Statistical-Mechanical Approach to the Thermodynamic Functions in

J. Phys. Chem. B 2001, 105, 6387-6395

6387

Statistical-Mechanical Approach to the Thermodynamic Functions in the Unfolding of Biomolecules Bruno Linder*,† and Robert A. Kromhout‡ Chemical Physics Program, The Florida State UniVersity, Tallahassee, Florida 32306-4390 ReceiVed: October 2, 2000; In Final Form: March 12, 2001

Our statistical-mechanical formulation for calculating heat-capacity and enthalpy functions in conformational changes of biomolecules is extended to include free-energy and entropy functions and generalized to treat multidomain systems. Each conformant may consist of one or several independent sections or domains. The treatment is based on a two-state model in which all residues within a domain are fully cooperative. The thermodynamic functions can be evaluated from the ratio of the effective partition functions of the residues in the different conformants, the difference in their zeros of energy, and the temperature derivatives of these partition functions. The former two parameters were evaluated from published optical density data of the conformants within the transition region; the last item was obtained from published heat-capacity data of the pure conformants outside the transition region. Three types of thermodynamic functions were considered: (1) total functions, which include contributions from both folded (conformant A) and unfolded (conformant B) states; (2) excess functions, which are the functions in excess of pure A; (3) conformational functions, which do not include contributions from either pure A or pure B. The theory was applied to solutions of ribonuclease, exemplifying single-domain systems, and to solutions of poly-γ-benzyl-L-glutamate, exemplifying multidomain systems. All thermodynamic functions, including the heat capacities, in the transition region were calculated. The results are compared with published experimental values and are in reasonably good agreement with the measured results. The excess and conformational thermodynamic functions are compared and discussed.

1. Introduction article,1

In a recent hereafter referred to as Paper 1, Linder and Kromhout developed a statistical-mechanical formulation for calculating thermal changes in the conformational transitions of simple biomolecules, such as small globular proteins. The treatment was specialized to a model in which there are two types of conformants: the ground-state conformant A (e.g., the native state of a protein) consisting only of residues of type “a” and an excited state conformant B (e.g., the denatured state of a protein) consisting only of residues of type “b”. It was assumed that complete cooperativity prevailed among the residues or subunits within each conformant. The subunits did not refer to the isolated residues, but rather to effective residues, modulated by their interaction with each other and with the solvent. This model was applied to calculating the heat-capacity and enthalpy functions of lysozyme and was in good agreement with published experimental data.2 Conformational changes in biomolecules are often likened to phase transitions,3 yet there is an important difference in the thermal behavior between biomolecular phase transitions and ordinary phase transitions. In a first-order phase transition, for example, the enthalpy and entropy functions are discontinuous at the (sharp) transition point, and the heat capacity is undefined there. In contrast, in biomolecular transitions there are no discontinuities in the entropy and enthalpy functions, and the heat-capacity functions are well defined and generally bellshaped.1 * To whom correspondence should be addressed. E-mail: linder@ chemmail.chem.fsu.edu. † Department of Chemistry. ‡ Department of Physics.

The use of statistical mechanics in the treatment of biomolecules is particularly advantageous, because fluctuations are automatically taken into account in the treatment of biomolecules, which from a macroscopic standpoint are small.4 Although statistical mechanics has been used to describe protein folding,5 the treatment has often been used to analyze thermodynamic functions obtained from measured heat capacities via deconvolution or other techniques.6 Only in recent years have attempts been made to calculate the thermodynamic functions, including the heat capacities, directly from statistical-mechanical formulations.1,7 Rigorous ab initio treatments would require not only the use of statistical mechanics but also of quantum mechanics, and such treatments are, as yet, not at hand. Fortunately, useful parameters for constructing the partition functions for evaluating thermal properties can often be inferred from nonthermodynamic data, such as optical density data, of the conformants in the coexistence region. Our approach, which is phenomenological, is based exclusively on optical measurements in the transition region; no heat-capacity or other thermodynamic data are used there. In this article, we extend the treatment to include entropy and free-energy functions and generalize it to multi- and singledomain systems. In a multidomain two-state system, which we are considering here, each conformant A or B consists of several independent, fully cooperative, sections or domains. To illustrate the usefulness of the method, the theory is applied to the thermal changes of solutions of recombinant Lys 25-ribonuclease T1 (ribonuclease, for short), exemplifying single-domain systems, and to solutions of poly-γ-benzyl-L-glutamate (PBG), exemplifying multidomain systems. The focus is on calculating the thermal properties rather than on mechanisms. Ribonuclease was

