Abstract We re-examine a site-binding approach independently proposed by Schellman (Schellman, J.A. (1958) Compt. rend. Lab. Carlsberg Ser. Chim. 30, 439-449) and Aune and Tanford (Aune, K.C. and Tanford, D. (1969) Biochemistry, 8, Abstract -4590) for explicitly including the denaturant concentration within the protein unfolding equilibrium. We extend and formalize the approach through development of a multidimensional analytical model in which the folding reaction coordinate is defined by the number of denaturant molecules bound to sites located on either the initially folded, or unfolded, states of the protein. We use the developed method to reexamine the mechanistic determinants underlying the sigmoidal shape of the unfolding transition. A natural feature of our method is that it presents a landscape picture of the denaturant induced protein unfolding reaction. Characterizing the stability of a protein in response to an unfolding stress is important in a variety of diverse fields ranging from manufacturing of enzymes used in detergent [1] , preventing spoilage in food processing [2] , optimizing storage conditions for protein pharmaceuticals [3] and understanding the causes of protein misfolding diseases [4] . In general, proteins can be denatured by either changing the temperature (5) (6) (7) (8) (9) , the pressure (9) (10) (11) (12) or by addition of increasing concentrations of chemical denaturants (9, 13) such as urea or guanidine hydrochloride (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) , various alcohols (25) (26) (27) and hydrogen/hydroxide ions (pH induced unfolding (28) (29) (30) ). Whilst the thermodynamics of thermally induced unfolding is relatively conceptually straightforward (6-9, 31-35), denaturant based unfolding is somewhat more difficult to describe due to the diversity of denaturants and their different modes of action (9, 17, 18, (35) (36) (37) (38) (39) (40) . A key juncture in the history of denaturant induced protein unfolding was the quantitative functionalization of the equilibrium constant, K FU , in terms of the denaturant concentration, C D i.e. K FU {C D }. At an early stage in the development of the field, two alternative conceptual treatments for achieving this analytical functionalization competed for popular usage [9, 17, 36] . The m-value approach by Greene and Pace [20] , which eventually achieved primacy, implicitly included the effects of denaturant within K FU via a linear perturbation treatment of the free energy governing the unfolding equilibrium [20, 41] (see Appendix 1) . A different approach that explicitly included the denaturant as a species in the unfolding reaction, was developed independently by Aune and Tanford [15, 28] and Schellman [16, 36] 1 . Although the explicit approach has faded in popularity, it potentially possesses a more straightforward linkage to mechanistic reality than the m-value approach due to its formulation in terms of measurable species [15, 16, 28, 36, 43, 44] . In the next section we first present the original development, closely following Aune and Tanford's approach. We then extend their formalism through development of a multidimensional analytical model in which the unfolding reaction coordinate is defined in terms of the number of denaturant molecules bound to sites located on the initially folded or unfolded state. This model is then used to provide a fresh look at some basic aspects of the protein unfolding transition. Approach of Schellman, Aune and Tanford Both Aune and Tanford [15, 28] and Schellman [16, 36] considered folded, F, and unfolded, U, forms of the protein as respectively possessing N FD and N UD specific independent and equivalent sites to which denaturant could bind, with respective site binding constants of k FD and k UD (Eqn. 1). All possible denaturant bound forms of folded and unfolded protein were accounted for via binding polynomials calculated using the binomial theorem (Eqn. 3a) [15, 41] . This description was further simplified (to Eqn. 3b) by identification of the equilibrium relation C U = K F → U {0}C F and invocation of the assumption that the site binding constant operating between the denaturant, the folded and unfolded forms of protein, was equal to a common value k PD [4b] Approach of Schellman, Aune and Tanford Taking account of the assumptions inherent within its derivation, Eqn. 4b provides a thermodynamically rigorous method for interrogating the dependence of the fraction of unfolded protein as a function of the denaturant concentration. We note that Eqn. 4b is based on just three 2 parameters, K F → U {0}, k PD and ∆N. Analog of the Aune-Tanford-Schellman Approach Our re-visitation of the Aune and Tanford [15] amd Schellman [16] approach begins with a more formal description of the folding/unfolding transition (Fig. 1) . A specific number, N FD , of denaturant binding sites are considered to exist on the exterior of the folded protein, their interaction with denaturant regulated by a site binding constant, k 1F . The same number of N FD sites also exist on the unfolded protein but are considered able to bind denaturant with a potentially different site binding constant, k 1U (Fig. 1b) . Previously hidden regions of the protein interior are accessible to solvent upon protein unfolding, with these newly exposed regions presenting a specific number, N UD , of binding sites for denaturant. Each interior site is considered to possess a common denaturant site binding constant, k 2U . This basic description can be more concisely expressed in two-dimensional matrix form ( Fig. 1b ) with each row /column entry defined by either an F or a U, with a two index subscript coordinate indicating the number of denaturant molecules bound to the initially folded or initially unfolded sites. In such an indexing scheme, F i,0 represents a folded species with i denaturant molecules bound to the folded denaturant sites, whereas U i,j describes an unfolded species with i and j denaturant molecules respectively bound to the initially folded and unfolded sites. The final class of parameters described within the model are the denaturant loading dependent unfolding constants (k FU ) i,0 which describe the