How to convert enrichment/depletion to frequency for comparing deep sequencing to sequence profile?
Although the question is kind of confusing at some places, what I understood is that you are trying to compare the relative amino acid enrichments in the two datasets.

As far as I know, you could construct the protein sequences from directed evolution experiment (presumably a time series data. Please clarify that.) and make a multiple sequence alignment (MSA) of that. In order to construct the sequences, there would be some technical procedures that would depend on the type of deep sequencing data you have. Factors such as the read length, protein length and coverage would need to be taken into consideration.

You could similarly make MSA for second datasets too.

Then using tools such as Rate4site (https://www.ncbi.nlm.nih.gov/m/pubmed/12169533/) you would be able to get evoutionary rates per site from MSAs. Then compare the evolutionary rates per sites for two datasets by correlating them.

If the correlation is high, the enrichments in both datasets are similar, otherwise not.