Some proteins produced in a cell have a regulatory role. A regulatory protein controls the transcription of specific genes in the genome. This control often involves the protein's binding to a particular DNA sequence in the vicinity of the regulated gene. The presence of the bound protein either promotes or inhibits transcription of the gene.

(a) Use information-theoretic arguments to obtain a lower bound on the size of a typical protein that acts as a regulator specific to one gene in the whole human genome. Assume that the genome is a sequence of 3 × 109 nucleotides drawn from a four letter alphabet {A, C, G, T}; a protein is a sequence of amino acids drawn from a twenty letter alphabet. [Hint: establish how long the recognized DNA sequence has to be in order for that sequence to be unique to the vicinity of one gene, treating the rest of the genome as a random sequence. Then discuss how big the protein must be to recognize a sequence of that length uniquely.]

(b) Some of the sequences recognized by DNA-binding regulatory proteins consist of a subsequence that is repeated twice or more, for example the sequence

is a binding site found upstream of the alpha-actin gene in humans. Does the fact that some binding sites consist of a repeated subsequence influence your answer to part (a)?