Fig. 6: The feedforward network motif owes its primary functions to a difference in regulatory delays.
From: Nonlinear delay differential equations and their application to modeling biological network motifs
a The feedforward motif with delays, in which a single output Z is controlled by an input X via two regulatory arms with differing delays. The straight, short arrow represents the “direct arm” with delay γ1 and the longer, curved arrow represents the “indirect arm” with delay γ2 > γ1. b The ODE model for an incoherent (type 1) feedforward motif, one of 8 possible networks in which the intermediate gene Y is modeled explicitly in the indirect arm. c Simulations of the four feedforward motifs with AND-type and OR-type logic (Fig. 5 and Supplementary Table 1) in response to short and long gain and loss of input signal. Blue curves: inputs (X), orange curves: reporter R activated with high cooperativity by Z. Note that the bottom two rows demonstrate pulse generation, while the top two rows filter short signals. d Response of an incoherent feedforward motif to oscillatory input after initial transients have died away. Z3 is a 3-frequency Fourier approximation of Z (see E). e Fourier decomposition of Z from (d) by Eq. (26) and by a numerical fit to the data in (d). f Frequency scan (Bode plot) of (d) for 3 values of Δγ, with the theoretical envelopes from Eq. (28). g The maximum amplitude of the motif in (d) over a range of Δγ and the corresponding frequencies at which the maxima occur. Z goes above 1 (activation threshold for R) for a small range of Δγ. For (c–f), η1 = 0.9, η2 = 0.7, n1 = 2, n2 = −2, n3 = −20, ηR = 2, A = 1. For d, f = 0.05, Δγ = 4.
