Immunochemistry, I977,

A

Vol. 14, pp. 573-576.

Pergamon Press.

SUGGESTED

Printed in Great Britain

METHOD OF

FOR

SCATCHARD

THE

RESOLUTION

PLOTS

S . I. R U B I N O W Graduate School of Medical Sciences, Cornell University, New York, NY 10021, U.S.A.

(First received 4 February 1977; in revised form 31 March 1977) Abstract--A simple scheme is proposed for analysing Scatchard curves that represent the equilibrium binding of a ligand to a macromolecule containing two or more different kinds of binding sites• The principal merit of the proposal is the utilization of a least-squares fitting procedure that is linear. The method is illustrated by the consideration of previously published data representing ligand binding with bovine serum albumin and with IgM.

INTRODUCTION

The equation representing the equilibrium binding of a ligand to a macromolecule containing many binding sites is represented canonically by the equation

• r =

ajc j

j=l

,

(1)

1 + ~ bjc j j=l

where r is the number of ligand molecules bound per macromolecule, n is the number of binding sites per macromolecule, c is the ligand concentration, and aj and bj are positive constants. A similar equation represents studies of enzymatic reactions, with the initial velocity v replacing r. At the present time, the method of choice in the presentation of the raw data of such studies is the Scatchard plot (Scatchard, 1949), in which r/c is plotted as a function of r. An objective of great practical importance in either equilibrium or initial velocity studies is the determination of the parameters aj, bj and n (Pauling et al., 1944; Sips, 1948; Karush & Sonenberg, 1949; Wyman, 1948; Karush, 1956; Nisonoff & Pressman, 1958; Klotz & Hunston, 1971; Endrenyi et al., 1971; Roholt et al., 1972; Werblin & Siskind, 1972; Mukkur et al., 1974). In this note we propose a simple scheme for determining these parameters from the consideration of the Scatchard plot.

in the first quadrant of the s-r plane, originating at the point (So, 0) and ending at the point (0, r~ ), where so=lims,

r~ = l i m r . c~

c~O

(3)

We assume here that, for a given investigation, r is determined experimentally as a function of c, so that s is likewise experimentally determined. Suppose for the moment that r~ can be found from the Scatchard plot. We note that from equation (1) this quantity fixes the value of an

--

bn

=

r~.

(4)

We form the function roo - - r u -

(5) S

It follows from equations (1)-(3) that u is essentially of the mathematical form of equation (1), n--1

r~ + ~ (r~bj - aj)ci u= j= 1_ 1

(6)

al + ~ aj+l Cj j=l

The advantage in changing variables from r to u is that n is replaced by n - 1. This reduction is most important when n = 2, because u is then linearly related to u/c, the same advantage that originally motivated the introduction of the Scatchard plot for I. F O R M U L A T I O N the case n = 1. We could, in fact make u to be exactly Let us define Scatchard's function of the form of equation (1) with n - 1 replacing n by defining u = (r~ - r)/s - r~/so. However, such a n-1 transformation requires knowledge of So as well as E aj+ i cj r~. As we shall see, such knowledge is unnecessary. r j=O s (2) Of course the transition from r to u requires knowlc ~ bjd edge of r~. In practice, it is often difficult to measure j=O r at very large (as well as very small) ligand concentrations, which makes the determination of r~, by As is well known, as c varies from zero to infinity, extrapolation uncertain. A straightforward applia curve which we call the Scatchard curve is traced cation of the method of least-squares to equation (1) 573

574

S. I. RUBINOW Table 1. Derived values of parameters Data for ligand binding with

Derived

Bovine serum albumin (Karush, 1950) IgM (Onoue et al., 1968)

Assumed r~

nl

22 19 9 10 11

4.99 4.72 4.08 4.07 ° 4.26

n2

17.01 14.28 4.92 5.93 6.74

is possible, although it requires computer-aided computations, engendered by the resulting n o n - h n e a r equations for a s a n d bj. O u r proposal, a variant application of the least-squares method, concerns itself with the best determination of the function u, a n d leads to linear algebraic equations for the desired parameters. In fact, the introduction of a linear curve-fitting procedure is the principal merit of our proposal. Assume we have a data set (r~, c~) which we can utilize to define a Scatchard curve. F r o m it, assume that r~ can be found, at least approximately. Inasm u c h as it is constrained by the theory to be an integer, we shall utilize an initial 'trial' integral value of r~, a n d determine subsequently Which value is most successful in representing the data. E q u a t i o n (6) can be rewritten in the form u

