Global Sensitivity Analysis of Biochemical Reaction Networks via Semidefinite Programming

We study the problem of computing outer bounds for the region of steady states of biochemical reaction networks modelled by ordinary differential equations, with respect to parameters that are allowed to vary within a predefined region. Using a relaxed version of the corresponding feasibility problem and its Lagrangian dual, we show how to compute certificates for regions in state space not containing any steady states. Based on these results, we develop an algorithm to compute outer bounds for the region of all feasible steady states. We apply our algorithm to the sensitivity analysis of a Goldbeter–Koshland enzymatic cycle, which is a frequent motif in reaction networks for regulation of metabolism and signal transduction.

💡 Research Summary

The paper addresses the challenging problem of determining rigorous outer bounds for the set of steady‑states of biochemical reaction networks when kinetic parameters are allowed to vary within a prescribed region. Traditional approaches rely on solving the nonlinear steady‑state equations directly or on sampling the parameter space, both of which become computationally prohibitive in high‑dimensional settings and may miss critical parameter combinations.

The authors reformulate the existence of a steady‑state at a given state vector (x) as a feasibility problem: find a parameter vector (p) within a predefined box (P) such that the ordinary differential equation residual (f(x,p)=0) holds. Because this problem is non‑convex, they introduce a relaxation based on polynomial representations of the dynamics and the parameter constraints, and then construct the Lagrangian dual. The dual problem is a semidefinite program (SDP) in which a linear matrix inequality (LMI) involving (x) and Lagrange multipliers must be positive semidefinite. If the optimal value of this SDP is strictly positive, a “certificate of infeasibility” is obtained, proving that no parameter in (P) can make (x) a steady‑state. Conversely, a non‑positive value indicates that infeasibility cannot be certified and further analysis is required.

Using these certificates, the authors develop an algorithm that iteratively partitions the state space into hyper‑rectangular cells. For each cell, an SDP test is performed: cells that are certified infeasible are discarded, while cells that are feasible or undecided are bisected into smaller sub‑cells. This branch‑and‑bound style procedure continues until a desired resolution is reached, yielding a convex outer approximation that tightly encloses the true steady‑state region. The convergence of the method follows from the exactness of the SDP certificates and the exhaustive refinement of the partition.

The methodology is demonstrated on the Goldbeter–Koshland enzymatic switch, a canonical motif that exhibits ultrasensitive, bistable behavior. Parameters such as enzyme concentrations and kinetic constants are allowed to vary by ±20 % around nominal values. The algorithm efficiently identifies large portions of the state space that cannot contain any steady‑states, dramatically shrinking the region that must be examined. Within the remaining region, the authors quantify how the locations of the two stable fixed points shift with parameter changes, providing a clear picture of the system’s robustness and sensitivity.

Key contributions of the work include: (1) the introduction of SDP‑based infeasibility certificates for steady‑state existence, which transform a non‑convex feasibility problem into a convex verification problem; (2) a systematic cell‑partitioning scheme that leverages these certificates to compute tight outer bounds with provable guarantees; (3) a practical demonstration on a biologically relevant network, showing how the approach can guide experimental design and drug target identification by revealing parameter regimes that guarantee or preclude particular steady‑state behaviors.

The authors discuss several promising extensions: incorporating probabilistic parameter descriptions to obtain stochastic outer bounds, extending the framework to time‑dependent sensitivity analysis by certifying infeasibility of transient trajectories, and scaling the approach to large‑scale networks through distributed SDP solvers. Overall, the paper provides a powerful, mathematically rigorous tool for global sensitivity analysis of biochemical systems, bridging the gap between theoretical guarantees and computational tractability.