pbsam-auto

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 multipole expansions. For details on the method, please see Yap, Head-Gordon (2010) and Yap, Head-Gordon (2013).

The current implementation of PB-SAM in APBS includes:

• Calculation of energies, forces and torques
• Calculation of electrostatic potentials
• Brownian dynamics simulations

Keywords for this calculation type include:

ELEC pbsam-auto keywords:

• 3dmap
• diff
• dx
• exp
• grid2d
• imat
• ntraj
• pbc
• pdie
• randorient
• runname
• runtype
• salt
• sdie
• surf
• temp
• term
• termcombine
• tolsp
• units
• xyz

Background information

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

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

Where \(H^{(i)}\) and :math`F^{(i)}` are multipole expansions, \(I_{E}^{(i,k)}\) is the exposed surface integral matrix for sphere \(k\) of molecule \(i\), and \(T\) is an operator that transforms the multipole expansion to a local coordinate frame.

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:

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

where \(\langle M, N \rangle\) denotes the inner product.

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 is discussed in Yap, Head-Gordon (2013).

PB-SAM files

Vertex/surface file

As part of the coarse-graining process a definition of the molecular surface is necessary.

Coarse-grained PQR file

The coarse-graining process will produce a new PQR file mol#_cg.pqr that contains the original PQR concatenated with coarse-graining spherical centers. The number # refers to the order the file was read during the READ input file section statements.

IMAT: surface integral file

The surface integrals are computed for the boundary element part of PB-SAM. Their calculation can be quite time-consuming, so the first time they are computed for a system, they are saved to the working directory with the name molmsphs.bin`. The m in molmsphs.bin` is the ordered ID of the molecule from the PQR section. The s in molmsphs.bin` refers to coarse-grained sphere s of the molecule.

Multipole expansion files

Much like the IMAT files, the expansion files are files generated from self-polarization that are useful and time-saving methods for running a system of full-mutual polarziation on many molecules. If no expansion path is provided, the program will calculate self-polarization for each type of molecule in the system and save files of the form molmH,F.s.exp, where m is the molecule ID, H and F refer to the respective expansion (see above), and s is the coarse-grained sphere number.