Examples of the Application of Nonlinear ... - ACS Publications

8 Examples of the Application of Nonlinear Regression Analysis to Chemical Data Y. C. MARTIN and J. J. HACKBARTH Downloaded by MICHIGAN STATE UNIV on February 18, 2015 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0052.ch008

Abbott Laboratories, North Chicago, IL 60064

Nonlinear regression analysis is a powerful mathematical tool which has been used by a few chemists (1-8), but which has not achieved widespread application in chemistry. It is the purpose of this communication to illustrate some of the circumstances in which we have found the method to be useful. The objective is to encourage others to use this technique for other data-fitting problems. Introduction to Nonlinear Regression Analysis What is nonlinear regression? How does it differ from linear regression? Linear regression analysis is the process of finding the least-squares best fit of a set of data to a uni- or multidimensional equation in which the parameters (coefficients) to be fit are linear functions of the observed properties. The simplest linear regression analysis involves fitting data to a single parameter; as the name implies, the equation is that of a straight line: Y

i

=

b

0

+

b

x

1i

+

ε

eq. 1

In equation 1, Y is the observed value of the dependent variable of observation i , is the value of the independent variable for observation i, bo is the intercept of the line on the Y axis, bi is the slope of the line, and ε is the error. 1

Nonlinear r e g r e s s i o n a n a l y s i s i s the process o f f i n d i n g the least-squares b e s t f i t o f a s e t o f data t o an equation which i s not l i n e a r i n the parameters t o be f i t . A very simple example is: b

log(Yi) = l o g ( l + a X ) + ε ±

eq. 2

Equation 2 i s n o n l i n e a r i n a and b, the parameters t o be f i t .

153

In Chemometrics: Theory and Application; Kowalski, B.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

154

CHEMOMETRICS:

THEORY AND APPLICATION

Most chemical r e l a t i o n s h i p s are n o n l i n e a r : one f a m i l i a r example i s the f r a c t i o n o f an a c i d i o n i z e d as a f u n c t i o n o f pH and pK : a

α

eq. 3

= pK -pH 10 +1

I f some p h y s i c a l property were l i n e a r l y r e l a t e d t o a, then the observed v a r i a b l e s would be the p h y s i c a l property i n question and pH. The p K would be the parameter t o be f i t . How does one determine what i s the "best" f i t i n the n o n l i n e a r case? As with l i n e a r r e g r e s s i o n , the l e a s t - s q u a r e s c r i t e r i o n f o r best f i t i s commonly used. I t i s d e f i n e d as t h a t choice o f value f o r the a d j u s t a b l e parameters (bo and b i i n Equation 1 o r a and b i n Equation 2) such t h a t the sum o f squared d i f f e r e n c e between the observed Y j / s and those c a l c u l a t e d on the b a s i s o f the X j / s i s a minimum. Mathematically, t h i s i n v o l v e s t a k i n g the p a r t i a l d e r i v a t i v e with r e s p e c t t o each o f the parameters o f the equation f o r the sum o f the squared d i f f e r e n c e s , s e t t i n g t h i s d e r i v a t i v e equal t b zero, and s o l v i n g f o r the parameter. In the l i n e a r case t h i s a l l works very w e l l — the values o f bg and b^ can be e x p l i c i t l y determined. Nonlinear equations do not y i e l d such an easy s o l u t i o n f o r the minimum sum o f squares: hence, i n order t o f i n d the best f i t t o a n o n l i n e a r equation an i t e r a t i v e procedure must be used. Hence one s t a r t s with a s e t o f b e s t guesses f o r the values o f the parameters t o be f i t . The sum o f squared d e v i a t i o n s between observed and c a l c u l a t e d Y j / s i s then c a l c u l a t e d . By some a l g o r i t h m another s e t o f (better) estimates i s chosen and the sum o f squared d e v i a t i o n s i s c a l c u l a t e d from these values. T h i s process continues u n t i l , by some p r e - e s t a b l i s h e d c r i t e r i o n , f u r t h e r changes i n the estimates do not decrease the sum o f squared d e v i a t i o n s from the f i t .

