## Abstract

A model which treats the denatured and the native conformers as being confined to harmonic Gibbs energy wells has been used to analyse the non-Arrhenius behaviour of spontaneously-folding fixed two-state systems. The results demonstrate that when pressure and solvent are constant: (*i*) a two-state system is physically defined only for a finite temperature range; (*ii*) irrespective of the primary sequence, the 3-dimensional structure of the native conformer, the residual structure in the denatured state, and the magnitude of the folding and unfolding rate constants, the equilibrium stability of a two-state system is a maximum when its denatured conformers bury the least amount of solvent accessible surface area (SASA) to reach the activated state; (*iii*) the Gibbs barriers to folding and unfolding are not always due to the incomplete compensation of the activation enthalpies and entropies; (*iv*) the difference in heat capacity between the reaction-states is due to both the size of the solvent-shell and the non-covalent interactions; (*v*) the position of the transition state ensemble along the reaction coordinate (RC) depends on the choice of the RC; and (*vi*) the atomic structure of the transiently populated reaction-states cannot be inferred from perturbation-induced changes in their energetics.

## Introduction

It was shown elsewhere, henceforth referred to as Papers I and II, that the equilibrium and kinetic behaviour of spontaneously-folding fixed two-state systems can be analysed by a treatment that is analogous to that given by Marcus for electron transfer.^{1-3} In this framework termed the parabolic approximation, the Gibbs energy functions of the denatured state ensemble (DSE) and the native state ensemble (NSE) are represented by parabolas whose curvature is given by their temperature-invariant force constants, α and ω, respectively. The temperature-invariant mean length of the reaction coordinate (RC) is given by *m*_{D-N} and is identical to the separation between the vertices of the DSE and the NSE-parabolas along the abscissa. Similarly, the position of the transition state ensemble (TSE) relative to the DSE and the NSE are given by *m*_{TS-D(T)} and *m*_{TS-N(T)}, respectively, and are identical to the separation between the *curve-crossing* and the vertices of the DSE and the NSE-parabolas, respectively. The Gibbs energy of unfolding at equilibrium, Δ*G*_{D-N(T)}, is identical to the separation between the vertices of the DSE and the NSE-parabolas along the ordinate. Similarly, the Gibbs activation energy for folding (Δ*G*_{TS-D(T)}) and unfolding (Δ*G*_{TS-N(T)}) are identical to the separation between the *curve-crossing* and the vertices of the DSE and the NSE-parabolas along the ordinate, respectively.

The purpose of this article is to use the framework described in Papers I and II to analyse the non-Arrhenius behaviour of the 37-residue FBP28 WW domain, at an unprecedented range and resolution.^{4}

## Equations

The expressions for the position of the TSE relative to the vertices of the DSE and the NSE Gibbs parabolas are given by
where the discriminant φ = λω + Δ*G*_{D-N(T)} (ω − α), and λ = α(*m*_{D-N})^{2} is the *Marcus reorganization energy* for two-state protein folding. The expressions for the activation energies for folding and unfolding are given by
where the parameters β_{T(fold)(T)} (= *m*_{TS-D(T)}/*m*_{D-N}) and β_{T(unfold)(T)} (= *m*_{TS-N(T)}/*m*_{D-N}) are according to Tanford’s framework.^{5} The expressions for the rate constants for folding (*k*_{f(T)}) and unfolding (*k*_{u(T)}), and Δ*G*_{D-N(T)} are given by
where, *k*^{0} is the temperature-invariant prefactor with units identical to those of the rate constants (s^{−1}), *R* is the gas constant, *T* is the absolute temperature. If the temperature-dependence of Δ*G*_{D-N(T)} and the values of α, ω, and *m*_{D-N} are known for any two-state system at constant pressure and solvent conditions (see **Methods**), the temperature-dependence of the *curve-crossing* relative to the ground states may be readily ascertained. The temperature-dependence of *curve-crossing* is central to this analysis since all other parameters can be readily derived by manipulating the same using standard kinetic and thermodynamic relationships.

The activation entropies for folding (Δ*S*_{TS-D(T)}) and unfolding (Δ*S*_{TS-N(T)}) are given by the first derivatives of Δ*G*_{TS-D(T)} and Δ*G*_{TS-N(T)} functions with respect to temperature
where *T _{S}* is the temperature at which the entropy of unfolding at equilibrium is zero (Δ

*S*

_{D-N(T)}= 0) and Δ

*C*

_{pD-N}is the temperature-invariant difference in heat capacity between the DSE and the NSE.

^{6}The activation enthalpies for folding (Δ

*H*

_{TS-D(T)}) and unfolding (Δ

*H*

_{TS-N(T)}) may be readily obtained by recasting the Gibbs equation: Δ

*H*

_{(T)}= Δ

*G*

_{(T)}+

*T*Δ

*S*

_{(T)}, or from the temperature-dependence of

*k*

_{f(T)}and

*k*

_{u(T)}to give

The difference in heat capacity between the DSE and the TSE (i.e., for the partial unfolding reaction [*TS*] ⇌ *D*) is given by

Similarly, the difference in heat capacity between the TSE and the NSE (for the partial unfolding reaction *N* ⇌ [*TS*]) is given by

The reader may refer to Papers I and II for the derivations.

## Results and Discussion

As mentioned earlier and discussed in sufficient detail in Papers I and II, the analysis we are going to perform has an explicit requirement for a minimal experimental dataset which are: (*i*) an experimental chevron obtained at constant temperature, pressure and solvent conditions (except for the denaturant); (*ii*) an equilibrium thermal denaturation curve obtained under constant pressure, and in solvent conditions identical to those in which the chevron was acquired but without the denaturant, using either calorimetry or spectroscopy; and (*iii*) the calorimetrically determined Δ*C*_{pD-N} value (i.e., the slope of the linear regression of a plot of model-independent Δ*H*_{D-N(Tm)} *vs T _{m}*, where Δ

*H*

_{D-N(Tm)}is the enthalpy of unfolding at the midpoint of thermal denaturation,

*T*; see Fig. 4 in Privalov, 1989).

_{m}^{7}Fitting the chevron to a modified chevron-equation using non-linear regression yields the values of

*m*

_{D-N}, the force constants α and ω, and the prefactor

*k*

^{0}(

*k*

^{0}is assumed to be temperature-invariant; see

**Methods**in Paper I). Fitting a spectroscopic sigmoidal equilibrium thermal denaturation curve using standard two-state approximation (van’t Hoff analysis using temperature-invariant Δ

*C*

_{pD-N}) yields van’t Hoff Δ

*H*

_{D-N(Tm)}and

*T*and enables the temperature-dependence of Δ

_{m}*H*

_{D-N(T)}, Δ

*S*

_{D-N(T)}and Δ

*G*

_{D-N(T)}functions to be ascertained across a wide temperature regime (Eqs. (A1)-(A3),

**Figure 1**and

**Figure 1−figure supplement 1**).

^{6}Once the values of

*m*

_{D-N}, the force constants, the prefactor, and the temperature dependence of Δ

*G*

_{D-N(T)}are known, the rest of the analysis is fairly straightforward. The values of all the reference temperatures that appear in this article are given in

**Table 1**.

### Temperature-dependence of *m*_{TS-D(T)} and *m*_{TS-N(T)}

Substituting the expression for the temperature-dependence of *G*_{D-N(T)} (Eq. (A3), **Figure 1**) in Eqs. (1) and (2) enables the temperature-dependence of the *curve-crossing* relative to the DSE and the NSE to be ascertained (**Figure 2**; substituted expressions not shown). Because by postulate the force constants, Δ*C*_{pD-N}, and *m*_{D-N} are temperature-invariant for any given primary sequence that folds in a two-state manner at constant pressure and solvent conditions, we get from inspection of Eqs. (1) and (2) that the discriminant φ, and must be a maximum when Δ*G*_{D-N(T)} is a maximum. Because Δ*G*_{D-N(T)} is a maximum at *T _{S}* (the temperature at which the entropy of unfolding at equilibrium, Δ

*S*

_{D-N(T)}, is zero),

^{6}a corollary is that φ and must be a maximum at

*T*; and any deviation in the temperature from

_{S}*T*will only lead to their decrease. Consequently,

_{S}*m*

_{TS-D(T)}and β

_{T(fold)(T)}(=

*m*

_{TS-D(T)}/

*m*

_{D-N}) are always a minimum, and

*m*

_{TS-N(T)}and β

_{T(unfold)(T)}(=

*m*

_{TS-N(T)}/

*m*

_{D-N}) are always a maximum at

*T*. This gives rise to two further corollaries: Any deviation in the temperature from

_{S}*T*can only lead to: (

_{S}*i*) an increase in

*m*

_{TS-D(T)}and β

_{T(fold)(T)}; and (

*ii*) a decrease in

*m*

_{TS-N(T)}and β

_{T(unfold)(T)}(

**Figure 2**and

**Figure 2−figure supplement 1**). In other words, when

*T*=

*T*, the TSE is the least native-like in terms of the SASA (solvent accessible surface area), and any deviation in temperature causes the TSE to become more native-like. A further consequence of

_{S}*m*

_{TS-D(T)}being a minimum at

*T*is that if for a two-state-folding primary sequence there exists a chevron with a well-defined linear folding arm at

_{S}*T*, then

_{S}*m*

_{TS-D(T)}> 0 and β

_{T(fold)(T)}> 0 for all temperatures (

**Figure 2A**and

**Figure 2−figure supplement 1A**). Since the

*curve-crossing*is physically undefined for φ < 0 owing to there being no real roots, the maximum theoretically possible value of

*m*

_{TS-D(T)}will occur when φ = 0 and is given by: where

*T*

_{α}and

*T*

_{ω}are the temperature limits such that for

*T*<

*T*

_{α}and

*T*>

*T*

_{ω}, a two-state system is not physically defined (see Paper II). Because

*m*

_{D-N}=

*m*

_{TS-D(T)}+

*m*

_{TS-N(T)}for a two-state system, and

*m*

_{D-N}is temperature-invariant by postulate, the theoretical minimum of

*m*

_{TS-N(T)}is given by: . Now, since

*m*

_{TS-N(T)}is a maximum and positive at

*T*but its minimum is negative, a consequence is that

_{S}*m*

_{TS-N(T)}= β

_{T(unfold)(T)}= 0 at two unique temperatures, one in the ultralow (

*T*

_{S(α)}) and the other in the high (

*T*

_{S(ω)}) temperature regime, and negative for

*T*

_{α}≤

*T*<

*T*

_{S(α)}and

*T*

_{S(ω)}<

*T*≤

*T*

_{ω}(

**Figure 2B**and

**Figure 2−figure supplement 1B**). Obviously,

*m*

_{TS-D(T)}=

*m*

_{D-N}and β

_{T(fold)(T)}is unity at

*T*

_{S(α)}and

*T*

_{S(ω)}. To summarize, unlike

*m*

_{TS-D(T)}and β

_{T(fold)(T)}which are positive for all temperatures and a minimum at

*T*

_{S},

*m*

_{TS-N(T)}and β

_{T(unfold)(T)}are a maximum at

*T*

_{S}, zero at

*T*

_{S(α)}and

*T*

_{S(ω)}, and negative for

*T*

_{α}≤

*T*<

*T*

_{S(α)}and

*T*

_{S(ω)}<

*T*≤

*T*

_{ω}.

The predicted Leffler-Hammond shift, which must be valid for any two-state system, is in agreement with the experimental data on the temperature-dependent behaviour of other two-state systems (Table 1 in Dimitriadis et al., 2004; Table 1 in Taskent et al., 2008; Fig. 5C in Otzen and Oliveberg, 2004),^{8-12} with the rate at which the *curve-crossing* shifts with stability (relative to the vertex of the DSE-parabola) being given by . Importantly, just as the Leffler-Hammond movement is rationalized in physical organic chemistry using Marcus theory,^{13-15} we can similarly rationalize these effects in protein folding using parabolic approximation (**Figures 3**, **4**, and **Figure 4−figure supplement 1**). When *T* = *T _{S}*, Δ

*G*

_{TS-D(T)}is a minimum, and Δ

*G*

_{D-N(T)}and Δ

*G*

_{TS-N(T)}are both a maximum; and any increase or decrease in the temperature relative to

*T*leads to a decrease in Δ

_{S}*G*

_{TS-N(T)}, and an increase in Δ

*G*

_{TS-D(T)}, consequently, leading to a decrease in Δ

*G*

_{D-N(T)}(

**Figures 1**,

**3B**and

**5**). Naturally at

*T*and

_{c}*T*, Δ

_{m}*G*

_{TS-D(T)}= Δ

*G*

_{TS-N(T)},

*k*

_{f(T)}=

*k*

_{u(T)}, and Δ

*G*

_{D-N(T)}= 0 (

**Figure 3C**). The reason why

*m*

_{TS-D(T)}=

*m*

_{D-N}, and

*m*

_{TS-N(T)}= 0 at

*T*

_{S(α)}and

*T*

_{S(ω)}is apparent from

**Figures 4A**,

**4C**and

**Figure 4−figure supplement 1A**: The right arm of the DSE-parabola intersects the vertex of the NSE-parabola leading to Δ

*G*

_{TS-D(T)}= α(

*m*

_{TS-D(T)})

^{2}= α(

*m*

_{D-N})

^{2}= λ, Δ

*G*

_{TS-N(T)}= ω(

*m*

_{TS-N(T)})

^{2}= 0, and Δ

*G*

_{D-N(T)}= − λ. Importantly, in contrast to unfolding which can become barrierless at

*T*

_{S(α)}and

*T*

_{S(ω)}, folding is barrier-limited at all temperatures, with the absolute minimum of Δ

*G*

_{TS-D(T)}occurring at

*T*; and any deviation in the temperature from

_{S}*T*will only lead to an increase in Δ

_{S}*G*

_{TS-D(T)}(

**Figure 5A**). Thus, a corollary is that if folding is barrier-limited at

*T*(i.e., the chevron has a well-defined linear folding arm with a finite slope at

_{S}*T*), then a protein that folds

_{S}*via*two-state mechanism can never spontaneously (i.e., unaided by ligands, co-solvents etc.) switch to a downhill mechanism (Type 0 scenario according to the Energy Landscape Theory; see Fig. 6 in Onuchic et al., 1997), no matter what the temperature, and irrespective of how fast or slow it folds. Although unfolding is barrierless at

*T*

_{S(α)}and

*T*

_{S(ω)}, it is once again barrier-limited for

*T*

_{α}≤

*T*<

*T*

_{S(α)}and

*T*

_{S(ω)}<

*T*≤

*T*

_{ω}, with the

*curve-crossing*occurring to the right of the vertex of the NSE-parabola (

**Figures 4A**,

**4B**,

**Figure 4−figure supplement 1B**and

**5B**), such that

*m*

_{TS-D(T)}>

*m*

_{D-N},

*m*

_{TS-N(T)}< 0, β

_{T(fold)(T)}> 1 and β

_{T(unfold)(T)}< 0 (

**Figure 2**and

**Figure 2−figure supplement 1**).

To summarize, for any two-state folder, unfolding is *conventional barrier-limited* for *T*_{S(α)} < *T* < *T*_{S(ω)} and the position of the TSE or the *curve-crossing* occurs in between the vertices of the DSE and the NSE parabolas. As the temperature deviates from *T _{S}*, the SASA of the TSE becomes progressively native-like, with a concomitant increase and a decrease in Δ

*G*

_{TS-D(T)}and Δ

*G*

_{TS-N(T)}, respectively. When

*T*=

*T*

_{S(α)}and

*T*

_{S(ω)}, the

*curve-crossing*occurs precisely at the vertex of the NSE-parabola, the SASA of the TSE is identical to that of the NSE, and unfolding is barrierless; and for

*T*

_{α}≤

*T*<

*T*

_{Sα}and

*T*

_{S(ω)}<

*T*≤

*T*

_{ω}, unfolding is once again barrier-limited but falls under the

*Marcus-inverted-regime*with the

*curve-crossing*occurring on the right-arm of the NSE-parabola, leading to the SASA of the NSE being greater than that of the TSE (i.e., the TSE is more compact than the NSE). Importantly, for

*T*<

*T*

_{α}and

*T*>

*T*

_{ω}, the TSE cannot be physically defined owing to being mathematically undefined for φ > 0. A consequence is that

*k*and

_{f(T)}*k*become physically undefined, leading to Δ

_{u(T)}*G*

_{D-N(T)}=

*RT*ln (

*k*/

_{f(T)}*k*) being physically undefined, such that all of the conformers will be confined to a single Gibbs energy well, which is the DSE, and the protein will cease to function.

