Biochimica et Biophysica Aeta, 427 (1976) 387-391

© Elsevier Scientific Publishing Company, Amsterdam- Printed in The Netherlands BBA 37287 T H E R M O D Y N A M I C S OF T H E I S O T H E R M A L D E N A T U R A T I O N OF LYSOZYME BY G U A N I D I N I U M C H L O R I D E

VOJKO VLACHY and SAVO LAPANJE Department of Chemistry, University of Ljubljana, Ljubljana (Yugoslavia)

(Received August 15th, 1975)

SUMMARY A thermodynamic analysis of the isothermal denaturation of lysozyme by guanidinium chloride has been performed. The analysis is based on the equation which relates the equilibrium constant for denaturation to the preferential binding of denaturant. The equation has been derived previously by thermodynamic methods, whereas in this article a derivation based on statistical mechanics is given. By application of the equation the free energy of denaturation is first calculated and from it, by subtracting the calorimetrically-determined enthalpy of denaturation, the entropy of denaturation is determined.

INTRODUCTION It has been shown by Tanford [1 ] that it is possible to analyze the interaction between a protein and a denaturant in terms of binding equilibria. An equation has been derived on the basis of specific assumptions about the mechanism whereby addition of denaturant perturbs the equilibrium between the native and denatured protein. The equation relates the equilibrium constant for denaturation to the amount of denaturant bound to the denatured and native forms of the protein, respectively. The same equation can be obtained by the application of the theory of linked functions, developed by Wyman [2], which deals with the binding of ligands to macromolecules. However, the equation derived does not consider the fact that the activity of denaturant, under the existing conditions, cf. below, and that of the principal solvent, e.g. water, are not independently variable. This in turn reflects the fact that there are actually two ligands [1, 3]. Therefore, an additional term has to be added to the equation. In the following we first present an alternative treatment of the denaturant binding based on statistical mechanics which leads to the same corrected expression. In a recent article ScheIlman [4] has reviewed the existing formulae and has derived some new formulae for the free energy of binding to a macromolecule. Both thermodynamic and statistical mechanical methods, have been dealt with, and the expression for the free energy of binding has been applied to various cases. The equivalence of thermodynamic and statistical mechanical approaches to the binding problem has

388 been demonstrated, so that our finding is not surprising. The corrected expression is then applied to the experimental results obtained in the study of the isothermal denaturation of lysozyme by guanidinium chloride. They include the preferential binding of denaturant to lysozyme as a function of denaturant concentration [5] and the calorimetrically determined enthalpies of binding [6], so that a complete thermodynamic description of the denaturation is feasible. THEORETICA L In the analysis we will assume that only two forms, native. N, and denatured, D, are present and the denaturation equilibrium can be described in terms of a single equilibrium constant Kt~ KD :-: [DI/[N] :- QD/QN

/I)

where QD and QN are the partition functions for the denatured and native forms, respectively. When binding of a denaturant X is involved, Q will in general be a product of two terms, one referring to all possible states at zero concentration of the solute QO and the other describing all possible states of binding, Qb. The average number of bound X molecules per protein molecule 5x, is then related to the activity of denaturant, ax as follows: dlnQ d In ax

i~x

(2)

The difference in binding between the denatured and native forms I~x is then given by: dlnKD d l n ax

- ,,li~x.

(3)

The effect of X is thus due to the difference in the number of binding sites between the two forms. Eqns. (2) and (3) have been proposed previously [7, 8]. Eqn. (3) is also the original equation derived by Tanford [3]. However, the actual solution we are interested in contains in addition to the denaturant and the protein also the principal solvent, water, W. If all binding sites on the native or denatured protein molecule are identical as well as independent, and there are nN such sites on the native molecule and no on the denatured one, then a grand partition function Q, oFen with respect to both X and W, can be defined Q(X,W,T) .... Q°(T)Qb(X,W,T)

(4)

where Q°(T) is the sum over all states when there is no binding, and Qb(X,W,T) is the sum over all states involving binding of X and W; therefore Qb(X,W.T) is given by [91. Qb(X.W.T) ~ (I : kxax

i kwaw) n

(5~

389 where aw is the activity of water, and kx and kw the intrinsic binding constants fol the reaction site ÷ X(W) ~ site-X(W) at each individual binding site. The average number of bound X, ~x, and W, ~w could then be obtained by differentiating Eqn. (3) with respect to ax and aw, respectively. However, the activities ax and aw are not independently variable. The relation between ax and aw is given by the Gibbs-Duhem equation. At constant temperature and pressure and infinite dilution of protein the following expression is obtained [3]: ?/x

d In aw --

d In ax

(6)

Hw

where nx and nw designate the total number of moles of each component in the solution. The derivative of In Qb with respect to In ax is thus d In Qb d In ax

nx nw

-- ~x . . . .

