Analysis of Two-State Folding Using Parabolic Approximation I: Hypothesis

Transcription

1 Analysis of Two-State Folding Using Parabolic Approxiation I: Hypothesis AUTHOR NAME: Robert S. Sade AUTHOR ADDRESS: Vinkensteynstraat 18, 56 TV, Den Haag, Netherlands AUTHOR ADDRESS: AUTHOR AFFILIATION: Independent Researcher KEY WORDS: Arrhenius rate law; Marcus theory; parabolic approxiation; transition state theory; two-state folding. Page 1 of 65

2 ABSTRACT A odel which treats the denatured and native conforers of spontaneously-folding fixed two-state systes as being confined to haronic Gibbs energy-wells has been developed. Within the assuptions of this odel the Gibbs energy functions of the denatured (DSE) and the native state (NSE) ensebles are described by parabolas, with the ean length of the reaction coordinate (RC) being given by the teperature-invariant denaturant value. Consequently, the enseble-averaged position of the transition state enseble (TSE) along the RC, and the enseble-averaged Gibbs energy of the TSE are deterined by the intersection of the DSE and the NSE-parabolas. The equations derived enable equilibriu stability and the rate constants to be rationalized in ters of the ean and the variance of the Gaussian distribution of the solvent accessible surface area of the conforers in the DSE and the NSE. The iplications of this odel for protein folding are discussed. Page of 65

3 INTRODUCTION Understanding the echanis(s) by which denatured or nascent polypeptides under folding conditions spontaneously fold to their unique three-diensional structures is one of the fundaental probles in biology. Although there has been treendous progress since the ground-breaking discovery of Anfinsen, and various theories and odels have been proposed for what has coe to be known as the Protein Folding Proble, our understanding of the sae is far fro coplete. 1 The purpose of this paper is to address issues that are pertinent to the folding proble using a treatent that is analogous to that given by Marcus for electron transfer. FORMULATION OF THE HYPOTHESIS Parabolic approxiation Consider the denatured state enseble (DSE) of a spontaneously-folding fixed two-state folder at equilibriu under folding conditions wherein the variables such as teperature, pressure, ph, ionic strength etc. are defined and constant. 3,4 The solvent accessible surface area (SASA) and the Gibbs energy of each one of the conforers that coprise the DSE, and consequently, the ean SASA and Gibbs energy of the enseble will be deterined by a coplex interplay of intra-protein and protein-solvent interactions (hydrogen bonds, van der Waals and electrostatic interactions, salt bridges etc.). 5-8 At finite but constant teperature, the incessant transfer of oentu fro the theral otion of water causes the polypeptide to constantly drift fro its ean SASA. 9 As the chain expands, there is a favourable gain in chain entropy due to the increased backbone and side-chain conforational freedo, and a favourable gain in solvation enthalpy due to the increased solvation of the backbone and the side-chains; however, this is offset by the loss of favourable chain enthalpy that stes fro the intra-protein backbone and the side-chain interactions, and the unfavourable decrease in solvent entropy, since ore water olecules are now tied down by the relatively ore exposed hydrophobic residues, hydrogen-bond donors and acceptors, and charged residues in the polypeptide. Conversely, as the chain attepts to becoe increasingly copact, there is a favourable gain in chain enthalpy due to an increase in the nuber of residual interactions, and a favourable increase in the solvent entropy due to the release of bound water olecules; however, this is opposed by the unfavourable decrease in both the backbone and the sidechain entropy (excluded volue entropy) and the enthalpy of desolvation. 10,11 Therefore, it is Page 3 of 65

4 postulated that the restoring force experienced by each one of the conforers in the DSE would be proportional to their displaceent fro the ean SASA of the DSE along the SASA-reaction coordinate (SASA-RC), or Fx ( ) x x i where x i is the SASA of the i th conforer in the DSE, x DSE is the ean SASA of the DSE, and F(x i ) is the restoring force experienced by it. Consequently, the Gibbs energy of the conforer, G(x i ), is proportional to the square of this displaceent, or i DSE DSE Gx ( ) x x. If the totality of forces that resist i expansion and copaction of the polypeptide chain are assued to be equal, then to a first approxiation the conforers in the DSE ay be treated as being confined to a haronic Gibbs energy-well with a defined force constant (Figure 1A). Once the Gibbs energies of the conforers are known, the probabilities of their occurrence within the enseble at equilibriu can be readily ascertained using the Boltzann distribution law (Figure 1B). We will coe back to this later. The native state enseble (NSE) in solution ay be treated in an analogous anner: Although the NSE is incredibly far ore structurally hoogeneous than the DSE, and is soeties treated as being equivalent to a single state (i.e., the conforational entropy of the NSE is set to zero) for the purpose of estiating the difference in conforational entropy between the DSE and the NSE, the NSE by definition is an enseble of structures In fact, this theral-noise-induced tendency to oscillate is so strong that native-folded proteins even when constrained by a crystal lattice can perfor this otion. 15 Thus, at finite teperature the NSE is defined by its ean SASA ( x NSE ) and its enseble-averaged Gibbs energy. As the native conforer attepts to becoe increasingly copact, its excluded volue entropy rises treendously since ost of the space in the protein core has already been occupied by the polypeptide backbone and the side-chains of the constituent aino acids. 16 In contrast, any attept by the polypeptide chain to expand and consequently expose ore SASA is et with resistance by the ultitude of interactions that keep the folded structure intact. Therefore, it is postulated that the restoring force would be proportional to the displaceent of the native conforer fro G( y ) y x Fy ( ) y x i i NSE i i NSE i xnse along the SASA-RC, or where y i is the SASA of the i th conforer in the NSE, F(y i ) is the restoring force experienced by it, and G(y i ) is the Gibbs energy of the native conforer. If the su total of the forces that resist copaction and expansion of the native conforer, respectively, are assued to be equal in agnitude, then the conforers in the Page 4 of 65

