Analysis of Two-State Folding Using Parabolic Approximation II: Temperature-Dependence

Transcription

1 Analysis of wo-tate Folding Using Parabolic Approxiation II: eperature-dependence AUHOR NAME: Robert. ade AUHOR ADDRE: Vinkensteynstraat 18, 56 V, Den Haag, Netherlands AUHOR ADDRE: AUHOR AFFILIAION: Independent Researcher KEY WORD: Arrhenius rate law; Marcus theory; parabolic approxiation; transition state theory; two-state folding. Page 1 of 44

2 ABRAC Equations that govern the teperature-dependence of the rate constants, Gibbs energies, enthalpies, entropies and heat capacities of activation for folding and unfolding of spontaneously-folding fixed two-state systes have been derived using a procedure that treats the denatured and the native conforers as being confined to haronic Gibbs energy wells. he notion that a two-state syste is physically defined only for a set teperature range is introduced. he iplications of this novel treatent for protein folding are discussed. Page of 44

3 INRODUCION It was shown previously, henceforth referred to as Paper-I, that the equilibriu and kinetic behaviour of spontaneously-folding fixed two-state proteins can be analysed by a treatent that is analogous to that given by Marcus for electron transfer. 1 In this fraework tered the parabolic approxiation, the Gibbs energy functions of the denatured (DE) and the native state (NE) ensebles are represented by parabolas whose curvature is given by their teperature-invariant force constants, and, respectively. he teperature-invariant ean length of the reaction coordinate (RC) is given by D-N and is identical to the separation between the vertices of the DE and the NE parabolas along the abscissa. iilarly, the position of the transition state enseble (E) relative to the DE and the NE are given by -D() and -N(), respectively, and are identical to the separation between the curvecrossing and the vertices of the DE and the NE parabolas along the abscissa, respectively. he Gibbs energy of unfolding at equilibriu, ΔG D-N(), is identical to the separation between the vertices of the DE and the NE parabolas along the ordinate. iilarly, the Gibbs activation energy for folding (ΔG -D() ) and unfolding (ΔG -N() ) are identical to the separation between the curve-crossing and the vertices of the DE and the NE parabolas along the ordinate, respectively (Figure 1). Further, it was shown that the curve-crossing relative to the DE and the NE Gibbs basins is given by -D( ) D-N λ GD-N( ) D-N φ (1) -N( ) λ G φ D-N( ) D-N D-N () where the discriinant G, and φ λ D-N( ) is the Marcus reorganization energy for two-state folding, and by definition is the Gibbs energy required to copress the denatured polypeptide under folding conditions to a state whose solvent accessible surface area (AA) is identical to that of the native folded protein but without the stabilizing native interactions. If the teperature-dependence of ΔG D-N() and the values of,, and D-N are known for any two-state syste at constant pressure and solvent conditions (see Methods in Paper-I), the position of its E relative to the ground states for any D-N Page 3 of 44

4 teperature ay be readily calculated for that particular solvent. he teperaturedependence of the curve-crossing is central to this analysis since all relevant therodynaic state functions can be readily derived by anipulating the sae. Because by postulate the force constants and D-N are teperature-invariant for a fixed two-state folder at constant pressure and solvent conditions, we get fro inspection of Eqs. (1) and () that the discriinant φ, and ust be a axiu when ΔG D-N() is a axiu. Because ΔG D-N() is a axiu when = ( is the teperature at which the entropy of unfolding at equilibriu, Δ D-N(), is zero), 3 a corollary is that φ and ust be a axiu when = ; and any deviation in the teperature fro will only lead to their decrease. Consequently, -D() and (fold)() ( -D( ) D-N ) are always a iniu, and -N() and (unfold)() ( -N( ) D-N ) are always a axiu at. his gives rise to two further corollaries: Any deviation in the teperature fro can only lead to: (i) an increase in - and ; and (ii) a decrease in and. A further consequence of D() (fold)() -N() (unfold)() - being a iniu at is that if for a two-state-folding priary sequence there exists a D() chevron with a well-defined linear folding ar at, then -D() > 0 for all teperatures (Figure 1B). ince the curve-crossing is physically undefined for φ < 0 owing to there being no real roots, the axiu theoretically possible value of -D() will occur when φ = 0 and is given by: (note that since the force constants are positive, -D( ) D-N 0 ax -D( ) D-N ). Because D-N = -D() + -N() for a two-state syste, ax and D-N is teperature-invariant by postulate, the theoretical iniu of -N() is given by: -N( ) in D-N. Iportantly, since -N() is a axiu and positive at but its iniu is negative, a consequence is that -N() = 0 at two unique teperatures, one in the ultralow and the other in the high teperature regie. We will discuss how this can lead to barrierless unfolding and inversion of the unfolding rate constant later. he teperature-dependent shift in the curve-crossing relative to the ground states along the RC is consistent with Haond oveent; and just as it is coonplace in physical organic cheistry to rationalize the physical basis of these effects using Marcus theory, we can siilarly rationalize these effects in protein folding using parabolic approxiation. 4-8 he Gibbs barrier heights for folding and unfolding are given by Page 4 of 44

5 φ G-D( ) D-N -D( ) β ( fol d) ( ) φ D-N G-N( ) -N( ) β (unfold)( ) (3) (4) Because the force constants are teperature-invariant by postulate, and -D() is a iniu, and -N() a axiu at, ΔG -D() and ΔG -N() are always a iniu and a axiu, respectively, at. hus, a corollary is that any deviation in teperature fro can only lead to an increase in ΔG -D() and a decrease in ΔG -N(). he Arrhenius expressions for the rate constants for folding (k f() ) and unfolding (k u() ), and ΔG D-N() are given by 0 D-N φ 0 β (fold)( ) kf ( ) k exp k exp R R (5) φ 0 D-N 0 β (unfold)( ) ku ( ) k exp k exp R R (6) σ DE( ) GD-N( ) (unfold)( ) (fold)( ) (unfold)( ) (fold)( ) σ NE( ) (7) where k 0 is the teperature-invariant prefactor with units identical to those of the rate constants (s -1 ), R is the gas constant, is the absolute teperature, and NE( ) σ R and DE( ) σ R are the variances of the Gaussian distribution of the AA of the conforers in the DE and the NE, respectively (see Paper-I). A consequence of Eq. (7) is that if for two-state systes at constant pressure and solvent conditions (fold)() 0.5 when =, then it is theoretically ipossible for such systes to be stable at equilibriu (ΔG D-N() > 0) unless σ σ. NE( ) DE( ) Page 5 of 44

