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

Abstract

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.

1. Introduction

Synthetic biology aims to engineer biological systems with predictable, robust, and programmable functionalities. However, rational design of gene and signaling networks remains challenging due to parameter uncertainty, environmental variability, and context dependence. A central difficulty is identifying kinetic parameter regimes that reliably produce desired behaviors. Even with fixed network topology, parameter variations can generate qualitatively distinct dynamics, including graded responses, ultrasensitivity, bistability, and oscillations. − Importantly, these qualitative behaviors are determined primarily by the underlying network architecture and kinetic parameter regimes rather than by the particular mathematical formulation used to describe the reaction kinetics. In many biochemical systems, reduced kinetic models obtained from mechanistic mass-action descriptions retain the same qualitative dynamical repertoire, including steady-state organization and bifurcation structure, while significantly reducing computational complexity. Such reduced formulations therefore provide an efficient and biologically meaningful framework for large-scale exploration of high-dimensional parameter spaces, motivating the modeling approach adopted in this work. This contributes to the persistent discrepancy between model predictions and experimental observationscommonly referred to as the DBTL gap.

Recent advances in synthetic biology have emphasized the design–build–test–learn (DBTL) cycle as a systematic framework for engineering biological systems. , In this paradigm, the design phase relies heavily on computational models to predict functional behaviors, which are then experimentally implemented and iteratively refined. However, a fundamental limitation of current DBTL implementations is that the design phase often focuses on identifying isolated parameter sets that yield desired behaviors, without quantifying the robustness or prevalence of these behaviors across the high-dimensional parameter space. As a result, designs that appear successful in silico frequently fail during the build and test phases due to parameter uncertainty, biological variability, and environmental fluctuations. − This limitation reflects an intrinsic gap in DBTL, where local model predictions do not translate into globally robust and experimentally realizable behaviors. In this work, we address this gap by introducing an ensemble-based bifurcation framework that systematically maps parameter space to dynamical regimes, enabling identification of behaviors that are not only theoretically feasible but also robust and experimentally accessible, as summarized in Figure .

1.

Design–build–test–learn (DBTL). Schematic overview of the EMBED framework as part of the design of DBTL for systematic exploration of parameter-dependent dynamics in dual phosphorylation–dephosphorylation (PdP) systems. Parameter ensembles are first generated using Latin Hypercube Sampling (LHS) to ensure a uniform coverage of the high-dimensional parameter space. Each sampled parameter set is then analyzed using a MATCONT-based workflow involving steady-state computation, numerical continuation, and detection of singularities such as limit points (LPs) and branch points (BPs). The resulting bifurcation structures are classified into distinct dynamical regimes, including saddle-node, pitchfork, and saddle-node–transcritical bifurcations, as well as nonsingular behaviors such as ultrasensitive and biphasic responses. This framework enables the mapping of the parameter space to functional outcomes, revealing that robust dynamical behaviors dominate, while bifurcation-driven regimes occur in narrowly constrained regions. The EMBED approach facilitates the design phase and iterative refinement of selecting the control parameter and performs dynamic targeted experimental exploration.

From a dynamical systems perspective, these behaviors are governed by bifurcation theory, which identifies parameter thresholds where qualitative changes occur. Bifurcations partition a high-dimensional parameter space into distinct regimes and underpin key synthetic biology functions such as switches, memory, and oscillators, typically associated with saddle-node (SN), transcritical (TC), pitchfork (PF), and Hopf bifurcations. Despite this framework, idealized bifurcations predicted by models are rarely observed in experiments. This discrepancy arises from parameter sloppiness, structural model approximations, and neglected effects such as stochasticity and hidden interactions. Importantly, bifurcation regimes often occupy narrow and finely tuned regions of parameter space, making them intrinsically fragile to perturbation. As a result, experimentally accessible systems are more likely to exhibit robust nonsingular behaviors, such as ultrasensitive and biphasic responses, rather than idealized bifurcation structures.