10.1021/jp003597d CCC: $20.00 © 2001 American Chemical Society Published on Web 06/16/2001

6388 J. Phys. Chem. B, Vol. 105, No. 27, 2001

Linder and Kromhout

chosen because good experimental data exist of the heat capacities,8 with which our calculated values can be compared, and also because it is a small protein. There is strong evidence that small globular proteins are single-domain systems.9 Similarly, measured heat-capacity data are available for the PBG helix-coil transformation10,11 with which we compare our calculated values.

functions (heat capacity, enthalpy, entropy, or free energy), we can define, symbolically, the excess (exc) and conformational (conf) thermodynamic functions

2. Procedure

where R a and R b are, respectively, the molar thermodynamic functions of the pure conformants and xa and xb are the mole fractions. The expressions for the pure conformant functions are the same as for the total functions shown in eq 3, except that the ∆’s and their derivatives are to be replaced by qa’s or qb’s (as the case may be), and their derivatives. The parameters needed to evaluate the partition function, ∆ (eq 2), are the ratio of the q’s, ∆h0, qa, and n and m. The second term in brackets in eq 2 clearly represents the ratio of the probability of finding “b” to the probability of finding “a” residues in a given domain; and because all conformants in that domain contain residues of either type “a” or type “b”, the ratio represents also the ratio of the probability of finding conformant B to the probability of finding conformant A in that domain, and if the m domains are equivalent (as assumed here), the ratio also represents the population ratio of the conformants in the entire system. Denoting the population ratio by xb/xa, we obtain

Paper 1 showed that the (isobaric-isothermal) partition function, ∆(T,P,N), of a single-domain molecule composed of N fully cooperative residues at temperature T and pressure P is

∆(T,P,N) ) qa(T)N {1 + [(qb(T)/qa(T)]N exp(-N∆h0/kT)} (1) where qa(T) and qb(T) are, respectively, the partition functions of the residues “a” and “b”, k is the Boltzmann constant, and ∆h0 is the difference of the zeros of energy (really enthalpy) of the “b” and “a” residues. In a two-state, multidomain molecule composed of m independent cooperative sections containing n1, n2, etc. residues, the molecular partition function is a product of the single-domain partition functions; and for a molecule of m equivalent domains, each containing n residues, the molecular partition function becomes

∆(T,P;n,m) ) {qa(T) [1 + [qb(T)/qa(T)] exp(-n∆h0/kT)} (2) n

n

Cp(T) ) 2RT∆′/∆ + RT [(∆′′/∆) - (∆′/∆) ]

(3a)

H(T) ) RT ∆′/∆

(3b)

S(T) ) R ln ∆ + RT∆′/∆

(3c)

G(T) ) -RT ln ∆

(3d)

2

2

2

where R is the gas constant, and ∆′ and ∆′′ are the first and second derivatives, respectively, of ∆:

∆′/∆ ) mn{∂ ln qa(T)/∂T + [qb(T)/qa(T)]n exp(-n∆h0/kT)[∂ ln qb(T)/∂T+ ∆h0/kT2]}/ {1 + [qb(T)/qa(T)]n exp(-n∆h0/kT)} (4a) -1 2

∆′′/∆ ) mn {(∂ ln qa(T)/∂T) + n ∂ ln qa(T)/∂T + 2

2

{1 + [qb(T)/qa(T)]n exp(-n∆h0/kT)} (4b) The thermodynamic functions shown in the foregoing equations represent the total functions; they include contributions from both conformants in the solution. We define two additional types of functions: excess functions, which are the functions in excess of the contributions of pure A, and conformational functions, which do not contain contributions from either pure A or pure B. (The designation “pure” is used to indicate that the solution contains one of the conformants and not both.) If we let R represent any of the above-mentioned thermodynamic

(5b)

[qb(Tm)/qa(Tm)]n ) exp(n∆h0/kTm)

(6)

(7)