6 he purpose of this article is to derive equations that will give a rigorous therodynaic description of fixed two-state systes across a wide teperature regie. Because this article is priarily an extension of Paper-I, any critical evaluation by the reader of the interpretation and the conclusions drawn here ust be done in conjunction with the fraework developed in Paper-I. In this article as well as in all subsequent papers, ters such as favourable and unfavourable will be used extensively. Although perhaps unnecessary, it is nevertheless entioned that their usage is inextricably linked to the specific reaction-direction being addressed. If the reaction-direction is reversed, the agnitude of the change in the relevant state functions will be invariant, but their algebraic signs will invert leading to a change in the interpretation. For exaple, if the forward reaction is endotheric and thus enthalpically disfavoured at a particular teperature, it naturally iplies that the reverse reaction will be exotheric and enthalpically favoured. iilarly, if ΔG D-N() > 0 at a certain teperature, then ΔG N-D() < 0; consequently, the unfolding reaction N Dis energetically disfavoured, while the folding reaction D N is energetically favoured at that particular teperature. Further, the ter equilibriu stability or stability, which we will often use is synonyous with ΔG D-N() and not ΔG N-D(). he reader will also note that in addition to the standard reference teperatures used in protein folding, 3 a few novel reference teperatures are introduced here to enable a physical description of the syste. A glossary of the sae is given in able 1. HERMODYNAMIC RELAIONHIP Activation entropy for folding (Δ -D() ) he activation entropy for the partial folding reaction D [ ] (D denotes DE and [] denotes E) is given by the first derivative of ΔG -D() with respect to teperature. Differentiating Eq. (3) gives G -D( ) -D( ) -D( ) -D( ) -D( ) -D( ) -D( ) (8) Page 6 of 44

7 Note that since -D() is a coposite function, we use the chain rule. he partial derivatives are to indicate that the derivative is at constant pressure and solvent conditions (constant ph, ionic strength, co-solvents etc.); however, we will oit the subscripts for brevity. ubstituting Eqs. (A6) and (A9) in (8) gives (see Appendix) C φ φ -D( ) D-N( ) -D( ) pd-n -D( ) ln (9) Although and ΔC pd-n are positive and teperature-invariant by postulate, since φ has no real roots for φ 0, -D() > 0 no atter what the teperature, and both φ and -D() vary with teperature, the teperature-dependence of Δ -D() would be coplex, with its algebraic sign being deterined purely by the ln ter. his leads to three scenarios: (i) Δ -D() > 0 for < ; (ii) Δ -D() < 0 for > ; (iii) Δ -D() = 0 when =. hus, the activation of denatured conforers to the E is entropically: (i) favourable for < ; (ii) unfavourable for > ; and (iii) neither favourable nor unfavourable when =. Activation entropy for unfolding (Δ -N() ) he activation entropy for the partial unfolding reaction N [ ] (N denotes NE) ay be siilarly obtained by differentiating ΔG -N() (Eq. (4)) with respect to teperature. hus, we ay write G -N( ) -N( ) -N( ) -N( ) -N( ) -N( ) -N( ) (10) ubstituting Eq. (A11) in (10) gives (see Appendix) C -N( ) D-N( ) -N( ) pd-n -N( ) ln φ φ (11) Despite the apparent siilarity between Eqs. (9) and (11), since -N() unlike -D() can be positive, zero, or even negative depending on the teperature, the variation in the algebraic sign of the Δ -N() function with teperature, and its physical interpretation is far ore Page 7 of 44

8 coplex. Although it is readily apparent that Δ -N() ust be zero when =, theoretically there can be two additional teperatures at which Δ -N() is zero, one in the ultralow teperature regie (designated () ), and the other at high teperature (designated () ); and these two additional teperatures at which Δ -N() is zero occur when -N() = 0. hus, the algebraic sign of the Δ -N() function across a wide teperature regie is deterined by both -N() and the ln( ) ters: (i) for < (), both -N() and ln( ) are negative, leading to Δ -N() > 0; (ii) when = (), -N() = 0, leading to Δ -N() = 0; (iii) for () < <, -N() > 0 but ln( ) < 0, leading to Δ -N() < 0; (iv) when =, ln( ) = 0, leading to Δ -N() = 0; (v) for < < (), both -N() and ln( ) are positive, leading to Δ -N() > 0; (vi) when = (), -N() = 0, leading to Δ -N() = 0; and (vii) for > (), -N() > 0 but ln( ) < 0, leading to Δ -N() < 0. Consequently, we ay state that the activation of native conforers to the E is entropically: (i) favourable for < () and < < () ; (ii) unfavourable for () < < and > () ; and (iii) neither favourable nor unfavourable at (),, and (). If we reverse the reaction-direction (i.e., the partial folding reaction[ ] N ), the algebraic sign of the Δ -N() function inverts leading to a change in the interpretation. herefore, we ay state that the flux of the conforers fro the E to the NE is entropically: (i) unfavourable for < () and < < () (Δ -N() > 0 Δ N-() < 0); (ii) favourable for () < < and > () (Δ -N() < 0 Δ N-() > 0); and (iii) neutral at (),, and (). Note that the ter flux is operationally defined as the diffusion of the conforers fro one reaction state to the other on the Gibbs energy surface. We know fro the pioneering work on the teperature-dependence of protein stability that at we have Δ D-N() = Δ -N() Δ -D() = 0. 3,9 However, this condition per se does not give us any inforation on the absolute values of Δ -D() or Δ -N() other than tell us that they ust be identical at. Eqs. (9) and (11) are rearkable because they deonstrate that Δ - and Δ are independently equal to zero at. Consequently, at we have = D() -N() D() () = N(). his relationship ust hold true for every two-state folder when pressure and solvent are constant. Page 8 of 44

