A Quadratic Programming Formulation for the Design of Reduced Protein Models in Continuous Sequence Space

The notion of optimization is inherent in the design of a sequence of amino acid monomer types in a long heteropolymer chain of a protein that should fold to a desired conformation. Building upon our previous work wherein continuous parametrization and deterministic optimization approach were introduced for protein sequence design, in this paper we present an alternative formulation that leads to a quadratic programming problem in the first stage of a two-stage design procedure. The new quadratic formulation, which uses the linear interpolation of the states of the monomers in Stage I could be solved to identify the globally optimal sequence(s). Furthermore, the global minimum solution of the quadratic programming problem gives a lower bound on the energy for a given conformation in the sequence space. In practice, even a local optimization algorithm often gives sequences with global minimum, as demonstrated in the examples considered in this paper. The solutions of the first stage are then used to provide an appropriate initial guess for the second stage, where a rescaled Gaussian probability distribution function-based interpolation is used to refine the states to their original discrete states. The performance of this method is demonstrated with HP (hydrophobic and polar) lattice models of proteins. The results of this method are compared with the results of exhaustive enumeration as well as our earlier method that uses a graph-spectral method in Stage I. The computational efficiency of the new method is also demonstrated by designing HP models of real proteins. The method outlined in this paper is applicable to very large chains and can be extended to the case of multiple monomer types.