Exploring Flory's isolatedpair hypothesis: Statistical mechanics of helix–coil transitions in polyalanine and the Cpeptide from RNase A
See allHide authors and affiliations

Edited by Harold A. Scheraga, Cornell University, Ithaca, NY, and approved September 16, 2003 (received for review July 8, 2003)
Abstract
To evaluate Flory's isolatedpair hypothesis in the context of helical peptides, we explore equilibrium conformations of αhelixforming polypeptides as a function of temperature by means of replica exchange molecular dynamics in conjunction with the CHARMM/GB implicit solvent force field and the weighted histogram analysis method. From these simulations, Zimm–Bragg parameters, s and σ, of AcAla_{n}NMe are computed as a function of temperature. The values obtained for s(T) and σ(T) remain unchanged along the length of the polypeptide except for very short chains and yield results consistent with measurements based on short helixforming peptides but suggest larger s values than anticipated from polymerbased measurements. From direct estimates of the density of states for AcAla_{n}NMe (n = 3–20) and peptide constructs based on the C peptide from RNase A, the conformational entropy is calculated versus temperature. The calculated S(T) shows a clear proportionality to the chain length over a wide range of temperature. This is observed in polypeptides with both significantly branched and simple methyl (alanine) side chains. These results provide evidence for the validity of Flory's isolated pair hypothesis, at least in the context of helical peptides and helixtocoil transitions in these peptides.
Flory suggested that it is safely assumed that the state of each pair of rotation angles, (ϕ, ψ), in a polypeptide is independent of the adjoining pair (1). This means that the total number of possible conformations of a polypeptide chain can be obtained as the product of the number of possible conformations of a N and Cterminally blocked peptide consisting of each residue in the full peptide. This “isolatedpair hypothesis” has been widely accepted because of its simplicity in relating local conformational properties to the number of possible conformations of longer peptides. However, this hypothesis also implies that the number of conformations grows exponentially with the increase of chain length, which leads to a longasked question of how a protein can fold to its native conformation from the vast majority of nonnative states on the time scale of minutes or less (2). One answer to this “paradox,” for the folding of protein molecules, comes from the theory of funneled energy landscapes (3), which suggests that the folding problem is a directed search on a sloped energy landscape. Support for this idea, as well as the notion that protein molecules adopt nativelike topologies, even in their denatured states (4–6), has become apparent from both experimental and theoretical studies during the past several years. The situation for peptides, however, remains unclear, although it has been suggested that the number of conformations is overestimated and that the isolatedpair hypothesis is not valid (7–9).
In one recent study, Rose and coworkers (8) evaluated the isolatedpair hypothesis by means of Monte Carlo exhaustive sampling. The results from their findings suggest a failure of the isolatedpair hypothesis and that the entropy cost on folding is much smaller than presently believed. Similar conclusions were reached by Sosnick and his colleagues (7) in studies of short peptides of length 1–3 aa. These results appear to be in accord with the findings of van Gunsteren and coworkers (9), who showed that a limited number of conformers were observed for peptides of various lengths during long molecular dynamics simulations. While Rose and coworkers argue that the failure of the hypothesis is caused by nonnearestneighbor, local steric clashes that occur mainly between residues i and i + 3, i + 4, or i + 5, their results are based on polypeptides of <10 residues. Their suggestions seem quite reasonable, because it is obvious that within this range (<10 residues) there are fewer residues that might clash for the smaller peptides (e.g., there is no possible clash between i and i + 3 in a tripeptide, and there are only two in a pentapeptide and four in a septapeptide). This “end effect” is expected, however, to decrease and die away as the length of the peptide increases. This leads us to ask: Will the isolatedpair hypothesis hold for longer chains, even if it appears to fail for short ones, or will nonlocal remote clashes also play a vital role in diminishing the available conformations such that Flory's ideas fail for peptides of any chain length? We can formulate these questions more quantitatively in terms of the entropy of a peptide chain, because the number of available conformations and entropy are intimately coupled.
