J. Phys. Chem. B 1999, 103, 10325-10330
10325
Statistical Mechanical Approach to the Conformational Heat Capacity and Enthalpy of Biomolecules Bruno Linder* and Robert A. Kromhout Chemical Physics Program, The Florida State UniVersity, Tallahassee, Florida 32306-4390 ReceiVed: July 13, 1999; In Final Form: September 16, 1999
A statistical mechanical treatment is presented for determining the heat capacity and enthalpy changes with temperature in conformational biomolecular (e.g. folding-unfolding of proteins) transitions. The theory is formulated in terms of subunit partition functions, which are the average partition functions of the residues modulated by their interactions with each other and the solvent. The theory is specialized to a system of two types of conformants: type A (for example, protein in the folded state) consisting of subunits of type “a”, and type B (for example, protein in the unfolded state) consisting of subunits of type “b”, assuming complete cooperativity. It is shown that for such a model, the heat capacity and enthalpy functions obtained from the isothermal-isobaric partition function of the biomolecular system can be calculated from the ratio of the partition functions of the “a” and “b” subunits, their temperature derivatives, and the difference between the lowest enthalpy levels of the subunits. The temperature variation of the partition functions of the subunits can be evaluated from thermal data of the pure conformants A and B outside the transition range. The other parameters may be inferred from the population ratio of the conformants inside the transition region. The theory is applied to lysozyme of pH 2.0, pH 2.25, and pH 3.5, using published optical density data (Khechineshvili, N. N.; Privalov, P. L.; Tictopulo, E. I. FEBS Lett. 1973, 30, 57) of the conformants within the transition range, and heat capacity data of the pure conformants outside the transition range. Kidokoro and Wada (Kidokoro, S.-I.; Wada, A. Biopolymers 1987, 26, 213) published heat capacity data for lysozyme at pH 2.0 with which we compared our calculated new capacity. The calculated heat capacity curve agrees well with the published one.
I. Introduction A problem of vital importance in biomolecular research is the understanding of conformational transitions between various forms of the macromolecules. Examples are denaturation or folding-unfolding of proteins, helix-coil transitions, etc. In recent years, it has become possible to study such transitions thermodynamically by differential scanning calorimetry as well as by other methods such as spectroscopic methods. For an excellent review see, for example, the chapter by Privalov in ref 1. Experimental heat capacity data based on scanning calorimetry methods have been used to analyze such transitions, using deconvolution techniques within the framework of statistical thermodynamic relationships.2,3 There have been earlier statistical-mechanical treatments of macromolecular conformational transitions,4-6 but only in recent years have attempts7,8 been made to obtain numerical results for heat capacities directly from statistical mechanical formulations. Our goal is the numerical evaluation of the thermal properties in conformational changes from statistical-mechanical principles. To this effect we have developed a technique amenable to simplified models which makes such calculations feasible. While our method has features in common with other treatments7,8 especially that of Ro¨sgen et al., we differ from them in general approach and in detailed analysis. Conformational transitions of macromolecular systems differ from ordinary first-order phase transitions not only quantitatively
