5.2 Simple ‘DV only’ model in a single cell
5.2.3 Non-dimensional model
To reduce the number of parameters in the model, it is nondimensionalised using the timescale of Goosecoid turnover, τ=µGt. Concentrations (Z) are scaled ˆZ=Z/θZ, where the following are defined for notational simplicity θM ≡ θM,G, θC ≡ θC,G, θV ≡ θV,G, θG ≡ θG,V, θBmp ≡ θBmp,V. Dimensionless parameters are defined as
ˆθZ,X ≡θZ,X/θZ, ˆλY,Z≡λY,Z/θZµG, µˆZ≡µZ/µG.
After applying these scalings and dropping the hats for ease of notation the non-dimensional model is dC dτ = −µCC, (5.2.7a) dG dτ = {λM,GH (M) +λC,GH (C)} 1− H V+ G θG,G −G, (5.2.7b) dV dτ =λBmp,VH (Bmp) {1− H (G)} −µVV, (5.2.7c) dBmp dτ =λV,BmpH V θV,Bmp −µBmpBmp. (5.2.7d)
We hope to find that (5.2.7) has two stable steady states representing dorsal and ventral fates, with the dorsal fate corresponding to Goosecoid being expressed in the absence of Vent and BMP4 and the ventral fate corresponding to Vent and BMP4 being expressed in the absence of
CHAPTER5: DORSAL-VENTRALPATTERNING IN ASINGLE-CELLMODEL OFMESENDODERM SPECIFICATION INXenopus
Variable Parameter Value Variable Parameter Value
C µC 0.1 V λBmp,V 5
G λM,G 10 µV 1
λC,G 50 Bmp λV,Bmp 1
θG,G 1 θV,Bmp 1
M 10 µBmp 0.1
Table 5.3:Dimensionless parameter values used to obtain numerical results for the system given in (5.2.7). Parameters were selected such that (5.2.7) is bistable with steady states corresponding to dorsal and ventral cell states, and so that the system evolves to these steady states dependent on the initial concentration of β-catenin in the cell.
Goosecoid. We seek solutions to (5.2.7) such that both dorsal and ventral fates are available. The model consists of a positive feedback loop between Vent and BMP, and mutual inhibition between Vent and Goosecoid. It has been shown that GRNs consisting of mutual inhibiting factors [37, 95] can be bistable provided that the cooperativity of binding is greater than unity. As a consequence of these observations we proceed to consider the case where the condition is satisfied, i.e. m>1.
5.2.4
Steady-state analysis
In this section, we consider the steady states of the system (5.2.7), satisfying the coupled system of equations C∗=0, (5.2.8a) G∗=λM,GH (M) 1− H V∗+ G ∗ θG,G , (5.2.8b) V∗= λBmp,V µV H (Bmp ∗) {1− H (G∗)}, (5.2.8c) Bmp∗= λV,Bmp µBmp H V∗ θV,Bmp . (5.2.8d)
Note that the steady state values of G∗, V∗ and Bmp∗are independent of the concentration of
β-catenin (C) since this decays to zero as the steady state is approached. However in section
5.2.5 we show that the system can evolve to the dorsal or the ventral steady state dependent on the initial condition of β-catenin.
Steady state solutions in the absence of Goosecoid (G∗=0)
In the absence of any factors to activate Goosecoid (M = 0 and C = 0), we expect that BMP and Vent can maintain themselves via mutual positive regulation and evolve to a non-trivial stable steady state. Figure 5.4 shows plots of the steady-state values of BMP and Vent for a range of λV,Bmp. The trivial steady state, where Vent and BMP are not expressed, is available for all values of λV,Bmp. As λV,Bmpincreases a saddle-node bifurcation marks the appearance of a two non-trivial steady states. The non-trivial stable steady state represents a cell adopting a ventral fate, with upregulated Vent and BMP. As the value of λV,Bmpincreases, the steady state concentration of BMP increases, while the steady state value of Vent is rather insensitive to
CHAPTER5: DORSAL-VENTRALPATTERNING IN ASINGLE-CELLMODEL OFMESENDODERM SPECIFICATION INXenopus
Figure 5.4:Numerical investigation of the Vent and BMP positive feedback loop, as described by (5.2.7c) and (5.2.7d) with G = 0. The trivial steady state (V∗ = 0, Bmp∗ = 0) is available for all values of λV,Bmp. As λV,Bmpincreases a saddle node bifurcation
occurs at λV,Bmp≈0.96, marking the appearance of a non trivial stable steady state
and an unstable steady state. Parameters other than λV,Bmpare as in table 5.3.
