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. 2020 Jul 24;9:e58165. doi: 10.7554/eLife.58165

Figure 2. A mathematical model of the RhoA-Rac1 network predicts dramatically distinct dynamic regimes for different DIA and ROCK abundances.

(A) Distinct dynamic regimes of the RhoA-Rac1 network dynamics for different DIA and ROCK abundances. Oscillations of RhoA and Rac1 activity exist within area 1 (regime 1). In area 3, sustained GTPase oscillations and a stable steady state with high RhoA and low Rac1 activities coexist. Regimes 0, 2, 5 and 6 have only one stable steady state. Notably, regime 2 is excitable. Steady state solutions with high RhoA activity exist in areas 2–4, and 6–8. Stable steady state solutions with high Rac1 activity exist in areas 0 and 5–8. Regimes 4, 7 and 8 are bistable with two stable steady states. (B, C) Wiring diagrams of the RhoA-Rac1 network for the cell leading edge (B) and the cell body and rear (C). Dashed blue lines indicate weak activating connections. (D–F) Typical time courses of RhoA and Rac1 activity in regimes 1 (D), and 2 (E). (F) In area 3, depending on the initial state, the GTPase network evolves either to a stable steady state (right) or a stable oscillatory regime (left).

Figure 2.

Figure 2—figure supplement 1. Distinct dynamic regimes of the RhoA-Rac1 network for different effector abundances.

Figure 2—figure supplement 1.

Domains of the distinct RhoA-Rac1 dynamics partition the planes of the abundances of (A) ROCK and DIA, (B) PAK and DIA, (C) PAK and ROCK. (D) Steady-state RhoA-GTP and Rac1-GTP dependences on the ROCK abundance are presented along a 1-D section of the PAK, ROCK plane (PAK/PAKtot=p=0.3, Equation 12) shown by white dashed line in (C). The amoeboid shape characterized by high RhoA activity emerges in mono-stable region 6 corresponding to the high RhoA-GTP and low Rac1-GTP when ROCK is non-inhibited. (E) Wiring diagram of the RhoA-Rac1 signaling network where the dimensionless multipliers, αYX, specify the regulatory influence of protein Y on protein X (see Star*Materials and methods for details).
Figure 2—figure supplement 2. Nullclines and vector fields describing the nine dynamic regimes of RhoA-GTP and Rac1-GTP shown in Figure 2A.

Figure 2—figure supplement 2.

(A–I) Nullclines and vector fields are calculated for a 2-D system given by Equation 12 for regimes 0–8, as indicated. The RhoA-GTP and Rac1-GTP nullclines are shown by red and blue curves, respectively. Projections of limit cycles of a 5-D system in Equation 6 into a 2-D space of the RhoA and Rac1 activities are shown by green curves. Circles show stable steady states; triangles represent unstable steady states. Inserts in panels (B–D, F, H) show the area near y-axis at a larger magnification.
Figure 2—figure supplement 3. One-parameter bifurcation diagrams for changing ROCK and DIA abundances separately in Figure 2A.

Figure 2—figure supplement 3.

(A–F) Minimum and maximum values of RhoA (A, C, E) and Rac1 (B, D, F) activity for the oscillatory regimes (dashed lines) and steady state values of RhoA (A, C, E) and Rac1 (B, D, F) activity (solid lines) are plotted against DIA (C–F) and ROCK (A–B) abundances. Black dashed lines represent borders of corresponding zones in Figure 2A.