propensity of an F i,0 folded species to transition to its U i,0 state. As will be subsequently demonstrated, we need only specify the value for the unfolding constant at zero denaturant concentration, (k FU ) 0,0 , in order to calculate all other (k FU ) i,0 on the basis of known k 1F , and k 1U . Analog of the Aune-Tanford-Schellman Approach Combinatorial determination of the number of ways, W(X,Y), of arranging Y objects within X available positions (Eqn. 5a) [45, 46] M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 6(7, 9) M A N U S C R I P T For a given value of C D , equation set 6 can be solved iteratively through incremental changes in C F0,0 , for a fixed value of the total protein concentration (C P ) TOT , until an acceptable minimum of Eqn. 7 is achieved. AR[7] With each species so defined, we are able to calculate the effective value of the unfolding equilibrium constant as a function of denaturant loading according to Eqn. 8a. Use of Eqn. 8a allows for identification of the dominant equilibrium pathway of unfolding via Eqn. 8b. 4 The interested reader may have noticed that we have not specified the unfolding equilibrium constants $,# → $,# for the case of i >0. We will address this point in the next section. 5 This latter term can nearly always be substituted for the total concentration of denaturant within the limits of the approximation, [NFD+NUD](CP)TOT << (CD)TOT. Equations 5-8 represent the foundation of our multi-dimensional reformulation of the Aune and Tanford/ Schellman model with explicit inclusion of the denaturant species within the unfolding equilibrium reaction. The difference between the original model (Eqn. 4) and our multi-dimensional analog (Eqns. 5-8) can be best appreciated by examining the types of output afforded by each approach (Fig. 2 ). Fig. 2a shows the traditional description of a denaturant induced unfolding transition in terms of the fraction unfolded, f U , versus the denaturant concentration, C D , for both the Aune and Tanford/ Schellman model equation (black line -Eqn. 4) and our modern multidimensional analog (red circles -Eqn. 5-8). Fig. 2b describes the additional data afforded by the multidimensional model in terms of logarithmic occupation density, log 10 (C Pi,j ) where each P species (either F or U) is defined in terms of the denaturant loading of the folded or unfolded denaturant sites, P i,j . Red dots represent the apex of each distribution for the given denaturant concentrations shown by the red circles in Fig. 2a . Fig. 2c represents the same information as Fig. 2b shown in a multi-dimensional heat map formatthe form we employ in the presentation of results from hereon. Fig. 2d shows, in isolation, the distribution of protein states corresponding to a single denaturant concentration, the C D equaling 8M point (labeled in Fig. 2A using a blue ring). AR[7] Having introduced the characteristic outputs of the model we now use it to revisit the determinants of perhaps the most basic feature of the two-state protein folding reaction -its apparent cooperativity manifested as a sigmoidal form of the transition. Results and Discussion The multi-dimensional reformulated Aune and Tanford/ Schellman model has six parameters K FU , k 1F , k 1U , k 2U , N FD and N UD . For cases involving proteins of known structure (or structural class) we can reduce the parameter space from six to five 6 by defining N FD in terms of the folded structure and N UD , on the basis of the number of amino acids in M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT the protein, N AA (the basis of this assumption is derived in Appendix 2) . Of this reduced model space we explore changes in each of the variable parameters in turn, examining five cases in all. For each case we compare the base simulation presented in Fig. 2 against two alternatives simulated using either a larger or smaller value of the parameter under examination. For the unit-less intrinsic stability partition constant, (k FU ) 0,0 , we simulate change over a million-fold range. For N AA we examine a 2.25 fold range, whereas for the three association constants, k 1F , k 1U and k 2U (having units of M −1 ) we assess the effect of a 25 fold range of values. To begin, we first study the effect of changes in the value of the intrinsic protein stability, (k FU ) 0,0 , i.e., the equilibrium unfolding constant operating at zero denaturant concentration. Case 1: Change in (k FU ) 0,0 The blue, red and green lines shown in Fig. 3A span a million-fold increase in the intrinsic stability as manifested within the (k FU ) 0,0 parameter (with lower values of (k FU ) 0,0 reflecting greater stability). Changes in (k FU ) 0,0 tend to displace the transition curve rather than modify its general slope characteristics (apparent cooperativity). From the trajectory plots shown in Figs. 3B-D we note that although the final state constitution (f U ~ 1.0) of the three simulations featuring different intrinsic stability is much the same, the early stages of unfolding, i.e., f U = 0.1, are significantly different in the sense that these unfolded states exhibit different numbers of bound denaturant molecules. This finding can be rationalized by the fact that the drive to populate denaturant sites on the protein is regulated via mass action principles such that lower concentrations of denaturant will lead to lower occupancy, with the converse case true for higher concentrations of denaturant. Lower intrinsic stabilities (higher values of (k FU ) 0,0 ) facilitate exposure of the U state denaturant sites to smaller concentrations of denaturant, thereby allowing U states with less than 10 bound denaturant molecules to exist. Figure 4 shows the effect of either a 1.5 fold decrease, or a 1.5 fold increase, in N AA (the number of amino acids in the protein chain) on the unfolding transition of a globular protein. Fig. 4A shows the change in stability with the change in polypeptide length over the range N AA ∈ [100, 150, 225]. The total number of denaturant binding sites, which change in the manner prescribed by Eqns. A2b and d, is listed in the figure legend. What seems initially to be a paradoxical result i.e., increasing N AA results in the