5 NSE ay be treated as being confined to a haronic Gibbs energy-well with a defined force constant. On the use of the denaturant D-N value as a global reaction coordinate The description of protein folding reactions in ters of reaction coordinates (RCs) and transition states is based on concepts borrowed fro the covalent cheistry of sall olecules. Because protein folding reactions are profoundly different fro reactions in covalent cheistry owing to their non-covalent and ulti-diensional nature, it is often argued that their full coplexity cannot be captured in sufficient detail by any single RC. 17 Nevertheless, it is not uncoon to analyse the sae using one-diensional RCs, such as the native-likeness in the backbone configuration, the fraction of native pair-wise contacts (Q i ) relative to the ground states DSE and NSE, the radius of gyration (R g ), SASA, P fold etc. 18,19 The use of SASA as a global RC in the proposed hypothesis poses a proble since it is very difficult, if not ipossible, to accurately and precisely deterine the ensebleaveraged length of the RC (ΔSASA D-N ) using structural and/or biophysical ethods. Although the ean SASA of the NSE and its fluctuations ay be obtained by applying coputational ethods to the available crystal or solution structures of proteins, 0 such approaches are not readily applicable to the DSE. 3 Although there has been considerable progress in odelling the SASA of the DSEs using siulations, 1, these ethods have not been used here for one predoinant reason: Unlike the NSE, the residual structure in the DSEs of ost proteins can be very sensitive to inor changes in the priary sequence and solvent conditions, which ay not be captured effectively by these theoretical ethods. Therefore, the experientally accessible D-N has been used as a proxy for the true ΔSASA D- N.19 Postulates of the odel The Gibbs energy functions of the DSE and the NSE, denoted by G DSE(r)(T) and G NSE(r)(T) respectively, have a square-law dependence on the RC, r, and are described by parabolas (Figure ). The curvature of parabolas is given by their respective force constants, and. As long as the priary sequence is not perturbed (via utation, cheical or post-translational odification), and pressure and solvent conditions are constant, and the properties of the solvent are teperature-invariant (for exaple, no change in the ph due to the teperature- Page 5 of 65

6 dependence of the pk a of the constituent buffer), the force constants and are teperatureinvariant (Figure 3), i.e., the conforers in the DSE and the NSE behave like linear-elastic springs. A corollary is that changes to the priary sequence, or change in solvent conditions (a change in ph, ionic strength, or addition of co-solvents) can bring about a change in either or or both. The vertices of the DSE and NSE-parabolas, denoted by G D(T) and G N(T), respectively, represent their enseble-averaged Gibbs energies. Consequently, in a parabolic representation, the difference in Gibbs energy between the DSE and NSE at equilibriu is given by separation between G D(T) and G N(T) along the ordinate (ΔG D-N(T) = G D(T) G N(T) ). A decrease or an increase in ΔG D-N(T) relative to the standard state/wild type upon perturbation is synonyous with the net oveent of the vertices of the parabolas towards each other or away fro each other, respectively, along the ordinate (Figure 3). Thus, a decrease in ΔG D- N(T) can be due to a stabilized DSE or a destabilized NSE or both. Conversely, an increase in ΔG D-N(T) can be due to a destabilized DSE or a stabilized NSE or both. The ean length of the RC is given by the separation between G D(T) and G N(T) along the abscissa, and is identical to the experientally accessible D-N (Figure C). For the folding reaction D N, since the RC increases linearly fro 0 D-N in the left-to-right direction, the vertex of the DSE-parabola is always at zero along the abscissa while that of the NSE-parabola is always at D-N. An increase or decrease in ΔSASA D-N, relative to a reference state or the wild type, in accordance with the standard paradig, will anifest as an increase or a decrease in D-N, respectively. 19 In a parabolic representation, an increase in D- is synonyous with the net oveent of vertices of the DSE and NSE-parabolas away N fro each other along the abscissa. Conversely, a decrease in D-N is synonyous with the net oveent of the parabolas towards each other along the abscissa (Figure 4). As long as the priary sequence is not perturbed, and pressure and solvent conditions are constant, and the properties of the solvent are teperature-invariant, x DSE and xnse are invariant with teperature, leading to ΔSASA D-N being teperature-independent; consequently, the ean length of the RC, D-N, for a fixed two-state folder is also invariant with teperature. A corollary is that perturbations such as changes to the priary sequence via utation, cheical or post-translational odification, change in pressure, ph, ionic strength, or Page 6 of 65

7 addition of co-solvents can bring about a change in either x DSE, or x NSE, or both, leading to a change in ΔSASA D-N, and consequently, a change in D-N. Because by postulate D-N is invariant with teperature, a logical extension is that for a fixed two-state folder, the enseble-averaged difference in heat capacity between DSE and the NSE (ΔC pd-n =C pd(t) C pn(t) ) ust also be teperature-invariant since these two paraeters are directly proportional to each other (see discussion on the teperature-invariance of ΔSASA D-N, D-N and ΔC pd-n ). 3,4 The ean position of the transition state enseble (TSE) along the RC, r (T), and the enseble-averaged Gibbs energy of the TSE (G TS(T) ) are deterined by the intersection of G DSE(r)(T) and G NSE(r)(T) functions. In a parabolic representation, the difference in SASA between the DSE and the TSE is given by the separation between G D(T) and the curvecrossing along the abscissa and is identical to TS-D(T). Thus, if the ean SASA of the TSE is denoted by x TSE( T ), then TS-D(T) is a true proxy for xdse xtse( T ) SASAD-TS T and is always greater than zero no atter what the teperature. Siilarly, the difference in SASA between the TSE and the NSE is given by the separation between G and the curve-crossing along N(T) the abscissa and is identical to TS-N(T), i.e., TS-N(T) is a true proxy for x TSE( T ) x NSE = ΔSASA TS-N(T). However, unlike TS-D(T) which is always greater than zero, TS-N(T) can approach zero (when x TSE( T ) x NSE ) and even becoe negative ( x TSE( T ) x NSE ) at very low and high teperatures for certain proteins. The enseble-averaged Gibbs activation energy for folding is given by the separation between G D(T) and the curve-crossing along the ordinate (ΔG TS-D(T) = G TS(T) G D(T) ), and the enseble-averaged Gibbs activation energy for unfolding is given by the separation between G N(T) and the curve-crossing along the ordinate (ΔG TS- = G G ). The position of the curve-crossing along the abscissa and ordinate N(T) TS(T) N(T) relative to the ground states is purely a function of the priary sequence when teperature, pressure and solvent conditions are defined. A corollary of this is that for any two-state folder, any perturbation that brings about a change in the curvature of the parabolas or the ean length of the RC can lead to a change in TS-D(T). Because D-N = TS-D(T) + TS-N(T) for a two-state syste, any perturbation that causes an increase in TS-D(T) without a change D-N will concoitantly lead to a decrease in TS-N(T), and vice versa. Consequently, the Page 7 of 65

