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Towards Quantitative Classification of Folded Proteins in Terms of Elementary Functions

A comparative classification scheme provides a good basis for several approaches to understand proteins, including prediction of relations between their structure and biological function. But it remains a challenge to combine a classification scheme that describes a protein starting from its well organized secondary structures and often involves direct human involvement, with an atomary level Physics based approach where a protein is fundamentally nothing more than an ensemble of mutually interacting carbon, hydrogen, oxygen and nitrogen atoms. In order to bridge these two complementary approaches to proteins, conceptually novel tools need to be introduced. Here we explain how the geometrical shape of entire folded proteins can be described analytically in terms of a single explicit elementary function that is familiar from nonlinear physical systems where it is known as the kink-soliton. Our approach enables the conversion of hierarchical structural information into a quantitative form that allows for a folded protein to be characterized in terms of a small number of global parameters that are in principle computable from atomary level considerations. As an example we describe in detail how the native fold of the myoglobin 1M6C emerges from a combination of kink-solitons with a very high atomary level accuracy. We also verify that our approach describes longer loops and loops connecting $α$-helices with $β$-strands, with same overall accuracy.