Theory of site-specific interactions of the combinatorial transcription factors with DNA

We derive a functional relationship between the mean first passage time associated with the concurrent binding of multiple transcription factors (TFs) at their respective combinatorial cis-regulatory module sites (CRMs) and the number n of TFs involved in the regulation of the initiation of transcription of a gene of interest. Our results suggest that the overall search time tau(s) that is required by the n TFs to locate their CRMs which are all located on the same DNA chain scales with n as tau(s) proportional to n(alpha) where alpha similar to (2/5). When the jump size k that is associated with the dynamics of all the n TFs along DNA is higher than that of the critical jump size k(c) that scales with the size of DNA N as k(c) similar to N-2/3, we observe a similar power law scaling relationship and also the exponent alpha. When k < k(c), alpha shows a strong dependence on both n and k. Apparently there is a critical number of combinatorial TFs n(c) similar to 20 that is required to efficiently regulate the initiation of transcription of a given gene below which (2/5) < alpha < 1 and beyond which alpha > 1. These results seem to be independent of the initial distances between the TFs and their corresponding CRMs and also suggest that the maximum number of TFs involved in a given combinatorial regulation of the initiation of transcription of a gene of interest seems to be restricted by the degree of condensation of the genomic DNA. The optimum number m(opt) of roadblock protein molecules per genome at which the search time associated with these n TFs to locate their binding sites is a minimum seems to scale as m(opt) proportional to Ln(alpha/2) where L is the sliding length of TFs whose maximum value seems to be such that L <= 10(4) bps for the E. coli bacterial genome.