If the ratio of the mole fractions of the conformants can be obtained from optical data (or other nonthermodynamic data, because our aim is to calculate the thermodynamic functions within the transition region), a plot of ln(xb/xa) vs 1/T will yield both the ratio of the q’s and ∆h0, as well as Tm. Outside the transformation region, the variation of the logarithms of qa and qb may be inferred from the behavior of the heat capacities of the pure conformants A and B. Let Ta be a sufficiently low temperature so that for all practical purposes only pure A is present, and Tb be a sufficiently high temperature so that only B conformants are present. Writing

ln qa(T) ) ln qa(Ta) + K ln(T/Ta) + A1(T - Ta) + A2(T - Ta)2 + ... (8a) ln qb(T) ) ln qb(Tb) + L ln(T/Tb) + B1(T - Tb) + B2(T - T2)2 + ... (8b)

n-1∂2 ln qb(T)/∂T2 + 2(∂ ln qb(T)/∂T)∆h0/kT2 2n-1∆h0/kT2 + (∆h0/kT2)2]}/

∆R conf ) R - xaR a - xbRb

and thus a plot of n-1 ln[xb(T)/xa(T)] vs T-1 will yield both the ratio qb(T)/qa(T) and ∆h0. At the midpoint, Tm, where xb(Tm) ) xa(Tm),

2

[qb(T)/qa(T)]n × exp(-n∆h0/kT)[(∂ ln qb(T)/∂T)2 +

(5a)

xb(T)/xa(T) ) [qb(T)/qa(T)]n exp(-n∆h0/kT)

m

The thermodynamic functions, per mole of molecules, of a very dilute system are readily obtained from ∆, and its temperature derivatives:

∆R exc ) R - Ra

and knowing the heat capacities of the pure conformants (either from experimental measurements or from empirical formulas), we can calculate the coefficients K, L, A1, B1 etc. and evaluate the derivatives of the q’s. Assuming that the molar heat capacities of the pure conformants A and B can be written respectively as a

Cp(T) ) aCp(Ta) + aC1(T - Ta) + aC2(T - Ta)2 + .....

(9a)

Cp(T) ) bCp(Tb) + bC1(T - Tb) + bC2(T - T2)2 + ....

(9b)

b

we obtain

Statistical-Mechanical Study of Thermal Folding

J. Phys. Chem. B, Vol. 105, No. 27, 2001 6389

RNK ) aCp(Ta) - TaaC1 + Ta2 aC2 + ...

(10a)

RNL ) bCp(Tb) - TbbC1 + Tb2 bC2 +.....

(10b)

RNA1 ) aC1/2 - (2/3)TaaC2 +

(10c)

RNB1 ) bC1/2 - (2/3)TbbC2 +

(10d)

RNA2 ) aC2/6 + ....

(10e)

RNB2 ) bC2/6

(10f)

yielding the derivatives

∂ ln qa(T)/∂T ) K/T + A1 + 2A2(T - Ta)2 + ....

(11a)

∂ ln qb(T)/∂T ) L/T + B1 + 2B2(T - Tb) + ....

(11b)

Figure 1. Plots of N-1 ln(xb/xa) vs 1/T for ribonuclease solutions S 1 (pH 2) and S 2 (pH 5) taken from optical density data of Kiefhaber et al. (ref 8; Figure 2A).

∂2 ln qa(T)/∂T2 ) -K/T2 + A2+ ....

(11c)

TABLE 1: Values of Tm and ∆h0/k of Ribonuclease Solutions at Varying pH Values; N ) 104

∂2 ln qb(T)/∂T2 ) -L/T2 + B2+ ....

(11d)

2

solution

pH

Tm (K)

∆h0/k

S1 S2

2 5

321.5 334.5

571.4 680

We now have the parameters needed to calculate all the thermodynamic functions of the single-domain molecules except the total entropy and total free energy, because these require knowledge of qa(Ta). For the multidomain molecules, we must also know the parameters m and n. Actually, there is really only one parameter, because the total number of residues, N, is generally known from the nature of the biomolecule and its molecular weight, and the product mn is of the order of N, and very likely as close to N as possible. Thus, if n is known, so is m. According to Privalov,9 the number of cooperative subunits in a domain is not likely to be less than 100 or greater than 200. In large proteins the domains are similar to those in small globular proteins.

other on Tm ) 324.0 K for easy comparison with the experimental heat-capacity curve of Kiefhaber et al. (The heat capacities are very sensitive to concentration, and the pH 2 solution the authors used to measure the optical density may not be exactly the same as the one used in their heat-capacity measurements.) The heat-capacity values of the pure conformants outside the transition range were taken from the semiempirical data of Gomez et al.12 For ribonuclease T1, they give the following expansions (in cal/deg‚mol)

3. Applications

unfolded state: Cp(T) ) 5130 + 15.43 (T-298) + ...

