🚀 go-pugleaf

Over the last year or so, I've been working on the application of experimental
design to the crystallization of proteins. This is an important problem,
because it's the ability of protein crystals to scatter X-rays which provides
the experimental data required to determine their structures at atomic
resolution.

It's probably necessary to provide a brief background to the problem.

Protein crystals form in supersaturated protein solutions (that is solutions
in which the concentration of of the protein exceeds its equilibrium
 solubility). Hence all the physical methods for protein crystallization
 involve bringing protein solutions into the supersaturated state by some
 means. A number of substances (typically salts and organic polymers) when
 added to protein solutions in high concentration, cause proteins to come out
 of solution. A typical protein crystallization trial will involve gradually
 increasing the concentration of such a substance in a protein solution
 ( by some kind of diffusion process) until the supersaturated state is
 achieved, and protein crystals can grow.

 Unfortunately, crystallization is only one of the outcomes of such an
 process. The other (and far more common) outcome is that the protein
 will simply crash out of solution and form a disordered precipitate.
 In fact, typically protein crystals will only form under a very limited
 range of solution conditions.

 Critical  variables to be considered are the substance used to exclude
 the protein from solution, it's final concentration,the concentration of
 protein,the temperature, and the solution pH. Additionaly, since pH in
 solution must be controlled by the presence of a chemical buffering system
 this leads to another two factors, the choice of buffering system and the
 buffer concentration.

So we have an experimental situation, with typically 7 or so factors,
several of which are qualitative and have a large number of potential classes.
Additionaly the process under study (crystallization) is often very
sensitive to changes in the levels of these factors, and is unlikely to be
observed at all over large areas of the potential experimental region.
Couple this with the fact that proteins, which have to be obtained from
biological systems, are typically available only in very small quantities,
and you have the makings of a very difficult experimental problem.

In fact the entire problem can probably be usefully broken down into two
stages. The first involves searching the experimental region for conditions
under which a protein will crystallize.This is usually the most critical
and difficult part of the problem. Then, once crystallization conditions
for a protein have been established conventional planned experiments can be
applied to establish treatment effects etc and crystal growth conditions
optimized via RSM techniques or the suchlike.

It's the first problem, that of searching for crystallization conditions
that is the subject of this posting.

 So how do scientists, working in this field, approach this problem?
 Most commonly they begin by doing some sort of limited full factorial
 experiment. However because these ideas have arisen independently from
 the statistics literature, these experiments are never explicitely
 identified as factorial experiments. A Typical description might be
 'We tested a matrix of crystallization conditions'. The essential problem
 with such an approach is the impractibly large size of such experiments.

 Other workers have proposed what they term the 'incomplete factorial
 method', which involves selecting, at random, a subset of the treatment
 combinations of a full factorial experiment. This would seem identical
 to the random balance experimentation proposed in the statistics
 literature by Satterthwaite in the late 1950's.This idea is frequently
 referenced in reviews of the subject, but few people actually seem to use it.
 

 Yet other workers have dispensed with a factorial structure altogether
 and proposed a screening experiment comprising 50 varied conditions which
 have 'worked before'. This was rather imaginatively described as
 a 'sparse matrix sampling method'. Rest assured that there's no shortage
 of baffling terminology in this branch of science.

And thats it really. So there seems to us to be a bit of room for development
here, as the state of experimental design in the field of protein
crystallization might be politely described as not very sophisticated.

Here at Massey we've been experimenting with the application of
Orthogonal fractional factorial designs of Resolution IV
( or equivalently Orthogonal arrays of strength 3, if you prefer).

Now in the context of the usual ANOVA/linear model fitting process, such
treatment plans evidently have a number of desirable properties (orthogonal
parameter estimates etc ). But in the initial stages of a protein
crystallization investigation, we're not really concerned with conducting a
conventional planned experiment, in which the objective is to assess
the treatment effects. Rather we're concerned with simply establishing
conditions under which the process of interest (crystallization)
occurs at all. So the question is (and it's taken me a long time to get here)
Do orthogonal arrays represent an efficient experimental approach under
such conditions?

Intuitively it seems to me that they probably do. And practically,
by the application of such arrays to a number of protein crystallization
problems, it also seems likely. In some of these cases, enough of the
treatments yielded crystals to make quantitative model fitting to
the results useful (based on variation in crystal size and regularity
of growth).

However intuition and a bit of practical work do not a thesis make,
and I'm stuck on how to rigorously justify their use in this problem.
Any knowledge of other problems of a similar nature, or
suggestions/criticism would be much appreciated.

Sorry for this rather long posting. If you've had the patience to read
and think about it then thankyou.


Richard Kingston
(R.L.Kingston@massey.ac.nz)
Protein Crystallography Research
Department of Chemistry and Biochemistry
Massey University
Palmerston North
New Zealand.

"I have yet to see any problem, however complicated, which, when you
looked at it in the right way, did not become still more complicated."
             Poul Anderson, New Scientist, 1969