自從生物學進化以來，離子通道是一種非常精妙的設計，而超離子狀態
是非常有效率的狀態。因此，我們希望將兩者結合起來，並希望測試超離
子狀態是否會出現在常見的離子通道中。在這個實驗中，我們選擇鉀通道
KcsA，僅取通道中每個原子的相對位置，我們將手動調節通道寬度（等倍
放大）、離子通道之原子攜帶的電荷和鹽（KCl）濃度。在文章中，我們使
用Poisson-Fermi 模型來模擬通道中的離子，並在該模型中將重新溶劑化
自由能添加到自由能中，以便可以組合兩個參考實驗模型。最後，觀察通
道中各個位置的空間位勢S^trc ，觀察是否有出現跳階的現象。

Since the evolution of biology, the ion channel is a very subtle design, and the superionic state is very effective. Therefore, we want to combine the two and want to test whether superionic states will appear in common ion channels. In this experiment we chose the potassium channel KcsA, taking only the relative position of each atom of the channel, and we will manually adjust the channel width (equal magnification), the charge carried by the atoms in the channel and the salt (KCl) concentration. In the article we
use Poisson-Fermi’s model to simulate the ion in the channel and add the re-solvation free energy to the free energy in this model, so that the two reference experimental models can be combined. Finally, observe the steric potential S^trc of each position in the channel, to see if there is a jump.

1 Introduction 1
2 The structure of KcsA 2
3 Poisson-Fermi model 4
4 Numerical results 10
5 Summary 15

[1] S Kondrat and A Kornyshev. Superionic state in double-layer capacitors
with nanoporous electrodes. Journal of Physics: Condensed Matter,
23(2):022201, 2010.
[2] Jinn-Liang Liu, Dexuan Xie, and Bob Eisenberg. Poisson-fermi formu-
lation of nonlocal electrostatics in electrolyte solutions. Molecular Based
Mathematical Biology, 5(1):116–124, 2017.
[3] Dexuan Xie, Jinn-Liang Liu, and Bob Eisenberg. Nonlocal poisson-fermi
model for ionic solvent. Physical Review E, 94(1):012114, 2016.
[4] Dexuan Xie, Yi Jiang, Peter Brune, and L Ridgway Scott. A fast solver
for a nonlocal dielectric continuum model. SIAM Journal on Scientific
Computing, 34(2):B107–B126, 2012.
[5] Dexuan Xie and Hans W Volkmer. A modified nonlocal continuum elec-
trostatic model for protein in water and its analytical solutions for ionic
born models. Communications in Computational Physics, 13(1):174–
194, 2013.
[6] Jinn-Liang Liu. Numerical methods for the poisson–fermi equation in
electrolytes. Journal of Computational Physics, 247:88–99, 2013.
[7] I-Liang Chern, Jian-Guo Liu, Wei-Cheng Wang, et al. Accurate eval-
uation of electrostatics for macromolecules in solution. Methods and
Applications of Analysis, 10(2):309–328, 2003.
[8] Weihua Geng, Sining Yu, and Guowei Wei. Treatment of charge sin-
gularities in implicit solvent models. The Journal of chemical physics,
127(11):114106, 2007.
[9] Ron S Dembo, Stanley C Eisenstat, and Trond Steihaug. Inexact newton
methods. SIAM Journal on Numerical analysis, 19(2):400–408, 1982.
[10] Michael J Holst and Faisal Saied. Numerical solution of the nonlin-
ear poisson–boltzmann equation: developing more robust and efficient
methods. Journal of computational chemistry, 16(3):337–364, 1995.
[11] Qin Cai, Meng-Juei Hsieh, Jun Wang, and Ray Luo. Performance of
nonlinear finite-difference poisson- boltzmann solvers. Journal of chem-
ical theory and computation, 6(1):203–211, 2009.
[12] Jen-Hao Chen, Ren-Chuen Chen, and Jinn-Liang Liu. A gpu
poisson-fermi solver for ion channel simulations. arXiv preprint
arXiv:1711.04060, 2017.
[13] Nathan Bell and Michael Garland. Efficient sparse matrix-vector mul-
tiplication on cuda. Technical report, Nvidia Technical Report NVR-
2008-004, Nvidia Corporation, 2008.