_{u(T)}^{16}Thus, from the view point of the physics of phase transitions,

*T*

_{α}≤

*T*≤

*T*

_{ω}denotes the

*coexistence temperature-range*where the DSE and the NSE, which are in a dynamic equilibrium, will coexist as two distinct phases; and for

*T*<

*T*

_{α}and

*T*>

*T*

_{ω}there will be a single phase, which is the DSE, with

*T*

_{α}and

*T*

_{ω}being the limiting temperatures for coexistence, or phase boundary temperatures from the view point of the DSE.

^{17-23}This is roughly analogous to the operating temperature range of a logic circuit such as a microprocessor; and just as this range is a function of its constituent material, the physically definable temperature range of a two-state system is a function of the primary sequence when pressure and solvent are constant, and importantly, can be modulated by a variety of

*cis-*acting and

*trans*-acting factors (see Paper-I). The limit of equilibrium stability below which a two-state system becomes physically undefined is given by: Δ

*G*

_{D-N(T)}|

_{T = Tα, Tω}= −λω/(ω − α). Consequently, the physically meaningful range of equilibrium stability for a two-state system is given by: Δ

*G*

_{D-N}

_{(TS)}+ [λω/(ω − α)], where Δ

*G*

_{D-N}

_{(TS)}is the stability at

*T*and is apparent from inspection of

_{S}**Figure 5−figure supplement 1**. This is akin to the stability range over which Marcus theory is physically realistic (see Kresge, 1973, page 494).

^{24}

Because by postulate *m*_{D-N}, *m*_{TS-D(T)} and *m*_{TS-N(T)} are true proxies for ΔSASA_{D-N}, ΔSASA_{D-TS(T)} and ΔSASA_{TS-N(T)}, respectively (see Paper I), we have three fundamentally important corollaries that must hold for all two-state systems at constant pressure and solvent conditions: (*i*) the Gibbs barrier to folding is the least when the denatured conformers bury the least amount of SASA to reach the TSE (**Figure 5−figure supplement 2A**); (*ii*) the Gibbs barrier to unfolding is the greatest when the native conformers expose the greatest amount of SASA to reach the TSE (**Figure 5−figure supplement 2B**); and (*iii*) equilibrium stability is the greatest when the conformers in the DSE are displaced the least from the mean of their ensemble along the SASA-RC to reach the TSE (the *principle of least displacement*; **Figure 5−figure supplement 1**).

### Temperature-dependence of the folding, unfolding, and the observed rate constants

Inspection of **Figures 6A** and **Figure 6−figure supplement 1A** demonstrates that Eq. (5) makes a remarkable prediction that *k _{f(T)}* has a non-linear dependence on temperature. Starting from the lowest temperature (

*T*

_{α}) at which a two-state system is physically defined,

*k*initially increases with an increase in the temperature and reaches a maximal value at

_{f(T)}*T*=

*T*

_{H(TS-D)}where ∂ ln

*k*/∂

_{f(T)}*T*= Δ

*H*

_{TS-D(T)}/

*RT*

^{2}= 0; and any further increase in temperature beyond this point will cause a decrease in

*k*until the temperature

_{f(T)}*T*

_{ω}is reached, such that for

*T*>

*T*

_{ω},

*k*is undefined. Inspection of

_{f(T)}**Figures 6B**and

**Figure 6−figure supplement 1B**demonstrates that the temperature-dependence of

*k*is far more complex: Starting from

_{u(T)}*T*

_{α},

*k*increases with temperature for the regime

_{u(T)}*T*

_{α}≤

*T*<

*T*

_{S(α)}(the low-temperature

*Marcus-inverted-regime*), reaches a maximum when

*T*=

*T*

_{S(α)}(

*k*=

_{u(T)}*k*

^{0}; the first extremum of

*k*), and decreases with further rise in temperature for the regime

_{u(T)}*T*

_{S(α)}<

*T*<

*T*

_{H(TS-N)}such that when

*T*=

*T*

_{H(TS-N)},

*k*is a minimum (the second extremum of

_{u(T)}*k*). And for

_{u(T)}*T*

_{H(TS-N)}<

*T*<

*T*

_{S(ω)}, an increase in temperature will lead to an increase in

*k*, eventually leading to its saturation at

_{u(T)}*T*=

*T*

_{S(ω)}(

*k*=

_{u(T)}*k*

^{0}; the third extremum of

*k*), and decreases with further rise in temperature for

_{u(T)}*T*

_{S(ω)}<

*T*≤

*T*

_{ω}(the high-temperature

*Marcus-inverted-regime*). Thus, in contrast to

*k*which has only one extremum,

_{f(T)}*k*is characterised by three extrema where ∂ ln

_{u(T)}*k*/∂

_{u(T)}*T*= Δ

*H*

_{TS-N(T)}/

*R*

^{2}= 0, and may be rationalized from the temperature-dependence of

*m*

_{TS-D(T)}and

*m*

_{TS-N(T)}, the Gibbs barrier heights for folding and unfolding, and the intersection of the DSE and the NSE Gibbs parabolas (

**Figures 2**-

**5**and their figure supplements). We will show in subsequent publications that the inverted behaviour at very low and high temperatures is not common to all fixed two-state systems and depends on the mean and variance of the Gaussian distribution of the SASA of the conformers in the DSE and the NSE.

Since the ultimate test of any hypothesis is experiment, the most important question now is how well do the calculated rate constants compare with experiment? Although Nguyen et al. have investigated the non-Arrhenius behaviour of the FBP28 WW, they find that the behaviour of its wild type is erratic, with its folding being three-state for *T* < *T _{m}* and two-state for

*T*>

*T*(Fig. 3A in Nguyen et al., 2003). Consequently, non-Arrhenius data for the wild type FBP28 WW are lacking. Incidentally, this atypical behaviour is probably artefactual since the protein aggregates and forms fibrils under the experimental conditions in which the measurements were made (see Figs. 2, 3 and 6 in Ferguson et al., 2003).

_{m}^{25,26}Nevertheless, data for ΔNΔC Y11R W30F, a variant of FBP28 WW are available between ∽ 298 and ∽357 K (Fig. 4A in Nguyen et al., 2003). Now since the relaxation time constants for the fast phase of wild type FBP28 WW (∽ 30 μs at 39.5 °C and < 15 μs at 65 °C, page 3950, Fig. 3A, Nguyen et al., 2003) are very similar to those of ΔNΔC Y11R W30F (∽ 28 μs at 40 °C and 11 μs at 65 °C, page 3952), a reasonable approximation is that the temperature-dependence of

*k*and

_{f(T)}*k*of the wild type and the mutant must be similar. Consequently, the temperature-dependence of the rate constants for the wild type FBP28 WW calculated using parabolic approximation must be very similar to the data for ΔNΔC Y11R W30F reported by Nguyen et al. The remarkable agreement between the said datasets is readily apparent from a comparison of Fig. 4A of Nguyen et al., and

_{u(T)}**Figure 6−figure supplement 2**, and serves an important test of the hypothesis.

Since the temperature-dependence of *k _{f(T)}* and

*k*across a wide temperature range is known, the variation in the observed rate constant (

_{u(T)}*k*

_{obs(T)}) with temperature may be readily ascertained using (see

**Appendix**)

Inspection of **Figure 7** demonstrates that ln(*k*_{obs(T)}) *vs* temperature is a smooth ‘W-shaped’ curve, with *k*_{obs(T)} being dominated by *k _{f(T)}* around

*T*

_{H(TS-N)}, and by

*k*for

_{u(T)}*T*<

*T*and

_{c}*T*>

*T*, which is precisely why the kinks in ln(

_{m}*k*

_{obs(T)}) occur around these temperatures. It is easy to see that at

*T*or

_{c}*T*,

_{m}*k*=

_{f(T)}*k*⇒

_{u(T)}*k*

_{obs(T)}= 2

*k*= 2

_{f(T)}*k*Δ

_{u(T)},*G*

_{D-N(T)}=

*RT*ln (

*k*/

_{f(T)}*k*)) = 0 or Δ

_{u(T)}*G*

_{TS-D(T)}= Δ

*G*

_{TS-N(T)}(

**Figures 3C**and

**Figure 7−figure supplement 1**). In other words, for a two-state system,

*T*and

_{c}*T*determined at equilibrium must be identical to the temperatures at which

_{m}*k*and

_{f(T)}*k*intersect. This is a consequence of the

_{u(T)}*principle of microscopic reversibility*, i.e., the equilibrium and kinetic stabilities must be identical for a two-state system at all temperatures.

^{27}It is precisely for this reason that the value of the prefactor in the Arrhenius expressions for the rate constants must be identical for both the folding and the unfolding reactions at all temperatures (Eqs. (5) and (6)). The steep increase in

*k*

_{obs(T)}for

*T*<

*T*and

_{c}*T*>

*T*is due to the Δ

_{m}*G*

_{TS-N(T)}approaching zero as described earlier. The argument that the shapes of the curves must be conserved across two-state systems applies not only to the temperature-dependence of

*m*

_{TS-D(T)},

*m*

_{TS-N(T)}, Δ

*G*

_{TS-D(T)}and Δ

*G*

_{TS-N(T)}described so far, but to the rest of the state functions that will be described in this article (see Paper-I).

An important conclusion that we may draw from these data is the following: Because we have assumed a temperature-invariant prefactor and yet find that the kinetics are non-Arrhenius, it essentially implies that one does not need to invoke a *super-Arrhenius temperature-dependence of the configurational diffusion constant* to explain the non-Arrhenius behaviour of proteins.^{28-32} Instead, as long as the enthalpies and the entropies of unfolding/folding at equilibrium display a large variation with temperature, and equilibrium stability is a non-linear function of temperature, both *k _{f(T)}* and

*k*will have a non-linear dependence on temperature. This leads to two corollaries: (

_{u(T)}*i*) since the large variation in equilibrium enthalpies and entropies of unfolding, including the pronounced curvature in Δ

*G*

_{D-N(T)}of proteins with temperature is due to the large and positive Δ

*C*

_{pD-N}, “

*non-Arrhenius kinetics can be particularly acute for reactions that are accompanied by large changes in the heat capacity*”; and (

*ii*) because the change in heat capacity upon unfolding is, to a first approximation, proportional to the change in SASA that accompanies it, and since the change in SASA upon unfolding/folding increases with chain-length,

^{33,34}“

*non-Arrhenius kinetics, in general, can be particularly pronounced for large proteins, as compared to very small proteins and peptides*.”

### Temperature-dependence of activation enthalpies

Inspection of **Figure 8** demonstrates that for the partial folding reaction *D* ⇌ [*TS*]: (*i*) Δ*H*_{TS-D(T)} > 0 for *T*_{α} ≤ *T* < *T*_{H(TS-D)}; (*ii*) Δ*H*_{TS-D(T)} < 0 for *T*_{H(TS-D)} < *T* ≤ *T*_{ω} and (*iii*) Δ*H*_{TS-D(T)} = 0 for *T* = *T*_{H(TS-D)}. Thus, the activation of the denatured conformers to the TSE is enthalpically: (*i*) unfavourable for *T*_{α} ≤ *T* < *T*_{H(TS-D)}; (*ii*) favourable for *T*_{H(TS-D)} < *T* ≤ *T*_{ω}; and (*iii*) neutral when *T* = *T*_{H(TS-D)}. Consequently, at *T*_{H(TS-D)}, Δ*G*_{TS-D(T)} is purely due to the difference in entropy between the DSE and the TSE (Δ*G*_{TS-D(T)} = −*T*Δ*S*_{TS-D(T)}) with *k _{f(T)}* being given by

Because *k _{f(T)}* is a maximum at

*T*

_{H(TS-D)}(∂ ln

*k*/∂

_{f(T)}*T*= 0), a corollary is that

*“for a two-state folder at constant pressure and solvent conditions, if the prefactor is temperature-invariant, then k*This statement is valid only if the prefactor is temperature-invariant. Now since Δ

_{f(T)}will be a maximum when the Gibbs barrier to folding is purely entropic.”*G*

_{TS-D(T)}> 0 for all temperatures (

**Figure 5A**and

**Table 1**), it is imperative that Δ

*S*

_{TS-D(T)}< 0 at

*T*

_{H(TS-D)}(see activation entropy for folding).

Unlike the Δ*H*_{TS-D(T)} function which changes its algebraic sign only once across the entire temperature range over which a two-state system is physically defined, the behaviour of Δ*H*_{TS-N(T)} function is far more complex (**Figure 9**): (*i*) Δ*H*_{TS-N(T)} > 0 for *T*_{α} ≤ *T* < *T*_{S(α)} and *T*_{H(TS-N)} < *T* < *T*_{S(ω)}; (*ii*) Δ*H*_{TS-N(T)} < 0 for *T*_{S(α)} < *T* < *T*_{H(TS-N)} and *T*_{S(ω)} < *T* ≤ *T*_{ω}; and (*iii*) Δ*H*_{TS-N(T)} = 0 at *T*_{S(α)}, *T*_{H(TS-N)}, and *T*_{S(ω)}. Consequently, we may state that the activation of native conformers to the TSE is enthalpically: (*i*) unfavourable for *T*_{α} ≤ *T* < *T*_{S(α)} and *T*_{H(TS-N)} < *T* < *T*_{S(ω)}; (*ii*) favourable for *T*_{S(α)} < *T* < *T*_{H(TS-N)} and *T*_{S(ω)} < *T* ≤ *T*_{ω}; and (*iii*) neutral at *T*_{S(α)}, *T*_{H(TS-N)}, and *T*_{S(ω)}. If we reverse the reaction-direction, the algebraic signs invert leading to a change in the interpretation. Thus, for the partial folding reaction [*TS*] ⇌ *N*, the flux of the conformers from the TSE to the NSE is enthalpically: (*i*) favourable for *T*_{α} ≤ *T* < *T*_{S(α)} and *T*_{H(TS-N)} < *T* < *T*_{S(ω)} (Δ*H*_{N-TS(T)} < 0); (*ii*) unfavourable for *T*_{S(α)} < *T* < *T*_{H(TS-N)} and *T*_{S(ω)} < *T* ≤ *T*_{ω} (Δ*H*_{N-TS(T)} > 0); and (*iii*) neither favourable nor unfavourable at *T*_{S(α)}, *T*_{H(TS-N)}, and *T*_{S(ω)} (**Figure 9−figure supplement 1A**). Note that the term “flux” implies “diffusion of the conformers from one reaction state to the other on the Gibbs energy surface,” and as such is an “operational definition.”

