1. Introduction 17

2. Dynamics of many-body systems 19

2.1 Effective dynamics of reduced systems 21

2.2 The constraint of thermodynamic equilibrium 24

2.3 Mean-field theories 25

3. Solvation free energy and electrostatics 27

3.1 Microscopic view of the Born model 27

3.2 Ion–Ion interactions in bulk solution 29

3.3 Continuum electrostatics and the PB equation 29

3.4 Limitations of continuum dielectric models 32

3.5 The dielectric barrier 33

3.6 The transmembrane potential and the PB-V equation 35

4. Statistical mechanical equilibrium theory 40

4.1 Multi-ion PMF 40

4.2 Equilibrium probabilities of occupancy 43

4.3 Coupling to the membrane potential 44

4.4 Ionic selectivity 48

4.5 Reduction to a one-dimensional (1D) free-energy profile 49

5. From MD toI–V: a practical guide 50

5.1 Extracting the essential ingredients from MD 51

5.1.1 Channel conductance from equilibrium and non-equilibrium MD 51

5.1.2 PMF techniques 52

5.1.3 Friction and diffusion coefficient techniques 53

5.1.4 About computational times 55

5.2 Ion permeation models 56

5.2.1 The 1D-NP electrodiffusion theory 56

5.2.2 Discrete-state Markov chains 57

5.2.3 The GCMC/BD algorithm 58

5.2.4 PNP electrodiffusion theory 62

6. Computational studies of ion channels 63

6.1 Computational studies of gA 65

6.1.1 Free-energy surface for K+ permeation 66

6.1.2 Mean-force decomposition 69

6.1.3 Cation-binding sites 69

6.1.4 Channel conductance 70

6.1.5 Selectivity 72

6.2 Computational studies of KcsA 72

6.2.1 Multi-ion free-energy surface and cation-binding sites 73

6.2.2 Channel conductance 74

6.2.3 Mechanism of ion conduction 77

6.2.4 Selectivity 78

6.3 Computational studies of OmpF 79

6.3.1 The need to compare the different level of approximations 79

6.3.2 Equilibrium protein fluctuations and ion distribution 80

6.3.3 Non-equilibrium ion fluxes 80

6.3.4 Reversal potential and selectivity 84

6.4 Successes and limitations 87

6.4.1 Channel structure 87

6.4.2 Ion-binding sites 87

6.4.3 Ion conduction 88

6.4.4 Ion selectivity 89

7. Conclusion 90

8. Acknowledgments 93

9. References 93

The 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.