Helix/coil nucleation: a local response to global demands

Abstract

A complete description of protein structure and function must include a proper treatment of mechanisms that lead to cooperativity. The helix/coil transition serves as a simple example of a cooperative folding process, commonly described by a nucleation-propagation mechanism. The prevalent view is that coil structure must first form a short segment of helix in a localized region despite paying a free energy cost (nucleation). Afterward, helical structure propagates outward from the nucleation site. Both processes entail enthalpy-entropy compensation that derives from the loss in conformational entropy on helix formation with concomitant gain in favorable interactions. Nucleation-propagation models inherently assume that cooperativity arises from a sequential series of local events. An alternative distance constraint model asserts there is a direct link between available degrees of freedom and cooperativity through the nonadditivity in conformational entropy. That is, helix nucleation is a concerted manifestation of rigidity propagating through atomic structure. The link between network rigidity and nonadditivity of conformational entropy is shown in this study by solving the distance constraint model using a simple global constraint counting approximation. Cooperativity arises from competition between excess and deficiency in available degrees of freedom in the coil and helix states respectively.

Figure 1

(a) Generic entropy spectrum. All interactions present in a network are rank-ordered by their entropy components (from top to bottom of the spectrum, j = 1 to N_int). A cutting line defines the Maxwell level, which defines the rigidity transition. Interactions ranked-ordered before the Maxwell level (i.e., j ≤ M_l) reduce S_conf, whereas interactions past it (i.e., j > M_l) do not. The type of interaction is not indicated in this schematic because sorting is based solely on entropy ranking. (b) The Maxwell counting approximation is explained by a simple edge-sharing quadrilateral in two-dimensions (rigid substructures are black, whereas flexible substructures are gray). In the top example, both quadrilaterals are isostatically rigid (meaning, rigid but no redundant constraints). In this case, two identical fluctuating interactions are present (shown as dashed lines), and each pays an entropic cost (e.g., a total cost of 2γ). Here, the entropy cost calculated by Maxwell equals the true network rigidity result due to uniformity. However, in the bottom example, the true entropy cost is only γ because one of the interactions is redundant. However, Maxwell assumes all interactions up to M_l are independent irrespective of their location in the network, which results in overprediction (again, 2γ) of the S_conf cost.

Figure 2

Schematic showing global dependence of Boltzmann factors due to Maxwell counting. Dashed lines represent torsion force constraints. Solid lines represent H-bond constraints, for which there are three per H-bond. Because H-bond constraints have greater entropy cost than torsion force constraints, they are placed in the network first. Six example cases (a–f) show different macrostates accessible to the polypeptide. Constraints are independent when the polypeptide is globally flexible (white) and redundant for a rigid polypeptide (gray). The left-hand bracket of Eq. 2 accounts for example cases a–e. (a) No constraints are present, which defines the random coil reference state. (b) No H-bonds formed, but some helical structure is present. (c) No helical structure formed, despite some H-bonds have formed. (d) Both helical structure and H-bonds have formed. In cases a–d the polypeptide is flexible with as many available degrees of freedom given by the gap between where the last independent constraint is shown, and the Maxwell level. (e) All H-bonds are independent, but the torsion force constraints associated with helical structure exhibit a mixture of being independent and redundant. (f) All torsion force constraints are redundant, but the H-bond constraints exhibit a mixture of being independent and redundant.

Figure 3

McDCM fits to excess heat capacity markedly well. Open circles denote experimental C_p data for the (a) A4, (b) V5, and (c) A6 polypeptides (29). In each case, the solid line is the corresponding McDCM best fit. (d) Light gray open circles denote experimental C_p data for peptide I from Scholtz et al. (30), and the solid line is the corresponding McDCM best fit.

Figure 4

One-dimensional free energy landscapes at fixed temperature (T = 302 K) are shown as a function of number of constraints. All curves were generated with McDCM parameters for the A6 polypeptide. The (black, gray) curves show free energy landscapes with a single basin centered on the (left, right) side when all constraints are modeled as either (independent, redundant). Open circles show a free energy landscape with a double basin predicted by the McDCM, indicating cooperativity arises from a competition between microstates that are primarily flexible in the coil state and rigid in the α-helical state. The vertical dash-dotted line denotes the Maxwell level (i.e., the number of constraints needed to make the polypeptide just rigid), which indicates the polypeptide is globally (flexible, rigid) to its (left, right).

Figure 5

Dependence of cooperativity on chain length. Based on parameters for the A6 polypeptide, the McDCM predicts the melting temperature initially increases with chain length, until a saturation level is reached.