8 noralized solvent RCs βt(fold)( T) TS-D( T) D-N and βt(unfold)( T) TS-N( T) D-N will also vary with the said perturbation. 5 Thus, fro the postulates of the parabolic hypothesis we have three fundaentally iportant equations for fixed two-state protein folders: T G (1) TS-D( T) TS-D( ) G () TS-N( T) TS-N( T) D-N TS-D( T) G G G (3) D-N( T) TS-N( T) TS-D( T) TS-N( T) TS-D( T) Consequently, for two-state proteins under folding conditions, as long as ΔG TS-N(T) > ΔG TS- (i.e., ΔG > 0 or ΔG < 0) and > D(T) D-N(T) N-D(T) TS-D(T) TS-N(T) > (Figure C). we have the logical condition Expression for the ean position of the TSE Consider the conventional barrier-liited interconversion of the conforers in the DSE and NSE of a two-state folder at any given teperature, pressure and solvent conditions (Figure C). Because by postulate the Gibbs energy functions G DSE(r)(T) and G NSE(r)(T) have a squarelaw dependence on the RC, r, whose enseble-averaged length is given by D-N, and since the RC increases linearly fro 0 D-N in the left to right direction, we can write 0 G r r (4) DSE( r)( T) ( T) ( T) G = r G (5) NSE( r)( T) D-N ( T) D-N( T) If the units of the ordinate are in kcal.ol -1 and the RC in kcal.ol -1.M -1, then by definition the force constants and have the units M.ol.kcal -1. The ean position of the TSE along the abscissa (r (T) ) is deterined by the intersection of G DSE(r)(T) and G NSE(r)(T). Therefore, at the curve-crossing we have G = G r = r G (6) DSE( r )( T) NSE( r )( T) ( T) D-N ( T) D-N( T) Page 8 of 65

9 r r + G = 0 (7) ( T) D-N ( T) D-N D-N( T) Solving for r (T) gives (see Appendix) r ( T) TS-D( T) TS-D( T ) G T D-N D-N D-N( ) D-N φ (8) (9) where the discriinant G, and the paraeter φ λ T D-N( ) λ D-N is analogous to the Marcus reorganization energy, and by definition is the Gibbs energy required to copress the denatured polypeptide under folding conditions to a state whose SASA is identical to that of the native folded protein but without the stabilizing native interactions (Figure 5). Since and D-N are by postulate teperature-invariant, is teperature-invariant by extension and depends purely on the priary sequence for a given pressure and solvent conditions. Since D-N = TS-D(T) + TS-N(T) for a two-state syste, we have TS-N( T ) φ D-N (10) If the values of the force constants and, D-N and ΔG D-N(T) of a two-state syste at any given teperature, pressure and solvent conditions are known, we can readily calculate the absolute Gibbs activation energies for the folding and unfolding (Eqs. (1) and ()). Equations for the folding and the unfolding rate constants The two theories that feature proinently in the analyses of protein folding kinetics are the transition state theory (TST) and the Kraers theory under high friction liit. 6-8 Despite their profound differences what is coon to both is the exponential ter or the Boltzann factor. Therefore, we will start with the conventional Arrhenius expression for the rate constants (the coplexity of the prefactor which here is assued to be teperature-invariant is addressed elsewhere). Substituting Eqs. (1) and (9), and () and (10) in the expressions for the rate constants for folding (k f(t) ) and unfolding (k u(t) ), respectively, gives Page 9 of 65

10 k k G TS-D( ) D φ T RT RT (11) 0 TS-D( T ) 0 0 -N f ( T) k exp k exp k exp G RT 0 TS-N( T ) 0 -N( ) 0 ut ( ) k exp k k RT TS φ T D-N exp exp RT RT (1) where k 0 is the pre-exponential factor with units identical to those of the rate constants (s -1 ). Because the principle of icroscopic reversibility stipulates that for a two-state syste the ratio of the folding and unfolding rate constants ust be identical to the independently easured equilibriu constant, the prefactors in Eqs. (11) and (1) ust be identical. 9 Eqs. (11) and (1) ay further be recast in ters of T(fold)(T) and T(unfold)(T) to give k f ( T) k β exp RT 0 T(fold) ( T ) (13) k 0 ut ( ) k exp β RT T(unfold) ( T ) (14) Eqs. (11) (14) at once deonstrate that the relationship between the rate constants, the equilibriu stability, and the denaturant value is incredibly coplex since the paraeters in the said equations can all change depending on the nature of the perturbation and will be explored in detail elsewhere. The force constants are inversely proportional to the variances of the Gaussian distribution of the conforers If G Di(T) and G Nj(T) (i, j = 1..n) denote the Gibbs energies of the conforers in the DSE and the NSE, respectively, then the probability distribution of their conforers along the RC at equilibriu is given by the Boltzann law. Because by postulate the Gibbs energies of the conforers in the DSE and the NSE have a square-law dependence on the RC, r, whose enseble-averaged length is given by D-N, and because the RC increases linearly fro 0 D-N in the left-to-right direction, we can write Page 10 of 65

