Hybrid Model (Model 3).
Assuming that the pools of Rac and Rho are constant in the cell fixes undesirable features of Model 2. (a) As in Fig 4(a) but with GTPase conservation Eq (1b). Other parameters: n = 3, δ = 1, ρT = RT = 2. Dashed lines indicate the range of bρ values that must be traversed to induce oscillation. (b) Bifurcation analysis with respect to γρ. Polarized (indicated by stable steady states, solid curves) and oscillatory (unstable steady states, dashed curves) regimes are present. (c) Two-parameter bifurcation diagrams depicting oscillatory and polarized parameter regimes separated by a locus of Hopf bifurcations, both with (lc = 0.1) and without (lc = 0) direct lamellipod coupling. Points (e,f) indicate parameter values at which simulations were performed (with lc = 0) to confirm dynamics. (d) Bifurcation diagram of the full model, with respect to basal Rac activation rate bR and ECM-induced Rho activation rate γE, concordant with such diagrams in . Solid (respectively dashed) lines indicate the boundary of different dynamical regimes without (resp. with) feedback from ECM onto Rac activation. (e,f) Sample simulations showing lamellipodial polarity and oscillations (ECM signaling levels in each lamellipod over time). Parameter values correspond to the labeled points in panel (c). For (e), lc = 0, γR = 0.2, γρ = 0.2. For (f), lc = 0, γR = 0.3, γρ = 0.4. All other parameters are given in the S1 Text.