My interest in chemistry and biology stems from my childhood experiences in my father's dental clinic. Observing numerous surgeries and operations triggered a deep curiosity in science. I was eventually led to applied mathematics, where mathematics meets the technology of today's world. I also have a great passion for teaching. I decided to join the science education program, offered in Middle East Technical University, where I received my B.Sc. as an honor student in mathematics teaching. My interest in mathematics grew in time and yielded a double major in mathematics. I am inclined toward interdisciplinary studies as well, and have broadened my background by taking various classes in computer science and astrophysics.
I participated the applied mathematics graduate program at University of California, Irvine (UCI). I was primarily involved in the numerical solutions of nonlinear partial differential equations (PDE) arising from engineering and physical problems. I completed courses on computational geometry and visualization, basic numerical approximation theory, and finite element methods (FEM) for PDEs. In order to complement the scientific computation aspect of my training, I also took some basic courses in data structures and software engineering. After my advisor Prof. Mike Holst moved to UCSD from UCI, I joined the Scientific Computation Group in the department of mathematics at UCSD. I have started working on developing multilevel FEM for the Poisson-Boltzmann equation (PBE), a nonlinear elliptic PDE arising in electrostatic models of biomolecules (described below), employing agglomeration and smoothed aggregation methods for building coarse problems. I have investigated geometry-based algebraic multigrid methods, non-nested-geometry-based coarsening methods, and smoothed aggregation for 3-D unstructured meshes. The above coarsening methods are crucial to enable the use of fast adaptive multilevel FEM solvers such as Manifold Code (MC) for the PBE (MC was developed by Prof. Holst over the last five years). In the last six months, I have begun to contribute to the multilevel solvers used in MC.
Recent breakthroughs in biological sciences have showed that computational methods can play an extremely important role in drug development. Progress depends heavily on understanding and predicting the complex behavior and function of biomolecules. A partial list of the devastating disorders in which a small number of biomolecules play key roles includes Acquired Immuno-Deficiency Syndrome (AIDS), Amyotrophic Lateral Sclerosis (ALS, from which both Stephen Hawking and my primary school teacher suffer), and Multiple Sclerosis (MS, from which my aunt suffers) as well as Alzheimer's Disease. Observing the tragic effects of MS on my aunt and ALS on a teacher that I had been close to, filled me with the desire that could lead to cures for such disorders.
Human immuno-deficiency virus (HIV), believed to cause AIDS, uses the so-called HIV protease for reproduction. In drug therapy, protease inhibitors are designed to interfere with the function of the HIV protease, thereby disrupting viral replication. By modeling the electrostatic behavior of the HIV protease on a computer, scientists can predict the effectiveness of a candidate drug molecule by simulating its interaction with the HIV protease. Molecules which exploit electrostatics to aid in binding to the active site in the protease would likely be the basis of the most effective drug. Other modeling candidates include superoxidedismutase, which has been implicated in ALS, and acetylcholinesterase (AChE), known to have a role in Alzheimer's disease. It has also been suggested recently [3] that dismylination of the central and peripheral nervous systems in humans, which leads to MS, might be understood through the use of electrostatic models, coupled to elasticity models of the myelin sheath.
