The theory of symmetric multivariate Lagrange interpolation is a beautiful but rather unknown tool that has many applications. Here we derive from it an Exchange Lemma that allows to explain in a simple and natural way the full description of the dou
ble sum expressions introduced by Sylvester in 1853 in terms of subresultants and their Bezout coefficients.
The twenty protein coding amino acids are found in proteomes with different relative abundances. The most abundant amino acid, leucine, is nearly an order of magnitude more prevalent than the least abundant amino acid, cysteine. Amino acid metabolic
costs differ similarly, constraining their incorporation into proteins. On the other hand, sequence diversity is necessary for protein folding, function and evolution. Here we present a simple model for a cost-diversity trade-off postulating that natural proteomes minimize amino acid metabolic flux while maximizing sequence entropy. The model explains the relative abundances of amino acids across a diverse set of proteomes. We found that the data is remarkably well explained when the cost function accounts for amino acid chemical decay. More than one hundred proteomes reach comparable solutions to the trade-off by different combinations of cost and diversity. Quantifying the interplay between proteome size and entropy shows that proteomes can get optimally large and diverse.