11 p D( it) 1 GD( it) 1 r ( T) exp exp QDSE( T) RT Q DSE( T) RT (15) p NjT 1 G NjT 1 D-N r( T) G D-N( T) = exp = exp Q NSE( T) RT Q NSE( T) RT (16) where p Di(T) and p Nj(T) denote the Boltzann probabilities of the conforers in the DSE and the NSE, respectively, with their corresponding partition functions Q DSE(T) and Q NSE(T) being given by Q DSE( T ) n G r ( ) πrt i1 RT RT DiT T exp exp dr (17) Q NSE( T ) n G NjT GD-N( T) D-N r ( T) exp exp dr j1 RT RT πrt G RT D-N( T ) exp (18) Because the equilibriu is dynaic, there is always a constant theral noise-driven flux of the conforers fro the DSE to the NSE, and fro the NSE to the DSE, via the TSE. Consequently, there is always a constant albeit incredibly sall population of conforers in the TSE at equilibriu. Now consider the first-half of a protein folding reaction as shown in Schee 1, where [D], [TS] and [N] denote the equilibriu concentrations of the DSE, the TSE, and the NSE, respectively, in olar. K (Reaction Schee 1) TS-D( ) [ D] T [ TS] [ N] Fro the perspective of a folding reaction, the conforers in the activated state or the TSE ay be thought of as a subset of denatured conforers with very high Gibbs energies. Therefore, we ay assue that the conforers in the TSE are in equilibriu with those conforers that are at the botto of the denatured Gibbs basin. If G D(T), G and G N(T) TS(T) denote the ean Gibbs energies of the DSE, NSE, and the TSE, respectively, then the ratio of the olar concentration of the conforers at the botto of the denatured Gibbs basin and those in the TSE is given by Page 11 of 65

12 RT [ D] [ D] RT [ TS] [ TS] GTS-D ( T ) TS-D( T ) ln GTS-D( T ) exp exp RT (19) Siilarly for the partial unfolding reaction (Reaction Schee ), the conforers in the TSE ay be thought of as a subset of native conforers with very high Gibbs energies. Therefore, we ay write K (Reaction Schee ) TS-N( ) [ N] T [ TS] [ D] RT [ N] [ N] RT [ TS] [ TS] GTS-N ( T ) TS-N( T ) ln GTS-N( T ) exp exp RT (0) Because the SASA of the conforers in the DSE or the NSE is deterined by a ultitude of intra-protein and protein-solvent interactions, we ay invoke the central liit theore and assue that the distribution of the SASA of the conforers is a Gaussian. If σ and DSE(T) σ NSE(T) denote the variances of the DSE and the NSE-Gaussian probability density functions (Gaussian-PDFs), respectively, along the SASA-RC which in our case is its proxy, the experientally easurable and teperature-invariant D-N, and x DSE, x NSE, and xtse( T ) denote the ean SASAs of the DSE, the NSE, and the TSE, respectively, then the ratio of the olar concentration of the conforers whose SASA is identical to the ean SASA of the DSE to those whose SASA is identical to the ean SASA of the TSE is given by πσdse( T ) D xtse( T) xdse TS-D( T) exp exp σ DSE( T ) DSE( T) σdse( T) [ TS] [ ] πσ (1) Siilarly, the ratio of the olar concentration of the conforers whose SASA is identical to the ean SASA of the TSE to those whose SASA is identical to the ean SASA of the NSE is given by πσnse( T ) N xtse( T) xn SE TS-N( T) exp exp σ NSE( T ) NSE( T) σnse( T) [ TS] [ ] πσ () Page 1 of 65

13 Because the ratio of the conforers in the TSE to those in the ground states ust be the sae whether we use a Gaussian approxiation or the Boltzann distribution (copare Eqs. (19) and (1), and Eqs. (0) and ()), we can write T T TS-D( ) TS-D( ) DSE( T ) RT DSE( T ) RT exp exp σ (3) σ T T TS-N( ) TS-N( ) exp RT exp σ NSE( T ) RT σ NSE( T ) (4) Thus, for any two state folder at constant teperature, pressure, and solvent conditions, the variance of the Gaussian distribution of the conforers in the DSE or the NSE along the D-N RC is inversely proportional to their respective force constants; and for a two-state syste with given force constants, the variance is directly proportional to the absolute teperature. Naturally, in the absence of theral energy (T = 0 K), all classical otion will cease and σ = DSE(T) σ = 0. The relationship between protein otion and function will be explored NSE(T) elsewhere. The area enclosed by the DSE and the NSE-Gaussians is given by π πrt DSE( T) exp πσdse( T) DSE( T) a I ax dx Q (5) π πrt GD-N( T ) INSE( T) expby dy πσnse( T) QNSE( T) exp (6) b RT where x x x, y y x i DSE, x i and y i denote the SASAs of the i th conforers in the i NSE DSE and the NSE, respectively, a 1σ, b 1σ DSE( T ) NSE( T ), and I DSE(T) and I NSE(T) denote the areas enclosed by the DSE and NSE-Gaussians, respectively, along the D-N RC. The reader will note that for a polypeptide of finite length, the axiu perissible SASA is deterined by the fully extended chain and the iniu by the excluded volue entropy. Thus, the use of the liits to + in Eqs. (17), (18), (5) and (6) is not physically justified. However, because the populations decrease exponentially as the conforers in both the DSE and the NSE are displaced fro their ean SASA, the difference in the agnitude of the partition functions calculated using actual liits versus and + will be Page 13 of 65

