BIOCHIMIE, 1973, 55, 277-282.

Simultaneous Determination of Equilibrium Constants and Thermodynamic Functions by Means of Relaxation

Amplitude Measurements.

Darwin THUSIUS.

Laboratoire d'Enzymologie Phgsico-chimique et Mol~culaire, Facult~ des Sciences, 91 Orsay, France.

(9-3-1972).

Abstract. - - Non-,linear regression is applied to the problem of determining thermo- dynamic parameters of one-st~p equilibria from chemical relaxation amplitudes. Details are worked out for the simultaneous evaluation of l,[ and AH for 1:1 complex formation and applications to two different systems of biological interest are cited. The method includes modified procedures which take into account signal changes due to non- chemical effects. The approach is extended to the analysis of the amplitudes of a rapid binding reaction followed by one or more slow isomerizations.

INTRODUCTION.

In recent years chemical re laxat ion methods for measur ing rates of very rapid react ions in solu- t ion have been applied to a variety of biological problems [1, 2i. In these techniques a react ion solution at equi l ib r ium is d is turbed by a sudden change in some external parameter such as tem- perature or pressure. If the physica l change is sufficiently rap id there will be a t ime lag du r ing which the chemical system approaches its new equi l ib r ium posit ion. For small pe r tu rba t ions the time course of reequi l ib ra t ion is described by a series of exponent ia l curves, the t ime constants of which are related to the e lementary rate constants of the unde r ly ing react ion mechanism.

In addi t ion to kinet ic data, re laxat ion experi- inents can also yield t he rmodynamic in format ion if the ampli tudes of the decay curves are analy- zed. Although in most re laxat ion studies to date the de te rmina t ion of e lementary kinet ic constants has been emphasized, there has recent ly been expand ing interest in the exploi tat ion of ampli- tude results [3-111. For example, Eigen and Wink- ler [3] have poin ted out that an ampli tude treat- ment can provide a more sensit ive means of deter- m i n i n g equi l ibr ium constants and t h e r m o d y n a m i c funct ions (e.g., A H, A V) than most usual methods. Classical and ampli tude t i t ra t ion curves have been compared [3], and a t r ia l -and-er ror method has been developed for de te rmin ing K and AH for a non-absorb ing system in the presence of an indi- cator [3, 4].

In the present article I explore the possibi l i ty of t reat ing ampli tude data for one-step react ion systems by non- l inear regression. The analysis for 1:1 complex format ion is discussed in detail and prat ical appl icat ions to two systems of biolo- gical interest are given. Theoret ical ampli tudes for a b i nd i ng react ion followed by one or more slow isomerizat ions are also considered. The value of an overall ampl i tude t i t ra t ion as a p r e l imina ry step in a relaxat ion study is emphasized.

Linearization of Amplitude Equation.

The general expression of eq 1 relates the expe- r imenta l ampli tudes to the equ i l ib r ium concentra- t ions of an overall one-step react ion system [8, 10!.

~po = (A~ ~lnK)F (1)

apo = P t : o - - P~-

¢ i - ~ P/~C~

K = HCiv,

r = l / Z (v,)~C~

When P is optical densi ty or percent t ransmis- sion, the ¢i are propor t iona l to molar ext inc t ion coefficients. The s toiehiometr ic coefficients v i are defined as posit ive for par t ic ipants on the r ight side of the react ion equat ion and negative for those on the left. The quant i ty MnK is proport io-

278 D. T h u s i u s .

nal to a the r lnodynamic function. For example, in a tempera ture- jump technique,

AH ~lnK _ ~,T (2)

RT ~

AH = E v i l l i

In prac t ice re laxat ion ampli tudes are measured at different analyt ical concent ra t ions of the reac- tants. If the equi l ib r ium constant is known, F va- lues can be calculated for a set of analyt ical con- centra t ions and the p roduc t Aq~81nK can be estim- ated f rom the l inear regress ion of 8P ° on F. If in addi t ion ~I) is known, 51nK can be calculated.

