prescribed in Gahler's method prevents complex building cations other than Cu(1) from reacting with neocuproine. From the values of the stability constants where the 'ligands are EDTA, citrate, nitrilotriacetate, and neocuproine (see S i l l h et al., 5 and 7), it follows that in the procedure given here, complex building cations other than Cu(1) are prevented from reacting with neocuproine by nitrilotriacetate or EDTA, by EDTA with respect to cations which have greater stability constants with EDTA than that of Zn(I1) and EDTA. So, with the procedure given here, it is reasonable to expect great selectivity toward copper. The pK, Value of tris(hydroxymethy1)aminomethane is about 8 and of hydroxylamine about 6. Thus, it is easy to maintain the solution a t a desirable pH value between 7.0 and 7.5 for the color reaction to take place. Smith and McCurdy, Jr. ( 2 ) found that halides and, in particular, nitrate precipitated the copper-neocuproine (7) L. G. Sillen, E. Hogfeldt, A. E. Marteil, and R. M. Smith, The Chemical Society, London, Spec Pub/.No. 25, 1971.
complex in an aqueous solution. In this study, it was found that the solubility of the copper-neocuproine complex was improved by adding a water-miscible alcohol. For this reason, neocuproine was added in a solution of 5wo 2-propanol. Table I shows that, with the procedure given here, it is possible to determine at least 0.1% copper in ammonium nitrate. With EDTA, some cations give colored solutions (West, 6). The cations that with EDTA or nitrilotriacetic acid give colored solutions absorbing light of the wavelength 450 nm, would interfere, if, in the measuring of absorbance, a solution containing an aliquot of the sample to be analyzed and all reagents used except neocuproine was not used in the reference cell. An example is given in Table I with a sample containing iron. Received for review June 1, 1973. Accepted January 21, 1974.
Model for Competitive Binding Assays-The Shape and Location of the Inhibition Curves Donald J. Laurence and Graeme Wilkinson institute of Cancer Research, Chester Beatty Research Institute, Fulham Road, London S W 3 6JB, England
The increasing application of competitive binding assays has required the development of mathematical models to represent the binding curves. Berson and Yalow ( I ) and Ekins, Newman, and O'Riordan (2) have discussed conditions for obtaining the maximum sensitivity from an assay. Feldman and Rodbard ( 3 ) and Feldman, Rodbard, and Levine ( 4 ) have presented graphical findings for a number of important special cases. The information obtained from the curves can come from their location on the concentration axis and from their shape. Studies of tumor-derived antigens ( 5 ) have posed the question of identity between substances in low concentration in body and tissue culture fluids and the materials purified from the tumor itself. In these cases, when the identity of the analyzed product is in question, the location of the curves is less informative than is their shape. Location depends both on the concentration and the affinity for protein of the test substance. If the substance cannot be assayed by an alternative method, it is not possible to separate these two aspects of the problem. The present theory was developed to determine conditions that are suitable for detecting affinity differences between competitors of unknown concentration. The conditions selected are not the same as those that would nor-
mally be chosen in order to establish a sensitive assay. When a test of identity with a sensitive assay appears to confirm the identity of competitors of different origins, it should be considered whether the test is capable of giving an alternative conclusion. The curves we have considered are plots of degree of inhibition as a function of logarithm of the ligand concentration. The degree of inhibition I is a normalized linear function of the ratio of bound to total ligand that increases from 0 to 1 as ligand concentration increases. On a log concentration scale, changes in absolute unitage of ligand, or affinity variations, cause a scale shift but no change in shape of the curves provided the maximum bound fraction ro is kept constant. It was found that the equations are of a simple analytical form if .the ligand concentration and shape parameters are expressed in terms of I and ro. As indicators of shape, the symmetry of the curves, the maximum slope, and the span are taken. The Inhibition Curve. For an equilibrium between ligand and protein with a single association constant K and no interaction between binding sites
(1) S. A. Berson and R. S. Yalow in "The Hormones," vol 4, G. Pincus. K. V. Thimann, and E. E. Astwood, Ed., Academic Press, New York, N . Y . , 1964, pp 567-72. (2) R. P. Ekins, G E. Newman and J. L. H. O'Riordan in "Radioisotopes in Medicine,'in Vitro Studies," R. L. Hayes, F. Goswitz, and B. E. P. Murphy, Ed., U . S . Atomic Energy Commission, Oak Ridge, Tenn.. 1968, pp 59-100. ( 3 ) H. Feldman and D. Rodbard in "Principles of Competitive Protein Binding Assays," W. D. Odell and W. H . Daughaday, Ed., J. E. Lippincott. Philadelphia, Pa., 1971, pp 158-216. ( 4 ) H. Feldman, D. Rodbard, and D. Levine, Anal. Biochern., 45, 530 (1972). ( 5 ) D. J. R . Laurence and A. M. Neville. Brit. J. Cancer, 26, 335 (1972).