ro~

c

atc

1 b - - ~ ajuc J - 2 alj= 2

n-1 j= 2

K2

(itM)- 1

0.257 × 10 - 2 0.357 x 10 -2 0.0293 0•0198 0.0131

X 10 - 2 × 10 - 2

E (~M)- 2 0.886 x 10 -5 0.587 x 10 -5 0.310 0.122 0.219

with the s u m m a t i o n extending over all points i of the data set (ri, ci). The e r r o r function E is to be minimized with respect to the 2n - 1 parameters, 5 s and fl~. By comparison of the residual expression in curly brackets in equation (8) with equation (7), we find that the parameters aj and bj are determined as 1 al = - - ,

bl =

51

as=-

(1 + i l l ) ,

0~1

roosj

51

, j = 2 , 3 . . . . . n,

bs = ~-1(,Ss - 5j), j = 2 .... , n - 1,

bn ~

Let us apply the least-squares m e t h o d to the determination of u/c from u a n d c. Thus, we define the error function E to be the sum of residuals

ci

9.76 10.25 5.97 6.24 5.31

1 n-~ - - ~ (r~b i - as)d -1 = 0 . alj= 1 (7)

[ ci

KI (#M)- 1

(9)

5n 5t

We emphasize that the quantity So is not required by the method, but in fact the 'best' value of So in the least-squares sense is determined as r~/:q.

t2

j

6O 6:

50

5~

40 s

4 s 3O

2O

'°

4

8

12

16

i

20

24

r

Fig. 1. Scatchard plot with data points representing the binding of bovine serum albumin by anionic dye (Karush, 1950). The solid line is the theoretical curve based on equation (13), an assumed value of r~, = 19, and the parameter values given in Table 1. The dashed curve, largely coincident with the solid curve, is similarly based on the value r~ = 22. The ordinate unit is 104M - 1

0

5

IO r

Fig. 2. Scatchard plot with data points representing the binding of 1-iodonaphthalene-4-sulfonate by IgM antibodies (Onoue et al., 1968)• The solid curve is theoretical, based on equation (13) with parameter values given in Table 1 for r,j = 10. The ordinate unit is ~ M ) - t

A Method for Scatchard Plots Resolution

575

U-alC I

"'?"-.~.......'", •

0

.

.

.

.

J

'

'

,

~'~.

I0

i

.

i

i

=

20

I

i

i

i

i

30

40

U-~ I

Fig. 3. The solid line represents the linear relationship between u - ~t and (u - ~q)/c, which is derived from equations (7) and (9) with n --- 2, r~o = ll, and parameter values given in Table 1. The circles represent the data points in such a plot. The dashed line is obtained when r~ = 9. The same data points are here represented by crosses.

We have applied the method in the next section to the case when n = 2. We have not applied the method to any example for n ~> 3. It is conceivable that variations in the procedure, such as the utilization of u-~ instead of u to reduce n by unity, could turn out to be more useful. The practical difference that arises in the choice between u and u-1 is that, in effect, different statistical weightings are given to the data points by different choices. A slight modification of the method is required when r is defined by an Adair function (Adair, 1925). Then aj = jb~,

r e = n,

(10)

and there are only n independent coefficients, bj appearing in equation (6). In this case the error function of equation (8) is to be replaced by E =

E~

~t,

"-'

~ - - - - (n - 1) -

i I ci

~ 0cj[(n - j)(ci) ~- 1 j=2

ci

-- jui(c~) J- 2] -- o~.nui(ci)~- 2~

(11)

J

and minimized with respect to the ~tj. F r o m their determination, we find n bl=--,

n~j bj=--,

OC1

j = 2 , 3 ..... n.

1I. APPLICATION TO BINDING STUDIES IN THE CASE n = 2 Karush (1950) investigated the binding by bovine serum albumin of the anionic dye p-(2-hydroxy-5methylphenylazo)-benzoic acid, and, inter alia, fitted the data by an equation of the form nlKtc

n2K2c

r = 1 + KI~ + 1 + K2~'

(13)

representing 2 sets of identical, independent binding sites, of integral numbers nl and n2, respectively. Here K 1 and K 2 are the association constants for the two kinds of binding sites. Karush tried different values of r.,,, chose s o = 52.8 × 10aM -1 by extrapolation of the Scatchard curve, and combined the resulting

o

i

)

(12)

~1

I0

20 U - ClI

Fig. 4. The same plot as in Fig. 3 for the case r~ = 10.

30

576

S.I. RUBINOW

constraints with the above equation evaluated at two particular points of the data set (ri, q). In this manner he found a good fit to the Scatchard curve with the parameter values nl = 4.82, n 2 --- 17.18, KI -- 9.93 x 104 M - 1, K2 = 0.29 × 104 M - i. Here we employ the paradigmatic method outlined above, which has the advantages of giving weight to all points of the data set, and avoiding the extrapolation procedure to determine So. We chose r~ = 22, initially as suggested by the analysis of Karush, but let all otl~er parameters be determined by the leastsquares method. By converting equation (13) to the canonical form equation (1), we see that n = 2 and ai = n l K x +

n2K2,

bl = K1 + K2,

a2 = (nl + n2)K1K2,

(14)

b2 = K 1 K 2 .