Importantly, although ∂ ln *k _{u(T)}*/∂

*T*= 0 ⇒ Δ

*H*

_{TS-N(T)}= 0 at

*T*

_{S(α)},

*T*

_{H(TS-N)}, and

*T*

_{S(ω)}, the behaviour of the system at

*T*

_{S(α)}and

*T*

_{S(ω)}is distinctly different from that at

*T*

_{H(TS-N)}: While

*m*

_{TS-N(T)}= Δ

*G*

_{TS-N(T)}= Δ

*H*

_{TS-N(T)}= Δ

*S*

_{TS-N(T)}= 0,

*m*

_{TS-D(T)}=

*m*

_{D-N}, Δ

*G*

_{TS-D(T)}= Δ

*G*

_{N-D(T)}= λ, and

*k*=

_{u(T)}*k*

^{0}at

*T*

_{S(α)}and

*T*

_{S(ω)}(note that if both Δ

*G*

_{TS-N(T)}and Δ

*H*

_{TS-N(T)}are zero, then Δ

*S*

_{TS-N(T)}must also be zero, see activation entropies),

*k*is a minimum (

_{u(T)}*k*≪

_{u(T)}*k*

^{0}) with the Gibbs barrier to unfolding being purely entropic (Δ

*G*

_{TS-N(T)}= −

*T*Δ

*S*

_{TS-N(T)}) at

*T*

_{H(TS-N)}. Consequently, we may write

Thus, a corollary is that *“for two-state system at constant pressure and solvent conditions, if the prefactor is temperature-invariant, then k _{u(T)} will be a minimum when the Gibbs barrier to unfolding is purely entropic.”* Since Δ

*G*

_{TS-N(T)}> 0 at

*T*

_{H(TS-N)}(

**Figure 5B**and

**Table 1**), it is imperative that Δ

*S*

_{TS-N(T)}be negative at

*T*

_{H(TS-N)}(see activation entropy for unfolding).

The criteria for two-state folding from the viewpoint of enthalpy are the following: (*i*) the condition that Δ*H*_{D-N(T)} = Δ*H*_{TS-N(T)} − Δ*H*_{TS-D(T)} must be satisfied at all temperatures; (*ii*) the intersection of Δ*H*_{TS-D(T)} and Δ*H*_{TS-N(T)} functions calculated directly from the temperature-dependence of the experimentally determined *k _{f(T)}* and

*k*, respectively, must be identical to the independently estimated

_{u(T)}*T*from equilibrium thermal denaturation experiments; and (

_{H}*iii*) the condition that

*T*

_{H(TS-N)}<

*T*<

_{H}*T*<

_{S}*T*

_{H(TS-D)}must be satisfied. A corollary of the last statement is that both Δ

*H*

_{TS-D(T)}and Δ

*H*

_{TS-N(T)}functions must be positive at the point of intersection. These aspects are readily apparent from

**Figure 9−figure supplement 1B**and

**Figure 9−figure supplement 2**.

### Temperature-dependence of activation entropies

Inspection of **Figure 10** shows that for the partial folding reaction *D* ⇌ [*TS*], Δ*S*_{TS-D(T)} which is positive at low temperature, decreases in magnitude with an increase in temperature and becomes zero at *T _{S}*, where the SASA of the TSE is the least native-like, Δ

*G*

_{TS-D(T)}is a minimum (∂Δ

*G*

_{TS-D(T)}/∂

*T*= −Δ

*S*

_{TS-D(T)}= 0) and Δ

*G*

_{D-N(T)}is a maximum (∂Δ

*G*

_{D-N(T)}/∂

*T*= −Δ

*S*

_{D-N(T)}= 0;

**Figures 1, 2, 5A, Figure 10−figure supplements 1**and

**2**); and any further increase in temperature beyond this point causes Δ

*S*

_{TS-D(T)}to become negative. Thus, the activation of denatured conformers to the TSE is entropically: (

*i*) favourable for

*T*

_{α}≤

*T*<

*T*; (

_{S}*ii*) unfavourable for

*T*<

_{S}*T*≤

*T*

_{ω}; and (

*iii*) neutral when

*T*=

*T*. At

_{S}*T*the Gibbs barrier to folding is purely due to the difference in enthalpy between the DSE and the TSE with

_{S}*k*being given by

_{f(T)}Inspection of **Figure 11** demonstrates that the behaviour of the Δ*S*_{TS-N(T)} function is far more complex than the Δ*S*_{TS-D(T)} function: (*i*) Δ*S*_{TS-N(T)} > 0 for *T*_{α} ≤ *T* < *T*_{S(α)} and *T _{S}* <

*T*<

*T*

_{S(ω)}; (

*ii*) Δ

*S*

_{TS-N(T)}< 0 for

*T*

_{S(α)}<

*T*<

*T*and

_{S}*T*

_{S(ω)}<

*T*≤

*T*

_{ω}; and (

*iii*) Δ

*S*

_{TS-N(T)}= 0 at

*T*

_{S(α)},

*T*, and

_{S}*T*

_{S(ω)}. Consequently, we may state that the activation of native conformers to the TSE is entropically: (

*i*) favourable for

*T*

_{α}≤

*T*<

*T*

_{S(α)}and

*T*<

_{S}*T*<

*T*

_{S(ω)}; (

*ii*) unfavourable for

*T*

_{S(α)}<

*T*<

*T*and

_{S}*T*

_{S(ω)}<

*T*≤

*T*

_{ω}; and (

*iii*) neutral at

*T*

_{S(α)},

*T*, and

_{S}*T*

_{S(ω)}. If we reverse the reaction-direction (

**Figure 11−figure supplement 1A**), the algebraic signs invert leading to a change in the interpretation. Consequently, we may state that for the partial folding reaction [

*TS*] ⇌

*N*, the flux of the conformers from the TSE to the NSE is entropically: (

*i*) unfavourable for

*T*

_{α}≤

*T*<

*T*

_{S(α)}and

*T*<

_{S}*T*<

*T*

_{S(ω)}(Δ

*S*

_{N-TS(T)}< 0); (

*ii*) favourable for

*T*

_{S(α)}<

*T*<

*T*and

_{S}*T*

_{S(ω)}<

*T*≤

*T*

_{ω}(Δ

*S*

_{N-TS(T)}> 0); and (

*iii*) neutral at

*T*

_{S(α)},

*T*, and

_{S}*T*

_{S(ω)}.

At *T* = *T _{S}*, the Gibbs barrier to unfolding is purely due to the difference in enthalpy between the TSE and the NSE (Δ

*G*

_{TS-N(T)}= Δ

*H*

_{TS-N(T)}) with

*k*being given by

_{u(T)}Although Δ*S*_{TS-N(T)} = 0 ⇒ *S*_{TS(T)} = *S*_{N(T)} at *T*_{S(α)}, *T _{S}*, and

*T*

_{S(ω)}, the underlying thermodynamics is fundamentally different at

*T*as compared to

_{S}*T*

_{S(α)}and

*T*

_{S(ω)}. While both Δ

*G*

_{TS-N(T)}and

*m*

_{TS-N(T)}are positive and a maximum, and Δ

*G*

_{TS-N(T)}is purely enthalpic at

*T*(Δ

_{S}*G*

_{TS-N(T)}= Δ

*H*

_{TS-N(T)}), at

*T*

_{S(α)}and

*T*

_{S(ω)}we have

*m*

_{TS-N(T)}= 0 ⇒ Δ

*G*

_{TS-N(T)}= ω(

*m*

_{TS-N(T)}

^{2}= 0 ⇒ Δ

*H*

_{TS-N(T)}= 0, and Δ

*G*

_{N-D(T)}= Δ

*G*

_{TS-D(T)}= λ; and because Δ

*G*

_{TS-N(T)}= 0 at

*T*

_{S(α)}and

*T*

_{S(ω)}, the rate constant for unfolding will reach an absolute maximum for that particular solvent and pressure at these two temperatures. To summarize, while at

*T*we have

_{S}*G*

_{TS(T)}≫

*G*

_{N(T)},

*S*

_{D(T)}=

*S*

_{TS(T)}=

*S*

_{N(T)}, and

*k*

_{u(T)}≪

*k*

^{0}, when

*T*=

*T*

_{S(α)}and

*T*

_{S(ω)}, we have

*G*

_{TS(T)}=

*G*

_{N(T)},

*H*

_{TS(T)}=

*H*

_{N(T)},

*S*

_{TS(T)}=

*S*

_{N(T)}, and

*k*

_{u(T)}=

*k*

^{0}(

**Figure 11−figure supplements 2**and

**3**). Thus, a fundamentally important conclusion that we may draw from these relationships is that “

*if two reaction-states on the folding pathway of a two-state system have identical SASA and Gibbs energy under identical environmental conditions, then their absolute enthalpies and entropies must be identical*.” This must hold irrespective of whether or not the two reaction-states have identical, similar or dissimilar structures. We will revisit this scenario when we discuss the heat capacities of activation and the inapplicability of the Hammond postulate to protein folding reactions.

The criteria for two-state folding from the viewpoint of entropy are the following: (*i*) the condition that Δ*S*_{D-N(T)} = Δ*S*_{TS-N(T)} — Δ*S*_{TS-D(T)} must be satisfied at all temperatures; (*ii*) the intersection of Δ*S*_{TS-D(T)} and Δ*S*_{TS-N(T)} functions calculated directly from the slopes of the temperature-dependent shift in the *curve-crossing* relative to the DSE and the NSE, respectively, must be identical to the independently estimated *T _{S}* from equilibrium thermal denaturation experiments (

**Figure 11−figure supplements 1B**,

**4**and

**5**); and (

*iii*) both Δ

*S*

_{TS-D(T)}and Δ

*S*

_{TS-N(T)}functions must independently be equal to zero at

*T*.

_{S}### Temperature-dependence of the Gibbs activation energies

Although the general features of the temperature-dependence of Δ*G*_{TS-D(T)} and Δ*G*_{TS-N(T)} were described earlier (**Figure 5** and its figure supplements), it is instructive to discuss the same in terms of their constituent enthalpies and entropies.

The determinants of Δ*G*_{TS-D(T)} in terms of its activation enthalpy and entropy may be readily deduced by partitioning the entire temperature range over which the two-state system is physically defined (*T*_{α} ≤ *T* ≤ *T*_{ω}) into three distinct regimes using four unique reference temperatures: *T*_{α}, *T _{S}*,

*T*

_{H(TS-D)}, and

*T*

_{ω}(

**Figure 12**and

**Figure 12−figure supplement 1**). (1) For

*T*

_{α}≤

*T*<

*T*, the activation of conformers from the DSE to the TSE is entropically favoured (

_{S}*T*Δ

*S*

_{TS-D(T)}> 0) but is more than offset by the endothermic activation enthalpy (Δ

*H*

_{TS-D(T)}> 0), leading to incomplete compensation and a positive Δ

*G*

_{TS-D(T)}(Δ

*H*

_{TS-D(T)}–

*T*Δ

*S*

_{TS-D(T)}. When

*T*=

*T*

_{S}, Δ

*G*

_{TS-D(T)}is a minimum (its lone extremum), and is purely due to the endothermic enthalpy of activation (Δ

*G*

_{TS-D(T)}= Δ

*H*

_{TS-D(T)}> 0. (2) For

*T*<

_{S}*T*<

*T*

_{H(TS-D)}, the activation of denatured conformers to the TSE is enthalpically and entropically disfavoured (Δ

*H*

_{TS-D(T)}> 0 and

*T*Δ

*S*

_{TS-D(T)}< 0) leading to a positive Δ

*G*

_{TS-D(T)}. (3) In contrast, for

*T*

_{H(TS-D)}<

*T*≤

*T*

_{ω}, the favourable exothermic activation enthalpy (Δ

*H*

_{TS-D(T)}< 0) is more than offset by the unfavourable entropy of activation (

*T*Δ

*S*

_{TS-D(T)}< 0), leading once again to a positive Δ

*G*

_{TS-D(T)}. When

*T*=

*T*

_{H(TS-D)}, Δ

*G*

_{TS-D(T)}is purely due to the negative change in the activation entropy or the

*negentropy*of activation (Δ

*G*

_{TS-D(T)}= –

*T*Δ

*S*

_{TS-D(T)}> 0), Δ

*G*

_{TS-D(T)}/

*T*is a minimum, and

*k*

_{f(T)}is a maximum (their lone extrema; see Massieu-Planck functions below). An important conclusion that we may draw from these analyses is the following: While it is true that for the temperature regimes

*T*

_{α}≤

*T*<

*T*and

_{S}*T*

_{H(TS-D)}<

*T*≤

*T*

_{ω}, Δ

*G*

_{TS-D(T)}is due to the incomplete compensation of the opposing activation enthalpy and entropy, this is clearly not the case for

*T*<

_{S}*T*<

*T*

_{H(TS-D)}where both these two state functions are unfavourable and complement each other to generate a positive Gibbs activation barrier.

