# tabi

This mode uses the TABI-PB integral equation software from Geng and Krasny to solve the linearized Poisson-Boltmzann equation. Boundary element methods offer the ability to focus numerical effort on a much smaller region of the problem domain: the interface between the molecule and the solvent. In this method, two coupled integral equations defined on the solute-solvent boundary define a mathematical relationship between the electrostatic surface potential and its normal derivative with a set of integral kernels consisting of Coulomb and screened Coulomb potentials with their normal derivatives. The boundary element method requires a surface triangulation, generated by a program such as MSMS or NanoShaper, on which to discretize the integral equations.

For more information, see the Geng & Krasny 2013 J Comput Phys paper.

ELEC tabi keywords

- ion
- mac
- mesh
- mol
- outdata
- pdie
- sdens
- sdie
- srad
- temp
- tree_n0
- tree_order

## Background information

The Treecode-Accelerated Boundary Integral Poisson-Boltzmann solver (TABI-PB; Geng, 2013 calculates electrostatics of solvated biomolecules governed by the linearized Poisson-Boltzmann equation. It uses a well-posed boundary integral Poisson-Boltzmann formulation to ensure rapid convergence. In addition, a fast treecode algorithm for the screened Coulomb potential (Li, 2009) is applied to speed up the matrix-vector products in each GMRES iteration. The molecular surfaces, which divide the entire domain into solute region and solvent region, are generated by MSMS or NanoShaper.

### TABI-PB algorithm

The coupled integral equations derived from the linearized Poisson-Boltzmann equation are

\[\begin{split}\frac{1}{2}(1+\epsilon)\phi({x})-\int_\Gamma(K_1({x},{y})\frac{\partial{\phi({y})}}{\partial{v}}+K_2({x},{y})\phi({y}))dS_{y} &=S_1({x}) \\ \frac{1}{2}(1+\frac{1}{\epsilon})\frac{\partial{\phi({x})}}{\partial{v}}-\int_\Gamma(K_3({x},{y})\frac{\partial{\phi({y})}}{\partial{v}}+K_4({x},{y})\phi({y}))dS_{y} &=S_2({x}), {x}\in\Gamma\end{split}\]

for the surface potential \(\phi\), and its normal derivative \(\frac{\partial\phi}{\partial v}\) on the surface :math`Gamma`. The kernels \(K_{1,2,3,4}\) are linear combinations of the Coulomb and screened Coulomb potentials:

\[\begin{split}G_0({x},{y}) &= \frac{1}{4\pi |{x}-{y}|} \\ G_{\kappa}({x},{y}) &= \frac{e^{-\kappa|{x}-{y}|}}{4\pi |{x}-{y}|}\end{split}\]

and their first and second derivatives.

The sums in the discretized form of the integral equations above have the form of N-body interactions,

\[V_i = \sum_{j=1,j \neq i}^{N} q_j G({x}_i,{x}_j), i=1,\ldots,N\]

where \(G\) is the screened Coulomb potential kernel, \({x}_i, {x}_j\) are the centroids of the triangles, and \(q_j\) is the charge at \({x}_j\). The particles (centroids of the triangles) are divided into a hierarchy of clusters having a tree structure. The treecode replaces the \(\mathcal{O}(N^2)\) particle-particle interactions by \(\mathcal{O}(N \log N)\) particle-cluster interactions and TABI-PB utilizes this feature efficiently.

## Output

The TABI-PB code produces an output file called `surface_potential.dat` containing:

- number of nodes, number of triangles
- node index, vertices, normal vector, surface potential (kJ mol-1 ec-1), surface potential normal derivatives (kJ mol-1 ec-1 A-1)
- connectivity data for MSMS surface triangulation

The format is given below:

    num_node num_triangle
    node_index x y z norm_x norm_y norm_z phi norm_phi
    (et cetera)
    node_index1 node_index2 node_index3

The TABI-PB code prints the free energy of solvation and Coulombic free energy in kJ/mol, along with some other information such as CPU time and the GMRES residuals at each step.

Additionally, TABI-PB can optionally output a VTK polygonal data file containing color mappable potentials and normal derivatives of potentials on the faces and vertices of the mesh. The VTK file can be visualized using ParaView.