Time-course of the fluctuation of the signaling molecules displays a chaos-like oscillation.

Diffusion of active cofactor binding signaling molecule (X) and of inactive cofactor binding signaling molecule (Z). The Appendix S1 presents the simulation parameters, with the notation of Eqs. (3.9). p is (a) 0.795, (b) 0.81, (c) 0.84, (d) 0.88, (e) 0.96, (f) 1.00, (g) 1.12, and (h) 1.16. The upper graph shows two parametric plots of X, and Z. Red, and blue lines in the lower graph represent the concentrations of X, and Z, respectively. The horizontal axis represents time (0 ≤ t ≤ 200) and the vertical axis represents the concentrations of X, and Z, respectively. When p exceeds 0.80, chaos-like oscillation is observed. Mathematica cord when p  =  0.795 (a) is shown below. Below is the simulation program when p  =  1.0253: D1  =  0.28 k2  =  0.00034580 a  =  800 b  =  656 c  =  100 d  =  100 e  =  100 f  =  100 p =  1.0523 D4  =  156 D5  =  156 R  =  1 X  =  k2/D1 Z  =  (k2 (D1∧2 R+ D4 k2))/(D1 (D1 p - D5 k2)) NDSolve[{Derivative[1][x][t]  =  =  -(R (D1 - a X) + 2 X D4 + D5 Z) x[t] + (R a - D4 + 2 c X + e Z) x[t]∧2 + (p - D5 X - b X - d X∧2 - f X Z) z[t] - (D5 + R b - e X + f Z) x[t] z[t] - (f X) z[t]∧2, Derivative[1][z][t]  =  =  (2 X D4 + D5 Z - c X∧2 - e X Z) x[t] + (D4 - 2 c X - e Z) x[t]∧2 + (D5 + 2 X d - e X + f Z) x[t] z[t] + (D5 X - p + d X∧2 + f X Z) z[t], x[0]  =  =  1.’*∧-6, z[0]  =  =  1.’*∧-6}, {x, z}, {t, 0, 30000}, MaxSteps -> 50000] g001  =  Plot[{X + x[t]}/. %, {t, 0, 200}, PlotRange -> All, PlotStyle -> {RGBColor[1, 0, 0]}, PlotRange -> ALL] g003  =  Plot[{Z + z[t]}/. %%, {t, 0, 200}, PlotRange -> All, PlotStyle -> {RGBColor[0, 0, 1]}, PlotRange -> All] g004  =  ParametricPlot[Evaluate[{X + x[t], Z + z[t]}/. %%%], {t, 0, 2000}, PlotRange -> All, Axeslabel -> {"X", "Z"}] Show [g001, g003, Axeslabel -> {"t", "X, Z"}].