We propose three complementary explanations for this discrepancy. First, bifurcation regimes require finely tuned parameters and symmetry conditions that are difficult to maintain in biochemical environments. Second, intrinsic stochasticity blurs sharp bifurcation boundaries, smoothing transitions and favoring robust responses. Third, methodological limitations are critical: widely used tools such as MATCONT and XPPAUT are inherently local, rely on single-trajectory continuation, and do not enable systematic exploration of high-dimensional parameter spaces. − We attempted to implement an automated detection approach based on the method described in automatic detection of saddle-node–transcritical interaction by. In practice, this implementation was not robust for our model: in some cases, it indicated SNTC interaction, while in many other cases, it failed to detect one, the global organization, prevalence, and robustness of dynamical regimes remain unresolved.

Dual phosphorylation–dephosphorylation (PdP) cycles, central to MAPK signaling pathways, provide an ideal platform to study these issues. These systems exhibit rich nonlinear behaviorsincluding bistability, hysteresis, oscillations, ultrasensitivity, and biphasic responsesarising from enzymatic nonlinearities and regulatory interactions. − Experimental studies confirm bistability and oscillations in MAPK systems, establishing them as canonical models of nonlinear biochemical computation. Even without explicit feedback, PdP systems generate nonlinearities through enzyme saturation, sequestration, and competition. Classical studies have demonstrated SN, TC, and PF bifurcations in biochemical networks. −

The dual phosphorylation–dephosphorylation (PdP) cycle considered in this work has been extensively studied using both detailed mechanistic mass-action models and reduced Michaelis–Menten (MM) formulations derived through the quasi-steady-state approximation (QSSA). While the mechanistic model explicitly represents enzyme–substrate complexes, the reduced MM model preserves the same biochemical network topology and reproduces the same qualitative dynamical behaviors over corresponding parameter regimes, including ultrasensitivity, graded and biphasic responses, multistability, and codimension-1 bifurcations. Consequently, the reduced MM formulation has become the standard representation for analyzing PdP dynamics because it retains the essential nonlinear mechanisms while substantially reducing computational complexity. Since the objective of the present work is to systematically map the organization of dynamical regimes across parameter space rather than compare alternative kinetic formulations, we adopt the MM model throughout this study.

However, existing studies focus on isolated parameter regimes and do not resolve how multiple dynamic behaviors are organized across the full parameter space. In particular, it remains unclear whether bifurcation-driven behaviors are robust and prevalent or confined to narrow, fragile regions relative to dominant nonsingular dynamics. Addressing this requires a systematic exploration of high-dimensional parameter spaces, which is not feasible with conventional continuation tools. Existing frameworks such as MATCONT and XPPAUT require manual initialization and do not support ensemble-scale analysis, preventing comprehensive sampling, automated classification, and global mapping of dynamical regimes. ,,

In this work, we address these limitations by introducing EMBED, an ensemble MATCONT-based bifurcation framework that integrates large-scale parameter sampling with automated continuation. This approach enables systematic mapping of SN, TC, and PF bifurcation regimes alongside nonsingular dynamics across a high-dimensional parameter space. By moving beyond single-trajectory continuation, EMBED quantifies the prevalence, robustness, and accessibility of dynamical behaviors and provides a mechanistic explanation for the rarity of classical bifurcations in experiments. Although demonstrated using a canonical dual phosphorylation–dephosphorylation network, the ensemble continuation strategy is broadly applicable to mechanistic nonlinear ODE models for signaling and gene regulatory systems, provided the corresponding dynamical system can be formulated for numerical continuation offering a scalable strategy to bridge nonlinear dynamical theory with the experimentally realizable biological design.

2. Methods

2.1. Dual Phosphorylation–Dephosphorylation (PdP) 2D Model

We consider a dual, processive phosphorylation–dephosphorylation (PdP) cycle described by Markevich et al. in which a substrate undergoes two sequential modification steps catalyzed by a kinase and reversed by a phosphatase. The substrate exists in three states: unphosphorylated (A), singly phosphorylated (AP), and doubly phosphorylated (APP) (see Figure A). These reactions collectively define a distributive, reversible dual phosphorylation–dephosphorylation cycle under standard MM kinetics. Each phosphorylation and dephosphorylation step proceeds through a distinct enzyme–substrate intermediate, capturing the mechanistic features of MAPK-like signaling cascades. The full model is given in Section (S1) of the Supplementary S1.7, along with the chemical reaction network theory (CRNT) analysis in Section (S1.1). The governing 2D-ordinary, nonlinear differential equations based on QSSA are given by

$$\frac{d[A]}{dt}=r_3-r_1$$ 1

$$\frac{d[APP]}{dt}=r_2-r_4$$ 2

where r _i (i = 1, 2, 3, 4) denotes the enzyme-mediated reaction rates.