Similarly, the determinants of Δ*G*_{TS-N(T)} in terms of its activation enthalpy and entropy may be readily divined by partitioning the entire temperature range into five distinct regimes using six unique reference temperatures: *T*_{α}, *T*_{S(α)}, *T*_{H(TS-N)}, *T _{S}*,

*T*

_{S(ω)}, and

*T*

_{ω}(

**Figure 13**and

**Figure 13−figure supplement 1**). (1) For

*T*

_{α}≤

*T*<

*T*

_{S(α)}, which is the ultralow temperature

*Marcus-inverted-regime*for unfolding, the activation of the native conformers to the TSE is entropically favoured (

*T*Δ

*S*

_{TS-N(T)}> 0) but is more than offset by the unfavourable enthalpy of activation (Δ

*H*

_{TS-N(T)}> 0) leading to incomplete compensation and a positive Δ

*G*

_{TS-N(T)}(Δ

*H*

_{TS-N(T)}– Δ

*T*Δ

*S*

_{TS-N(T)}> 0). When

*T*=

*T*

_{S(α)}, Δ

*S*

_{TS-N(T)}= Δ

*H*

_{TS-N(T)}= 0 ⇒ Δ

*G*

_{TS-N(T)}= 0. The first extrema of Δ

*G*

_{TS-N(T)}and Δ

*G*

_{TS-N(T)}/

*T*(which are a minimum), and the first extremum of

*k*

_{u(T)}(which is a maximum,

*k*

_{u(T)}=

*k*

^{0}) occur at

*T*

_{S(α)}. (2) For

*T*

_{S(α)}<

*T*<

*T*

_{H}_{(TS-N)}, the activation of the native conformers to the TSE is enthalpically favourable (Δ

*H*

_{TS-N(T)}< 0) but is more than offset by the unfavourable negentropy of activation (

*T*Δ

*S*

_{TS-N(T)}< 0) leading to Δ

*G*

_{TS-N(T)}> 0. When

*T*=

*T*

_{H(TS-N)}, Δ

*H*

_{TS-N(T)}= 0 for the second time, and the Gibbs barrier to unfolding is purely due to the negentropy of activation (Δ

*G*

_{TS-N(T)}= –

*T*Δ

*S*

_{TS-N(T)}> 0. The second extrema of Δ

*G*

_{TS-N(T)}/

*T*(which is a maximum) and

*k*

_{u(T)}(which is a minimum) occur at

*T*

_{H(TS-N)}. (3) For

*T*

_{H(TS-N)}<

*T*<

*T*, the activation of the native conformers to the TSE is entropically and enthalpically unfavourable (Δ

_{S}*H*

_{TS-N(T)}> 0 and

*T*Δ

*S*

_{TS-N(T)}< 0) leading to Δ

*G*

_{TS-N(T)}> 0. When

*T*=

*T*, Δ

_{S}*S*TS-N(

*T*) = 0 for the second time, and the Gibbs barrier to unfolding is purely due to the endothermic enthalpy of activation (Δ

*G*

_{TS-N(T)}= Δ

*H*

_{TS-N(T)}> 0). The second extremum of Δ

*G*

_{TS-N(T)}(which is a maximum) occurs at

*T*. (4) For

_{S}*T*<

_{S}*T*<

*T*

_{S(ω)}, the activation of the native conformers to the TSE is entropically favourable (

*T*Δ

*S*

_{TS-N(T)}> 0) but is more than offset by the endothermic enthalpy of activation (Δ

*H*

_{TS-N(T)}> 0) leading to incomplete compensation and a positive Δ

*G*

_{TS-N(T)}. When

*T*=

*T*

_{S(ω)}, Δ

*S*

_{TS-N(T)}= Δ

*H*

_{TS-N(T)}= 0 for the third and the final time, and Δ

*G*

_{TS-N(T)}= 0 for the second and final time. The third extrema of Δ

*G*

_{TS-N(T)}and Δ

*G*

_{TS-N(T)}/

*T*(which are a minimum), and the third extremum of

*k*

_{u(T)}(which is a maximum,

*k*

_{u(T)}=

*k*

^{0}) occur at

*T*

_{S(ω)}. (5) For

*T*

_{S(ω)}<

*T*≤

*T*

_{ω}, which is the high-temperature

*Marcus-inverted-regime*for unfolding, the activation of the native conformers to the TSE is enthalpically favourable (Δ

*H*

_{TS-N(T)}< 0) but is more than offset by the unfavourable negentropy of activation (

*T*Δ

*S*

_{TS-N(T)}< 0), leading to Δ

*G*

_{TS-N(T)}> 0. Once again we note that although the Gibbs barrier to unfolding is due to the incomplete compensation of the opposing enthalpies and entropies of activation for the temperature regimes

*T*

_{α}≤

*T*<

*T*

_{S(α)},

*T*

_{S(α)}<

*T*<

*T*

_{H(TS-N)},

*T*<

_{S}*T*<

*T*

_{S(ω)}, and

*T*

_{S(ω)}<

*T*≤

*T*

_{ω}, both the enthalpy and the entropy of activation are unfavourable and collude to generate the Gibbs barrier to unfolding for the temperature regime

*T*

_{H(TS-N)}<

*T*<

*T*. Thus, a fundamentally important conclusion that we may draw from this analysis is that “

_{S}*the Gibbs barriers to folding and unfolding are not always due to the incomplete compensation of the opposing enthalpy and entropy*.”

In a *protein folding* scenario where the activated conformers diffuse on the Gibbs energy surface to reach the NSE, the algebraic signs of the state functions invert leading to a change in the interpretation (**Figure 13−figure supplements 2** and **3**). Thus, for the partial folding reaction[*TS*] ⇌: (1) For *T*_{α} ≤ *T* < *T*_{S(α)}, the flux of the conformers from the TSE to the NSE is entropically disfavoured (*T*Δ*S*_{TS-N(T)} > 0 ⇒ *T*Δ*S*_{N-TS(T)} < 0) but is more than compensated by the favourable change in enthalpy (Δ*H*_{TS-N(T)} > 0 ⇒ Δ*H*_{N-TS(T)} < 0), leading to Δ*G*_{N-TS(T)} < 0. (2) For *T*_{S(α)} < *T* < *T*_{H(TS-N)}, the flux of the conformers from the TSE to the NSE is enthalpically unfavourable (Δ*H*_{TS-N(T)} < 0 ⇒ Δ*H*_{N-TS(T)} > 0) but is more than compensated by the favourable change in entropy (*T*Δ*S*_{TS-N(T)} < 0 ⇒ *T*Δ*S*_{N-TS(T)} > 0) leading to Δ*G*_{N-TS(T)} < 0. When *T* = *T*_{H(TS-N)}, the flux is driven purely by the positive change in entropy (Δ*G*_{N-TS(T)} = –*T*Δ*S*_{N-TS(T)} > 0). (3) For *T*_{H(TS-N)} < *T* < *T _{S}*, the flux of the conformers from the TSE to the NSE is entropically and enthalpically favourable (Δ

*H*

_{N-TS(T)}< 0 and

*T*Δ

*S*

_{N-TS(T)}> 0) leading to Δ

*G*

_{N-TS(T)}< 0. When

*T*=

*T*, the flux is driven purely by the exothermic change in enthalpy (Δ

_{S}*G*

_{N-TS(T)}= Δ

*H*

_{N-TS(T)}< 0). (4) For

*T*<

_{S}*T*<

*T*

_{S(ω)}, the flux of the conformers from the TSE to the NSE is entropically unfavourable (

*T*Δ

*S*

_{TS-N(T)}> 0 ⇒

*T*Δ

*S*

_{N-TS(T)}< 0) but is more than compensated by the exothermic change in enthalpy (Δ

*H*

_{TS-N(T)}> 0 ⇒ Δ

*H*

_{N-TS(T)}< 0) leading to Δ

*G*

_{N-TS(T)}< 0. (5) For

*T*

_{S(ω)}<

*T*≤

*T*

_{ω}, the flux of the conformers from the TSE to the NSE is enthalpically unfavourable (Δ

*H*

_{TS-N(T)}< 0 ⇒ Δ

*H*

_{N}

_{TS(T)}> 0) but is more than compensated by the favourable change in entropy (

*T*Δ

*S*

_{TS-N(T)}< 0 ⇒

*T*Δ

*S*

_{N-TS(T)}> 0), leading to Δ

*G*

_{N-TS(T)}< 0.

Thus, the criteria for two-state folding from the viewpoint of Gibbs energy are the following: (*i*) the condition that Δ*G*_{D-N(T)} = Δ*G*_{TS-N(T)} – Δ*G*_{TS-D(T)} must be satisfied at all temperatures; (*ii*) the cold and heat denaturation temperatures estimated from equilibrium thermal denaturation must be identical to independently determined temperatures at which *k*_{f(T)} and *k*_{u(T)} are identical, i.e., the temperatures at which Δ*G*_{TS-D(T)} and Δ*G*_{TS-N(T)} functions intersect must be identical to the temperatures at which Δ*H*_{D-N(T)} – *T*Δ*S*_{D-N(T)}= Δ*G*_{D-N(T)} = 0. The basis for these relationships, as mentioned earlier, is the *principle of microscopic reversibility*;^{27} (*iii*) Δ*G*_{TS-D(T)} and Δ*G*_{TS-N(T)} must be a minimum and a maximum, respectively, at *T _{S}*; and (

*iv*) the condition that

*T*

_{H(TS-N)}<

*T*<

_{H}*T*<

_{S}*T*

_{H(TS-D)}must be satisfied. A far more detailed explanation in terms of chain and desolvation entropies and enthalpies is given in the accompanying article.

### Massieu-Planck functions

The Massieu-Planck function, Δ*G*/*T*, or its equivalent –*R*ln*K* (*K* is the equilibrium constant) predates the Gibbs energy function by a few years and is especially useful when analysing temperature-dependent changes in protein behaviour (see Schellman, 1997, on the use of Massieu-Planck functions to analyse protein folding, and why the use of Δ*G versus T* curves can sometimes lead to ambiguous conclusions).^{6,35} Comparison of **Figure 6−figure supplement 1A** and **Figure 14A** demonstrates that although Δ*G*_{TS-D(T)} is a minimum at *T _{S}* (

**Figure 5A**),

*k*

_{f(T)}will be a maximum not at

*T*but instead at

_{S}*T*

_{H(TS-D)}where the Massieu-Planck activation potential for folding (Δ

*G*

_{TS-D(T)}/

*T*≡ –

*R*ln

*K*

_{TS-D(T)}) is a minimum, and is readily apparent if we recast the Arrhenius expression for

*k*

_{f(T)}in terms of the equilibrium constant for the partial folding reaction

*D*⇌ [

*TS*].

Eq. (19) shows that the rate determining *K*_{TS-D(T)} ([*TS*]/[*D*]) or the population of activated conformers relative to those that nestle at the bottom of the denatured Gibbs energy well is a maximum not at *T _{S}* but at

*T*

_{H(TS-D)}(

**Figure 14−figure supplement 1A**). Similarly, comparison of

**Figure 6−figure supplement 1B**and

**Figure 14B**shows that although Δ

*G*

_{TS-N(T)}is a maximum at

*T*(

_{S}**Figure 5B**), the minimum in

*k*

_{u(T)}will occur not at

*T*but instead at

_{S}*T*

_{H(TS-N)}where the Massieu-Planck activation potential for unfolding (Δ

*G*

_{TS-N(T)}/

*T*≡ –

*R*ln

*K*

_{TS-D(T)}) is a maximum (Eq. (20)).

Thus, for the partial unfolding reaction *N* ⇌ [*TS*], the rate determining *K*_{TS-N(T)} ([*TS*]/[*N*]) or the population of activated conformers relative to those at the bottom of the native Gibbs basin is a minimum not at *T _{S}* but at

*T*

_{H(TS-N)}(

**Figure 14−figure supplement 1B**). Similarly, we see that although the Δ

*G*

_{N-D(T)}is a minimum or the most negative at

*T*

_{S}(

**Figure 1−figure supplement 1**),

*K*

_{N-D(T)}([

*N*]/[

*D*]) is a maximum not at

*T*but at

_{S}*T*where Δ

_{H}*H*

_{N-D(T)}= 0 and

*k*

_{f(T)}/

*k*

_{u(T)}is a maximum (

**Figure 14−figure supplement 2A**).

^{6}Because the ratio of the solubilities of any two reaction-states is identical to the equilibrium constant, we may state that for any two-state folder at constant pressure and solvent conditions: (

*i*) the solubility of the TSE as compared to the DSE is the greatest when the Gibbs barrier to folding is purely entropic, and this occurs precisely at

*T*

_{H(TS-D)}(

**Figure 14−figure supplement 3A**); (

*ii*) the solubility of the TSE as compared to the NSE is the least when the Gibbs barrier to unfolding is purely entropic and occurs precisely at

*T*

_{H(TS-N)}(

**Figure 14−figure supplement 3B**); (

*iii*) the solubilities of the TSE and the NSE are identical at

*T*

_{S(α)}and

*T*

_{S(ω)}where Δ

*S*

_{TS-N(T)}= Δ

*H*

_{TS-N(T)}= Δ

*G*

_{TS-N(T)}= 0, and

*k*

_{u(T)}=

*k*

^{0}(

**Figure 14−figure supplement 3B**); and (

*iv*) the solubility of the NSE as compared to the DSE is the greatest when the net flux of the conformers from the DSE to the NSE is driven purely by the difference in entropy between these two reaction-states and occurs precisely at

*T*(

_{H}**Figure 14−figure supplement 2B**). The notion that “certain aspects of the temperature-dependent protein behaviour are greatly simplified when the Massieu-Planck functions are used in preference to the Gibbs energy” is readily apparent from inspection of

**Figure 14−figure supplements 4**and

**5**: While the natural logarithms of

*k*

_{f(T)}and

*k*

_{u(T)}have a complex dependence on their respective Gibbs barriers, a simple linear relationship exists between the rate constants and their respective Massieu-Planck functions.

### Temperature-dependence of Δ*C*_{pD-TS(T)} and Δ*C*_{pTS-N(T)}

In order to provide a rational explanation for the temperature-dependence of the Δ*C*_{pD-TS(T)} and Δ*C*_{pTS-N(T)} functions, it is instructive to first discuss the inter-relationships between ΔSASA_{D-N}, *m*_{D-N}, and Δ*C*_{pD-N}. According to the “*liquid-liquid transfer*” model (LLTM) the greater heat capacity of the DSE as compared to the NSE (i.e., Δ*C*_{pD-N} > 0 and substantial) is predominantly due to anomalously high heat capacity and low entropy of water that surrounds the exposed non-polar residues in the DSE (referred to as “microscopic icebergs” or “clathrates”; see references in Baldwin, 2014).^{36} Because the size of the solvation shell depends on the SASA of the non-polar solute, it naturally follows that the change in the heat capacity must be proportional to the change in the non-polar SASA that accompanies a reaction. Consequently, protein unfolding reactions which are accompanied by large changes in non-polar SASA lead to large and positive changes in the heat capacity.^{33,37,38} Because the denaturant *m* values are also directly proportional to the change in SASA that accompanies protein unfolding reactions, the expectation is that *m*_{D-N} and Δ*C*_{pD-N} values must also be proportional to each other: The greater the *m*_{D-N} value, the greater is the Δ*C*_{pD-N} value and *vice versa* (Figs. 2, 3 and 5 in Myers et al., 1995). However, since the residual structure in the DSEs of proteins under folding conditions is both sequence and solvent-dependent (i.e., the SASAs of the DSEs two proteins of identical chain lengths but dissimilar primary sequences need not necessarily be the same even under identical solvent conditions),^{39,40} and because we do not yet have reliable theoretical or experimental methods to accurately quantify the SASA of the DSEs of proteins under folding conditions (i.e., the values are model-dependent),^{41-43} the data scatter in plots that show correlation between the experimentally determined *m*_{D-N} or Δ*C*_{pD-N} values (which reflect the true ΔSASA_{D-N}) and the calculated values of ΔSASA_{D-N} can be significant (Fig. 2 in Myers et al., 1995, and Fig. 3 in Robertson and Murphy, 1997). Now, since the solvation shell around the DSEs of large proteins is relatively greater than that of small proteins even when the residual structure in the DSEs under folding conditions is taken into consideration, large proteins on average expose relatively greater amount of non-polar SASA upon unfolding than do small proteins; consequently, both *m*_{D-N} and Δ*C*_{pD-N} values also correlate linearly with chain-length, albeit with considerable scatter since chain length, owing to the residual structure in the DSEs, is unlikely to be a true descriptor of the SASA of the DSEs of proteins under folding conditions (note that the scatter can also be due to certain proteins having anomalously high or low number of non-polar residues). The point we are trying to make is the following: Because the native structures of proteins are relatively insensitive to small variations in pH and co-solvents,^{44} and since the number of ways in which foldable polypeptides can be packed into their native structures is relatively limited (as inferred from the limited number of protein folds, see SCOP: www.mrc-lmb.cam.ac.uk and CATH: www.cathdb.info databases), one might find a reasonably good correlation between chain lengths and the SASAs of the NSEs of proteins of differing primary sequences under varying solvents (Fig. 1 in Miller et al., 1987).^{45,46} However, since the SASAs of the DSEs under folding conditions, owing to residual structure are variable, until and unless we find a way to accurately simulate the DSEs of proteins, and if and only if these theoretical methods are sensitive to point mutations, changes in pH, co-solvents, temperature and pressure, it is almost impossible to arrive at a universal equation that will describe how the ΔSASA_{D-N} under folding conditions will vary with chain length, and by logical extension, how *m*_{D-N} and Δ*C*_{pD-N} will vary with SASA or chain length. Nevertheless, if we consider a single two-state-folding primary sequence under constant pressure and solvent conditions and vary the temperature, and if the properties of the solvent are temperature-invariant (for example, no change in the pH due to the temperature-dependence of the p*K _{a}* of the constituent buffer), then the manner in which the Δ

*C*

_{pD-TS(T)}and Δ

*C*

_{pTS-N(T)}functions vary with temperature must be consistent with the temperature-dependence of

*m*

_{TS-D(T)}and

*m*

_{TS-N(T)}, respectively, and by logical extension, with ΔSASA

_{D-TS(T)}and ΔSASA

_{TS-N(T)}, respectively.

