Research Interests

Mathematical and computational crystallography

My research interests lie in the area of mathematical biology, especially the development of novel mathematical models and optimal procedures for determining three-dimensional crystal structures from X-ray and neutron diffraction data. The so-called “phase problem” in crystallography is the problem of determining the phase angles of the diffracted rays. The phase information, which is lost in the diffraction experiment, is in fact recoverable from the measurable intensities alone. The methods devised to solve the phase problem are known as direct methods.

Mathematical Models for Crystallographic Phase Problem

In the minimal principle method, the crystallographic phase problem is formulated as a problem of constrained global optimization. Traditionally, a probabilistic model is used to derive the so-called cosine minimal function that plays a pivotal role for determining three-dimensional structures. We have recently established a novel statistical model and corresponding statistical minimal function based on the empirical distributions of the structure invariants. This statistical function serves as the foundation of an optimization procedure called statistical Shake-and-Bake. Favorable applications of this procedure have shown the overall improvement over the existing procedures.

Optimal Procedures for Crystallographic Computing

Crystallographic computing is one of the areas that would benefit greatly from mathematical optimization. In collaboration with Prof. Nikolaos Sahinidis (Univ. of Illinois), we are currently investigating novel procedures that combine the necessary complementary backgrounds from mathematical optimization and macromolecular crystallography.  Specifically, we are developing mathematical optimization algorithms to solve the crystallographic phase problem in a reliable and efficient way. This work promises to lay the foundations of a new generation of crystallographic computing systems that will reveal structures important in the understanding of life, material science, and drug design.

Selected  Publications
Dr. Xu’s publications include total of 24 papers & chapters, 42 abstracts

• Xu, H. & Hauptman, H.A. (2006). Recent advances in direct phasing methods for protein structure determination, Acta cryst. D62. (In press).
• Xu, H., Hauptman, H. A. & Langs, D.A. (2005). Novel statistical approach to the phase problem in neutron crystallography, In “Hydrogen- and Hydration-Sensitive Structural Biology”, pp 197-205. KubaPro Co., Ltd., Tokyo, Japan.
• Xu, H., Weeks, C.M. & Hauptman, H.A. (2005). Optimizing statistical Shake-and-Bake for Se-atom substructure determination. Acta Cryst. D61, 976-981.
• Xu, H. & Hauptman, H.A. (2004). Statistical approach to the phase problem. Acta Cryst. A60, 153-157.
• Xu, H. & Hauptman, H.A. (2003). On integrating the techniques of direct methods and SIRAS: the probabilistic theory of doublets and its applications. Acta Cryst. A59, 60-65.
• Xu, H., Hauptman, H.A. & Weeks, C.M. (2002). Sine-enhanced Shake-and-Bake: the theoretical basis and applications to se-atom substructures. Acta Cryst. D58, 90-96.
• Xu, H. & Hauptman, H.A. (2000). On the extrapolation of the magnitudes |E| of the normalized structure factors E. Acta Cryst. A56, 284-287.
• Xu, H., Weeks, C.M., Deacon, A., Miller, R. & Hauptman, H.A. (2000). Ill-conditioned Shake-and-Bake: the trap of the false minimum. Acta Cryst. A56, 112-118.
• Xu, H., Hauptman, H.A., Weeks, C.M. & Miller, R. (2000). P1 Shake-and-Bake: can success be guaranteed? Acta Cryst. D56, 238-240.