but also qualitatively. In solid-liquid, liquid-vapor, etc. phase transitions, there are discontinuities in the enthalpy and entropy so that the heat capacity is undefined at the transition. In conformational transitions of macromolecules, there are no discontinuities of the enthalpy and entropy functions, and the heat capacity vs temperature curve is bell-shaped, going through a maximum. There are examples, such as electronic heat capacities of systems with two or a few low-lying energy levels, which exhibit maxima, but these bell-shaped curves are much broader than those in macromolecular transitions; the latter have commonly widths in degrees kelvin of the order of a few percent of the temperature at the peak. In this paper we present a technique for calculating conformational heat capacity changes of dilute solutions of macromolecules. The treatment, based on statistical mechanics, is applicable to a system of small dimensions, such as a macromolecule, as well as to a system of macroscopic dimensions owing to the fact that fluctuations are automatically included in the formalism. In small systems, fluctuations become important as emphasized by Hill,6 and great caution must be exercised in drawing conclusions from purely thermodynamic relations. Ideally, the fundamental quantity in a statistical treatment of thermal properties, namely the partition function, ought to be calculated from first principles. This represents a formidable task. Fortunately, useful information about the partition function in the transition region can often be obtained from spectroscopic data. In addition, information about the partition function outside
10.1021/jp9923724 CCC: $18.00 © 1999 American Chemical Society Published on Web 11/03/1999
10326 J. Phys. Chem. B, Vol. 103, No. 46, 1999
Linder and Kromhout
the transition region can be obtained from experimental calorimetric data or from semiempirical formulas9 for the specific heat capacity of the isolated conformants. As a working model, we consider a solution of macromolecules, sufficiently dilute so that the macromolecules do not interact. Each macromolecule consists of N subunits (or residues). The properties of the subunits are not the intrinsic properties of the isolated residues but average properties of the residues modulated by their interaction with each other and with the solvent. Although a macromolecule (the “system”) can, in principle, contain an assortment of residues, in our model of biomolecules the subunits of a given conformant will be either of type “a” or of type “b” or will contain large segments of “a” and “b”. This is due to the cooperative nature of the conformational transition in biomolecules.1 Of ultimate importance to the understanding of conformational behavior of biomolecules is an interpretation from a molecular point of view. A quantitative exposition of this subject would require the full complexity of quantum mechanics, which is not tractable. Numerous qualitative and semiquantitative treatments have been published by various authors, not all in agreement with each other.1,9 The general consensus seems to be that hydrogen bonding, van der Waals interactions, and electrostatic interactions, as well as hydrophobic effects, are important; interaction with water plays a vital role. Our aim is to try to understand and predict the thermal behavior of the conformational changes from a statistical mechanical rather than from a molecular point of view. In section II, we develop in some detail the statistical mechanical formulas in order to delineate the nature of the assumed or implied approximations. We conclude that section with a recipe for calculating the enthalpy and heat capacity curves of a “two-state” system. In section III, we apply the theory to lysozyme and display the calculated enthalpy and heat capacity curves versus temperature in the range within which the conformants change from native to denatured. The results are compared with published data. We conclude with a discussion in section IV. II. Theory
where the label j distinguishes the system energies belonging to given Na and Nb, where Na is the number of a subunits, and Nb is the number of b subunits. The total number of subunits, N, in a macromolecule is fixed independent of j: N ) ∑rNrja + ∑sNsjb ) Na + Nb. Denoting Nb as n, then Na ) N - n, and so Ej is a function not only of N and V but also of n which in the general treatment is allowed to run from 0 to N. Accordingly N
∆)
∑V n)0 ∑∑j Ωj(n,N,V)e-βH (N,V) jn
(2)
where Hj ) Ej + PV
β ) (kT)-1
(3)
and Ωj is the system degeneracy of the jth molecular level a
Ωj ) (N - n)!
∏r
Narj
b
ωr
Nrja !
n!
∏s
Nbsj
ωs
(4)
Nsjb !
where aωr and bωs are the degeneracies of the a and b subsystems respectively. Let aVr and bVs represent the volume of a subunit, noting that V ) ∑rNrja aVr + ∑sNrjb bVs. We can write
∆(N,P,T) )
[
∑V ∑n ∑j a
(N - n)!
a
(aωre-β hr)
∏r
a
hr ) br + PaVr
Narj
(bωse-β hr) b
n! Nr!
