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Figure 4 | BMC Systems Biology

Figure 4

From: Regulation of cytoplasmic polyadenylation can generate a bistable switch

Figure 4

Steady state solution characteristics of molecular loop with respect to parameter λ. This contain an analytical solution (a, c) and numerical/analytical bifurcation diagrams (b, d). Approximate analytical solution is developed through graphing equation 6, whereas the numerical/analytical bifurcation diagrams are developed through tracking the steady state behavior of all three models with respect to the degradation rate. Two different set of parameters are compared. For first set of parameters the approximate solution and numerical/analytical bifurcation diagrams from three alternative models converge to same upper, stable steady state solution branch (a, b). For the second set of parameters the approximate solution and numerical/analytical bifurcation diagrams from three alternative models does not converge to same upper, stable steady state and solution branch(c, d). The approximate analytical solution for first set of parameters (a) is located at two different degradation rates (The solid blue line#"1" represent λ = 0.0001s-1 and other solid blue line#"2" represent λ = 0.0003 s-1 ). This method locates the upper stable steady state at XT = 95, while the lower and unstable steady state are at XT = 0 and XT = 9.4 when degradation rate is set at 0.0001 s-1 (curve 1). As degradation rate is increased to 0.0003 s-1 (curve 2) the new solution is located at XT = 31 (upper stable steady state), XT = 0.0001 (lower stable steady state) and XT = 9.2 (unstable steady state). The numerical and analytical bifurcation diagrams for first set of parameters (b) are developed through tracking the steady state behavior of all three models. Three bifurcation diagrams exactly match with each other for the entire range of degradation parameter (dotted blue line represent the bifurcation diagram based on full model, whereas solid blue lines represent the bifurcation diagrams based on reduced three differential equation and a single equation model). The approximate analytical solution for second set of parameters (c) is also located at two different degradation rates (The solid blue curve "1" represent λ = 0.0001s-1, and other solid blue curve "2" represent λ = 0.0003 s-1). This method locates the upper stable steady state at XT = 95, while the lower and unstable steady state are at XT = 0.0001 and XT = 9.4 when degradation rate is set at 0.0001s-1 (curve 1). As degradation rate is increased to 0.0003 s-1(curve 2) the new solution is located at XT = 31 (upper stable steady state), XT = 0.0001 (lower stable steady state) and XT = 9.2 (unstable steady state). The numerical and analytical bifurcation diagrams for second set of parameters (d) are developed. The bifurcation diagram from full model does not match with the bifurcation diagrams from reduced and single equation model.

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