The prediction of standard curves & errors for the assay of estradiol by competitive protein binding

77

THE PREDICTION OF STANDARD CURVES & ERRORS FOR THE ASSAY OF ESTRADIOL BY COMPETITIVE PROTEIN BINDING by D. Wilson, G. Sarfaty, B. Clarris, Margaret Douglas* and K. Crawshaw Endocrine Research Unit, Cancer Institute & *Honeywell Computer Time-Sharing Service Melbourne, Victoria, Australia. Received:

11/30/70 ABSTRACT

Standard curves for competitive protein binding analysis of steroids are usually arrived at by empirical methods. It is possible from a knowledge of the equilibrium constant, concentration of binding site and radioactive tracer as well as practical inefficiencies of the system, to predict and experimentally confirm standard curves for the estradiol-uterine cytosol assay. This has been done by modifying the mass action equation for the non-ideal conditions that arise in the actual assay. Estimating the error in the bound fraction when tracer only is present allows the 95% confidence interval to be predicted at each hormone concentration on the curve. This estimate of error is used to define a working range and a limit of sensitivity. Because of the complexity of the equations for the relevant variables a computer program has been utilised to solve equations and to provide tabular output for standard curves, error, working range and sensitivity. Recently introduced steroid assays in which binding proteins have been used to measure the extent of radioisotopic steroid dilution (i) have gained wide acceptance.

(2,3).

The setting up of standard curves

with this technique, herein called competitive protein binding assay, can be arrived at either empirically,

that is by practical experiment,

or by application of the law of mass action in relation to the practical assays (4).

The practical approach requires progressive dilution of the

steroid binding protein.

At each dilution the fraction of radioactivity

bound (Rb), in the presence of a constant amount of radioactively labelled tracer (h*) for each hormone level (h) of interest is measured.

78

S T ER O I D S

18:1

From the series of curves obtained one is chosen, either by intuition or experience, which appears to have the appropriate working range and sensitivity needed by the experimenter. An alternative method is based on competitive protein binding theory which assumes the steroid and binding protein interaction to be an example of the law of mass action (5).

Expounded by Ekins et al (6),

this approach has resulted in the prediction of standard curves.

For

this method to be usefully applied over a wide range of practical curves, the theoretical predictions need experimental confirmation. To do this requires a modification of the mass action equation allowing for the non-equilibrium state that could result during the separation of free and bound hormone.

If the non-equilibrium state

during separation is not taken into account a false estimate of the equilibrium constant (K) and the binding site concentration

(q), could

result in substantial errors when predicting assay curves.

Thus, the

best estimates of K and q cannot be determined without previous study of variables and constants associated with the separation of free and bound hormone.

When constants associated with the separation procedure

are experimentally determined they can be incorporated into the mass action equation (7) to obtain corrected values for K and q necessary for prediction of standard curves and associated errors of measurement. Both predicted curves and errors can be then confirmed by practical experiment. This report describes equations and experimentally determined constants used to forecast standard curves and errors for the estradiol/ uterine cytosol assay system.

The predicted standard curves and errors

have been tested by practical experiment and are in good agreement.

July 1971

s T ER O I D S

A method has also been devised each assay

curve.

This utilizes

in hormone concentration region error

of the

assay

for determining

a standardized

a n d an a c c e p t a b l e

curve is

then obtained

of hormone concentrations

79 a working range of

959 c o n f i d e n c e

nominated value

interval

of error.

in which variations

A

in the

are minimised.

METHODS AND MATERIALS Definition of Terms h, hi, h 2 and h*

h, the concentration of non-radioactive ligand hormone selected at each level on the standard curve, hl, the value of h at the lower limit of optimum working range and is the minimum value of h for a chosen error constant c, as defined below under optimum working range. h2, the upper limit of the range as defined by c. h* is the concentration of radioactively labelled hormone of known specific activity.

q and K

q, the concentration of protein binding site per assay tube and K, the equilibrium constant for the protein-ligand reaction.

Rb and Rob s

are experimentally determined values such that, Rb is the observed fraction of h* bound at each h level and Rob s is the calculated ratio of free to bound h* at the same h level. is an index of efficiency of adsorption by charcoal of free hormone.

kI and k2

are respective slopes of equations 3 and 4.

t

duration of charcoal contact with incubate.

c2

ordinate intercept of the dilution curve in equation 4.

Vp and V m

Vp is the variance, due to radioactive decay, of the sample containing the assumed fraction bound. It is given by the variance in counting radioactivity of the sample. V m is the remaining variance of the method.

