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 NanoShaper, on which to discretize the integral equations.
For more information, see the Geng & Krasny 2013 J Comput Phys paper.
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 NanoShaper.
TABI-PB algorithm¶
The coupled integral equations derived from the linearized Poisson-Boltzmann equation are
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:
and their first and second derivatives.
The sums in the discretized form of the integral equations above have the form of N-body interactions,
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 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.