Minimization of the error function, equation (8), leads to the following linear algebraic equations for the determination of ~l, ct2, and ill,

~l Y, (c,)-2 + ~ 57 u~(ci)-I + fll Y, (c,)-i = 57 u,(c,)- 2, i

i

i

i

(~1 E Ui(Ci)- I -~- ~2 E (Ui)2 "~-fll E Ul ~--"E (Ui)2(Ci)- 1, i

i

(15)

i

cq E (c,)- i + a2 57 u, + fll Z 1 = 57 ui(q)- i,

i

i

with the solution in this case cq = 0.414, ~2 = 1.04, fix = 3.15. The corresponding parameter values, derived from equations (12) and (14), are shown in the first line of Table 1. The values shown there are similar to those found by Karush. We have also applied the method for neighboring choices of r~. In each case we computed the associated error, defined by equation (11). We find that a minimum in the error function occurs for the value r~ = 19. The parameter values associated with this case are also given in Table 1. Figure 1 displays the theoretical Scatchard curve based on equations (13) and (14), for ro~ = 19 and 22, together with the observed data points. Both curves are seen to give a good fit to the data points, and a rational decision as to which is better cannot be determined 'by eye' from the two curves. The fact that the error is smaller for the case r® = 19, makes that choice preferable. On the other hand, the choice r~ = 22 is favored by the fact that nl and n2 are very close to integral values, which more strongly supports the hypothesis (13). Some additional measurements, especially at large concentrations, would readily tip the scales in favor of one or another value of r~. As a second example, we consider the binding of 1-iodonaphthalene-4-sulfonate by IgM antibodies (Onoue et al., 1968). In this case, as can be seen from the data points shown in theScatchard plot of Fig. 2, extrapolation to determine So would be very hazardous, and all that can be certain about the value of

r~ is that it is in the neighborhood of 10. Using our procedure, and assuming that equations (8) and (13) are applicable, we chose r, = 9, 10, and 11, and determined in each case the values of nl, n2, K1, and K2, shown in Table 1. Table 1 also gives the associated errors as calculated from equation (8). We note that the value r~ = 10 yields the smallest error and is therefore the value of choice. This conclusion is made visual by a plot of (u - ~1)/c vs u - cq, shown in Fig. 3 for the cases r~ = 9 and 11, and in Fig. 4 for the case r~ = 10. According to equation (7) with n = 2, the relationship between these variables is linear. By comparing Figs. 3 and 4, it can be seen that the variance of the data points about the regression line determined by the least-squares fitting procedure is least for the case r~_ = 10. The associated theoretical Scatchard curve based on the choice r,~ = 10 and on equation (13) is also displayed in Fig. 2. In this example too, it would be very difficult to decide by inspection of the derived Scatchard curves as to which value of r~ is best. We see from Table 1 that the derived values of nl and n2 are closest to integral values when r~ = 10, and further supports the choice r~ = 10. Moreover, this conclusion is consistent with the theoretical conception (Miller & Metzger, 1966), supported by structural studies, that IgM contains precisely 10 binding sites. The fact that nl and n2 are close to 4 and 6, respectively, instead of 5 and 5, as might be expected from the pentameric nature of the IgM molecule, is of course unexplained by these considerations. Acknowledgement--The author thanks Fred Karush for helpful discussions of this work.

REFERENCES

Adair C. S. (1925) J. biol. Chem. 63, 529. Endrenyi L., Chan M.-S. & Wong J. T.-F. (1971) Can. J. Biochem. 49, 581. Karush F. & Sonenberg M. (1949) J. Am. chem. Soc. 71, 1369. Karush F. (1950) J. Am. chem. Soc. 72, 2705. Karush F. (1956) J. Am. chem. Soc. 78, 5519. Klotz I. M. & Hunston D. L. (1971) Biochemistry 10, 3065. Miller F. & Metzger H. (1966) J. biol. Chem. 241, 1732. Mukkur T. K. S., Szewczuk M. R. & Schmidt D. E. (1974) lmmunochemistry 11, 9. Nisonoff A. & Pressman D. (1958) J. lmmun. 80, 417. Onoue K., Grossberg A. L., Yagi Y. & Pressman D. (1968) Science 162, 574. Pauling L., Pressman D. & Grossberg A. L. (1944) J. Am. chem. Soe. 66, 784. Roholt O. A., Grossberg A. L., Yagi Y. & Pressman D. (1972) lmmunochemistry 9, 961. Scatchard G. (1949) Ann. N.Y. Acad. Sci. 51, 660. Sips R. (1948) J. chem. Phys. 16, 490. Werblin T. P. & Siskind G. W. (1972) lmmunochemistry 9, 987.