A statistical mechanical model for β-hairpin kinetics
Abstract
Understanding the mechanism of protein secondary structure formation is an essential part of the protein-folding puzzle. Here, we describe a simple statistical mechanical model for the formation of a β-hairpin, the minimal structural element of the antiparallel β-pleated sheet. The model accurately describes the thermodynamic and kinetic behavior of a 16-residue, β-hairpin-forming peptide, successfully explaining its two-state behavior and apparent negative activation energy for folding. The model classifies structures according to their backbone conformation, defined by 15 pairs of dihedral angles, and is further simplified by considering only the 120 structures with contiguous stretches of native pairs of backbone dihedral angles. This single sequence approximation is tested by comparison with a more complete model that includes the 215 possible conformations and 15 × 215 possible kinetic transitions. Finally, we use the model to predict the equilibrium unfolding curves and kinetics for several variants of the β-hairpin peptide.
Footnotes
-
↵ * To whom reprint requests may be addressed: e-mail: vmunoz{at}helix.nih.gov or eaton{at}helix.nih.gov.
-
This paper was presented at the colloquium “Computational Biomolecular Science,” organized by Russell Doolittle, J. Andrew McCammon, and Peter G. Wolynes, held September 11–13, 1997, sponsored by the National Academy of Sciences at the Arnold and Mabel Beckman Center in Irvine, CA.
-
↵ † When pairs of dihedral angles are used instead of single dihedral angles, the specification of a pair of angles produces a problem in phasing between the loss of entropy and the compensating decrease in interaction free energy. Either choice of φ,ψ pairs represents a compromise. This can be illustrated by considering the formation of a six-residue β-hairpin with a side–chain interaction between residues two and five. To form the backbone–backbone hydrogen bond requires native values for four dihedral angles, φ 3,ψ 3,φ 4,ψ 4. If we were only concerned with hydrogen bond formation, as in helix-coil theory for homopolypeptides, then the natural choice for the dihedral angle pairs would be the φ and ψ associated with the same residue—in this case the two pairs φ 3,ψ 3 and φ 4,ψ 4. With this choice, however, formation of the two- to five-side–chain interaction requires that eight dihedral angles assume native values—when only six, i.e., ψ 2,φ 3,ψ 3,φ 4,ψ 4, and φ 5, actually are required. So, in choosing ψ i,φ i+1 instead of φ i,ψ i pairs, we overestimate the loss in entropy associated with formation of the first hydrogen bond, in favor of accurately representing the compensation between entropy loss and formation of side–chain interactions in subsequent steps.
-
↵ ‡ The system of equations is stiff and was integrated using an iterative multi-step backward differentiation formula method (37), as implemented in the cvode package (36, 38). This algorithm requires the solution of a set of nonlinear algebraic equations by Newton iteration at each time step. Each Newton iteration in turn requires solving an NxN linear system AΔP = residual, where the matrix A is derived from the rate matrix K. For n = 32,768, this problem is rather too large to solve using standard methods (39). However, the matrix A is sparse, containing only ≈500,000 nonzero elements of a possible 109. Therefore, an iterative generalized minimal residual method (40) appropriate for large sparse linear systems, as implemented in the cvode package (36, 38), was used. The performance of the algorithm was improved dramatically in this application by Jacobi (diagonal) preconditioning or very simple block-diagonal preconditioning (40).