destabilization of the protein, can be explained on the basis of the following two points relating to factors which either promote or retard protein unfolding, (i) In the model, the driving forces promoting unfolding with increasing N AA , derive jointly from both the opportunity for lowering the system potential energy (gained as a consequence of forming a greater number of productive bonds between the denaturant and the internal binding sites) and the gain in system entropy (obtained from the greater number of possible ways of distributing bound denaturant molecules within a larger number of available sites). For the globular geometry assumed in the present case (see Appendix 2) the number of internal denaturant sites is approximately linearly dependent on N AA , whereas the number of external solvent exposed sites shows a (N AA ) 2/3 dependence (c.f. Eqn. A2a and Eqn. A2c). As such both of these two 'forces' driving unfolding will, for globular proteins at least, increase with longer polypeptide chain length. Case 2: Change in N AA (ii) In the model, the features helping to prevent protein unfolding stem jointly from favorable denaturant interaction with the initially solvent exposed binding sites, i.e. (k 1F , N FD ) > 0, and the intrinsic stability of the protein at zero denaturant concentration, i.e. (k FU ) 0,0 < 1. As discussed in point (i) above, the relationship between N FD and N UD (determined by the compact shape adopted by globular proteins (N UD > N FD )) will produce a situation whereby increasing denaturant concentration ultimately promotes unfolding 7 . With regard to the effect of changes in N AA on intrinsic protein stability, we note that the intrinsic stability is generally thought to reflect a balance between a combination of differential intra-and inter-molecular interaction potential energies 7 Interestingly, in a manner that is dependent upon the relative values of NFD, NUD, k1F, k1U and k2U, a case exists such that small concentrations of denaturant, below the TD, will stabilize the protein against other types of denaturing pressure, such as temperature induced unfolding. Such denaturant induced stabilization behavior has been observed experimentally by Goto et al. (47, 48) and others (49, 50) . As the relationship between NFD and NUD is also determined by protein shape there is also significant potential for antithetical denaturant action for proteins featuring relatively exotic, non-globular structure (51, 52) . We will revisit this situation in a future publication. M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT between the folded and unfolded states (i.e. solvent-solvent, solvent-protein and amino acidamino acid within the polypeptide chain), along with differential entropies reflecting the degeneracy/multitude of states for which a particular potential energy state can exist (i.e. number of states reflecting equivalent modes of solvent binding to the folded and unfolded states of the protein along with the number of iso-energetic configurational states in which the folded and unfolded protein states can separately exist -53). An equation for intrinsic stability as a function of amino acid number has been previously given by Dill (35, 53) and is included here in slightly modified form as Eqn. 9 (note this is discussed in more detail in Appendix 3). M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT Here δµ AA,tr (C D ) is the average change in free energy (per mole of amino acid) upon partition of the amino acid from a hydrophobic phase (reflective of the interior of a protein) to an aqueous solution phase (containing a particular concentration of denaturant) and Z is the degeneracy of configurational states for a polypeptide given h different allowable orientations per amino acid φ and ψ angle in the polypeptide chain. Z is written as an inequality as it must be less than the equivalent value for a freely rotating chain composed of N AA monomers (35, 53) 8 . Dill has suggested that increasing N AA over the size range of 50 to 300 amino acids tends towards greater intrinsic stability (smaller (k FU ) 0,0 ) (53) 9 . M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT In order to isolate the effects of particular parameters, in the present simulation, only the size has been altered, with no inclusion of a compensatory change in the intrinsic stability parameter, (k FU ) 0,0 . As such the observation of destabilization accompanying increases in N AA , can be taken as providing a window into the extent of factors promoting protein destabilization only, whereas in the majority of experimental cases, increases in N AA will tend to stabilize proteins via 8 A lower estimate of Z ≈ h (2NAA−1) .exp[−(2NAA−1)] has been suggested by Dill (53) for the representation of a polypeptide chain by an excluded volume chain with h energetic minima per bond rotation. 9 Up to a certain size limit of ~300 residues at which point larger proteins tend to form multiple domains which then assemble as intact units via a different governing equation. M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT concomitant changes in the (k FU ) 0,0 parameter. In this light Figs. 4B-D show that the greater occupation of unfolded (but not folded) denaturant binding sites drives an apparent decrease in stability with increasing protein size. Case 3: Change in k 1F In Fig. 5A we note a rather modest change in protein stability [as judged by change in the (C D ) MP ], along with the negligible change to the apparent cooperativity (as judged by change in the steepness of the transition) when k 1F , the site binding constant for denaturant binding to solvent accessible regions in the folded state, is changed over a 25 fold range, in isolation from the other variable parameters. As is directly reconcilable from mass action principles, increases in k 1F tend to drag the equilibrium towards the folded state, thereby leading to denaturant induced stabilization (47-50 -see also Footnote 4) . In the present case, due to the greater number of binding sites in the unfolded state and the much greater multiplicity of states in which these U site denaturant molecules may be complexed, the protein still unfolds at some