Now cons ider the more general case w h e r e K is not known. When the F t e rm can be expressed expl ic i t ly in terms of analyt ical concent ra t ions and K, eq 1 can be approx imated by a Taylor ' s series expansion in w h i c h h igher than first o rder terms are d ropped [121.

~po ~ ~1 X, + ~ X~ (3)

~i = A O ~lnK X I ---- F•.

~ = ~,(K, - - K.) X~ = (~ F/D K)K.

Here K o is an estimate of the equ i l ib r ium cons- tant and K 1 is an i m p r o v e d e s t i m a t e ; FK° and (DF/DK),~° represent the pe r t inen t F t e rm and its first der iva t ive expressed in terms of K 0 and ana- lyt ical concentra t ions . If an approx imate K value is available, the regress ion coefficients (11 and a2. can be calculated by a mul t i l inear least-squares analysis of the ampl i tude data. A second estimate can be calculated f rom the re la t ion K 1 : o~2/a t + K 0. If K 1 is then te rmed K0, the process can be cont inued unti l the quant i ty (K 1 - - K0)/K 1 satis- fies a test for convergence. In this way statisti- cally refined values of K and AO51nK are obtained, together wi th their s tandard errors [121. The lat ter can p rov ide an object ive means of dec id ing whe the r the ampl i tude results are consis tent wi th a par t icu la r reac t ion s to ich iomet ry [13!.

It can be added that the p rob lem of we igh t ing factors in the above p rocedure wil l usually be s t ra ight forward . If the absolute e r ro r in the expe- r imenta l ampl i tudes is constant, the weigh t ing factor is uni ty ; if the relat ive e r ror is constant, the we igh t ing factor is (1/5po)2 [12]. Final ly , r ap id data p rocess ing for most appl ica t ions does not requ i re a digital compute r but can be real ized wi th a p rogrammable e lec t ronic calculator.

Even if the equi l ib r ium constant of an overal l one step system can be de te rmined independent ly , it is always of interest to analyze the re laxat ion ampli tudes wi th a stat is t ical ly sound i tera t ive me- thod. A significant di f ference be tween the value of an ampl i tude-der ived equi l ib r ium constant and the value de te rmined by a static measurement suggests an under ly ing react ion mechan i sm more complex than ind ica ted by the overal l s toichio- metry and the observat ion of a single re laxat ion c u r v e .

Format ion o[ 1:I Complexes.

Now cons ider the appl ica t ion of the above curve-f i t t ing p rocedure to the fo l lowing one-step system of general interest .

A - - ~ B ~ _~_ AB (4)

AO = • AB -- (I~B -- ¢I) A

AH = H-AB -- H-B - - 'H--. (5)

K = AB/A" B

( r A B

In the general case whe re no res t r ic t ions are placed on the relat ive concent ra t ions of A and B, it can be shown from mass act ion and mass con- servat ion cons idera t ions that the independen t va- r iables of eq 3 are

1 - - 1/(l--4p/s~-)l/~ x, = (r)~o = - - K ; ~ 2 (6)

D F

_ _ X j _ _ K:(s --4p),12 [(s~--4p)t /2--2(K'~-~-X,) ] ( 7 )

w e r e s = (A ° + B ° + K °-1) a n d p = A ° "B ° [14].

To assure convergence and to obtain accurate va- lues for K and AOMnK, the quant i ty X 1 must be var ied over a sufficiently large range. In prac t ice this can be achieved in an << ampl i tude t i t ra t ion >> where the concent ra t ion of, say, componen t A is held constant and the ampl i tudes are measured after succesive addi t ions of component B. The F te rm in these condi t ions is a bel l -shaped funct ion of the analyt ical concen t ra t ion of B ; the profile of the t i t ra t ion curve becomes sharper as the quant i ty A ° .K increases [3]. Fur the rmore , the fo l lowing simple re la t ion has been der ived for the value of B ° co r re spond ing to the max imum

ampli tude.

(m)~P%~, = A ° + K-' (8)

A provis iona l estimate of K for the i tera t ive calcu- lation can therefore be readi ly obta ined by gra-

BIOCHIMIE, 1973, 55, n ° 3.

