Basic Molecular Dynamics
- Key Factors : The bond lengths, bond dihedral angles, bond valence angles, and non-bonded interactions like van der Waals and electrostatic interactions contribute to the total energy of the systems
- Various Methods : Periodic Boundary Conditions, Ewald Summation Techniques, Particle Mesh Ewald Method, Thermostats in MD, Solvent Models, Energy-Minimization Methods in MD Simulations
Verlet algorithm
From the Newton’s equation for N-body problem
$$F_{i}=m_{i}a_{i}$$
$$F_{i}=-\frac{\partial \mathcal{V}(r_{1},r_{2},\cdots,r_{N} )}{\partial r_{i}}$$
Numerically approach
$$\begin{align*}
x(t+\Delta t) &= x(t) +\dot{x}(t)\Delta t + \frac{1}{2}\ddot{x}(t)\Delta t^{2} + \frac{1}{6}\dddot{x}(t)\Delta t^{3} + \frac{1}{24}\ddddot{x}(t)\Delta t^{4} + \cdots \\
x(t-\Delta t) &= x(t) -\dot{x}(t)\Delta t + \frac{1}{2}\ddot{x}(t)\Delta t^{2} - \frac{1}{6}\dddot{x}(t)\Delta t^{3} + \frac{1}{24}\ddddot{x}(t)\Delta t^{4} + \cdots \\
\end{align*}$$
$$
\therefore x(t+\Delta t) = 2x(t) - x(t-\Delta t) + \ddot{x}\Delta t^{2} + \mathcal{O}(\Delta t^{4})
$$
Shadow Trajectory
- Molecular Chaos : Lyapunov instability
Lagrangian Classical Mechanics
$$J=\int_{t_{b}}^{t_{e}} \mathcal{L} dt $$
$$\mathcal{L} = \mathcal{L}(\dot{r}, r) = \sum_{i=1}^{N} \frac{1}{2}m_{i}\dot{r}_{i}^{2} - \mathcal{V}(r_{1},r_{2},\cdots,r_{N})$$
Monte Carlo (MC) simulation
Langevin dynamics
Functional Form in Molecular System
[REF|1]
$$\begin{align*}
E & = {\color{red}{E_{covalent}}} + {\color{blue}{E_{noncovalent}}} \\
& = {\color{red}{E_{bond}}} + {\color{red}{E_{angle}}} + {\color{red}{E_{dihedral}}} + {\color{blue}{E_{electrostatic}}} + {\color{blue}{E_{van der Waals}}} \\
\end{align*}$$
- Force Fields : AMBER, GROMOS, and CHARMM
Polarizable Force Fields
AMOEBA Polarizable Potential Energy Model
The AMOEBA potential function
$$\mathcal{V} = \mathcal{V}_{bond} + \mathcal{V}_{angle} + \mathcal{V}_{b\theta} + \mathcal{V}_{oop} + \mathcal{V}_{torsion} + \mathcal{V}_{vdW} + \mathcal{V}_{e}^{perm} + \mathcal{V}_{e}^{ind}$$
Energy by empirical functions
$$\begin{align*}
\mathcal{V}_{bond} &= K_{b}(b-b_{0})^{2} \left [ 1-2.55(b-b_{0})+3.793125(b-b_{0})^{2} \right ] \\
\mathcal{V}_{angle} &= K_{b}(b-b_{0})^{2} \left [
1-0.014(\theta-\theta_{0})
+ 5.6\times 10^{-5}(\theta-\theta_{0})^{2}
- 7.0 \times 10^{-7} (\theta - \theta_{0})^{3}
+ 2.2 \times 10^{-8} (\theta-\theta_{0})^{4} \right ] \\
\mathcal{V}_{b\theta} &= k_{b\theta}\left [ (b-b_{0}) - (b^{\prime} - b^{\prime}_{\theta} ) \right ] (\theta - \theta_{0}) \\
\end{align*}$$
Energy by Wilson–Decius–Cross function
$ \mathcal{V}_{oop} = k_{\chi} \chi^{2} $
Bell torsion Energy
$ \mathcal{V}_{torsion} = \sum_{n} {K_{n\phi}[1+cos(n\phi \pm \delta)]} $
The pairwise additive vdW interaction
$ \mathcal{V}_{vdW}(ij) = \epsilon_{ij} + \left ( \frac{1.07}{\varrho_{ij} + 0.07} \right )^{7} \left ( \frac{1.12}{\varrho_{ij}^{7}+0.12} -2 \right ) $
Permanent Electrostatic Interactions in Cartesian polytensor formalism