Figure 5.5:Numerical investigation of the Vent and BMP positive feedback loop, as described by (5.2.7c) and (5.2.7d) with G = 0. The trivial steady state (V∗ = 0, Bmp∗ = 0) is available for all values of λBmp,V. As λBmp,Vincreases a saddle node bifurcation
occurs at λBmp,V≈0.96, marking the appearance of a non trivial stable steady state
(thick solid line) and an unstable steady state (thick dashed line). Parameters other than λBmp,Vare as in table 5.3.
changes in λV,Bmp. Plotted in figure 5.5 are the steady-state values of BMP and Vent for various values of λBmp,V. As λBmp,V increases a saddle-node bifurcation marks the appearance of the non-trivial steady states.
Steady state solutions for fixed Mix (M) concentration
We now investigate steady-state solutions of (5.2.7) in the presence of a fixed concentration of Mix (M>0). Steady-state concentrations of Vent, BMP and Goosecoid are plotted as functions of λM,Gin figure 5.6. When λM,G=0, there are three steady states for the concentration of BMP and Vent: the trivial stable steady state, a (non-trivial) unstable steady state and a (non-trivial) stable steady state. At λM,G =0, the only steady state for Goosecoid is the trivial steady state, as λM,G increases Goosecoid becomes bistable, with stable steady states corresponding to up- regulated and downregulated levels. For further increases in λM,G, the downregulated steady
CHAPTER5: DORSAL-VENTRALPATTERNING IN ASINGLE-CELLMODEL OFMESENDODERM SPECIFICATION INXenopus
Figure 5.6:Steady state concentrations of BMP (Bmp∗), Vent (V∗) and Goosecoid (G∗) as func- tions of λM,G. As λM,Gincreases a fold bifurcation marks the disappearance of the
stable steady state representing ventral fates (thin solid line) and the unstable steady state (thick dashed line). The stable steady state representing dorsal fates (thin solid line) is present for all non-negative values of λM,G.
state disappears via a fold bifurcation, leaving a monostable steady state with upregulated Goosecoid.
5.2.5
Time-dependent solutions
In this section we explore the time-dependent behaviour of our model in different conditions and compare these with the experimental results of [124]. Figure 5.7 shows how the initial condition of β-catenin, C(0) =C0, (i.e. the strength of the dorsalising signal a cell is subjected to) determines the fate of a cell. In the absence of β-catenin (figure 5.7(i)), the Vent/BMP positive feedback loop becomes established and the activation of Goosecoid by Mix is suppressed by Vent, meaning the system evolves to the ventral steady state. For C0less than a critical value (C0C) the solutions will evolve to the ventral steady state as shown in figures 5.7(i) and (ii). In figure 5.7(ii), where C0is close to the critical value, there exists a phase where Goosecoid, BMP and Vent are coexpressed at non-negligible levels before the Vent/BMP feedback loop becomes established and the system evolves to the ventral steady state. This co-expression of all three factors corresponds to the solution passing close to the unstable steady state. For C0greater than the critical value (C0 >C0C), the concentration of Goosecoid in the system grows rapidly, leading to the repression of Vent, with the BMP initially present decaying to negligible levels.