Downloaded by MICHIGAN STATE UNIV on February 18, 2015 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0052.ch008

a

From t h i s b r i e f d e s c r i p t i o n i t can be seen t h a t n o n l i n e a r r e g r e s s i o n a n a l y s i s s u f f e r s from s e v e r a l apparent disadvantages compared t o l i n e a r r e g r e s s i o n . An i n i t i a l estimate o f the parameter values must be s u p p l i e d , an a l g o r i t h m f o r f i n d i n g the minimum sum o f squares must be provided, and many c a l c u l a t i o n s o f t h i s sum o f squares are r e q u i r e d f o r the s o l u t i o n t o one problem. In view o f these d i f f i c u l t i e s , the t r a d i t i o n a l method f o r f i t t i n g a n o n l i n e a r equation has been t o transform the equation i n t o a l i n e a r form and f i t the data t o t h i s transformed equation. The disadvantages o f such a l i n e a r i z a t i o n s t r a t e g y are t h a t i t may i n v o l v e hours o f a l g e b r a i c manipulation, that f r e q u e n t l y assumptions must be made as t o the range o f the data o r the importance o f terms i n a sum, and t h a t the r e s u l t i n g equation i m p l i c i t l y weights the data i n a manner which may not be c o n s i s t e n t with experiment.


8.

MARTIN AND HACKBARTH

155

Nonlinear Regression Analysis

On the o t h e r hand, advances i n computer technology have made these disadvantages o f n o n l i n e a r r e g r e s s i o n a n a l y s i s r e l a t i v e l y unimportant. I n i t i a l estimates are e a s i l y determined with standard g r a p h i c s techniques i n which the data and c a l c u l a t e d curve are d i s p l a y e d on a cathode ray tube. Algorithms t o f i n d minima and maxima are easy t o implement. F i n a l l y , the computations o f many sums o f squares i s a t r i v i a l task f o r a computer. The major advantages o f n o n l i n e a r r e g r e s s i o n a n a l y s i s are t h a t one f i t s data t o the equation as d e r i v e d and t h a t s i m p l i f y i n g assumptions are not necessary.


Computer Programs As noted above, i n the a n a l y s i s o f n o n l i n e a r problems we have found i t convenient t o be able t o d i s p l a y the data and a c a l c u l a t e d curve on a cathode ray tube. The parameters o f the c a l c u l a t e d curve may then be a l t e r e d u n t i l v i s u a l l y there i s a reasonable f i t t o the data. Hence, i n i t i a l s e a r c h i n g o f the parameter space f o r estimates i s not d i f f i c u l t , i n f a c t i t o f t e n r e v e a l s unsuspected f a c e t s o f the data. A companion program takes the same data f i l e and generates i n s t r u c t i o n s f o r a Calcomp p l o t t e r t o make a hard copy o f what was seen on the screen. We have w r i t t e n programs f o r two dimensions (one independent v a r i a b l e ) and three dimensions (two independent v a r i a b l e s ) . We use a simplex method t o f i n d the minimum sum o f squares (9,10). The a l g o r i t h m i n c l u d e s expansion and c o n t r a c t i o n o f dimensions o f the simplex. The s t a t i s t i c a l p r o p e r t i e s o f the b e s t f i t are c a l c u l a t e d by the equations used i n the BMD P - s e r i e s (11). General D e s c r i p t i o n o f Examples The f o l l o w i n g d i s c u s s i o n i l l u s t r a t e s the two p r i n c i p l e types of a p p l i c a t i o n which we have made o f n o n l i n e a r r e g r e s s i o n analysis. The f i r s t type o f a p p l i c a t i o n i s the c a l c u l a t i o n o f p h y s i c a l p r o p e r t i e s o f a molecule from experimental o b s e r v a t i o n s . We have chosen as an example the c a l c u l a t i o n o f the p K ' s o f a d i b a s i c substance from measurements o f absorbance vs pH. The other two examples are from our work on the q u a n t i t a t i v e r e l a t i o n s h i p between p h y s i c a l p r o p e r t i e s and b i o l o g i c a l potency of drug analogs. V a r i a t i o n i n a n t i b a c t e r i a l potency o f n i t r o p h e n o l s and erythromycins as a f u n c t i o n o f s t r u c t u r e and pH of the t e s t are examined. a

Example 1: pK 's o f a D i b a s i c Substance from Measurements a

Absorbance


156

CHEMOMETRICS:


