PMC

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 X_T = 95, while the lower and unstable steady state are at X_T = 0 and X_T = 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 X_T = 31 (upper stable steady state), X_T = 0.0001 (lower stable steady state) and X_T = 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 X_T = 95, while the lower and unstable steady state are at X_T = 0.0001 and X_T = 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 X_T = 31 (upper stable steady state), X_T = 0.0001 (lower stable steady state) and X_T = 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.

(a) Two dimensional steady state distribution at ω = 100. (b) Two dimensional steady state distribution at ω = 0.1. (c) Two dimensional steady state distribution at ω = 0.001. (d) Three dimensional steady state distribution at ω = 100. (e) Three dimensional steady state distribution at ω = 0.1. (f) Three dimensional steady state distribution at ω = 0.001. (g) Three dimensional landscape at ω = 100. (h) Three dimensional landscape at ω = 0.1. (i) Three dimensional landscape at ω = 0.001.

Steady state solution characteristics of molecular loop with respect to CPEB1 activation parameter k₅. The parametric steady state solution characteristics are developed through an approximate analytical solution (a, c) and numerical/analytical bifurcation diagrams (b, d). The 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 k₅. 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 values of rate constant k5 (The dotted red curve "1" represent k5 = 0.0072 μM^-1.s^-1, second dotted red curve "2" represent k5 = 0.0001 μM^-1 .s^-1 ). This method locates the upper stable steady state at X_T = 95, while the lower and unstable steady state are at X_T = 0.0001 and X_T = 9.4 when k₅ is set at 0.0072μM^-1 .s^-1 (curve "1"). As k₅ is decreased to 0.0001 new solution is located at X_T = 70 (upper stable steady state), X_T = 0.0001 (lower stable steady state) and X_T = 16 (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 with respect to k5. The three bifurcation diagrams are exactly matching with each other for the entire range of activation 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 values of rate constant k₅ (The dotted red curve "1" represent k₅ = 0.0072 μM^-1.s^-1, second dotted red curve "2" represent k5 = 0.0001 μM^-1.s^-1). The numerical and analytical bifurcation diagrams for second set of parameters (d) are developed through tracking the steady state behavior of all three models with respect to k5. Full model bifurcation diagram does not match with reduced and single equation model.

Each panel shows the probability of following the desired anterior state trajectory against the probability of following the desired posterior state trajectory for all networks that we considered. In each panel, we highlight in red networks that contain a particular combination of interactions. All other bad networks are marked with black crosses, all other good networks are marked with blue pluses. (A) In red are networks containing nodes with no regulators. These entered the anterior steady state or posterior steady state but not both. (B) In red are networks with Fgf8 only downstream of the four TFs (Fgf8 Emx2, Fgf8 Pax6, Fgf8 Coup-tfi and Fgf8 Sp8 all absent). Because the only difference in the starting state between the two compartments was Fgf8 activity, these networks could not enter both the anterior and posterior steady states with probability. (C) Marked in red are networks with auto-induction. Networks with Emx2, Pax6, Coup-tfi or Sp8 auto-induction entered the anterior steady state or the posterior steady state but not both. Networks with Fgf8 auto-induction could reliably enter the posterior steady state but not the anterior steady state. To enter the anterior steady state, they required Sp8 to become and remain active before the state of Fgf8 was updated. Because nodes were updated asynchronously in a random order, this could not occur with probability. (D) In red are networks containing inductive loops. These also could not enter the anterior steady state with probability by similar reasoning to C. (E) In red are networks containing isolated repressive loops (that is, X repressing Y was the only regulation of Y and Y also repressed X). These also could not reproduce the average gradients observed experimentally.