Dynamics of ion channels via non-Hermitian quantum mechanics

TL;DR

This paper explores the dynamics and thermodynamics of ion channels modeled as non-Hermitian quantum systems, introducing a semiclassical framework and algebraic topology tools to analyze complex Hamiltonians and their physical implications.

Contribution

It develops a novel semiclassical approach for non-Hermitian quantum mechanics applied to ion channels, using algebraic topology to handle complex action integrals.

Findings

Framework for semiclassical calculations of non-Hermitian Hamiltonians

Application of algebraic topology to evaluate complex WKB integrals

Insights into thermodynamics and correlations of multivalent ions in channels

Abstract

We study dynamics and thermodynamics of ion channels, considered as effective 1D Coulomb systems. The long range nature of the inter-ion interactions comes about due to the dielectric constants mismatch between the water and lipids, confining the electric filed to stay mostly within the water-filled channel. Statistical mechanics of such Coulomb systems is dominated by entropic effects which may be accurately accounted for by mapping onto an effective quantum mechanics. In presence of multivalent ions the corresponding quantum mechanics appears to be non-Hermitian. In this review we discuss a framework for semiclassical calculations for corresponding non-Hermitian Hamiltonians. Non-Hermiticity elevates WKB action integrals from the real line to closed cycles on a complex Riemann surfaces where direct calculations are not attainable. We circumvent this issue by applying tools from algebraic topology, such as the Picard-Fuchs equation. We discuss how its solutions relate to the thermodynamics and correlation functions of multivalent solutions within long water-filled channels.