The algorithms constitute an adaptive finite-element computer programme called the Manifold Code, developed by Dr. Holst, an associate professor in the Scientific Computation Group at the Math Department, and an Adaptive Poisson-Boltzmann Solver (APBS), developed by Nathan Baker. The codes allow the calculation of the electric potential of arbitrarily complex structures. Baker and his colleagues in the group of J. Andrew McCammon, Department of Chemistry and Biochemistry, collaborated with Dr. Holst and other members of the Scientific Computation Group to adapt the Manifold Code via APBS for biological problems.
Intermolecular interaction is strongly constrained by the electrostatics: the way in which the landscape of electrical charge is laid out in the molecular environment. Knowledge about the electric potential is needed across the full domain of interest, which may contain one or many molecules, usually in solvent. "To find the potential around a typical charged biological structure immersed in a salt solution, computational biochemists must solve a non-linear second-order partial differential equation called the Poisson-Boltzmann equation", Baker stated. "This equation has been a formidable challenge, and many techniques for approximating the solutions have been developed over the past decade, with widely varying degrees of accuracy and efficiency."
The equation (PBE for short) is difficult to solve both speedily and accurately because of the several numerical problems it presents, including sharp, non-linear discontinuities of its parameters at the interfaces between solvent and charged structure. APBS was written by Baker to allow the Manifold Code to cope with the specifics of the PBE. Nathan Baker added that the method is so promising and has such wide application that he has been joined by the postdoctoral researcher David Sept to extend the research further. Next to Dr. Holst, the mathematicians include professor Randolph E. Bank, UC Los Angeles postdoctoral fellow Feng Wang, and UC San Diego math graduate student Burak Aksoylu. Baker is supported by a pre-doctoral fellowship from the Howard Hughes Medical Institute (HHMI).
The melding of APBS and the Manifold Code into a powerful engine of multi-molecular simulation took nearly a year. With Wang, Baker and Dr. Holst first used the APBS-Manifold combination to examine four bio-molecules having distinct electrostatic properties:
- a neutral HIV integrase dimer,
- mouse acetylcholinesterase which has an overall negative charge,
- fasciculin-2, a component of snake venom with a net positive charge, and
- a 36-unit polymer of DNA.
The Manifold Code, developed by Dr. Holst over several years at Caltech and UC San Diego, is an adaptive, multi-level finite-element method, applicable to a general class of problems which include the PBE. It is similar in spirit to a code developed earlier by Randolph Bank which solves problems in two dimensions. Dr. Holst's code however can handle 3D problems. The finite elements which are being used, are the simplest and are called, collectively, "simplices": triangles in 2D problems and tetrahedra in 3D problems.
It is here that Baker's APBS code is essential for setting up the problem, Dr. Holst notes. The code first solves a small version of the problem on a coarse mesh of simplices. A procedure to estimate the errors in the solution is then used to partition the problem into sub-domains whose error is about the same. The sub-domains are further divided into new meshes of tetrahedra. Solutions are calculated on the new meshes, the errors are re-estimated, and, if necessary, the problem is further parcelled out. All these iterative procedures insure, as Baker puts it, "that the calculation takes place where the action is, where there are large gradients or discontinuities in the potential, where proteins are folded or overlaid on one another." Thus the adaptive part of both codes enables them to calculate on as refined a mesh as needed, just where it is needed and nowhere else.
Each iteration produces a system of non-linear finite-element equations to be solved, that is done using a variant of the well-known Newton method via an algebraic multi-level algorithm. When the solutions have reached a desired degree of accuracy, the sub-domains are knit back together in an overall solution. A parallel version of the multi-level adaptive finite-element method was recently developed by Dr. Holst and Randolph Bank. "We have immediately gone on to construct larger problems with APBS that can make use of massively parallel machines like SDSC's Blue Horizon", Baker said. "With the Linux box able to handle structures containing some 200.000 atoms, we wanted to tackle much bigger problems using the parallel version."
Those problems presented themselves in the form of actin and tubulin. Actin monomers, the most abundant protein in the cell, and tubulin dimers, which constitute the micro-tubules which transport proteins throughout a cell, are both highly conserved globular proteins. "Since they are important components of all eukaryotic cells, we wanted to take a computational look at them", Baker commented. Actin filaments, together with associated proteins, are important in controlling cell shape, motility, and transport. They were first found in muscle tissue and, together with myosin and other proteins, supply the mechanism of muscle contraction. Baker and Sept used APBS and the Manifold Code to investigate the electrostatic properties of an actin filament.
By contrast with actin filaments, micro-tubules appear to be constituted to resist compressive forces, and they perform dynamic intracellular functions. Micro-tubules, for example, pull at the spindles formed during cell division (mitosis). "Since a drug like taxol operates by binding to micro-tubules and disrupting the division of cancer cells", Sept explained, "a more complete understanding of these structures should be helpful in the structure-based drug discovery." Sept and Baker used Blue Horizon to calculate the electrostatic potential of a 15-protofilament micro-tubule. Some of these calculations involved nearly a million atoms and runs on up to 32 processors permitted the use of a mesh refined into more than six million simplices. "The calculational box was 90 nanometers on a side", Baker stated, "and the smallest simplex had a longest edge of 0.088 nanometers."
"These are very exciting calculations", commented group leader McCammon, "and we are now analysing them with a view to pushing the envelope further and tackling a number of large-scale problems in the intracellular activity." "A faster PBE solver which can maintain solution accuracy has certainly been needed, especially as we extend our domains to accommodate multiple molecules, multi-molecular interactions, and processes occurring on sub-cellular scales", according to Baker. "Many investigators have been using scaled-up molecular simulation codes to look at multi-molecular interactions", added McCammon, "and now we can contribute mightily to the realism and accuracy of such studies, with rapid and accurate determinations of the electrostatic potential of structures and processes of major pharmacological as well as biological importance."
The approach taken by Dr. Bank and Dr. Holst to the parallelisation reduced interprocessor communication to a minimum and improved load balancing. "Our codes were written as sequential codes, but they can run in a parallel environment without a large investment in re-coding", Dr. Holst stated. Most important is that the use of multi-level finite-element methods yields orders of magnitude reductions in solution time compared to uniform mesh approaches. "We had been working for years on new and better computational solutions for the elliptical partial differential equations that arise in many scientific problems", Dr. Holst explained, "and we were just delighted to find that Andy McCammon's group was pursuing exactly the right problems to test our ability to deliver accurate numerical solutions in a computationally efficient fashion, despite the well-known difficulty of the problems."