∏s
b
Ns!
hs ) bs + PbVs
b
and
]
Nbsj
(5)
(6)
Using the multinomial expansion gives
∆)
∑V ∑n [(∑r aωre-β h )N-n(∑s bωse-β h )n] a
r
b
s
(7)
or
A. Partition Functions. We shall refer to the partition function of a macromolecule as the system partition function, and to the partition function of a residue (averaged as mentioned before) as the subunit partition function. Most experimental studies are done under conditions of constant pressure, so it is advantageous to use as an ensemble the isobaric-isothermal ensemble, since in this ensemble there are no fluctuations in temperature, pressure, or number of subsystems, N, in a molecule. The partition function, ∆(P,T,N), of the isothermal-isobaric ensemble, which is conceptually a collection of a large number of systems having different energies (Ej), volumes (V), and enthalpies (Hj), will ultimately be developed in terms of the subunit partition functions. We assume that the subunits are of two types, a or b with partition functions qa* and qb*. Let ar and bs represent the energy levels of the subunits, and aωr and bωs represent their corresponding degeneracies; let the indices r and s label which level is intended among the manifold available to a subunit of type a or b, respectively. The assumption of effectively independent subsystems allows us to write the energy of the macromolecule as
Ej(Na,Nb,V) )
∑r Nrja ar + ∑s Nsjb bs
(1)
N
∆)
[(q/a )N-n(q/b)n] ∑ n)0
(8)
where q/a and q/b are the subsystem isobaric-isothermal partition functions defined in eq 7. Although n can, in principle, take on any value between 0 and N, in practice it will be restricted to a small number of specific values denoting, for example, that all subunits of a molecule are a or b, or alternatively, may contain large segments within which the subunits are of the same type (resulting from the cooperative nature of the conformational transitions). The latter could represent a macromolecule (system) which can exist in more than two conformations or “states”. In this paper, we focus on two-state systems (conformant A in which n ) 0 and conformant B in which n ) N), which appears to be adequate for small globular proteins, and write
∆ ) (q/a )N + (q/b)N
(9)
The enthalpy levels of a and b must be measured from a common reference point. However, it is customary and convenient to label the lowest state of each subunit to be zero and refer the q’s to these zeros. To accomplish this and still have a
Conformation Heat Capacity and Enthalpy of Biomolecules common reference state we replace q/a and q/b, respectively, by
qa ) q/a e-βha0, qb ) q/be-βhb0
(10)
J. Phys. Chem. B, Vol. 103, No. 46, 1999 10327
{( [(
e-N∆h0/kT
and obtain
∆ ) (qae-βha0)N + (qbe-βhb0)N
(11)
or
[ ()
qb ∆ ) (qae-βha0)N 1 + qa
N
e-β∆h0N
]
(12)
where ∆h0 ) hb0 - ha0 is the enthalpy difference between the lowest levels of the a and b subunits. Since the common reference level is at our disposal, we choose it at the lowest state of the a subsystem, rendering ha0 equal to zero. B. Enthalpy and the Heat Capacity Functions. Formula 12 is the fundamental equation from which the enthalpy and heat capacitysin fact all thermodynamic functionsscan be derived. The q’s represent the partition functions of the subunits, and are, in general, temperature dependent since they embody various kinds of motions (vibration, hindered rotation, etc.) as well as interactions. This fact is borne out by experimental data such as temperature-dependent heat capacities measured under conditions wherein only conformant A or conformant B is present (referred to hereafter as “pure A” or “pure B”). The derivatives of ln ∆ define the enthalpy, H h , and heat capacity, C h p, per molecule (the system).