where B and F are concentrations of bound and free ligand, respectively, and Q is the concentration of free protein binding sites. If the ligand or protein are multivalent, the equivalent concentrations should be considered. Taking the total concentration of ligand and protein sites as p and q, respectively, and defining
1132
ANALYTICAL CHEMISTRY, VOL. 46, NO. 8 , JULY 1974
BIF
=
KQ
Y = ( K q ) - ' , r = B j p . and m = q / p then:
r2
- d1
+ m(1 + y ) ] + m=O
(2)
when q
> > p , m is large and from Equation 2
t.0-
0.9-
(3) 04-
This defines the maximum bound fraction ro as the ratio of bound to total ligand at low ligand concentration-ie., with no competition. During the setting up of an assay, a plot of ro us. q (in relative concentration units) leads to the selection of an ro value. It is supposed that in any single assay, ro and, hence, y will be known and held constant. The degree of inhibition is defined.as
I =1 - (r/ro)
(4 1
If m is replaced by q / p in Equation 2 , this equation may be rearranged to express p in terms of q, r, and y. An expression derived from Equations 3 and 4, uiz.
+ y)
r =(1 - I ) / ( 1
0.7-
,E
0.6-
f
05-
0
0.4-
s
-
Ea OB-
a
0.20.1I
01
I
5
0
,
I
I , / / I
5
1
1
1 1 1 1 1
5
1
10
,
1
I
I
I I , , ,
50
100
(5)
and the definition of y in terms of q and K are then employed to express p in terms of I, K , and y as follows:
In Figure 1 are plotted the values relating p and I a t constant y(r0). Slope. The slope of the curve can be obtained from Equation 6
+
dI = (y 1)(1 - I ) I d In p (y 1')
+
This has zero slope at I = 0 and 1 and must therefore have a maximum slope between these values as I is an increasing function of p . To obtain the slope in a log2 (doubling dilution) or loglo scale, the slope given by Equation 7 is multiplied by In 2 or In 10, respectively. Point of Inflection. To determine the point of maximum slope
- dZ F(I) --
d2Z --
d In p ( y
(d In p)'
+ 1')'
+
(y+l-i) (y+i)
i (l-i)
Slope and span
Symmetry ro
Small 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .o
Irn
1 .oo 1 .oo 1 .oo 0.99 0.98 0.97 0.95 0.92 0.87 0.80 0 .oo
Go.o
1 .oo 1 .oo 1 .oo 1.01 1.02 1.04 1.06 1.10 1.18 1.32 1.67
S,
1.00 1.03 1.06 1.10 1.14 1.20 1.28 1.38 1.53 1.84 4.00
No.0
1.00 0.92 0.84 0.76 0.67 0.58 0.49 0.40 0.30 0.21 0.11
No.9
1 .oo 0.94 0.88 0.81 0.74 0.67 0.59 0.51 0.43 0.34 0.25
The actual values when ro is small are I , = 0.5; Go.o = 1; Srn = 0.25(ln) 0.173(10gz), 010.576(10gio); No.0 = 81, andNo.8 = 16.
