Biology Monte Carlo method
[3] The first one uses Monte Carlo method to solve the Boltzmann equation, while the later splits the electrostatic forces into short-range and long-range components.
In full-atomic molecular dynamics simulations of ion channels, most of the computational cost is for following the trajectory of water molecules in the system.
In addition to that, the protein atoms of the ion channel are also modeled as static point charges embedded in a finite volume with a given dielectric coefficient.
The ensemble of ions in the simulation region, are propagated synchronously in time and 3-D space by integrating the equations of motion using the second-order accurate leap-frog scheme.
Solving this equation provides a self-consistent way to include applied bias and the effects of image charges induced at dielectric boundaries.
The ion and partial charges on protein residues are assigned to a finite rectangular grid using the cloud-in-cell (CIC) scheme.
[8] As a result, the dielectric constant of water inside an ion channel could be quite different from the value under bulk conditions.
This study showed that water undergoes distinct transitions in structure, dielectric properties, and mobility as the tube diameter is varied.
Solving the Poisson equation based on an anisotropic permittivity has been incorporated into BioMOCA using the box integration discretization method,[9] which has been briefly described below.
The finite size of ions is accounted for in BioMOCA using pairwise repulsive forces derived from the 6–12 Lennard-Jones potential.
Availability of high-resolution X-ray crystallographic measurements of complete molecular structures provides information about the type and location of all atoms that forms the protein.
In BioMOCA the protein atoms are modeled as static point charges embedded in a finite volume inaccessible to the ions and associated with a user-defined dielectric coefficient.
Moreover, a number of force-field parameters are available that provide information about the charge and radii of atoms in different amino-acid groups.
The conjunction of the molecular structure and force fields provide the coordinates, radii, and charge of each atom in the protein channel.
Many recent open source molecular biology packages have built-in facilities that determine the volume accessible to ions in a protein system.
As a reduced particle approach, BioMOCA replaces the explicit water molecules with continuum background and handles the ion-water interactions using BTMC method, in which, appropriate scattering rates should be chosen.
The free flight times, Tf, are generated statistically from the total scattering rate according to where r is a random number uniformly distributed on the unit interval.
As the equation indicates, reduced diffusivity of ions inside the lumen of the channel renders to increased incidence of scattering events.
The theory of hydration shells is well developed in the physical chemistry literature however a simple model is required that captures the essential effects with as little computational overhead as possible.
In continuum models, for instance, where ionic density exist rather than explicit ions, the electrostatic potential is calculated in a self-consistent manner by solving the Poisson equation.
By solving the Poisson equation it is possible to self-consistently include the forces arising from the bias to the system, while this is a difficult issue to be addressed in MD simulations.
As discussed earlier box integration method is used in the pCG solver, which treats the Poisson equation in the most accurate way.
Even though a full multigrid solver based on box-integration method has been under development, there is a neat way to reuse the already exiting code and treat the ion channel systems.
Ion channel simulations require the presence of large bath regions for accurate treatment of screening.
Besides the ultimate interest is to study the channel behavior in terms of ion permeability, selectivity, gating, density, etc....
is a relatively much finer mesh that spans a sub-domain of the system containing the region that requires fine resolution like the channel pore.
The Poisson equation is first solved on the coarse mesh with all the Dirichlet and Neumann boundary conditions, taking into account the applied bias.
Part of this potential energy barrier is due to the interaction between the crossing ion and the permanent/partial charges on the protein residues.
Another issue that additionally impacts the measurements is absence of water hydration molecules, which move with the ion and shield part of its charge.
The downside might be that programming and decoding the data could become very tricky, but once it's done correctly and with care, the advantages of using binary format are well worth the extra effort.
