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

[+] Author and Article Information
Sung K. Koh

Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104

G. K. Ananthasuresh

Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104 and Department of Mechanical Engineering, Indian Institute of Science, Bangalore-560 012, India

Christopher Croke

Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104

J. Mech. Des 127(4), 728-735 (Feb 25, 2005) (8 pages) doi:10.1115/1.1901705 History: Received November 01, 2004; Revised February 25, 2005

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.

Copyright © 2005 by American Society of Mechanical Engineers
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Figure 1

Nomenclature of proteins. (a) Schematic of three residues in a protein chain, (b) ball and stick model showing the 3-D arrangement.

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Figure 2

(a) H-P models of proteins (a) a 2-D protein model on an irregular lattice, (b) a 3-D protein on a 3×3×3 regular lattice. Black dots represent hydrophobic (H) residues and white dot polar (P) residues.

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Figure 3

Two 4×4 lattice conformations. (a) Conformation A: number of sequences in native state=1022; (b) Conformation B: Number of sequences in native state=459.

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Figure 4

(a) Ribbon representation of 1SRL protein. (b) Minimum energy as found by the new method (circles) and exhaustive enumeration (black dots) for a different number of H residues. (c) Improvement energy in the minima found by the new method over the previous method.

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Figure 5

The improvement in the energy beyond what the graph spectral-based old method gave plotted against the number of H residues for a (a) 4×4×4 lattice; (b) 5×5×5 lattice

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Figure 6

(a) Ribbon representation of Ubiquitin (1UBQ). (b) Energy improvement over the old method.

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Figure 7

(a) Ribbon representation of Csk Homologous Kinase (1JWO). (b) Energy improvement over the old method.

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Figure 8

(a) Ribbon representation of triophosphate isomerase complex with sulfate (5TIM). (b) Energy improvement over the old method.

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Figure 9

(a) Ribbon representation of tobacco ringspot virus capsid protein (1A6C). (b) Energy improvement over the old method.




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