Top-level dynamics and the regulated gene response of feed-forward loop transcriptional motifs

Feed-forward loops are hierarchical three-node transcriptional subnetworks, wherein a top-level protein regulates the activity of a target gene via two paths: a direct-regulatory path, and an indirect route, whereby the top-level proteins act implicitly through an intermediate transcription factor. Using a transcriptional network of the model bacterium Escherichia coli, we confirmed that nearly all types of feed-forward loop were significantly overrepresented in the bacterial network. We then used mathematical modeling to study their dynamics by manipulating the rise times of the top-level protein concentration, termed the induction time, through alteration of the protein destruction rates. Rise times of the regulated proteins exhibited two qualitatively different regimes, depending on whether top-level inductions were "fast" or "slow." In the fast regime, rise times were nearly independent of rapid top-level inductions, indicative of biological robustness, and occurred when RNA production rate-limits the protein yield. Alternatively, the protein rise times were dependent upon slower top-level inductions, greater than approximately one bacterial cell cycle. An equation is given for this crossover, which depends upon three parameters of the direct-regulatory path: transcriptional cooperation at the DNA-binding site, a protein-DNA dissociation constant, and the relative magnitude of the top-level protien concentration.