2.

Dual phosphorylation–dephosphorylation (PdP) reaction network and its reduced representation. (A) Schematic of the distributive dual PdP cycle. Here, A denotes the unphosphorylated substrate, AP denotes the singly phosphorylated substrate, and APP denotes the doubly phosphorylated substrate. K and P represent the kinase and phosphatase enzymes, respectively. The forward phosphorylation steps (A → AP and AP → APP) and the reverse dephosphorylation steps (APP → AP and AP → A) are represented by the reduced Michaelis–Menten reaction rates r ₁, r ₂, r ₄, and r ₃, respectively, as defined in Equations , , , and . For clarity, only the substrate states are shown in the schematic. The complete reaction network consists of nine species (A, AK, AP, APK, APP, APPC, APC, K, and P), where the enzyme–substrate complexes (AK, APK, APC, and APPC) are omitted from the figure for visual simplicity but are included in the full mechanistic model described in Supporting Information SI-7 (Section S1.7). (B) Network representation of the reduced two-dimensional model illustrating the effective double-negative feedback loop and negative autoregulation between the dynamical variables A and APP. (C) Sign structure of the Jacobian matrix for the reduced two-dimensional model. All Jacobian elements are negative irrespective of the parameter values, indicating the effective double-negative feedback architecture (see for mechanistic explanation for double-negative feedback). The symbolic Jacobian calculation was performed in Mathematica, and the corresponding notebook is provided in Supporting Information (SI-6). The derivation of the reduced two-dimensional model from the full mechanistic model using the quasi-steady-state approximation (QSSA) is presented in Supporting Information SI-7 (Section S1.2).

Using the conservation relation

$$ATOT=A+AP+APP$$ 3

the intermediate species AP can be eliminated as

$$AP=ATOT-(A+APP)$$ 4

thereby reducing the system to two dynamical variables, A and APP. The phosphorylation reactions catalyzed by the kinase are modeled using the Michaelis–Menten kinetics with competitive substrate binding, and the rates are given below

$$r_1=\frac{k_{2}KTOT\frac{A}{K_{m1}}}{1+\frac{A}{K_{m1}}+\frac{ATOT-A-APP}{K_{m3}}}$$ 5

$$r_2=\frac{k_{4}KTOT\frac{ATOT-A-APP}{K_{m3}}}{1+\frac{A}{K_{m1}}+\frac{ATOT-A-APP}{K_{m3}}}$$ 6

Similarly, the dephosphorylation reactions catalyzed by the phosphatase are given by

$$r_3=\frac{k_{8}PTOT\frac{ATOT-A-APP}{K_{m7}}}{1+\frac{ATOT-A-APP}{K_{m7}}+\frac{APP}{K_{m5}}}$$ 7

$$r_4=\frac{k_{6}PTOT\frac{APP}{K_{m5}}}{1+\frac{ATOT-A-APP}{K_{m7}}+\frac{APP}{K_{m5}}}$$ 8

The kinetic constants are

$$K_{m1}=\frac{k_{-1}+k_2}{k_1}{K}_{m3}=\frac{k_{-3}+k_4}{k_3}{K}_{m5}=\frac{k_{-5}+k_6}{k_5}{K}_{m7}=\frac{k_{-7}+k_8}{k_7}{V}_{\max 1}={k_2}^{*}KTOT{V}_{\max 2}={k_4}^{*}KTOT{V}_{\max 3}={k_6}^{*}\mathrm{P}\mathrm{T}\mathrm{O}\mathrm{T}{V}_{\max 4}={k_8}^{*}PTOT$$

In this formulation, KTOT and PTOT denote the total kinase and phosphatase concentrations, respectively, while k _i and K _mi represent the catalytic rate constants and Michaelis constants, respectively. The shared denominators in the rate expressions capture competition among substrate species for a common enzyme pool, thereby introducing nonlinear coupling through enzyme saturation and sequestration effects.