In Section A, we illustrate the technique by calculating the thermodynamic functions of ribonuclease T1. In Section B, we consider the thermal changes of PBG. A. Ribonuclease T1. Ribonuclease T1 has a molecular weight of 11 085 with 104 residues per molecule. Because it is a singledomain system N ) n and m ) 1. Figure 1 shows plots of N-1 ln(xb/xa) vs. 1/T for ribonuclease solutions S 1 (pH 2) and S 2 (pH 5) taken from the optical density data of Kiefhaber et al.8 The points represent values read from the curves of Kiefhaber et al. This is true of all our figures containing data taken from published experimental curves. We deduce from these plots the values of Tm and ∆h0/ k, listed in Table 1, for each solution. The straightness of the plots indicates that the ratio of qb to qa (but not necessarily their derivatives) is essentially temperature independent (see eq 6). The Tm values listed here agree well with the Tm values of Kiefhaber et al.8 who report Tm ) 48.5 °C for pH 2 and Tm ) 61.2 °C for pH 5 which they deduced from their optical data plot shown in their Figure 2. On the other hand, they display in their Figure 3 a peak for the pH 5 heat-capacity curve which is essentially the same as the peak at Tm they obtained from optical data, but the pH 2 heat-capacity peak is about 2.5° higher. All our calculations are based on the values of Tm listed in Table 1, except the heat-capacity curves for pH 2 shown in Figure 3. One of the calculated curves was based on Tm ) 321.5 K, the

Transforming these expressions in terms of Ta and Tb, which we arbitrarily set equal to Ta ) 308 K and Tb ) 360 K, we obtain the heat capacities (in J/deg‚mol)

native state: Cp(T) ) 3851.7 + 17.3 (T-298) (12a) (12b)

native state: Cp(T) ) 1.6823 × 104 + 72.31(T-308) (13a) unfolded state: Cp(T) ) 2.5442 × 104 + 64.49 (T-360) (13b) (The semiempirical expressions of Gomez et al.12 also include a quadratic term for the unfolded but not the folded state. We use two-term expressions throughout.) The use of eqs 10, 11, and 13 gives

K ) -6.30732; L ) 2.57315;

A1 ) 4.1849 × 10-2 B1 ) 3.7325 × 10-2

(14a) (14b)

The parameters that characterize the heat capacities outside the transition range are those given by eq 14a for the native state, and by eq 14b for the unfolded state. In the transition region, all the states of subunits “a” must eventually go over into those of “b”. Rather than speculate on how this occurs, we find it expedient to treat the background heat capacities of both conformants to behave like A below Tm and to behave like B


Linder and Kromhout

Figure 2. Plot of Cp(T) in kJ/deg‚g of S 2 (pH 5) of ribonuclease calculated from ∆h0 and Tm obtained from the slope and the equimolar temperature of the curve in Figure 1, listed in Table 1, and the semiempirical heat-capacity data of the pure conformants (ref 8). The points marked by circles are taken from the experimental heat-capacity curve of Kiefhaber et al. (ref 8, Figure 3). The dashed sigmoidal curve represents the calculated background heat-capacity contribution.

above Tm. In other words, we replace the derivatives of ln qb in eqs 4 and 5 by the derivatives of ln qa below Tm and the derivatives of ln qa by the derivatives of ln qb above Tm. This underestimates the contribution of B and overestimates the contribution of A below Tm; above Tm, it is the other way around. We compensate for the inaccuracies by subtracting the enthalpy and heat capacity of pure A below Tm and of pure B above Tm and replace them by the background enthalpy and heat capacity, defined as

H(b′g) ) xaHa + xbHb

(15a)

Cp(b′g) ) xaaCp + xbbCp

(15b)

Figure 3. Plots of Cp(T) in kJ/deg‚g of pH 2 ribonuclease solutions S 1 (Tm ) 321.5 K) and S 1′ (Tm ) 324.0 K). The points marked by circles are taken from the experimental pH 2 heat-capacity curve of Kiefhaber et al. (ref 8, Figure 3).

where Ha and Hb are the enthalpies of the pure conformants,

aC p

Ha ) NRT2(K/T + A1)

(16a)

Hb ) NRT2(L/T + B1)

(16b)

and bCp are the heat capacities of the pure conformants, a

Cp ) 2Ha/T - NRK

(17a)

Cp ) 2Hb/T - NRL

(17b)

b

and xb and xa ) 1 - xb are the mole fractions of the conformants related to the residue partition functions in accordance with eq 6. Figure 2 gives a plot of the heat capacity versus temperature of solution S 2 (pH 5). The solid curve is calculated using eqs 2-5, 15b, 17a, and 17b. The sigmoidal dashed curve represents the background contribution calculated from eqs 15b, 17a, and 17b. The points marked by circles are taken from the experimental heat-capacity curve of Kiefhaber et al. (ref 8; Figure 3). The theoretical curve agrees well with experimental points. Figure 3 shows plots of two pH 2 solutions, one characterized as S 1 with Tm ) 321.5 K and one characterized as S 1′ with Tm ) 324.0 K. The points marked by circles are from the experimental curves of Kiefhaber et al. (ref 8, Figure 3).