Equilibrium Constants and Thermodynamic Functions. 279

phical means (cf. Figs. 1-3). However, it should be added that in the appl icat ions given below an order of magni tude est imation of K was sufficient to obtain convergence.

Usually re laxat ion ampli tudes are estimated by the t ime-consuming process of extrapolat ing plots of ln5P t versus t ime t o t : 0. If only one relaxa- t ion process is observed, its ampl i tude can be obtained directly by measur ing the square-wave signal change produced at oscilloscope sweep times slow relative to the chemical re laxat ion t ime [3, 4, 10]. In addit ion, this type of measure- ment is quite precise since one can work wi th the smallest electronic bandwid th of a given instru- ment ' s detection system. Often, however, the external per turba t ion produces a significant change in volume or has a direct effect on the va- lue of Aq). Since physical effects are usual ly much more rap id than chemical relaxations, they can often be suppressed if the re laxat ion ins t rumen t is equipped wi th an << automatic offset >>. When such a device is not available, or when the physi- cal and chemical effects overlap on the time axis, it is necessary to correct the overall ampli tude for the physical cont r ibu t ions [3, 10~.

For 1:1 complex format ion where only A and AB absorb, eq 3 can be wr i t ten as

~P%o,= ~ X~ + a~ X~ (9)

where XI and X 2 are given in eqs 6 and 7 and 8P¢°o is the overall ampli tude corrected for the physical effects of the external per turbat ion .

2p ~P%o,. = ~P°L~--~A~" (10)

s + ( s ~ -- 4pP/~

~P°~o,~ = BP°totat -- ~P~= o (11)

~A¢" = (~A¢/A¢ + ~V/V) A

Ao

Here V = volume and ~i~_o and 8P~=~ are the total ampli tudes observed in the absence and at sa tura t ing concent ra t ions of B. If sufficiently pre- cise values for ~P~ :o and ~P~== can be deter- mined (so that the unce r t a in ty in ~P~o~ reflects essentially the exper imenta l error in ~ P ~ ) , a modif ied non- l inear regression can be employed in which the dependent variables ~P~o~ of the ith i terat ion are calculated with the equi l ibr ium constant of the preeeeding i terat ion.

Alternatively, if K is k n o w n but for reasons such as solubil i ty or self-association of B the para- meter 5P~=~ cannot be de termined exper iment-

ally, the constants A¢51nK and 8A¢* can be esti- mated wi th a non-i terat ive regression,

~P%'ot~a = a~Xt --[- a~X~ (12)

aj ---- A ¢ ~lnK X, = F

2p X~

s -~- (s ~ - - 4p)1/2

where X 1 and X 2 are calculated wi th the experi- mental equi l ib r ium constant.

Final ly , all three parameters ACMnK, K, and 5Aq)* can be estimated by ana lyz ing the overall ampl i tude data wi th a mult iple i terative method. Firs t eq 12 is rea r ranged to

2 p ~P~ , = A~ ~lnK(F + a s + (s ~ - - 4p)1/~) (13)

a = ~Aq)'/A(I)MnK

Then eq 13 is expanded in a Taylor 's series by taking first derivatives wi th respect to K and a. The result is

~P,gt~, = a,X, + a., X. ,+ % X3 (14)

~1 = h ~ l n K

( K? ao)X:, X, : : \ (s~__-~-p)t/2 -+-

~.~ = ~l(K, -- Ko) X._, = (Xa/(s~-- 4p)t/'0

( 2X~K~ ~ 2 X:~ K'~ 1 - - ao s ~ _ 4 p , s~__4 p

~ = ~,(a, -- ao) 2p

X:~ s + (s ~ - - 4p)t/"~

Successive l inear regressions are per formed unt i l in i t ia l estimates of a and K converge to constant values. A program of this type has been wr i t t en and found to satisfactori ly converge for different sets of ar t i f icial data. In this appl ica t ion it is essential to obtain a considerable number of data points and to approach the highest pract icable concent ra t ions of B.