For a set of capped polyalanine peptides, AcAla_{n}NMe, at a certain temperature, the isolatedpair hypothesis may be formulated as [1] with the equality denoting that the hypothesis holds. S_{n} is the entropy of AcAla_{n}NMe, and S_{n}_{,}_{i} is that of the residue i of AcAla_{n}NMe (e.g., S_{1,1} is the entropy of residue 1 of AcAlaNMe; in this case, S_{1,1} = S_{1}, because there is only one residue). If nonnearestneighbor, local steric clashes are the only factor that lead to the inequality in Eq. 1, we would still anticipate proportionality between n and S_{n} for some larger n: [2] where n_{min} is the minimum chain length at which the “end effect” becomes insignificant, c is a positive constant, is the average of S_{n}_{,}_{i} as S_{n}/n, and . If nonlocal steric clashes (i.e., any clash between residue i and i + 6 or larger) are not negligible, the lefthand side of Eq. 2 falls more and more short of the equality as n increases. Rose and coworkers (8) have suggested that in their systems the equality in Eq. 1 does not hold. Our question here is whether the hypothesis holds perfectly (Eq. 1), partially (Eq. 2), or not at all, and if it holds partially, for what length of peptide, n_{min}?
By employing one of the currently most efficient computational search methods [replica exchange molecular dynamics, or REMD (10)] and an implicit solvent force field that is efficient to compute (CHARMM/GB) (11), we explore the validity of Flory's hypothesis for polypeptides up to 26 residues in length, which are long enough to observe nonnearestneighbor steric effects.
To examine the general quality of our force field modeling and sampling methodology, we calculate (the average) thermostatistical properties of the helixforming polypeptides and compare their features, as given by the CHARMM/GB force field, with those from experiment. To do this, we employed the standard (μ = 1) Zimm–Bragg (ZB) model (12) to estimate the helix nucleation and propagation parameters σ and s, respectively, for peptides AcAla_{n}NMe of different chain length. The standard ZB model is isomorphic with the onedimensional Ising model, because of its assumption of local interaction between adjacent peptide units only. If the isolatedpair hypothesis holds and nonnearestneighbor steric effects are not significant, the parameters σ and s should be independent of the chain length.
We continue our analysis by calculating the entropy of equilibrated conformations of each polypeptide as a function of temperature. To examine the adherence to Eq. 1 and Eq. 2, the density of states of the peptide is estimated by combining REMD sampling at different temperatures by using the WHAM method (13, 14). If the isolatedpair hypothesis is valid, the calculated S(T) should increase with increasing chain length (Eq. 1). Otherwise, S(T) is expected to fall short of the proportionality at larger chain length.
Materials and Methods
Capped Polyalanine and CPeptide Homologues.Nacetyl(lalanil)_{n}N′methylamide, or AcAla_{n}NMe, where n = 6–15 and 20, were generated using CHARMM (15) with the param19 polar hydrogen force field (16). The peptide related to the Cpeptide from ribonuclease A (KETAAAKFLRQHM, Cp13) was prepared in the same way. Three homologues of Cp13, the tandem (Cp26), the Nterminal seven residue peptide fragment (CpN7), and the Cterminal six residue peptide (CpC6) were prepared similarly. Cp26 has the same composition of side chains as Cp13, but CpN7 and CpC6 do not. Nevertheless, the average of the thermostatistical properties of CpN7 and CpC6 can be regarded as the properties of a hypothetical peptide of 6.5 residues whose side chain composition is the same as that of Cp13.
The Definition of a Hydrogen Bond. We use the criteria of Ravishanker et al. (17) to define an αhelical hydrogen bond (i.e., between residues i and i + 4). Specifically we employ the definition: 1.5 Å ≤ r_{O—H} ≤ 3.0 Å and 120° ≤ ∠ C—O—H ≤ 180°.