Inspection of **Figures 15** and **Figure 15−figure supplements 1**, **2** and **3** demonstrate that: (*i*) both Δ*C*_{pD-TS(T)} and Δ*C*_{pTS-N(T)} vary with temperature; and (*ii*) their gross features stem primarily from the second derivatives of the temperature-dependence of the *curve-crossing* with respect to the DSE and the NSE. The prediction that the change in heat capacities for the partial unfolding reactions, *N* ⇌ [*TS*] and [*TS*] ⇌ *D*, must vary with temperature is due to Eqs. (12) and (13). Although this may not be readily apparent from a casual inspection of the equations, even a cursory examination of **Figures 8** and **9** shows that it is simply not possible for Δ*C*_{pD-TS(T)} and Δ*C*_{pTS-N(T)} functions to be temperature-invariant since the slopes of the Δ*H*TS-D(*T*) and the Δ*H*_{TS-N(T)} functions are continuously changing with temperature. If we recall that the force constants are temperature-invariant, it becomes readily apparent that the second terms in the brackets on the right-hand-side (RHS) of Eqs. (12) and (13) i.e., ω*T*(Δ*S*_{D-N(T)} and α*T*(Δ*S*_{D-N(T)})^{2}, respectively, will be parabolas with a minimum (zero) at *T _{S}*. This is due to Δ

*S*

_{D-N(T)}being negative for

*T*<

*T*, positive for

_{S}*T*>

*T*, and zero for

_{S}*T*=

*T*. Furthermore, since φ, and

_{S}*m*

_{TS-N(T)}are a maximum, and

*m*

_{TS-D(T)}a minimum at

*T*, the expectation is that Δ

_{S}*C*

_{pD-TS(T)}must be a minimum (or Δ

*C*

_{pTS-D(TS)}is the least negative), and Δ

*C*

_{pTS-N(T)}must be a maximum at

*T*. Thus, for

_{S}*T*=

*T*, Eqs. (12) and (13) become

_{S}The prediction that the extrema of Δ*C*_{pD-TS(T)} and Δ*C*_{pTS-N(T)} functions must occur at *T _{S}* is readily apparent from

**Figure 15**and

**Figure 15−figure supplement 1B**. Importantly, consistent with the relationship between

*m*

_{D-N}and Δ

*C*

_{pD-N}values, comparison of these two figures with

**Figure 2**and

**Figure 2−figure supplement 1**demonstrates that just as

*m*

_{TS-D(T)}and

*m*

_{TS-N(T)}are a minimum and a maximum at

*T*, respectively, so too are Δ

_{S}*C*

_{pD-TS(T)}and Δ

*C*

_{pTS-N(T)}functions. This leads to two obvious corollaries: (

*i*) the difference in heat capacity between the DSE and the TSE is a minimum when the difference in SASA between the DSE and the TSE is a minimum; and (

*ii*) the difference in heat capacity between the TSE and the NSE is a maximum when the difference in SASA between the TSE and the NSE is a maximum. Because Δ

*S*

_{TS-D(T)}= Δ

*S*

_{TS-N(T)}= 0, Δ

*G*

_{TS-D(T)}is a minimum, and both Δ

*G*

_{TS-N(T)}and Δ

*G*

_{D-N(T)}are a maximum, at

*T*(

_{S}**Figures 1, 5**and

**Figure 11−figure supplement 1B**), a fundamentally important conclusion is that the

*Gibbs barriers to folding and unfolding are a minimum and a maximum, respectively, and equilibrium stability is a maximum, and are all purely enthalpic when*Δ

*C*

_{pD-TS(T)}

*and*Δ

*C*

_{pTS-N(T)}

*are a minimum and a maximum, respectively*.

Inspection of **Figure 15** and **Figure 15−figure supplement 1** demonstrates that unlike Δ*C*_{pD-TS}(*T*) which is positive across the entire temperature range, Δ*C*_{pTS-N(T)} which is a maximum and positive at *T _{S}*, decreases with any deviation in temperature from

*T*, and is zero at

_{S}*T*

_{CpTS-N(α)}and

*T*

_{CpTS-N(ω)}; consequently, Δ

*C*

_{pTS-N(T)}< 0 for

*T*

_{α}≤

*T*<

*T*

_{CpTS-N(α)}and

*T*

_{CpTS-N(ω)}<

*T*≤

*T*

_{ω}. The reason for this behaviour is apparent from inspection of

**Figures 9**and

**11**: The slope of the Δ

*H*

_{TS-N(T)}and Δ

*S*

_{TS-N(T)}functions becomes zero at

*T*

_{CpTS-N(α)}and

*T*

_{CpTS-N(ω)}; and any further decrease or increase in temperature, respectively, causes the slope to invert. This can be mathematically shown as follows: Since

*m*

_{TS-N(T)}= 0 at

*T*

_{S(α)}and

*T*

_{S(ω)}, we have and φ = (α

*m*

_{D-N})

^{2}at

*T*

_{S(α)}and

*T*

_{S(ω)}. Substituting these relationships in Eq. (13) leads to

Further, since Δ*C*_{pD-N} = Δ*C*_{pD-TS(T)} + Δ*C*_{pTS-N(T)} for a two-state system, we have

Because Δ*C*_{pTS-N(T)} < 0 at *T*_{S(α)} and *T*_{S(ω)}, and the lone extremum of Δ*C*_{pTS-N(T)} (which is algebraically positive and a maximum) occurs at *T _{S}*, it implies that there will be two unique temperatures at which Δ

*C*

_{pTS-N(T)}= 0, one in the low temperature (

*T*

_{CpTS-N(α)}) such that

*T*

_{S(α)}<

*T*

_{CpTS-N(α)}<

*T*, and the other in the high temperature regime (

_{S}*T*

_{CpTS-N(ω)}) such that

*T*<

_{S}*T*

_{CpTS-N(ω)}<

*T*

_{S(ω)}. Thus, at the these two unique temperatures

*T*

_{CpTS-N(α)}and

*T*

_{CpTS-N(ω)}, we have Δ

*C*

_{pD-TS(T)}= Δ

*C*

_{pD-N}⇒ β

_{H(fold)(T)}= 1 and

*β*H(unfold)(

*T*) = 0; and for the temperature regimes

*T*

_{α}≤

*T*<

*T*

_{CpTS-N(α)}and

*T*

_{CpTS-N(ω)}<

*T*≤

*T*

_{ω}, we have Δ

*C*

_{pD-TS(T)}> Δ

*C*

_{pD-N}⇒ β

_{H(fold)(T)}> 1, and Δ

*C*

_{pTS-N(T)}< 0 ⇒ β

_{H(unfold)(T)}< 0 (see heat capacity RC below for the definition of β

_{H(fold)(T)}and β

_{H(unfold)(T)}).

Although the prediction that Δ*C*_{pTS-N(T)} must approach zero at very low and high temperatures may not be readily verified by experiment for the low-temperature regime owing to technical difficulty in making a measurement, the prediction for the high-temperature regime is strongly supported by the data on CI2 from the Fersht lab: Despite the temperature-range not being substantial (320 to 340 K), and the data points that define the Δ*H*_{TS-N(T)} function being sparse (7 in total), it is apparent even from a cursory inspection that it is clearly non-linear with temperature (Fig. 5B in Tan et al., 1996).^{47} Although Fersht and co-workers have fitted the data to a linear function and reached the natural conclusion that the heat capacity of activation for unfolding is temperature-invariant, they nevertheless explicitly mention that if the non-linearity of Δ*H*_{TS-N(T)} were given due consideration, and the data are fit to an empirical-quadratic instead of a linear function, Δ*C*_{pTS-N(T)} indeed becomes temperature-dependent and is predicted to approach zero at ∽ 360 K (see text in page 382 in Tan et al., 1996).^{47} Now, since Δ*C*_{pTS-N(T)} > 0 and a maximum, and Δ*C*_{pD-TS(T)} is a minimum and positive at *T _{S}*, and decrease and increase, respectively, with any deviation in temperature from

*T*, and since Δ

_{S}*C*

_{pTS-N(T)}becomes zero at

*T*

_{CpTS-N(α)}and

*T*

_{CpTS-N(ω)}, the obvious mathematical consequence is that Δ

*C*

_{pD-TS(T)}and Δ

*C*

_{pTS-N(T)}functions must intersect at two unique temperatures. Because at the points of intersection we have the relationship: Δ

*C*

_{pD-TS(T)}= Δ

*C*

_{pTS-N(T)}= Δ

*C*

_{pD-N}/2, a consequence is that Δ

*C*

_{pTS-N(T)}must be positive at the said temperatures, with the low-temperature intersection occurring between

*T*

_{CpTS-N(α)}and

*T*, and the high-temperature intersection between

_{S}*T*and

_{S}*T*

_{CpTS-N(ω)}. This is readily apparent from inspection of

**Figure 15−figure supplement 1B:**Both Δ

*C*

_{pD-TS(T)}and Δ

*C*

_{pTS-N(T)}are identical at 214.1 K and 345.9 K. An equivalent interpretation is that at these temperatures, the absolute heat capacity of the TSE is exactly half the algebraic sum of the absolute heat capacities of the DSE and the NSE. As we shall show in subsequent publications, the intersection of various state functions is a source of interesting relationships that may be used as constraints in simulations (see also

**Figure 9−figure supplement 2**).

### The position of the TSE along the heat capacity RC

Inspection and comparison of **Figure 2−figure supplement 1** and **Figure 15−figure supplement 1B** demonstrates that although the manner in which the Δ*C _{p}*D-TS(

*T*) and Δ

*C*

_{pTS-N(T)}functions vary with temperature is consistent with the relationship between

*m*

_{D-N}and Δ

*C*

_{pD-N}values, there is nevertheless an intriguing anomaly that is at odds with the LLTM for heat capacity. If we consider the partial folding reaction

*D*⇌ [TS], it is readily apparent from these figures that although the denatured conformer diffuses > ∽ 70% along the normalized SASA-RC to reach the TSE for 240 K <

*T*< 320 K, Δ

*C*

_{pD-TS(T)}≪ Δ

*C*

_{pTS-N(T)}throughout this regime. Conversely, if we consider the total unfolding reaction

*N*⇌

*D*, a large fraction of Δ

*C*

_{pD-N}is accounted for not by the second-half of the unfolding reaction ([

*TS*]) ⇌

*D*but by the first-half (

*N*⇌ [

*TS*]), despite the native conformer diffusing less than ∽30% along the SASA-RC to reach the TSE. To put things into perspective, we will need to normalize the heat capacities of activation. Adopting Leffler’s framework for the relative sensitivities of the activation and equilibrium enthalpies in response to a perturbation in temperature,

^{48}we may write where β

_{H(fold)(T)}= β

_{S(fold)(T)}and β

_{H(unfold)(T)}= β

_{S(unfold)(T)}(see Paper-II) are classically interpreted to be a measure of the position of the TSE along the heat capacity RC.

^{49}Naturally, for a two-state system the algebraic sum of β

_{H(fold)(T)}and β

_{H(unfold)(T)}is unity. Recasting Eqs. (24) and (25) in terms of (12) and (13) gives

When *T* = *T _{S}*, Δ

*S*

_{D-N(T)}= 0 and Eqs. (26) and (27) reduce to

As explained earlier, because Δ*C*_{pD-N} is temperature-invariant by postulate, and Δ*C*_{pD-TS(T)} is a minimum, and Δ*C*_{pTS-N(T)} is a maximum at *T*_{S}, β_{H(fold)(T)} and β_{H(unfold)(T)} are a minimum and a maximum, respectively, at *T*_{S}. How do β_{H(fold)(T)} and β_{H(unfold)(T)} compare with their counterparts, β_{T(fold)(T)} and β_{T(unfold)(T)}? This is important because a statistically significant correlation exists between *m*_{D-N} and Δ*C*_{pD-N}, and both these two parameters independently correlate with ΔSASA_{D-N}. Recasting Eqs. (28) and (29) gives

Since *m*_{TS-N(T)} > 0 and a maximum, and *m*_{TS-D(T)} > 0 and a minimum, respectively, at *T _{S}*, it is readily apparent from inspection of Eqs. (1) and (2) that and at

*T*. Consequently, we have: β

_{S}_{T(fold)(T)}|

_{T=TS}> β

_{H(fold)(T)}|

_{T=TS}and β

_{T(unfold)(T)}|

_{T=TS}< β

_{H(unfold)(T)}|

_{T=TS}.

In agreement with the predictions of Eqs. (30) and (31), inspection of **Figure 16** demonstrates that although the denatured conformer advances by > ∽ 70% along the SASA-RC to reach the TSE when *T* = *T _{S}*, it accounts for < ∽20% of the total change in Δ

*C*

_{pD-N}(i.e., β

_{T(fold)(T)}|

_{T=TS}> β

_{H(fold)(T)}|

_{T=TS}), with the rest of the change (> ∽ 80%) in heat capacity coming from a mere ∽ 30% diffusion of the activated conformer along the SASA-RC to reach the bottom of the native Gibbs basin (i.e., β

_{T(unfold)(T)}|

_{T=TS}< β

_{H(unfold)(T)}|

_{T=TS}). The theoretical prediction that β

_{T(fold)(T)}> β

_{H(fold)(T)}across a substantial temperature range is supported by the finding by Gloss and Matthews (1998) that the position of the TSE relative to the DSE along the heat capacity RC is consistently lower than the same along the SASA-RC (see also page 178 in Bilsel and Matthews, 2000, and references therein).

^{50,51}

Now, if we accept the long held premise that the greater heat capacity of the DSE as compared to the NSE is purely or predominantly due to structured water around the exposed non-polar residues in the DSE, then the only way we can explain why Δ*C*_{pD-TS(T)} ≪ Δ*C*_{pTS-N(T)} despite β_{T(fold)(T)} > ∽70% for the partial folding reaction *D* ⇌ [*TS*] is that the non-polar SASA of both the DSE and the TSE are very similar at *T _{S}*. Because it is physically near-impossible for the denatured conformer to advance by > ∽ 70% along the SASA-RC to reach the TSE, and yet keep the non-polar SASA fairly constant such that Δ

*C*

_{pD-TS(T)}is just about 20% of Δ

*C*

_{pD-N}, the natural conclusion is that “

*the large and positive difference in heat capacity between the DSE and the NSE cannot be only due to the clathrates of water molecules around exposed non-polar residues in the DSE*.”