9 hese analyses lead to two fundaentally iportant conclusions: First, for any two-state folder at constant pressure and solvent conditions, ΔG D-N() will be the greatest when the activation entropies for folding and unfolding are both zero, and this occurs precisely at. A corollary is that ΔG D-N() is a axiu when ΔG -D() and ΔG -N() are both purely enthalpic; and because ΔG -D() and ΔG -N() are both positive at, it iplies that the activation enthalpy for folding (ΔH -D() ) and unfolding (ΔH -N() ) are both endotheric at. Because by postulate D-N, -D() and -N() are true proxies for ΔAA D-N, ΔAA D- and ΔAA, respectively, equilibriu stability of a two-state syste at constant () -N() pressure and solvent conditions will be the greatest when the conforers in the DE are displaced the least fro the ean of their enseble along the AA-RC to reach the E (or bury the least aount of AA to reach the E), and this occurs precisely at. his principle of least displaceent ust be valid for every two-state syste. A corollary is that ΔG D-N() will be the greatest when the native conforers expose the greatest aount of AA to reach the E. econd, although Δ -N() = 0 () = N() at (),, and (), the underlying therodynaics is fundaentally different at as copared to () and (). While both ΔG -N() and -N() are a axiu and ΔG -N() is purely enthalpic at (ΔG -N() = ΔH -N() ), we have -N() = 0 G 0H 0 at () and -N( ) -N( ) -N( ) (). In addition, because ΔG -N() = 0 at () and (), the rate constant for unfolding will reach a axiu for that particular solvent at these two teperatures. o suarize, while G () >> G N() and D() = () = N() at, we have G () = G N(), H () = H N(), () = N(), and k u() = k 0 at () and (). A corollary is that if two reaction-states on a protein folding pathway have identical AA and Gibbs energy under identical environental conditions (teperature, pressure, ph, co-solvents etc.), then their enthalpies and entropies, ust also be identical. his ust hold irrespective of whether or not they have identical, siilar, or dissiilar olecular structures. he iplications of this stateent for the applicability of the Haond postulate to protein folding will be addressed in the subsequent publication. Page 9 of 44

10 eperature liits of a two-state syste A singularly iportant consequence of φ being undefined for φ 0 has a deeper physical eaning. It iplies that a barrier-liited two-state syste (i.e., the notion that the conforers are confined to two intersecting haronic Gibbs energy wells) is physically defined only for a set teperature-range given by α ω, where α and ω are the ultralow and the high teperature liits, respectively. hus, the prediction is that for < α and > ω, the E cannot be physically defined, and all of the conforers will be confined to a single haronic Gibbs energy well, which is the DE (glass transition is discussed elsewhere) A consequence is that although one could, in principle, use the Gibbs-Helholtz equation (Eq. (A51)) and calculate equilibriu stability for any teperature above absolute zero, equilibriu stability per se has physical eaning only for α ω. his is because equilibriu stability is an equilibriu anifestation of the underlying theral-noise-driven flux of the conforers fro the DE to the NE, and vice versa, via the E; consequently, if the position of the E along the RC becoes physically undefined owing to φ being atheatically undefined for φ 0, k f() and k u() becoe physically undefined, leading to D-N( ) f ( ) u( ) G Rln k k being physically undefined. hus, the liit of equilibriu stability below which a two-state syste becoes physically undefined is given by: D-N( ), G and is purely a function of the priary sequence when pressure and solvent are defined. Consequently, the physically eaningful range of equilibriu stability for a two-state syste is given by: D-N( ) D-N( ) D- N ( ), G G G. his is akin to the stability range over which Marcus theory is physically realistic (see Kresge, 1973, page 494). 16 hese arguents apply to equilibriu enthalpies and entropies as well, i.e., although the values Δ D-N() and ΔH D-N() can be calculated for any teperature above absolute zero using Eqs. (A8) and (A4), respectively, they do not have any physical eaning for < α and > ω. Now, fro the view point of the physics of phase transitions, α ω denotes the coexistence teperature-range where the DE and the NE, which are in a dynaic equilibriu, will coexist as two distinct phases; and for < α and > ω there will be a single phase, which is the DE, with α and ω being the liiting teperatures for Page 10 of 44

11 coexistence (or phase boundary teperatures fro the view point of the DE), and the protein will cease to function. 17 A consequence is that as long as the covalent structure of spontaneously-folding priary sequences are not altered on exposure to < α (as in the case of glacial periods in an ice age) and > ω (as in the case of intense forest fires), their behaviour will be identical to that of untreated proteins when the teperature returns to α ω. Further discussion on the teperature-range over which two-state systes are physically defined and how this range can be odulated by living systes to cope with a wide variety of environents, the parallels between two-state proteins and Boolean circuits, and why higher intelligence ay not be possible without teperature control and biological ebranes is beyond the scope of this article and will be explored in subsequent publications. 18- Activation enthalpy for folding he activation enthalpy for the partial folding reaction D [ ] is given by the Gibbs equation H G (1) -D( ) -D( ) -D( ) ubstituting Eqs. (3) and (9) in (1) and recasting gives C pd-n H-D( ) -D( ) 1 ln -D( ) φ (13) Inspection of Eq. (13) shows that for <, ΔH -D() > 0, but decreases with a rise in H G, i.e., teperature. When =, Eq. (13) reduces to -D( ) -D( ) -D( ) 0 the Gibbs barrier to folding is purely enthalpic at, with k f() being given by k G H 0 -D( ) 0 -D( ) f ( ) k exp k exp (14) R R Because ln( ) 0 for > and becoes increasingly negative with a rise in 1 0, ΔH -D() will continue to be teperature, as long as CpD-N ln( ) -D( ) φ Page 11 of 44

12 positive. Precisely when ln( ) -D( ) φ CpD-N, ΔH -D() = 0; and if we designate this teperature as H(-D), then k f() is given by G k f ( ) exp exp H (-D) R R 0 -D( ) 0 -D( ) k k (15) H(-D) H(-D) ubstituting Eq. (9) in (15) gives k ( ) k C 0 -D( ) pd-n exp ln f H (-D) R φ H (-D) (16) As the teperature increases beyond H(-D), CpD-N ln( ) -D( ) φ 1 0, leading to ΔH -D() < 0 for > H(-D). Although we have discussed the teperature-dependence of the algebraic sign of the ΔH -D() function fro the view point of the su of the ters inside the square bracket on the RH of Eq. (13), there is another equally valid set of logical arguents that enables us to arrive at the sae. ince > 0, we have ΔG -D() > 0 for all teperatures (except for the special case of barrierless folding, see below). Because Δ -D() > 0 for <, (Eq. (9)) the only way the condition that G-D( ) H-D( ) -D( ) 0 is satisfied is if ΔH -D() > 0 or endotheric. In addition, when =, Δ -D() = 0 (Eq. (9)), and the positive Gibbs activation energy for folding is entirely due to endotheric activation enthalpy for folding, i.e., G-D( ) H-D( ) 0. hus, we ay state that ΔH -D() > 0 or endotheric for. At H(-D) we have G-D( ) -D( ) 0 which is satisfied if and only if Δ -D() < 0. Because Δ -D() < 0 only when > (Eq. (9)), we have the second logical condition: H(-D) >. Now since ΔG -D() has only one extreu, which is a iniu at along the teperature axis ( G-D( ) -D( ) 0), there will also be one extreu which is a iniu for G-D( ) across the teperature axis (i.e., the first derivative of the Massieu-Planck activation potential for folding is zero, or G -D( ) -D( ) 0 H ; see chellan, 1997, on the use of Massieu-Planck functions in protein therodynaics). 3 In other words, there exists only one teperature, H(-D), at which ΔH -D() = 0. hus, a corollary is that for two-state folders at constant Page 1 of 44

