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. 2006 Sep;15(9):2062–2070. doi: 10.1110/ps.062297006

Fundamental processes of protein folding: Measuring the energetic balance between helix formation and hydrophobic interactions

Wujing Xian 1, Peter J Connolly 2, Marcela Oslin 3, Andrew C Hausrath 4, John J Osterhout 4
PMCID: PMC2242613  PMID: 16882995

Abstract

Theories of protein folding often consider contributions from three fundamental elements: loops, hydrophobic interactions, and secondary structures. The pathway of protein folding, the rate of folding, and the final folded structure should be predictable if the energetic contributions to folding of these fundamental factors were properly understood. αtα is a helix-turn-helix peptide that was developed by de novo design to provide a model system for the study of these important elements of protein folding. Hydrogen exchange experiments were performed on selectively 15N-labeled αtα and used to calculate the stability of hydrogen bonds within the peptide. The resulting pattern of hydrogen bond stability was analyzed using a version of Lifson-Roig model that was extended to include a statistical parameter for tertiary interactions. This parameter, x, represents the additional statistical weight conferred upon a helical state by a tertiary contact. The hydrogen exchange data is most closely fit by the XHC model with an x parameter of 9.25. Thus the statistical weight of a hydrophobic tertiary contact is ∼5.8× the statistical weight for helix formation by alanine. The value for the x parameter derived from this study should provide a basis for the understanding of the relationship between hydrophobic cluster formation and secondary structure formation during the early stages of protein folding.

Keywords: hydrogen exchange, Lifson-Roig, hydrophobicity, protein folding, peptide model, helix

It has been known since the seminal paper by Anfinsen (1973) that, given the correct conditions, proteins can fold spontaneously using only the information contained in the primary amino acid sequence. Levinthal's calculation ruled out random folding because it would take longer than the age of the universe for even a small protein to sample all of its conformational space (Levinthal 1968). The obvious conclusion was that protein folding must be directed and that the pathway to the final folded conformation must reduce the search space so as to allow folding to proceed in seconds or even microseconds. During the ensuing 30 years, researchers have been working to determine the mechanism of folding, to characterize the intermediates in the folding process, to delineate the pathway of folding, and to find a method of calculating the rate of folding and/or the final folded structure.

Protein folding theories consider secondary structures, loops, hydrophobic interfaces, and interactions among them as fundamental pieces of the folding puzzle. A clear understanding of the energetic balance among these fundamental parts is necessary to be able to predict protein folding and structure from first principles. The study of the energetic balance among the component parts is confounded by the complexity of real proteins, which are very rich in the interactions that stabilize folded structures. In a protein a given residue may participate in multiple types of interactions simultaneously, and so the interpretation of the energetic consequences of mutations in terms of fundamental physical interactions is problematic. Yet it is precisely this information that is needed in order to create a predictive theory of protein folding.

An approach to this problem is to use peptide models to simplify the problem or to isolate specific interactions (Osterhout 2005). However, the essential characteristic of real protein energetics that is not captured by simple peptide models is that of coupling between different types of interactions. The energetic contribution to stability of a particular interaction may vary depending on its structural context. This means that distinct interactions can influence one another so that it will ultimately be necessary to study interactions in the presence of other interactions rather than in isolation.

A peptide model system has been developed that is intermediate in complexity between peptide models and whole proteins, simple enough that effects can be interpreted in terms of fundamental interactions and yet with sufficient complexity that its energetics display the coupled character of protein domains. This model system is αtα, a 38-residue peptide of de novo design that consists of two interacting helices connected by a turn region (Fezoui et al. 1994a,b, 1997).

The results of hydrogen–deuterium exchange measurements on αtα are reported here. The pattern of stability of the hydrogen bonds is determined and compared with predictions from a statistical mechanical theory that takes into account long-range interactions between sites distant in the sequence. A statistical mechanical theory has been developed that takes into account inter-helix hydrophobic interactions. This theory is an extension of the Lifson-Roig helix-coil transition theory (Lifson and Roig 1961) and includes a statistical weight for the hydrophobic interactions in the helix–helix interface. Using this theory, it is possible to arrive at an initial estimate for the energetic balance between helix formation and hydrophobic stabilization in the context of an early folding intermediate.