Optimum working range

extends from a level h I to h 2. These limits satisfy the equation, (95% confidence width of h) " h = c. A value of c is chosen to minimise differences in the standardized confidence width at each hormone concentration.

80

S T E R O ID S Mathematical

18:1

Analysis and Computer Solutions

The basic equation used to characterise the estradiol/uterine cytosol interaction derived from the law of mass action as applied to univalent reacting species (6,8).

I

--i

R2 + R where,

h* + h q

I--

Kq

1

- K-~ = 0

• ..

1

R = ratio of free to bound h* existing before the separation procedure.

This ideal eqaation needs modification to incorporate the expected inefficiency in separation of free and bound hormone and instability of the bound complex (7,9) as follows: Efficiency

of Adsorption of h* RX Robs = 1 + (i - X) R

... 2

X, experimentally found for estradiol, = 0.980 -+ 0.005, S.D, n=8 The value X was independent of h at h=0pg and h=800pg. Dissociation

of Bound Complex Rb = Rb,t=O + klt

• ..

3

where, k I = -0.005 min -I calculated from duplicate estimations at 4, 8, 12 and 16 minutes following charcoal addition with q=SS.gpg ml -I. Similar kl'S were found for q=89.6 and 27.95pg ml -I Dilution of Binding Protein kl = k2q + c2 k2, experimentally c 2, experimentally

... 4

found for estradiol = 1.36 x 10-5ml min -I pg-i found for estradiol = 4.8 x 10 -3 min -I

Considering the above variables the following equation can be derived which relates the experimentally observed fraction bound with the fraction bound (r) prior to the addition of charcoal.

Robs = (R + I) [i - (k2q + c2) t] - 1 - (1 - X) R (R + i) (k2q + c2) t + i + (i - X) R

where, Rob s -

1 - Rb Rb

and R -

1 - r r

... 5

July 1971

s T ER O I D S

81

Equation 1 can now be expressed in terms of Robs, from equation S, and corrected estimates of q and K obtained by the Scatchard (i0) equation. By suitable substitutions, corrected estimates of K can be used for a given h* and a range of h, to find Rb at any dilution of binding protein, i.e., different levels of q. Method Errors

(a)

P r e d i c t e d Values o f Rb

The 95% confidence interval in Rb is given by, ± 1.96

/V m +

Vp

From h=0pg to h=800pg, V~d~-~ =constant and the calculated value obtained by experiment for estradiol was S0.02. dVp/Rb is a variable depending on counts measured in the fraction bound. The value of /Vp/Rb ranges from 0.004 at h=0pg to 0.011 at h=800pg. By calculating the upper and lower limits of the 95% confidence width for predicted Rb at h=0pg, two values of Rob s (in equation 5) can be obtained and hence two values of R. Equation 1 can now be solved for h at each limit over the range of h=Opg to h=800pg where /Vm/Rb =constant. (b)

Experimentally found Values of Rb

The 95% confidence limits for these values at each h level were calculated by conventional statistical techniques (ii). A correction was applied to standardize differences between experimental values of Rb and those theoretically predicted from equations 1 and 2. The standardized Rb allows a direct comparison of the confidence intervals obtained by the predicted and experimental methods.

Equation 5 enables the law of mass action given by equation 1 to be combined with the observed properties of the assay system, thus providing for prediction of Rb and the 95% confidence interval at h and other features discussed in the main text. The equations for predicting Rb and errors are difficult to solve manually. A computer program a" was written in Fortran for the Honeywell GE 265 Time-Sharing System computer. Output of computer solutions, obtained in both graphical and tabular form, were compared with practical results. Input data of this computer program are corrected values

a'workers requiring a copy can obtain a paper punch tape and listing of the tabular form. Address requests to Gordon A. Sarfaty M.D., Head, Endocrine Research Unit, Cancer Institute, 278 William Street, Melbourne 3000, Victoria, Australia. Costs make it necessary to charge a nominal amount for each copy.

