Method overview

PB-SAM is a semi-analytical solution to the linearized Poisson-Boltzmann equation for multiple molecules of arbitrary charge distribution in an ionic solution. The solution is an extension of the analytical method, leveraging Fast-Multipole methods as well as boundary elements. Each molecule is coarse-grained as a system of overlapping spheres, whose surface charges are represented by the multipole expansions \(H^{(i)}\) and \(F^{(i)}\). To solve for the potential, the following interactions are considered:

  • Intra-molecular interactions between overlapping spheres are treated numerically
  • Intra-molecular interactions between non-overlapping spheres are treated analytically
  • Inter-molecular interactions between spheres on different molecules

With these interactions, the multipole expansions are solved with an iterative SCF method, briefly given as

\[ H^{(i,k)} = I_{E}^{(i,k)} \cdot \left ( H^{(i,k)} + F^{(i,k)} + T \cdot H^{(j,l)} \right ) \] \[ F^{(i,k)} = I_{E}^{(i,k)} \cdot \left ( H^{(i,k)} + F^{(i,k)} + T \cdot F^{(j,l)} \right ) \]

For details on the method, please see [YaHe10] and [YaHe13].

Physical calculations

Interaction energies

From the above formulation, computation of the interaction energy (\(\Omega^{(i)}\)) for molecule i, is given as a sum of all the interactions of spheres \(k\) within it with all external spheres (in a simplified form) as follows:

\[\Omega^{(i)}=\frac{1}{\epsilon_s} \sum_{k \, in\, i} \sum_{j \ne i}^N \sum_{l\, in \, j} \left \langle T \cdot H^{(j,l)} , H^{(i,k)} \right \rangle \]

Where \(\langle . . . \rangle\) denotes an inner product.

Forces and Torques

When energy is computed, forces follow as:

\[ \textbf{F}^{(i)} = \nabla_i \Omega^{(i)}=\frac{1}{\epsilon_s} [ \langle \nabla_i \,T \cdot H^{(j,l)} , H^{(i,k)} \rangle + \langle T \cdot H^{(j,l)} , \nabla_i \, H^{(i,k)} \rangle ]\]

The method to calculate the torque \(\boldsymbol{\tau}^{(i)}\) on molecule is outside the scope of this manual, but is discussed extensively in [YaHe13]