limiting denaturant concentration. This however, need not necessarily always be true and could be taken as marking the boundary of transition from destabilizing denaturant to stabilizing ligand (43, 47-50). One interesting point not seen in the other case examples is that as k 1U becomes larger we see a change in the F → U transition pathway ( Fig. 6B -D) such that the transition proceeds via a more denaturant loaded F state. As for the initial case example considering changes in intrinsic stability, the unfolded denaturant sites of proteins with decreased stability (corresponding to lower k 1F values) exhibit lower extents of denaturant occupancy at fractional extents of unfolding (~f U < 0.1). This behavior makes the unfolding path trajectories significantly different, from a chemical species perspective, for the different cases of k 1F (Figs. 5B-D). Case 4: Changes in k 1U A 25-fold change in k 1U , the association equilibrium constant governing binding of denaturant to the folded sites in the unfolded U state, produces significant changes in both the stability and the apparent cooperativity ( Fig. 6A ). It also produces an interesting effect not seen for the other cases in that increases in k 1U dramatically change both the extent of denaturant binding as M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT well as the direction of the unfolding trajectory (c.f. Fig. 6B and D). As the k 1U parameter determines the strength of the interaction between denaturant and the range of N FD sites in the unfolded state of the protein (regions of the protein that were initially exposed to solvent in the folded state and continue to be exposed to solvent in the unfolded state). For a globular protein the number of N FD sites should be much less than the number of N UD sites (i.e. N FD < N UD ). The system should therefore be less sensitive to the k 1U parameter for an identical increase in the numerical value of k 1U and k 2U and indeed this is what is seen (c.f. green line in Fig. 6A vs. Fig. 7A ). Case 5: Changes in k 2U The blue, red and green lines in Although this point will be further addressed in the discussion, the notional concentration independence of the protein unfolding reaction (on the protein concentration itself) potentially opens the door for examining unfolded state separate from the folded state by simply increasing (C P ) TOT . If the well-separated nature of the unfolded and folded state ensembles shown in Fig. 7B can be captured via use of an appropriate experimental technique, then the relationship between the protein and denaturant concentrations may prove to be a more powerful tool for tracking the progress of the unfolding reaction than previously thought. Conclusions Summary of the model In this paper we have extended an existing model of protein unfolding that explicitly includes denaturant in the unfolding reaction [15, 16, 43] . Our extension is based on consideration of each denaturant laden unfolded species as separate and countable -allowing their determination using a constraint based iterative solution method. The model produces a non-typical output stream that describes the unfolding reaction coordinate in terms of the density of protein states with denaturant bound to the initially solvent exposed and initially solvent hidden regions. Due to the additional dimensionality, these 'new look' unfolding trajectory plots contain more information than their f U vs C D forebears (c.f. Fig. 2A vs. 2B), potentially offering greater insight into the unfolding reaction. Amongst these improvements are the following; Summary of the model (i) A more realistic accounting of the properties displayed by internal and surface located amino acid types by allowing for denaturant k 1F sites to be described by a k 1U binding constant in the unfolded state -where both k 1F , k 1U and k 2U may take on unique values. Summary of the model (ii) Ability to identify the denaturant-laden folded state from which the unfolding trajectory is most likely to take place (see Eqn. 8b) due to absolute evaluation of the number of denaturant molecules bound rather than simply the difference (as prescribed by Eqn. 4. In the following sections we compare our model against other explicit approaches and describe possible means for its experimental testing and extension. Other analytical denaturant site-binding models Our approach belongs to a class of mechanistic based models that feature independent and groupwise equivalent denaturant binding sites existing on the folded and unfolded states of the protein [15, 16, 43, 46] . Before discussing the literature associated with this class of models it is helpful to discuss what we actually mean when invoking the term 'denaturant binding site'. With reference to the denaturants urea and guanidine hydrochloride, results of molecular dynamics simulations (54) (55) (56) (57) (58) (59) (60) (61) (62) suggest that the energetic basis of these 'sites' derive from a mixture of; Other analytical denaturant site-binding models (a) Specific direct interactions between the denaturant and protein driven by hydrogen bond formation between denaturant and backbone amide and side chain moieties (55, 57, 59, 61). Other analytical denaturant site-binding models (b) Non-specific direct interactions between the denaturant and protein driven by a combination of short-range attractive dispersive interactions (54, 57, 59, 60). Other analytical denaturant site-binding models (c) Non-specific indirect interactions between the denaturant and protein driven by disruption of water structure by the denaturant (55, 56, 62 vs. 58) 10 . Other analytical denaturant site-binding models Irrespective of the exact nature of the binding site their invocation is central to models seeking explicit inclusion of denaturant into the unfolding mechanism on the basis of proximal contact between the protein and the denaturant. Schellman [36] and Fersht [9] provide introductions to the quantitative description of denaturant induced protein unfolding in terms of site binding theory. Two general approaches have been adopted, models based on the existence of distinct binding sites displaying a positional energetic minimum (15, 