2.1.1. Model Reduction for Bifurcation and Phase-Line Analysis

To facilitate both numerical continuation and geometric interpretation of the system dynamics, we consider three complementary representations of the dual phosphorylation–dephosphorylation (PdP) model: (i) the full mechanistic reaction network, (ii) a reduced two-dimensional (2D) model obtained using the quasi-steady-state approximation (QSSA), and (iii) a further reduced one-dimensional (1D) model derived from rate-balance conditions. The successive model reductions preserve the steady-state solutions and qualitative bifurcation structure of the original system while substantially improving analytical and computational tractability.

Notably, this minimal PdP model incorporates intrinsic biochemical nonlinearities without requiring explicit feedback loops. As shown in Figure C, the Jacobian calculated for the two-dimensional model is negative for all elements, irrespective of the parameter values chosen, indicating the presence of a double-negative feedback loop, as illustrated in Figure B. The Mathematica notebook used for the symbolic Jacobian calculation is provided theSupporting Information (SI-6). The combination of enzyme saturation, competitive binding, and conservation constraints enables the system to exhibit a wide range of dynamical behaviors, including ultrasensitivity, bistability, and biphasic responses, depending on the choice of kinetic parameters. The detailed derivation of the two-dimensional model from the full reaction network using the quasi-steady-state approximation (QSSA) is provided in Supporting Information SI-7 (Section S1.2).

To further simplify the dynamical analysis, the reduced two-dimensional system is transformed into an equivalent one-dimensional formulation by considering the net production and consumption dynamics of the doubly phosphorylated product (APP). The reduced two-dimensional system can be written in the general form

$$\frac{d[A]}{dt}=f_1(A,APP)$$ 9

$$\frac{d[APP]}{dt}=f_2(A,APP)$$ 10

Assuming that the concentration of the unphosphorylated substrate rapidly relaxes to its steady-state value, we set

$$\frac{d[A]}{dt}=0$$ 11

which yields

$${[A]}_{\mathrm{s}\mathrm{s}}=f_3(APP)$$ 12

Substituting this steady-state expression into the APP equation gives

$$\frac{d[APP]}{dt}=f_4(f_3(APP),APP)$$ 13

resulting in a one-dimensional ordinary differential equation governing the dynamics of the product APP. The complete derivation of the one-dimensional model is provided in Supporting Information SI-3 .

The resulting 1D rate-balance plots are constructed by plotting the net rate $\frac{d[APP]}{dt}$ as a function of APP. The zero crossings correspond to steady states, while the local slopes determine their stability. Consequently, single, double, and triple intersections correspond to mono-, bi-, and multistable regimes, respectively, whereas tangential intersections indicate bifurcation points. These plots provide an intuitive geometric visualization of how steady states emerge, disappear, and exchange stability as kinetic parameters vary. Unless otherwise stated, the vector-field plots presented in Figures ,, , and are generated using this reduced one-dimensional formulation, whereas the bifurcation analysis itself is performed on the reduced two-dimensional model.

3.

Saddle-node bifurcation (full and 2D), rate-balance plot, and vector-field plot. (A) Codimension-1 bifurcation of the PdP reaction with ATOT as the bifurcation parameter, exhibiting bistability using the full model. SS and USS are, respectively, the stable and unstable steady states. SN1 and SN2 are the two saddle-node bifurcations. (B) Codimension-1 bifurcation of the PdP reaction with ATOT as the bifurcation parameter exhibiting bistability using 2D model. (C) Vector-field plot for three different ATOT apart from the limit points. The vector-field plot captures the mono- and bistability. For monostability, ATOT = 53 (green), for bistability ATOT = 58 (red). Again, monostability for ATOT = 63 (purple). (D) Vector-field plots at limit points where ATOT = 56.455 (yellow) and ATOT = 60.96 (orange). (E) Two-parameter bifurcation between k4 and ATOT using the parameter set for saddle-node bifurcation. It is divided into three regions: “I” and “III” have one steady state, and “II” has three steady states. The vector field in each region is shown in the subset figures. (F) Two-parameter bifurcation between K _m5 and ATOT using the parameter set for saddle-node bifurcation. It is divided into three regions: “I” and “III” have one steady state and “II” has three steady states. The vector field at each region is shown in the subset figures. The XPPAUT files for the full and 2D model are given in the Supporting Information SI-2 and SI-1, respectively. The MATLAB code for the vector-field plots is given in the Supporting Information SI-3.