It is of interest that optical re laxat ion ampli- tudes can be convenien t ly measured over a wide f requency range by independen t ly thermosta t ing the cells of a double beam speetrophotometer . The ampli tude t i t ra t ion for 1:1 conlplex format ion can be realized by placing ident ical solutions of com- ponent A in the two cell compar tments and re- cording the tempera ture difference spectra pro- duced by adding aliquots of B to the reference

o .

cell. Under these condi t ions 5Pt~ot~ = 8Pto~, and an i ndependen t de te rmina t ion of 8P°r= o is not

necessary.

BIOCHIMIE, 1973, 55, n ° 3.

280 D. Thusius.

Practical Applications.

I n t h e f o l l o w i n g a p p l i c a t i o n s t h e i t e r a t i v e p r o - c e d u r e of eq 9 h a s b e e n u s e d to e s t i m a t e equ i l i - b r i u m c o n s t a n t s a n d e n t h a l p i e s f o r 1:1 c o m p l e x f o r m a t i o n . C a l c u l a t i o n s w e r e c a r r i e d o u t w i t h a W a n g 370 e l e c t r o n i c c a l c u l a t o r . T h e a b s o l u t e e r r o r i n t he a m p l i t u d e m e a s u r e m e n t s w a s e q u a l to t h e i n s t r u m e n t ¢ n o i s e ~ a n d w a s e s s e n t i a l l y c o n s t a n t . A w e i g h t i n g f a c t o r of u n i t y w a s t h e r e f o r e a s s u m e d i n t he l e a s t - s q u a r e s a n a l y s i s .

Trypsin-proflavin. - - T h e b i n d i n g of t h e a c r i - d i n e d y e p r o f l a v i n to t r y p s i n to f o r m a 1:1 com- p l e x is a c c o m p a n i e d b y a p e r t u r b a t i o n of t h e p r o - f l a v i n a b s o r p t i o n s p e c t r u m [5]. A l i q u o t s of t r y p - s i n w e r e a d d e d to a p r o f l a v i n s o l u t i o n at n e u t r a l p H a n d t h e o v e r a l l a m p l i t u d e s w e r e m e a s u r e d in

+ .02

+,01

• Q

o I t I

X I OaM E

Fro. 1. - - Tryps in-prof lav in ampl i tude t i t ra t ion . Con- di t ions : Prof lavin = 0.24 × 10-4 M, 0.14 M Tris CI-, pH 8.1, 0.02M CaC12, Ti~ti~l ---- 11.0°C, ST ----- 4.4°C. Overall ampl i tudes are repor ted as changes in percent t r ansmiss ion . The l ine was e~leulated w i t h the expe- r imen ta l (S I°/I°)E°=O value and the K and Ae4~.: SInK values ob ta ined by the i tera t ive procedure of eq 9 ( (SI° / I°)~,= ~ assumed to be negligible).

t h e m i l l i s e c o n d r a n g e w i t h a M e s s a n l a g e n t em- p e r a t u r e - j u m p s p e c t r o m e t e r (Fig. 1). T h e c o n d i - t i o n s 2 X 10 .4 M ~ E ° ~ po w e r e m a i n t a i n e d to p r e v e n t a g g r e g a t i o n of e n z y m e o r l i g a n d . T h e

s m a l l s i g n a l c h a n g e i n t h e a b s e n c e of e n z y m e is v e r y p r o b a b l y due to t h e d i r e c t e f fec t of t e m p e -

