A method for optimizing potential-energy functions by a hierarchical design of the potential-energy landscape: application to the UNRES force field

Abstract

A method for optimizing potential-energy functions of proteins is proposed. The method assumes a hierarchical structure of the energy landscape, which means that the energy decreases as the number of native-like elements in a structure increases, being lowest for structures from the native family and highest for structures with no native-like element. A level of the hierarchy is defined as a family of structures with the same number of native-like elements (or degree of native likeness). Optimization of a potential-energy function is aimed at achieving such a hierarchical structure of the energy landscape by forcing appropriate free-energy gaps between hierarchy levels to place their energies in ascending order. This procedure is different from methods developed thus far, in which the energy gap and/or the Z score between the native structure and all non-native structures are maximized, regardless of the degree of native likeness of the non-native structures. The advantage of this approach lies in reducing the number of structures with decreasing energy, which should ensure the searchability of the potential. The method was tested on two proteins, PDB ID codes and, with an off-lattice united-residue force field. For, the search of the conformational space with the use of the conformational space annealing method and the newly optimized potential-energy function found the native structure very quickly, as opposed to the potential-energy functions obtained by former optimization methods. After even incomplete optimization, the force field obtained by using located the native-like structures of two peptides, and betanova (a designed three-stranded beta-sheet peptide), as the lowest-energy conformations, whereas for the 46-residue N-terminal fragment of staphylococcal protein A, the native-like conformation was the second-lowest-energy conformation and had an energy 2 kcal/mol above that of the lowest-energy structure.

Figure 1

Schematic illustration of the energy levels, which is the goal of the algorithm for optimizing the potential function proposed in this work, using the 1FSD peptide (24) as an example. The energies of the conformations should decrease with their increasing “native likeness.” The highest energy level (Level 0) is occupied by structures with either no or non-native secondary structure. The next level (Level 1) is occupied by the structures with one native secondary structure element (the N-terminal β-hairpin or the C-terminal α-helix; the native-like structure fragments are indicated by thicker lines). Yet lower energy (Level 2) has structures with both α-helix and β-hairpin but no or incorrect packing of these two substructures and/or shifted turn in the β-hairpin. Finally, the native-like structures, with α-helix and β-hairpin packed correctly, occupy the lowest energy level (Level 3). Because the number of structures with more and more defined native-like elements decreases, such ordering of structures leads to diminishing conformational entropy following the energy decrease, which is highly desirable to find the native structure quickly in a spontaneous energy-driven search of the conformational space.

Figure 2

Illustration of the progress of optimization of the UNRES energy function for 1FSD [a designed α/β peptide (24)]. The crosses represent conformations obtained in CSA runs after optimizing parameters in a given iteration: (a) initial parameters; (b) iteration 1; (c) iteration 4 (the last iteration). See text for description.

Figure 3

Superposition of the lowest-energy structures (green) of 1FSD (Left) and 1IGD (Right) obtained with the force fields optimized on these proteins on their experimental structures (22, 24) (red). For 1FSD, residues 3–26 (which are well-defined in the NMR structure) are superposed. The C^α-rmsd over residues 3–26 is 1.8 Å, whereas the C^α-rmsd over all 28 residues is 2.8 Å. For 1IGD, residues 25–47 (the central α-helix and the second loop) are superposed.

Figure 4

Superposition of the lowest-energy structure of betanova (18), 1FSD (24), and the second lowest in energy structure of protein A (19) (green) on the corresponding NMR structures (red), left to right. The C^α rmsds are 0.9, 2.9, and 3.1 Å, respectively.