14 insignificant. Eqs. (3) and (4) allow k f(t) and k u(t) to be recast in ters of the variances of the DSE and the NSE-Gaussians k k 0 TS-D( T ) f ( T) k exp σdse( T ) 0 TS-N( T ) ut ( ) k exp σnse( T ) (7) (8) We will show elsewhere when we deal with non-arrhenius kinetics in protein folding in detail that although the variance of the DSE and the NSE-Gaussians increases linearly with absolute teperature, the curve-crossing and the Gibbs barrier heights for folding and unfolding are non-linear functions of their respective variances. Equations for equilibriu stability The relationship between the partition functions, the area enclosed by DSE and the NSE Gaussians, and the Gibbs energy of unfolding ay be readily obtained by dividing Eq. (18) by (17) G Q σ Q Q RT ln RT ln RT ln NSE( T) DSE( T) NSE( T) NSE( T) D-N( T ) α Q DSE( T) σnse( T) Q DSE( T) I NSE( T) (9) I I DSE( T ) NSE( T ) π π σ DSE( T ) σdse( T ) σnse( T ) σnse( T ) (30) α where σ DSE(T) and σ NSE(T) denote the standard deviations of the DSE and NSE-Gaussians, respectively, along the D-N RC. There are any other ways of recasting the equation for equilibriu stability (not shown), but the siplest and perhaps the ost useful for is σ DSE( T ) GD-N( T) T(unfold)( T) T(fold)( T) T(unfold)( T) T(fold)( T) σ NSE( T ) (31) Eq. (31) deonstrates that when pressure, teperature, and solvent conditions are constant, the equilibriu and kinetic behaviour of those proteins that fold spontaneously without the need for any accessory factors is deterined purely by three priary-sequence-dependent Page 14 of 65

15 variables which are: (i) the enseble-averaged ean and variance of the Gaussian distribution of the conforers in the DSE along SASA-reaction-coordinate; (ii) the ensebleaveraged ean and variance of the Gaussian distribution of the conforers in the NSE along the SASA-reaction-coordinate; and (iii) the position of the curve-crossing along the abscissa. A necessary consequence of Eq. (31) is that: (i) if for spontaneously-folding fixed two-state systes at constant pressure and solvent conditions ΔSASA D-N is positive and teperatureinvariant (i.e., D-N and C pd-n are teperature-invariant), and T(fold)(T) 0.5 when T = T S (the teperature at which stability is a axiu), 30 then it is ipossible for such systes to be stable at equilibriu (ΔG D-N(T) > 0) unless σ σ DSE( T) NSE( T) no atter what the teperature; (ii) if two related or unrelated two-state systes have identical pair of force constants, and if their ΔSASA D-N as well as the absolute position of the DSE and the NSE along the SASA-RC are also identical, then the protein which folds through a ore solvent-exposed TSE will be ore stable at equilibriu; and (iii) if D-N and C pd-n are teperature-invariant, a spontaneously-folding two-state syste at constant pressure and solvent conditions, irrespective of its priary sequence or 3-diensional structure, will be axially stable at equilibriu when its denatured conforers are displaced the least fro the ean of their enseble to reach the TSE along the SASA-RC (the principle of least displaceent). Because equilibriu stability is the greatest at T S, a logical extension is that TS-D(T) or β T(fold)(T) ust be a iniu, and TS-N(T) or β T(unfold)(T) a axiu at T S (Figure 3). A corollary is that the Gibbs activation barriers for folding and unfolding are a iniu and a axiu, respectively, when the difference in SASA between the DSE and the TSE is the least. Matheatical foralis for why the activation entropies for folding and unfolding ust both be zero at T S will be shown in the subsequent publication. The correspondence between Gibbs parabolas and Gaussian-PDFs for two well-studied twostate proteins: (i) CI; and (ii) the B doain of staphylococcal protein A (BdpA Y15W) are shown in Figures 6 and 7, respectively. The paraeters required to generate these figures are given in their legends. As entioned earlier, the logical condition that as long as ΔG D-N(T) > 0 and TS-D(T) > TS-N(T) then > is readily apparent fro Figures 6A and 7A. Because the Gaussian variances of the DSE and the NSE are inversely proportional to the force constants, > iplies σ < NSE(T) σ (Figures 6B and 7B). A detailed discussion of the theory DSE(T) underlying the procedure required to extract the values of the force constants fro the Page 15 of 65

16 chevrons and its inherent liitations is beyond the scope of this article since it involves a radical reinterpretation of the chevron. A brief description is given in ethods. On the teperature-invariance of ΔSASA D-N, D-N and ΔC pd-n One of the defining postulates of the parabolic hypothesis is that for a spontaneously-folding fixed two-state folder, as long as the priary sequence is not perturbed via utation, cheical or post-translational odification, and pressure and solvent conditions are constant, and the properties of the solvent are invariant with teperature, the enseble-averaged SASAs of the DSE and NSE, to a first approxiation, are teperature-invariant; consequently, the dependent variables D-N and ΔC pd-n will also be teperature-invariant. Consider the DSE of a two-state folder at equilibriu under folding conditions: Within the steric and energetic constraints iposed by intra-chain and chain-solvent interactions, the SASA of the denatured conforers will be norally distributed with a defined ean ( x DSE ) and variance (σ ). Now, if we raise the teperature of the syste by tiny aount δt DSE(T) such that the new teperature is T+δT, a tiny fraction of the conforers will be displaced fro the ean of the enseble, soe with SASA that is greater than the ean, and soe with SASA that is less than the ean; and the agnitude of this displaceent fro the enseble-ean will be deterined by the force constant. Consequently, there will be a tiny increase in the variance of the Gaussian distribution, and a new equilibriu will be established. Thus, as long as the integrity of the spring (i.e., the priary sequence) is not coproised, and pressure and solvent conditions are constant, the distribution itself will not be biased in any one particular way or another, i.e., the nuber of conforers that have becoe ore expanded than the ean of the enseble, on average, will be identical to the nuber of conforers that have becoe ore copact than the enseble-ean, leading to xdse being invariant with teperature. A siilar arguent ay be applied to the NSE leading to the conclusion that although its variance increases linearly with teperature, its ean SASA ( x NSE ) will be teperature-invariant. However, if the olecular forces that resist expansion and copaction of the conforers in the DSE are not equal or approxiately equal and change with teperature, then the assuption that the conforers in the DSE are confined to a haronic Gibbs energy-well would be flawed. What is iplied by this is the distribution of the conforers in the DSE along the SASA-RC is no longer a Gaussian, but instead a skewed Gaussian. For exaple, if the change in teperature causes a shift in the Page 16 of 65