(8)
where F(I) = I4 + 4yF 2y(y - 1)1- y 2 . The point of inflection is given by F(Zm) = 0. I , lies between 0.4 and 0.5 for ro between 0.1 and 0.9. The maximum slope S, is found by substituting into Equation 7 the value of I , determined by graphical solution of the quartic. Span. Defining the span N , as p , / p I - ,, where p Lis the value o f p at I = i etc., we find from Equation 6
N' =
Table I. Relative Values for Different M a x i m u m Bound Fractions ro of the S y m m e t r y P a r a m e t e r s ( I , and GO.$) and t h e Slope P a r a m e t e r s S,, No.$ and No.*.All Values Are Normalized by P u t t i n g the Values for ro Small as Unitye
(9)
d ' I l ( d In p)? = I(1
- 1x1 - 2 1 )
(12)
The point of maximum slope is at I m = 0.5 and the value of this slope S m = 0.25. Equation 11 was derived by Weber (6). When ro is nearly unity, there is a stoichiometric relationship between total ligand and protein site occupancy. Free ligand appears only if ligand exceeds number of available sites: when p < < q, r = ro = 1, and when p > q, r = m.Also
d I / d In p = - d 2 1 / ( d In p I 2 Geometric Mean Ratio. As an alternative measure of =l- I (13) curve symmetry we may calculate G, = @,/PI - ~ ) ~ ' ~ / p 0 . 5 . Numerical Values. Table I gives values of the curve This parameter is unity if the distance from p L to p0.5 is parameters calculated by the formulas presented in the the same as that between pO.5 and p1 - on a log scale of text. The most sensitive parameters are the span and the P. slope. The curve symmetry is not as sensitive to changes The Case of Weak and Strong Binding. When ro is in ro between 0.1 and 0.9. The main conclusion to be small, the concentration of free ligand F may be replaced drawn from these results is that to obtain a significant byp in Equation 1and then: difference of slope between two samples differing in affini(10) I = (1 yrn)-' ty constant, then the one of greater affinity must usually be presented on a curve with ro greater than 0.8 and prefThe functional similarity between Equations 3 and 10 is apparent as y = ( K q ) - l and y m = ( K p ) - l . To obtain the erably between 0.9 and' 1.0. This is a region of ro values first and second derivatives, take y large in Equations 7 that is not commonly used in routine binding assays as and 8.
+
d I / d In p
=
f(1
- I)
(11)
(6) G. Weber in "Molecular Biophysics," B. Pullman and M . Weissbluth, Ed., Academic Press, New York and London, 1965,pp 369-96.
ANALYTICAL CHEMISTRY, VOL. 46, NO. 8, JULY 1974
1133
-
1.0
-
0.8
06
li
-
0.4-
6
t
1
I
1
O1
Label or c a p t l l a c m m t r o t m
Figure 2. Model curves for comparison of a weakly binding competitor (0)with the authentic equivalent of the label ( 0 ) : case of f o = 0.4. The curves shown are the theoretical ones and not those corresponding to the points that have been attached
the inhibition curves are not a t maximum sensitivity in this region. Competitors Differing from Label. If the label differs in affinity from the competitor, the label acts as an indicator of a reaction that is occurring at a different but unknown ro value. If the label is present in low concentration so that it occupies few of the protein binding sites, then it seems likely that the binding curve would be that appropriate to the roof the competitor. To see this for the case of a weakly binding competitor, consider Equation 1 for the reaction of label and competitor (suffixes L and C ) and eliminate Q between the two equations
rL 1
- rL
- KK ,c l -rc rc
(14)
As the competitor is weakly binding, we may ignore rc in comparison with unity. Writing a corresponding equation suffix zero to represent the maximum bound situation, combining the two equations and rearranging: (15) Substituting for IC using Equation 10 we find
+ (1 - rOL)-'yrnI-l
(16)
Comparison with Equation 10 shows that the binding curve has the characteristic hyperbolic shape, typical of a weakly binding substance. The only difference is that the effective binding constant K'c is determined according to the equation:
Kfc=(l
- roL)Kc
(17)
The use of label to test identity would correctly identify a weak inhibitor. Inhibition by the true identity of the label-and this may be tested by added label as inhibitor and also the unlabeled authentic product-would give the curve appropriate to the high ro value chosen. As illustrations of these comparisons, Figures 2 and 3 show the inhibition curves that would be obtained if a weakly binding competitor were compared with the authentic equivalent of the label. The points were placed on the theoretical curves with twofold increases in concentration between points and were then subjected to random 1134
ANALYTICAL CHEMISTRY, VOL. 46, N O . 8, JULY 1974