∂ ln ∆ H h ) kT ∂T P,N
( ) ∂ ∂ ln ∆ C h ) [ kT ( ∂T ∂T ) ] 2
(13)
2
p
P,N P,N
(14)
The bars serve to denote that these quantities are determined statistically as ensemble averages. Fluctuations, as noted before, can be substantial in small systems, but are taken account of statistically. To distinguish the thermal values per molecule from the values per mole which are usually cited in the literature, we must multiply H h and C h p by Avogadro’s number (NA) or simply replace k in eqs 13 and 14 by R. The molar quantities will be denoted as H ˜ and C ˜ p. Since the q’s are temperature-dependent, it is convenient to rewrite eqs 13 and 14 in a form which explicitly displays the derivatives of the q’s. Let us denote the derivatives of ∆ as ∆′ ) (∂∆/∂T)P,N and ∆′′ ) (∂2∆/∂T2)P,N. The molar enthalpy and heat capacity become
[ ]
) )
∂ ln qa(T) 2 1 ∂2 ln qa(T) qb(T) ∆′′ + + ) N2 2 ∆ ∂T N qa(T) ∂T
(
N
×
)
∂ ln qb(T) 2 1 ∂2 ln qb(T) ∂ ln qb(T) ∆h0 + +2 2 ∂T N ∂T ∂T kT2 ∆h0 2 qb(T) N -N∆h0/kT 2 ∆h0 + 1 + e (18) N kT3 qa(T) kT2
( ) ]}/{ [ ]
}
C. Estimation of Statistical Parameters. It is seen that in order to evaluate the thermodynamic functions H ˜ and C ˜ p one must have knowledge of ∆h0, the enthalpy difference between the zero levels of the subunits a and b, as well as the ratio of the partition functions of the subunits and their individual temperature dependences. The quantities ∆h0 and qb/qa can often be obtained by extrathermodynamic measurements, for example, from spectroscopic or optical density data. It follows from eq 11 or 12 that the fraction of molecules with subunits b (i.e., in conformation B) will be
[ ] [ ] qb(T)
N
qa(T)
xjb(T) )
1+
qb(T) qa(T)
e-N∆h0/kT N
(19) e-N∆h0/kT
Since xja(T) + xjb(T) ) 1, where xja is the fraction of molecules with subunits a, we have
xjb(T) xja(T)
)
[ ] qb(T) qa(T)
N
e-N∆h0/kT
(20)
If in the region of conformational transitions xjb/xja can be obtained from optical density data, then a plot of (1/N)ln[xjb(T)/ xja(T)] vs 1/T yields both ∆h0 and qb(T)/qa(T). Of particular importance is the midpoint, Tm, i.e., the temperature at which xjb/xja ) 1, for qb(Tm)/qa(Tm) ) e∆h0/kTm. Outside of the transformation region the q’s and their derivatives may be approximated by developing them in a series. Let Ta and Tb be temperatures sufficiently far from the transition region, where it can be reasonably assumed that near Ta, all molecules consist of only a subunits, and that near a high temperature, Tb, all molecules consist of only b subunits. We assume that when only conformant A is present, the subunit a partition function can be expanded in the following series.
ln qa(T) ) ln qa(Ta) + K ln T/Ta + A1(T - Ta) + A2(T - Ta)2 + ... (21a) and similarly, when only conformant B is present
H ˜ ) RT
2∆′
(15)
∆
∆′ ∆′′ ∆′ 2 C ˜ p ) 2RT + RT2 - RT2 ∆ ∆ ∆
( )
(16)
with
∆′ )N ∆
{
[ ] [ [ ]
∂ ln qa(T) qb(T) + ∂T qa(T) 1+
N
e-N∆h0/kT
qb(T) qa(T)
N
e
]}
∂ ln qb(T) ∆h0 + 2 ∂T kT
-N∆h0/kT
(17)
ln qb(T) ) ln qb(Tb) + L ln T/Tb + B1(T - Tb) + B2(T - Tb)2 + ... (21b) where K, A1, A2, ..., L, B1, B2, ...are constants. These coefficients can be determined from the temperature variation of the heat capacities of the pure conformants, which are frequently known from measurements or empirical formulas (9). By writing the molar heat capacity as
˜ ap(Ta) + Ca1(T - Ta) + Ca2(T - Ta)2 + ... C ˜ ap(T) ) C ˜ bp(Tb) + Cb1(T - Tb) + Cb2(T - Tb)2 + ... (22) C ˜ bp(T) ) C
10328 J. Phys. Chem. B, Vol. 103, No. 46, 1999
Linder and Kromhout
Figure 1. Plots of ln(xjb/xja)1/N versus (1/T) for lysozyme from optical absorption data of Khechinashvili et al.11 The slope is the negative of ∆h0/k. The intercepts with the zero ordinate give the equimolar temperature Tm.
