The Flory isolated-pair hypothesis is not valid for polypeptide chains: implications for protein folding

Abstract

Using an all-atom representation, we exhaustively enumerate all sterically allowed conformations for short polyalanyl chains. Only intrachain interactions are considered, including one adjustable parameter, a favorable backbone energy (e.g., a peptide hydrogen bond). The counting is used to reevaluate Flory's isolated-pair hypothesis, the simplifying assumption that each phi,psi pair is sterically independent. This hypothesis is a conceptual linchpin in helix-coil theories and protein folding. Contrary to the hypothesis, we find that systematic local steric effects can extend beyond nearest-chain neighbors and can restrict the size of accessible conformational space significantly. As a result, the entropy price that must be paid to adopt any specific conformation is far less than previously thought.

Figure 1

The labeled coarse-grain bins (mesostates) for a φ,ψ pair are superimposed on a Ramachandran map (gray) of the alanine dipeptide. The 14 populated mesostates are: {A,G,M,R,L,F,E,K,Q,J,P,O,I,o}, and only the L mesostate is fully allowed. The fractional occupancy of each mesostate is listed in Table 1. The Ramachandran map was computed by generating 150,000 independent conformations within each mesostate by using backbone-dependent values for the N–C_α–C′ bond angle (Table 1).

Figure 2

“Blurograms” of four mesostate strings in chains of length n = 5: (a) OOOOO, (b) PPPPP, (c) LLLLL, and (d) IoJOP. Within a given mesostate string, sterically allowed conformers are structurally similar, like an NMR solution set. For each string illustrated here, 30 sterically allowed conformers were selected at random and superimposed.

Figure 3

Testing the isolated pair hypothesis. The 14 populated mesostates were subdivided into two sets: (a) {A,G,M,R,L,F,E,K,Q} and (b) {J,P,O,I,o}. The isolated-pair hypothesis holds only for higher-order conformations derived from set a. In the experiment illustrated, 200 mesostate strings of length n = 5 were generated, half using random combinations of letters chosen from set a, the other half using random combinations of letters from set b. For each string, 1.75 × 10⁶ independent conformations were generated. The number of conformers expected, Γ_expected, is plotted against the number allowed, Γ_A, for the 100 strings in sets a [+] and b [*]. The correlation coefficient between expected and allowed conformations is 0.99 in set a and 0.3 in set b.

Figure 4

Non-nearest-neighbor steric clashes involving the five contracted mesostates in strings of length n = 9: (a) J9, (b) P9, (c) O9, (d) I9, and (e) o9. For each mesostate string, 2 × 10⁷ independent conformers were generated. The fraction of non-nearest-neighbor steric clashes (a proper superset of the steric map for an alanine dipeptide) is plotted as a function of separation between chain monomers. For example, approximately two-thirds of the steric clashes in α helices (O₉) are between residues at sequence separations of i and i + 4, an intuitively reasonable result in an α helix, which has 3.6 residues per helical repeat. Similarly, most of the steric clashes in 3₁₀ helices (P₉) are between residues at separations of i and i + 3.

Figure 5

Distribution of allowed helical φ,ψ values in environments of interest: (a) the O mesostate in an alanine dipeptide; (b) the O mesostate in the central residue of Ac-Ala₉-N′-methylamide, with other residues held fixed at (−63°, 45°); and (c) the central residue of Ac-Ala₉ -N′-methylamide, with all residues allowed to vary uniformly within the O mesostate. In the 9-mer, the allowed region is winnowed substantially by higher-order local steric effects not present in an alanine dipeptide (b). Such effects persist, even when the 9-mer is allowed to relax (c). For comparison, the distribution of O mesostate φ,ψ values for all non-glycine, non-proline residues from 236 proteins of known structure (39) is shown in d. The 10,879 residues are from the December 1998 release of PDB_Select (40). A subset of these database residues was excised from the middle of 9-mers of polyO mesostate strings (shown in red); they cluster tightly around canonical α-helical φ,ψ values. The overall distribution in d is a subset of a relaxed polyalanine peptide constrained to the O mesostate (c), whereas the points in red are a subset of allowed values in a canonical α helix (b).

Figure 6

Mean radii of gyration, <R_g>, as a function of hydrogen bond strength, ɛ, for polyalanyl a chain of length n = 7. The mean radii of gyration used here track with other thermodynamic averages of experimental interest. All behave in a discernable “two-state manner,” with contracted conformers favored at stronger hydrogen bond strengths and extended conformers favored at weaker hydrogen bond strengths. Mean radii are calculated from Boltzmann-weighted contributions over all mesostate strings for given values of n and ɛ, i.e., 〈R_g〉 (ɛ) = ∑_i=1¹⁴ⁿ R_gⁱρ_i(ɛ). Here R_gⁱ is the average radius of gyration for allowed conformers in mesostate string i, and ρ_i(ɛ) is the Boltzmann weight of mesostate string i with hydrogen bond strength = ɛ and T = 300 K. The structures that make significant contributions to the Boltzmann-weighted population at key positions along the two-state curve are shown. When hydrogen bonds are strong (ɛ = −4.0 kcal/mol), 3₁₀ helices dominate, although some α helix is also present. At the midpoint (ɛ = −2.00 kcal/mol), other conformations are also seen, including type II turns (turn/loop) and extended conformers (β). When hydrogen bonds are weak (ɛ = −1.0 kcal/mol), extended conformers predominate.

Figure 7

Fraction of the 14ⁿ mesostate strings needed to account for at least 90% of the Boltzmann-weighted equilibrium population, plotted as a function of hydrogen bond strength, ɛ, for chains of length n = 3 (x), 4 (square), 5 (diamond), and 6 (triangle). Each data point was calculated as follows: for a chain of length n and energy ɛ, the normalized Boltzmann weight of a given mesostate string is w_i(ɛ), with 0 < w_i(ɛ) ≤ 1. The sum of w_i(ɛ) over all 14” such strings is unity. We compute a fraction f(ɛ), 0 < f(ɛ) ≤ 1, such that the sum over 14ⁿ f(ɛ) mesostate strings is ≥ 0.9. In detail, strings are sorted in descending order by population and then summed until a threshold of 0.9 is attained. This fraction, represented as a percentage, is plotted as a function of ɛ. The figure shows that polyalanyl chains visit only two distinct regions in conformational space: a smaller island of contracted conformers and a larger island of extended conformers as shown in Fig. 6. The former can be stabilized by favorable backbone interactions, whereas the latter cannot. Entropy favors the larger island. However, the energy of hydrogen bonds is weighted exponentially, and their contributions to equilibrium quickly outpace entropy. Thus, the smaller island is populated preferentially as either chain length or hydrogen bond strength increases.