17 balance of olecular forces such that it is relatively easier for the denatured conforer to expand rather than becoe copact, in a parabolic representation, the left ar of the DSEparabola will be shallow as copared to the right ar, and the Gaussian distribution will be negatively skewed, leading to a shift in the ean of the distribution to the left. In other words, xdse will increase, and assuing that xnse is teperature-invariant, will lead to an increase in ΔSASA D-N, and by extension, an increase in D-N. In contrast, if the change in teperature akes it easier for the denatured conforer to becoe copact rather than expand, then the right ar of the DSE-parabola will becoe shallow as copared to the left ar, and the Gaussian distribution will becoe positively skewed; consequently, x DSE will decrease leading to a decrease in ΔSASA D-N and D-N. Siilar arguents apply to the NSE. Thus, as long as the Gibbs energy-wells are haronic and their force constants are teperatureinvariant, ΔSASA D-N, D-N, and ΔC pd-n will be teperature-invariant. The approxiation that the ean length of the RC is invariant with teperature is supported by both theory and experient: (i) The R g of the DSE and the NSE (after 00 pico seconds of siulation) of the truncated CI generated fro all ato olecular dynaic siulations (MD siulations) varies little between K(see Table 1 and explanation in page 14 in Lazaridis and Karplus, 1999); 31 (ii) Studies on the theral expansion of native etyoglobin by Petsko and colleagues deonstrate that the increase in the SASA and the volue of the folded protein on heating fro K is not ore than 3%; 3 (iii) In cheical denaturation experients as a function of teperature, D-N is, in general, teperature-invariant within experiental error In addition, it is logically inconsistent to argue about possible teperature-induced changes in D-N when its counterpart, ΔC pd-n, is assued to be teperature-invariant in the analyses of theral denaturation data. 30 The widely accepted explanation for the large and positive ΔC pd-n of proteins is based on Kauzann s liquid-liquid transfer odel (LLTM) which likens the hydrophobic core of the native folded protein to a liquid alkane, and the greater heat capacity of the DSE as copared to the NSE is attributed priarily to the anoalously high heat capacity and low entropy of the clathrates or icroscopic icebergs of water that for around the exposed non-polar residues in the DSE (see Baldwin, 014, and references therein). 37,38 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 ust be proportional to the change in the non-polar Page 17 of 65

18 SASA that accopanies a reaction. Consequently, protein unfolding reactions which are accopanied by large changes in non-polar SASA, also lead to large and positive changes in the heat capacity. 39,40 Because the denaturant values are also directly proportional to the change in SASA that accopany protein (un)folding reactions, the expectation is that D-N and ΔC pd-n values ust also be proportional to each other: The greater the D-N value, the greater is the ΔC pd-n value and vice versa. 3,4 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 dissiilar priary sequences need not necessarily be the sae even under identical solvent conditions), 3,4 and because we do not yet have reliable theoretical or experiental ethods to accurately and precisely quantify the SASA of the DSEs of proteins under folding conditions (the values are odeldependent), 1, the data scatter in plots that show correlation between the experientally deterined D-N or ΔC pd-n values (which reflect the true ΔSASA D-N ) and the theoretical odel-dependent values of ΔSASA D-N can be significant (see Fig. 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 sall proteins even when the residual structure in the DSEs under folding conditions is taken into consideration, large proteins on average expose relatively greater aount of non-polar SASA upon unfolding than do sall proteins; consequently, both D-N and ΔC 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 anoalously high or low nuber of non-polar residues). The point we are trying to ake is the following: Because the native structures of proteins are relatively insensitive to sall variations in ph and co-solvents, and since the nuber of ways in which foldable polypeptides can be packed into their native structures is relatively liited (as inferred fro the liited nuber of protein folds, see SCOP: and CATH: databases), one ight find a reasonably good correlation between chain lengths and the SASAs of the NSEs for a large dataset of proteins of differing priary sequences under varying solvents (see Fig. 1 in Miller et al., 1987). 16,44 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 siulate the DSEs of proteins, and if and only if these theoretical ethods are sensitive to point utations, changes in ph, co-solvents, neutral crowding agents, Page 18 of 65

19 teperature and pressure, it is alost ipossible 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 D-N and ΔC pd-n will vary with SASA or chain length. Analyses of ΔC pd-n values for a large dataset of proteins show that they generally vary between 10-0 cal.ol -1.K -1.residue -1. 3,4 Now that we have suarised the inter-relationships between ΔSASA D-N, D-N, and ΔC pd-n, it is easy to see that when ΔSASA D-N is teperature-invariant, so too ust ΔC pd-n, i.e., the absolute heat capacities of the DSE and the NSE ay vary with teperature, but their difference, to a first approxiation, can be assued to be teperature-invariant. The reasons for this approxiation are as follows: (i) the variation in ΔC pd-n(t) over a substantial teperature range is coparable to experiental noise; 39 and (ii) the variation in equilibriu stability that stes fro sall variation in ΔC pd-n(t) is once again coparable to experiental noise. 30 Consequently, the use of odified Gibbs-Helholtz relationships with a teperature-invariant ΔC pd-n ter is a coon practice in the field of protein folding, and is used to ascertain the teperature-dependence of the enthalpies, the entropies, and the Gibbs energies of unfolding/folding at equilibriu (Eqs. (3) (34)). However, what is not justified is the use of experientally deterined ΔC pd-n of the wild type/reference protein for all its utants for the purpose of calculating the change in enthalpies, entropies and the Gibbs energies of unfolding upon utation (i.e., ΔΔH D-N(wt-ut)(T), ΔΔS D-N(wt-ut)(T) and ΔΔG D- ; the subscripts wt and ut denote the wild type and the utant protein, N(wt-ut)(T) respectively). This is especially true if the D-N values of the utants are significantly different fro that of the wild type, since those utants with increased D-N values will be expected to have increased ΔC pd-n values, and vice versa, for identical solvent conditions and pressure, as copared to the wild type or the reference protein. These considerations are iplicit in the Schellan approxiation: GD-N(wt-ut)( T ) HD-N(wt)( T ) T(wt-ut) T(wt) (see Fig. 8 in Becktel and Schellan, 1987, and discussion therein). T D-N( T) D-N( T ) + D-N( ) D-N( ) + T p T T pd-n H H C dt H C T T (3) Page 19 of 65