13 pressure and solvent conditions, k f() is a axiu when the Gibbs barrier to folding is purely entropic. hus, we ay conclude that for a two-state syste at constant pressure and solvent conditions: ΔH -D() > 0 at very low teperature, decreases in agnitude with a rise in teperature, and eventually reaches zero at H(-D) ; and any further increase in teperature beyond H(-D) causes ΔH -D() to becoe negative or exotheric. o suarize, we have three iportant scenarios: (i) ΔH -D() > 0 for < H(-D) ; (ii) ΔH -D() < 0 for H(-D) < ; and (iii) ΔH -D() = 0 for = H(-D). hus, the activation of the denatured conforers to the E is enthalpically: (i) unfavourable for < H(- ; (ii) favourable for < ; and (iii) neutral when =. D) H(-D) H(-D) Activation enthalpy for unfolding he activation enthalpy for the partial unfolding reaction N [ ] is given by H G (17) -N( ) -N( ) -N( ) ubstituting Eqs. (4) and (11) in (17) and recasting gives C pd-n H-N( ) -N( ) 1 ln -N( ) φ (18) Unlike the ΔH -D() function which is positive for < H(-D), negative for H(-D) <, and iportantly, changes its algebraic sign only once across the entire teperature range over which a two-state syste is physically defined, the behaviour of ΔH -N() function is far ore coplex; and just as the Δ -N() function can becoe zero at three distinct teperatures, so too can the ΔH -N() function. tarting fro the lowest teperature ( α ) at which a two-state syste is physically defined, for α < (), both -N() and ln( ) ters are negative, leading to ΔH -N() > 0. When = (), -N() = 0, leading to a unique scenario wherein ΔG -N() = ΔH -N() = Δ -N() = 0, and k u() = k 0. hus, () is the first and the lowest teperature at which ΔH -N() = 0. In addition, the first extreu of ΔG -N() ( G-N( ) -N( ) 0), the first extreu of G-N( ) (i.e., the first derivative of the Massieu-Planck activation potential for unfolding is zero, or Page 13 of 44

14 G -N( ) H ), and the first extreu of k u() ( -N( ) 0 ) occur at () ; and while ΔG -N() and G-N( ) are ln ku ( ) H-N( ) R 0 both a iniu, k u() is a axiu. Because -N() > 0 for () < < (), and ln( ) 0 for <, as long as CpD-N ln( ) -N( ) φ 1, ΔH -N() will be negative. ince for <, ln( ) becoes less negative with a rise in teperature, at soe point CpD-N -N( ) ln( ) φ 1(or when ln( ) -N( ) φ CpD-N ). Naturally, at this teperature the algebraic su of the ters inside the square bracket on the RH of Eq. (18) is zero leading to ΔH -N() becoing zero for the second tie. If we designate the teperature at which this occurs as H(-N), then k u() is given by G ku ( ) exp exp H (-N) R R 0 -N( ) 0 -N( ) k k (19) H(-N) H(-N) ubstituting Eq. (11) in (19) gives k u ( ) H (-N) C R φ 0 -N( ) pd -N k exp ln (0) H (-N) Because ln ku ( ) H-N( ) R 0, a corollary is that the second extreu of k u() ust occur when = H(-N). In addition, since the first derivative of the Massieu-Planck activation potential for unfolding ust also be zero, i.e., -N( ) H-N( ) 0 when = H(-N), and since G-N( ) -N( ) G 0 for () and (), and is a axiu when = s, the conclusion is that G-N( ) ust be a axiu at H(-N). Because k u() ust be a iniu when G-N( ) is a axiu, a corollary is that for two-state systes at constant pressure and solvent conditions, k u() is a iniu when the Gibbs barrier to unfolding is purely entropic. Iportantly, in contrast to the first extrea of k u(), ΔG -N() and G-N( ) which occur at (), and are characterised by k u() being a axiu (k u() = k 0 ), and both ΔG -N() and G-N( ) being a iniu, the second extrea of k u() and G-N( ) which occur at H(-N) are characterised by k u() Page 14 of 44

15 being a iniu and G-N( ) being a axiu (the reader will note that the second extreu of ΔG -N() which is a axiu occurs not at H(-N) but at ). Now, the obvious question is: Where is H(-N) relative to () and? Because -N( ) -N( ) G 0 for () and (), G-N( ) -N( ) 0when = H(-N) ; and the only way this condition can be satisfied is if Δ -N() < 0. ince Δ -N() < 0 only for () < < (Eq. (11)), the logical conclusion is that H(-N) ust lie between () and, or () < H(-N) <. When = Eq. (18) reduces to H G, i.e., the Gibbs barrier to unfolding is purely -N( ) -N( ) -N( ) 0 enthalpic at, with k u() being given by k G H 0 -N( ) 0 -N( ) u ( ) k exp k exp (1) R R Now, since -N() > 0 for () < < (), and ln( ) 0 for >, ΔH -N() > 0 for >. However, since -N() is a axiu at and decreases with any deviation in teperature, the ΔH -N() function initially increases with an increase in teperature, reaches saturation, and decreases with further rise in teperature; and precisely when = (), - = 0 and ΔH = 0 for the third and the final tie. Consequently, we ay state that N() -N() ΔH -N() > 0 for H(-N) < < (). Akin to (), when = (), we note that ΔG -N() = ΔH -N() = Δ -N() = 0, and k u() = k 0. his is also the teperature at which both ΔG -N() G and -N( ) are a iniu, and k u() is a axiu (the third and the final extreu). With further rise in teperature, i.e., for > (), -N() < 0 and ln( ) 0, leading to ΔH -N() < 0. o suarize, we have three interesting scenarios: (i) ΔH -N() > 0 for < () and H(-N) < < () ; (ii) ΔH -N() < 0 for () < < H(-N) and () < ; and (iii) ΔH - = 0 at,, and. hus, the activation of the native conforers to the E N() () H(-N) () is enthalpically: (i) unfavourable for < () and H(-N) < < () ; (ii) favourable for () < < H(-N) and () < ; and (iii) neither favourable nor unfavourable at (), H(-N), and (). If we now reverse the direction of the reaction (i.e., for the partial folding Page 15 of 44

