Extended Data Fig. 8: Mathematical modeling of the WOX5/HAN/CDF4 cFFL suggests a mechanism to buffer CSC maintenance against input noises.
a: the coherent feed-forward loop (cFFL) with the input signal s. b: Results of the simulation of the network motif with noisy input. The dashed line denotes the periodic and noisy WOX5 abundance. CDF4 NOR: wiring of the cFFL by a NOR-gate, CDF4 NAND: wiring of the cFFL by a NAND-gate, CDF4 WOX5-only: no HAN inhibition. The parameters are the same for all cases and read: k1 = k2 = k3 = k4 = k5 = k6 = 0.1 h−1, K1 = K2 = 6. The signal s(t) into WOX5 is turned off at t = 120 h. To explore the effect of noisy input signals and a loss of signal on the cFFL motif, we modelled the network shown in panel (a) using Ordinary Differential equations: \(\frac{{dx}}{{dt}}={k}_{1}s\left(t\right)-{k}_{2}x\). \(\frac{{dy}}{{dt}}={k}_{3}x-{k}_{4}y\). \(\frac{dz}{dt}={k}_{5}\overrightarrow{{\bf{1}}}\cdot \overrightarrow{{\bf{c}}}-{k}_{6}z\), with x \(\stackrel{\wedge}{=}\) [WOX5], y \(\stackrel{\wedge}{=}\) [HAN], and z \(\stackrel{\wedge}{=}\) [CDF4]. The input signal s is modelled as a log-normally distributed stochastic variable constructed from the stochastic process: \(d\mu \left(t\right)=\alpha \cos \omega t-{\tau }^{-1}\mu (t)+\sqrt{2/\tau \varepsilon {dW}(t)}s(t)={e}^{\mu \left(t\right)-{\varepsilon }^{2}/2}\) with τ = 3 h and ε = 0.2, α = 0.15 h−1, ω = 0.26 h−1 (which corresponds to a period of 24 h). Note, that for α = 0 h-1 μ is an Ornstein-Uhlenbeck process and s has mean 1 and variance \({e}^{{\varepsilon }^{2}}\) – 1. The vector \(\vec{{\boldsymbol{c}}}\) = (c00, c10, c01, c11) represents the four different states of the promoter for CDF4. c00 is the state of free binding sites, that is, neither WOX5 nor HAN is bound, c10 denotes the state of only WOX5 bound, etc. Using a quasi-steady state approximation, we can write for the states: \(\begin{array}{cc}{c}_{00}=\scriptstyle\frac{1}{(1+{K}_{1}x)(1+{K}_{2}y)} & {c}_{10}=\scriptstyle\frac{{K}_{1}x}{(1+{K}_{1}x)(1+{K}_{2}y)}\\ {c}_{01}=\scriptstyle\frac{{K}_{2}y}{(1+{K}_{1}x)(1+{K}_{2}y)} & {c}_{11}=\scriptstyle\frac{{K}_{1}x{K}_{2}y}{(1+{K}_{1}x)(1+{K}_{2}y)}\end{array}\). K1 and K2 are the equilibrium constants for the binding of WOX5 and HAN, resp. The NOR-gate logic is given by \(\vec{{\boldsymbol{l}}}\) = (1, 0, 0, 0) and a NAND-gate logic by \(\vec{{\boldsymbol{l}}}\) = (1, 1, 1, 0). The WOX5 only response is modelled via setting K2 to zero and using the NOR-Gate logic. The results of the simulation can be seen in panel (b). Due to the noisy input signal s(t) WOX5 fluctuates. We compare three different scenarios: i) inhibition by WOX5 only (CDF4 only WOX5), ii) combining the WOX5 and HAN signal in a NOR gate (CDF4 NOR), iii) combining the WOX5 and HAN signal in a NAND gate (CDF4 NAND). In all three cases the motif acts as a low-pass filter, smoothing the response of CDF4. The striking difference between the different wirings is the response to a loss of WOX5: while the WOX5-only and the NAND-gate wiring behave similarly, the NOR-gate wiring shows a delayed response to the decay of WOX5. The NAND-gate exhibits the opposite behaviour; the response is faster compared to the WOX5-only network.