~w

(7)

Application of Eqn. (2) yields finally: d In K o d In ax

z

ZI~x

//x

Zl~w z

A~x.pref"

(8)

nw

which is identical with the corrected equation derived by Tanford [3]. A~x,pr,f is the preferential binding of denaturant X to a protein molecule. It is defined as the amount of X bound in excess of its proportion in the solvent mixture. Preferential binding can be positive as well as negative, the latter case representing excess hydration. RESULTS AND DISCUSSION Eqn. (8) can apparently be used for the experimental determination of the standard free energy of denaturation if preferential binding is known as a funttion of denaturant activity. By integration of Eqn. (8) we obtain: den

f

den

d(logKD)=

nat~

f

A~x.~.~fdlogax

(9)

RT|" ( m = 0Av× pref (m) d log ax ) '

(10)

ilal-

and .(m)

/IGD° :

--2,3

since A G ° at the lower limit of integration is zero. By integrating the right-hand side of Eqn. (10) A G ° can be calculated if log ax is known as a function of molality, m. For the upper limit of integration one naturally chooses a value of m at which denaturation is completed. Thus it is immediately evident that A G ° is a function of denaturant molality. If preferential binding were given as an analytical function of denaturant molality, a straightforward integration would be possible. However, usually the binding is given at only a few concentrations and therefore graphical integration has to be applied. In a previous article [5] we have published the values of preferential binding

390 o f g u a n i d i n i u m chloride to lysozyme at 2 5 . 0 ' C and p H 5.7. Thus it is possible to calculate using Eqn. (10) the free energy of d e n a t u r a t i o n for lysozyme, since log a as a function o f g u a n i d m i u m chloride c o n c e n t r a t i o n is known [10]. The dependence is actually given as a p o l y n o m i a l in terms o f m o l a r c o n c e n t r a t i o n and tkerefore had to be converted to the molality scale. TABLE I THERMODYNAMIC PARAMETERS OF THE 1NTERA(TION OF GUANIDINIUM ('HI.ORIDE WITH LYSOZYME AT 25 '(" AN[) pH 5.7 The values of I1"-.'o IH~ and T IS~ refer lo the transfer of lysozyme from aqueous ,,olutions to ~1)~ guanidinium chloride sohltions of designated molarity or molality. IG°, 1HI] and T l~S'~, are in kcal.'mol. G uanidinium chloride conch. (M)

(molal)

4.0 6.0 7.0

5.6 10.6 14.2

/x,, 8.3" 10'; 3.8 +10~'' 3.8. I(P -~

Itt,'i

I(;(i 10 15 17

5 7 8

59.5 74.1) 85.0

T 1.S','I 50 59 68

In Table 1 the ,IG ° and KD o b t a i n e d using Eqn. 10) and tile experimental d a t a are presented. We have chosen three different molarities: 4, 6 a n d 7. In 4 M solution lysozyme is not yet c o m p l e t e l y unfolded [5]; in 6 and 7 M solution unfolding is complete. All the values o f : I G ° are negative, a n d they become m o r e negative with increasing d e n a t u r a n t c o n c e n t r a t i o n . Moreover, it can be inferred from the fact that the d e n a t u r a n t is preferentially b o u n d t h r o u g h o u t the whole c o n c e n t r a t i o n range studied, i.e., t¥om 0 to 7 M [5], that at lower d e n a t u r a n t c o n c e n t r a t i o n s 1Gis also negative. This finding is in agreement with previous o b s e r v a t i o n s [11]. However. in the present instance it has been arrived at by a direct t h e r m o d y n a m i c method. F r o m Table I it can also be inferred that the errors involved are relatively large: lhus in (I 5 7) kcal/mol. The errors naturally reflect the large error 6 M solution IG oD is a c c o m p a n y i n g the d e t e r m i n a t i o n of preferential binding. But this should not be d r a m a t i s e d considering the fact that the a p p r o a c h to the p r o b l e m has been based on m a n y simplifications [I ], and we really c a n n o t m a k e a statement of how close to the true values o!" IG oD the values in Table ! really are. Even less can such a statement be made with respect to the a p p a r e n t values o f