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Crystallization of Nucleic Acids and ProteinsA Practical Approach$
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Arnaud Ducruix and Richard Giegé

Print publication date: 1999

Print ISBN-13: 9780199636792

Published to Oxford Scholarship Online: November 2020

DOI: 10.1093/oso/9780199636792.001.0001

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PRINTED FROM OXFORD SCHOLARSHIP ONLINE (oxford.universitypressscholarship.com). (c) Copyright Oxford University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 27 October 2021

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

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

Chapter:
4 (p.75) Experimental Design, Quantitative Analysis, and the Cartography of Crystal Growth
Source:
Crystallization of Nucleic Acids and Proteins
Author(s):

C. W. Carter

Publisher:
Oxford University Press
DOI:10.1093/oso/9780199636792.003.0008

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).

Keywords:   mathematical models, multiple regression, orthogonal arrays, response surface, sampling, sparse matrix kits, stationary point identification, statistics packages

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