r a t u r e o n e469 [10]. A l t h o u g h a r e l i a b l e 6 ~ - v a l u e c a n n o t b e e s t i m a t e d u n d e r t h e p r e s e n t e x p e r i m e n t a l c o n d i t i o n s , u s e f u l i n f o r m a t i o n w a s o b t a i n e d b y a n a l y z i n g t h e o v e r a l l a m p l i t u d e s a c c o r d i n g to e q 9 a n d a s s u m i n g 8 P ~ = ~ to b e ne- g l ig ib le , T h e i t e r a t i o n gave ,K ---- (8.6 _ 0.3) 103 M -1 a n d hs~69 blp.K = (2.10 __ 0.02) 103 OD M -1 c m -1. T h e s m a l l s t a n d a r d e r r o r s i n d i c a t e t h a t t h e s toi - c h i o m e t r y i n t h e c o n c e n t r a t i o n r a n g e s t u d i e d is a d e q u a t e l y d e s c r i b e d b y eq 4. T h e p a r a m e t e r s /~e469 SinK a n d he469 [5] a l l o w o n e to e s t i m a t e AH; a v a l u e of - - 3 . 0 _ 0.4 k c a l / m o l e w a s c a l c u l a t e d , w h i c h is i n s a t i s f a c t o r y a g r e e m e n t w i t h e a r l i e r r e s u l t s [5, 10]. F u r t h e r m o r e , t h e e q u i l i b r i u m c o n s - t a n t is i n good a g r e e m e n t w i t h t h e c o n s t a n t eva lu - a t e d b y a s p e c t r o p h o t o m e t r i c t i t r a t i o n [5]. An e n t r o p y of - - 8 e.u. w a s c a l c u l a t e d f r o m K a n d AH.

. 03

-£ 0 .02

.01

i I I S 10 15

e'a2o3 x 104 M

Fro. 2. - - Th iosu l f a t e -aquocoba lamin ampl i tude tiLration. Condi t ions : coba lamin ° ---- 1.8 × 10-4 M, 0.5 M KNO3, pH 5, Ts~mp~e =- 26°C, (Ts,mp~e-Tr.~, . . . . . ) = 13.5°C. Ident ical solut ions of aquocoba lamin were placed in independen t ly t he rmos t a t ed ceils of a Gary Model 14 double -beam spee t rophotomete r and a l iquots of a concent ra ted th iosu l fa te solut ion ~vere added to the reference cell. The b roken l ine represents SOD°4~ at sa tura t ion . The solid l ine was calculated wi th th is pa rame te r and the K and Ae~,~SInK values obtained by the i te ra t ive procedure of eq 9.

Aquocobalamin-Thiosul[ate. In t h e t r y p s i n - p r o - f l a v i n a m p l i t u d e t i t r a t i o n t h e l i m i t i n g c o n d i t i o n K -1 > > A o w a s a p p r o a c h e d . I t is a lso of i n t e r e s t to e x a m i n e a m o r e g e n e r a l ease w h e r e K-1 ,~ A o. T h e r e p l a c e m e n t of vca te r i n a q u o c o b a l a m i n b y t h i o s u l f a t e [15, 16] w a s c h o s e n as a n e x a m p l e . Due to t h e s l o w r e a c t i o n r a t e at l o w l i g a n d c o n c e n - t r a t i o n s , t h e o v e r a l l a m p l i t u d e s w e r e m e a s u r e d b y

BIOCHIMIE, 1973, 55, n ° 3.

Equilibrium Constants and Thermodynamic Functions. 281

i n d e p e n d e n t l y t h e r m o s t a t i n g the s a m p l e and re- f e r e n c e cel ls of a Cary Model 14 s p e c t r o p h o t o - m e t e r and r e a d i n g 8OD~95 w i t h a 0-0.1 s l i d e w i r e (Fig. 2). T h e e x t i n c t i o n coef f ic ien t s of c o b a l a m i n s a re qu i te s ens i t i ve to t e m p e r a t u r e [16, 17]. H o w - ever , in th is t y p e of t i t r a t i o n i t is no t n e c e s s a r y to m e a s u r e 8P~= o (i.e., 8P~t~ : 8P~ota I ) and 8 P ~ = ~ cou ld be eas i ly d e t e r m i n e d at s a t u r a t i n g c o n c e n t r a t i o n s of th iosu l fa te . The resu l t s of the i t e r a t i ve f i t t ing w e r e : K-1 = (1.3.6 _+ 0.11) 10 -4 M, Ae495 81nK = 769 -4- 18 O,D M-I cm-L T h e p a r a - m e t e r A~49 ~ was ob ta ined by sa tu ra t ing aquoco- b a l a m i n w i t h l i g a n d and m e a s u r i n g the d i f f e r e n c e spec t ra . The t h e r m o d y n a m i c f u n c t i o n s w e r e esti- m a t e d to be AH = - - 3 . 2 (_+ 0.5) k c a l / m o l e and AS = + 7 e.u. These resu l t s a re in good agree- m e n t w i t h l i t e r a t u r e va lues for the e q u i l i b r i u m c o n s t a n t [15, 161 and an a p p r o x i m a t e AH ca lcu l - a ted f r o m the a q u a t i o n and a n a t i o n a c t i v a t i o n ene rg ie s [163.