16 reaction N ), we ay state that the flux of the conforers fro the E to the NE is enthalpically: (i) favourable for < () and H(-N) < < () (ΔH -N() > 0 ΔH N- < 0); (ii) unfavourable for < < and < (ΔH < 0 ΔH () () H(-N) () -N() N- > 0); and (iii) neither favourable nor unfavourable at,, and. In () () H(-N) () addition, k u() < k 0 for () and (), k u() = k 0 when = () and = (), and a iniu when = H(-N). Further, while protein unfolding occurs via a conventional barrier-liited process for the teperature regie () < < (), and is barrierless when = () and = (), the ultralow teperature regie < (), and the high teperature regie () < are once again barrier-liited but fall under the Marcus-invertedregie. 4 Barrierless and Marcus-inverted-regies Fro the perspective of the parabolic hypothesis, barrierless folding is characterised by the condition: 0 G 0. Consequently, for a barrierless folder -D( ) -D( ) -D( ) ΔG D-N() = ΔG -N() and -N() = D-N. In a parabolic representation, the left-ar of the NE-parabola intersects the vertex of the DE-parabola. For the special case of the Marcusinverted-regie associated with the folding reaction, the curve-crossing occurs to the left of the vertex of the DE-parabola, i.e., the left-ar of the NE-parabola intersects the left-ar of the DE-parabola leading to -N() > D-N (see Figure 6 in Marcus, 1993). 4 In contrast, barrierless unfolding is characterised by the condition: 0 0, leading to G D-N( ) G-D( ) -D( ) G -N( ) -N( ) -N( ) ; and because -D() = D-N for this scenario, it iplies G. In a D-N( ) D-N parabolic representation, the right-ar of the DE-parabola intersects the vertex of the NEparabola. For the special case of the Marcus-inverted-regie associated with the unfolding reaction, the curve-crossing occurs to the right of the vertex of the NE-parabola (i.e., the right-ar of the DE-parabola intersects the right-ar of the NE-parabola) leading to - >. D() D-N hus, although fro the perspective of the parabolic hypothesis barrierless unfolding and the Marcus-inverted-regies associated with the unfolding reaction are a theoretical possibility Page 16 of 44

17 for two-state-folding proteins, as long as -D() 0 and -N() 0, the protein cannot fold or unfold, respectively, via downhill echanis, irrespective of whether or not it is an ultrafast folder. Because the extreu of -D() which is a iniu occurs at, it essentially iplies that as long as there exists a chevron with a well-defined linear folding ar at, it is theoretically ipossible for a two-state folder at constant pressure and solvent conditions to spontaneously (i.e., unaided by ligands, co-solvents etc.) switch to a downhill echanis, no atter what the teperature. A corollary is that chevrons with well-defined linear folding and unfolding ars are fundaentally incopatible with downhill scenarios. 5 he reader will note that the theoretically ipossible downhill folding scenario that is being referred to here is not the one wherein the denatured conforers spontaneously fold to their native states via a first-order process with kf ( ) 0 k (anifest when ΔG -D() is approxiately equal to theral noise, i.e., G-D( ) 3R), but the controversial ype 0 scenario according to the Energy Landscape heory, (see Figure 6 in Onuchic et al., 1997) wherein the conforers in the DE ostensibly reach the NE without encountering any barrier (ΔG -D() = 0). 6-9 A detailed discussion on barrierless folding is beyond the scope of this article and will be addressed in subsequent publications. Deterinants of Gibbs activation energies for folding and unfolding he deterinants of ΔG -D() in ters of its activation enthalpy and entropy ay be readily deduced by partitioning the entire teperature range over which the two-state syste is physically defined ( α ω ) into three distinct regies using four unique reference teperatures:,, H(-D), and. 1. For <, the activation of conforers fro the DE to the E is entropically favoured (Δ -D() > 0) but is ore than offset by the endotheric activation enthalpy (ΔH -D() > 0), leading to incoplete copensation and a positive ΔG -D() ( H ). When =, the positive Gibbs barrier to folding is purely due to -D( ) -D( ) 0 the endotheric enthalpy of activation ( G-D( ) H-D( ) 0) and ΔG -D() is a iniu (its lone extreu). Page 17 of 44

18 . For < < H(-D), the activation of denatured conforers to the E is enthalpically and entropically disfavoured (ΔH -D() > 0 and Δ -D() < 0) leading to a positive ΔG -D(). 3. In contrast, for H(-D) <, the favourable exotheric activation enthalpy for folding (ΔH -D() < 0) is ore than offset by the unfavourable entropy of activation for folding (Δ -D() < 0), leading once again to a positive ΔG -D(). When = H(-D), ΔG -D() is purely due to the negative change in the activation entropy or the negentropy of activation ( G-D( ) -D( ) 0 ), G-D( ) is a iniu and k f() is a axiu (their lone extrea). An iportant conclusion that we ay draw fro these analyses is the following: While it is true that for the teperature regies < and H(-D) <, the positive Gibbs barrier to folding is due to the incoplete copensation of the opposing activation enthalpy and entropy, this is clearly not the case for < < H(-D) where both these two state functions are unfavourable and collude to yield a positive Gibbs activation barrier. In short, the Gibbs barrier to folding is not always due to the incoplete copensation of the opposing enthalpy and entropy. iilarly, the deterinants of ΔG -N() in ters of its activation enthalpy and entropy ay be readily divined by partitioning the entire teperature range into five distinct regies using six unique reference teperatures:, (), H(-N),, (), and. 1. For < (), which is the ultralow teperature Marcus-inverted-regie for protein unfolding, the activation of the native conforers to the E is entropically favoured (Δ - > 0) but is ore than offset by the unfavourable enthalpy of activation (ΔH > 0) N() -N() leading to incoplete copensation and a positive ΔG -N() ( H-N( ) -N( ) 0 ). When = (), Δ -N() = ΔH -N() = 0 ΔG -N() = 0. he first extrea of ΔG -N() and G -N( ) (which are a iniu), and the first extreu of k u() (which is a axiu, k u() = k 0 ) occur at ().. For () < < H(-N), the activation of the native conforers to the E is enthalpically favourable (ΔH -N() < 0) but is ore than offset by the unfavourable negentropy of activation (Δ -N() < 0) leading to ΔG -N() > 0. When = H(-N), ΔH -N() = 0 for the Page 18 of 44