$ \mathcal{V}_{e}^{perm} = \begin{bmatrix} q_{i} & \\ d_{ix} & \\ d_{iy} & \\ d_{iz} & \\ Q_{ixx} & \\ \vdots & \end{bmatrix}^{t} \begin{bmatrix} 1 & \frac{\partial}{\partial x_{j}} & \frac{\partial}{\partial y_{j}} & \frac{\partial}{\partial z_{j}} & \cdots \\ \frac{\partial}{\partial x_{i}} & \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} & \frac{\partial^{2}}{\partial x_{i} \partial y_{j}} & \frac{\partial^{2}}{\partial x_{i} \partial z_{j}} & \cdots \\ \frac{\partial}{\partial y_{i}} & \frac{\partial^{2}}{\partial y_{i} \partial x_{j}} & \frac{\partial^{2}}{\partial y_{i} \partial y_{j}} & \frac{\partial^{2}}{\partial x_{i} \partial z_{j}} & \cdots \\ \frac{\partial}{\partial z_{i}} & \frac{\partial^{2}}{\partial z_{i} \partial x_{j}} & \frac{\partial^{2}}{\partial z_{i} \partial y_{j}} & \frac{\partial^{2}}{\partial x_{i} \partial z_{j}} & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \\ \end{bmatrix} \left ( \frac{1}{r_{ji}} \right ) \begin{bmatrix} q_{j} & \\ d_{jx} & \\ d_{jy} & \\ d_{jz} & \\ Q_{jxx} & \\ \vdots & \end{bmatrix} $
Polarization Energy
$ \mathcal{V}_{e}^{ind} = -\frac{1}{2} \sum_{i} \alpha_{i} \left ( \sum_{j \ne i} T^{1}_{ij}M_{j} - \sum_{k \ne i} T^{11}_{ik}\mu_{k} \right )^{T}E_{i} $
$ \mathcal{V}_{oop} = k_{\chi} \chi^{2} $
Bell torsion Energy
$ \mathcal{V}_{torsion} = \sum_{n} {K_{n\phi}[1+cos(n\phi \pm \delta)]} $
The pairwise additive vdW interaction
$ \mathcal{V}_{vdW}(ij) = \epsilon_{ij} + \left ( \frac{1.07}{\varrho_{ij} + 0.07} \right )^{7} \left ( \frac{1.12}{\varrho_{ij}^{7}+0.12} -2 \right ) $
Permanent Electrostatic Interactions in Cartesian polytensor formalism
$ \mathcal{V}_{e}^{perm} = \begin{bmatrix} q_{i} & \\ d_{ix} & \\ d_{iy} & \\ d_{iz} & \\ Q_{ixx} & \\ \vdots & \end{bmatrix}^{t} \begin{bmatrix} 1 & \frac{\partial}{\partial x_{j}} & \frac{\partial}{\partial y_{j}} & \frac{\partial}{\partial z_{j}} & \cdots \\ \frac{\partial}{\partial x_{i}} & \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} & \frac{\partial^{2}}{\partial x_{i} \partial y_{j}} & \frac{\partial^{2}}{\partial x_{i} \partial z_{j}} & \cdots \\ \frac{\partial}{\partial y_{i}} & \frac{\partial^{2}}{\partial y_{i} \partial x_{j}} & \frac{\partial^{2}}{\partial y_{i} \partial y_{j}} & \frac{\partial^{2}}{\partial x_{i} \partial z_{j}} & \cdots \\ \frac{\partial}{\partial z_{i}} & \frac{\partial^{2}}{\partial z_{i} \partial x_{j}} & \frac{\partial^{2}}{\partial z_{i} \partial y_{j}} & \frac{\partial^{2}}{\partial x_{i} \partial z_{j}} & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \\ \end{bmatrix} \left ( \frac{1}{r_{ji}} \right ) \begin{bmatrix} q_{j} & \\ d_{jx} & \\ d_{jy} & \\ d_{jz} & \\ Q_{jxx} & \\ \vdots & \end{bmatrix} $
Polarization Energy
$ \mathcal{V}_{e}^{ind} = -\frac{1}{2} \sum_{i} \alpha_{i} \left ( \sum_{j \ne i} T^{1}_{ij}M_{j} - \sum_{k \ne i} T^{11}_{ik}\mu_{k} \right )^{T}E_{i} $
Dynamics in Fluid System
Reference
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