A local approximation to supersaturation affords a useful coordinate transformation for the study of crystal growth

Two of the most important experimental variables in the search for appropriate crystallization conditions are the initial concentrations of macromolecule and crystallizing agent. Previously, it has been suggested that the coordinate transformation { [crystallizing agent], [macromolecule] } --> { [macromolecule] x [crystallizing agent], [macromolecule]} be used to sample crystal growth conditions. Here, it is shown that this transformation is a special case of a generally applicable transformation. The initial supersaturation can be represented locally by a rectangular hyperbola involving multiples of the product of macromolecule and crystallizing agent concentrations. The coordinate system for the solubility diagram, ([crystallizing agent] versus [macromolecule]), can thus be transformed analytically to an alternative coordinate system in which the independent variables are local approximations to the initial supersaturation and the reservoir of soluble macromolecule available to feed a growing seed. In the new coordinate system the 'nucleation zone' is 'orthogonalized', so it can be sampled efficiently on a rectangular grid, with greater assurance that experiments will give rise to crystals. Moreover, since these new coordinate directions segregate fundamental effects on nucleation from effects on growth, using them in experimental designs should improve data analysis for response surface experiments.