Modelling coupled oscillations in the Notch, Wnt, and FGF signaling pathways during somitogenesis: a comprehensive mathematical model

Abstract

Somite formation in the early stage of vertebrate embryonic development is controlled by a complicated gene network named segmentation clock, which is defined by the periodic expression of genes related to the Notch, Wnt, and the fibroblast growth factor (FGF) pathways. Although in recent years some findings about crosstalk among the Notch, Wnt, and FGF pathways in somitogenesis have been reported, the investigation of their crosstalk mechanisms from a systematic point of view is still lacking. In this study, a more comprehensive mathematical model was proposed to simulate the dynamics of the Notch, Wnt, and FGF pathways in the segmentation clock. Simulations and bifurcation analyses of this model suggested that the concentration gradients of both Wnt, and FGF signals along the presomitic mesoderm (PSM) are corresponding to the whole process from start to stop of the segmentation clock. A number of highly sensitive parameters to the segmentation clock's oscillatory pattern were identified. By further bifurcation analyses for these sensitive parameters, and several complementary mechanisms in respect of the maintenance of the stable oscillation of the segmentation clock were revealed.

Figure 1

The schematic diagram of the model. The diagram is created using CellDesigner [18]. In the diagram, the outer yellow ellipse represents cytomembrane; the inner yellow ellipse represents nuclear membrane; the light green rectangles represent protein; the bottle green parallelograms represent mRNA; the white rectangles which contain light green rectangles represent complex; Φ represents the resultants of degradation reactions or reactants of combination reactions. The arrow lines represent biochemical reactions.

Figure 2

The simulation results of the model. (a) The oscillatory expression of the target genes under the condition of a constant extracellular Wnt and FGF signal. (b) The expression patterns of the target genes after the induction of the Wnt signal to Msgn1 was removed at time point 240 minutes. (c) The expression patterns of the target genes after the induction of the Wnt signal to the Hes7 gene was removed at time point 240 minutes. (d) The expression patterns of the target genes after the FGF protein was removed at time point 240 minutes. (e) The expression patterns of the target genes after the repression of the Hes7 protein to the Dusp4 gene was removed at time point 240 minutes. (f) The expression patterns of the target genes after the repression of the Hes7 protein to the Dll1 gene was removed at time points 240 minutes.

Figure 3

The results of the bifurcation scanning to the Wnt signal. The X-axis represents time. The Y-axis represents the concentration of the Wnt signal. The Z-axis and the colour bar both represent the concentration of the target genes' expression. (a) The expression level of the Hes7 gene in two periods during scanning the Wnt signals. (b) The expression level of the Lfng gene in two periods during scanning the Wnt signals. (c) The expression level of the Axin2 gene in two periods during scanning the Wnt signals. (d) The expression level of the Dusp4 gene in two periods during scanning the Wnt signals. (e) The expression level of the Dll1 gene in two periods during scanning the Wnt signals. (f) The expression level of the Nkd1 gene in two periods during scanning the Wnt signals.

Figure 4

The results of one-parameter bifurcation analysis for the parameters in the feedback loops. The X-axis represents time. The Y-axis represents the values of the analyzed parameter. The Z-axis and the colour bar both represent the concentration of the target genes' expression. (a) The expression level of the Hes7 gene in respect to k _mHes7 with the length of time 600 minutes. (b) The expression level of the Hes7 gene in respect to EX _Hes7 with the length of time 600 minutes. (c) The expression level of the Hes7 gene in respect to K _aDll1 with the length of time 600 minutes. (d) The expression level of the Hes7 gene in respect to k _mLfng with the length of time 600 minutes. (e) The expression level of the Hes7 gene in respect to k _NICD with the length of time 600 minutes. (f) The expression level of the Hes7 gene in respect to K _ILfng with the length of time 600 minutes. (g) The expression level of the Hes7 gene in respect to k _mDusp4 with the length of time 600 minutes. (h) The expression level of the Hes7 gene in respect to IM _Erka with the length of time 600 minutes. (i) The expression level of the Hes7 gene in respect to EX _Erkn with the length of 600 minutes.

Figure 5

The results of one-parameter bifurcation analysis to the parameters in the reactions of the Hes7 gene's expression. The X-axis represents time. The Y-axis represents the values of the analyzed parameter. The Z-axis and the colour bar both represent the concentration of the target genes' expression. (a) The expression level of the Hes7 gene in respect to K _NsmHes7 with the length of time 600 minutes. (b) The expression level of the Hes7 gene in respect to K _BsmHes7 with the length of time 600 minutes. (c) The expression level of the Hes7 gene in respect to K _EsmHes7 with the length of time 600 minutes. (d) The expression level of the Hes7 gene in respect to K _HsmHes7 with the length of time 600 minutes.

Figure 6

The results of two-parameter bifurcation analysis of the parameter pairs from the feedback loops centred on Hes7 and Lfng. The X-axis and Y-axis represent two parameters analyzed, respectively. The colour bar represents the normalized stable degree (see Equation (1)) of the Hes7 gene's oscillation at some a parameter value pair and it has no unit. (a) k _mHes7-K _aDll1. (b) EX _Hes7-K _aDll1. (c) k _mHes7-k _mLfng. (d) EX _Hes7-k _mLfng. (e) k _mHes7-k _NICD. (f) EX _Hes7-k _NICD. (g) k _mHes7-K _ILfng. (h) EX _Hes7-K _ILfng.

Figure 7

The expression of the Hes7 gene under the different conditions of k _mHes7, k _mLfng, k _mDusp4 and K _aDll1. (a) k _mHes7 = 0.1 min⁻¹. (b) k _mLfng = 0.01 min⁻¹. (c) k _mHes7 = 0.1 min⁻¹ and k _mLfng = 0.01 min⁻¹. (d) k _mHes7 = 0.27 min⁻¹. (e) k _mLfng = 0.03 min⁻¹. (f) k _mHes7 = 0.1 min⁻¹ and k _mLfng = 0.03 min⁻¹. (g) k _mHes7 = 0.15 min⁻¹. (h) k _mDusp4 = 0.05 min⁻¹. (i) k _mHes7 = 0.15 min⁻¹ and k _mDusp4 = 0.05 min⁻¹. (j) k _mHes7 = 0.27 min⁻¹. (k) k _mDusp4 = 0.11 min⁻¹. (l) k _mHes7 = 0.27 min⁻¹ and k _mDusp4 = 0.11 min⁻¹. (m) k _mHes7 = 0.1 min⁻¹. (n) K _aDll1 = 0.5 nM. (o) k _mHes7 = 0.1 min⁻¹ and K _aDll1 = 0.5 nM. (p) k _mHes7 = 0.27 min⁻¹. (q) K _aDll1 = 1.1 nM. (r) k _mHes7 = 0.27 min⁻¹ and K _aDll1 = 1.1 nM.