...The transport of DNA molecule with counterions and coions is studied
where a narrow nanopore along the z-direction seperates wide downside
and upside regions (Ref. 1). The cylinder of the pore is assumed 1.5 nm wide
and 4.0 nm high, embedded in 8.1 nm in x and y directions and 16.8 nm in
the z direction. The short-range Coulomb and Lennard-Jones forces are
treated in the right-hand side of the equation, i.e.,

.....m dv/dt = (Gamma*q'q/r^2) (grad r/r) - fgm *(2 r(i)-r(i+1)-r(i-1))
.....+ 48*(epsil_LJ/kT) grad[(sigma/r_ij)^12 -(sigma/r_ij)^6] + q E 　(1).

The spring beads model for the DNA is assumed in the second term and
the E(i,j,k)= - grad pot(i,j,k) stands for the electric field in the fourth term.
The long-range potential forces with the mesh (i,j,k) coordinates are solved
by the Poisson equation, i.e.,

.....div(eps(i,j,k) [grad pot(i,j,k)]) = - 4*pi Gamma*rho(i,j,k) 　(2).

The dielectric constant eps(i,j,k) is highly changed from eps=79 in water
to eps=3 within the pore region.

...There are large potentials of positive and negative drops at end plates, and
they are small otherwise. The simulation code is named @nanoporAPG.f03
(ca 9,900 lines with graphics), and the parameter file paramAPG.h and the
configuration file PORV21_config.start3. The major subroutines are: RUN_MD,
moldyn, sht-forces, LJ-forces, sprmul, reflect_endpl, init, poissn, emcof3,
cresmd, and graphics. There are many input parameters to run the code, like
the nanopore sizes, the number of DNA, counterions and coions, the Cartesian
meshes for the Poisson solver, a time step dt, the potential values of top and
bottom plates, and Gamma at 300 K (the Bjerrum length), etc. It has L_x=L_y=
8.1 nm and L_z= 16.8 nm (80 x 80 x 100 meshes), ca. 14,000 particles. A run
of t= 500 takes 15 minutes by the 6 cores/3.0 GHz computer with the time step
dt=0.01 (x 10^-14 s, Ref. 2).

...The file porv21.773a.pdf for very small dielectric constant in the pore region
shows four plots of potentials, particles of DNA and ions, those of all particles
(every 5 water molecules), and the velocity distributions. One can see that
the DNA chain moves toward the positive z direction into the cell volume.
Moreover, the lowr dielectric constant eps(\r) in the pore region makes the
DNA blob more concentrated because counterions find negatively-charged
DNA easily, which accelerates it to inside the positive cell region (Ref. 3).

References
1. Y. Rabin and M. Tanaka, Phys.Rev.Lett., vol.94, 148103 (2005).
2. The simulation codes of this directory were updated in April, 2025.
3. M.Tanaka, https://github.com/Mtanaka77/Electrostatic_molecular_
dynamics_simulation_for_living_human_cells, April, 2025.

細胞の静電的な分子動力学法
　https://github.com/Mtanaka77/
Electrostatic_molecular_dynamics_simulation_for_living_human_cells　
この領域に入ると，分子動力学コードを理解できるだろう。GPL-3.0ライセンス
のもとで，コードをダウンロードして使用できる（Fortran 2003で）。

　皮膚の膜孔を通ってDNAが通過する問題を扱っている。静電粒子コードに
おいて，基本方程式の運動方程式は，

.....m dv/dt = (Gamma*q'q/r^2) (grad r/r) - fgm *(2 r(i)-r(i+1)-r(i-1))
.....+ 48*(epsil_LJ/kT) grad[(sigma/r_ij)^12 -(sigma/r_ij)^6]　+ qE　(1),

である。(1)ではクーロン力，ばね復元力、LJ復元力を考え，E(i,j,k) は静電場
である。またz方向には値 kT程度に大きく変動するので，ポテンシャル場は

.....div(eps(i,j,k) [grad pot(i,j,k)]) = - 4*pi*Gamma*rho(i,j,k)　(2)

を用いる。ベクトル演算子 div, grad, およびeps(i,j,k) のところに気を付ける。
静電定数は場所の関数であり，27度Cの水では 79の値であるが，膜孔では
小さくなり eps=3 となる。

　コード名は @nanoporAPG.f03 であり，パラメータファイルに paramAPG.h，
設定ファイルに PORV21_config.start3 を用いる。皮膚の垂直方向に電場を
かけることで，DNAが膜孔を通過してセル内へ輸送されることが示される。

*) 水平方向が境界連続の @nanoporAPR.f03 は， porv11.773.pdf で
　示すが，L_x= 8.1 nm では大きな距離が取れないので一部で人工的な
　結果が見られる。

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