Figure 4. Plots of the conformational heat capacities, ∆Cp,conf., for solutions S 1 (pH 2; Tm ) 321.5 K) and S 2 (pH 5; Tm ) 334.5 K), obtained from the total heat capacities, Cp, by subtracting the contributions from pure A and pure B.

In Figure 4 we show the conformational heat capacities, ∆Cp,conf, for solution S 1 (pH 2; Tm ) 321.5 K) and solution S 2 (pH 5; Tm ) 334.5 K) obtained by subtracting the background heat capacities (eq 15b) from the total. The results are virtually identical with the heat capacities obtained by setting K ) L ) A1 ) B1 ) 0, indicating the self-consistency of the procedure. (The heat-capacity curves in Figures 2 and 3 are expressed in



Figure 6. Plots of the conformational enthalpies, ∆Hconf, of the two solutions S 1 and S 2.

Figure 5. Plots of the enthalpies of the solutions S 1 and S 2. The H values are the total enthalpies, inclusive of the background; the ∆Hexc are the total enthalpies minus the contributions from pure conformant A; the ∆Hconf are the total enthalpies minus the contributions from pure A and pure B.

Joules per degree per gram, like the experimental curves,8 rather than in kilojoules per degree per mole as in Figure 4.) Figure 5 shows plots of enthalpy functions of S 1 (pH 2) and S 2 (pH 5). We distinguish three types of enthalpy functions: (1) H, the total enthalpy; (2) ∆Hexc, the total enthalpy minus the enthalpy of pure conformant A, i.e., ∆Hexc ) H Ha; and (3) ∆Hconf, the conformational enthalpy, i.e., the total enthalpy minus the contribution from the background, ∆Hconf ) H - (xaHa + xbHb). The mole fractions, xa ) 1 - xb, and xb, are the mole fractions corresponding to the composition of the conformants in the solution, as expressed by eq 6. It is easy to show that ∆Hexc and ∆Hconf can be written as

∆Hexc ) RTxbN[∆h0/kT + ∂ ln qb/∂T - ∂ ln qa/∂T ]

(18) (19)

Figure 7. Plots of the excess entropies, ∆Sexc, of solutions S 1 and S 2.

Note that the total enthalpies, as shown in Figure 5, are much larger than either the excess or conformational enthalpies, indicating that most of the heat is used in heating the pure background conformants A and B. Moreover, the conformational enthalpies are smaller than the excess enthalpies (roughly by a factor of 5), indicating that most of the enthalpy change is associated with the change in enthalpies of pure A and B. The enthalpies that we refer to as excess enthalpies, ∆Hexc, correspond to the enthalpies usually determined by deconvolution methods from experimental heat-capacity data.13 The conformational enthalpies, as is apparent from eq 19, are associated with the promotion from the ground to the excited state, as measured by the change of the zeros of energy of “a” and “b”.

This enthalpy for xb ) 1 should correspond to what is generally referred to as van’t Hoff enthalpy of a single-domain, two-state system. Figure 6 shows enlarged plots of ∆Hconf vs T for the two solutions. Kiefhaber and co-workers8 report values for “van’t Hoff enthalpies” of ∆HvH ) 448 kJ/mol at Tm ) 48.5 °C (pH 2) and ∆HvH ) 531 kJ/mol at Tm ) 61.2 °C (pH 5), in reasonable agreement with our high temperature (xb ) 1) values of ∆Hconf ) 496 kJ/mol for solution S 1 (pH 2) and ∆Hconf ) 588 kJ/mol for solution S 2 (pH 5). Figures 7 and 8, respectively, depict the excess entropy, ∆Sexc, defined as the total entropy in excess of pure conformant A, i.e., ∆Sexc ) S - Sa, and the conformational entropy, ∆Sconf,

∆Hconf ) RxN∆h0/k


Linder and Kromhout

Figure 8. Plots of the conformational entropies, ∆Sconf, of solutions S 1 and S 2.

defined as the total entropy minus the contributions from the background, i.e., ∆Sconf ) S - xaSa - xbSb.

