

to Q, and the extrapolation

Complications

is not performed.

also arise if Mi >> F bmax; pb may not be approximated

accurately, resulting in difficulty K O, and Fb max; and the subroutine

in making first approximations for a, used to solve Eq. (2) may not converge

MAKING

FIRST

APPROXIMATIONS

537

to give Ff, because as Mi increases relative to Fb n,ity, the solution lies close to the singularity. The computation of Ff from Eq. (2) can be accomplished in several ways when ML is large. The best method is t-o use double precision mathematics in the subroutine where Ff calculation occurs (see Fig. 21. Double precision does not facilitate making first estimates for pb, however, so to ensure pb and Ffi determinations can be made and to give the greatest accuracy in the final values for the equation constants of prime interest, a, K,,, and Fa max, it is usually necessary to obtain data for which and Fbmax may not be known a priori, Mi ? Fb max. Unfortunately, several titrations in different concentration ranges may be necessary bcfore suitable conditions are obtained. Requests for copies of the computer program should be directed to Harriette C. Schapiro at California State University at San Diego, San Diego, California. ACKNOWLEDGMENTS This work was supported in part by National Science Foundation Grant, Xo. GB31611, and by a Grant from the San Diego State University Foundation. REFERENCES W. B. AND DE SAUSSURE, V. A. (1970) Immunochemistry 7, 799-828. W. B., SCHAPIRO, H. C., MEDUSKI, J. W., ALONSO, R., FEIGEN, G. A., AND HAMRICK, J. R. (1964) Immunochemistry 1, 165-191. 3. DANDLIKER, W. B. (1971) in Methods Immunology and Immunochemistry (Curtis A. Williams and Merrill W. Chase, eds.), Vol. 3, pp. 43562, Academic Press, New York. 4. KIERSZENBAUM, F., DANDLIKER, J., AND DANDLIKER. W. B. (1969) Immunechemistry 6, 125-137. 5. DANDLIKER, W. B., HALBERT, S. P., FLORIN, M. C., ALONSO, R., .4ND SCHAPIRO, H. C. (1965) J. Erp. Med. 122, 1029-104s. 6. PORTMANN, A. J., LEVISON, S. A., AND DANDLIKER, W. B. (1971) Biochem. Biophy,~. Res. Commun. 43, 267-212. 7. MCCRACKEN, D. D. (1965) A Guide to FORTRAN Programming, John Wile> and Sons, Inc., p. 71, New York. 1. DANDLIKER, 2. DANDLIKER,