20 T C pd-n( T) T SD-N( T) SD-N( T ) + D-N( ) + D-N ln dt S T T C p T T H D-N( T ) T + C pd-n ln T T (33) T T GD-N( T) HD-N( T ) 1 + C D-N + D-N ln p T T TCp T T (34) where ΔH D-N(T), ΔH and ΔS, ΔS D-N(T denote the equilibriu enthalpies and the ) D-N(T) D-N(T ) entropies of unfolding, respectively, at any given teperature, and at the idpoint of theral denaturation (T ), respectively, for a given two-state folder under defined solvent conditions. The teperature-invariant and the teperature-dependent difference in heat capacity between the DSE and NSE is denoted by ΔC pd-n and ΔC pd-n(t), respectively. TESTS OF HYPOTHESIS A logical way of testing hypotheses in epirical sciences is to ake quantitative predictions and verify the via experient. 45 The greater the nuber of predictions, and the ore risky they are, the ore testable is the hypothesis and vice versa; and the greater is the agreeent between theoretical prediction and experient in such tests of hypothesis, the ore certain are we of its veracity. Naturally, any hypothesis that insulates itself fro falsifiability, or refutability, or testability, is either pseudoscience or pathological science. 45,46 The theory described here readily lends itself to falsifiability because it akes certain quantitative predictions which can be iediately verified via experient. The variation in TS-D(T) and TS-N(T) with D-N A general observation in two-state protein folding is that whenever utations or a change in solvent conditions cause statistically significant changes in the D-N value, a large fraction or alost all of this change is anifest as a variation in TS-D(T), with little or alost no change in TS-N(T) (see Figs. 7 and 9 in Sanchez and Kiefhaber, 003). 19 Although these effects were analysed using self-interaction and cross-interaction paraeters, 19 the question is Why ust perturbation-induced changes in D-N predoinantly anifest as changes in TS-D(T)? Is there any theoretical basis for this epirical observation? Iportantly, can we predict how TS-D(T) varies as a function of D-N for any given two-state folder of a given equilibriu Page 0 of 65

21 stability when teperature, pressure and solvent are constant? To siulate the behaviour two hypothetical two-state systes one with force constants =1 and = 10 M.ol.kcal -1 (Figure 8A), and the other with = 1 and = 100 M.ol.kcal -1 (Figure 8B) were chosen. Within each one of these pair of parent two-state systes are six sub-systes with the sae pair of force constants as the parent syste but with a unique and constant G D-N(T). We now ask how the curve-crossings for each of these systes change when the separation between the vertices of DSE and NSE-parabolas along their abscissae are allowed to vary (i.e., a change in D-N as in Figure 4). Siply put, what we are doing is taking a pair of intersecting parabolas of differing curvature such that > and systeatically varying the separation between their vertices along the abscissa and ordinate, and calculating the position of the curve-crossing along the abscissa for each case according to Eqs. (9) and (10). Despite the odel being very siplistic (because the curvature of the parabolas can change with structural or solvent perturbation), the siulated behaviour is strikingly siilar to that of 1064 proteins fro 31 two-state systes: A perturbation-induced change in D-N is predoinantly anifest as a change in TS-D(T) with little or no change in TS-N(T) (Figure 9 and Figure 9 figure suppleent 1). Although the apparent position of the TSE along the RC as easured by T(fold)(T) changes, the absolute position of the TSE along the RC ay not change significantly, and this effect can be particularly pronounced for systes with high T(fold)(T) or late TSEs (Figure 8B). This ability to siulate the behaviour of real systes serves as the first test of the hypothesis. Non-Arrhenius kinetics Unlike the teperature-dependence of the rate coefficients of ost cheical reactions of sall olecules, protein folding reactions are characterised by non-arrhenius kinetics, i.e., at constant pressure and solvent conditions, k f(t) initially increases with an increase in teperature and reaches a plateau; and any further increase in teperature beyond this point causes k f(t) to decrease. This anoalous non-linear teperature-dependence of k f(t) has been observed in both experient and coputer siulations. 17,36,47-56 Two predoinant explanations have been given for this behaviour: (i) non-linear teperature-dependence of the prefactor on rugged energy landscapes; 17 and (ii) the heat capacities of activation, ΔC pd- and ΔC, which in turn lead to teperature-dependent enthalpies and entropies of TS(T) pts-n(t) activation for folding and unfolding. 36,48,50,51 Arguably one of the ost iportant and Page 1 of 65

22 experientally verifiable predictions of the parabolic hypothesis is that as long as the enthalpies and the entropies of unfolding/folding at equilibriu display a large variation with teperature, and as a consequence, equilibriu stability is a non-linear function of teperature, both k f(t) and k u(t) will have a non-linear dependence on teperature. The equations that describe the teperature-dependence of k f(t) and k u(t) of two-state systes under constant pressure and solvent conditions ay be readily derived by substituting Eq. (34) in (11) and (1). T T D-N HD-N( T ) 1 CpD-N TTTCpD-N ln T T 0 RT kf ( T) k e (35) T T HD-N( T ) 1 CpD-N TTTCpD-N ln D-N T T 0 RT kut ( ) k e (36) Thus, if the paraeters ΔH, T, ΔC, D-N(T, the force constants and, and k0 ) pd-n D-N (assued to be teperature-invariant) are known for any given two-state syste, the teperature-dependence of k f(t) and k u(t) ay be readily ascertained. Why does the prediction of non-arrhenius kinetics constitute a rigorous test of the parabolic hypothesis (see confiring evidence, Popper, 1953)? As is readily apparent, the values of the constants and variables in Eqs. (35) and (36) coe fro two different sources: While the values of and, k 0, and D-N are extracted fro the chevron, i.e., fro the variation in k f(t) and k u(t) with denaturant at constant teperature, pressure and solvent conditions (i.e., all solvent variables excluding the denaturant are constant), ΔH and T D-N(T are deterined fro theral ) denaturation at constant pressure and identical buffer conditions as above but without the denaturant, using either calorietry or van t Hoff analysis of a sigoidal theral denaturation curve, obtained for instance by onitoring the change in a suitable spectroscopic signal with teperature (typically CD 17 n for β-sheet proteins, CD n for α-helical proteins or CD 80 n to onitor tertiary structure). 30,57 The final paraeter, ΔC pd-n, is once again deterined independently (i.e., the slope of a plot of odelindependent calorietric ΔH versus T D-N(T, see Fig. 4 in Privalov, 1989). What this )(cal) essentially iplies is that if Eqs. (35) and (36) predict a non-linear teperature-dependence of k f(t) and k u(t), and iportantly, if their absolute values agree reasonably well with Page of 65

