Exploring the origins of topological frustration: design of a minimally frustrated model of fragment B of protein A

Abstract

Topological frustration in an energetically unfrustrated off-lattice model of the helical protein fragment B of protein A from Staphylococcus aureus was investigated. This G-type model exhibited thermodynamic and kinetic signatures of a well-designed two-state folder with concurrent collapse and folding transitions and single exponential kinetics at the transition temperature. Topological frustration is determined in the absence of energetic frustration by the distribution of Fersht phi values. Topologically unfrustrated systems present a unimodal distribution sharply peaked at intermediate phi, whereas highly frustrated systems display a bimodal distribution peaked at low and high phi values. The distribution of phi values in protein A was determined both thermodynamically and kinetically. Both methods yielded a unimodal distribution centered at phi = 0.3 with tails extending to low and high phi values, indicating the presence of a small amount of topological frustration. The contacts with high phi values were located in the turn regions between helices I and II and II and III, intimating that these hairpins are in large part required in the transition state. Our results are in good agreement with all-atom simulations of protein A, as well as lattice simulations of a three- letter code 27-mer (which can be compared with a 60-residue helical protein). The relatively broad unimodal distribution of phi values obtained from the all-atom simulations and that from the minimalist model for the same native fold suggest that the structure of the transition state ensemble is determined mostly by the protein topology and not energetic frustration.

Figure 1

Folding landscape frustration diagram. The axes denote energetic and topological frustration. In the upper left corner of the figure (point 1), no energetic or topological frustration is present and the folding landscape is a smooth funnel. At the other extreme (point 2), energetic and topological frustration are prevalent and the landscape is very rugged. Proteins will fold on a landscape that lies between these extremes. The degree of energetic and topological frustration will vary with the design of the protein and the choice of the native topology. As frustration increases along either axis the distribution of φ values becomes less ideal and tends to spread out until reaching a limiting bimodal distribution denoting strong pathway dependence of folding.

Figure 2

Temperature dependence of the specific heat (a) and average number of native contacts (b) for the optimized PA model. Together these provide an indication of the first-order-like folding of this model protein.

Figure 3

Thermodynamic functions at the transition temperature. (a) Free energy (G, in kcal/mol) as a function of energy (V in kcal/mol). (b) Number of native contacts, Q, as a function of energy.

Figure 4

Distribution of φ values, N(φ), at the transition temperature for thermodynamically and kinetically determined φ values.

Figure 5

Correlation between thermodynamic and kinetic φ values.