Data Generation and Collection: REMD. The replicaexchange method (18) is one of the generalizedensemble algorithms that yield simulations with nonBoltzmann weight factors and enable a randomwalk search throughout a specific phase space. Sugita and Okamoto (10) have combined this method with molecular dynamics as REMD to provide sampling within the canonical ensemble over a wide range of temperatures for protein folding studies. In the present study, we also use REMD, with only a few minor changes, to generate the ensembles of conformations for the polypeptides mentioned above. We assign a replica of each peptide to one of sixteen temperature windows, which are exponentially distributed between 200 and 600 K (200.0 K, 215.2 K, 231.6 K, 249.1 K, 268.1 K, 288.4 K, 310.4 K, 334.0 K, 359.3 K, 386.6 K, 416.0 K, 447.6 K, 481.6 K, 518.2 K, 557.6 K, and 600.0 K). To ensure sufficiently complete conformational sampling in the molecular dynamics, a continuum solventbased generalized Born force field, CHARMM/GB (11), is used. Furthermore, the SHAKE algorithm (19) is used to constrain hydrogenheavy atom bonds, and the time step for dynamics integration is set to 2.0 fs. Every one thousand steps (2.0 ps), a conformation is recorded in each temperature window and a conformational exchange is attempted. Each simulation was started with replicas in a fully extended conformation that was generated with all the ϕ, ψ, and ω set to 180° and then energyminimized. Ten thousand conformations from each window, between 20 and 40 ns, were used for analysis. Temperature adjustment was implemented by way of assigning velocities from a Gaussian distribution appropriate for the temperature window. This also applies to the end of each cycle, where the replicas are exchanged. We believe random reassignment of velocities from a Gaussiandistributed sample may be a better scheme than uniform scaling of the atomic velocities. Reassignment enables the search to move more rapidly into a different hyperplane of the phase space while maintaining correct coupling to the canonical temperature bath. Thus, equation 12 in ref. 10 is not adopted here. All simulations were carried out using the MMTSB Tool Set (20).
ZB Model. We employ the standard (μ = 1) ZB model (12) without the infinite chain length (largen) approximation, because the polypeptides used here are not very long. We outline the equations in this limit below. The details of the model have been well described elsewhere (12, 21, 22). The partition function, q_{n}, by means of the standard (μ = 1) ZB model is [3] where Ω_{j}_{,}_{k} is the number of ways to put k αhelical hydrogen bonds into j segments, σ is the helix nucleation parameter, and s is the helix propagation parameter. Note that n – 2 is the maximum possible number of αhelical hydrogen bonds in a peptide with both termini capped, such as AcAla_{n}NMe. Because of the assumption of μ = 1 (local interaction between adjacent peptide units only), q_{n} can also be described in a format that is more suitable for computation, [4] where ; λ_{1} ≥ λ_{2}.
The average number of αhelical hydrogen bonds, 〈k〉, and the average number of αhelical segments, 〈j〉, are [5] The fractional helicity, θ, is simply 〈k〉/(n – 2), [6] The average length of a helical segment, 〈k〉/〈j〉, is [7] Because the coordinates of the molecules at each temperature are recorded during the REMD simulation, every αhelical hydrogen bond in each molecule is located. The availability of this microscopic information enables us to obtain not only the fractional helicity but also the average length of helical segments directly, in order to fit s and σ at each temperature by Eqs. 6 and 7. We note that such is not the case in most experiment and therefore simultaneous determination of the parameters σ and s as a function of temperature are generally precluded (23–25). We obtain optimal values for s and σ by bruteforce exploration of the (s, σ) space for the pair of s and σ that minimizes the sum of the squared relative errors of θ and 〈k〉/〈j〉: where θ^{obs} and (〈k〉/〈j〉)^{obs} are from our REMD results. θ^{(}^{s}^{,σ)} and (〈k〉/〈j〉)^{(}^{s}^{,σ)} are those from Eqs. 6 and 7, given a set of s and σ. Typical values of the error function after optimization were on the order of 10^{–8}.