Relaxation Amplitudes as Probes for Mechanism. Because t h e y a re p r e c i s e and r ap id , o v e r a l l am-

p l i t u d e t i t r a t i ons can p r o v i d e use fu l p r e l i m i n a r y i n f o r m a t i o n on p l a u s i b l e m e c h a n i s m s . Befo re one beg ins a de t a i l ed k i n e t i c s tudy of an a p p a r e n t one- s tep r e a c t i o n , fo r example , it is of i n t e re s t to d e t e r m i n e w h e t h e r the ove ra l l a m p l i t u d e da ta a re cons i s t en t w i t h the s t o i c h i o m e t r y d e d u c e d f r o m s ta t ic e q u i l i b r i u m m e a s u r e m e n t s .

T h e va lue of a p r e l i m i n a r y a m p l i t u d e t i t r a t i o n wi l l be i l l u s t r a t ed fo r the f o l l o w i n g mul t i - s t ep me- c h a n i s m ,

A + B ~ " (AB), ~ ) " (AB).2 ~( ~- . . . . . . (AB).-~ ~ > " (AB)n (15)

w h e r e the (AB) i r e p r e s e n t an a r b i t r a r y n u m b e r of d i f f e r en t i somers . I t is w e l l k n o w n tha t the m i c r o - s c o p i c s t o i c h i o m e t r y of th is sys tem c a n n o t be de- t e r m i n e d by e q u i l i b r i u m m e a s u r e m e n t s a lone ; the o b s e r v e d ove ra l l s t o i c h i o m e t r y is i d e n t i c a l to tha t of eq 4. The ove ra l l e q u i l i b r i u m c o n s t a n t fo r the mu l t i s t ep r e a c t i o n can be w r i t t e n as

(AB)i

Kapp - - = K t (1 "Jr- "() A ' B

(AB)I K I - - - -

A ' B

(AB)i

y - -

(AB),

N o w assume tha t 1) the b i n d i n g step is obser - vab le a n d m u c h fas te r t h a n the i s o m e r i z a t i o n s and

2) the r e - e q u i l i b r a t i o n of the i s o m e r i z a t i o n s teps is too s low to be de t ec t ed w i t h a p a r t i c u l a r re- l axa t ion t e c h n i q u e . The resu l t s of a c lass ica l t i t ra- t ion and the d e t e c t i o n of a s ing le r e l a x a t i o n c u r v e w o u l d a lone suggest the p r o c e s s of eq 4 as a l ike ly m e c h a n i s m . The ques t i on is t hen : w h e n feas ib le , w o u l d an ove ra l l a m p l i t u d e t i t r a t i o n u n d e r these c o n d i t i o n s a l l ow one to d i s t i n g u i s h b e t w e e n the m o d e l s of eqs 4 and 15 ?

The f o l l o w i n g gene ra l e x p r e s s i o n can be der i - v e d fo r t he a m p l i t u d e of t he fast r e l a x a t i o n p r o - cess in the above example .

~Pi' = (Aq),~lnK,)F, (16)

',-~ ~ ) i : ( ~ ( A B ) i - - (]) A - - (I) B (17)

AH, 8T MnK l __ R.l.:~

( s - - (s 2 - - K a p p - !