Results

Protection factors for amide proton exchange

15N was incorporated into αtα at selected positions (Table 1; Fig. 1) by solid phase peptide synthesis. The protection factors for the exchange of the 15N-labeled amide protons of 15NαtαI and 15NαtαII are shown in Table 1. Measurements were performed at pH values between 2.1 and 3.2, since it was previously determined that αtα shows increased stability in this pH range (Fezoui et al. 1994a). Decreasing the pH at 18°C resulted in an increase in the protection factors, which is consistent with the increase in helicity observed previously (Fezoui et al. 1994a). The protection factors increased significantly as the temperature was decreased to 5°C (Table 1). Some of the protons exchanged completely before data acquisition began. These included Gly18 and Ala23 in all experiments and Leu3 and Ala16 in 15NαtαI at 0°C. According to Bai et al. (1993), all protons should be exchanging slowly enough to measure under these conditions of temperature and pH. The data imply either that the exchange for the protons in these positions is being catalyzed or the calculation of the intrinsic rates (Bai et al. 1993) is slightly in error because of a heretofore unrecognized sequence dependence of the exchange. All of the rapidly exchanging protons are located at the ends of the helices or the turn (Gly18). Leu3 has a measurable exchange rate at 18°C at all pH values but the calculated protection factors are less than one, again implying catalysis or slight inaccuracies in the estimation of the intrinsic exchange rate, kint.

Table 1.

Variation of protection factors (Pf) in αtα with pH and temperature

graphic file with name 2062tbl1.jpg

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Figure 1.

Figure 1.

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15N labels in the three-dimensional structure of αtα. The 15N-labeled nitrogens and their attached hydrogens are rendered in a dark-gray space-filling representation and the side chains of the amino acids of the hydrophobic interface are in light gray. The structure was determined by two-dimensional NMR as reported previously (Fezoui et al. 1997), accession code 1ABZ.

The pattern of amide proton protection as a function of position and pH is shown in Figure 2. The data for 15NαtαI and 15NαtαII have been combined for each pH range to provide a pattern for the entire peptide. The largest protection factors occur in the central parts of both helices and very little protection near the ends of the helices (Fig. 2). The protection factors measured for αtα (Table 1; Fig. 2) indicate that the amide protons are stabilized compared with peptides corresponding to the individual helices, consistent with earlier experiments using CD and non-site-specific hydrogen exchange experiments (Fezoui et al. 1999).

Figure 2.

Figure 2.

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Positional dependence of protection factors (Pf). The results from 15NαtαI and 15NαtαII have been combined: pH 2.1 (•), pH 2.7 and 2.8 (▴), pH 3.0 and 3.2(▪) (see also Table 1). In this figure the data points marked “fast” in Table 1 are set to 1 for the purpose of providing a reasonable representation (see Results).

Protection factors and the three-dimensional structure

The protection factors are shown color-coded onto the three-dimensional structure of αtα in Figure 3. This figure shows that the central part of the helices are the most stable while the residues in the loop or near the ends of the helices are the least stable. It is notable that the most stable positions in each helix are actually on the outside of the helices, opposite the hydrophobic interface (Ala13 in the first helix and Ala27 and Ala30 in the second helix).

Figure 3.

Figure 3.

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Protection factors of the amide nitrogens mapped onto the three-dimensional structure. The pH 2.1 protection factors are color-coded onto space-filling representations of the amide nitrogens and their attached hydrogens according to the key (inset).

Comparison to extended helix-coil theory

Extended helix-coil transition theory (XHC theory; see Materials and Methods) was used to calculate the protection factors for all the amide protons of αtα. The calculated and measured protection factors (pH 2.1, 18°C) are plotted in Figure 4. The lines show the calculated protection factors at values of the tertiary interaction parameter, x, ranging from 7 to 11. It can be seen that, for all values of x, XHC theory predicts that the hydrogen bonds in the second helix should be more stable than those in the first helix. The curve for x = 9.25 (solid line) best approximates the hydrogen exchange pattern (Fig. 4, see inset).

Figure 4.

Figure 4.

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Protection factors predicted from XHC theory. The protection factors predicted from XHC (lines) are compared with measured protection factors from hydrogen exchange (•) (pH 2.1). The dotted lines correspond to x values ranging from 7 to 11 in increments of one. The solid line represents x = 9.25, which is the x value with the lowest residual (inset, arrow). (Inset) Residuals (sum of the differences squared) plotted against the x parameter. Gly18 and Ala23, which exchanged too quickly to measure (Table 1), were arbitrarily set to one (no protection) for this plot.