CHAPTER5: DORSAL-VENTRALPATTERNING IN ASINGLE-CELLMODEL OFMESENDODERM SPECIFICATION INXenopus
Figure 5.7:Time evolution of (5.2.7) for various initial conditions of β-catenin, showing Vent (dotted line), BMP (solid line) and Goosecoid (dot-dashed line). (i) In the absence of β-catenin (C(0) = 0) the system evolves to the ventral steady state branch. (ii) For value of β-catenin close to some critical value, the solution passed close to the unstable steady state. (iii) For a larger value of β-catenin (C(0) =5) the cell evolves to the dorsal steady state branch.
In this case the system has evolved to the dorsal steady state (see figure 5.7(iii)).
β-catenin dose response
As shown in figure 5.7 the initial dose of β-catenin can determine whether solutions to (5.2.7) evolve to the ventral or dorsal steady state. Solutions are shown in figure 5.8 as functions of the initial concentration of β-catenin. Initially both BMP and Goosecoid are expressed for all concentrations of β-catenin. At later times the expression becomes more refined, and Vent/BMP are expressed at low concentrations and Goosecoid is expressed at high concentrations. In figure 5.9 the effect of varying the concentration of Mix (M) in the system is shown in the absence (figure 5.9(i)) and presence (figure 5.9(ii)) of β-catenin. In the absence of β-catenin the system shows the same qualitative behaviour for all values of M, with Vent and BMP reaching an upregulated steady state and any initial expression of Goosecoid becomes repressed. In the presence of β-catenin, Vent and BMP are repressed by Goosecoid and reach the trivial steady state for all values of M. The steady state value of Goosecoid (G∗) is dependent on the value of
CHAPTER5: DORSAL-VENTRALPATTERNING IN ASINGLE-CELLMODEL OFMESENDODERM SPECIFICATION INXenopus
(a)τ=0.1 (b)τ=1
(c)τ=10 (d)τ=100
Figure 5.8:Numerical solutions to (5.2.7) plotted against initial β-catenin concentration (C0) at
time τ=T. β-catenin can induce Goosecoid (dot-dashed line) at high concentration, but at low concentrations Goosecoid becomes repressed by Vent (dotted line) and BMP (solid line).
M: as M increases G∗also increases (see equation (5.2.8)). However the qualitative behaviour is similar for all M.
Figure 5.10 plots the time evolution of Vent, BMP and Goosecoid for several initial values of BMP. In the absence of BMP, the system evolves to the dorsal steady state independent of the initial concentration of β-catenin.
Goosecoid and Vent knockouts
By setting parameters on the right hand side of (5.2.7b) to be zero, so that dG/dt = 0, we investigate the effect of a Goosecoid knockout on our system. Experimental observations in [124] state that in animal caps (consisting of a uniform population of cells) treated with Activin and a Goosecoid MO, the expression of Vent1 and Vent2 are upregulated. In our model the knockout of Goosecoid results in the system evolving to the ventral steady state (figure 5.11 (iv)-(vi)), which is consistent with experimental observations. Conversely a knockout of Vent results in the system evolving to the dorsal steady state for all values of C0(figure 5.11 (i)-(iii)), consistent with experimental data showing that the double knockdown of Vent1/Vent2 results in dorsalised embryos [124].
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Figure 5.9:Numerical solutions to (5.2.7) for various values of M (concentration of Mix) for two cases: (i) C(0) =0 and (ii) C(0) =5. (i) The qualitative behaviour is the same for all values of M, with the system evolving to a state with upregulated Vent and BMP. (ii) In the presence of β-catenin the qualitative behaviour is the same for all values of M, with the system evolving to a state with no Vent and BMP, and the steady state value of Goosecoid dependent on the value of M.
Figure 5.10:Numerical solutions to (5.2.7) for various values of initial BMP concentration (Bmp(0) =BMP0) for two cases: (i) C(0) =0 and (ii) C(0) =5. (i) For all Bmp0≥1
the system evolves to the ventral steady state. When Bmp0=0 the system evolves
to the dorsal steady state. (ii) When β-catenin is present the system evolves to the dorsal state for all Bmp0≤6. For large Bmp0, the system evolves to a ventral fate.