∆Sexc(T) ) R ln{1 + [q(T)/qa(T)]N exp(-N∆h0/kT)} + RxbNT[∆h0/kT2 + ∂ ln qb/∂T - ∂ ln qa/∂T] (20) ∆Sconf(T) ) R ln{1 + [qb(T)/qa(T)]N exp(-N∆h0/kT)} + RxbN[∆h0/kT - ln qb(T)/qa(T)] (21) In the event the ratio of the q’s is independent of the temperature, which is often the case, the last term in eq 21 may be approximated by ln(qb/qa) ) ∆h0/kTm. Total entropies and free energies are not included, because (as noted before) their calculation requires knowledge of ln qa, which we do not have. Notice that the excess entropy curves (Figure 7) resemble the excess enthalpy curves (Figure 5), but the conformational entropies (Figure 8) and conformational enthalpies (Figure 6) differ markedly. As shown in Figure 8, the conformational entropies go through a maximum and are exceedingly small. Comparing the excess and conformational entropy functions shows that virtually the entire entropy change in going from A to B can be accounted for by the difference between the entropies of the pure conformants with virtually no promotional contribution arising from the difference in the zeros of energy. It is known from elementary statistical mechanics that the zeros of energy affect the enthalpy (and free energy) functions but not the entropy functions. Finally, the excess free energy, ∆Gexc, and conformational free energy, ∆Gconf, defined, respectively, as ∆Gexc ) -RT ln ∆ - Ga and ∆Gconf ) -RT ln ∆ - xaGa - xbGb, can be written

∆Gexc ) -RT ln[1 + (qb/qa)N exp(-N∆h0/kT)] (22) ∆Gconf ) -RT ln[1 + (qb/qa)N exp(-N∆h0/kT)] + RTNxb ln(qb/qa) (23) (If the ratio of the q’s is independent of the temperature, the last term in eq 23 may be replaced by RTNxb∆h0/kTm.) Figure 9 shows the variation of ∆Gexc with temperature. The free energies are very much smaller (roughly by 2 orders of

Figure 9. Plots of the excess Gibbs free energies, ∆Gexc, of solutions S 1 and S 2.

magnitude) than the corresponding enthalpies, ∆Hexc, indicating a high degree of excess enthalpy-entropy compensation. On the other hand, the values of ∆Gconf, shown in Figure 10, are relatively large and very nearly equal to ∆Hconf reflecting the very small contributions from ∆Sconf. B. Poly-γ-benzyl-L-glutamate. We examined two solutions of PBG, using the optical rotation data of Ackermann and Ruterjans10 and of Karasz et al.11 We shall refer to these solutions as S 3 and S 4, respectively. The former is 0.438 g of a solution of 0.257 mol/kg of PBG (MW ∼235,000) in a 81:19 mixture of dichloroacetic acid (DCA) and 1,2-dichloroethane (DCE); the latter is 100 g of a solution of 2 g of PBG (MW ∼270,000) in a 75:25 mixture of DCA and DCE. We estimate that the total number of residues, N, of PBG in S 3 is 1007, and in S 4 it is 1157. The optical rotation data provide information on the ratio of helix to coil conformants, and in a system of equivalent multidomains, as we are assuming here, this ratio also represents the ratio of the “b” to “a” residues in a single domain. Thus, a plot of 1/n ln(xb/xa) vs 1/T should provide ∆h0 and the ratio of the q’s. We arbitrarily set n of both solutions equal to 130, a value comparable with the number of residues in small proteins (lysozyme has n ) 129; ribonuclease has n ) 104). We have used values of n both smaller and larger than 130 and found that the heat capacities varied slowly with n. We could have used n as a fitting parameter, but decided not to. Clearly, it would have been useful to have data that independently provided a value of n or m. For n ) 130 and the N values listed previously we obtain for solution S 3 the domain number m ) 7, and for solution S 4, m ) 9. Figure 11 shows plots of 1/n ln(xb/xa) vs 1/T for S 3 and S 4 solutions taken from ref 10 and ref 11, respectively. The results obtained from these plots are summarized in Table 2.



Figure 12. Plots of ∆Cp,conf in cal/deg‚mol of PBG residues of solutions S 3 and S 4.

Figure 10. Plots of the conformational Gibbs free energy, ∆Gconf, of solutions S 1 and S 2.