82

S T E R O I D S

18:l

o f K, q, and v a l u e s f o r X, k2 and c2 d e t e r m i n e d f r o m e x p e r i m e n t . Also i n p u t e d a r e a p p r o p r i a t e v a l u e s o f h*, t h e s p e c i f i c a c t i v i t y o f h * , volume o f i n c u b a t i o n m i x t u r e , volume o f and t i m e t a k e n f o r c o u n t i n g t h e bound f r a c t i o n , and an e s t i m a t e o f t h e s t a n d a r d d e v i a t i o n i n Rb a t h=0pg o b t a i n e d f r o m 8 t o I0 r e p l i c a t e s . Practical

Assays

B i n d i n g o f e s t r a d i o l b" (E2) by t h e 10S,000G f r a c t i o n o f c a l f u t e r u s was u s e d t o c o n s t r u c t p r a c t i c a l standard curves. Cytosol prep a r a t i v e t e c h n i q u e s w e r e s i m i l a r t o t h o s e o f Korenman ( 1 7 ) . Values of Rb, n=2, w e r e f o u n d u s i n g h * = l l , 3 0 0 c p m 3H(---57.15pg E 2 ) , a t h=20, 50, 200 i 400 and 800pg o f E2 r e s p e c t i v e l y f o r e a c h d i l u t i o n where K=0.26ml pg- . ( 7 . 1 x 101°L/tool). To f i n d Rb a t h=0pg, 8 t o i0 r e p l i c a t e s were assayed. A l l g l a s s w a r e was s i l a n i z e d . Samples w e r e p r e p a r e d b y s t a n d a r d t e c h n i q u e s (17) w i t h d e x t r a n / c h a r c o a l to separate the free f r a c t i o n a f t e r s a m p l e i n c u b a t i o n f o r 16-20 h r s . a t 4C. At t h i s t e m p e r a t u r e a minimum o f 16 h r s . i s r e q u i r e d t o r e a c h e q u i l i b r i u m o v e r t h e s t a t e d r a n g e o f hormone c o n c e n t r a t i o n s . Radioactivity in 2/3 of t h e s u p e r n a t a n t a f t e r s e p a r a t i o n was c o u n t e d i n a Model 3375 P a c k a r d liquid scintillation s p e c t r o m e t e r f o r 20 m i n u t e s . RESULTS Figure

1 compares c o m p u t e r p l o t t e d

f o r Rb ( a ) , w i t h t h o s e limitations

f o u n d by p r a c t i c a l

of the computer plotting

e x t e n d as f a r

as 600pg o n l y .

initial

binding

plotted

figures)

site

It

concentration,

b.

is

Because of

q,

(written

at each level

h levels

This excellent

(h) g r e a t e r

of sixty

1 (a)

of

as Q by t h e c o m p u t e r closely

approxi-

agreement between

f o u n d o v e r a wide r a n g e o f Rb v a l u e s

and c o n f i r m e d w i t h a t o t a l levels

(b).

solutions

program the curves of Figure

v a l u e s o f Rb a t c o r r e s p o n d i n g

t h e two a p p r o a c h e s

at estradiol

experiment

can b e s e e n t h a t

mate t h o s e o b s e r v e d e x p e r i m e n t a l l y .

the theory

curves of theoretical

practical

forecast

by

assay points

than zero.

Both non-radioactive estradiol (i, 3,5, (i0) -estratriene-3,178-dioi), from Sigma Chemical Co., and estradiol-6,7-3H; s.a. 48mC/mM from New England Nuclear Corporation, were used as supplied.

July 1971

s T ER O I D S

83

..... ~ >

//]lJ 0° //f// i

~

a

4J

'd

,

6"

~1

i

8"

L"

9"

S"

I,. £" GNn09 -~I

Z"

I.

O"

84

S T ER O I D S

4-

e4

18:1

['~

O~ ~ ~'~

0 ~0 ~

L'~ ~ ~

0 O~ 0

0 "~ 0

o

o

o

c~

o

o

7 O u')

~D O

~q-~ 0

O II

,-1 ~.1

o u ~

~ ~J

o\o o\O o~0~

.r-i

0

O.8

~o

0

I11 O

g

~ 0

il

¢ L

0

0 O0

0 0

0 0 '~

0

0 0 O0

.'~ ~

0'~

ffl ~

,"'(

0

0

o

~

'~

L~O

~

,.--(

,--4

0

c~

o

c~

o

o

o

,e~

N ~

• N c'M ¢'~

"~~

~l o\o O

~i~~

~

•

II

~

I

O

oo ~ ~.,-o ..

ffl

0

~

N 0

f~ q..( 0

O

.;-4 0r.t • r'.t . M ~ O

U ~

4-J

~

•M

~

O m ~

0.) ,.-3

0

0 '~"

0 oO

0 0 0,1

0 0 ~

0

0 0 O0 q-

~

J u l y 1971

s T E R O I D S

To p r e d i c t

errors

o f hormone c o n c e n t r a t i o n ,

limits

o f Rb a t h=0pg ( e q u a t i o n

limits

at each level

predicted

estimates

dence l i m i t s trivial,

T a b l e I compares t h e

values obtained

determined Rb'S.