Recently we wanted t o know the pK 's o f the f o l l o w i n g compound: a


H

Since i t i s not s o l u b l e enough t o t i t r a t e , we measured i t s u l t r a v i o l e t spectrum with changes i n degree o f p r o t o n a t i o n , ie_ as a f u n c t i o n o f pH. E s s e n t i a l l y the method d e s c r i b e d by A l b e r t and S e r j e a n t were followed (12). That i s , the u l t r a v i o l e t spectrum o f the compound was recorded i n b u f f e r s o l u t i o n a t s e v e r a l pH's near the suspected pK 's. F i g u r e 1 shows the absorbance a t 304 nm as a f u n c t i o n o f pH. Because measurable changes occurred over such a wide pH i n t e r v a l , we concluded t h a t the absorbance change r e f l e c t e d more than one pK . The f i r s t n o n l i n e a r f u n c t i o n we attempted t o f i t was t h a t given by A l b e r t and Serjeant (12): a

a

+

c a [H ] t

2

d

ο^ Κ Κ

+ c^tH+lK! +

η

λ

2

A =1

eq. 4 +

[H ]

2

+ [H+l^ + Κ Κ χ

2

i n which c i s the t o t a l c o n c e n t r a t i o n o f the compound; a , a^, and a are the molar a b s o r b t i v i t i e s o f the d i c a t i o n , monocation, and n e u t r a l forms r e s p e c t i v e l y ; and and K 2 are the f i r s t and second a c i d d i s s o c i a t i o n constants. T h i s form of the equation l e d t o problems i n f i t t i n g , p a r t i c u l a r l y with data from t r i b a s i c analogs. S o l u t i o n o f Equation 4 i n terms o f [H ] l e d t o an a l g e b r a i c a l l y very complex r e l a t i o n s h i p . So we manipulated Equation 3 f u r t h e r u n t i l we r e a l i z e d the f o l l o w i n g : t

d

n

+

+

A([H ]

2

+ [Η+ΙΚχ + K]K ) 2

+

= c a [H ] t

d

2

+ CtamtH+ΙΚχ + c a K i K 2 t

n

or 0 = (c a t

+

d

- A) [H ]2 + (

C t

a

m

- A) t H + l ^ + ( c a t

n

- Α)Κ Κ χ

2

Since a l l terms add up t o a sum o f zero, the sum o f those with a p o s i t i v e s i g n must equal the sum o f those with a negative s i g n . The former sum i s l a b e l l e d "POS", and the l a t t e r , "NEG". Hence :


8.



157

POS » -NEG or POS 1 -NEG or 0 «


Hence we chose t o f i t the f o l l o w i n g f u n c t i o n : POS H

P calc '

l o

9 -NEG

eq. 5

+ pH obs

The computer program t e s t s the s i g n o f each term i n the equation f o r each data p o i n t , sums the p o s i t i v e and negative terms s e p a r a t e l y , and p l a c e s the sum o f the p o s i t i v e values i n the numerator and the sum o f the negative values i n the denominator. From t h i s type o f a n a l y s i s o f the data, the f o l l o w i n g values were c a l c u l a t e d : ρΚ pK c a c a χ

2

c

a

w

a

t

d

t

m

« » «

2.31 ± .02 4.45 ± .02 0.508 ± .001 0,316 ± .002

s

t n e x p e r i m e n t a l l y e s t a b l i s h e d from measurements a t high pH t o be 0,181. The s t a t i s t i c s o f f i t a r e : R2 - .9997,

s = .0253, w i t h 6 degrees o f freedom.

F i g u r e 1 shows the curve c a l c u l a t e d on the b a s i s o f t h i s f i t . From every standpoint, the use o f n o n l i n e a r r e g r e s s i o n a n a l y s i s allowed the maximum i n f o r m a t i o n t o be gained from the data. The p r e c i s i o n o f f i t i s s a t i s f a c t o r y c o n s i d e r i n g the low number o f experimental p o i n t s i n v o l v e d and the c l o s e n e s s o f the two pK 's. We have used t h i s method o f c a l c u l a t i o n t o f i t the pKa's o f t r i b a s i c substances from absorbance measurements a t s e v e r a l pH's, and a l s o t o f i t the p K ' s o f the i n d i v i d u a l a

a


158

CHEMOMETRICS: THEORY AND APPLICATION

A-44296 ABSORBANCE AT 304 V3 PH


.S5

.AS

U

-3S

U

Figure 1. The variation in absorbance of Compound I at 304 nm as a function of pH. The curve is calculated from Eq. 4, pK, = 2.31, pK = 4.45, and the absorptivity of the diprotonated species times the concentration equal to 0.580, that of the monoprotonated species equal to 0.316, and that of the nonprotonated species equal to 0.181. 2

SCHEME I Compartment-. Aqueous no. 1

Nonaqueous

Aqueous no. 2

Receptor

Equilibria

jr jr jDr CDjr


8.


