Fig. 3: Sustaining a cellular network.

a We apply our framework to the regulatory dynamics captured by Michaelis–Menten model⁴⁴. b Simulations (symbols) and theory (lines, Eq. (10) with ρ = 0) results for a free system of Erdős-Rényi (ER) structure with N = 10⁴ and κ = 40 and for the parameters’ values a = 1 and h = 2. There is a bi-stable region (gray shade). In case that a ≥ h, there is no bi-stability³⁸, thus here, we consider h > a to analyze system sustaining. c Demonstration of the activities of the system with β = 3.1 for ρ = 0. Both states are stable, therefore the system is unsustainable. d For a forced system by a fraction ρ = 0.03 of random nodes with activity Δ = 5, Eq. (8) provides a phase diagram (thick curve), exhibiting an s-shape curve which has now also a regime with only a single active state (blue shade). This regime is a sustainable phase. Note that simulations (symbols) are in agreement with the theory (lines). The network is the same as in (b). e The same as (d) with a larger fraction of controlled nodes, ρ = 0.11. Here we see that the unsustainable region almost vanishes, and the transition becomes almost continuous. This agrees with Eq. (14). f Activities for β = 1.5, 2.5, 3.1 in three regions, inactive (red diamond), unsustainable (cyan circle), and sustainable (blue triangle) correspondingly. The dark blue nodes are the forced nodes. The red nodes represent x₀, the cyan nodes represent unsustainable x₁, and the light blue nodes represent sustainable x₁. g The new phase diagram in (β, ρ)-space for Δ = 5. The simulations were done on ER networks with N = 10⁴ for 50 values of ρ, 50 values of β, for κ = 20, 60, 100, and averaged over ten realizations. In the color bar, the value 0 represents an unsustainable system, and 1 represents the other cases. The black lines are obtained from Eqs. (10) and (11). h The same as (g) with lower intervention force Δ = 1. As expected, in this case, a larger fraction of controlled nodes is needed to make the network sustainable. i (κ, λ)-space for a free system. j The sustaining phase diagram for ρ = 0.01 and Δ = 5. The light blue is the sustainable phase when forcing a fraction ρ = 0.01 of nodes, and the dark blue is the sustainable regime for holding a single node. The white lines represent the theory, Eqs. (10) and (11). k Horizontal trajectory in (κ, λ)-space for fixed λ = 0.1 and varying κ. Symbols are simulations and the line is theory obtained from Eqs. (10) and (11). The slope is according to Eq. (12). l Vertical trajectory in (κ, λ)-space for fixed κ = 20, 100 and varying λ. Note that the critical fraction for sustaining, ρ_c, for a given λ approaches 0, where it reaches the single-node sustainable phase in simulations (symbols). The theory, Eqs. (10) and (11) (continuous lines), deviate from the simulation results for small ρ. The dashed lines are from a different theory, see Supplementary Note 3, which captures also the limit of small ρ.