Fig. 3: A delayed model of cascades recapitulates the implicit delays in ODE models.
From: Nonlinear delay differential equations and their application to modeling biological network motifs
a A cascade is a linear sequence of regulation steps, here X regulating Y regulating Z. b Standard models of cascades use ODEs (top), in which each step leads to a characteristic delay in the products Y and Z, which increases with each step based on the half-maximal inputs and degradation rates for each step. In an equivalent DDE model (middle), similar behavior is accomplished by replacing the explicit cascade of implicit delays with a single-step regulation including an explicit delay. A cascade in which each step contains an explicit delay (bottom) behaves analogously to delayed direct regulation (as in middle), with the final step delayed by the sum of delays in each step. ηX = 1.5, ηY = ηZ = 2.17, βX = βY = 1, βZ = 0.667, nX = nY = − 2. For the bottom graph in (b), each step is governed by Eq. (9) with identical parameters.
