EMBED: Ensemble MATCONT-Based Detection and Classification of Bifurcations in Dual Phosphorylation-Dephosphorylation Reaction Networks.

We present EMBED, an ensemble-based extension of MATCONT that addresses the persistent design-build-test-learn (DBTL) gap in synthetic biology from a mechanistic, dynamical systems perspective. While the DBTL gap is widely attributed to biological complexity, context dependence, and incomplete models, here we show that it also arises fundamentally from the geometry of high-dimensional parameter spaces, where desired dynamical regimes occupy narrow and fragile regions. Conventional use of MATCONT is limited to single-parameter or single-trajectory bifurcation analysis, restricting its ability to capture this global organization. To overcome this, we integrate MATCONT with large-scale parameter sampling and automated continuation, enabling high-throughput and reproducible ensemble bifurcation analysis. Using dual phosphorylation-dephosphorylation (PdP) systems as a model, we demonstrate that although saddle-node, pitchfork, and saddle-node-transcritical bifurcations are theoretically possible, they are confined to finely tuned parameter regions and are therefore difficult to realize experimentally. In contrast, robust nonsingular dynamics such as ultrasensitive and biphasic responses dominate under biologically relevant conditions. We further show that the choice of control parameter fundamentally constrains accessible dynamics, establishing a key design axis for biochemical systems. By quantifying the prevalence, robustness, and accessibility of dynamical regimes, EMBED enables the identification of parameter regions that are not only functionally desirable but also experimentally realizable. Thus, EMBED complements existing DBTL approaches by providing a global, structure-based framework that links kinetic parameters to dynamical behavior. In doing so, it transforms MATCONT from a local analysis tool into a scalable platform for global dynamical mapping and offers a principled strategy for the rational design of robust biological circuits across signaling and gene regulatory networks.