...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 1.5 nm in x and y directions and 15.5 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_{i} dv_{i}/dt = Sum_i (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_{i} 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=
7.5 nm and L_z= 15.5 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 five-atom TIP5P water simulation code with the Poisson equation is
available. It solves water dynamics whose code is 11,000 lines, and runs
for about 10 hours by 6 core/3.0 GHz (Ref. 3). In the z-direction, the DNA
blobs are moved in the positive z-direction, where water molecules are
changed to have the double-top distributions by long-range potentials.

...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 blobs more concentrated and accelerated because counterions find
negatively-charged DNA easily, which accelerates them 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/Molecular_dynamics_of_
DNA_transport_for_living_cells, July, 2025.

細胞の静電的な分子動力学法
　https://github.com/Mtanaka77/
Molecular_dynamics_simulation_for_living_human_cells　
この領域では，分子動力学コードを見ることにより，理解できるだろう。
GNU General Public License v3.0 のライセンスを承認した上で，
当該コードをダウンロードして使用できる（Fortran 2003を使用）。

　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，i-th番目の粒子には

.....m_{i} dv_{i}/dt = Sum_i (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_{i} E　(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 @nanoAPG.f03 および @nanoWPa.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 などを用いる。
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。

　3次元の水分子コードを用いた膜のDNA輸送は完成している。電場Ezにより
DNAはZ方向に移動している様子が見られるが，水分子も垂直方向に加速されて
いる（https://github.com/Mtanaka77/ のDNA directoryのporv31.773a.pdf 参照）。
@nanoWPa.f03コードはgithub/Mtanaka77/ で見ることができる。

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