The biological world is composed of folded linear molecules of bewildering topological complexity and diversity. The topology of folded biomolecules such as proteins and ribonucleic acids is often subject to change during biological processes. Despite intense research, we lack a solid mathematical framework that summarizes these operations in a principled manner. Circuit topology, which formalizes the arrangements of intramolecular contacts, serves as a general mathematical framework to analyze the topological characteristics of folded linear molecules. In this work, we translate familiar molecular operations in biology, such as duplication, permutation, and elimination of contacts, into the language of circuit topology. We show that for such operations there are corresponding matrix representations as well as basic rules that serve as a foundation for understanding these operations within the context of a coherent algebraic framework. We present several biological examples and provide a simple computational framework for creating and analyzing the circuit diagrams of proteins and nucleic acids. We expect our study and future developments in this direction to facilitate a deeper understanding of natural molecular processes and to provide guidance to engineers for generating complex polymeric materials.
We examine the olfactory discrimination of left- and right-handed enantiomers of chiral odorants based on the odorant-mediated electron transport from a donor to an acceptor of the olfactory receptors embodied in a biological environment. The chiral odorant is effectively described by an asymmetric double-well potential whose minima are associated to the left- and right-handed enantiomers. The introduced asymmetry is considered an overall measure of chiral interactions. The biological environment is conveniently modeled as a bath of harmonic oscillators. The resulting spin-boson model is adapted by a polaron transformation to derive the corresponding Born-Markov master equation with which we obtain the elastic and inelastic electron tunneling rates. We show that the inelastic tunneling through left- and right-handed enantiomers occurs with different rates. The discrimination mechanism depends on the ratio of tunneling frequency to localization frequency.
We explore the interplay between tunneling process and chiral interactions in the discrimination of chiral states for an ensemble of molecules in a biological environment. Each molecule is described by an asymmetric double-well potential and the environment is modeled as a bath of harmonic oscillators. We carefully analyze different time-scales appearing in the resulting master equation at both weak-and strong-coupling limits. The corresponding results are accompanied by a set of coupled differential equations characterizing optical activity of the molecules. We show that, at the weak-coupling limit, chiral interactions prohibit the coherent racemization induced by decoherence effects and thus preserve the initial chiral state. At the strong-coupling limit, considering the memory effects of the environment, Markovian behavior is observed at long times.
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