Theoretical and computational models of biological ion channels

Abstract

1. Introduction 172. Dynamics of many-body systems 192.1 Effective dynamics of reduced systems 212.2 The constraint of thermodynamic equilibrium 242.3 Mean-field theories 253. Solvation free energy and electrostatics 273.1 Microscopic view of the Born model 273.2 Ion–Ion interactions in bulk solution 293.3 Continuum electrostatics and the PB equation 293.4 Limitations of continuum dielectric models 323.5 The dielectric barrier 333.6 The transmembrane potential and the PB-V equation 354. Statistical mechanical equilibrium theory 404.1 Multi-ion PMF 404.2 Equilibrium probabilities of occupancy 434.3 Coupling to the membrane potential 444.4 Ionic selectivity 484.5 Reduction to a one-dimensional (1D) free-energy profile 495. From MD toI–V: a practical guide 505.1 Extracting the essential ingredients from MD 515.1.1 Channel conductance from equilibrium and non-equilibrium MD 515.1.2 PMF techniques 525.1.3 Friction and diffusion coefficient techniques 535.1.4 About computational times 555.2 Ion permeation models 565.2.1 The 1D-NP electrodiffusion theory 565.2.2 Discrete-state Markov chains 575.2.3 The GCMC/BD algorithm 585.2.4 PNP electrodiffusion theory 626. Computational studies of ion channels 636.1 Computational studies of gA 656.1.1 Free-energy surface for K+ permeation 666.1.2 Mean-force decomposition 696.1.3 Cation-binding sites 696.1.4 Channel conductance 706.1.5 Selectivity 726.2 Computational studies of KcsA 726.2.1 Multi-ion free-energy surface and cation-binding sites 736.2.2 Channel conductance 746.2.3 Mechanism of ion conduction 776.2.4 Selectivity 786.3 Computational studies of OmpF 796.3.1 The need to compare the different level of approximations 796.3.2 Equilibrium protein fluctuations and ion distribution 806.3.3 Non-equilibrium ion fluxes 806.3.4 Reversal potential and selectivity 846.4 Successes and limitations 876.4.1 Channel structure 876.4.2 Ion-binding sites 876.4.3 Ion conduction 886.4.4 Ion selectivity 897. Conclusion 908. Acknowledgments 939. References 93The goal of this review is to establish a broad and rigorous theoretical framework to describe ion permeation through biological channels. This framework is developed in the context of atomic models on the basis of the statistical mechanical projection-operator formalism of Mori and Zwanzig. The review is divided into two main parts. The first part introduces the fundamental concepts needed to construct a hierarchy of dynamical models at different level of approximation. In particular, the potential of mean force (PMF) as a configuration-dependent free energy is introduced, and its significance concerning equilibrium and non-equilibrium phenomena is discussed. In addition, fundamental aspects of membrane electrostatics, with a particular emphasis on the influence of the transmembrane potential, as well as important computational techniques for extracting essential information from all-atom molecular dynamics (MD) simulations are described and discussed. The first part of the review provides a theoretical formalism to ‘translate’ the information from the atomic structure into the familiar language of phenomenological models of ion permeation. The second part is aimed at reviewing and contrasting results obtained in recent computational studies of three very different channels; the gramicidin A (gA) channel, which is a narrow one-ion pore (at moderate concentration), the KcsA channel from Streptomyces lividans, which is a narrow multi-ion pore, and the outer membrane matrix porin F (OmpF) from Escherichia coli, which is a trimer of three β-barrel subunits each forming wide aqueous multi-ion pores. Comparison with experiments demonstrates that current computational models are approaching semi-quantitative accuracy and are able to provide significant insight into the microscopic mechanisms of ion conduction and selectivity. We conclude that all-atom MD with explicit water molecules can represent important structural features of complex biological channels accurately, including such features as the location of ion-binding sites along the permeation pathway. We finally discuss the broader issue of the validity of ion permeation models and an outlook to the future.