Space-efficient and exact system representations for the nonlocal protein electrostatics problem
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Abstract
The study of proteins and protein interactions, which represent the main constituents of biological functions, plays an important role in modern biology. Application settings such as the development of pharmaceutical drugs and therapies rely on an accurate description of the electrostatics in biomolecular systems. This particularly includes nonlocal electrostatic contributions of the water-based solvents in the cell, which have a significant impact on the long-range visibility of immersed proteins. Existing mathematical models for the nonlocal protein electrostatics problem can be approached with numerical standard techniques, including boundary element methods (BEM). However, the typically large, dense, and asymmetric matrices involved in discretized BEM formulations previously prevented the application of the method to realistically-sized biomolecular systems. Here, we overcome this obstacle by developing implicit yet exact representations for such BEM matrices, capturing trillions of floating-point values in only a few megabytes of memory and without loss of information. We present generalized reference implementations for our implicit matrix types alongside specialized matrix operations for the Julia and Impala programming languages and demonstrate how they can be utilized with existing linear system solvers. On top of these implementations, we develop full-fledged CPU- and CUDA-based BEM solvers for the nonlocal protein electrostatics problem and make them available to the scientific community. In this context, we show that our solvers can perform dozens of matrix-vector products for the previously inaccessible BEM systems within a few seconds or minutes and thus allow, for the first time, to solve the employed BEM formulations with exact system matrices in the same time frame.