^{38,52-54}This brings us to two studies on the heat capacities of proteins, one by Sturtevant almost four decades ago, and the other by Lazaridis and Karplus.

^{55,56}While Sturtevant identified six possible sources of heat capacity which are: (

*i*) the hydrophobic effect; (

*ii*) electrostatic charges; (

*iii*) hydrogen bonds; (

*iv*) conformational entropy; (

*v*) intramolecular vibrations; and (

*vi*) changes in equilibria, and concluded that the most important of these are the

*hydrophobic*,

*conformational*and

*vibrational*effects, Lazaridis and Karplus concluded from their molecular dynamics simulations on truncated CI2 that the heat capacity can have a significantly large and a positive contribution from intra-protein non-covalent interactions. What these two studies essentially imply is that when the pressure and solvent properties are defined and temperature-invariant, the ability of the conformers in a protein reaction-state to absorb thermal energy and yet resist an increase in temperature is dependent on: (

*i*) its molecular structure; and (

*ii*) the size and the character of its molecular surface (i.e., the relative proportion of polar and non-polar SASA). While the first variable determines the capacity of the reaction-state to absorb thermal energy and distribute it across its various internal modes of motion (the vibrational, rotational, and to some extent, the translational entropy from elements such as the N and C-terminal regions, loops etc. that can flap around in the solvent), the second variable determines not only the size and thickness of the solvent shell but also how tightly or loosely the solvent molecules are bound to the protein surface and to themselves (i.e., the dynamics of water in the solvation shell as compared to bulk water; see Fig. 1 in Frauenfelder et al., 2009), and by extension, the amount of excess thermal energy needed to disrupt the solvent shell as the reaction-states interconvert due to thermal noise.

^{36,52,57-61}Further discussion on the determinants of heat capacity is beyond the scope of this article and will be addressed elsewhere.

### On the inapplicability of the Hammond postulate to protein folding

Although it is difficult to provide a detailed physical explanation for the temperature-dependence of the heat capacities of activation without deconvoluting the activation enthalpies and entropies into their constituent *chain* and *desolvation* enthalpies and entropies (shown in the accompanying article), it is instructive to give one extreme example to emphasize why both the solvent shell and the non-covalent interactions make a significant contribution to heat capacity (note that as long as the difference in the number of covalent bonds between the reaction-states is zero, to a first approximation, their contribution to the *difference in heat capacity* between the reaction-states can be ignored; see Lecture II in Finkelstein and Ptitsyn, 2002, and references therein).^{38,56,62,63}

It was shown earlier that when *T* = *T*_{S(α)} and *T*_{S(ω)}, we have *m*_{TS-N(T)} = 0 ⇒ ΔSASA_{TS-N(T)} = 0, leading to a unique set of relationships: *G*_{TS(T)} = *G*_{N(T)}, *H*_{TS(T)} = *H*_{N(T)}, *S*_{TS(T)} = *S*_{N(T)}, and *k*_{u(T)} = *k*^{0} (**Figures 2B, Figure 2−figure supplement 1B, 4C, 5B, 6B, 9,** and **11**). However, we note from Eq. (22) that Δ*C*_{pTS-N(T)} < 0 at these two temperatures and is ∽ −6.2 kcal.mol^{-1}.K^{-1} for FBP28 WW (**Figure 15B**). Since the molar concentration of the TSE is identical to that of the NSE at *T*_{S(α)} and *T*_{S(ω)}, what this physically means is that if we were to take a mole of NSE and a mole of TSE and heat them at constant pressure under identical solvent conditions, we will find that the NSE, relative to the TSE, will absorb thermal energy equivalent to ∽6.2 calories before both the TSE and the NSE will independently register a 10^{−3} K rise in temperature. Because at these two temperatures the SASA, the Gibbs energy, the enthalpy, and the entropy of the TSE and the NSE are identical, this large difference in heat capacity which is ∽15-fold greater than Δ*C*_{pD-N} (6.2/0.417 = 14.8) must stem from a complex combination of: (*i*) a difference in the number and kinds of non-covalent interactions;^{64} (*ii*) the precise 3D-arrangement of the non-covalent interactions (i.e., the network of interactions) leading to a difference in their fundamental frequencies;^{55,56} and (*iii*) the character of the surface exposed to the solvent (i.e., polar *vs* non-polar SASA) between the said reaction-states.^{65-67} Thus, a fundamentally important conclusion that we may draw from this behaviour is that “*two reaction-states on a protein folding pathway need not necessarily have the same structure even if their interconversion proceeds with concomitant zero net-change in SASA, enthalpy, entropy, and Gibbs energy*.” A corollary is that the reaction-states on a protein folding pathway are distinct entities with respect to both their internal structure and the character of their molecular surface. What this implies is that the Hammond postulate which states that “*if two states, as for example, a transition state and an unstable intermediate, occur consecutively during a reaction process and have nearly the same energy content, their interconversion will involve only a small reorganization of the molecular structures*,”^{68} although may be applicable to reactions of small molecules, is inapplicable to protein folding. The inapplicability stems primarily from the profound differences between non-covalent protein folding reactions and covalent reactions of small molecules. In the simplest reactions of small molecules, except for the one or two bonds that are being reconfigured, the rest of the reactant-structure, to a first approximation, usually remains fairly intact as the reaction proceeds (this need not necessarily hold for all simple chemical reactions and probably not for complex reactions). Consequently, if we were to use the bond-length of the bond that is being reconfigured as the RC, and find that the difference in Gibbs energy between any two reaction-states that occur consecutively along the RC are very similar, a reasonable assumption/expectation would be that their structures must be very similar.^{69-77} However, such an assumption cannot be valid for protein folding since an incredibly large number of chain and solvent configurations can lead to conformers having exactly the same Gibbs energy. Consequently, it is difficult to imagine how one can infer the structure of the transiently populated protein reaction-states, including the TSEs, to a near-atomic resolution purely from energetics (see Φ-value analysis later).^{78-80}

### The position of the TSE along the entropic RC

The Leffler parameters for the relative sensitivities of the activation and equilibrium Gibbs energies in response to a perturbation in temperature are given by the ratios of the derivatives of the activation and equilibrium Gibbs energies with respect to temperature.^{13-15,81} Thus, for the partial folding reaction *D* ⇌ [*TS*], we have
where β_{G(fold)(T)} is classically interpreted to be a measure of the position of the TSE relative to the DSE along the entropic RC.^{49} Recasting Eq. (32) in terms of (8) and (A4) and rearranging gives

Similarly for the partial unfolding reaction *N* ⇌ [*TS*] we have
Where β_{G(unfold)(T)} is a measure of the position of the TSE relative to the NSE along the entropic RC. Substituting Eqs. (9) and (A6) in (34) gives

Inspection of Eqs. (32) and (34) shows that β_{G(fold)(T)} + β_{G(unfold)(T)} = 1 for any given reaction-direction. Now, since Δ*S*_{D-N(T)} = Δ*S*_{TS-D(T)} = Δ*S*_{TS-N(T)} = 0 at *T _{S}*, β

_{G(fold)(T)}and β

_{G(unfold)(T)}will be undefined for

*T*=

*T*. However, these are removable discontinuities as is apparent from Eqs. (33) and (35); consequently, curves simulated using the latter set of equations will have a hole at

_{S}*T*. If we ignore the hole at

_{S}*T*to enable a physical description and their comparison to other RCs, the extremum of β

_{S}_{G(fold)(T)}(which is positive and a minimum) and the extremum of β

_{G(unfold)(T)}(which is positive and a maximum) will occur at

*T*(

_{S}**Figure 17**and

**Figure 17−figure supplement 1**) and is a consequence of

*m*

_{TS-D(T)}being a minimum, and both

*m*

_{TS-N(T)}and φ being a maximum, respectively, at

*T*. This can also be demonstrated by differentiating Eqs. (32) and (34) with respect to temperature (not shown). Comparison of Eqs. (28) and (33), and Eqs. (29) and (35) demonstrate that when

_{S}*T*=

*T*, we have β

_{S}_{H(fold)(T)}= β

_{G(fold)(T)}and β

_{H(unfold)(T)}= β

_{G(unfold)(T)}, i.e., the position of the TSE along the heat capacity and entropic RCs are identical at

*T*, and non-identical for

_{S}*T*≠

*T*(

_{S}**Figure 17**). Further, since

*m*

_{TS-N(T)}= β

_{T(unfold)(T)}= 0 at

*T*

_{S(α)}and

*T*

_{S(ω)}(

**Figure 2B**and

**Figure 2−figure supplement 1B**), β

_{G(unfold)(T)}≡ β

_{T(unfold)(T)}= 0 and β

_{G(fold)(T)}≡ β

_{T(fold)(T)}= 1, and not identical for

*T*≠

*T*

_{S(α)}and

*T*

_{S(ω)}; and for

*T*

_{α}≤

*T*<

*T*

_{S(α)}and

*T*

_{S(ω)}<

*T*≤

*T*

_{ω}(the ultralow and high temperature

*Marcus-inverted-regimes*, respectively), β

_{G(fold)(T)}and β

_{T(fold)(T)}are greater than unity, and β

_{G(unfold)(T)}and β

_{T(unfold)(T)}are negative (

**Figure 18**). Note that although β

_{G(fold)(T)}is unity at

*T*

_{S(α)}and

*T*

_{S(ω)}, the structures of the TSE and the NSE cannot be assumed to be identical as explained earlier.

Although it is beyond the scope of this manuscript to perform a large-scale survey of literature for corroborating evidence, the notion that these equations must hold for any two-state folder (as long as they conform to the postulates laid out in Paper-I) is readily apparent from the experimental data of Kelly, Gruebele and colleagues.^{25,82-84} However, the reader will note that what Gruebele and coworkers refer to as Φ_{T}(*T*, *P*) (see Eq. (8) in Crane et al., 2000 and Jäger et al., 2001, Eq. (5) in Ervin and Gruebele, 2002, and Eq. (3) in Nguyen et al., 2003) is equivalent to β_{G(T)} in this article. We will reserve the letter Φ for Φ-value analysis which we will address later.^{79} Inspection of Fig. 7a in Crane et al., 2000 demonstrates that β_{G(fold)(T)} increases with temperature for *T* > *T _{S}* for both the wild type hYAP WW domain and its mutant W39F (∽0.4 at 38 °C and ∽0.8 at 78 °C). This pattern is once again repeated for the wild type and several mutants of Pin WW domain (Fig. 8 in Jäger et al., 2001) and more importantly for ΔNΔC Y11R W30F, a variant of FBP28 WW (inset in Fig. 4B in Nguyen et al., 2003). Nevertheless, all is not in agreement since the shapes of their β

_{G(fold)(T)}curves are distinctly different from what is expected from the formalism discussed in this article. This discrepancy most probably has to do with their use of Taylor expansion with three adjustable parameters to calculate the temperature-dependence of equilibrium stability and the Gibbs activation energies. While it is stated that the use of this non-classical model and the associated adjustable parameters in preference to the physically realistic Schellman formalism (which requires the model-independent calorimetrically determined value of Δ

*C*

_{pDN})

^{6}makes little or no difference to the temperature-dependence of equilibrium stability over an extended temperature range, this may not be true for the activation energy. Once again in good agreement with prediction that β

_{G(unfold)(T)}must decrease with temperature for

*T*>

*T*, Tokmakoff and coworkers find that β

_{S}_{G(unfold)(T)}for ubiquitin decreases with temperature (0.77 at 53 °C and 0.67 at 67 °C).

^{85}Note that although raw data of the said groups and their conclusion that the position of the TSE shifts closer to the NSE as the temperature is raised for

*T*>

*T*is in agreement with the predictions of the equations derived here, their Hammond-postulate-based inference of the structure of the TSE is flawed from the perspective of the parabolic approximation.

_{S}Now, at the midpoint of thermal (*T _{m}*) or cold denaturation (

*T*), Δ

_{c}*G*

_{D-N(T)}= 0; therefore, Eqs. (1) and (2) become

Substituting Eqs. (36) and (37), and in (33) and (35), respectively, and simplifying gives

Simply put, at the midpoint of cold or heat denaturation, the position of the TSE relative to the DSE along the entropic RC is identical to the position of the TSE relative to the NSE along the SASA-RC (**Figure 19A**). Similarly, the position of the TSE relative to the NSE along the entropic RC is identical to the position of the TSE relative to the DSE along the SASA-RC (**Figure 19B**). Dividing Eq. (38) by (39) gives

This seemingly obvious relationship has far deeper physical meaning. Simplifying further and recasting gives

Thus, at the temperatures *T _{c}* and

*T*where the concentration of the DSE and the NSE are identical, the ratio of the slopes of the folding and unfolding arms of the chevron determined at the said temperatures are a measure of the ratio of the change in entropies for the partial folding reactions [

_{m}*TS*] ⇋

*N*and

*D*⇋[

*TS*], or the square root of the ratio of the Gaussian variances of the DSE and the NSE along the SASA-RC, or equivalently, the ratio of the standard deviations of the DSE σ

_{DES(T)}and the NSE σ

_{NES(T)}Gaussians

**(Figure 19−figure supplement 1**; see Paper-I for the relationship between force constants, Gaussian variances and equilibrium stability). A corollary is that irrespective of the primary sequence, or the topology of the native state, or the residual structure in the DSE, if for a spontaneously folding two-state system at constant pressure and solvent conditions it is found that at a certain temperature the ratio of the distances by which the denatured and the native conformers must travel from the mean of their ensemble to reach the TSE along the SASA RC is identical to the ratio of the standard deviations of the Gaussian distribution of the SASA of the conformers in the DSE and the NSE, then at this temperature the Gibbs energy of unfolding or folding must be zero.

