Collaborators
Proofs
- Derive MSA in an understandable way. We have shown that we can reproduce MSA with NLBC.
- TOM: Prove convergence of the Picard iteration for the NLBC BIE in the physical parameter range
Metadata
Difficulty Low Timeframe 2 months Impact Medium Motivation
We would like to model the asymmetry in the induced electric charge around a biomolecule arising from the molecular nature of water. We have formulated a nonlinear Boundary Integral Equation (BIE) for the induced surface charge. We would like to solve these equations, and also prove the existence and hopefully the uniqueness of solutions.
Background
Jay and I have a paper specifying the asymmetric model. We also have a long and rambling unpublished document which might be helpful. I also have notes for a paper on the Picard iteration and a proof using the Banach Fixed-point Theorem. This should also provide existence and uniqueness of the solution.
Project
- Derive the BIE from the PDE. I learned how to do this from http://arxiv.org/abs/1203.5997. Helsing is a really good writer.
- Derive the nonlinear BIE from the PDE with nonlinear BC
- Setup Born ion problem in pointbem and write a description
- Prove that the Picard iteration converges for this BIE for some parameter range
- Writeup paper for submission to SISC or CompMath (Ketch likes this one)
- TOM: Prove existence and uniqueness for the NLBC PDE system
Metadata
Difficulty Medium Timeframe 6 months Impact Medium Motivation
Proving this in a different way from the BIE might give insight into the physics. Even just making the correspondence between the PDE and BIE rigrous would be worth a paper I think.
- I NO LONGER THINK THIS IS POSSIBLE: Derive an effective BC for the NLBC+Gauss' Law Correction system
Metadata
Difficulty High Timeframe 6 months Impact High Motivation
We would like a single, nonlinear effective boundary condition which captures both charge-hydration asymmetry as well as charge conservation. This is likely to come from a limiting argument as the Gauss' Law Correction surface approaches the original NLBC surface.
Calculations
Accurate Molecular Solvation Models
ProblemImplicit solvent models for molecules dissolved in fluid offer a solution to the enormous computational demands of molecular dynamics (MD) simulation. However, the dominant Poisson model currently requires tuning of hundreds of parameters, which can consume months and even years of researcher effort, and has cannot reproduce cruical phenomena such as charge-hydration asymmetry and entropies of solution.
ApproachWe have developed a simple 4 parameter implicit solvent model based upon a nonlinear boundary integral formulation of the fundamental physics. Our model correctly predicts charge-hydration asymmetry and energies of mixtures, as well as thermodynamic measures, such as entropy, that traditional Poisson models are incapable of reproducing.
ImpactAn accurate, robust, therodynamically viable implicit solvent model could revolutionize computational drug design, currently at the mercy of the extreme computational and interpretive demands of MD. Moreover, our model is compatible with the workhorse of computational chemistry, the Polarizible Comtinuum Model (PCM), which possesses the same model shortcomings as the Poisson model.
PlanWe have developed a proof-of-concept code, and published numerous stuides indicating the superior accuracy and efficiency of this method. We are currently working to integrate our model into large scale community code in computational chemistry, such as Psi4 and NWChem.
- Get a die-off radius for high freq molecular EM effects on molecule surface using Lap estimate from SIAM Rev paper
- We can calculate maximum polarization field, so we can guess alpha
- Extend pointbem-petsc to the Stokes kernel
- Ellipsoidal shape approximations
Motivation
We would like to efficiently calculate approximate eigenmodes for the potential operator on complex molecular surfaces. We plan to use the analytic eigenmodes of the potential operator expressed as ellipsoidal harmonics (the convergence of spherical harmonics is much much too slow, but comparing would be nice).
Background
There is a nice Wikipedia page on this quadrature and it has a link to David Bailey's writeup. It truly is faster than Gaussian quadrature for integrands which are not polynomial. The theory is laid out in the Mori paper. The Wikipedia page also has example code for this and other quadrature rules.
Jay and I have a paper on the use of ellipsoidal harmonics for the potential problem on surfaces. Along with this paper, we produced a repository with all the code for calculations.
This paper could be useful "Numerical problems in the computation of ellipsoidal harmonics" by G. Sona
Project
Smaller paper on accurate quadrature- Implement the quadrature in C
- Develop a test set (you could use on of those on the Wikipedia page)
- Compare to trapezoid rule and Gauss-Legendre from PETSc
- Demonstrate convergence order on the test problems
- Use a work-precision diagram to quantify the comparison
- Replace the quadrature in the ellipsoidal repository (Python code) with this new version and replicate the examples.
- Write up new version of paper with tanh-sinh quadrature, showing work/precision comparison of different quadratures
- Mesh an ellipsoid using meshmaker
- Calculate the low modes of S2S operator for the ellipsoid. Check against the analytical modes.
- Calculate the low modes of S2S for ARG and check against the effective ellipsoid.
- Develop pipeline from PDB, to meshmaker, to ellipsoid approximation/full BEM, to solution comparison
- Solving the NLBC Model
Motivation
We would like to model the asymmetry in the induced electric charge around a biomolecule arising from the molecular nature of water. We have formulated a nonlinear Boundary Integral Equation (BIE) for the induced surface charge. We would like to solve these equations.
Background
Jay and I have a paper specifying the asymmetric model. We also have a long and rambling unpublished document which might be helpful. I also have notes for a paper on the Picard iteration and a proof using the Banach Fixed-point Theorem.Project
- Solve the problem for the sphere using Galerkin boundary elements. You can use the PETSc code here to do that, with some help from me.
- Add the asymmetric term to the point or panel bem code, whichever is easier.
- Use the PETSc Picard solver to solve the nonlinear BIE
- Compare to Jay's MD solutions for the sphere
- Run using an amino acid, like ARG
- Try Anderson Acceleration, and validate against Kelley convergence theory
- Writeup paper for submission to SISC
Formulations
- Williard Miller (Symmetry and Separation of Variables, p.204) about 3D transform of solutions outside general shapes to spheres