\ F I __ (18)

(s'-' - - 4p)1/2 -~- Kanp -1 .y

w h e r e s = AO + B o + Kam ) 1 and p is de l i ned b e l o w eq 7. I t is c l ea r t ha t i f v - - O , t h e n Kap p - - > K 1 and the a m p l i t u d e p ro f i l e w i l l a p p r o a c h tha t of a s i m p l e b i n d i n g r eac t i on . In Fig. 3 a set of t h e o r e t i c a l a m p l i t u d e c u r v e s h a v e been c a l c u l a t e d for d i f f e ren t 7 values .

r,

Y:o

0 2

o i Y:5

o

B °

FIG. 3. - - Theoretical ampli tudes for a fast binding reaction follo~ved by slow isomerizations. The curves were calculated from eq 18 for the case A ° = 1, K.vp-1 = 5.

T h e m a x i m u m of t he t i t r a t i o n c u r v e b e c o m e s b r o a d e r and is sh i f t ed to h i g h e r B o va lues as 7 inc reases . If in an a m p l i t u d e t i t r a t i o n of A by B the a m p l i t u d e p ro f i l e is be l l - shaped , but the p a r a -

m e t e r (B° )~po~ is f o u n d to be s ign i f i can t ly lar- ger t h a n the s u m A0 + Kapo-1 , one w o u l d c o n c l u d e tha t the sys t em ' s u n d e r l y i n g r e a c t i o n m e c h a n i s m must be m o r e c o m p l e x t h a n the s c h e m e of eq 4. F u r t h e r m o r e , the fit of the a m p l i t u d e resu l t s to the

BIOCHIMIE, 1973, 55, n ° 3.

2 8 2 D. Thusius.

m o d e l of eq 15 can be e s t i m a t e d by n o n - l i n e a r r e g r e s s i o n . If Kap p is k n o w n , bes t va lues of A¢~81nK 1 a n d v c a n be e v a l u a t e d by e x p a n d i n g eq 16 in t e r m s of t he n o n - l i n e a r p a r a m e t e r V, to g ive

~Pf ~ % X 1 q- ~., X., (19)

a I _~_ h~l~lnK~

X, = ( P 0 : o

~ = ~ ( ~ l - - ~,,) Kapp -i

(s ~ - - 4p)l/~ + Kapp .~o

It is e v i d e n t t h a t e s t i m a t e s of Kap p a n d V a l low one to ca l cu l a t e K~. Of c o u r s e c o n f i r m a t i o n of eq 15 u l t i m a t e l y r e s t s on t h e d e t e c t i o n of o n e o r m o r e s l o w r e l a x a t i o n e f fec ts w h o s e t i m e c o n s t a n t s a n d a m p l i t u d e s s h o w t h e c o n c e n t r a t i o n d e p e n - d e n c e c h a r a c t e r i s t i c of a m o n o m o l e c u l a r p r o c e s s c o u p l e d to a r a p i d b i n d i n g r e a c t i o n . A p r o g r a m for the i t e r a t i o n of eq 19 has b e e n w r i t t e n a n d t e s t e d fo r c o n v e r g e n c e .

CONCLUSION.

By a s t r a i g h t f o r w a r d a p p l i c a t i o n of n o n - l i n e a r r e g r e s s i o n b o t h t h e e q u i l i b r i u m c o n s t a n t a n d e n t h a l p y fo r 1:1 c o m p l e x f o r m a t i o n c a n be est i - m a t e d f r o m a s ing le r e l a x a t i o n a m p l i t u d e t i t r a t i on . The f o r m a l i s m d e v e l o p e d h e r e is gene ra l i n t h a t no r e s t r i c t i o n s are p l a c e d on the r a t i o A0/B 0 [18]. In t he a p p l i c a t i o n s c i ted , p r e c i s e K a n d AeSlnK va lues w e r e o b t a i n e d even t h o u g h the e n t h a l p i e s w e r e m o d e r a t e a n d t h e t e m p e r a t u r e c h a n g e s sma l l c o m p a r e d to t he overa l l AT r e q u i r e d fo r a v a n ' t Hoff d e t e r m i n a t i o n of c o m p a r a b l e p r e c i s i o n . As ide f r o m t h e e v a l u a t i o n of t h e r m o d y n a m i c p a r a - m e t e r s , a s t a t i s t i c a l t r e a t m e n t of a m p l i t u d e d a t a o f fe rs an e l egan t m e a n s of t e s t i n g t h e v a l i d i t y of a m e c h a n i s m s u g g e s t e d by k i n e t i c a n d e q u i l i b r i u m m e a s u r e m e n t s .