from t h e c o n f i -

Adjustment,

because of a small difference

and p r e d i c t e d

Both p r e d i c t e d

than zero.

with the adjusted

of experimentally

t h e 95% c o n f i d e n c e

6) w e r e u s e d t o f i n d t h e c o r r e s p o n d i n g

of h greater

is necessary

experimental

85

though

i n t h e means o f

v a l u e s o f Rb ( 1 2 ) .

confidence intervals

and t h o s e d e r i v e d

from e x p e r i -

ment a r e i n good a g r e e m e n t . F i g u r e I I shows t h e v a r i a t i o n confidence width of h)-h,

from zero,

It

demonstrates

size

of error

diminishes

mum v a l u e and i s t h e n f o l l o w e d b y a g r a d u a l

L-1)

in Figure I,

width

(Figure II)

standard

curve,

depends on t h e p a r t i c u l a r

from infinity

increases to a mini-

rise.

q = 5 5 . 9 p g ml-

t h e minimum v a l u e 0 . 1 1 3 ,

(95%

Standardization

t h a t when hormone c o n c e n t r a t i o n

the magnitude of error

For a typical

error,

w i t h hormone c o n c e n t r a t i o n .

is needed because the relative v a l u e o f h.

of a standardized

( 2 . 0 5 x 10 -10 mols

of standardized

confidence

can b e u s e d t o d e t e r m i n e an optimum w o r k i n g r a n g e .

The limits of range h I and h 2 are points on either side of the minimum value. The limits of any nominated working range depends on the chosen value of the constant c and such a range is only optimum for the particular value of c. 0.12.

In Figure II, the value for the constant c is

It corresponds to a lower limit, hl=180pg and the upper limit,

h2E700pg.

Thus, as defined, the optimum working range for the curve

q=55.gpg ml -I, is 180-700pg.

In terms of minimised relative error

the lower limit of sensitivity for this curve is 180pg.

86

ST ER O I D S

18:1

Z O

O 0

m

ln,. lz i.i.i (.~ Z 0

LI,J

Z O Z

N

u'i

-r

~1

:I= I--

o.~

l,.iJ Z 14.1 I.,I=

Z O (D

I,.i.I I,/'I

yi

=

N

Q::

°

Z

I.,,•

•

•

LI'0 91"0 •I'0

|

ITI'0

41p!M a0uap!ju00

•

£I'0

I

Zl'0

pas!pJopuD1s

ll'0

July 1971

s T E R O I D S

87

DISCUSSION The competitive protein binding technique is a simple, rapid and potentially specific means of steroid assay.

Although the mass action

law has been studied for the purpose of defining standard curves and maximum sensitivity (6), they have not included the known efficiencies of the practical competitive protein binding assay of steroids.

With-

out attention to these factors, it would not be possible to meaningfully forecast standard curves. There are many possible inefficiencies in the competitive protein binding assay of steroids.

These include, non-equilibrium conditions,

instability of the bound complex, inefficiency of adsorption of free steroid by charcoal; adsorption of the binding protein, as well as steroid and protein-steroid complex to the reaction vessel.

In the

present treatment the reaction is carried out at equilibrium (16 hr.) and all glassware is silanized.

Standardizing these conditions and

also pH and ionic strength, it was sufficient to consider only the instability of the protein-steroid complex and the inefficiency of adsorption of the free steroid. Because corrections for instability of bound complex (kl) and efficiency of adsorption of free steroid (X) are minimal in the uterine cytosol estradiol system, good agreement between predicted and experimental curves is readily obtained.

On the other hand, these corrections

are more important for predicting standard curves in the assay of testosterone, using human plasma as the source of binding protein (13). It is likely that any system in which instability of the bound complex and inefficient adsorption of the free hormone are major factors would result in increased difficulty in accurately predicting standard curves.

88

S T ER O ID S

18:1

As experimental values for Rb closely approximated those predicted by theory over a wide range of experimentally determined estimates (n=70), K and q can be considered as accurate.

Otherwise agreement

between the different techniques could not be found. It is also important to be aware that other physico-chemical

con-

ditions, both practical and theoretical may affect the validity of using the mass action equations in its simple form and as modified by equation 5 in the present work.

For example, multivalent interactions,

ionic

strength, the nature of the ionic species (14) and allosteric binding could require similar study in other systems. The errors of standard curves can only be predicted when the modified mass action equation (the proposed model) provides a satisfactory description of the experimental curves.

Likewise, the estimation of

the constants of the mass action equations have to be sufficiently accurate and determined with a minimum of error.