(arbitrary units)
Figure 3. A s for Figure 2 but with f o = 0.9
error with a standard deviation in I of 0.025. The set of errors consisted of all positive and negative deviations in the range -2.9 to +2.9 standard deviations, in intervals of 0.1 standard deviation. Selection of errors was by mapping into the error set, a set of all integers from 00 to 99 which were then used as an index. Reference to the index was by a list of two digit random numbers. It was arranged that the number of integers mapped onto a given error was proportional, within the limitation of the small numbers employed, to the ordinate of the normal distribution for that error. In this way, the random selection of errors using the index would be in accordance with the normal law of errors. Finally the errors were multiplied by the chosen standard deviation and applied as duplicate points to the points on the curve in a predetermined manner, not related to the error values. In Figure 2 for ro = 0.4, the difference in shape between the label and competitor curves is small and in the absence of theoretical curves the central points could have been approximated by two parallel lines. In Figure 3, for ro = 0.9, the error in the determinations does not invalidate the conclusion that the curves represent differential substances. Location of the Curves. For the curves shown in Figure 1, their location is sufficiently determined by the value of ~ 0 . and 5 from Equation 6, replacingy by ro po,s= 2/[(1
I , =[I
XXI
10
10
or c a p t i t o r cDmntratm m m r y untts)
-
roX2 - r,4KI
(18)
The curve is moved by variation in K by a scale shift of -In K . The more strongly binding the ligand-protein complex, the further to the left is the curve on the concentration axis. As ro increases from 0 to 1, the value of p0.5 will increase from 1 / K to infinity. Effects of Heterogeneity, If the protein or ligand is heterogeneous so that there are a number of different association constants, then the slope of the curve as viewed from the concentration axis is decreased (6). The extent of heterogeneity may be represented by the constant c, less than unity, in the equation:
z = [l + (?'m)']-!
(19)
provided ro is small. The corresponding slope (6) is: d I / d In p = c ( l - Z)Z
(20)
Studies of antibody-ligand complexes have revealed c values from 0.2 to 0.9 ( 7 ) . With the same antibody sys( 7 ) H. N. Eisen and G. W. Siskind, Biochemistry. 3, 1003 (1964)
The binding protein concentration should be at least 10 times that required for routine assays to ensure PO > 0.9. The range of ligand concentrations chosen would also need to be increased so that a complete inhibition curve is traced. This may require a partial purification and concentration of the inhibitor but does not require its isolation. The label concentration is kept low to avoid selfcompetition when the curve of an unknown substance is being followed. As the binding site concentration is high, this would not mean a decrease for most assays in the number of counts available. Heterogeneity could also interfere with the sharpening of the curves that is the basis of discrimination in the homogeneous case. Unless heterogeneity had its chance discriminatory effect, this could make the proof of identity more uncertain. It is recommended that the inhibition curves should be examined a t several different ro values to ascertain whether the curves obtained are typical of homogeneous or heterogeneous binding.
tem, different ligands may give c values differing by a factor of two (7).So relatively large differences in slope might be expected even in the region of small ro where the homogeneous system would have exhibited relatively small differences. Given an extreme heterogeneity of the binding affinities, a n incomplete inhibition could result. This is equivalent to a resolution of binding into discrete classes of association constant. Incomplete inhibition may give a good discrimination if the label and competitor differ in this respect, as the upper bound is a well-defined property of the inhibition curve. CONCLUSIONS Lack of identity between inhibitors can be verified by differences of shape of the inhibition curves when the system is homogeneous with respect to affinity constant. The interpretation of shape in a log concentration scale does not require prior knowledge of absolute concentration of the inhibitors. The method is therefore suitable for examination of unknown inhibitors present in biological fluids and in vitro systems. Dilutions should be done as far as possible in a medium similar to the test medium but lacking the inhibitory substance (e.g., control culture medium or the plasma of normal healthy individuals).
