Quantitative analysis in the characterization and optimization of protein crystal growth

Protein crystal growth often depends on the combination of many different factors. Some affect protein solubility directly; others may act indirectly by causing conformational changes. Systematic characterization of these factors can be important for generating good crystals. It can also provide useful insight into the biochemical behavior of the protein being crystallized. Here we focus on statistical methods to achieve these two objectives. (1) Characterization of a protein system by analyzing patterns of crystal polymorphism under different levels of biochemical parameters, such as ligands and pH. Tests of the reproducibility of crystal growth experiments indicate that quantitative scales of crystal quality can be statistically significant. Analysis of variance for a replicated, full-factorial design in which four factors were tested at two levels has been used to demonstrate highly significant, biochemically relevant, two-factor interactions strongly implicating pH and ligand-dependent conformational changes. (2) Optimization of crystal growth via response-surface methods. `Minimum predicted variance' designs provide for efficient response-surface experiments aimed at constructing quadratic models in several dimensions. We have used such models to improve crystal size and quality significantly for three forms of Bacillus stearothermophilus tryptophanyl-tRNA synthetase. In one case we can now avoid having to increase the size by repeated seeding, a difficult procedure that also produces unwanted growth of satellite crystals. Graphs of two-dimensional level surfaces reveal a number of ridges, where the same result is obtained for many combinations of the factors usually varied when trying to improve crystals. An important inference is that it may be better to sample simultaneously for the effects of protein concentration and supersaturation. For a system involving only one crystallizing agent, supersaturation can be approximated as the product of protein and precipitant concentrations. Use of this search direction significantly improves the performance of response-surface experiments. Advantages of growing crystals at stationary points of their response surfaces include better crystals and higher reproducibility, since crystal growth at stationary points is insulated from the deleterious effects of experimental fluctuations. This arises because the derivatives of the response are by definition zero with respect to the experimental variables. Quantitative analysis of appropriately designed crystal growth experiments can thus be a powerful way to characterize complex and interacting biochemical dependencies in macromolecular systems and optimize parameters important to the crystallography.