The inverse protein folding problem on 2D and 3D lattices

Abstract

In this paper we investigate the inverse protein folding (IPF) problem under the Canonical model on 3D and 2D lattices [W.E. Hart, On the computational complexity of sequence design problems, Proceedings of the First Annual International Conference on Computational Molecular Biology 1997, pp. 128–136; E.I. Shakhnovich, A.M. Gutin, Engineering of stable and fast-folding sequences of model proteins, Proc. Natl. Acad. Sci. 90 (1993) 7195–7199]. In this problem, we are given a contact graph G = ( V , E ) of a protein sequence that is embeddable in a 3D (respectively, 2D) lattice and an integer 1 ⩽ K ⩽ | V | . The goal is to find an induced subgraph of G of at most K vertices with the maximum number of edges. In this paper, we prove the following results: • An earlier proof of NP-completeness of the IPF problem on 3D lattices [W.E. Hart, On the computational complexity of sequence design problems, Proceedings of the First Annual International Conference on Computational Molecular Biology 1997, pp. 128–136] is based on the NP-completeness of the IPF problem on the 2D lattices. However, the reduction was not correct and we show that the IPF problem for 2D lattices can be solved in O ( K | V | ) time. But, we show that the IPF problem on 3D lattices is indeed NP-complete by a providing a different reduction from a different NP-complete problem. • We design a polynomial-time approximation scheme for the IPF problem on 3D lattices using the shifted slice-and-dice approach in [P. Berman, B. DasGupta, S. Muthukrishnan, Approximation algorithms for MAX-MIN tiling, J. Algorithms 47(2) (2003) 122–134; D. Hochbaum, Approximation Algorithms for NP-hard Problems, PWS Publishing Company, MA, 1997; D.S. Hochbaum, W. Mass, Approximation schemes for covering and packing problems in image processing and VLSI, J. ACM 32(1) (1985) 130–136], thereby improving the previous best polynomial-time approximation algorithm which had a performance ratio of 1 2 [W.E. Hart, On the computational complexity of sequence design problems, Proceedings of the First Annual International Conference on Computational Molecular Biology 1997, pp. 128–136].