16, 43) and models based on the concept of partition of denaturant into a protein surface phase (36, 44, 57, 60, 63-67). Other analytical denaturant site-binding models (i) Models based on energetic sites: Schellman and Aune and Tanford (15, 16) independently produced binding models of the general type shown by Eqns. 1-4. Aune and Tanford (16) extended this approach to include displacement 11 type reactions and denaturation based on pH titration in which the binding ligand was a proton. In a series of publications from the 70s to the late 90s Schellman refined the site binding model to account for multiple adsorbing ligands [36, 44] . Other analytical denaturant site-binding models Although not directed at studying denaturant induced unfolding Miller and Dill [43] applied ligand binding theory to their two dimensional HP system 12 to generate a model of denaturant unfolding based on a two state equilibrium (F ⇌U) 13 . Other analytical denaturant site-binding models (ii) Models based on partition and surface phase: A different approach for mechanistic inclusion of the denaturant concentration into the quantitative 10 Note that this may be extremely dependent upon solvent composition [see 58 vs. 62] 11 Referring to displacement of solvent molecules. 12 Here two-dimensional refers to where H = hydrophobic amino acid and P = polar amino acid). 13 We note that Dill and coworkers developed another class of models based on the non-mechanistic m-value formulation for description of both the denaturant and temperature dependence of the unfolding equilibrium constant [35] . Other analytical denaturant site-binding models formulation of denaturant unfolding based on the general concept of differential partition of solute between bulk and local surface-phase (immediately adjoint to the protein surface) has been variously advanced by the groups of Bolen [63] , Parsegian [64] , Winzor [65] , Minton [66] and Schellman [36, 44] and Matubayasi [67] . Differential partition of solute into this local surface phase can signify an attractive (K P > 1 more solute than expected) or a repulsive (K P < 1 less solute than expected) potential of mean force (67) The designations local and bulk respectively refer to the volume regions defined by the normal projection from the surface a distance approximately equal to the molecular distance of a denaturant molecule [60, 68-70] and the volume regions starting from this end of the local phase extending further out into solution (below the protein-protein free volume approach distance defined by the protein concentration. Schellman has advanced this approach, extending it to take into account various forms of non-ideality in the bulk solution phase [69] . It is interesting to note that this partition/surface excess approach is also amenable to implementation in our multidimensional modelling strategy. A further point for consideration is that all chemical site binding models should be based on chemical activity, the mass action analog of concentration, which includes any additional non-deal work associated with maintaining the listed concentration in the specified environment 14 . In the present work we have assumed ideal behavior of the denaturant but note that Aune and Tanford [15] did present an empirical activity correction based on vapor pressure data. In future work we will explore improved experimental and theoretical methods for making this activity correction. M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT Experimental testing Used in combination with a non-linear least squares regression routine [71] , the developed model may be used to fit denaturant induced protein unfolding curves and evaluate the variable parameters within the model. Experimental dependencies for each parameter could be constructed, upon such phenomenon as size, shape and internal secondary structure content [e.g. 18]. The model could be used to fit standard f U vs C D type transition curves (e.g. Fig. 2A ) or alternatively fit data of the type shown in Fig. 2B [24] . Experimental testing Interestingly, use of the model in an analytical capacity suggests a second line of experimental investigation based on a protein engineering /mutation approach [75] [76] [77] . Evaluation of a particular amino acid's contribution to one of the characteristic parameters (when located at a particular position) will provide mechanistic insight in regard to the mode of its effect upon stability [75, 76] thereby complementing modern techniques [77, 78] . For instance, with regard to the intrinsic stability parameter, (k FU ) 0,0 , a broad based alanine scan [75, 78] could identify key stabilizing amino acids, with their particular roles assigned through analysis of covariance with other parameters (k 1F , k 1U , k 2U , N FD or N UD ). It is also conceivable, that designed mutation could be used to experimentally contrive a series of differently sized proteins for which intrinsic protein stability was kept constant with increasing size, thereby allowing the effects on apparent cooperativity predicted in this simulation ( Fig. 4 ) to be tested (ref. 79 is interesting in regard to this point). Furthermore, identification of a strong/weak experimental correlation between a particular amino acid and any of the parameter set [(k FU ) 0,0 , k 1F , k 1U , k 2U , N FD or N UD ] could assist in more judicious design of mutations chosen with alternative aims, of say increasing enzymatic catalytic efficiency (80) or selection of a nonstructural amino acid as a site for covalent attachment of a useful molecule (such as a FRET donor/acceptor (81) or a PEG/biotin molecule for further stabilization/purification (82) . Possible Extensions to the Model Despite (or perhaps because of) its relative simplicity we believe that the 'new look' model is extensible and there are a number of directions we would like to take it. Without being exhaustive we discuss a few of these areas below. Possible Extensions to the Model One future avenue would be the inclusion of more structural information in the assignment of the number of N FD and N UD sites. Appendix 2 describes this calculation for transition between a folded sphere (globular protein) and an