19 second tie, and the Gibbs barrier to unfolding is purely due to the negentropy of activation ( G-N( ) -N( ) 0). he second extrea of G-N( ) k u() (which is a iniu) occur at H(-N). (which is a axiu) and 3. For H(-N) < <, the activation of the native conforers to the E is entropically and enthalpically unfavourable (ΔH -N() > 0 and Δ -N() < 0) leading to ΔG -N() > 0. When =, Δ -N() = 0 for the second tie, and the Gibbs barrier to unfolding is purely due to the endotheric enthalpy of activation ( G-N( ) H-N( ) 0). he second extreu of ΔG -N() (which is a axiu) occurs at. 4. For < < (), the activation of the native conforers to the E is entropically favourable (Δ -N() > 0) but is ore than offset by the endotheric enthalpic of activation (ΔH -N() > 0) leading to incoplete copensation and a positive ΔG -N(). When = (), Δ -N() = ΔH -N() = 0 for the third and the final tie, and ΔG -N() = 0 for the second and final tie. he third extrea of ΔG -N() and G-N( ) (which are a iniu), and the third extreu of k u() (which is a axiu, k u() = k 0 ) occur at (). 5. For () <, which is the high teperature Marcus-inverted-regie for protein unfolding, the activation of the native conforers to the E is enthalpically favourable (ΔH -N() < 0) but is ore than offset by the unfavourable negentropy of activation (Δ - < 0), leading to ΔG > 0. N() -N() Once again we note that although the Gibbs barrier to unfolding is due to the incoplete copensation of the opposing enthalpies and entropies of activation for the teperature regies < (), () < < H(-N), < < (), and () <, both the enthalpy and the entropy of activation are unfavourable and collude to generate the Gibbs barrier to unfolding for the teperature regie H(-N) < <. However, in a protein folding scenario where the activated conforers diffuse on the Gibbs energy surface to reach the NE, the interpretation changes because the algebraic signs of the state functions invert. hus, for the reaction[ ] N we ay state: Page 19 of 44

20 1. For < (), the flux of the conforers fro the E to the NE is entropically disfavoured (Δ -N() > 0 Δ N-() < 0) but is ore than copensated by the favourable change in enthalpy (ΔH -N() > 0 ΔH N-() < 0 ), leading to ΔG N-() < 0.. For () < < H(-N), the flux of the conforers fro the E to the NE is enthalpically unfavourable (ΔH -N() < 0 ΔH N-() > 0) but is ore than copensated by the favourable change in entropy (Δ -N() < 0 Δ N-() > 0) leading to ΔG N-() < 0. When = H(-, the flux is driven purely by the positive change in entropy ( G N) N-( ) N-( ) 0 ). 3. For H(-N) < <, the flux of the conforers fro the E to the NE is entropically and enthalpically favourable (ΔH N-() < 0 and Δ N-() > 0) leading to ΔG N-() < 0. When =, the flux is driven purely by the exotheric change in enthalpy ( G ). N-( ) HN-( ) 0 4. For < < (), the flux of the conforers fro the E to the NE is entropically unfavourable (Δ -N() > 0 Δ N-() < 0) but is ore than copensated by the exotheric change in enthalpy (ΔH -N() > 0 ΔH N-() < 0) leading to ΔG N-() < For () <, the flux of the conforers fro the E to the NE is enthalpically unfavourable (ΔH -N() < 0 ΔH N-() > 0) but is ore than copensated by the favourable change in entropy (Δ -N() < 0 Δ N-() > 0), leading to ΔG N-() < 0. A detailed analysis of the deterinants of the Gibbs activation energies in ters of the chain and desolvation enthalpies, and the chain and desolvation entropies is beyond the scope of this article and will be addressed in subsequent publications. Heat capacities of activation for folding and unfolding he heat capacity of activation for partial folding reaction D [ ] (ΔC p-d() ) is given by the derivative of Δ -D() (Eq. (8)) with respect to teperature. -D( ) C p-d( ) -D( ) () Page 0 of 44

21 he solution for Eq. () is given by (see Appendix) -D( ) -D( ) φ p pd-n D-N D-N( ) C C φ φ (3) Because a negative change in heat capacity is less intuitive, we ay recast Eq. (3) by changing the reaction-direction (i.e., [ ] D ) to give pd-( ) -D( ) φ pd-n D-N D-N( ) C C φ φ (4) ubstituting anford s relationship -D( ) β (fold)( ) D-N in Eq. (4) and recasting gives 30 D-N Cp D-( ) (fold)( ) φcp D-N D- N( ) φ φ (5) he heat capacity of activation for the partial unfolding reaction N [ ] (ΔC p-n() ) is given by the derivative of Δ -N() (Eq. (10)) with respect to teperature. -N( ) C p-n( ) -N( ) (6) he solution for Eq. (6) is given by (see Appendix) -N( ) -N( ) φ p pd-n D-N D-N( ) C C φ φ (7) ubstituting -N( ) β(unfold)( ) D-N in Eq. (7) and recasting gives D-N Cp -N( ) (unfold)( ) φcp D-N D -N( ) φ φ (8) Although not shown, identical expressions for the heat capacities of activation for folding and unfolding ay be obtained by differentiating the activation enthalpies for folding and unfolding with respect to teperature. It is instructive to further analyse Eqs. (4) and (7): If we recall that the force constants and D-N are teperature-invariant, it becoes readily apparent that the second ters in the Page 1 of 44

22 square brackets on the right-hand-side (RH) i.e., D-N( ) and D-N( ) will be parabolas, and their values positive for, and zero when =. his is due to Δ D-N() being negative for <, positive for >, and zero for =. 3 Furtherore, we note that φ, φ and -N() are a axiu, and -D() a iniu at. Consequently, for a twostate folder at constant pressure and solvent conditions, ΔC pd-() is a iniu (or ΔC p- is the least negative), and ΔC is a axiu when =. hus, Eqs. (4) and (7) D() p-n() becoe C pd - ( ) -D( ) C p D-N φ 0 (9) C pn - ( ) -N( ) C p D-N φ 0 (30) ince Δ -D() and Δ -N() are zero, ΔG D-N() and ΔG -N() are a axiu, and ΔG -D() is a iniu when =, a corollary is that the Gibbs barriers to folding and unfolding are a iniu and a axiu, respectively, and equilibriu stability is a axiu, and are all purely enthalpic when ΔC pd-() and ΔC p-n() are a iniu and a axiu, respectively. Consistent with the olecular interpretation of change in heat capacity, ΔC pd-() is a iniu when the conforers in the DE travel the least distance fro the ean AA of their enseble along the AA-RC to reach the E (see the principle of least displaceent above), and ΔC p-n() is a axiu when the conforers in the NE expose the greatest aount of AA to reach the E. ee Appendix for what becoe of Eqs. (4) and (7) at () and (), and their iplications. Coparison of RCs Heat capacity RC Adopting Leffler s fraework, the relative sensitivities of the activation and equilibriu enthalpies in response to a perturbation in teperature is given by the ratio of the derivatives of the activation and equilibriu enthalpies with respect to teperature. 4,31 hus, for the partial folding reaction D [ ], we have Page of 44

