"Statistical complexity of protein folding: application of computational mechanics to molecular dynamics"
Complex behaviour can emerge from inherently simple systems; a well-known and vitally important example is the ability of many proteins to spontaneously fold into a consistent and reproducible three-dimensional shape fundamental to their biological function. In a poorly understood manner the sequence of amino acids that make up the protein as well as their interactions with solvent encodes a highly reproducible folding process. Our approach consists of considering the protein as a dynamic, self-organizing system that exhibits an emergent behaviour. To understand the fundamental basis of the folding process we developed a methodology for the quantitative estimation of the dynamic complexity of an MD simulated peptide in explicit water. For our purposes we have adopted the approach by Crutchfield et. al. termed “computational mechanics” . This approach combines and implements the ideas from Shannon entropy and Kolmogorov-Chaitin algorithmic complexity theories. It describes the system in terms of “symbolic dynamics”: a sequence of symbols from an “alphabet” of finite size. A symbolic sequence is used to reconstruct an algorithmic automaton that propagates the system from one state (the so called “causal state”) to the next one. “Computational” signifies that the complexity of the system is equal to the complexity of this automaton. Being well developed from the formal mathematical point of view this approach provides a practical algorithm for calculating the complexity of real systems. One of the advantages of this approach is that it is based on an informatic-theoretical analysis of the dynamical evolution of the system and opens up the possibility of quantifying the emergent behaviour of the system. We have demonstrated that this approach can be applied to low-dimensional projections of a molecular systems trajectory and provides new information about the system’s dynamics. Considerably different complexity of the orientational as well as the translational motion of water molecules in an electrolyte solution at different locations with respect to the ion has been found . Additionally, a zwitterion which is made up of two oppositely charged groups separated by an aliphatic chain in vacuum has a specific “loop”-like conformation that the molecule, if allowed to dynamically evolve, takes regardless of the initial configuration. This system, being simple, nevertheless demonstrates a “folding” behaviour and elements of self-organization. Precisely at the moment of “folding” the complexity of individual atom three-dimensional trajectories shows a considerable drop, and then rises to a higher level when the molecule stabilizes in the “folded” conformation. We have now turned our attention to the whole 2N-dimensional trajectory in larger solvated molecular systems, where N is the number of atoms. Obviously, a straightforward calculation of the statistical complexity of this prohibitively high-dimensional signal (e.g. a protein in water) is impossible. However, the local character of interactions in molecular systems allows the calculation of the statistical complexity of the whole system by virtue of the copulas formalism. We have demonstrated that it is sufficient to estimate the local dynamics of small subsets of directly interacting degrees of freedom of the system to reconstruct (without any approximations) the complete 2N-dimensional trajectory properties. The limiting cases of a large number of identical molecules (for example, bulk water) are investigated and their complexity is analysed. For the more complicated case of a peptide in water, it is shown that the useful information is confined within a relatively small subset of atoms, consisting of the protein atoms and their immediate neighbouring water molecules. We have simulated the β-turn formation process in the pentapeptide leu-enkephalin in explicit water and applied computational mechanics analysis to the estimation of complexity of various aspects of the dynamics. A decisively important role of the water network has been recognized and attention has been concentrated on the complexity of water dynamics around the peptide before, at, and after the moment of turn formation. Analysis of various characteristics of the water dynamics support the hypothesis that a simplification of water reorientation takes place in the second solvation shell of the peptide at the moment of the turn formation. 1. J.P. Crutchfield, D.P. Feldman, and C.R. Shalizi, Phys. Rev. E, 62, 2996 (2000) 2. D. Nerukh, G. Karvounis, and R. Glen, J. Chem. Phys., 117(21), 9611-9617 (2002) and J. Chem. Phys., 117(21), 9618-9622 (2002)