Discussion

An accurate understanding of the pathway of protein folding will lead to both the prediction of protein tertiary structure and an understanding of the mechanism of protein misfolding in disease. A route toward the understanding of the folding pathway is to understand the energetic and kinetic relationships among the fundamental elements of folding. Peptide models have long been employed to study these fundamental processes (Osterhout 2005), usually with peptides that attempt to isolate particular elements. These studies, although informative, cannot tell the whole story because they are missing the interdependence of effects that are exhibited by proteins.

In order to understand the complementary roles of secondary structure formation and the formation of hydrophobic clusters, it is necessary to use more complicated systems. αtα is a peptide that was designed de novo as a model system to study the stabilization of secondary structures through a hydrophobic interface (Fezoui et al. 1994a,b). It contains two interacting α-helices, a connecting loop, and a hydrophobic interface, constituting a minimal system for the analysis of the interplay of these effects. The goal is to study the energetic relationships among these elements within the same peptide so as to preserve context-dependent relationships.

Lifson-Roig and XHC theories

The Lifson-Roig (LR) theory and its subsequent modifications have provided an intellectual framework to determine the energies of interactions that stabilize α-helices from experimental data (Doig 2002; Osterhout 2005). This work is well-advanced. While some questions remain, it is likely that the chief determinants of helical structure have been quantitatively analyzed, and it is now possible to make fairly accurate predictions of helical content for any peptide sequence, using the knowledge of the intrinsic properties of the amino acids. The next challenge is to apply this knowledge to the study of situations containing tertiary structure. As a first step toward this goal, we have developed an extended helix-coil transition theory (XHC theory) in which both secondary structure formation and tertiary contacts are treated on an equal basis.

The Lifson-Roig formulation of helix-coil theory assigns each residue in the chain either helix (h) or coil (c) conformation depending on whether its Ramachandran angles fall within helical values or not, so that any conformation of the chain can be represented as a string of hs and cs. Weights are assigned to each residue depending on its state and those of its nearest neighbors: c states receive weight u, and h states receive weight v if they have one or two c neighbors and weight w if they have only h neighbors. The statistical weight for a state is the product of the weights for the individual residues, and so will be a product of powers of u, v, and w. A convenient, flexible, and powerful matrix formalism introduced by Lifson and Roig (1961) enables calculation of the partition function and a variety of experimentally observable ensemble averages.

In order to balance the effect of helix formation with the formation of hydrophobic interactions in the interface, Lifson-Roig helix-coil transition theory was extended to include tertiary interactions. In this model, residues are divided into two classes: hydrophobic residues of the inter-helical interface that make long-range contacts (X) and residues not participating in the interface that make only short range contacts (-). The extended helix-coil model postulates that X residues may form a contact with another X residue distant in the sequence. Each such contact between hydrophobic residues confers an additional weight x to the states in which it occurs. Such contacts can only take place between helical (w state) residues, and all such contacts, whether present in the final structure or not, are assumed to be energetically identical. Therefore, each state of the molecule is represented by a term that is a product of powers of u, v, w, and x. When x is zero, the XHC model becomes identical with LR theory. Like LR, the XHC model does not explicitly consider the spatial arrangement of the chain. Rather, a list of contact sets must be supplied, which represents the tertiary contact states accessible to the molecule. This choice of contact sets represents the ensemble of three-dimensional configurations considered by the model, which is otherwise only concerned with energetic relationships. The choice of contact sets defines the unfolded state ensemble and also is used as a means to exclude from consideration the physically unrealistic states. A modification of the Lifson-Roig matrix formalism enables the enumeration of all helix-coil states consistent with the particular contact sets used. The contact sets used in the calculations in this manuscript are illustrated in Figure 5.

Figure 5.

Figure 5.

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XHC interaction states. Graphical representation of the hydrophobic interaction states for αtα in the extended helix-coil transition model. “S = 0,” “S = 1,” etc. are the register shifts of the hydrophobic interface. The states in S = 0 are in register while the states in S = 1 are shifted by one position. LR, Lifson-Roig, is the state with no interchain hydrophobic interactions, which reduces to the standard Lifson-Roig theory.

