Protein Folding in the 2D Hydrophobic-Hydrophilic (HP) Square Lattice Model is Chaotic

Abstract: Among the unsolved problems in computational biology, protein folding is one of the most interesting challenges. To study this folding, tools like neural networks and genetic algorithms have received a lot of attention, mainly due to the NP-completeness of the folding process. The background idea that has given rise to the use of these algorithms is obviously that the folding process is predictable. However, this important assumption is disputable as chaotic properties of such a process have been recently highlighted. In this paper, which is an extension of a former work accepted to the 2011 International Joint Conference on Neural Networks (IJCNN11), the topological behavior of a well-known dynamical system used for protein folding prediction is evaluated. It is mathematically established that the folding dynamics in the 2D hydrophobic-hydrophilic (HP) square lattice model, simply called "the 2D model" in this document, is indeed a chaotic dynamical system as defined by Devaney. Furthermore, the chaotic behavior of this model is qualitatively and quantitatively deepened, by studying other mathematical properties of disorder, namely: the indecomposability, instability, strong transitivity, and constants of expansivity and sensitivity. Some consequences for both biological paradigms and structure prediction using this model are then discussed. In particular, it is shown that some neural networks seems to be unable to predict the evolution of this model with accuracy, due to its complex behavior.