TABLE 1: Values of Tm, Ta, Tb, and ∆h0 and q Ratios of Lysozyme at Varying pH’s pH 2.0 2.25 3.5
Tm (K) 321.5 330.5 342.5
Ta 283 292 304
Tb 353 362 374
∆h0/k 347 407 455.5
qb(Tm)/qa(Tm) 2.9427 3.4262 3.7808
it is easy to show that
K)
C ˜ ap(Ta) TaCa1 Ta2Ca2 + + ... RN RN RN A1 )
Ca1 2TaCa2 + ... 2RN 3RN
A2 ) Ca2/6RN + ...
(23)
with similar expressions for L, B1, and B2. The derivatives of qa and qb are now easily determined from eqs 21. Now all parameters are listed which are needed to evaluate eqs 17 and 18 and thus the heat capacity over a range of temperatures varying from Ta to Tb. III. Application to Lysozyme Khechinashvili et al.11 measured the optical density of lysozyme vs temperature for various solutions at three different pH values. Figure 1 is a plot of (1/N)ln(xjb/xja) vs T-1 obtained from their Figure 5. The three curves are remarkably linear, implying that qb(T)/qa(T) is independent of T, to good approximation, in the transition region. From these curves we deduce the values of ∆h0/k and Tm shown in Table 1. The values of Ta and Tb given in the table are arbitrarily set at Ta ) Tm 38.5 K and Tb ) Tm + 31.5 K for each of the lysozyme solutions. The qb/qa ratios given for T ) Tm in the table are, for these particular examples, the same for any temperature in the
Figure 2. C ˜ p(T) (in kJ K-1 mol-1) calculated using eqs 16-18 and 21-23 with (∆h0/k) and Tm as determined from the optical data in Figure 1 and listed in Table 1, as well as the parameters describing the thermal properties of the pure conformants, determined from thermal data as described in the text.
transition region because of the linearity of the curves. This will not necessarily be true for other solutions. Kidokoro and Wada3 show that the heat capacities of the pure conformants in these solutions increase linearly with temperature. From their data we deduce that for the native conformant ˜ ap(283 K) ) 42 200 J K-1 mol-1 and for the at Ta ) 283 K, C ˜ bp(353) ) 57 400 J K-1 denatured conformant at Tb ) 353 K, C -1 mol . The coefficient of the linear term in the series, eq 22, is the same for both: Ca1 ) Cb1 ) 130 J K-2 mol-1; the higher order terms are zero. Using eq 23, we find that K ) 5.0441, L ) 10. 7315, and A1 ) B1 ) 6.0604 × 10-2 K-1 and the coefficients of the higher order terms are zero. These parameters describe accurately the behavior of the heat capacities of the pure conformants outside the transition region. It is obvious that the manifold of states r, which comprise the partition function qa, must eventually go over into the manifold of states s, which are the states of qb. Rather than speculate how the subsystem would behave in the transition region, we arbitrarily treat the heat capacity of both pure conformants as behaving like pure A below Tm (we set K ) L ) 5.0441) and like pure B above Tm (we set K ) L ) 10.7315). We are now in a position to calculate the enthalpy and heat capacity functions, including the background, of lysozyme (N ) 129). Figure 2 shows plots of the molar heat capacity of lysozyme vs temperature for the three values of pH listed above. The heat capacity curves are obtained directly from formulas 16-18 and 21-23. Kidokoro and Wada3 published in their Figure 2b (curve 1) a curve of “test data” for lysozyme in which the equimolar point is Tm ) 320 K. In order to compare our calculated C ˜ p curve with their test data, we have recalculated C ˜ p for the first case, pH 2.0, but using Tm ) 320 K; the results are shown in Figure 3 where the circle points are obtained from
Conformation Heat Capacity and Enthalpy of Biomolecules
Figure 3. C ˜ p(T) (in kJ K-1 mol-1) calculated using eqs 16-18 and 21-23, with ∆h0/k ) -347 K, but using the Tm value of 320 K given by Kidokoro and Wada.3 The points marked by circles are taken from their Figure 2b (curve 1).