A cknotoledgments.

The author is grateful to Dr. F. Seydoux for nume- rous discussions and to Prof. J. Yon for her interest in this work. Fiua*neial s~pport f rom a C.N.R.S. grant (Groupe de Recherche No. 13) is also gratefully acknowledged.

R~SVM~.

Une mdthode de regression non-l indaire est appli- qude au prol~l~me de la ddterminat ion des paramdtres the rmodynamiques h par t i r des ampli tudes de relaxa- t ion ehimique. En particulier, des re lat ions mathd- mat iques sont ddvelopp6es en rue de la ddterminat ion de la consiante d 'dquil ibre et de la variat ion d 'en- thalpie relatives il la format ion des complexes bi- naires. Ces relat ions o n t d t d appliqudes h 1'analyse des ampl i tudes de deux syst~mes diffdrents d ' intdr~t bio- logique. La mdthode t ient compte des ehangements de signal dus h des effets purement physiques. La m~me approche est employde pour analyser los ampl i tudes d 'une rdaction caract6risde par une 6tape de fixation rapide d 'un ligand, suivie par une ou p lus ieurs drapes d ' i somerisa t ions lentes.

BEFERENCES.

1. M. Eigen, in << Nobel Symposium 5>>, S. Claesson, Ed., Interscience, New York, 1967.

2. Hammes, G. G. 1968. Advances in Protein Che- mistry, 23, 1.

3. Winkler, R. Doctoral Thesis, Gottingen, 1969. 4. a) Eigen, M. & Winkler , R. in << The Neuroseiences :

Second Study Programm >>, F. O. Schmitt , Ed., Rockefeller UniversKy Press, Ne~v York, 1970, p. 685; b) Eigen, M. in ¢ Probes of Structure and Function of Macromolecules and Mem- branes >>, Vol. 1, Chance, B., Lee, C. & Blasie, J. K. Eds., Academic Press, New York, 1971, p. 535.

5. Guillain, F. & Thusius., D. 1970. J. Am. Chem. Soc., 92, 5534.

6. Chock, P. B. 1971. Biochimie, 53, 161. 7. Czerlinski, G. H. in ¢ Theoretic~,l and Exper imen-

tal Biophysics >>, Vol. 2, A. Cole, Ed., Marcel Dekker, New York, 1969, p. 106.

8. Hammes, G. G. a Schimmel, P. R. in <<The En- zymes >>, Vol. II, P. D. Boyer, Ed., Academic Press, New York, 1970, p. 90.

Schimmel, P. R. 1971. J. Phys. Chem., 54, 4136. Thusius, D. 1972. J. Am. Chem. Soc., 94, 356. Thusius, D. Submit ted for publicati,on. Non-l inear regression analyses of this type have

recently been reviewed by Clvland ; W. W. Cle- land, Adv. in Enzym~l., 29, 1 (1'967).

13. It is noted tha t .the in terpre ta t ion of s tandard errors in non- l inear regression is not always as s t ra ight forward as in l inear analyses. See, e.g., E. J. Wil l iams, ¢ Regression Analysis >), John Wiley, New York, 1959, Chapter 4.

14. The superscripts (°) indicate analytical concen- t rat ions.

15. Prat t , J. M. ~, Thorp, R. G. 19.6,6. J. Chem. Soc., Sect. A, 187.

16. Thusius, D. 1971. J. Am. Chem. Soe., 93, 2629. 17. Fir th , R. A., Hill, H. A. O., Mann, B. E., Prat t , J. M.,

Thorp, R. G. & Will iams, R. J. P. 1968, J. Chem. Soc. A, 2419.

1,8. The present method has not, however, been tested for the l imit ing case K-1 <~,( A ° (B °, variable). Under these condit ions a << classical >) ti . tradon is insensi t ive to the value of K.3,4.

9. 1O. 11. 12.

BIOCHIMIE, 1973, 55, n ° 3.