Estimation of Conformational Entropy. The WHAM method was utilized to build an estimate of the density of states, Ω_{n}(i_{Hb}, E_{j}), for the peptides as a function of the number of αhelical hydrogen bonds i_{Hb} and potential energy level E_{j}. The partition function, q_{n}, of AcAla_{n}NMe at a given temperature, therefore, in terms of the estimated Ω_{n} (i_{Hb}, E_{j}) is simply [8] where i_{Hb} corresponds to i hydrogen bonds formed, E_{j} is the jth energy level, T is the temperature, and k_{B} is the Boltzmann constant. From the partition function the remaining thermodynamic properties follow simply. The total entropy of the peptide at T is given as S^{total}n (T) = (E_{n}(T) – A_{n}(T))/T, where the energy is E_{n}(T) = k_{B}T^{2} (∂ ln q_{n}/∂T) and the free energy is A_{n}(T) = –k_{B}T ln(q_{n}). The conformational entropy of the peptide at T is [9] with the vibrational entropy Θ_{νk} = hν_{k}/k_{B} is the vibrational temperature and h is Planck's constant (26). Each normal mode, ν_{k}, is obtained from an energyminimized, fully αhelical conformation of the molecule. Alternatively, one could consider the ensemble of peptide conformations at each temperature and carry out vibrational analysis for each member of these ensembles to remove the vibrational contribution to the entropy. Such a calculation is very intensive, and if (as we verified in the present case) little difference is seen between the vibrational entropy of helical and extended conformations, this additional computational effort is unnecessary.
It should be noted that the estimated Ω_{n}(i_{Hb}, E_{j}), and therefore E_{n}(T), A_{n}(T), and S_{n}(T), are independent of the bin size used in i and j, as long as i and j are not too small and the bins span the entire reaction coordinate and energy range. We use n – 1 for i binning as a natural choice, because the possible number of αhelical hydrogen bonds in AcAla_{n}NMe ranges from 0 to n – 2. For j, we trace the lowest and highest energies among the REMD trajectory and use fifty equally spaced bins between these two values.
Results and Discussion
Analysis of ZB Parameters. In Fig. 1, we show the temperature dependence of the helicity of AcAla_{15}NMe. The results from our model for the fraction of αhelical hydrogen bonds, θ (Fig. 1a), is slightly higher than experiment and the “folding transition” is shifted to a higher temperature (≈350 K). However, the overall shape of this temperaturedependent curve mimics experiment. The average length of helical segments is shown in Fig. 1b and suggests that as the temperature is lowered below 300 K, the length of contiguous helix increases significantly. The number of αhelical hydrogen bond segments is shown in Fig. 1c. This property is defined as those hydrogenbonded segments separated by a series of nonhydrogenbonded peptide units of three or more, because at least three broken hydrogen bonds are needed to break a helix completely into two segments. Although there is an overall temperature shift, the CHARMM/GB model yields a reasonable distribution of polypeptide conformations compared to experiments on similar polypeptides (27, 28).
The fitted values of the ZB parameter s(T) for the peptide AcAla_{15}NMe from sixteen temperature windows are plotted in Fig. 2 against θ, 〈k〉/〈j〉, and 〈j〉. Eqs. 5–7 are also plotted for fixed σ as solid lines for guidance. The red curves are those from the commonly used largen approximation, in which θ and 〈k〉/〈j〉 are nindependent. In this approximation 〈k〉/〈j〉 shoots up beyond n – 2 (= 13 in this case), the longest possible length of a helix in the peptide, whereas the “exact” 〈k〉/〈j〉 (Eq. 7) with small σ quickly converges to n – 2 as s increases. Also, the approximate 〈j〉 does not converge to 1 along s as it is supposed to do. It is clear that for the peptides used here the largen approximation is not appropriate or applicable.
In Fig. 3 a and b, the fitted values of the s(T) and σ(T) of AcAla_{n}NMe are plotted against temperature. Both s(T) and σ(T) are basically unchanged for different n, consistent with the isolatedpair hypothesis, although some deviations are observed for n = 6 and7in s(T) and 6–9in σ(T). The monotonic decrease of s(T) with temperature and the specific value of s = 1.5 at room temperature agrees well with experiments on short peptides (27, 29) but is counter to the findings for s of alanine by polymerbased experiments (30) and datamining methods (31). The σ(T) increases until the transition temperature has a bump there, and then reaches a plateau. At lower temperatures σ(T) is of the same order of magnitude as the fixed σ determined by other groups (29, 30) but is an order of magnitude too large at other temperatures.