J. Phys. Chem. B, Vol. 103, No. 46, 1999 10329
Figure 4. ∆C ˜ p(T) (in kJ K-1 mol-1) calculated using eqs 16-18 and 21-23, with (∆h0/k) and Tm as determined from Figure 1 and given in Table 1 but with the pure conformant parameters (K, L, A1, A2, B1, B2) set equal to zero.
the Kidokoro curve. The small shift of Tm by 1.5 K could result from a minor change in a molecular environment such as a very small pH shift from 2.0, so we have assumed that ∆h0/k is 347 K. The actual pH of the test data was not given by Kidokoro and Wada. The size of the circles represents the estimated reading errors in the points. In Figure 4 we display the conformational heat capacity, ∆C ˜ p, calculated without including the temperature dependence of the heat capacities of the two conformants (i.e., setting K, L, A1, and B1 equal to zero). These curves agree well with values ˜ p(back) obtained by subtracting from C ˜ p the background C
C ˜ p(back) ) xja(T)C ˜ ap(T) + xjb(T)C ˜ bp(T)
(24)
the heat capacity of a mixture of pure conformants. The mole fractions xja(T) and xjb(T) are calculated from eqs 19 and 20. Figure 5 shows the conformational enthalpy calculated from eqs 15, 17, 18, and 21 by setting the pure conformant parameters (K, L, A1, A2, B1, B2) equal to zero. As an illustration of the enthalpy including the effects of the thermal characteristics of the pure conformants, Figure 6 shows the variation with temperature of the total enthalpy, H ˜ , the background enthalpy, H ˜ (back), and of the conformational enthalpy, ∆H ˜ , for solution 1, pH 2.0. The solid curve for H ˜ was calculated directly from eqs 15, 17-18, and 21-23. The dashed curve was calculated by the following formula
˜ a(T) + xjb(T)H ˜ b(T) H ˜ (back) ) xja(T)H
(25)
( )
with
( )
H ˜ a ) NRT2
∂ ln qa ∂T
p
Figure 5. Conformational enthalpy ∆H ˜ (T) (in kJ mol-1) calculated for the three lysozyme solutions calculated by setting the pure conformant parameters (K, L, A1, A2, B1, B2) equal to zero.
H ˜ b ) NRT2 (26a)
∂ ln qb ∂T
p
(26b)
The points indicated by squares on the ∆H ˜ curve are the differences between H ˜ and H ˜ (back). The solid curve ∆H ˜ in
10330 J. Phys. Chem. B, Vol. 103, No. 46, 1999
Linder and Kromhout be negligible in the case of lysozyme. This is shown in Figures 3 and 6 by the agreement of the points and the solid curves. Outside the transition range contributions from the derivatives are essential. If we compare eqs 19 and 20 for xja and xjb with eq 17 for ∆′/∆, we observe that eq 15 for the enthalpy, H ˜ , can be written
[
]
∂ ln qNa ∂ ln qNb H ˜ ) RT2 xja + NaN∆h0xjb + xjb ∂T ∂T
Figure 6. Enthalpy, H ˜ (T) (in kJ mol-1) calculated using eqs 17-23 with (∆h0/k) ) 347 K and Tm ) 321.5 K and the thermal parameters of the pure conformants as described in the text. The dashed curve marked H ˜ (back) was calculated according to eqs 25 and 26. The curve marked ∆H ˜ was calculated by setting the pure conformant parameters (K, L, A1, A2, B1, and B2) to zero. The points indicated by circles are the differences H ˜ and H ˜ (back). The values plotted for H ˜ and H ˜ (back) are divided by 25 for convenience in plotting on the same scale as ∆H ˜.