An important characteristic of the LR theory is that it can accommodate heteropolymers, so that calculations can be done for peptides of any sequence. This is accomplished in the LR theory by assigning a particular w value to each type of residue. These values have been measured by various means, including host-guest copolymers, peptides, coiled coils, and proteins (O'Neil and DeGrado 1990; Wójcik et al. 1990; Horovitz et al. 1992; Blaber et al. 1993, 1994; Chakrabartty et al. 1994; Myers et al. 1998a,b). Although some context dependence has been observed, the finding that w values are the same in proteins and peptides (Myers et al. 1998a,b) suggests that the underlying idea of a helical propensity is valid if medium range side chain interactions are properly accounted for. The current version of XHC theory makes use of w parameters that are residue specific (Chakrabartty et al. 1994) but uses a single x value for all sites.

It seems likely that a more accurate representation of the coupling between secondary and tertiary structures could be obtained using residue-specific x values, much as improved predictive power of helix-coil theory was achieved for heteropolymers by using site-specific helical propensities. These have not been implemented at this time for two reasons. First, the agreement between theory and experiment is good, suggesting that site-specific x values will serve mostly to modulate the main effects. Second, the introduction of additional parameters at the present time would almost certainly result in better fits to the data, but more experimental data would be necessary to constrain additional model parameters. Experiments utilizing hydrophobic variants of αtα are presently underway.

The x parameter and its physical meanings

Formally, the x parameter is simply the additional statistical weight provided to a helical state by a tertiary contact. It is important to note that this weight is applied equally within native and non-native states so that native contacts are not inherently different from non-native contacts. Tertiary contacts also stabilize non-native states, and so interpreting x as an equilibrium constant for native tertiary contact formation is incorrect and will significantly overestimate the net gain in stability of the native state due to formation of a tertiary contact.

Despite the statistical nature of x, it is natural to want to correlate x with physical properties. Our favored interpretation of the physical basis of the x parameter is that it primarily results from the energetically favorable transfer of a hydrophobic side chain from water to the hydrophobic interior. The formulation of the model in terms of pairs of interacting residues is intended to ensure the existence within a state of such an interior. The identity of the other residues constituting the environment into which the side chain partitions may vary between states, but the primary effect is due to transfer out of water and into a hydrophobic environment (Tanford 1962).

The value of x obtained here, 9.25, corresponds to an energy of −1.3 kcal/mol. For comparison, the side chain solvent transfer energies between water and octanol vary between −2.04 and −1.18 kcal/mol for leucine, isoleucine, valine, and methionine (Radzicka and Wolfenden 1988), the residue types that constitute the hydrophobic core of αtα. Therefore, the measured x value is consistent with such an interpretation.

Other physical effects might also be operative. These include differences in buried surface area, spatial complementarity favoring the interdigitation of particular pairs of amino acids (as in coiled coils; Harbury et al. 1993, 1994), and conformational restrictions of side chains. It is easily imagined that these effects could result in different pairs of amino acids having different interaction energies (Stapley et al. 1995). The good correlation of the theory using a single x value with the amide proton exchange data argues that there is an overriding effect, such as the solvent transfer effect, which may be modulated by other effects to define the structural details. Further experimentation will be required to distinguish these possibilities and to quantify their contributions.

Differential stabilization of the N- and C-terminal helices

In an earlier paper, it was shown that the stability of αtα was substantially greater than the stabilities of the peptides, α1 and α2, corresponding to its constituent helices (Fezoui et al. 1999). This observation implies that the interaction between the helices leads to their mutual stabilization. An expected consequence of mutual stabilization is that the interacting secondary structures must stabilize each other equally, so the formation of one helix reinforces the formation of another by the same degree. However, the results of the present study show that the segments of secondary structure of αtα are differentially stabilized relative to the intrinsic stability of the corresponding isolated peptides α1 and α2. Under all the conditions tested, the N-terminal helix of αtα was less stable than the C-terminal helix. CD and HX measurements of α1 and α2 indicated that they were very nearly equal in their stabilities, 34% and 36% helical by CD respectively (Fezoui et al. 1999). These results show that there is not a large difference in the intrinsic stability of the two helices. However, in the context of αtα both helices are increased in stability, and the differential stabilization of the C-terminal helix is greater than the differential stabilization of the N-terminal helix. If the two helices were stabilizing each other, the stabilization would be reciprocal, but this is not observed here.

This apparent paradox can be resolved by abandoning the mental picture of two helices that form through local interactions and then subsequently interact through tertiary interactions. The stabilization due to tertiary contacts must indeed be reciprocal. However, this reciprocal stabilization operates not on the helices, but within individual states.

