(a) Structure of T-cell survival network: each node is labelled with its generic name, and the arrowhead and diamond-head edges represent activation and inhibition regulations, respectively. The inhibitory edges from ‘Apoptosis' to other nodes are not shown (for clarity). (b) Attractor network of the T-cell network, which contains three nodes: two cancerous states denoted as C₁ and C₂ and a normal state denoted as N. The two directed edges in the attractor network are multiple, each containing altogether 48 individual edges corresponding to controlling the 48 edges in the original network, which are indicated by the dark dashed lines, whereas the remaining edges in the original network are signified by the light solid lines. Our detailed computations reveal that parameter perturbation on any one of the 48 edges can drive the system from a cancerous state to the normal state. That is, regardless of whether the initial state is C₁ or C₂, with a proper modification to one of the 48 parameters, the system can be driven to the normal state N.

For the T-cell network, (a) an inverted rectangular control signal of duration and amplitude , where μ₀ is the original parameter value and μ_n is the control parameter value. A saddle-node bifurcation occurs for μ=μ_c, so Δ_e=μ_c−μ_n is the excessive amount of the parameter change over the critical value μ_c. (b,c) Minimal control time versus μ_n, where parameter control is applied to the activation edge from node ‘S1P' to node ‘PDGFR' and to the inhibitory edge from ‘DISC' to ‘MCL1', respectively. These four nodes are indicated with the solid black circles in . The corresponding plots on a logarithmic scale in the insets of (b,c) suggest a power-law scaling behaviour between and Δ_e (). The fitted power-law scaling exponents are β≈−0.44 and −0.55, respectively, for (b,c).

(a) Success rate to control the T-cell network from the cancerous state C₁ to the normal state N using a combination of parameter perturbation and external noise (of amplitude) σ, where Δ_d≡μ_n−μ_c is the parameter deficiency. Warm colours indicate higher probability values of successful control. The perturbation duration is . The results are averaged over 1,000 realizations. (b) A three-dimensional plot: success rate versus Δ_d and σ.

(a) Simplified model of the two-node GRN, where the arrowhead and bar-head edges represent activation and inhibition regulations, respectively, and the sawtooth lines denote the strength of the tunable edge. (b) Bifurcation diagram with respect to the control parameter a₁, where the red and grey solid lines denote the stable and unstable steady states, respectively. In the two parallel cross-sections (with dashed line boundaries) for and , the yellow and brown dots represent the corresponding stable attractors, respectively. (c) Control signals required to drive the system from attractor A to attractor B. In d–f, grey dashed lines represent the basin boundaries; black solid circles and green crosses denote attractors and unstable steady states, respectively. (d) For the initial parameter setting, , the system is at a low concentration state A, and the target state is B. (e) By changing a₁ from to , attractor A is destabilized and its original basin is absorbed into that of the intermediate attractor B′, so the system approaches B′. (f) When control perturbation upon a₁ is released, the landscape recovers to that in d. Once the system has entered the basin of the target state B during the process in e it will evolve spontaneously towards B. Parameters in simulation are , , t₀=0, t₁=23 and t₂=40.

(a–c) Bifurcation diagrams with respect to the coupling parameters a₁, a₂ and b₂, respectively, where each bifurcation point can be exploited to design control. (d) The corresponding attractor network, in which a directed edge corresponds to an elementary control that is designed to steer the system from the original attractor to the directed one. The solid and dashed edges, respectively, denote the positive and negative changes in the corresponding control parameters. (e) Sequential control signals required to drive the system from attractor A to attractor C through the path A→B→C. In simulation, the original parameter values are and . We set , followed by setting , and the respective durations of the parameter perturbation are and .

‘Pseudo' potential of the two-node GRN system (a) for a₁=1.0 (Δ_d≈0.3549), σ=0.05 and (b) for a₁=1.3 (Δ_d≈0.0549), σ=0.05. Regions of warm and cold colours indicate the states with large and small pseudo energies, respectively. (c) For fixed σ=0.02, two-dimensional representation of for a number of values of a₁. (d) For fixed a₁=1.3, two-dimensional representation of for a number of values of σ.

(a) Schematic illustration of a three-node GRN system. The arrowhead and bar-head edges represent activation and inhibitory regulations, respectively. The sawtooth lines specify that the corresponding edge strength is experimentally adjustable. (b) Coexisting attractors (A to H) in the phase space. (c) The underlying attractor network, where each node represents an attractor and each weighted directed link indicates that its strength can be experimentally tuned to steer the system from the starting attractor to the pointed attractor. Each grey directed link with a single arrow indicates that only one parameter is needed to achieve control, and each purple link with a double-arrow and the label 3 represent the case where three parameters are required to achieve control.