Scaling behavior of stochastic minimization algorithms in a perfect funnel landscape

Abstract

We determined scaling laws for the numerical effort to find the optimal configurations of a simple model potential energy surface (PES) with a perfect funnel structure that reflects key characteristics of the protein interactions. Generalized Monte-Carlo methods(MCM, STUN) avoid an enumerative search of the PES and thus provide a natural resolution of the Levinthal paradox. We find that the computational effort grows with approximately the eighth power of the system size for MCM and STUN, while a genetic algorithm was found to scale exponentially. The scaling behaviour of a derived lattice model is also rationalized.Despite recent successes in the description of the molecular structure [1,2] and the folding process of small polypeptides [3,8] the ab-initio prediction of the molecular structure for larger proteins remains an elusive goal. Since sequencing techniques presently outperform available experimental techniques for protein structure prediction (PSP) by a wide margin, the reservoir of sequenced proteins of unknown structure represents an ever growing pool of available, but as of yet inaccessible, biological information. These observations motivate the search for ab-initio techniques to predict the molecular structure of proteins from the amino acid sequence alone as one of the outstanding challenges to biological physics.In one widely pursued theoretical approach to PSP, the native structure of the protein is sought as the global minimum of an appropriate potential/free energy-function of the molecule [2,9-11] often including interactions with the solvent in an approximate, implicit fashion. As the folding process in nature takes place on a long time scale (10 −3 − 10 s), its direct simulation cannot be accomplished with the presently available computational resources. It is therefore desirable to determine the global minimum of the potential function without recourse to the folding dynamics. It has been argued that the resulting minimization problem is NP-hard [12-14], i.e. that the number of low-energy local minima grows exponentially with the number of amino acid residues. For this reason stochastic minimization procedures [15] are widely believed to be the most promising avenue to avoid an exponential increase of the numerical effort for the probabilistic "solution" to this problem. Since the available computational resources fall short by orders of magnitude to treat large proteins, it is important to obtain an order-of-magnitude estimation of the numerical effort required. This question can be answered by addressing the scaling laws [16,17]:governing the dependence of the computational effort (n CPU ) on the system size (N ). In this investigation we determined the scaling exponents for four different global minimization methods, for a very simple, idealized a model that reflects some key characteristics of the realistic problem. Our results demonstrate that the Levinthal paradox [18][19][20], which arises from the enourmous number of low-lying conformations of the protein, is naturally resolved in ...

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