As an aside, the reader will note that β_{G(fold)(T)} and β_{G(unfold)(T)} are equivalent to the Brønsted exponents alpha and beta, respectively, in physical organic chemistry; and their classical interpretation is that they are a measure of the structural similarity of the transition state to either the reactants or the products.^{81} If the introduction of a systematic perturbation (often a change in structure *via* addition or removal of a substituent, pH, solvent etc.) generates a reaction-series, and if for this reaction-series it is found that alpha is close to zero (or beta close to unity), then it implies that the energetics of the transition state is perturbed to the same extent as that of the reactant, and hence inferred that the structure of the transition state is very similar to that of the reactant. Conversely, if alpha is close to unity (or beta is almost zero), it implies that the energetics of the transition state is perturbed to the same extent as the product, and hence inferred that the transition state is structurally similar to the product. Although the Brønsted exponents in many cases can be invariant with the degree of perturbation (i.e., a constant slope leading to linear free energy relationships),^{70,86} this is not necessarily true, especially if the degree of perturbation is substantial (Fig. 3 in Cohen and Marcus, 1968; Fig. 1 in Kresge, 1975).^{y14,72,81} Further, this seemingly straightforward and logical Hammond-postulate-based conversion of Brønsted exponents to similarity or dissimilarity of the structure of the transition states to either of the ground states nevertheless fails for those systems with Brønsted exponents greater than unity and less than zero (see page 1897 in Kresge, 1974).^{24,81,87-91}

To summarise, a comparison of the position of the TSE along the solvent (β_{T(T)}), heat capacity (β_{H(T)}), and entropic (β_{G(T)}) RCs leads to three important general conclusions (**Figure 20**): (*i*) as long as ΔSASA_{D-N} is large, and by extension Δ*C*_{pD-N} is large and positive, the position of the TSE relative to the ground states along the various RCs is neither constant nor a simple linear function of temperature when investigated over a large temperature range; (*ii*) for a given temperature, the position of the TSE along the RC depends on the choice of the RC; and (*iii*) although the algebraic sum of β_{T(fold)(T)} and β_{T(unfold)(T)}, β_{H(fold)(T)} and β_{H(unfold)(T)}, and β_{G(fold)(T)} and β_{G(unfold)(T)} must be unity for a two-state system for any particular temperature, individually they can be positive, negative, or zero. Consequently, the notion that the atomic structure of the transiently populated reaction-states in protein folding can be inferred from their position along the said RCs is flawed.^{78}

### Temperature-dependence of Φ-values

Φ-value analysis is a variation of the Brønsted procedure introduced by Fersht and coworkers which when properly implemented claims to provide a near-atomic-level description of the transiently populated reaction-states in protein folding.^{79,80} In this procedure, the primary sequence of the target protein is modified using protein engineering, and the effect of these perturbations are quantified through a parameter Φ (0 ≤ Φ ≤ 1) which by definition is the ratio of mutation-induced change in the Gibbs activation energy of folding/unfolding to the corresponding change in equilibrium stability. According to the canonical formulation, when Φ_{F(T)} = 0 (Φ-value for folding), it implies that the energetics of the TSE is perturbed to the same extent as that of the DSE upon mutation, and hence *inferred* that the said reaction-states are structurally identical with respect to the site of mutation. In contrast, when Φ_{F(T)} = 1, it implies that the energetics of the TSE is perturbed to the same extent as that of the NSE, and hence *inferred* that the structure at the site of mutation is identical in both the TSE and the NSE. Partial Φ-values are difficult to interpret and are thought to be due to partially developed interactions in the TSE, or multiple routes to the TSE. Thus, while Φ *per se* is the slope a two-point Brønsted plot, the conversion of this value to relative-structure is based on the Hammond postulate and the canonical range: The Hammond postulate provides the licence to infer structure from energetics, and the canonical scale enables one to infer how similar or dissimilar the TSE is to either the DSE or the NSE. Assuming that the prefactor is identical for the wild type and the mutant proteins, we may write for the partial folding (*D* ⇌ [*TS*]) and unfolding (*N* ⇌ [*TS*]) reactions
where the subscripts “wt” and “mut” denote the reference or the wild type, and the structurally perturbed protein, respectively, and Φ_{U(T)} is the Φ-value for unfolding. Inspection of Eqs. (42) and (43) shows that for a two-state system, Φ_{F(T)} + Φ_{U(T)} = 1. Now, although the primary sequence is intact in thermal denaturation experiments, we can readily calculate the temperature-dependence of Φ values for folding and unfolding using the protein at one unique temperature as the internal reference or the wild type, and protein at all the rest of the temperatures as the mutants. Thus, if the protein at *T _{S}* is defined as the internal reference or the wild type, Eqs. (42) and (43) become

Similarly, if the protein at *T _{m}* is defined as the internal reference or the wild type, Eqs. (42) and (43) become
Where

*x*= Δ

*G*

_{TS-D(Tm)}= Δ

*G*

_{TS-N(Tm)}and

*y*= Δ

*G*

_{N-D(T)}≡ −Δ

*G*

_{D-N(T)}(the denominator reduces to a single quantity since Δ

*G*

_{D-N(Tm)}≡ −Δ

*G*

_{N-D(Tm)}= 0). The parameters Φ

_{F(internal)(T)}and Φ

_{U(internal)(T)}(which are obviously undefined for the reference temperatures) when interpreted according to the canonical Φ-value framework (i.e., the notion that 0 ≤ Φ ≤ 1) are a measure of the

*global*similarity or dissimilarity of the structure of the TSE to either the DSE or the NSE. Thus, if Φ

_{F(internal)(T)}= 0, it implies that the energetics of the TSE is perturbed to the same extent as that of the DSE upon a perturbation in temperature, and hence inferred that the global structure of the TSE is identical to that of the DSE. Conversely, if Φ

_{F(internal)(T)}= 1, it implies that the energetics of the TSE is perturbed to the same extent as the NSE upon a perturbation in temperature, and hence inferred that the global structure of the TSE is identical to that of the NSE.

Inspection of **Figures 21** and **Figure 21−figure supplements 1**, **2**, **3** and **4** immediately demonstrates that: (*i*) irrespective of which temperature is defined as the internal reference (i.e., the wild type), Φ_{F(internal)(T)} must be a minimum and Φ_{U(internal)(T)} must be a maximum at *T _{S}* (see

**Appendix)**; (

*ii*) the magnitude of Φ

_{F(internal)(T)}is always the least, and the magnitude of Φ

_{U(internal)(T)}is always the greatest when the protein at

*T*is defined as the reference or the wild type protein, and any deviation in the definition of the reference temperature from

_{S}*T*must lead to a uniform increase in Φ

_{S}_{F(internal)(T)}and a uniform decrease in Φ

_{U(internal)(T)}for all temperatures; (

*iii*) although the algebraic sum of Φ

_{F(internal)(T)}and Φ

_{U(internal)(T)}is unity for all temperatures, the notion that they must independently be restricted to 0 ≤ Φ ≤ 1 is flawed; and (

*iv*) although both Leffler β

_{G(T)}and Fersht Φ values are derived from changes in Gibbs activation energies for folding and unfolding relative to changes in equilibrium stability upon a perturbation in temperature, their response is not the same since the equations that govern their behaviour are not the same. While the magnitude of the Leffler β

_{G(T)}is independent of the reference owing to it being the ratio of the derivatives of the change in Gibbs energies with respect to temperature, the magnitude of Φ(internal)(

*T*) is dependent on the definition of the reference state. For example, if the protein at

*T*is defined as the wild type, then β

_{S}_{G(fold)(T)}≈ Φ

_{F(internal)(T)}and β

_{G(unfold)(T)}≈ Φ

_{U(internal)(T)}around the temperature of maximum stability; but as the temperature deviates from

*T*, β

_{S}_{G(fold)(T)}increases far more steeply than Φ

_{F(internal)(T)}, and β

_{G(unfold)(T)}decreases far more steeply than Φ

_{U(internal)(T)}such that for

*T*≠

*T*we have β

_{S}_{G(fold)(T)}> Φ

_{F(internal)(T)}and β

_{G(unfold)(T)}< Φ

_{U(internal)(T)}(

**Figure 21−figure supplement 3**). In contrast, if the protein at

*T*is defined as the wild type, then we have: (

_{m}*i*) β

_{G(fold)(T)}< Φ

_{F(internal)(T)}for

*T*<

_{c}*T*<

*T*and β

_{m}_{G(fold)(T)}> Φ

_{F(internal)(T)}for

*T*<

*T*and

_{c}*T*>

*T*; and (

_{m}*ii*) β

_{G(unfold)(T)}> Φ

_{U(internal)(T)}for

*T*<

_{c}*T*<

*T*and β

_{m}_{G(unfold)(T)}< Φ

_{U(internal)(T)}for

*T*<

*T*and

_{c}*T*>

*T*(

_{m}**Figure 21−figure supplement 4**). The point we are trying to make is that a comparison of the position of the TSE along Leffler β

_{G(T)}and Φ

_{(internal)(T)}RCs is not straightforward since both β

_{G(T)}and Φ

_{(internal)(T)}are temperature-dependent, and importantly respond differently to temperature-perturbation; and even if we restrict the comparison to one particular temperature, the answer we get is still subjective since the magnitude of Φ

_{(internal)(T)}is dependent on how we define the wild type.

^{92}

Although the mathematical formalism for why the extrema of Φ_{F(internal)(T)} (which is a minimum) and Φ_{U(internal)(T)} (which is a maximum) must always occur precisely at *T _{S}* has been shown in the appendix, it is instructive to examine the same graphically. Inspection of

**Figure 21−figure supplements 5, 6**and

**7**demonstrates that this is a consequence of Δ

*G*

_{TS-D(T)}and Δ

*G*

_{N-D(T)}being a minimum, and Δ

*G*

_{TS-N(T)}and Δ

*G*

_{D-N(T)}being a maximum at

*T*

_{S}. Subtracting the reference Gibbs energies from the numerator and the denominator (Eq. (44)) has the effect of lowering the Δ

*G*

_{TS-D(T)}curve and raising the Δ

*G*

_{N-D(T)}, such that the value of the said curves are zero at the reference temperature, but the shapes of the curves are not altered in any way (

**Figure 21−figure supplement 5**). On the other hand, for Δ

*G*

_{TS-N(T)}and Δ

*G*

_{D-N(T)}curves (Eq. (45)), apart from the value of the curves becoming zero at the reference, it causes them to flip vertically (

**Figure 21−figure supplement 6**). Consequently, if we divide the transformed Gibbs activation energies by the transformed equilibrium Gibbs energies, we end up with Φ

_{F(internal)(T)}and Φ

_{U(internal)(T)}which are a minimum and a maximum, respectively, at

*T*(

_{S}**Figure 21−figure supplement 7**).

Now that the process that leads to the temperature-dependence of Φ has been addressed, the question is “Can we infer the structure of the TSE as being similar to either the DSE or the NSE from these data?” The answer is “no” for several reasons. First, as argued earlier, the Hammond postulate cannot be valid for protein folding; and because the structural interpretation of Φ values is based on the Hammond postulate, it too must be deemed fallacious. Second, even if we accept the premise that Hammond postulate is applicable to protein folding, the inference that the global structure of the TSE as being denatured-like for Φ_{F(internal)(T)} = 0, and native-like for Φ_{F(internal)(T)} = 1 is flawed since Φ values need not necessarily be restricted to 0 ≤ Φ ≤ 1 (**Figure 21−figure supplement 2**). Third, even if we summarily exclude those wild types that lead to anomalous Φ values as being unsuitable for Φ analysis, we still have a problem since even within the restricted set of wild types that yield 0 ≤ Φ ≤ 1, their magnitude depends on the definition of the wild type; consequently, for the same temperature, the degree of structure in the TSE relative to that in the DSE appears to increase as the definition of the wild type deviates from *T _{S}* (

**Figure 21−figure supplement 1**). If we try to circumvent this interpretational problem by arguing that the “inference of the structure of the TSE” is always relative to the residual structure in the DSE, and that changing the definition of what constitutes the wild type will invariably affect Φ values, then we can’t really say much about the structure of the TSE without first solving the structure of the DSE. Fourth, even if through a judicious combination of various structural and biophysical methods (residual dipolar couplings, paramagnetic relaxation enhancement, small angle X-ray scattering, single molecule spectroscopy etc.), and computer simulation, we are able to determine the residual structure in the DSE,

^{93-96}the structural interpretation of Φ values leads to physically unrealistic scenarios. For example, inspection of

**Figure 21A**shows that around room temperature (298 K) Φ

_{F(internal)(T)}≈ 0.18. A canonical interpretation of this number implies that the global structure of the TSE is very similar to that of the DSE. However, inspection of

**Figure 2−figure supplement 1A**shows that the denatured conformer has buried ∽70% of the total SASA to reach the TSE (i.e., advanced by about 70% along the SASA-RC). Similarly, inspection of

**Figure 5A**shows that Δ

*G*

_{TS-D(T)}= 2.6 kcal.mol

^{-1}at 298 K (note that this is not a small number that can be ignored since Δ

*G*

_{D-N(T)}= 2.1 kcal.mol

^{-1}at 298 K). Further, we have shown earlier in the section on the “Inapplicability of the Hammond postulate to protein folding,” that even when two reaction-states have identical SASA, Gibbs energies, enthalpies, and entropies, there need not necessarily have identical structure. Thus, the question is: How can we conclude with any measure of certainty that the global structure of the TSE is very similar to that of the DSE at 298 K when they have such a large difference in SASA, and a substantial difference in Gibbs energy? To illustrate why it is difficult to rationalize the theoretical basis of Φ analysis, it is instructive to directly examine the ratio of the Gibbs activation energies and the difference in Gibbs energy between the ground states (

**Figure 21−figure supplement 8**). It is immediately apparent that the ratios are a complex function of temperature; and although we can readily provide an explanation for the particular features of these complex dependences, it is difficult to see how subtracting reference energies from the numerator and denominator of the ratios Δ

*G*

_{TS-D(T)}/Δ

*G*

_{N-D(T)}and Δ

*G*

_{TS-N(T)}/Δ

*G*

_{D-N(T)}allows us to divine the structure of the TSE to a near-atomic resolution. This is once again readily apparent from the complex non-linear relationship between equilibrium stability and the rate constants (

**Figure 21−figure supplement 9**).

To further illuminate the difficulty in rationalizing the Φ-value procedure, it is instructive to apply Eqs. (42) and (45) to treat enthalpies. Thus, for the partial folding (*D* ⇌ [*TS*]) and unfolding (*N* ⇌ [*TS*]) reactions we have
Where the parameters Φ_{HF(internal)(T)} and Φ_{HU(internal)(T)} are the “*enthalpic analogues*” of Φ_{F(internal)(T)} and Φ_{U(internal)(T)}, respectively (the subscript “H” indicates we are using enthalpy instead of Gibbs energy), when the protein at the temperature *T _{S}* is defined as the wild type. Now, if we apply an analogous version of the canonical interpretation given by Fersht and coworkers, it implies that when Φ

_{HF(internal)(T)}= 0, the enthalpy of the TSE is perturbed to the same extent as that of the DSE upon a perturbation in temperature; and when Φ

_{HF(internal)(T)}= 1, it implies that the enthalpy of the TSE is perturbed to the same extent as that of the NSE. It is easy to see that just as Φ

_{F(internal)(T)}and Φ

_{U(internal)(T)}are the

*Fersht-analogues*of the Leffler β

_{G(fold)(T)}and β

_{G(unfold)(T)}, respectively (see entropic RC), the parameters Φ

_{HF(internal)(T)}and Φ

_{HU(internal)(T)}are similarly the Fersht-analogues of the Leffler β

_{H(fold)(T)}and β

_{H(unfold)(T)}, respectively (see heat capacity RC).

Inspection of **Figure 22** and its supplements immediately demonstrates that the same anomalies that prevent a straightforward structural interpretation of Φ_{F(internal)(T)} and Φ_{U(internal)(T)} are also emerge if we try to assign structure to their enthalpic analogues, Φ_{HF(internal)(T)} and Φ_{HU(internal)(T)}. First, although the algebraic sum of Φ_{HF(internal)(T)} and Φ_{HU(internal)(T)} is unity for all temperatures, they need not independently be restricted to a canonical range of 0 ≤ Φ ≤ 1 (**Figure 22**). Second, the magnitude of Φ_{HF(internal)(T)} and Φ_{HU(internal)(T)} are dependent on the definition of the wild type (**Figure 22−figure supplement 1**). Third, changing the definition of the wild type has a dramatic effect on the relationship between the Leffler β_{H(T)} and its analogue, the Fersht Φ_{H(internal)(T)}. Consequently, the question of whether Leffler β_{H(T)} underestimates or overestimates structure is dependent on how we analyse the system (**Figure 22−figure supplements 2** and **3**). Fourth, just as the temperature-dependent position of the TSE relative to the ground states depends on the choice of the RC (**Figure 20**), we see that Φ_{(internal)(T)} and its enthalpic analogue, Φ_{H(internal)(T)}, change at different rates upon a perturbation in temperature (**Figure 22−figure supplement 4**). The difficulty in rationalizing how subtracting reference values from the numerator and the denominator of Eqs. (42) and (49) can yield residue-level information is once again apparent from the complex dependence of the ratios ∂ ln *k*_{f(T)}/∂ ln *K*_{N-D(T)} = Δ*H*_{TS-D(T)}/Δ*H*_{N-D(T)} and ∂ ln *k*_{u(T)}/∂ ln *K*_{D-N(T)} = Δ*H*_{TS-N(T)}/Δ*H*_{D-N(T)} on temperature (**Figure 22−figure supplement 5**).