The hydrogen exchange data indicate the fraction of states that are protected from exchange by formation of hydrogen bonds in the helices. But at different sites this protection may be achieved by different combinations of states. In this way, the influence of tertiary contacts can differ from site to site. Some contacts may stabilize states in which particular sites distant from the contact are unprotected and so diminish rather than increase protection at those sites. (In contrast, in a model in which tertiary contacts act upon preformed helices, the tertiary interactions could only result in increases in protection.) The helical regions identified by hydrogen exchange are the result of the net-combined influence of both the tertiary contacts and secondary structure interactions. The tertiary interactions are inextricably bound up in the stabilization of regions of secondary structure. Both types of interactions participate in a single equilibrium. In other words, the nonlocal interactions are coupled to the local interactions.

The reciprocal stabilization of secondary structure elements is equivalent to the additivity of the energies for secondary and tertiary interactions. In this case, the energies are not additive, which is the thermodynamic signature of coupling. Many experimental studies have found that the thermodynamic effects of distinct point mutations on the thermodynamic stability of proteins may not be additive (Zhang et al. 1995; Baldwin et al. 1996; Lipscomb et al. 1998). A thermodynamic analysis can identify such non-additivity, but a statistical mechanical analysis is necessary to sort out the relative contributions of the interacting effects. In general, the complexity of most proteins that display such coupling precludes such a theoretical treatment. The αtα peptide system is advantageous in this respect because it has sufficient complexity to display coupling but is simple enough to admit a statistical mechanical treatment.

x, w, and folding mechanisms

Since the tertiary interaction parameter, x, and the helix propensity, w, are both statistical weights, they can be directly compared. A rough estimate of the relative contributions of these effects is simply the ratio of the two parameters, x/w. Where the value of x is 9.3 (Fig. 4) and w 1.6 (the helix propensity of Ala; Chakrabartty et al. 1994), this ratio is ∼5.8. This ratio represents the first direct measurements of one of the fundamental aspects of protein folding: the balance between hydrophobic collapse and secondary structure formation.

This balance is central to most theories of protein folding. For instance, it has been recently argued that the nucleation-condensation (Fersht 1997) and diffusion-collision (Karplus and Weaver 1976) models represent opposing extremes of the same mechanism (Fersht 2000), the essential difference between these models being that secondary structures are more stable in the case of the diffusion-collision mechanism. The hydrophobic/secondary structure balance also plays a role in trying to understand the relationship between contact order and the rate of protein folding (Makarov et al. 2002). It would be helpful in this case to be able to estimate more accurately the strength of individual “contacts” and to correct for the effects of secondary structure formation. While the present paper is not concerned with prediction of these processes, the initial estimate obtained here for the balance between hydrophobicity and secondary structure formation represents an advance on the pathway toward this goal.

Future directions—Extensions of the model

This version of extended helix-coil theory does not account for capping effects (Lyu et al. 1993; Doig and Baldwin 1995) or medium-range side chain interactions such as salt bridges (Marqusee and Baldwin 1987) and hydrophobic interactions (Padmanabhan and Baldwin 1994a, b; Stapley et al. 1995). These kinds of interactions have been incorporated into different versions of the Lifson-Roig formalism (Scholtz et al. 1993; Doig et al. 1994; Shalongo and Stellwagen 1995; Stapley et al. 1995) at the expense of greater complexity. In the long term it will be of interest to compare the magnitude of these effects to those of the tertiary contacts by incorporating such interactions into the basic XHC formalism.

Conclusions

A new implementation of Lifson-Roig helix-coil theory, XHC, has been presented in which an additional parameter, x, is introduced to account for long-range interactions. By comparing predictions from XHC to amide proton exchange data from a model peptide, it has been possible to estimate x for hydrophobic interactions as ∼9.25. The ratio of the statistical weight of a hydrophobic interaction to the statistical weight of the alanine helix propensity, x/w, is ∼5.8. This ratio represents the balance between two of the fundamental processes in protein folding. The energetic balance between these types of interactions has important consequences for the prediction of protein folding pathways and final folded structure. Particularly, it is a start toward understanding and predicting the relative importance of the formation of a particular hydrophobic cluster relative to the formation of a particular secondary structural motif.