From s(T) we can estimate thermodynamic properties of αhelix formation for alanine. In Fig. 3c, the free energy change of an alanine residue from a coil manifold of states to αhelical states, ΔG = –RTln(s(T)), is plotted against temperature. The regression line to this data over a range of temperatures gives the enthalpy change, ΔH, as the intercept and the entropy change, ΔS, as the negative slope, assuming the ΔS is constant within the temperature range. From the range between 200 and 386.6 K, ΔH and ΔS are estimated to be –0.83 kcal·mol^{–1} and –2.11 cal·mol^{–1}·K^{–1}, respectively. By calculating σ ≈ exp(3ΔS/k_{B}), σ is found to be 0.042, which matches well with σ(T) in the same temperature range (see Fig. 3b). The ΔH by calorimetry is –0.9 ± 0.1 kcal·mol^{–1} (32), and as Kallenbach and coworkers (33) calculated, this leads to ΔS of –2.20 ± 0.37 cal·mol^{–1}·K^{–1}. Our results are in agreement with these. We note that unfolded conformations from our simulations are also observed to sample the polyproline II helical region observed by Kallenbach and coworkers; however, the population of such conformations is small. The meaning of the regression line for high temperatures is not clear. It gives ΔS of 1.05 cal·mol^{–1}·K^{–1} for ≥400 K, which results in a disturbingly small σ of 1.3 × 10^{–7}.
Similar values for s(T) and largerthanexperiment σ(T) were also observed in other simulations. Using the CHARMM force field with explicit solvent, a large equilibrium constant (equivalent to σs) was obtained for AcAla_{3}NMe (34), and similar values of s were found for AcAla_{n}NMe (3 ≤ n ≤ 15) at ambient temperature (35). Also, Mitsutake and Okamoto (36) calculated s(T) and σ(T) by using the largen approximation for short peptides by a force field based on ECEEP/2. Their σ(T) is even larger than that we observe, but it would be about the same if fitted without this approximation (cf. Fig. 2b). Furthermore, Garcia and coworkers (37, 38) observed a similarly larger σ value for polyalanines in both explicit and implicit (GB) water when using the AMBER force field. Because similar s and σ values were obtained from a number of different workers using different force fields, the energy landscapes representing polypeptides from these force fields are likely to be similar, furthering the suggestion that our findings regarding the isolatedpair hypothesis (see next section) are not CHARMM/GB specific.
The experimentally obtained σ itself varies by more than an order of magnitude (29, 30); consequently, it is difficult to judge the causes of the difference between the simulated σ and experimental values. One possible reason is that these force fields favor helical conformations, consistent with findings by Garcia and Sanbonmatsu (37), who modified the AMBER force field to reduce helical propensity and found σ values closer to experiment. This finding found further support when Zaman et al. (7) showed that the modified AMBER, as well as OPLSAA, favors helical conformations less than other force fields. These findings suggest that OPLSAA would probably yield a σ value for helix initiation closer to experiment. In addition, our observation that the current force fields underestimate the entropy cost of helix initiation could be a result of inadequate representations of the hydrophobic effect in current models with implicit (38) and explicit solvent representations (34, 35, 38).
Conformational Entropy. To examine the length and composition dependence of the peptide entropy, we calculate the conformational entropy at a given temperature T, S_{n}(T), by Eq. 9. This is shown in Fig. 4 for both the AcAla_{n}NMe peptides and the Cpeptide homologues. The curves are shifted so that the minimum value matches zero, which is appropriate if only one (or a few) states are sampled at the lowest temperatures, as we see in our simulations. These curves should reach plateau values at low and high temperatures, and this behavior does occur. However, because of our approximate removal of vibrational entropy as described above, some additional curvature occurs at these extremes. The general feature that is quite clear is that the longer the molecule the larger the entropy, and for each molecule, the higher the temperature the larger the entropy, as expected. Fig. 5 a and b display S_{n}(T) versus n at 300, 400, and 500 K. For all temperatures, S_{n}(T) for both AcAla_{n}NMe and the Cpeptide homologues is linear in n (correlation coefficient of 0.95 or higher, except 0.71 of AcAla_{n}NMe at 300 K).
