Sorry for such a question,if it is too naive.
Is there any definition of complexity about gene and protein according their function?If so,what is the relation between them?
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Proteins and genes usually are classified in superfamilies and families acording with its inner structure (mainly its domain organization). Domains are parts of the protein with its own 3D structure that usually have their own function and acts more or less in an independent manner.
However, this doesn't talk very much about their complexity. Myoglobin and hemoglobin, for instance, are very related proteins and thus are classified in the same family. However, hemoglobin forms tetramers, while myoglobin is a single molecule. It's not unheard that very close-related proteins may differ in their association with other proteins, polymeration pattern, addition of new domains, etc. Very similar poteins may have very different functions, i.e. catalyzing similar reaction in opposite directions or in very different pathways, or acting in different parts of the cell or the organism.
In biology, it is desirable for any classification to be based in evolutionary criteria, since not only it is a more neutral and unbiased way to classify biological things, but also provide information about the origin and structure of those. You could classify proteins by their structure, number of motifs, number of domains, number of different domains, size, function (in fact, there is a very useful and widely used classification for enzymes), etc; and you could classify the genes according to its number of exons, structure of its promoter/enhancer, policystronic or not, etc. However, this classification will usually prove itself to be inferior to an evolutionary classification.