Figure 11. Plots of 1/n ln(xb/xa) vs 1/T for PBG solutions S 3 and S 4, taken from optical rotation data of Ackermann and Ruterjans [ref 10, Figure 1] and Karasz et al. [ref 11, Figure 2], respectively.

TABLE 2: Values of Tm and ∆h0/k of PBG Solutions; N ) 130 solution

m

Tm (K)

∆h0/k

S3 S4

7 9

305.9 299.9

270.43 305.75

Figure 12 shows plots of the conformational heat capacities in calories per degree per mole of PBG residues in solutions S 3 and S 4. The difference in height and shape of the two curves is largely caused by the difference in the ∆h0/k values. The experimental heat capacities given in refs 10 and 11, with which we compare our calculated results in Figures 13 and 14, were determined by micro- but not differential calorimetry measurements, and no accurate information on the pure

Figure 13. Plot of ∆Cp,conf in cal/deg of 438 g of solution S 3. The drawn curve is calculated; the points are the experimental values obtained from data of ref 10.

conformants outside the transition region can be inferred. Our focus in Section B of this article is on the conformational heatcapacity and enthalpy values. The conformational free-energy values can be expected to be close to the conformational enthalpies, and the conformational entropies can be expected to be close to zero. Figure 13 displays the variation of the conformational heat capacity in calories per degree of solution S 3. The drawn curve is the theoretical curve; the points are taken from the experimental values of Ackermann and Ruterjans10 (their Figure 1). As seen, the two curves are quite similar although there is a slight shift in the Tm values, and the half-width of the theoretical curve is broader.


Figure 14. Plot of ∆Cp,conf in J/deg‚g of solution S 4. The drawn curve is calculated. The points are the experimental values obtained from ref 11.

Figure 15. Plots of ∆Hconf in cal/mol of PBG residues in solutions S 3 and S 4.

Figure 14 depicts the heat capacity in Joules per degree per gram of S 4 solution. The drawn curve is the calculated one; the points are taken from the experimental values of Karasz et al.11 (their Figure 2). Again, the theoretical curve is slightly broader than the experimental, although there is no noticeable shift in Tm. Figure 15 depicts the conformational enthalpy changes in calories per mole of PBG residues in S 3 and S 4. The sigmoidal curves are typical of the enthalpy curves in the transition region. The difference between the high- and low-temperature values of the curve in S 3 is 483.0 cal/mol of residues and for S 4 it is 623.6 cal/mol of residues. These are comparable with the experimental values of 430 cal/mol of residues of Ackermann and Ruterjans and 525 cal/mol residue of Karasz et al. obtained by the authors from the areas under the heat-capacity curves. 4. Summary and Conclusions 1. The formulation of paper 1 for calculating heat capacities and enthalpies in the unfolding of biomolecules proves to be a viable technique for extension to other thermodynamic functions and for generalization to multidomain systems.

Linder and Kromhout 2. Application to ribonuclease and PBG predict the characteristic features (bell-shaped heat capacities, sigmoidal enthalpies, etc.) in the thermal unfolding, and the results are in reasonably good agreement with published experimental data. The calculated results are based on a two-state model. In the case of multidomain systems, the domains are treated as equivalent. 3. The calculated excess functions, ∆Hexc, and T∆Sexc, of ribonuclease are virtually equal to each other showing a great deal of entropy-enthalpy compensation, in harmony with a large class of studies of protein folding.14-16 4. The conformational functions behave quite differently. These functions are indicative of the promotional change from the ground state to the excited state, largely because of the difference in the zeros of energy, which has a negligible effect on the entropy. The denaturation of the protein may be viewed as a two-step process: (1) involving the promotion from the native ground state to the excited denatured state, and (2) the activation of the degrees of freedom of the excited state. The excess functions comprise both processes, but the conformational functions are largely caused by the promotional process. Comparing the excess and conformational functions suggests that although part of the heat supplied is used as promotional enthalpy, virtually all the entropy change is associated with activating the degrees of freedom of the excited state. 5. The excess free-energy functions, ∆Gexc, are very small, consistent with the large enthalpy-entropy compensation of the excess functions. On the other hand, ∆Gconf is essentially equal to ∆Hconf reflecting the small contribution of ∆Sconf. Although ∆Gconf is positive, ∆Gexc is negative, clearly implying that what makes the transition feasible is the higher entropy value of the denatured state. 6. The functions analogous to the thermodynamic functions in true first-order transitions are the excess functions. From the standpoint of statistical thermodynamics, the near equality of ∆Hexc and T∆Sexc is to be expected for our model. The continuous, although steep rise of the entropy and enthalpy curves with temperature is a consequence of the smallness of the domain. 7. The information that we obtain from the optical data are the characteristic values (∆h0, Tm, qb/qa) for effective residues, which takes account phenomenologically of the interaction between residues and between residues and solvent. By doing so, we lose detailed knowledge of the effects of residue structure and heterogeneity of residues. An advantage of our approach is that changes in solvent, including pH, are included insofar as they affect the folding. The characteristic parameters are found to change substantially with pH resulting in different thermodynamic functions, and this is borne out by experiment. 8. Although the characteristic parameters of our model, referred to above, are obtained phenomenologically, the data used for the calculation are nonthermodynamic data. Once the parameters are fixed, the model predicts the conformational thermodynamic functions. To obtain the full thermodynamic functions, including background contributions, also requires data of the thermodynamic heat capacities of the pure conformants outside the transition region. 9. Finally, the simple, statistical-mechanical approach to the conformational changes in biomolecules, presented here and in paper 1,1 provides a unified treatment of the thermal properties of varied systems. The theory predicts correctly the thermal properties in several biomolecular systems (two ribonuclease solutions, two PBG solutions, three lysozyme solutions1). The present treatment is predicated on the assumption that all