The biochemistry community is intensely interested in the complicated range of interactions between enzymes and their substrates such as reactions between AChE and acetylcholine (ACh), due to their role in these and other diseases. These reactions can be characterized as diffusion-influenced bimolecular reactions. To simulate such complex systems, researchers rely on a number of techniques, including molecular dynamics and Brownian dynamics (BD) simulations. In BD, the stochastic trajectory of a particle is simulated by a series of small displacements chosen from a distribution which is the short time solution of Smoluchowski equation. This method is sufficiently general to model hydrodynamic interactions and arbitrary inter- and intramolecular forces. In order to make such simulations possible, the discrete nature of the ionic solutions in which these interactions take place is often approximated with continuum mechanics, leading to the PBE. In many cases, the accuracy of the simulations is limited by the enormous amount of computer time required to solve the PBE, in order to account for the long-range electrostatic forces in the simulation. Recent progress has been made on improving the efficiency of numerical methods for PBE [4],[5],[6] but much work remains. My research will provide the interdisciplinary link between biochemical problems and high performance numerical solvers.Multilevel methods are optimal or nearly optimal complexity numerical methods for solving large sparse linear systems arising from the discretization of PDEs, such as the PBE. In particular, multilevel method can be shown to have O(N) or O(N logN) complexity, where N is the number of unknowns in the discrete problem. In contrast, other popular methods, such as the conjugate gradient method, have complexity O(N^p), for p > 1. Although this seems like a minor difference, it manifests itself in terms of solution times on the order of seconds for multilevel methods compared to several hours (or worse) for non-optimal methods applied to a typical problem. The main power of the multilevel methods comes from utilizing coarser (rough representative of the original linear system together with few unknowns) and the finest (original system with many unknowns) grid representations of the problem. The transportation between fine and coarse systems are performed by the prolongation (coarse to fine) and restriction (fine to coarse and usually obtained from prolongation operator) operations. Without appropriate prolongation, the methods would lose superior complexity or even fail. Having a mechanism for constructing prolongation operators enables us to build multilevel methods. Given such prolongation operators, MC implements a complete adaptive multilevel numerical solution strategy, with guaranteed error bounds. However, both adaptivity and prolongation aspects of the solver are quite problematic in the case of the PBE due to the complex nature of the problem. The adaptivity difficulties are being addressed by Prof. Holst in a related research project, and will be incorporated into MC. I plan to focus on the remaining difficulty, namely the multilevel solver and its implementation in MC. Some of the challenges to be faced are as follows. For accurate representation of the solvent accessibility surface around charged structures such as proteins (i.e., the ''molecular skin'' of the structure), an extremely fine triangular surface mesh is required, leading to a correspondingly fine tetrahedral volume mesh. This precludes the use of standard multilevel methods for these problems, in that it is not possible to create a useful multilevel hierarchy through subdivision. My approach to this problem will be to design a prolongation operator specifically for the PBE which has provably good (described below) approximation and stability, through the use of agglomeration and aggregation techniques [1], [2], [7]. Such a prolongation operator is the missing link which will lead to an optimal complexity multilevel solver for the PBE. Combined with adaptivity, such a multilevel method will provide orders of magnitude improvement in solution time over current methods. I will use AChE and HIV protease data for test problems using MC.
Electostatic potential around the enzyme plays the key role as the driving force field in packages like University of Houston Brownian Dynamics (UHBD), and other related codes such as Simulation of Diffusional Association of Proteins (SDAP), University of Colombia's DelPhi, and the Object-Oriented Model for Probing Assemblages of Atoms (OOMPAA). To get a precise estimate of the rate constant several thousand trajectories of the molecule are calculated, requiring enormous computational resources. These packages are in need of a high performance solver which is possible only through adaptivity and multilevel methods. MC provides a solid implementation base for both of these requirements. I will incorporate MC with BD simulation packages of diffusion-influenced bimolecular reactions and electrostatic simulators such as UHBD, SDAP or OOMPAA.
If one can rigorously prove complexity statements and error estimates by employing inequalities about a numerical method, then it is called provably good. This characteristic is extremely crucial from the scientific computational point of view, in the sense that we can predict the run time and the error within prescribed bounds. These bounds eliminate the randomness of the computation time and error. Current approaches to the PBE lack this feature. However, the MC adaptive multilevel finite element framework provides provably good numerical solutions. The methods I will design will be based on this approach taken in MC.
The expertise of the McCammon group provides an environment where I can share my insights with and implement my approaches to their mathematics oriented problems. I will contribute to the mathematical framework of the existing biomolecular models. We are planning to improve the efficiency of the electrostatics and Brownian/molecular dynamics programs in particular for the linearized and non-linear PBE.
On the other hand, MC's capabilities, adaptivity, and multilevel approach, are just what is needed by the biology community, particularly the McCammon group. The main challenge is to create coarse problems which will represent the original problem. Therefore, constructing prolongation operators which will transport the solution from coarse to fine levels for the multilevel methods (e.g. multigrid and domain decomposition) is needed in solving the PDE involved in biomolecular modeling processes. The construction will be done with regard to producing provably good prolongation operators. Those operators and the theory can also be utilized in the Holst group for future MC implementations because of the generality of the coarsening techniques employed, and approximation and stability properties of the prolongation operators. I believe that I can contribute to the McCammon group by incorporating MC into their existing simulation packages.
Both Prof. J. Andrew McCammon and Prof. Michael Holst have exceptional experience in and tremendous enthusiasm for their respective areas of study, areas which have come together in various ways. I firmly believe that this collaboration between the groups will result in a fruitful project that is truly at the interface of biochemical and mathematical sciences.
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