If Eq. 2, Flory's hypothesis, does not hold, or nonlocal steric clashes are not negligible, Eq. 2 should fall short of the equality as n increases. This failure should be more obvious in the Cpeptide homologues than in AcAla_{n}NMe, because the larger side chains may cause clashes even if the main chain portions do not. These plots show a linear n dependency; hence, the effects of nonlocal steric clashes must be minimal. Although n is limited to 20 (AcAla_{n}NMe) or 26 (the Cpeptide homologues) and is not evaluated for some of the integers in those ranges, the clear proportionality between S_{n}(T) and n in Fig. 5 a and b indicates that Eq. 2 holds and that n_{min} is not within this range of n.
What, then, is n_{min}? Its value must be ≤6, because n_{min} is the smallest n for which we observe proportionality between S_{n} and n. On the other hand, if nonnearestneighbor effects are obvious, n_{min} should appear somewhere around 10, because n_{min} is the minimum length at which the “end effect” of nonnearestneighbor local clashes becomes negligible. To further explore this question, the same calculations were carried out for n = 3–5, and S_{n}(T) of AcAla_{n}NMe at 300 and 400 K are displayed as the red filled and small black circles with the red regression lines in Fig. 5 c. The black lines are identical to those in Fig. 5a. All the red circles fall on or below the black regression lines; however, the points of n = 4 and 5 are closer to the regression lines than those of n = 3. Note that the red lines have steeper slopes than the black lines, just like S_{1,1} of Eq. 1 and of Eq. 2. Also, the intercepts of these red lines are close to zero, whereas those of the black lines have positive intercepts (>0.02 kcal·mol^{–1}·K^{–1}), like c of Eq. 2. At 500 K, the regression line for n = 6–20 has a near zero intercept (Fig. 5 a), which makes sense because the majority of conformations are rather extended and local steric effects become less significant at this high temperature. These observations suggest that, basically, Eq. 1 holds for the region of n ≤ 5 and Eq. 2 holds for 6 ≤ n; thus, n_{min} is 6.
This finding is in general agreement with the results from Rose and coworkers that indicate the failure of Flory's hypothesis (Eq. 1) in Ala_{n} of n = 5 (8). However, their results suggest a growing divergence from Eq. 2 as n increases. Our findings suggest that this is not the case. The observations by van Gunstern and coworkers (9), that the number of conformers observed will not exponentially increase with the chain length, is counterintuitive and interesting. In table 1 of their article, however, there seems to be a correlation between the number of conformers and simulation length for (almost) the same chain length, possibly indicating insufficient conformational sampling.
From the findings we present above, we conclude that Flory's isolatedpair hypothesis does not fail in the form of Eq. 2, at least for AcAla_{n}NMe and the Cpeptide homologues with the CHARMM/GB force field. There do exist nonnearestneighbor, local steric effects (i.e., c in Eq. 2 is not zero), but nonlocal steric effects are not significant. However, we note that calculations for AcAla_{n}NMe of even larger n or proteins with topologies other than αhelical have not been examined, and there remains the possible presence of the nonlocal, remote steric effects in these systems. Clearly, such effects will occur as a result of the finite volume of the peptide chain, as peptides long enough to achieve their persistence length [20 Å for polyalanine (39)] under specific solvent conditions are accessed.
The slope of the regression lines in Fig. 5 a and b give the average conformational entropy per residue. In Fig. 5 d, we display S(T) per residue for AcAla_{n}NMe and the C peptide. With the assumption of additivity between the entropy of the main chain and the side chain, the entropy difference between the two curves gives the average side chain entropy per residue of the C peptide. For instance, at 300 K the entropy difference of 0.00475 kcal·mol^{–1}·K^{–1} corresponds to about 11 conformations per residue. Because there are a total of 31 χ angles in the C peptide, the average number of dihedral angles per residue is 2.38, leading to about 2.7 conformations per χ angle.
