The study of unfoldable self-avoiding walks - Application to protein structure prediction software

Self-avoiding walks (SAWs) are the source of very difficult problems in probability and enumerative combinatorics. They are of great interest as, for example, they are the basis of protein structure prediction (PSP) in bioinformatics. The authors of this paper have previously shown that, depending on the prediction algorithm, the sets of obtained walk conformations differ: For example, all the SAWs can be generated using stretching-based algorithms whereas only the unfoldable SAWs can be obtained with methods that iteratively fold the straight line. A deeper study of (non-)unfoldable SAWs is presented in this paper. The contribution is first a survey of what is currently known about these sets. In particular, we provide clear definitions of various subsets of SAWs related to pivot moves (unfoldable and non-unfoldable SAWs, etc.) and the first results that we have obtained, theoretically or computationally, on these sets. Then a new theorem on the number of non-unfoldable SAWs is demonstrated. Finally, a list of open questions is provided and the consequences on the PSP problem is proposed.

[1]  Gordon Slade,et al.  The self-avoiding walk: A brief survey , 2011 .

[2]  Ron Unger,et al.  Genetic Algorithm for 3D Protein Folding Simulations , 1993, ICGA.

[3]  Mihalis Yannakakis,et al.  On the complexity of protein folding (extended abstract) , 1998, STOC '98.

[4]  Iwan Jensen.  Improved lower bounds on the connective constants for self-avoiding walks , 2004 .

[5]  Paul J. Flory,et al.  The Configuration of Real Polymer Chains , 1949 .

[6]  R. Unger,et al.  Chaos in protein dynamics , 1997, Proteins.

[7]  Camelia Chira,et al.  Simplified chain folding models as metaheuristic benchmark for tuning real protein folding algorithms? , 2010, IEEE Congress on Evolutionary Computation.

[8]  Philippe Flajolet,et al.  Some New Self-avoiding Walk and Polygon Models , 2012, Fundam. Informaticae.

[9]  Jacques M. Bahi,et al.  Is protein Folding Problem Really a NP-Complete One? First Investigations , 2013, J. Bioinform. Comput. Biol..

[10]  Abdul Sattar,et al.  Genetic algorithm feature-based resampling for protein structure prediction , 2010, IEEE Congress on Evolutionary Computation.

[11]  Axel Bacher,et al.  Weakly directed self-avoiding walks , 2011, J. Comb. Theory, Ser. A.

[12]  Rolf Backofen,et al.  Algorithmic approach to quantifying the hydrophobic force contribution in protein folding , 1999, German Conference on Bioinformatics.

[13]  A. Sokal,et al.  The pivot algorithm: A highly efficient Monte Carlo method for the self-avoiding walk , 1988 .

[14]  N. Madras,et al.  THE SELF-AVOIDING WALK , 2006 .

[15]  I. Jensen.  Improved lower bounds on the connective constants for two-dimensional self-avoiding walks , 2004, cond-mat/0409381.

[16]  Holger H. Hoos,et al.  An ant colony optimisation algorithm for the 2D and 3D hydrophobic polar protein folding problem , 2005, BMC Bioinformatics.

[17]  K. Dill.  Theory for the folding and stability of globular proteins. , 1985, Biochemistry.

[18]  Abdul Sattar,et al.  Genetic Algorithm inAb Initio Protein Structure Prediction Using Low Resolution Model: A Review , 2009, Biomedical Data and Applications.

[19]  Madhu Chetty,et al.  Clustered memetic algorithm for protein structure prediction , 2010, IEEE Congress on Evolutionary Computation.

[20]  P. G. de Gennes,et al.  Exponents for the excluded volume problem as derived by the Wilson method , 1972 .

[21]  Jacques M. Bahi,et al.  Protein Folding in the 2D Hydrophobic–Hydrophilic (HP) Square Lattice Model is Chaotic , 2011, Cognitive Computation.

[22]  George Weiss,et al.  Random walks and random environments, volume 1: Random walks , 1996 .

[23]  Jacques M. Bahi,et al.  Chaos of protein folding , 2011, The 2011 International Joint Conference on Neural Networks.

[24]  Jinbo Xu,et al.  Raptorx: Exploiting structure information for protein alignment by statistical inference , 2011, Proteins.

[25]  Jacques M. Bahi,et al.  Computational investigations of folded self-avoiding walks related to protein folding , 2013, Comput. Biol. Chem..

[26]  P. M. Lee,et al.  Random Walks and Random Environments: Volume 1: Random Walks , 1995 .

[27]  Iwan Jensen.  Enumeration of self-avoiding walks on the square lattice , 2004 .

[28]  A R Conway,et al.  Algebraic techniques for enumerating self-avoiding walks on the square lattice , 1993 .