Complexity of polypeptide dynamics: chaos, Brownian motion and elasticity in aqueous solution

Abstract

Is the dynamics in solution of flexible molecules Brownian and chaotic? And, could it be related to the elastomeric properties of macromolecules? In this framework molecular dynamics simulations and nonlinear theory analysis was used to examine the complexity of polypeptide motions in aqueous solutions. Global indicators of classical chaos and self-organised criticality are investigated and a model for elasticity of rubber proteins is proposed. The glycine-rich polypeptide Ac-Phe-Gly-Gly-Met-Gly-Gly-Gly-Asn-Ala-Gly-NMe, the repeat motif of the rubber protein Abductin, has been modelled in aqueous solution by nearly 30 ns of parallel molecular dynamics simulations and motions carefully analysed using nonlinear theory of complex systems. The most important conformational lengths describing the peptide size as end-to-end distance, gyration radius and the longest interatomic distance have been considered. Moreover, the configurational 3 N-dimensional vector R, whose components are the Cartesian coordinates of peptide atoms, was considered as the global motion descriptor at the molecular scale. The system complexity was analysed in terms of time correlation functions, fast Fourier transforms, attractor dimension, Hurst and Lyapunov critical exponents. The global dynamics of R is Brownian with typical autocorrelation function and Fourier spectrum and Hurst exponent H=1/2 was estimated. In contrast, the conformational lengths undergo anomalous diffusion with antipersistent fractional Brownian motion ( H=1/3) according to the Self-Organised Criticality—SOC—theory. The delay coordinate embedding shows the existence of a low dimensional chaotic attractor with five positive Lyapunov exponents typical of dissipative and chaotic systems. In the framework of the chaotic hypothesis, which assumes that the properties of statistical mechanics can be predicted by treating the systems as chaotic, and the Gaspard's microscopic motion observations, the Brownian dynamics of the peptide appear to be an emergent behaviour due to the chaos of an unstable state. The SOC regime of the polypeptide defines the metastable chaotic region of poor-solvent, observed for macromolecules between ideal θ solvation and phase separation, in agreement to the soft solution and protein rubber state. The proposed soft solution model can be crucial in the expression of the entropic elastic force exhibited by rubber proteins as Elastin and Abductin.