Thousands of protein linear motif classes may still be undiscovered.

Linear motifs are short protein subsequences that mediate protein interactions. Hundreds of motif classes including thousands of motif instances are known. Our theory estimates how many motif classes remain undiscovered. As commonly done, we describe motif classes as regular expressions specifying motif length and the allowed amino acids at each motif position. We measure motif specificity for a pair of motif classes by quantifying how many motif-discriminating positions prevent a protein subsequence from matching the two classes at once. We derive theorems for the maximal number of motif classes that can simultaneously maintain a certain number of motif-discriminating positions between all pairs of classes in the motif universe, for a given amino acid alphabet. We also calculate the fraction of all protein subsequences that would belong to a motif class if all potential motif classes came into existence. Naturally occurring pairs of motif classes present most often a single motif-discriminating position. This mild specificity maximizes the potential number of coexisting motif classes, the expansion of the motif universe due to amino acid modifications and the fraction of amino acid sequences that code for a motif instance. As a result, thousands of linear motif classes may remain undiscovered.

Amino Acid metabolism conflicts with protein diversity.

The 20 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, a diverse set of protein sequences is necessary to build functional proteomes. 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 are remarkably well explained when the cost function accounts for amino acid chemical decay. More than 100 organisms reach comparable solutions to the trade-off by different combinations of proteome cost and sequence diversity. Quantifying the interplay between proteome size and entropy shows that proteomes can get optimally large and diverse.