23 H(fold)( ) H C C H C C -D( ) p-d( ) pd-( ) N-D( ) pn-d pd-n (31) where H(fold)() is classically interpreted to be a easure of the position of the E relative to the DE along the heat capacity RC. iilarly, for the partial unfolding reaction N [ ] we ay write H(unfold)( ) H C C H C C -N( ) p-n( ) pn-( ) D-N( ) pd-n pn-d (3) where H(unfold)() easure of the position of the E relative to the NE along the heat capacity RC. Naturally, for a two-state syste H(fold)() + H(unfold)() = 1 for any given reaction-direction. iilarly, the relative sensitivities of the activation and equilibriu entropies to a perturbation in teperature are given by the ratio of the derivatives of the activation and equilibriu entropies with respect to teperature. hus, we ay write (fold)( ) C C C C C C -D( ) p-d( ) p-d( ) pd-( ) N-D( ) pn-d pn-d pd-n (33) (unfold)( ) C C C C C C -N( ) p-n( ) p-n( ) pn-( ) D-N( ) pd-n pd-n pn-d (34) where (fold)() and (unfold)() are identical to H(fold)() and H(unfold)(), respectively (copare Eqs. (33) and (34) with (31) and (3), respectively). Dividing Eqs. (4) and (7) by ΔC pd-n gives H(fold)( ) -D( ) φ CpD-N D-N D-N( ) φ φc pd-n D-N (fold)( ) φ CpD-N D-N( ) C pd-n φ φ H(unfold)( ) -N( ) φ CpD-N D-N D-N( ) φ φc pd-n φ φ D-N (unfold)( ) φ Cp D-N D-N( ) C pd-n (35) (36) Page 3 of 44

24 When =, Δ D-N() = 0 and Eqs. (35) and (36) reduce to -D( ) (fold)( ) D-N H(fold)( ) (fold)( ) (37) φ φ -N( ) (unfold)( ) D-N H(unfold)( ) (unfold)( ) (38) φ φ As explained earlier, because ΔC pd-n is teperature-invariant by postulate, and ΔC pd-() is a iniu, and ΔC p-n() is a axiu at, H(fold)() and H(unfold)() are a iniu and a axiu, respectively, at. How do H(fold)() and H(unfold)() copare with their counterparts, (fold)() and (unfold)()? his is iportant because a statistically significant correlation exists between D-N and ΔC pd-n, and both these two paraeters independently correlate with ΔAA D-N. 3,33 Recasting Eqs. (37) and (38) gives H(fold)( ) (fold)( ) D-N 1 φ (fold)( ) (fold)( ) (39) H(unfold)( ) (unfold)( ) D-N 1 φ (unfold)( ) (unfold)( ) (40) ince -N() > 0 and a axiu, and -D() > 0 and a iniu, respectively at, it is readily apparent fro inspection of Eqs. (1) and () that φ and D-N φ at. D-N Consequently, (fold)( ) H(fold)( ) and. We will (unfold)( ) H(unfold)( ) deonstrate using experiental data in subsequent publications that while H(fold)() and H(unfold)() have qualitatively siilar dependences on teperature as do (fold)() and (unfold)(), respectively, and both H(fold)() and (fold)() are a iniu, and both H(unfold)() and (unfold)() are a axiu, respectively, at, H(fold)() (fold)() and H(unfold)() (unfold)() except for two unique teperatures, one in the low teperature, and the other in the high teperature regie, and that H(fold)() < (fold)() and H(unfold)() > (unfold)() across a wide teperature regie (see page 178 in Bilsel and Matthews, 000). 34 his has certain Page 4 of 44

25 iplications for the origin of the large and positive heat capacity for protein unfolding at equilibriu and is addressed in subsequent articles. Entropic RC he Leffler paraeters for the relative sensitivities of the activation and equilibriu Gibbs energies in response to a perturbation in teperature are given by the ratios of the derivatives of the activation and equilibriu Gibbs energies with respect to teperature. 6-8 hus, for the partial folding reaction D [ ], we have G(fold)( ) G-D( ) -D( ) -D( ) (41) G N-D( ) N-D( ) D-N( ) where G(fold)() is classically interpreted to be a easure of the position of the E relative to the DE along the entropic RC. ubstituting Eqs. (9) and (A8) in (41) and rearranging gives G(fold)( ) -D( ) D-N( ) D-N( ) φ -D( ) (4) φ iilarly for the partial unfolding reaction N [ ] we have G(unfold)( ) G G -N( ) -N( ) D-N( ) D-N( ) (43) where G(unfold)() is a easure of the position of the E relative to the NE along the entropic RC. ubstituting Eqs. (11) and (A8) in (43) gives G(unfold)( ) -N( ) D-N( ) D-N( ) φ -N( ) (44) φ Inspection of Eqs. (41) and (43) shows that G(fold)() + G(unfold)() = 1 for any given reactiondirection. Now, since Δ D-N() = Δ -D() = Δ -N() = 0 at, G(fold)() and G(unfold)() calculated using Eqs. (41) and (43), respectively, will be undefined or indeterinate for =, with plots of the teperature-dependence of the sae having vertical asyptotes at. However, these are reovable point discontinuities as is apparent fro Eqs. (4) and (44); consequently, graphs generated using the latter set of equations will be devoid of the vertical Page 5 of 44