unfolded thin rod (polypeptide chain) but it would be interesting to repeat this calculation for asymmetric proteins both at the mesoscopic level of approximation (51) as well as using structural information featuring atomic detail. Ultimately we wish to study site numbers as a function of shape and to use this model extension to examine how shape can determine stability (18) . Possible Extensions to the Model Another area we would like to develop the model involves using it to examine competitive binding between two different types of denaturant, between denaturant and ligand and even that between denaturant and water (15, 43, 47-50, 57, 60) . The model could be reformulated to accommodate displacement reactions occurring between two denaturants/ligands which could prove useful in addressing the dispute between models which assign differential weights to hydrophilic and hydrophobic forces [83, 84] . Possible Extensions to the Model We believe that our model also holds great promise as the base vehicle for factoring in both denaturant and temperature dependence of the unfolding equilibrium -leading to a generalized analytical formulation. Similar to the approach attempted by Miller and Dill (43) we suggest that each site binding constant could be parameterized in terms of an Arrhenius type dependence. Final Remarks The original motivation underlying this study was the development of a model of denaturant induced protein unfolding into which we could introduce additional mechanistic factors affecting the denaturant protein interaction. Three such cases of particular interest to us are (i) role of denatured state in contributing to amyloid formation [85] [86] [87] [88] , (ii) effects of competition between denaturants and weakly stabilizing ligands, and (iii) competing effects of macromolecular crowding on protein folding and denaturant binding events [89] [90] [91] . As we realized the potential pedagogical power of our developed approach we decided to put these initial plans on hold and investigate the basic capabilities of the model to inform on the process of denaturant induced protein folding. The model provides a different perspective on the protein folding problem M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT and a fresh set of parameters with which to classify the process. It has (re)kindled our interest in the subject and we believe that it might provide others with a new viewpoint from which to ponder the protein folding problem -one of the biggest scientific questions of the last century [92, 93] . A C C E P T E D ACCEPTED MANUSCRIPT Acknowledgements: DH would like to acknowledge a lively/heated discussion with Dr. Allen P. Minton that prompted a deeper look at the equations used for characterizing the protein unfolding transition. DH would also like to thank Prof. Thomas Huber and Prof. Gottfried Otting for helpful discussions and comments on an early version of this paper. The work of DH was jointly funded by an Australian National University Senior Research Fellowship, an Osaka University Cross-Appointment as an Associate Professor. The work of AK was funded by the Protein Data Base of Japan (PDBj). The work of YG was funded under the Cooperative Research Program for the Institute for Protein, Osaka University, ICR-15-02 and was supported by JSPS KAKENHI Grant Number 15H04362. Appendix 1: The m-value approach to the description of protein folding by Pace et al. In a 1974 publication, Greene and Pace [A1A] showed that the measured free energy of unfolding was linear with respect to denaturant concentration for four different proteins. This empirical finding was the basis for the m-value approach which has later been rationalized by comparison against other theories by Pace In equation set A1a-e the symbol G refers to the free energy of the system at constant temperature and pressure, R is the universal gas constant and T is the absolute temperature. For a two-state folding equilibrium the value of the denaturant concentration at which C U = C F (i.e. K F → U = 1) is termed the denaturant midpoint (C D ) mp . If (C D ) mp is taken as a reference state then the difference in chemical potential between F and U at some value ∆C D [where ∆C D = C D − (C D ) mp ] can be calculated via a first order linear perturbation in free energy, with the constant derivative term named as the m value (Eqn. a1f-g). Functionally the m value can be used as fitting parameter [A1B, A1E] or estimated ab initio using various levels of statistical mechanics based approaches [A1F, A1G] to produce an equation of the form given by Eqn. a1h-i. In a 1974 publication, Greene and Pace [A1A] With only two variable parameters (m and (C D ) mp ), the first-order linear perturbation treatment (Eqn. a1h) has been shown to provide an excellent empirical description of the two state protein unfolding equilibrium for a large number of proteins [A1a]. [A1G] Ghosh, K. and Dill, K.A., 2009. Computing protein stabilities from their chain lengths. Proceedings of the National Academy of Sciences, 106 (26) , pp.10649-10654. Appendix 1 References Appendix 2: Defining N UD and N FD in terms of N AA for a particular protein shape We demonstrate the basis of our simplifying assumption that N UD and N FD may be directly determined by a single parameter N AA for a protein of known structure by treating the relatively simple example of a globular (and therefore approximately spherical) protein. To make our case we first assume that the total number of denaturant binding sites on the protein is proportional to the total normal projected surface area of the stretched polypeptide chain, In a similar fashion the normal projected surface area of the folded protein, A F , relative to a spherical denaturant of radius R D , can be determined on the basis of the assumptions of spherical geometry, volume additivity and set volume packing fraction, α, reflecting the degree of close packing of amino acids within the folded protein [see appendix of A2E] (Eqn. a2c). As for the preceding case, N FD is considered proportional to A F and is determined via Eqn. a2d on the basis of assumptions reflecting (i) an assumed area of projection of denaturant onto the