A gene regulatory motif that generates oscillatory or multiway switch outputs

Abstract

The pattern of gene expression in a developing tissue determines the spatial organization of cell type generation. We previously defined regulatory interactions between a set of transcription factors that specify the pattern of gene expression in progenitors of different neuronal subtypes of the vertebrate neural tube. These transcription factors form a circuit that acts as a multistate switch, patterning the tissue in response to a gradient of Sonic Hedgehog. Here, by simplifying aspects of the regulatory interactions, we found that the topology of the circuit allows either switch-like or oscillatory behaviour depending on parameter values. The qualitative dynamics appear to be controlled by a simpler sub-circuit, which we term the AC-DC motif. We argue that its topology provides a natural way to implement a multistate gene expression switch and we show that the circuit is readily extendable to produce more distinct stripes of gene expression. Our analysis also suggests that AC-DC motifs could be deployed in tissues patterned by oscillatory mechanisms, thus blurring the distinction between pattern-formation mechanisms relying on temporal oscillations or graded signals. Furthermore, during evolution, mechanisms of gradient interpretation might have arisen from oscillatory circuits, or vice versa.

Figure 1.

(a) Diagram of the gene circuit controlling the specification of neural progenitor domains in the vertebrate neural tube. The diagram shows the interactions between the morphogen Shh and the three TFs Pax6, Olig2 and Nkx2.2. (Pointed arrowheads indicate induction, while blunt arrowheads indicate repression.) (b) An illustration of how Hill functions approach a Heaviside (step) function, as the Hill coefficient tends to infinity. T represents the concentration of a repressor. The blue lines show repressive Hill functions, R(T), with Hill coefficients 1, 2, 5 and 50. The red line shows a repressive Heaviside function, H(T).

Figure 2.

Numerical profiles of the simplified model, illustrating the existence of a sharp switch (a,d) or oscillations at intermediate S levels (b,c,e) in the expression profiles of Pax6 (blue), Olig2 (red) and Nkx2.2 (green) in the transition B₁ → B₂. Model parameters are: α = 5, β = 5, γ = 5, h₁ = 2, h₂ = 2, O_crit = 1, P_crit1 = 0.5, O_crit1 = 5, N_crit1 = 2, k₁ = 1, k₂ = 1, k₃ = 1 and in (a) and (d) N_crit = 0.9 and (b), (c) and (e) N_crit = 1.1. In (a) and (b), the profiles are generated at t = 60, at which time the profiles have converged to steady state (to within a tiny tolerance), if they ever will do so. If they do not, we simply leave a gap in the plot. In (c), we show the steady-state value (red line) or maximum (black line) and minimum (cyan line) values of O for parameters as in (b), once the system has converged either to steady state or to a limit cycle (we use t = 500), plotted against S. In (d) and (e), we plot time courses of P, O and N for parameters as in (a) and (b), respectively, and for fixed S. The values of S used in (d) and (e) are marked with arrows in (a) and (b), respectively.

Figure 3.

Numerical profiles of the simplified model, illustrating the existence of a sharp switch (a,d) or oscillations at intermediate S levels (b,c,e) in the expression profiles of Pax6 (blue), Olig2 (red) and Nkx2.2 (green) in the transition B₁ → B₃ → B₂. Model parameters are: α = 3, β = 5, γ = 4, h₁ = 2, h₂ = 2, O_crit = 1, P_crit1 = 1, O_crit1 = 5, N_crit1 = 2, k₁ = 1, k₂ = 1, k₃ = 1 and in (a) and (d) N_crit = 1 and in (b), (c) and (e) N_crit = 2. In (a) and (b), the profiles are generated at t = 60, at which time the profiles have converged to steady state (to within a tiny tolerance), if they ever will do so. If they do not, we simply leave a gap in the plot. In (c), we show the steady state value (red line) or the maximum (black line) and minimum (cyan line) values of O for parameters as in (b), once the system has converged either to steady state or to a limit cycle (we use t = 500), plotted against S. In (d) and (e), we plot time courses of P, O and N for parameters as in (a) and (b), respectively, and for fixed S. The values of S used in (d) and (e) are marked with arrows in (a) and (b), respectively.

Figure 4.

Numerical profiles of the full model, with parameters as in figure 2, but with Hill coefficients h₃ = h₄ = h₅ = 2. In this case, as in figure 2, when N_crit = 0.9 there is a straight switch from Pax6 dominance at low Shh, to Olig2 dominance at intermediate Shh, to Nkx2.2 dominance at high Shh, although Nkx2.2 and Olig2 are coexpressed for a large range of Shh concentration (a,c). For N_crit = 1.1 (b,d), unlike the simplified model, there is no intermediate regime of oscillations, and Nkx2.2 never dominates Olig2. In (a) and (b), the profiles are generated at t = 60, at which time the profiles have converged to steady state (to within a tiny tolerance), if they ever will do so. In (c) and (d), we plot time courses of P, O and N for parameters as in (a) and (b), respectively, and for fixed S. The values of S used in (c) and (d) are marked with arrows in (a) and (b), respectively.

Figure 5.

Numerical profiles of the full model, with parameters as in figure 2, but with Hill coefficients h₃ = h₄ = h₅ = 5. In this case, as in figure 2, when N_crit = 1.1 there is an intermediate range of Shh concentrations for which the system exhibits oscillations (c,d,f). This range is extended in size relative to the simplified model. In addition, there is also an intermediate range of Shh concentration for which the system exhibits oscillations, when N_crit = 0.9 (see (a) and (c)). The diagrams of the steady state values (red line, solid when the state is stable and dashed when it is unstable), maximum (black line) and minimum (cyan line) values of O, once the system has converged to either a steady state or a limit cycle (we use t = 500), are shown in (c) for parameters as in (a), and in (d) for parameters as in (b). We note that there can be more than one unstable steady state, but, for clarity, we only show the continuation of the state which was stable before the bifurcation to the limit cycle. We show for comparison the temporal profiles of the TFs for S = 5 for N_crit = 0.9 (e) and N_crit = 1.1 (f). Only in the latter case is S = 5 within the range of S leading to oscillations. In (a) and (b), the profiles are generated at t = 60, at which time the profiles have converged to steady state (to within a tiny tolerance), if they ever will do so. The values of S used in (e) and (f) are marked with arrows in (a) and (b), respectively.

Figure 6.

The AC–DC circuit. Depending on whether the green or the red interactions are stronger, the circuit behaves as a positive or negative feedback loop. It can either display bistability and hysteresis or oscillations.

Figure 7.

The logic of multipartite expression. A natural circuit to give rise to two domains of gene expression in response to morphogen is given in (a). S denotes the morphogen, O the TF expressed at high morphogen concentration and P the TF expressed at low morphogen concentration. (b) Alternative circuits that may be able to give rise to three domains of gene expression. Once again S denotes the morphogen, N is the TF expressed at high, O at intermediate and P at low morphogen concentration. As detailed in the text, (i) requires differential sensitivity of O and N to the morphogen to give three domains of TF expression. (ii) The AC–DC circuit, which can give rise to three domains of gene expression for a wide range of parameters.

Figure 8.

A diagram of a network capable of generating four stripes of gene expression. Q is expressed at the highest levels of the morphogen S, followed by N, followed by O, while P is expressed at the lowest levels.