### Comparison of theoretical and experimental Φ-values obtained from structural perturbation across 31 two-state systems

Given that the framework of Φ-value analysis was primarily developed to be used in conjunction with structural rather than temperature perturbation, and despite its anomalies has been used extensively for more than twenty years to divine the structures of the TSEs of not just globular but also membrane proteins, it is imperative to demonstrate that the notion that the structure of the TSE cannot be inferred from Φ-values is also valid for structural perturbation.^{97-101} Although a detailed reappraisal is beyond the scope of this article and will be presented elsewhere, because we have questioned the validity of Φ analysis, one is compelled to provide some justification in this article.

Consider the wild type of a hypothetical two-state folder whose equilibrium stability and the mean length of the RC at constant temperature, pressure and solvent conditions are given by Δ*G*_{D-N(T)} = 6 kcal.mol^{-1} and *m*_{D-N} = 2 kcal.mol^{-1}.M^{-1}, respectively. Although not necessarily true and addressed elsewhere, to limit the number of hypothetical scenarios to a manageable number, we will assume that the force constants of the DSE and the NSE-parabolas of the wild type and all its mutants are given by α = 1 M^{2}.mol.kcal^{-1} and ω = 30 M^{2}.mol.kcal^{-1}. The effect of single point mutations on the wild type may be classified into a total of five unique scenarios (**Figure 23A**).

**Case I (Quadrant x2):** The introduced mutation causes a concomitant decrease in both the stability and the mean length of the RC (i.e., Δ

*G*

_{D-N(T)(wt)}> Δ

*G*

_{D-N(T)(mut)}and

*m*

_{D-N(wt)}>

*m*

_{D-N(mut)}). This is equivalent to the introduced mutation causing the separation between the vertices of the DSE and the NSE-parabolas along the abscissa and ordinate to decrease (

**Figure 23−figure supplement 1A**).

**Case II (Quadrant y1):** The introduced mutation causes a decrease in stability but concomitantly causes an increase in the mean length of the RC (i.e., Δ

*G*

_{D-N(T)(wt)}> Δ

*G*

_{D-N(T)(mut)}and

*m*

_{D-N(wt)}<

*m*

_{D-N(mut)}). This is equivalent to the mutation causing a decrease in the separation between the vertices of the DSE and the NSE-parabolas along the ordinate, but an increase along the abscissa (

**Figure 23−figure supplement 1B**).

**Case III (Quadrant x1):** The introduced mutation leads to an increase in stability but concomitantly causes a decrease in the mean length of the RC (i.e., Δ

*G*

_{D-N(T)(wt)}< Δ

*G*

_{D-N(T)(mut)}and

*m*

_{D-N(wt)}>

*m*

_{D-N(mut)}). This is equivalent to the mutation causing an increase in the separation between the vertices of the DSE and the NSE-parabolas along the ordinate, but a decrease along the abscissa (

**Figure 23−figure supplement 1C**).

**Case IV (Quadrant y2):** The introduced mutation leads to a concomitant increase in both the stability and the mean length of the RC (i.e., Δ

*G*

_{D-N(T)(wt)}< Δ

*G*

_{D-N(T)(mut)}and

*m*

_{D-N(wt)}<

*m*

_{D-N(mut)}). This is equivalent to the mutation causing an increase in the separation between the vertices of the DSE and the NSE-parabolas along the ordinate and the abscissa (

**Figure 23−figure supplement 1D**).

**Case V:** The introduced mutation leads to a change in stability but has no effect on the mean length of the RC (*m*_{D-N(wt)} = *m*_{D-N(mut)}). This is equivalent to the mutation causing an increase or a decrease in the separation between the vertices of the DSE and the NSE-parabolas along the ordinate, but the separation along the abscissa is invariant (**Figure 23−figure supplement 2**).

In summary, what we done is taken a pair of intersecting parabolas of differing curvature (ω > α), and systematically varied the separation between their vertices along the abscissa (*m*_{D-N}) and ordinate (Δ*G*_{D-N(T)}) without changing the curvature of the parabolas. Once this is done, we can calculate *a priori* the position of the *curve-crossings* relative to the vertex of the DSE-parabola along the abscissa (i.e., *m*_{TS-D(T)}; Eq. (1)) and ordinate (i.e., Δ*G*_{TS-D(T)}; Eq. (3)). Once the Δ*G*_{TS-D(T)} values for all combinations of Δ*G*_{D-N(T)} and *m*_{D-N} are obtained (each combination is equivalent to a point mutation), Φ_{F(T)} values can be readily calculated using Eq. (50) by arbitrarily choosing one particular combination of Δ*G*_{D-N(T)} (= 6 kcal.mol^{-1}) and *m*_{D-N} (= 2 kcal.mol^{-1}.M^{-1}) as the wild type.

**Figure 23A** which has been generated by plotting the theoretical Φ_{F(T)} values as a function of ΔΔ*G*_{D-N(wt-mut)(T)} leads to two important conclusions: (*i*) Φ_{F(T)} values are not restricted to 0 ≤ Φ ≤ 1, and that the perceived unusualness of anomalous or non-classical Φ values is a consequence of flawed canonical limits; and (*ii*) the magnitude of Φ_{F(T)} values decrease as the difference in stability between the wild type and the mutant proteins increase, and at once debunks the idea that one must use an arbitrary ΔΔ*G*_{D-N(wt-mut)(T)} cut-off (± 0.6 kcal.mol^{-1} according to the Fersht lab, and ± 1.7 kcal.mol^{-1} according to Sanchez and Kiefhaber) for Φ_{F(T)} values to be interpretable.^{98,102} While it is true that Φ values would be error prone when |ΔΔ*G*_{D-N(wt-mut)(T)}| is less than the error with which one can determine Δ*G*_{D-N(T)} of both the wild type and the mutant proteins (typically about ± 5-10% of Δ*G*_{D-N(T)}),^{103} the increase in the magnitude of Φ_{F(T)} values when ΔΔ*G*_{D-N(wt-mut)(T)} approaches zero (the vertical asymptotes) is a mathematical certainty and not because of error as is commonly argued. Nevertheless, because these conclusions are based on the results of a model that is purely hypothetical, they would naturally be meaningless without experimental validation. Thus, as a test of the hypothesis, experimental Φ_{F(T)} values in water were calculated according to Eq. (51) using published kinetic data of a total of 1064 proteins (1035 mutants + 29 wild types) from 31 two-state systems (details of the systems analysed will be provided elsewhere).

The remarkable agreement between theoretical prediction and experimental Φ_{F(T)} values is immediately apparent from an overlay of the said datasets (**Figure 23B**), and serves as arguably one of the most rigorous tests of the hypothesis for the following reasons: (1) The space enclosed by the curves in **Figure 23A** is complex and restricted. Therefore, if the experimental Φ_{F(T)} values fall within this restricted theoretical space it would be highly unlikely for it to be purely due to some dramatic coincidence. (2) The sample size of experimental dataset is sufficiently large (1035 mutations), and the two-state systems investigated include α, β, and α/β proteins (note that α and β refer to secondary structure in this context and not to the force constant of the DSE or the Tanford beta value, respectively), with size ranging from 37 to 107 residues. (3) The published kinetic data used to calculate experimental Φ_{F(T)} values were acquired by various labs under varying solvent conditions (buffers, co-solvents and pH; denaturant is either guanidine hydrochloride or urea) and temperature (as low as 278 K to as high as 301.16 K), over a period of about two decades using a variety of experimental methods, including infrared laser-induced and electrical discharge temperature-jump relaxation measurements, stopped flow and manual mixing experiments, and lineshape analysis of exchange-broadened NMR resonances. These results, including those on the temperature-dependence of Φ_{F(T)} values lead to an important conclusion: Because the canonical scale itself has no basis, Φ-value-based interpretation of the structure of the transiently populated protein reaction-states is dubious.

## Concluding Remarks

Although the temperature-dependent behaviour of FBP28 WW was analysed in great detail using the theory developed in the Papers I and II, and novel conclusions have been drawn, this is by no means sufficient since we have barely addressed the physical chemistry underlying the effect of temperature on the Gibbs energies, the enthalpies, the entropies, and the heat capacities of activation for folding and unfolding. These aspects will be dealt with in the accompanying articles. Further, there is a good reason why we have given little importance to the actual values of the reference temperatures and instead focussed on what they actually mean and how they relate to each other. Although the remarks in Table 1 are valid for all reference temperatures, except for the values of the equilibrium reference temperatures (*T _{c}*,

*T*,

_{H}*T*, and

_{S}*T*), the values for the rest of them can change depending on the values of the force constants. However, what will not change is the inter-relationship between them. The nature of this limitation will be addressed when the mechanism of action of denaturants is investigated.

_{m}## Methods

The temperature-dependence of Δ*G*_{D-N(T)} of FBP28 WW wild type (**Figure 1**) was simulated according to Eq. (A1) using *T _{m}* = 337.2 K, Δ

*H*

_{D-N(Tm)}= 26.9 kcal.mol

^{-1}and Δ

*C*

_{pD-N}= 417 cal.mol

^{-1}.K

^{-1}(Table 1 in Petrovich et al., 2006).

^{4}The values of

*k*

^{0}= 2180965 s

^{-1}, α = 7.594 M

^{2}.mol.kcal

^{-1}, ω = 85.595 M

^{2}.mol.kcal

^{-1}, and

*m*

_{D-N}= 0.82 kcal.mol

^{-1}.M

^{-1}were extracted from the chevron of FBP28 WW (acquired at 283.16 K in 20 mM 3-[morpholino] propanesulfonic acid, ionic strength adjusted to150 mM with Na

_{2}SO

_{4}, pH 6.5) by fitting it to a modified chevron-equation using non-linear regression as described in Paper-I. The data required to simulate the chevron (

*k*

_{f(H2O)(T)},

*k*

_{u(H2O)(T)},

*m*

_{TS-D(T)}and

*m*

_{TS-N(T)}) were taken from Table 4 in Petrovich et al., 2006.

^{4}Once the parameters Δ

*H*

_{D-N(Tm)},

*T*, Δ

_{m}*C*

_{pD-N},

*m*

_{D-N}, the force constants α and ω, and

*k*

^{0}are known, the left-hand side of all the equations in this article may be readily calculated for any temperature. Note that the spring constants,

*k*

^{0},

*m*

_{D-N}, and Δ

*C*

_{pD-N}are temperature-invariant.

## Competing Financial Interests

The author declares no competing financial interests.

## Appendix

### The temperature-dependence of Δ*G*_{D-N(T)}, Δ*H*_{D-N(T)}, and Δ*S*_{D-N(T)} functions

The temperature-dependence of the change in Gibbs energy, enthalpy and entropy of two-state systems upon unfolding at equilibrium are given by^{6}

Where Δ*H*_{D-N(T)}, Δ*H*_{D-N(Tm)} and Δ*S*_{D-N(T)}, Δ*S*_{D-N(Tm)} denote the equilibrium enthalpies and entropies of unfolding, respectively, at any given temperature, and at the midpoint of thermal denaturation (*T _{m}*), respectively, for a given two-state folder under defined solvent conditions. The temperature-invariant and the temperature-dependent difference in heat capacity between the DSE and NSE are denoted by Δ

*C*

_{pD-N}and Δ

*C*

_{pD-N(T)}, respectively.

### The first derivatives of *m*_{TS-D(T)}, *m*_{TS-N(T)}, β_{T(fold)(T)} and β_{T(unfold)(T)} with respect to temperature

The first derivative of *m*_{TS-D(T)} is given by

Because β_{T(fold)(T)} = *m*_{TS-D(T)}/*m*_{D-N}, we also have

Since ∂*m*_{TS-D(T)}/∂*T* and ∂β_{T(fold)(T)}/∂*T* are physically undefined for φ < 0, their algebraic sign at any given temperature is determined by the ln(*T*/*T*_{S}) term. This leads to three scenarios: (*i*) for *T* < *T _{S}* we have ∂

*m*

_{TS-D(T)}/∂

*T*> 0 and ∂β

_{T(fold)(T)}/∂

*T*> 0; and (

*iii*) for

*T*=

*T*

_{S}we have ∂

*m*

_{TS-D(T)}/∂

*T*= 0 and ∂β

_{T(fold)(T)}/∂

*T*= 0.

Because *m*_{TS-N(T)} = (*m*_{D-N} − *m*_{TS-D(T)}) for a two-state system, and β_{T(unfold)(T)} = *m*_{TS-N(T)}/*m*_{D-N}, we have

Eqs. (A6) and (A7) once again lead to three scenarios: (*i*) for *T* < *T _{S}* we have ∂

*m*

_{TS-N(T)}/∂

*T*> 0 and ∂β

_{T(unfold)(T)}/∂

*T*> 0; (

*ii*) for

*T*>

*T*

_{S}we have ∂

*m*

_{TS-N(T)}/∂

*T*< 0 and ∂β

_{T(unfold)(T)}/∂

*T*> 0; and (

*iii*) for

*T*=

*T*we have ∂

_{S}*m*

_{TS-N(T)}/∂

*T*= 0 and ∂β

_{T(unfold)(T)}/∂

*T*= 0.

### The second derivatives of *m*_{TS-D(T)} and *m*_{TS-N(T)} with respect to temperature

Differentiating Eq. (A4) with respect to temperature gives

Simplifying Eq. (A8) yields

Similarly, we may show that

### Expression for the temperature-dependence of the observed rate constant

The observed rate constant *k*_{obs(T)} for a two-state system is the sum of *k*_{f(T)} and *k*_{u(T)}.^{104} Therefore, we can write

### Expressions to demonstrate why the extrema of Φ_{F(internal)(T)} and Φ_{U(internal)(T)} must occur at *T*_{S}

_{S}

Differentiating Eq. (44) with respect to temperature gives
where the protein at the temperature *T _{Ref}* is by definition the wild type protein. Because Δ

*S*

_{N-D(T)}and Δ

*S*

_{TS-D(T)}are both zero at

*T*, irrespective of

_{S}*T*, the derivative of Φ

_{Ref}_{F(internal)(T)}will be zero at

*T*. Similarly, we can show by differentiating Eq. (45) that

_{S}Once again, since Δ*S*_{D-N(T)} and Δ*S*_{TS-N(T)} are both zero at *T _{S}*, irrespective of

*T*, the derivative of Φ

_{Ref}_{U(internal)(T)}will be zero at

*T*.

_{S}## Footnotes

Vinkensteynstraat 128, 2562 TV, Den Haag, Netherlands, robert.sade{at}gmail.com