It should be empha-

sised that in the estradiol/uterine cytosol assay the magnitude of the constants kl, k 2 and c2 are small, hence their associated errors are negligible.

Thus, the validity of the approach used here assumes negli-

gible errors for the instability and inefficiency constants of the system.

While this is true for estradiol/uterine cytosol assay,

not true in other systems

it is

(13).

Optimising indicates the limits which can be achieved with a particular set of conditions stant)

(tracer, binding protein and equilibrium con-

and also provides the ability to measure hormones at a desired

level in which the significance of the results in respect of error are defined.

This has been achieved for estradiol by combining theoretical

aspects of protein-ligand interaction with known errors that occur in

July 1971 the p r a c t i c a l assay.

s T ER O I D S 89 Application of these p r i n c i p l e s to the e s t r a d i o l

calf uterine-cytosol system has provided a prediction of quantitative features of estradiol standard curves, error, an optimum working range and levels of sensitivity for this steroid.

Previous studies of error

have demonstrated its relationship to hormone concentration

(12,15) but

have not applied the information to range and sensitivity for predicted series of standard curves. Optimum working range as defined here is the region of the standard curve in which the standardized error of estimating each value within the range has been minimised.

This result has been obtained by con-

sidering a standardized error band, a region in which variations are minimised, thus allowing the experimenter to interpret hormone concentrations free of large relative errors. Although standard curves can be developed for any number of q and h* values, it is not immediately apparent as to what constitutes a useful curve in respect of working range.

Intuitive approaches

(16)

suggest the position of maximum slope of any curve is an optimum region in which to assay.

However,

from consideration of the standardized

error in Figure II it is apparent that maximum slope of the standard curve is associated with large errors at low hormone concentration. Indeed, at concentrations near to zero, the magnitude of standardized error tends to infinity. A preferred approach is to choose a region of the assay curve in which errors approximate a minimum value.

This region for the estradiol

curve, q=55.9pg ml -I (Figure I), has been defined by the constant c and are the points hl and h2 in Figure If. approximately 0.2 and 0.1.

This corresponds to Rb values of

It should be noted that for fixed values of

the constant c the working range will vary in magnitude and direction as a function of q, h*, K and other discussed variables of the system.

90

S T ER O I D S

18:1

REFERENCES i.

Rosenblum, C., Anal. Chem. 29, 1740, (1957).

2.

Murphy, B.P., Engelberg, W., and Pattee, C.J., J. Clin. Endocr. 23, 293, (1963).

3.

Steroid Assay by Protein Binding, Acta Endocr. Suppl. 147 (1970).

4.

Ekins, R.P., Protein and Polypeptide Hormones, ed. M. Margonties, Excerpta Med., Intl. Congr. Series 161, p.630, Amsterdam, (1969).

S.

Schellman, J.A., Lumry, R., and Samuels, L.T., J. Am. Chem. Soc., 7_.66, 2808, (1954).

6.

Ekins, R.P., Newman, G.B., and O'Riordan, I . L . H . , Radioisotopes in Medicine - In Vitro Studies, Atomic Energy Comm. Symp. 13, 59, (1968).

7.

Wilson, D., Strachan, J.K., Humfray, A.A., Douglas, M., and Sarfaty, G., (in preparation).

8.

Berson, S.E., and Yallow, R.S., Proc. 49th Annual Meeting, Am. Soc. Clin. Invest., Atlantic City, N.J., (1957).

9.

Korenman, S., Tulchinsky, D., Eaton, L.W. Jr., Acta Endocr. Suppl. 147, 298, (1970).

i0.

Scatchard, G., Ann. N.Y. Acad. Sci. 51, 660, (1949).

ii.

Snedecor, G.W., and Cochran, W.G., Statistical Methods, 6th Ed. (1968), Ames, Iowa, U.S.A.

12.

Meinert, C.L., and McHugh, R.B., Math. Biosci. 2, 319, (1968).

13.

Wilson, D., Sarfaty, G., Clarris, B., and Crawshaw, K., Fourth Asia and Oceania Congr. Endocr; Abstract 69, (1971).

14.

0akey, R.E., and Emanuel, M.B., Nature 223, 5201, (1969).

15.

Rodbard, D., and Cooper, J.A., In Vitro Procedures with Radioisotopes in Medicine, Int. Atomic Energy Agency, p.659, Vienna, (1970).

16.

Murphy, B.E.P., Recent Progr. Hormone Res. 25, 563, (1969).

17.

Korenman, S., Perrin, L.E., and McCallum, T.P., J. Clin. Endocr., 29, 879, 1969.