^{1}, George C. Schatz

^{2,a)}and Mark A. Ratner

^{2,b)}

### Abstract

An algorithm in which kinetic lattice grand canonical Monte Carlo simulations are combined with mean field theory (KLGCMC/MF) is presented to calculate ion currents in a modelion channel system. In this simulation, the relevant region of the system is treated by KLGCMC simulations, while the rest of the system is described by modified Poisson-Boltzmann mean field theory. Calculation of reaction field due to induced charges on the channel/water and membrane/water boundaries is carried out using a basis-set expansion method [Im and Roux, J. Chem. Phys.115, 4850 (2001)]. Calculation of ion currents, electrostatic potentials, and ion concentrations, as obtained from the KLGCMC/MF simulations, shows good agreement with Poisson-Nernst-Planck (PNP) theory predictions when the channel and membrane have the same dielectric constant as water. If the channel and membrane have a lower dielectric constant than water, however, there is a considerable difference between the KLGCMC/MF and PNP predictions. This difference is attributed to the reaction field, which is missing in PNP theory. It is demonstrated that the reaction field as well as fixed charges in the channel play key roles in selective ion transport. Limitations and further development of the current KLGCMC/MF approach are also discussed.

This work was supported by the Network for Computational Nanotechnology (NCN) through a grant from the National Science Foundation, and by the Northwestern Materials Research Center (NSF Grant No. DMR-0520513).

I. INTRODUCTION

II. METHODS

A. KLGCMC/MF simulation with the local control method

B. Calculation of the ion interactionenergy

C. Definition of the time step and calculation of the ion current

III. DATA AND RESULTS

A. Description of the model system and details of the KLGCMC simulations

B. Test of the basis-set expansion method for the reaction field in the model system with

C. Ion current calculations for the model system where the bulk concentrations at the bottom and top are both

D. Ion current calculations for the model system where the bulk concentrations at the bottom and top are and , respectively

IV. DISCUSSION AND CONCLUSION

### Key Topics

- Ion channels
- 37.0
- Electrostatics
- 30.0
- Dielectric constant
- 25.0
- Mean field theory
- 17.0
- Monte Carlo methods
- 15.0

## Figures

Schematic of a model ion channel. The whole system is composed of a lattice grand canonical Monte Carlo (LGCMC) region and two mean field (MF) regions. In the MF region I and II, the ions are described by the screening factor and the explicit ions appear only in the LGCMC region. The LGCMC region comprises two local control (LC) regions and one diffusion region. LC regions I and II in the LGCMC region are located at the bottom and the top , respectively.

Schematic of a model ion channel. The whole system is composed of a lattice grand canonical Monte Carlo (LGCMC) region and two mean field (MF) regions. In the MF region I and II, the ions are described by the screening factor and the explicit ions appear only in the LGCMC region. The LGCMC region comprises two local control (LC) regions and one diffusion region. LC regions I and II in the LGCMC region are located at the bottom and the top , respectively.

Reaction field energy calculations for 50 independent configurations taken from the basis-set expansion results and from the PB equation results for the model system. For the basis-set expansion method, both Eqs. (18) and (19) are used.

Reaction field energy calculations for 50 independent configurations taken from the basis-set expansion results and from the PB equation results for the model system. For the basis-set expansion method, both Eqs. (18) and (19) are used.

Electrostatic potentials of the ions (ion field) and induced charges (reaction field) from the basis-set expansion method for three configurations: (a) configuration 27, (b) configuration 31, and (c) configuration 39. Comparison with the PB calculations is also shown.

Electrostatic potentials of the ions (ion field) and induced charges (reaction field) from the basis-set expansion method for three configurations: (a) configuration 27, (b) configuration 31, and (c) configuration 39. Comparison with the PB calculations is also shown.

Ion current-voltage curves of the and ions from the KLGCMC/MF simulations in the model ion channel system in the case of . The bulk ion concentrations at the bottom and top are both . Comparison with the PNP calculations is also shown.

Ion current-voltage curves of the and ions from the KLGCMC/MF simulations in the model ion channel system in the case of . The bulk ion concentrations at the bottom and top are both . Comparison with the PNP calculations is also shown.

(a) Electrostatic potentials due to the total, ions (ion field), induced charges (reaction field), and fixed dipoles on the channel (static field) and (b) and concentrations at . The membrane and channel dielectric constant and bulk ion concentrations are the same as in the previous figure. The dotted lines represent the boundaries between the LGCMC and MF regions at and the channel entrances at . Comparison with the PNP calculations is also shown. Note the semilogarithmic scale on the axis in (b).

(a) Electrostatic potentials due to the total, ions (ion field), induced charges (reaction field), and fixed dipoles on the channel (static field) and (b) and concentrations at . The membrane and channel dielectric constant and bulk ion concentrations are the same as in the previous figure. The dotted lines represent the boundaries between the LGCMC and MF regions at and the channel entrances at . Comparison with the PNP calculations is also shown. Note the semilogarithmic scale on the axis in (b).

Ion current-voltage curves of the and ions from the KLGCMC/MF simulations in the model ion channel system in the case of . The bulk ion concentrations at the bottom and top are both . Comparison with the PNP results is also shown.

Ion current-voltage curves of the and ions from the KLGCMC/MF simulations in the model ion channel system in the case of . The bulk ion concentrations at the bottom and top are both . Comparison with the PNP results is also shown.

(a) Electrostatic potentials due to the total, ions (ion field), induced charges (reaction field), and fixed dipoles on the channel (static field) and (b) and concentrations at . The membrane and channel dielectric constant and bulk ion concentrations are the same as in the previous figure. The dotted lines represent the boundaries between the LGCMC and MF regions at and the channel entrances at . Comparison with the PNP calculations is also shown. Note the semilogarithmic scale on the axis in (b).

Ion current-voltage curves for the and ions from the KLGCMC/MF simulations in the model ion channel system in the case of . The bulk ion concentrations at the bottom and top are and , respectively. Comparison with the PNP results is also shown.

Ion current-voltage curves for the and ions from the KLGCMC/MF simulations in the model ion channel system in the case of . The bulk ion concentrations at the bottom and top are and , respectively. Comparison with the PNP results is also shown.

Ion current-voltage curves of the and ions from the KLGCMC/MF simulations in the model ion channel system in the case of . The bulk ion concentrations at the bottom and top are and , respectively. Comparison with the PNP calculations is also shown.

Ion current-voltage curves of the and ions from the KLGCMC/MF simulations in the model ion channel system in the case of . The bulk ion concentrations at the bottom and top are and , respectively. Comparison with the PNP calculations is also shown.

## Tables

Ion parameters.

Ion parameters.

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