Rationalizing the abundance statistics of feed-forward loop motifs: an evolutionary strategy mediated by control on fluctuations

Bacteria rely on gene-transcription regulatory network motifs to harness dynamic environmental signals, crucial for their continued sustenance in evolutionary time-scales. Feed-forward loop (FFL) is the only motif naturally selected from a set of several random patterns and even within this group, there are only two out of possible eight motifs, which are statistically abundant. These circuits are indispensable in controlling metabolism and chemotaxis in E. coli. We construct a theoretical framework hypothesizing the rationale behind this phenomenon. Motivated by the E. coli dual-reporter technique (Elowitz et. al., Science, 2002), we derive a closed-form analytical expression of extrinsic noise supplied by the coregulator to the target gene in a generic FFL. This size-independent metric clearly demonstrates contributions of promoter occupancy, cooperativity, time-scale separation, regulatory modality (activation / repression), expression levels, and signal integration logic (additive / multiplicative). We demonstrate that the abundant coherent and incoherent type-1 FFLs have the highest amount of extrinsic noise compared to other FFLs in both fixed and variable input gene-expression conditions.