23 experiental data, then the success of such a prediction cannot be fortuitous since it is statistically iprobable for these paraeters obtained fro fundaentally different kinds of experients to collude and yield the right values. We are then left with the alternative that at least to a first approxiation, the hypothesis is valid. The predictions of Eqs. (35) and (36) are shown for three well-studied two-state folders: (i) BdpA Y15W, the 60-residue three-helix B doain of Staphylococcal protein A (Figure 10 and its figure suppleents); 58 (ii) BBL H14W, the 47-residue all-helical eber of the Peripheral-subunit-binding-doain faily (Figure 11 and its figure suppleents); 59 and (iii) FBP8 WW, the 37-residue Forin-binding three-stranded β-sheet WW doain (Figure 1 and its figure suppleents). 60 Inspection of Figures 10A, 11A and 1A (see also Figure suppleent A for each of these figures) shows that Eq. (35) akes a rearkable prediction that k f(t) has a non-linear dependence on teperature. Starting fro a low teperature, k f(t) initially increases with an increase in teperature and reaches a axial value at T = T H(TS-D) where ln k T = H RT 0 H 0 ; and any further increase in f ( T) TS-D( T) TS-D( T) teperature beyond this point will cause a decrease in k f(t). The reader will note that the partial derivatives are purely to indicate that these relationships hold if and only if the pressure, solvent variables and the prefactor are constant. In contrast, inspection of Figures 10B, 11B and 1B (see also Figure suppleent B for each of these figures) shows that k u(t) starting fro a low teperature, decreases with a rise in teperature and reaches a iniu at T = T H(TS-N) where ln k T = H RT 0 H 0 ; and any further increase in teperature ut ( ) TS-N( T) TS-N( T) beyond this point will cause an increase in k u(t). This behaviour which is dictated by Eq. (36) at once provides an explanation for the origin of a isconception: It is soeties stated that non-arrhenius kinetics in protein folding is liited to k f(t) while k u(t) usually follows Arrhenius-like kinetics. 36,49,53,61 It is readily apparent fro these figures that if the experiental range of teperature over which the variation in k u(t) investigated is sall, Arrhenius plots can appear to be linear (see Fig. 5A in Tan et al., 1996, Fig. 3 in Schindler and Schid, 1996, and Fig. 6c in Jacob et al., 1999). 49,53,6 In fact, even if the teperature range is substantial, but owing to technical difficulties associated with easuring the unfolding rate constants below the freezing point of water, the range is restricted to teperatures above K, k u(t) can still appear to be have a linear dependence on Page 3 of 65

24 teperature in an Arrhenius plot since the curvature of the libs in Figures 10B, 11B and 1B is rather sall. This can especially be the case if the nuber of experiental data points that define the Arrhenius plot is sparse. Consequently, the teperature-dependence of k u(t) can be fit equally well within statistical error to a linear function, and is apparent fro inspection of the teperature-dependence of k u(t) of CI protein (see Fig. 4 in Tan et al., 1996). 6 Because T H(TS-N) << K for psychrophilic and esophilic proteins, it is technically deanding, if not ipossible, to experientally deonstrate the increase in k u(t) for T < T H(TS-N) for the sae. Nevertheless, the levelling-off of k u(t) at lower teperatures (see Fig. 3 in Schindler and Schid, 1996), and extrapolation of data using non-linear fits (see Fig. 6B in Main et al., 1999) indicates this trend. 49,63 In principle, it ay be possible to experientally deonstrate this behaviour for those proteins whose T H(TS-N) is significantly above the freezing point of water. It is interesting to note that lattice odels consisting of hydrophobic and polar residues (HP+ odel) also capture this behaviour (see Fig. B in Chan and Dill, 1998). 50 As entioned earlier, the cause of non-arrhenius behaviour is a atter of soe debate. However, because we have assued a teperature-invariant prefactor and yet find that the kinetics are non-arrhenius, it essentially iplies that one does not need to invoke a super-arrhenius teperature-dependence of the configurational diffusion constant to explain the non-arrhenius behaviour of proteins. 17,36,48,50,55 Once the teperature-dependence of k f(t) and k u(t) across a wide teperature range is known, the variation in the observed or the relaxation rate constant (k obs(t) ) with teperature ay be readily ascertained using (see Appendix) 64 ln k 0 D-N φ φ 0 D-N obs( T ) ln k exp k exp RT RT (37) Inspection of Figure suppleent-1 of Figures 10, 11 and 1 deonstrates that ln(k obs(t) ) vs teperature is a sooth W-shaped curve, with k obs(t) being doinated by k f(t) around T H(TS-, and by k for T < T and T > T, which is precisely why the kinks in ln(k ) occur N) u(t) c obs(t) around these teperatures. It is easy to see that at T c or T, k f(t) = k u(t) k obs(t) = k f(t) = k u(t) and GD-N( T) RT k f ( T) ku( T) ln 0. In other words, for a two-state syste, T c and T easured at equilibriu ust be identical to the teperatures at which k f(t) and k u(t) intersect. Page 4 of 65