Materials and methods

Peptide synthesis and purification

15N-labeled acids, alanine, glycine, and leucine were purchased from Cambridge Isotope Laboratories. The FMOC (fluoren-9-ylmethoxycarbonyl) derivatives of these amino acids were synthesized according to a protocol from Stewart and Young (1984) with minor modifications. The amino acid sequence of αtα is Succinyl-DWLKARVEQELQALEARGTDSNAELRAMEAKLKAEIQK-NH2. Two peptides were synthesized, 15NαtαI and 15NαtαII containing different sets of labeled amino acids (see Table 1 for the individual locations). The backbone of αtα from the NMR structure (Fezoui et al. 1997) is shown in Figure 1 with the locations of the labels indicated. The peptides were synthesized on an Applied Biosystems 431A peptide synthesizer using Rink amide MBHA resin from NovaBiochem and amino acids from both NovaBiochem and Applied Biosystems. The peptides were purified by reverse phase HPLC as described previously (Fezoui et al. 1994a). The amino acid composition and molecular mass of both peptides were confirmed by amino acid analysis and electrospray mass spectroscopy.

NMR spectroscopy and data processing

All NMR experiments were performed on a JEOL GX-400 NMR spectrometer at the Rowland Institute for Science (now the Rowland Institute at Harvard). Spectra were collected using 3 mM αtα at 18°C and 5°C and the pH of the samples were measured after completion of the NMR experiments. The amide proton resonances were identified by comparison of the previous assignments for αtα (Fezoui et al. 1994a, 1997) with PCOSY (Marion and Bax 1988) and 1H-15N HSQC (Bodenhausen and Ruben 1980; Cavanagh et al. 1991) experiments on the labeled peptides. Hydrogen exchange was monitored with one-dimensional 1H-15N HSQC experiments. Thirty-two transients were accumulated for each spectrum. NMR data were processed using the Rowland NMR ToolKit (Hoch and Stern 1993).

Hydrogen/deuterium exchange

Hydrogen exchange was initiated by dissolving lyophilized peptide in 20 mM NaCl in D2O that was pre-adjusted with DCl to the desired pH. The amide proton exchange rates were determined by fitting the decay of the peak heights of each proton resonance with single or double exponential functions using KaleidaGraph (Synergy Software). The intrinsic exchange rates were calculated by the method of Bai and colleagues (Bai et al. 1993). Pf is the calculated intrinsic exchange rate divided by the observed exchange rate, Pf = kint/kobs. The equilibrium constant for the closed (hydrogen bonded) form is Kcl = Pf − 1 (Fezoui et al. 1999).

Extended helix-coil transition theory

Extended helix-coil theory (XHC) is a model for the helix-coil transition which includes long-range tertiary contacts in addition to the local interactions of helix formation (see the accompanying paper in this issue by Hausrath). In this model, residues of the hydrophobic interface that are considered to make long range tertiary contacts are designated as “X” in the binary pattern. The remainder of the residues are not considered to have long range interactions and are designated as “-”. The binary pattern used for αtα is - - X - - - X - - - X - - X - - - - - - - - - - X - - X - - - X - - - X - -. Note that this is not strictly a hydrophobic pattern. The Ala residues are not considered part of the interface and Trp2, while hydrophobic, forms a short range intra-helix cluster with Leu3 and Arg6 and does not form long range contacts.

The allowed tertiary contacts must be specified with a list of contact sets. The sets used here are illustrated in Figure 5. H residues in a helical conformation state can interact with other H residues in helical conformations, conferring an additional statistical weight represented within the model by a tertiary contact parameter, x. Site-specific hydrogen exchange protection factors for a given site were calculated with the XHC model from the probability of the site participating in a hydrogen bond (see the accompanying article by Hausrath). This probability was converted to a protection factor by the equation [Phbond/(1 − Phbond)] + 1 = Pf. The values for the helix propagation parameters (w) used for the XHC calculation are taken from Chakrabartty et al. (1994), and the values for v (the statistical weight for having a coil neighbor) and u (the statistical weight for the coil state) are 0.048 and 1, respectively. The XHC model was implemented in Mathematica (Wolfram Research, Inc.) and calculations were done on a PC running Linux.

Acknowledgments

Funding for this work was provided by the Rowland Institute for Science, Cambridge, MA, and the Department of Biochemistry and Molecular Biophysics, The University of Arizona, Tucson, AZ.

Footnotes

Reprint requests to: John J. Osterhout, Department of Biochemistry and Molecular Biophysics, BSW 352, P.O. Box 210088, University of Arizona, Tucson, AZ 85721; e-mail: johno@email.arizona.edu; fax: (520) 626-9204.

Article published online ahead of print. Article and publication date are at http://www.proteinscience.org/cgi/doi/10.1110/ps.062297006.

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