Statistical-Mechanical Study of Thermal Folding residues of a given conformant in a domain are fully cooperative and that all cooperative segments are equivalent. The good agreement between the calculated and experimental values indicates that the model is reasonable for the systems considered. References and Notes (1) Linder, B.; Kromhout, R. A. J. Phys. Chem. B 1999, 103, 1032510330. (2) Kidokoro, S.-I.; Wada, A. Biopolymers 1987, 26, 213-229. (3) Karplus, M.; Shakhnovich, E. In Protein Folding; Creighton, T. E., Ed.; W. H. Freeman: New York, 1992; p 137. (4) Hill, T. L. Thermodynamics of Small Systems, Part 1; W.A. Benjamin: New York, 1963. Thermodynamics of Small Systems, Part II; W.A. Benjamin: New York, 1964. (5) (a) Hardin C.; Luthey-Schulten Z.; Wolynes, P. G. Protein Struct Funct. Genet. 34, 281-294. (b) Wilder, J.; Shakhanovich, E. I. Phys. ReV. E 2000, 62, 7100-7110. (c) Plotkin, S. S.; Onuchic J. N. Proc. Natl. Acad. Sci. USA 2000, 97, 6509-6514.

J. Phys. Chem. B, Vol. 105, No. 27, 2001 6395 (6) Freire, E.; Bilthoven, R. L. Biopolymers 1978, 17, 463-479. (7) (a)Hansen, A.; Jensen, M. H.; Sneppen, K.; Zocchi, G. Eur. Phys. J. B 1998, 6, 157-161. b) Rosgen, J.; Hellerbach, B.; Hinz, H. J. Biophys. Chem. 1998, 74, 153-161. c) Dinner, A. R.; Sali, A.; Smith, L. J.; Dobson, C. M.; Karplus, M. Trends Biochem. Sci. 2000, 25, 331-339. (8) Kiefhaber, T.; Schmid, F. X.; Renner, M.; Hinz, H. J.; Hahn, U.; Quaas, R. Biochemistry 1990, 29, 8250-8257. (9) Privalov, P. L. In Protein Folding; Creighton, T. E., Ed.; W. H. Freeman: New York, 1992; Chapter 3. (10) Ackermann, V. T.; Ruterjans, H. Ber. Bunseng. 1964, 68, 850. (11) Karasz, F. E.; O’Reilly, J. M.; Bair, H. E. Nature 1964, 202, 693. (12) Gomez, J.; Hilser, V. J.; Xie, D.; Freire, E. Proteins 1995, 22, 404412. (13) Freire, E. In Protein Stability and Folding; Shirley, B. A., Ed.; Humana Press: Totowa, NJ, 1995; pp 191-218. (14) Liu, L.; Yang, C.; Guo, Q. -X. Biophys. Chem. 2000, 84, 239251. (15) Lee, E. Biopolymers 1991, 31, 993-1008. (16) Grunwald, E.; Comeford, L. L. In Protein-SolVent Interactions; Gregory, R. B., Ed.; Marcel Dekker: New York, 1995; Chapter 10.

Statistical-Mechanical Approach to the Thermodynamic Functions in

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