Experimental Design, Quantitative Analysis, and the Cartography of Crystal Growth

This chapter is about practical uses of mathematical models to simplify the task of finding the best conditions under which to crystallize a macromolecule. The models describe a system’s response to changes in the independent variables under experimental control. Such a mathematical description is a surface, whose two-dimensional projections can be plotted, so it is usually called a ‘response surface’. Various methods have been described for navigating an unknown surface. They share important characteristics: experiments performed at different levels of the independent variables are scored quantitatively, and fitted implicitly or explicitly, to some model for system behaviour. Initially, one examines behaviour on a coarse grid, seeking approximate indications for multiple crystal forms and identifying important experimental variables. Later, individual locations on the surface are mapped in greater detail to optimize conditions. Finding ‘winning combinations’ for crystal growth can be approached successively with increasingly well-defined protocols and with greater confidence. Whether it is used explicitly or more intuitively, the idea of a response surface underlies the experimental investigation of all multivariate processes, like crystal growth, where one hopes to find a ‘best’ set of conditions. The optimization process is illustrated schematically in Figure 1. In general, there are three stages to this quantitative approach: (a) Design. One must first induce variation in some desired experimental result by changing the experimental conditions. Experiments are performed according to a plan or design. Decisions must be made concerning the experimental variables and how to sample them. (b) Experiments and scores. Each experiment provides an estimate for how the system behaves at the corresponding point in the experimental space. When these estimates are examined together as a group, patterns often appear. For example, a crystal polymorphism may occur only in restricted regions of the variable space explored by the experiment. (c) Fitting and testing models. Imposing a mathematical model onto such patterns provides a way to predict how the system will behave at points where there were no experiments. The better the predictions, the better the model. Adequate models provide accurate interpolation within the range of experimental variables originally sampled; occasionally a very good model will correctly predict behaviour outside it (1).