INSPIRE: Assessing feasible regions of configuration spaces for macromolecular crystals

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This INSPIRE project is jointly funded by Algorithmic Foundations in CISE/CCF, Analysis in MPS/DMS, and the NSF Office of Integrative Activities.Knowledge of the 3-dimensional structure of protein molecules supports scientific understanding of how proteins perform their functions within cells. Structures of over 100,000 proteins in the Protein Data Bank have been determined by macromolecular x-ray crystallography: measuring the diffraction of x-ray beams from crystal composed of many symmetrically arranged copies of one or more protein molecules gives partial information (the amplitudes of the Fourier transform) that must be filled in (solving the "phase problem," often by molecular replacement -- taking phases from related molecules) to complete the 3d structure. Molecular replacement works quite well for simple single-domain proteins, but breaks down for multi-domain proteins and large complexes; one needs to explore the possible combinations of domains and their diffraction patterns as replacement candidates. This cross-disciplinary project brings together experts in robotics and in pure mathematics to address the ''phase problem'' of macromolecular x-ray crystallography. The mathematical and computational framework developed in this project will enable many more protein structures to be solved in a less laborious way than can be done now. The project also introduces Baltimore City high school students to mathematics and molecular biophysics through unique visualization activities.The essence of combining domains is geometric. The team can use articulated multi-rigid-body models from the field of robotics to combine rigid portions of structures, from domains with similar sequences. The relative rigid-body motions between the domains become the unknown degrees of freedom in these articulated models. Crystal packing constraints will rule out the majority of possible configurations for these domains, and reduce the otherwise high-dimensional nature of the search space. The project will develop new algebro-geometric and computational methods for rapidly discarding the large collision regions in configuration space, so that searches will focus on the remaining small-volume feasible regions in this high-dimensional search space. Computer code will be prepared integrating the resulting methods into existing molecular crystallography software packages.