Conclusions
We have shown that the conformational entropy of capped polyalanine, AcAla_{n}NMe, is proportional to the chain length, n, over a wide range of temperatures and for chain lengths of n = 3–20. The proportionality between chain entropy and chain length is also observed in a set of polypeptides with side chains, the Cpeptide homologues that maintain the same composition of residue types. The ZB parameters s(T) and σ(T) of AcAla_{n}NMe are basically independent to n, in accordance with the observed entropychain length proportionality. These results support the validity of the isolatedpair hypothesis with local steric effects.
Acknowledgments
Y.Z.O. thanks J. Karanicolas and K. V. Damodaran for insightful discussions and technical assistance. The REMD algorithms used here are based on methods developed by M. Feig and J. Karanicolas. These methods are available through the National Institutes of Health funded research resource (RR12255) Multiscale Modeling Tools in Structural Biology at http://mmtsb.scripps.edu. Financial support from the National Institute of Health (Grant GM48805) is greatly appreciated.
Footnotes

↵* To whom correspondence should be addressed. Email: brooks{at}scripps.edu.

This paper was submitted directly (Track II) to the PNAS office.

Abbreviations: REMD, replica exchange molecular dynamics; ZB, Zimm–Bragg.
 Received July 8, 2003.
 Copyright © 2003, The National Academy of Sciences
References
 ↵
Flory, P. J. (1969) Statistical Mechanics of Chain Molecules (Wiley, New York).
 ↵
Levinthal, C. (1969) in Mossbauer Spectroscopy in Biological Systems, eds. Debrunner, P., Tsibris, J. C. M. & Muenck, E. (Univ. of Illinois Press, Urbana), pp. 22–24.
 ↵
 ↵
Shortle, D. & Ackerman, M. S. (2001) Science 293, 487–489.pmid:11463915
 ↵
 ↵
 ↵
Pappu, R. V., Srinivasan, R. & Rose, G. D. (2000) Proc. Natl. Acad. Sci. USA 97, 12565–12570.pmid:11070081
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
Vijakumar, S., Vishveshwara, S., Ravishanker, G. & Beveridge, D. L. (1994) Am. Chem. Soc. Symp. Ser. 569, 175–193.
 ↵
 ↵
 ↵
Feig, M., Karanicolas, J. & Brooks, C. L., III (2003) J. Mol. Graph. Model, in press.
 ↵
Cantor, C. R. & Schimmel, P. R. (1980) Biophysical Chemistry (Freeman, New York), Vol. 3, pp. 1041–1074.
 ↵
 ↵
 ↵
 ↵
McQuarrie, D. A. (2000) Statistical Mechanics (University Science Books, Sausalito, CA).
 ↵
Marqusee, S., Robbins, V. H. & Baldwin, R. L. (1989) Proc. Natl. Acad. Sci. USA 86, 5286–5290.pmid:2748584
 ↵
 ↵
 ↵
 ↵
Ooi, T. & Oobatake, M. (1991) Proc. Natl. Acad. Sci. USA 88, 2859–2863.pmid:2011595
 ↵
Lopez, M. M., Chin, D.H., Baldwin, R. L. & Makhatadze, G. I. (2002) Proc. Natl. Acad. Sci. USA 99, 1298–1302.pmid:11818561
 ↵
Shi, Z., Olson, C. A., Rose, G. D., Baldwin, R. L. & Kallenbach, N. R. (2002) Proc. Natl. Acad. Sci. USA 99, 9190–9195.pmid:12091708
 ↵
 ↵
 ↵
 ↵
Garcia, A. E. & Sanbonmatsu, K. Y. (2002) Proc. Natl. Acad. Sci. USA 99, 2782–2787.pmid:11867710
 ↵
Nymeyer, H. & Garcia, A. (2002) Prot. Sci. 11, Suppl. 1, 205.
 ↵
Cantor, C. R. & Schimmel, P. R. (1980) Biophysical Chemistry (Freeman, New York), Vol. 3, pp. 1013.