26 asyptotes, albeit with a hole at. If we ignore the hole at to enable a physical description and their coparison to other RCs, the extreu of G(fold)() (which is positive and a iniu) and the extreu of G(unfold)() (which is positive and a axiu) will occur at. his is because -D() is a iniu, and -N() and φ are a axiu, respectively, at. Further, since -N() = 0 at () and (), G(fold)() (fold)() = 1 and G(unfold)() (unfold)() = 0 at the sae; and for < () and () < (the ultralow and high teperature Marcus-inverted-regies, respectively), G(fold)() and (fold)() are greater than unity, and G(unfold)() and (unfold)() are negative. Coparison of Eqs. (37) and (4), and Eqs. (38) and (44) deonstrate that when =, we have (45) H(fold)( ) (fold)( ) G(fold)( ) (46) H(unfold)( ) (unfold)( ) G(unfold)( ) hus, the position of the E along the heat capacity and entropic RCs are identical at, and non-identical for all. In physical organic cheistry, the ters G(fold)() and G(unfold)() are equivalent to the Brønsted exponents alpha and beta, respectively, and are interpreted to be a easure of the structural siilarity of the transition state to either the reactants or products. If the introduction of a systeatic perturbation (often a change in structure via addition or reoval of a substituent, ph, solvent etc.) generates a reactionseries, and if for this reaction series it is found that alpha is close to zero (or beta close to unity), then it iplies that the structure of the transition state is very siilar to that of the reactant. Conversely, if alpha is close to unity (or beta is alost zero), it iplies that the transition state is structurally siilar to the product. Although the Brønsted exponents in any cases ay be invariant with the degree of perturbation (i.e., a constant slope leading to linear free energy relationships), 35,36 this is not necessarily true, especially if the degree of perturbation is substantial (Figure 3 in Cohen and Marcus, 1968; Figure 1 in Kresge, 1975). 6,37,38 Further, this seeingly straightforward and logical Haond-postulate-based conversion of Brønsted exponents to siilarity or dissiilarity of the structure of the transition states to either of the ground states nevertheless fails for those systes with Brønsted exponents greater than unity and less than zero. 16,39-4 Page 6 of 44

27 In suary, there are three iportant general conclusions that we ay draw fro coparison of solvent ( () ), entropic ( G() ) and heat capacity ( H() ) RCs: (i) as long as ΔAA D-N is large, positive and teperature-invariant, and by logical extension, ΔC pd-n and D-N are positive and teperature-invariant, the position of the E along the various RCs is neither constant nor a siple linear function of teperature when investigated over a large teperature range; (ii) for a given teperature, the position of the E along the RC depends on the choice of the RC; and (iii) although the algebraic su of (fold)() and (unfold)(), H(fold)() and H(unfold)(), and G(fold)() and G(unfold)() ust be unity for a two-state syste for any particular teperature, individually they can be positive, negative, or zero. his is strongly supported by the occurrence of anoalous Brønsted exponents ( < 0 and > 1) in physical organic cheistry, and anoalous Φ-values (Φ < 0 and Φ > 1) in protein folding. 43,44 What this eans for the Φ-value-based canonical interpretation of the structure of the protein folding Es (a variation of the Brønsted procedure introduced by Fersht and coworkers wherein 0 Φ 1) 45 and the classical interpretation of the position of the E in reactions of sall olecules is beyond the scope of this article and will be addressed elsewhere. CONCLUDING REMARK A syste of equations that relate the position of the E along the RC to the various state functions has been derived for a spontaneously-folding fixed two-state syste at constant pressure and solvent conditions using a treatent that is analogous to that given by Marcus for electron transfer. hese equations allow the position of the E, the rate constants, the Gibbs energies, enthalpies, entropies, and heat capacities of activation for the folding and the unfolding to be ascertained at an unprecedented range and resolution of teperature provided a single theral denaturation curve, a chevron, and the calorietrically deterined value of ΔC pd-n are available. Although these equations have been developed for two-state proteins, they can further be extended to ulti-state systes if the conforers that coprise the interediate reaction-states, akin to those in the ground states, also behave like linear elastic springs, i.e., the Gibbs energies of the reactants and products have a square-law dependence on a suitable reaction coordinate, which incidentally, need not necessarily be AA. Obviously, when the force constants for the reactants and products are identical and the ean length of the RC is noralized to unity, the equations reduce to those developed by Marcus. Page 7 of 44

28 As ephasized in Paper-I, the approxiation that the ean length of the RC, ΔC pd-n, and the force constants are invariant for a given solvent at constant pressure applies only when teperature is the perturbant and need not be true, as we will show in subsequent publications, for other kinds of perturbations such as the change in priary sequence, addition of co-solvents, a change in the pressure, ph, priary sequence, etc. Consequently, the equations that describe the behaviour of two-systes when subjected to these perturbations will be far ore coplex, invariably requiring the use of ultivariate calculus. COMPEING FINANCIAL INERE he author declares no copeting financial interests. APPENDIX he first derivatives of -D() and -N() with respect to teperature If we recall that the force constants and D-N are teperature-invariant, the first derivative of -D() (Eq. (1)) with respect to teperature is given by φ φ -D( ) D-N 1 = ( ) ( ) (A1) φλ G ( ). Now since φ is a coposite function we use the chain where D-N( ) rule to get φ 1 φ φ (A) φ GD-N( ) λ GD-N( )( ) ( ) (A3) ubstituting Eq. (A3) in (A) gives φ ( ) G φ D-N( ) (A4) ubstituting the fundaental relationship GD-N( ) D-N( ) in (A4) gives Page 8 of 44

29 φ ( ) φ D-N( ) (A5) ubstituting Eq. (A5) and (A1) yields φ -D( ) D-N( ) (A6) he teperature-dependence of Δ D-N() is given by 3 C pd-n( ) D-N( ) = D-N( ) + d= D-N( ) + D-N ln C p (A7) where Δ D-N() and Δ D-N( denote the equilibriu entropies of unfolding, respectively, at ) any given teperature, and at the idpoint of theral denaturation ( ), respectively, for a given two-state folder under defined solvent conditions. he teperature-invariant and the teperature-dependent difference in heat capacity between the DE and NE is denoted by ΔC pd-n and ΔC pd-n(), respectively. If (the teperature at which Δ D-N() = 0) is used as the reference teperature, Eq. (A7) reduces to D-N( ) = CpD-N ln (A8) ubstituting Eq. (A8) in (A6) gives the final for 0 for -D( ) Cp D-N ln 0 for φ 0 for (A9) Because, β(fold)( ) -D( ) D-N we also have β(fold)( ) 1 -D( ) CpD-N ln D- N D-N φ (A10) ince / -D( ) and β (fold)( ) / are physically undefined forφ 0, their algebraic sign at any given teperature is deterined by the ln ter. his leads to three scenarios: (i) for < we have / 0 -D( ) and β (fold)( ) / 0; (ii) for > we have Page 9 of 44