spherical surface of the folded protein, A D , and (ii) a two dimensional packing fraction, ε, reflecting fractional usage of the polypeptide surface area by denaturant. Appendix 2: Defining N UD and N FD in terms of N AA for a particular protein shape The values of the additional supporting parameters, β and ε, reflecting the degree of two dimensional close packing of denaturant to the normal projected surface area of the unfolded polypeptide chain and the folded spherical protein, were set to their two dimensional random close packing limit of 0.7 15 i.e. ε = β = 0.5. A numerical value for α 15 The chosen fractional value should be lower than both that for random (0. . As denaturant molecules come in a range of sizes e.g. much smaller than an amino acid (such as guanidine hydrochloride and urea) or much larger than an amino acid (e.g. various long chain alcohols, sodium dodecyl sulfate) we have taken a conservative estimate and set the denaturant size equal to the amino acid size (i.e. R D = R AA ) 16 . Appendix 2: Defining N UD and N FD in terms of N AA for a particular protein shape As the values of N UD and N FD are calculated on assumption of fractional attainment of the close packing relation of denaturant on the surface of the folded and unfolded structures (with no consideration paid to the location or geometrical relationship between particular chemical groups) it is expected that their values represent upper bounds. Appreciation of the importance of this parameter can be gained from inspection of the case (ii) simulations in which the protein size is varied. Within the confines of the model presented, this case is equivalent to varying the value of β and ε to produce more or less denaturant sites. Appendix 2: Defining N UD and N FD in terms of N AA for a particular protein shape Particular binding locations of various denaturant molecules have been considered previously via computer simulation based on molecular dynamics [A2G]. Appendix 2: Defining N UD and N FD in terms of N AA for a particular protein shape Although we have considered the relationship between protein size and denaturant binding site number (in the folded and unfolded states) for the particular case of spherical geometry, it is a trivial matter to recalculate this relationship for both (i) Approximate regular shapes commonly used for general description of protein structures (such as ellipsoids or cylinders [A2H]), (ii) Particular protein structures obtained from the protein data bank (e.g. PDBj [A2I]) when used in conjunction with a denaturant binding site model of the type described here. 16 None of the conclusions drawn in this paper depend upon the exact size of the denaturant relative to the amino acid size. Appendix 2 References [A2A] Hall, D. 2008. Kinetic models describing biomolecular kyle77@example.org. In Handbook of Surface Plasmon Resonance (pp. . Royal Society of Chemistry. Appendix 2 References [A2B] Tozzini, V., 2005. Coarse-grained models for proteins. Current opinion in structural biology, 15(2), pp.144-150. Appendix 2 References [A2C] Nielsen, S.O., Lopez, C.F., Srinivas, G. and Klein, M.L., 2004. Coarse grain models and the computer simulation of soft materials. Journal of Physics: Condensed Matter, 16 (15) , p.R481. Appendix 2 References [A2D] Hall, D., Li, S., Yamashita, K., Azuma, R., Carver, J.A. and Standley, D.M., 2014. A novel protein distance matrix based on the minimum arclength between two amino-acid residues on the surface of a globular protein. Biophysical chemistry, 190, pp.50-55. Appendix 2 References [A2E] Sasahara, K., Hall, D. and Hamada, D., 2010. Effect of lipid type on the binding of lipid vesicles to islet amyloid polypeptide amyloid fibrils. Biochemistry, 49 (14) , pp.3040-3048. Appendix 2 References [A2F] Cantor, C.R. and Schimmel, P.R., 1980. Biophysical chemistry: Part III: the behavior of biological macromolecules. Macmillan. Appendix 2 References [A2G] Stumpe, M.C. and Grubmüller, H., 2007. Interaction of urea with amino acids: implications for urea-induced protein denaturation. Journal of the American Chemical Society, 129 (51) , pp.16126-16131. Appendix 2 References [A2H] Taylor, W.R., Thornton, J.T. and Turnell, W.G., 1983. An ellipsoidal approximation of protein shape. Journal of Molecular Graphics, 1(2), pp. 30-38. [A2I] Kinjo, A.R., Suzuki, H., Yamashita, R., Ikegawa, Y., Kudou, T., Igarashi, R., Kengaku, Y., Cho, H., Standley, D.M., Nakagawa, A. and Nakamura, H., 2012. Protein Data Bank Japan (PDBj): maintaining a structural data archive and resource description framework format. Nucleic acids research, 40(D1), pp.D453-D460. Appendix 3: Combination of the Linear Extrapolation Method (LEM) and the Tanford Transfer Free Energy Method. An anonymous reviewer of the manuscript has asked us to point out an alternative physical description of the two state protein unfolding reaction that is derived by combination of the mvalue approach [A3A] and the Tanford transfer free energy method [A3B]. This can be realized by first extending Eqn. a1h to create Eqn. a3a which A similar functional form can be determined for the two-state protein unfolding reaction when described in terms of the Tanford transfer free energy method which is based on the concept of a free energy cycle [a3b]. In this model the free energy per mole associated with the transfer of either the folded, or unfolded, species from solvent without denaturant to solvent with denaturant is respectively denoted as ∆u , •° and ∆u , Appendix 3: Combination of the Linear Extrapolation Method (LEM) and the Tanford Transfer Free Energy Method. In Eqn. a3d the difference in transfer free energy is approximated by a linear summation of the differential free energies for transferring a number N i of a particular i type of amino acid (between the unfolded and folded states) from solvent without denaturant to solvent containing denaturant, δµ i,tr . Appendix 3: Combination of the Linear Extrapolation Method (LEM) and the Tanford Transfer Free Energy Method. An additional parameter, α i , reflecting the fractional exposure of a particular amino acid in the folded and unfolded state was found necessary to reconcile constituent data with experiments