Equilibrium deviations and orbit structure changes caused by enzyme deletions. Impacts of enzyme deletions on dynamical stability are shown in terms of equilibrium deviations and orbit structure changes. The curves are drawn under the principle of topology conjugacy. They show the qualitative dynamics but they are not real-value trajectories. Lines in different colors represent different curves initiated at different values (init 1-3). In each subfigure, the 3 dimensions represent the metabolites that differ most from the original kinetics. (A) The original equilibrium (Xeq), denoted by (0,0,0). All trajectories converge to it. (B) The unstable equilibrium after deleting TKb. The dimension sed7p is divergent, (ribu5p, xyl5p) form an unstable limit cycle. The 3 colored orbits initiated from different values lead to convergence, limit cycle and divergence on the 2D plane (ribu5p, xyl5p). (A) and (B) are separately drawn in order to show the limit cycle clearly. Orbits are centred at (0,0,0) to achieve a better visual effect. (C) Deleting R5PI causes a long-distance equilibrium deviation and alters the system stability. The source and target ends of the grey arrow mark Xeq and the re-established equilibrium (Xd) respectively, with the distance (in the metric unit of the state space) marked in the box. Initiated from identical values, Xeq attracts the orbits and Xd repels the orbits along the dimensions fdp and 3pg. (D) Deleting TIS causes the equilibrium to deviate an even larger distance. Legends are the same. (E) Deleting PGK makes the system have no equilibrium. The deviation is denoted as infinite (∞). (F) Deleting PGI does not cause obvious changes in system dynamics. The re-established equilibrium is also asymptotically stable and it is very near to Xeq. See Additional file 6 for abbreviations of metabolite names.
Li and Liu BMC Systems Biology 2012 6(Suppl 1):S11 doi:10.1186/1752-0509-6-S1-S11