Received for review September 5, 1973. Accepted January 11, 1974. The investigation was supported by the Medical Research Council (Grants 970/656/B and 971/817/B).
Support-Bonded Phases for Gas Chromatography Derived from Alkyl and Phenyl Chlorosilanes A.
H. AI-Taiar, J. R. Lindsay Smith, and D. J. Waddington
Department of Chemistry, University of York, Heslington, York YO7 5DD, England
In a previous paper ( I ) , we described the preparation of some thermally stable gas-chromatographic packing materials which were prepared by the hydrolytic polymerization of trichlorosilanes with or without added silicon tetrachloride. The materials which contained a considerable excess of the chlorosilane (98%) allowed the elution of a wide range of nonpolar and polar compounds and were thermally stable up to a t least 350 “C; however, their efficiencies were less than those obtained for many conventional columns. In another approach to making useful stable chromatographic phases, reported first by Abel, Pollard, Uden, and Nickless (2) and investigated in some detail by Aue and Hastings (3-8) and others (9-I2),chlorosilanes are poly-
merized on the surface of typical chromatographic supports. The materials obtained from this procedure are potentially useful in both gas and liquid chromatography, the organic phase being chemically or strongly physically bonded to the inorganic support. Apart from the obvious advantage of thermostability that these bonded phases have over conventional chromatographic materials, it should be possible to produce a wide range of closely related materials to study solute-solvent interactions in chromatography. In this paper, we describe the preparation of a number of chromatographic materials prepared from phenyl- and substituted-phenylchlorosilanes (and, for comparison, octadecyltrichlorosilane) on Gas Chrom Q as the inert support. EXPERIMENTAL
(1) A. H. AI-Taiar, J . R. Lindsay Smith, and D. J. Waddington, Anal. Chem., 42,935 (1970). (2) E. W . Abel, F. H . Pollard, P. C. Uden, and G.Nickless, J. Chromatogr., 22,23 (1966). (3) W . A . A u e and C. R. Hastings, J. Chromatogr., 42,319 (1969). (4) C. R. Hastings, W. A. Aue, and J. M. Augi, J. Chrornatogr.. 53,487 (1970). (5) W. A . Aue, C. R. Hastings, J . M . Augl, M . K . Norr, and J . V. Larsen, J. Chromatogr., 56,295 (1971). (6) C. R. Hastings, W. A. Aue, and F. N . Larsen, J. Chromatogr., 60, 329 (1971). (7) W . A . Aue and P. M. Teii, J. Chromatogr., 62,15 (1971). (8) W. A . Aue, S. Kapila, and C. R. Hastings, J. Chromatogr., 73, 99 (1972). (9) H. N . M. Stewart and S. G . Perry, J . Chrornatogr., 37,97 (1968). (10) J. B. Sorrel1 and R. Rowan, Jr.,Anal. Chern., 42,1712 (1970). (11) R. Rowan, J r . , and J. B. Sorrel1,Anal.Chem., 42, 1716 (1970). (12) M. Novotny, S. L. Bektesh, K . B. Denson, K . Grohmann. and W. Parr, Anal. Chem., 45,971 (1973).
Operating Conditions of the Gas Chromatograph. All results were obtained using glass columns (1.5-m x 4-mm i.d.) in a Pye 104 gas chromatograph equipped with a flame ionization detector coupled to an RE 511 Goertz Servoscribe recorder. The carrier gas, nitrogen (BOC, “oxygen free”), was deoxygenated and dried by passing it through successively a solution of chromium(I1) chloride in hydrochloric acid over zinc amalgam, concentrated sulfuric acid, potassium hydroxide pellets, silica gel, and molecular sieve 5A. Measurement of Thermal Stability of Packing Materials. The detector ionization current was measured for columns packed with the new materials and with Celite (AW)-silicone elastomer E301 over a wide range of temperature, and compared with that of a n empty column under identfcal conditions. A current of A gave a full scale deflection when the recorder was set at 10 mV with the amplifier attenuation a t 1 x 103. ANALYTICAL CHEMISTRY, VOL. 46, NO. 8, JULY 1974
1135
