Noise decomposition algorithm and propagation mechanism in feed-forward gene transcriptional regulatory loop

Feed-forward gene transcriptional regulatory networks, as a set of common signal motifs, are widely distributed in the biological systems. In this paper, the noise characteristics and propagation mechanism of various feed-forward gene transcriptional regulatory loops are investigated, including （i） coherent feed-forward loops with AND-gate, （ii） coherent feed-forward loops with OR-gate logic, and （iii） incoherent feed-forward loops with AND-gate logic. By introducing logarithmic gain coefficient and using linear noise approximation, the theoretical formulas of noise decomposition are derived and the theoretical results are verified by Gillespie simulation. From the theoretical and numerical results of noise decomposition algorithm, three general characteristics about noise transmission in these different kinds of feed-forward loops are observed, i） The two-step noise propagation of upstream factor is negative in the incoherent feed-forward loops with AND-gate logic, that is, upstream factor can indirectly suppress the noise of downstream factors, ii） The one-step propagation noise of upstream factor is non-monotonic in the coherent feed-forward loops with OR-gate logic, iii） When the branch of the feed-forward loop is negatively controlled, the total noise of the downstream factor monotonically increases for each of all feed-forward loops. These findings are robust to variations of model parameters. These observations reveal the universal rules of noise propagation in the feed-forward loops, and may contribute to our understanding of design principle of gene circuits.