on whole proteins [A3C]. In practical usage an averaged value, Ÿ ¡ , is adopted for the fractional contribution term allowing Eqn. a3d to be further simplified [a3e]. Appendix 3: Combination of the Linear Extrapolation Method (LEM) and the Tanford Transfer Free Energy Method. Bolen and colleagues [A3D, A3E] have used the apparent linear dependence of the unfolding free energy on the concentration of denaturant (Eqn. a3a) to demonstrate a direct equivalence for the transfer free energy term [a3f]. Appendix 3: Combination of the Linear Extrapolation Method (LEM) and the Tanford Transfer Free Energy Method. Ÿ ¡ ∑ n u , • G = −ƒ. Appendix 3: Combination of the Linear Extrapolation Method (LEM) and the Tanford Transfer Free Energy Method. [a3f] Appendix 3: Combination of the Linear Extrapolation Method (LEM) and the Tanford Transfer Free Energy Method. Such a formulation has allowed for electrostatic models of the free energy of transfer to be used in the reconciliation of experimental data of protein unfolding in solutions containing high concentrations of osmolytes such as trimethylamine oxide (TMAO) [A3D]. However due to its requirement for prior evaluation of the m value and its specification of a large number of individual transfer parameters the described method lacks the capabilities of the procedure described in the current paper with regard to, (i.) Accounting for the form of the dependence of the unfolding curve on denaturant concentration when it differs from the ideal case. Appendix 3: Combination of the Linear Extrapolation Method (LEM) and the Tanford Transfer Free Energy Method. (ii.) Providing useful information on the distributed nature of the unfolded state as a function of denaturant concentration. A number of points can be noted from Appendix Figs. 1A-F and Eqn. a4a, using the true value of the parameter, λtrue. Typically parameter confidence is estimated using a large number of Monte Carlo cycles to add equivalent extents of normal error to the data, followed by non-linear regression to evaluate the best fit value of the parameter for evaluation. An frequency plot of obtained parameter values is then constructed and confidence values for the parameter are determined by integration of this distribution. For reasons relating to computational simplicity we have adopted a slightly different approach in which the calculated errors are derived from comparison of a distribution of absolute residuals calculated between the data and best fit (correct) solution and a distribution of residuals between with the correct values and solutions for each altered parameter value with added normal error. Appendix 3 References (i) Highest probability of acceptance of the null hypothesis is around the input value. Appendix 3 References (ii) Acceptance probability for the null hypothesis broadens with regard to the magnitude of ∆λ for larger amounts of added normal error. Appendix 3 References (iii) Determinability of the parameter is related to the number of data points (i.e. n) in the data set to be fitted. Appendix 3 References (iv) Error distributions reflect the sensitivity of f U , C D plot results to change in the parameter. As such we note that the parameters relating to the number and affinity of the UD sites are the most sensitive to relative change in parameter value. Appendix 3 References Typically, quoted errors in parameter values are given as the value of ∆λ at which P(λ + Δ£) = 0.05 . As noted from Appendix Fig. 1A -F, all parameter values have a discernible maxima at their true value. Furthermore, even with a very low number of data points (n = 5) all parameters are determinable to within 50% of their true value when extents of added normal error have a standard deviation below 0.075 (in relation to a range of f u vaues of [0, 1.0]. This value of error is easily achievable in protein folding experiments providing sufficient care is taken (e.g. as for [A4B]). As such, we suggest that provided a suitably efficient non-linear regression routine is employed for the fitting cycle then our model will be be capable of determining parameter values with sufficient accuracy and acceptable levels of error for a data set with low noise and a large number of data points. In the next paper in this series we will explore this point in more detail. [20] Greene, R.F. and Pace, C.N., 1974. Urea and guanidine hydrochloride denaturation of ribonuclease, lysozyme, α-chymotrypsin, and βlactoglobulin. Journal of Biological Chemistry, 249 (17) , pp.5388-5393. Appendix 4 References [21] Appendix 4 References Nozaki, Y. and Tanford, C., 1971. The solubility of amino acids and two glycine peptides in aqueous ethanol and dioxane solutions establishment of a hydrophobicity scale. Journal of Biological Chemistry, 246 (7) , pp.2211-2217. Nagarajan, S., Amir, D., Grupi, A., Goldenberg, D.P., Minton, A.P. and Haas, E., 2011. Modulation of functionally significant conformational equilibria in adenylate kinase by high concentrations of trimethylamine oxide attributed to volume exclusion. Biophysical journal, 100 (12) , pp.2991-2999. the site binding model in which a particular sequence of denaturant equilibria is displayed with each equilibria governed by a particular site binding constant described using a lower case k (either k 1F , k 1U or k 2U ). Blue and yellow parts of the polypeptide chain represent external (solvent exposed) and internal (hidden from solvent) segments. Red dots represent denaturant molecules bound to a region of the polypeptide chain that initially exists in an external region. Black dots represent denaturant molecules bound to polypeptide regions not initially exposed to the solvent. k 1F , k 1U and k 2U respectively represent site binding constants for denaturant binding to initially solvent exposed regions in the folded state, initially solvent exposed regions in the unfolded state and initially internalized regions in the unfolded state. Unfolding constants (k FU ) m,0 represent the site specific equilibrium constants for the unfolding of proteins loaded with m number of denaturant molecules bound to the initially solvent exposed polypeptide regions.
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