Local stabilizability implies global controllability in catalytic reaction systems

Abstract

Controlling complex reaction networks is a fundamental challenge in the fields of physics, biology, and systems engineering. Here, we prove a general principle for catalytic reaction systems with kinetics where the reaction order and the stoichiometric coefficient match: the local stabilizability of a given state implies global controllability within its stoichiometric compatibility class. In other words, if a target state can be maintained against small perturbations, the system can be controlled from any initial condition to that state. This result highlights a tight link between the local and global dynamics of nonlinear chemical reaction systems, providing a mathematical criterion for global reachability that is often elusive in high-dimensional systems. The finding illuminate the robustness of biochemical systems and offers a way to control catalytic reaction systems in a generic framework.

Figures (12)

• Figure 1: (a) A schematic illustration of the Sel'kov model. The black arrows represents the substrate-product relationship of each reaction, whereas the red arrow is the feedback activation. (b) The stoichiometric vectors of the reactions (red, blue, and green arrows) in the phase space. The stoichiometric compatibility class for $\bm x^{(i)}\, (i=1,2,3)$ of the model consisting only of the reaction $R_2$ are depicted as the broken lines.
• Figure 2: The phase space of the Sel'kov model (Eq. \ref{['eq:example']}) are partitioned by the balance manifolds ${\cal M}_r$ (red, blue, and green lines) into cells $C(\bm \sigma_A), C(\bm \sigma_B),\ldots, C(\bm \sigma_G)$. The directed stoichiometric vectors $\{\sigma_i \bm S_i\}_{i=1}^3$ are shown in each cell. The colors of the directed stoichiometric vectors are the same as the corresponding reactions. The yellow shaded region is the state reachable from $\bm x^{\rm src}$ by the non-negative controls. One target state $\bm x^{{\rm tgt},1}$ is thus controllable from $\bm x^{\rm src}$, while the other target $\bm x^{{\rm tgt},2}$ is not. In this model, only $C(\bm \sigma_D)$ is a free cell.
• Figure 3: A visual comparison of controllability in free versus non-free cells. Yellow region is the stoichiometric cone inside the cell. (a) A perturbation from $\bm x$ to $\bm x'$ (dashed arrow) in a free cell can be counteracted, restoring $\bm x$ (solid red arc), as the conical combinations span the cell. (b) In a non-free cell, the stoichiometric cone of $\bm x'$ (yellow region) is insufficient to return the system to $\bm x$.
• Figure 4: A graphical description of the control procedure. The state $\bm x^{\rm src}$ is driven to the target state $\bm x^{\rm tgt}$ via three steps. (Step 1) Control is applied to reach the equilibrium state $\bm x^{\mathrm{eq}}$ by activating the relevant control parameters (the others are set to zero). (Step 2) The control is adjusted to $\bm u^*$, controlling the state to $\bm x^{*}$. (Step 3) Finally, control within the free cell brings the state to $\bm x^{\rm tgt}$. The free cell is highlighted in yellow. The states $\bm x^{\rm src}$, $\bm x^{\rm eq}$, $\bm x^*$, and $\bm x^{\rm tgt}$ are depicted by an open circle, green square, blue diamond, and orange cross, respectively. Balance manifolds intersecting at $\bm x^{\mathrm{eq}}$ (e.g., ${\cal M}_i$ and ${\cal M}_k$) are shown in red, while the other balance manifold (${\cal M}_j$) is depicted in blue.
• Figure 5: A schematic diagram of the toy model of metabolism. The substrate chemical S is taken up from the external environment via the reaction $R_1$ with consumption of $n$ ATPs. The reaction $R_2$ generates $(n+1)$ ATP by secreting the chemical S, respectively. The reaction $R_3$ is the reaction for balancing ATP and ADP.

Theorems & Definitions (34)

• Definition 1: Controllable
• Definition 2: Local stabilizability
• Definition 3: Balance Manifold
• Definition 4: Cell
• Definition 5: Free Cell
• Proposition 1: Stabilizability in free cells
• Proposition 2: Steady-states and freeness
• Definition 6: Stoichiometrically Compatibe Kinetics
• Theorem 1: Main Theorem
• Proposition S1: Stabilizability of states in free cells