159


amines o f p o l y b a s i c molecules from the pH dependence o f the change i n the carbon-13 NMR chemical s h i f t .


Example 2;

A n t i b a c t e r i a l Potency o f Nitrophenols

Our p r i n c i p l e r e s e a r c h i s i n the a n a l y s i s o f the r e l a t i o n s h i p s between chemical and b i o l o g i c a l p r o p e r t i e s o f compounds. We have r e c e n t l y become aware o f the importance o f the form o f the equation t o which the data are f i t . S p e c i f i c a l l y , a c o n s i d e r a t i o n o f the g e n e r a l p r o p e r t i e s o f the b i o l o g i c a l system i n which the data was generated coupled with a statement o f the l i n e a r f r e e energy assumptions which may be made about the s t r u c t u r e - a c t i v i t y r e l a t i o n s h i p s o f each p a r t o f t h i s b i o l o g i c a l system can l e a d t o some very u s e f u l i n s i g h t s i n t o the form o f the equation which should be used t o analyze the data. Our f i r s t p u b l i c a t i o n on such model-based equations concerned b i o l o g i c a l systems i n which the drug i s e q u i l i b r a t e d between s e v e r a l compartments; t h i s model a p p l i e s p r i n c i p a l l y to i n v i t r o t e s t s (13). The equation d e r i v e d f o r a f o u r compartment model (Scheme I) i s s u f f i c i e n t t o c o r r e l a t e the data from the examples d i s c u s s e d i n t h i s paper. Compartment one i s the aqueous compartment o u t s i d e o f the b a c t e r i a , t h a t i s , the medium; compartment two i s the aqueous compartment i n s i d e o f the b a c t e r i a ; compartment three i s a nonaqueous, nonreceptor compartment; and compartment four i s the r e c e p t o r . Compartments one and two become i d e n t i c a l i f t h e i r pH's are identical. From simple l i n e a r f r e e energy assumptions about the r e l a t i o n s h i p between b i n d i n g i n a compartment and h y d r o p h o b i c i t y o f the v a r i o u s analogs as measured by the octanol-water p a r t i t i o n c o e f f i c i e n t (Ρ), the f o l l o w i n g equation may be d e r i v e d : α 1 + Ζ — — I î1-ou + X log(1/C) = l o g e(

+ dP

1 + ζ l-Olo

c

1-Oh

ι-α

2

The symbol a i n d i c a t e s the f r a c t i o n o f the compound i n the i o n i z e d form a t the pH o f the p a r t i c u l a r compartment. The dependent v a r i a b l e i s the negative l o g a r i t h m o f the c o n c e n t r a t i o n , C., o f the compound r e q u i r e d t o produce the d e f i n e d b i o l o g i c a l response. The independent v a r i a b l e s are and P. F i n a l l y , the parameters t o be f i t i n the r e g r e s s i o n a n a l y s i s are Z, a, b, c, and d. From the model i t may be shown t h a t Ζ i s the r e l a t i v e a f f i n i t y f o r the r e c e p t o r o f the i o n


6

CHEMOMETRICS:

160



-6.0

1.5 LOG Ρ

υ*

2.0

2.5

1

2.5

Figure 2. Tu?o uieu* of the variation in antibacterial activity of nitrophenoh as a function of foaV and log(l — a). The curve is calculated from Eq. 7. compared with t h a t o f the n e u t r a l form o f a drug. A, b, c, and d are r e l a t e d t o the extrathermodynamic r e l a t i o n s h i p s between b i n d i n g constants t o the nonaqueous and receptor compartments and logP. A s e r i e s o f s i x n i t r o p h e n o l s had been t e s t e d vs E. c o l i a t pH's 5.5, 6.5, 7.5, and 8.5 (14). A l l 24 data p o i n t s f i t the f o l l o w i n g equation (15): 1 + Ζ 1-α + χ

log(1/C) • l o g 1 + 1-α

+ dP° -κ1-α


eq. 7

8.



161

i n which: log(Z) c log(d) X* pK a

» -3.60 » 4.95 = -6,96 = 5.41 = 3.26

± ± ± ± ±

0.17 0.81 1.61 0.10 0.73

The s t a t i s t i c s o f f i t a r e :


R

2

+ 0.909,

s » 0.209, and η » 24.

