Amino acids posterior probabilities

There is some confusion in using Bayes rule on what are sometimes called explanatory variables. As an example, we can try to use Bayesian statistics to derive the probabilities of each secondary structure type for each amino acid type, that is p( x r), where J. is a, P, or Y (for coil) secondary strucmres and r is one of the 20 amino acids. It is tempting to writep( x r) = p(r x)p( x)lp(r) using Bayes rule. This expression is, of course, correct and can be used on PDB data to relate these probabilities. But this is not Bayesian statistics, which relate parameters that represent underlying properties with (limited) data that are manifestations of those parameters in some way. In this case, the parameters we are after are 0 i(r) = p( x r). The data from the PDB are in the form of counts for y i(r), the number of amino acids of type r in the PDB that have secondary structure J.. There are 60 such numbers (20 amino acid types X 3 secondary structure types). We then have for each amino acid type a Bayesian expression for the posterior distribution for the values of xiiry. 

A similar formalism is used by Thompson and Goldstein [90] to predict residue accessibilities. What they derive would be a very useful prior distribution based on multiplying out independent probabilities to which data could be added to form a Bayesian posterior distribution. The work of Arnold et al. [87] is also not Bayesian statistics but rather the calculation of conditional distributions based on the simple counting argument that p(G r) = p(a, r)lp(r), where a is some property of interest (secondary structure, accessibility) and r is the amino acid type or some property of the amino acid type (hydro-phobicity) or of an amino acid segment (helical moment, etc). 

Note that if only a single amino acid type was observed in any profile position among the defining set, the posterior conditional probabilities can be approximated simply by the prior conditional probabilities of observing each of the other 19 amino acid types given that one was observed, i.e.,... 

Other substances in addition to the serum proteins that have presented adsorption problems are lysozyme, hemoglobin, pea proteins, phycoeryth-rin, ACTH, some basic materials prepared from tissues, egg white proteins, lactase, and milk proteins. In general, amino acids and small peptides do not present the adsorptive problems encountered with most proteins. The basic peptides of the posterior pituitary showed no detectable adsorption (44,74). Flodin and Tiselius (21) have modified the filter paper by altering the surface charge and this has aided in the elimination of adsorption in specific instances. Other methods, such as soaking the paper in a protein solution to saturate the adsorbing groups, have proved of some aid. It is probable that in the future a special paper will be developed for protein work which overcomes this obstacle. However, at present it remains a serious limitation of the method, particularly for preparative work. 

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