Figure 6 was calculated without including the temperature dependencies of the pure conformers (i.e., setting K, L, A1, and B1 equal to zero in eqs 21-23). IV. Discussion Our calculated heat capacity curve for Tm ) 320 K closely resembles the experimental curve of Kidodora and Wada, as exemplified by Figure 3. The three curves for different values of pH shown in Figure 2 behave similarly. It is not surprising that the asymptotic values of the heat capacities of the pure conformants coincide with the experimental values, since the parameters K, L, A1, and B1 were chosen so that this would be the case. But no thermal data were used to determine the level spacing ∆h0, the partition function ratio qb(T)/qa(T), and the equimolar temperature Tm which are the essential features of the heat capacity calculation in the transition region. Those features were obtained entirely from optical density data. A study of the optical data of Kechinashvili et al.11 reveals that the latter three parameters are functions of the molecular environment, especially of pH. For lysozyme, a plot of ln(xjb/xja) vs 1/T (Figure 1) is surprisingly linear, indicating that in the transition region qb(T)/ qa(T) is essentially temperature independent. This does not imply that qa and qb, the partition functions of the subunits, or their derivatives, are temperature independent. Contributions from the derivatives are included in the calculations of C ˜ p and H ˜ in the transition region, but their contributions to the conformational heat capacity and conformational enthalpy are found to
(27)
Notice that the first term in this result is H ˜ (back) as defined by eq 26; this is the contribution of the pure conformants to H ˜ as can be seen by its dependence on the temperature derivatives of the subsystem partition functions, q. The last term above is independent of the temperature derivatives of the subsystem partition functions but not of temperature which enters through ˜ calculated the factor xjb, as defined by eq 19. It is the same as H by setting all the parameters defining the approximate temperature dependence of the pure conformants (K, L, A1, A2, B1, B2) to zero, and is the conformational enthalpy provided that the temperature dependence of qb/qa is negligible in the transition region, which is the case for lysozyme according to the optical data used to determine the temperature behavior of ∆ in the transition region. Our procedure for calculating the enthalpy is valid, however, even for a case in which qb/qa depends on temperature in the transition region. The enthalpy functions displayed in Figure 6 show clearly that in conformational transitions the bulk of the heat required to denature the protein goes into increasing the enthalpies of the pure conformants interacting with the solvent. The prescription for calculating C ˜ p and H ˜ , based on the twostate model presented in this paper, proves to be adequate for lysozyme. It is anticipated that for other proteins, especially large ones, the two-state model will be inadequate, and the theory will have to be generalized to include more states. What the present calculations clearly show is that the heat capacity curve is bell-shaped as observed experimentally. From a theoretical point of view, the bell-shaped curve is due essentially to the smallness of the system (the system being a macromolecule), and to the cooperative nature of the transitions in biomolecules which ensures that configurations in which only a small number (,N) of subunits are of one type, a or b, contribute negligibly to the system partition function, ∆, whereas those in which number is of order N contribute most of the partition function. References and Notes (1) Creighton, T. E., Ed. Protein Folding; W. H. Freeman and Co.: New York, 1992. (2) Freire, E.; Biltonen, R. L. Biopolymers 1978, 17, 497. (3) Kidokoro, S.-I; Wada, A. Biopolymers 1987, 26, 213. (4) Gibbs, J. H.; DiMarzio, E. A. J. Chem. Phys. 1959, 30, 271. (5) Zimm, B. H.; Bragg, J. K. J. Chem. Phys. 1959, 31, 526. (6) Hill, T. L. Thermodynamics of Small Systems; W. A. Benjamin: New York, Part I, 1963; Part II, 1964. (7) Hansen, A.; Jensen; M. H.; Sneppen, K.; Zocchi, G. Euro. Phys. J. B 1998, 6, 157. (8) Ro¨sgen, J.; Hellerbach, B.; Hinz, H. J. Biophys. Chem. 1998, 74, 153. (9) Shirley, B. A., Ed. Protein Stability and Folding; Humana Press: Totowa, NJ, 1995. (10) Privalov, P. L.; Khechinashvili, N. N. J. Mol. Biol. 1974, 86, 665. (11) Khechinashvili, N. N.; Privalov, P. L.; Tiktopulo, E. I. FEBS Lett. 1973, 30, 57.