F i g u r e 2 i s a p l o t o f l o g ( l / C ) as a f u n c t i o n o f logP and log(l-a). These parameters correspond t o a case i n which there i s no v a r i a t i o n i n hydrophobic bonding t o the r e c e p t o r w i t h i n the s e r i e s . However, there i s a v a r i a t i o n i n hydrophobic bonding t o the nonaqueous compartment. Only one aqueous compartment (at the pH o f the e x t e r n a l medium) i s needed t o f i t the data. A s l i g h t l y b e t t e r f i t i s found when the p K o f p i c r i c a c i d i s allowed t o vary; t h i s p K i s i n d i c a t e d with an a s t e r i s k above. The value o f Ζ i n d i c a t e s t h a t the i o n i c form o f any n i t r o p h e n o l has 4000 X l e s s potency than the n e u t r a l form o f the same compound; t h i s value i s s i g n i f i c a n t l y d i f f e r e n t from zero. a

a

Example 3: A n t i b a c t e r i a l Potency o f N-benzyl Erythromycins (15) In t h i s s e r i e s one o f the hydrogen atoms o f the dimethylamino group o f erythromycin, s t r u c t u r e below, was r e p l a c e d w i t h a s u b s t i t u t e d phenyl group. CH

3

CH,

CH

IA *=OH Erythromycin A IB R=H Erythromycin 6

The l o g r e l a t i v e p o t e n c i e s o f these compounds as measured a t pH 6.00, 7.00, and 7.65 i s w e l l f i t by the f o l l o w i n g equation:



162

eq. 8 pK -6.0 1 + 10 pKa-pHjL pK -6.0 1 + 10 + a ( l + 10 a

log(1/C) « l o g

+ eEs + X

a

i n which: log(a) = 1,30 * 0.23 X = 2.64 ± 0.17 e = 0.303 ± 0.065


The s t a t i s t i c s o f f i t a r e : R

2

= .844,

s = 0.186,

η = 38.

Thé s u b s t i t u e n t constant Es i s the T a f t s t e r i c parameter f o r the r e l a t i v e s i z e o f the s u b s t i t u e n t a t p o s i t i o n 4 o f the phenyl r i n g . F i g u r e 3 i s a p l o t o f potency as a f u n c t i o n o f the p K o f the compound and the pH o f the t e s t . Both the numerator and denominator o f Equation 6 are dominated by a s i n g l e term because the amount o f i o n i c form o f the drug bound t o the r e c e p t o r i s l a r g e r than the amount o f n e u t r a l form o f the drug bound t o the receptor, and because most o f the drug added t o the system remains i n the e x t e r n a l aqueous compartment. As a consequence o f t h i s i t was not p o s s i b l e t o independently f i t a Z, and a . Therefore, pH was a r b i t r a r i l y s e t a t 6.0, and a h y b r i d constant, a, was fit: H a

l f

2

2

P

2

a = Ζ Within the l i m i t s s t a t e d p r e v i o u s l y , the values o f the constants found by the n o n l i n e a r r e g r e s s i o n a n a l y s i s l e a d t o c e r t a i n t e n t a t i v e c o n c l u s i o n s with r e s p e c t t o the determinants o f potency i n t h i s s e r i e s . F i r s t , two aqueous compartments o f d i f f e r e n t pH must be considered i n the a n a l y s i s . Additionally, the pH o f the aqueous phase w i t h i n the b a c t e r i a remains r e l a t i v e l y constant when the pH o f the e x t e r n a l phase i s v a r i e d from 6.0 to 7.65. Second, the i o n i c form o f these compounds c o n t r i b u t e s s i g n i f i c a n t l y t o potency. I t i s not p o s s i b l e t o e s t a b l i s h the r e l a t i v e potency o f the i o n vs t h a t o f the n e u t r a l form, however, because o f the problem d i s c u s s e d above. T h i r d , there i s no evidence f o r hydrophobic bonding o f the phenyl r i n g and i t s s u b s t i t u e n t s t o the r e c e p t o r o r t o an i n e r t phase. Fourth, s u b s t i t u e n t s a t the para p o s i t i o n decrease potency by a s t e r i c e f f e c t .



8.



Figure 3. Two views of the variation in antibacterial activity of N-benzyl erythromycins as a function of pH of the medium and pK of the compound. The curve is calculated from Eq. 8. a

These examples show some o f the uses which we have made o f nonlinear regression analysis. Once one has a l i t t l e experience w i t h a program i t can e a s i l y become a r o u t i n e l y used and extremely h e l p f u l t o o l f o r the mathematical a n a l y s i s o f chemical data.


163

164


Literature

Cited


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