^{1,a)}

### Abstract

How many steps are required to model permeation through ion channels? This question is investigated by comparing one- and two-step models of permeation with experiment and MD simulation for the first time. In recent MD simulations, the observed permeation mechanism was identified as resembling a Hodgkin and Keynes knock-on mechanism with one voltage-dependent rate-determining step [Jensen *et al.*, PNAS107, 5833 (2010)]. These previously published simulation data are fitted to a one-step knock-on model that successfully explains the highly non-Ohmic current–voltage curve observed in the simulation. However, these predictions (and the simulations upon which they are based) are not representative of real channel behavior, which is typically Ohmic at low voltages. A two-step association/dissociation (A/D) model is then compared with experiment for the first time. This two-parameter model is shown to be remarkably consistent with previously published permeation experiments through the MaxiK potassium channel over a wide range of concentrations and positive voltages. The A/D model also provides a first-order explanation of permeation through the *Shaker*potassium channel, but it does not explain the asymmetry observed experimentally. To address this, a new asymmetric variant of the A/D model is developed using the present theoretical framework. It includes a third parameter that represents the value of the “permeation coordinate” (fractional electric potential energy) corresponding to the triply occupied state *n* of the channel. This asymmetric A/D model is fitted to published permeation data through the *Shaker*potassium channel at physiological concentrations, and it successfully predicts qualitative changes in the negative current–voltage data (including a transition to super-Ohmic behavior) based solely on a fit to positive-voltage data (that appear linear). The A/D model appears to be qualitatively consistent with a large group of published MD simulations, but no quantitative comparison has yet been made. The A/D model makes a network of predictions for how the elementary steps and the channel occupancy vary with both concentration and voltage. In addition, the proposed theoretical framework suggests a new way of plotting the energetics of the simulated system using a one-dimensional permeation coordinate that uses electric potential energy as a metric for the net fractional progress through the permeation mechanism. This approach has the potential to provide a quantitative connection between atomistic simulations and permeation experiments for the first time.

The author wishes to thank Sabrina Sanchez, Alexis Wadowski, Jaqui Lynch, and the boys for helpful comments on an earlier draft of the manuscript. Support from the National Science Foundation (Grant No. 0836833) is gratefully acknowledged.

I. INTRODUCTION

II. SINGLE-FILE PERMEATION MODELS

A. Knock-on model

B. Connecting the knock-on model with simulation

C. Association/dissociation (A/D) model

D. Connecting the A/D model with experiment

E. Association barrier model

F. Connecting the asymmetric A/D model with simulation and experiment

III. DISCUSSION

A. Symmetric A/D model

B. Asymmetric A/D model

C. Connections with simulation and experiment

D. Modeling outlook

IV. SUMMARY AND CONCLUSION

### Key Topics

- Dissociation
- 19.0
- Molecular dynamics
- 19.0
- Experiment design
- 16.0
- Ion channels
- 15.0
- Potassium
- 13.0

## Figures

Hodgkin and Keynes knock-on mechanism. Permeation is a one-step process—after Bernèche and Roux. (See Ref. 17.)

Hodgkin and Keynes knock-on mechanism. Permeation is a one-step process—after Bernèche and Roux. (See Ref. 17.)

Energetics of the knock-on model for a positive transmembrane voltage *V*. The permeation coordinate represents the fractional progress through outward permeation. *qV* is the electrical work done on the ions during a complete outward permeation transition. (1 − χ)*qV* is the increase in the transition state energy as a result of the transmembrane voltage *V*. *E* _{ a } is the activation energy for *V = 0*.

Energetics of the knock-on model for a positive transmembrane voltage *V*. The permeation coordinate represents the fractional progress through outward permeation. *qV* is the electrical work done on the ions during a complete outward permeation transition. (1 − χ)*qV* is the increase in the transition state energy as a result of the transmembrane voltage *V*. *E* _{ a } is the activation energy for *V = 0*.

Comparing the asymmetric knock-on model Eq. (3) with simulated single-channel channel Kv1.2 permeation data. The theoretical curve was fitted to all the nonzero current data resulting in χ = 0.95 and *k* _{ a } = 4.5 × 10^{6} s^{−1} M^{−1}. Simulation data (symbols) reproduced from Jensen *et al.* (See Ref. 14.)

Comparing the asymmetric knock-on model Eq. (3) with simulated single-channel channel Kv1.2 permeation data. The theoretical curve was fitted to all the nonzero current data resulting in χ = 0.95 and *k* _{ a } = 4.5 × 10^{6} s^{−1} M^{−1}. Simulation data (symbols) reproduced from Jensen *et al.* (See Ref. 14.)

Energetics of the A/D model for positive membrane voltage *V*. The two states of the selectivity filter are labeled *n* (triple occupancy) and *m* (double occupancy). The permeation coordinate represents the fractional progress through outward permeation. *qV* is the electrical work done on all of the ions during a compete outward permeation event. (1 − χ)*qV* is the increase in state *n*'s energy as a result of the transmembrane voltage *V*. *E* _{ d } is the dissociation energy for *V* = 0.

Energetics of the A/D model for positive membrane voltage *V*. The two states of the selectivity filter are labeled *n* (triple occupancy) and *m* (double occupancy). The permeation coordinate represents the fractional progress through outward permeation. *qV* is the electrical work done on all of the ions during a compete outward permeation event. (1 − χ)*qV* is the increase in state *n*'s energy as a result of the transmembrane voltage *V*. *E* _{ d } is the dissociation energy for *V* = 0.

