Extrinsic noise of the target gene governs abundance pattern of feed-forward loop motifs

Feed-forward loop (FFL) is found to be a recurrent structure in bacterial and yeast gene transcription regulatory networks. In a generic FFL, transcription factor (TF) S regulates production of another TF X while both of these TFs regulate production of final gene-product Y. Depending upon the regulatory programs (activation or repression), FFLs are grouped into two broad classes: coherent (C) and incoherent (I), each class containing four distinct types (C1–C4 and I1–I4). These FFL types are experimentally observed to occur with varied frequencies, C1 and I1 being the abundant ones. Here we present a stochastic framework singling out the absolute value of the normalized covariance of X and Y to be the determining factor behind the abundance of FFLs while considering differential promoter activities of X and Y. Our theoretical construct employs two possible signal integration mechanisms (additive and multiplicative) to synthesize Y while steady-state population level of S remains fixed or becomes tunable reflecting two possible environmental signaling scenarios. Our model categorically points out that abundant FFLs exhibit higher amount of the designated metric which has a biophysical connotation of extrinsic noise for the target gene Y. Our predictions emanating from an overarching analytical expression utilizing biologically plausible parametric conditions are substantiated by stochastic simulation.

Figure 1

Schematic diagrams of coherent (C1–C4) and incoherent (I1–I4) FFL motifs. Here S regulates X and both of them regulate Y [19, 23].

Figure 2

Absolute value of the normalized covariance of X and Y ($|{\mathrm{\Sigma }}^N(X,Y)|=:|\mathrm{\Sigma }(X,Y)/\left(\langle X\rangle \langle Y\rangle \right)|$) in units of ${\text{c.n.}}^{-1}$ for (a) C1–C4 (additive), (b) C1–C4 (multiplicative), (c) I1–I4 (additive), and (d) I1–I4 (multiplicative) with respect to variation of $k_s$ (in units of ${\min }^{-1}$). We maintain $\langle S\rangle =10,\langle X\rangle =\langle Y\rangle =100$ copies in a unit effective cellular volume. Relaxation rate parameters are ${\mu }_x=0.5{\text{min}}^{-1}$, ${\mu }_y=1{\text{min}}^{-1}$, and ${\mu }_s=k_s/\langle S\rangle$. The downstream synthesis parameters for C1 FFL are as follows: $k_{sx}={\mu }_x\langle X\rangle {[{\langle S\rangle }^n/\left({{\langle S\rangle }^n+K_{sx}^{+}}^n\right)]}^{-1}$. For additive signal integration, $k_{sy}^A=0.5{\mu }_y\langle Y\rangle {[{\langle S\rangle }^n/({\langle S\rangle }^n+{K_{sy}^{+}}^n)]}^{-1}$ and $k_{xy}^A=0.5{\mu }_y\langle Y\rangle {[{\langle X\rangle }^n/({\langle X\rangle }^n+{K_{xy}^{+}}^n)]}^{-1}$ ensure equal contributions from direct and indirect paths to produce total $\langle Y\rangle$ copies. In the multiplicative case, $k_y^M={\mu }_y\langle Y\rangle {[{\langle S\rangle }^n/\left({{\langle S\rangle }^n+K_{sy}^{+}}^n\right)]}^{-1}{[{\langle X\rangle }^n/({\langle X\rangle }^n+{K_{xy}^{+}}^n)]}^{-1}$. Synthesis rate parameters for other FFL types were chosen according to the corresponding combinations of regulatory edges. For the activating edges we keep the activation coefficient $K_{..}^{+}=1000$ copies. Otherwise, the operating points are set to ensure half-maximal repression by setting $K_{sx}^{-}=K_{sy}^{-}=\langle S\rangle =10$, and $K_{xy}^{-}=\langle X\rangle =100$ copies. $n=1$ is used for all the regulatory edges in different FFLs. Lines are steady-state analytical results supported by simulation data from Gillespie's SSA [51, 52] (symbols) with ensemble averaging over ${10}^5$ independent samples at steady state.

Figure 3

Absolute value of the normalized covariance of X and Y ($|{\mathrm{\Sigma }}^N(X,Y)|=:|\mathrm{\Sigma }(X,Y)/\left(\langle X\rangle \langle Y\rangle \right)|$) in units of ${\text{c.n.}}^{-1}$ for (a) C1–C4 (additive), (b) C1–C4 (multiplicative), (c) I1–I4 (additive), and (d) I1–I4 (multiplicative) with respect to variation of $\langle S\rangle$ (in units of c.n.). We maintain $\langle X\rangle =\langle Y\rangle =100$ copies in a unit effective cellular volume. Keeping ${\mu }_s=0.1{\text{min}}^{-1}$ and varying $k_s$, we ensure $\langle S\rangle$ to increase by 1 copy at each instant. Other parametric conditions are identical with Fig. 2. Analytical results are designated by lines while results from Gillespie SSA [51, 52] are presented with symbols. These latter data sets are obtained with ensemble averaging over ${10}^5$ independent samples at steady state.