COMPUTATIONAL MODELING AND SIMULATION AS ENABLERS FOR BIOLOGICAL DISCOVERY 147
For understanding the dynamics of molecular regulatory systems, bifurcation theory is a powerful
complement to numerical simulation. The bifurcation diagram in Figure 5.7 presents recurrent solutions
(steady states and limit cycle oscillations) of the differential equations as functions of cell size. The
control system has three characteristic steady states: low cdc2 activity (G1 = pre-replication), medium
cdc2 activity (S/G2 = replication and post-replication), and high cdc2 activity (M = separation of repli-
cated DNA molecules). G1 and S/G2 are stable steady states; M is unstable because of a negative
feedback loop as shown in Figure 5.5 (cdc2-cdc13 activates Slp1, which degrades cdc13).
When the time courses of size and cdc2 activity from Figure 5.6 are superimposed on the bifurcation
diagram (curve labeled “size”), one sees how progress through the cell cycle is governed by the bifurca-
tions that turn stable steady states into unstable steady states and/or stable oscillations. A mutation
changes a specific rate constant, which changes the locations of the bifurcation points in Figure 5.7,
which changes how cells progress through (or halt in) the cell cycle. By this route one can trace the
dynamical consequences of genetic information all the way to observable cell behavior.
FIGURE 5.6 Simulated time courses of cdc2 and related proteins during the cell cycle of fission yeast. Numerical
integration of the full set of differential equations that describe the wiring diagram in Figure 5.5 yields these time
courses. Time is expressed in minutes; all other variables are given in arbitrary units. “Size” refers to the number of
ribosomes per nucleus. Notice the brief G1 phase, when ste9 is active and rum1 is abundant. After a long S/G2 phase,
during which cdc2 is tyrosine phosphorylated, the cell enters M phase, when cdc25 removes the inhibitory phosphate
group. After some delay, slp1 activates and degrades cdc13. As cdc2–cdc13 activity falls, the cell exits mitosis. Size
decreases twofold at nuclear division. SOURCE: J.J. Tyson, K. Chen, and B. Novak, “Network Dynamics and Cell
Physiology,” Nature Reviews of Molecular Cell Biology 2(12):908-916, 2001. Figure and caption reproduced with permis-
sion from Nature Reviews of Molecular Cell Biology. Copyright 2001 Macmillan Magazines Ltd.