A/D mechanism. Permeation is a two-step process. During the association step all three ions move together in a concerted manner as the entering ion shunts-on the two helper ions to form the triply occupied state *n*. During the dissociation step all three ions move together in a concerted manner as the two helper ions shunt-off the permeant ion to return the channel to state *m*.

A/D mechanism. Permeation is a two-step process. During the association step all three ions move together in a concerted manner as the entering ion shunts-on the two helper ions to form the triply occupied state *n*. During the dissociation step all three ions move together in a concerted manner as the two helper ions shunt-off the permeant ion to return the channel to state *m*.

(a) Comparing the two-parameter symmetric A/D model Eq. (19) with single-channel MaxiK permeation data. Theoretical curves (solid lines) were fitted to the *V* > 0 experimental data using Eq. (19), resulting in *k* _{ d } = 2.2 × 10^{8} s^{−1} and *K* _{ d } = 310 mM. Experimental data (symbols) and previous 15-parameter fit (dotted lines) reproduced (with permission) from Schroeder and Hansen. (See Ref. 45.) (b) Corresponding occupancy states plot of the positive-voltage data in Fig. 6(a). Within the A/D model, the experimental values of the scaled flux *j** represent the occupancy θ_{ n } (degree of saturation) of the channel—see Eq. (16).

(a) Comparing the two-parameter symmetric A/D model Eq. (19) with single-channel MaxiK permeation data. Theoretical curves (solid lines) were fitted to the *V* > 0 experimental data using Eq. (19), resulting in *k* _{ d } = 2.2 × 10^{8} s^{−1} and *K* _{ d } = 310 mM. Experimental data (symbols) and previous 15-parameter fit (dotted lines) reproduced (with permission) from Schroeder and Hansen. (See Ref. 45.) (b) Corresponding occupancy states plot of the positive-voltage data in Fig. 6(a). Within the A/D model, the experimental values of the scaled flux *j** represent the occupancy θ_{ n } (degree of saturation) of the channel—see Eq. (16).

Comparing the symmetric A/D model Eq. (19) with single-channel *Shaker* K^{+} channel permeation data. (a) Theoretical curves (solid lines) were fitted to the *V* > 0 experimental data resulting in *K* _{ d } = 150 mM, *k* _{ d } = 1.4 × 10^{7} s^{−1}. Experimental data (symbols) reproduced (with permission) from Heginbotham and MacKinnon. (See Ref. 37.) (b) Corresponding occupancy states plot of the positive-voltage data in Fig. 6(a). Solid circles represent the zero-volt conductances reported by Heginbotham and MacKinnon that have been fitted to Eq. (22).

Comparing the symmetric A/D model Eq. (19) with single-channel *Shaker* K^{+} channel permeation data. (a) Theoretical curves (solid lines) were fitted to the *V* > 0 experimental data resulting in *K* _{ d } = 150 mM, *k* _{ d } = 1.4 × 10^{7} s^{−1}. Experimental data (symbols) reproduced (with permission) from Heginbotham and MacKinnon. (See Ref. 37.) (b) Corresponding occupancy states plot of the positive-voltage data in Fig. 6(a). Solid circles represent the zero-volt conductances reported by Heginbotham and MacKinnon that have been fitted to Eq. (22).

Two-step asymmetric “shunt-on pop-off” variant of the A/D permeation model proposed here based on the MD simulations of Jensen *et al.* (See Ref. 14.)

Two-step asymmetric “shunt-on pop-off” variant of the A/D permeation model proposed here based on the MD simulations of Jensen *et al.* (See Ref. 14.)

Comparing the asymmetric A/D model Eq. (12) with single-channel *Shaker* K^{+} channel permeation data. (a) Theoretical curves (solid lines) were fitted to the experimental data resulting in *K* _{ d } = 89 mM, *k* _{ d } = 1.3 × 10^{7} s^{−1}, and χ = 0.67. Experimental data (symbols) reproduced (with permission) from Heginbotham and MacKinnon. (See Ref. 37.) (b) Corresponding occupancy states plot of the positive-voltage data in Fig. 9(a).

Comparing the asymmetric A/D model Eq. (12) with single-channel *Shaker* K^{+} channel permeation data. (a) Theoretical curves (solid lines) were fitted to the experimental data resulting in *K* _{ d } = 89 mM, *k* _{ d } = 1.3 × 10^{7} s^{−1}, and χ = 0.67. Experimental data (symbols) reproduced (with permission) from Heginbotham and MacKinnon. (See Ref. 37.) (b) Corresponding occupancy states plot of the positive-voltage data in Fig. 9(a).

Comparing the asymmetric A/D model Eq. (12) with single-channel *Shaker* K^{+} channel permeation data. (a) Theoretical curves (solid lines) were fitted to both positive and negative experimental data resulting in *K* _{ d } = 49 mM, *k* _{ d } = 9.5 × 10^{6} s^{−1}, and χ = 0.60. Experimental data (symbols) reproduced (with permission) from Heginbotham and MacKinnon. (See Ref. 37.) (b) Corresponding occupancy states plot of all the data in Fig. 10(a).

Comparing the asymmetric A/D model Eq. (12) with single-channel *Shaker* K^{+} channel permeation data. (a) Theoretical curves (solid lines) were fitted to both positive and negative experimental data resulting in *K* _{ d } = 49 mM, *k* _{ d } = 9.5 × 10^{6} s^{−1}, and χ = 0.60. Experimental data (symbols) reproduced (with permission) from Heginbotham and MacKinnon. (See Ref. 37.) (b) Corresponding occupancy states plot of all the data in Fig. 10(a).

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