4.

Saddle-node–transcritical–SNTC bifurcation (full and 2D), vector-field plot, and two-parameter bifurcation. (A) Codimension-1 bifurcation of the PdP reaction with ATOT as the bifurcation parameter exhibiting bistability using full model. SN is the saddle-node bifurcation, and TC is the transcritical bifurcation. (B) Codimension-1 bifurcation of the PdP reaction with ATOT as the bifurcation parameter exhibiting bistability using 2D model. (C) Vector-field plot for three different values of ATOT apart from the branch and limit points. Monostability for ATOT = 50 (green). Three steady states for ATOT = 75 and 100 (red and purple, respectively). (D) The vector-field plot at the branch and limit points. At ATOT = 60.1 (orange), it is a limit point. At ATOT = 84.84 (yellow), it is a branch point. (E) Two-parameter bifurcation between k4 and ATOT using the parameter set for saddle-node transcritical bifurcation. It is divided into three regions: “I” and “III” have one steady state and “II” has three steady states. The vector field at each region is shown in the subset figures. (F) Two-parameter bifurcation between K_m5 and ATOT using the parameter set for saddle-node transcritical bifurcation. It is divided into four regions, “IV” has one steady state and “I”–“III” have three steady states. The vector field at each region is shown in the subset figures. The XPPAUT files for the full and 2D models are given in the Supporting Information SI-2 and SI-1, respectively. The MATLAB code for the vector-field plots is given in the Supporting Information SI-3.

5.

Pitchfork bifurcation (full and 2D), vector-field plot, and two-parameter bifurcation. (A) Codimension-1 bifurcation of the PdP reaction with ATOT as the bifurcation parameter exhibiting bistability using full model. SS and USS are, respectively, the stable and unstable steady states. PF is the pitchfork bifurcation. (B) Codimension-1 bifurcation of the PdP reaction with ATOT as the bifurcation parameter exhibiting bistability using 2D model. (C) Vector-field plot for two different values of ATOT other than the branch point. One steady state for ATOT = 125 (green). For bistability due to pitchfork bifurcation ATOT = 140. (D) The vector-field plot at the branch point where ATOT = 132.76. (E) Two-parameter bifurcation between k4 and ATOT using the parameter set for pitchfork bifurcation. It is divided into three regions: “I” and “II” have one steady state and “II” has three steady states. The vector field at each region is shown in the subset figures. (F) Two-parameter bifurcation between K _m5 and ATOT using the parameter set for pitchfork bifurcation. It is divided into four regions, “I”–“III” have three steady states and “IV” has one steady state. The vector field at each region is shown in the subset figures. The XPPAUT files for the full and 2D models are given in the Supporting Information SI-2 and SI-1, respectively. The MATLAB code for the vector-field plots is given in (SI-3).

6.

SNTC with K _m5 = 60 and K _m5 = 40. (A) The bifurcation diagram at K _m5 = 60 using the full model. The red and black lines correspond to the stable and unstable steady states, respectively. The blue and green markers correspond to the branch point (BP) and the limit point (LP), respectively. (B) The bifurcation diagram at K _m5 = 40 using the full model. The red and black lines correspond to the stable and unstable steady states, respectively. The blue and green markers correspond to the branch point (BP) and the limit point (LP), respectively. (C) The bifurcation diagram at K _m5 = 60 using the 2D model. In the subset figure, the lines correspond to the values of ATOT for which the vector-field plots were generated. The green, yellow, violet, gray, and maroon lines correspond to the ATOT values, 142, 142.2, 142.4, 142.6316, and 143, respectively. (D) The bifurcation diagram at K _m5 = 40 using the 2D model. In the subset figure, the lines correspond to the values of ATOT for which the vector-field plots were generated. The green, yellow, violet, gray, and maroon lines correspond to the ATOT values, 120.5, 120.9356, 121.2, 121.5789, and 122, respectively. (E) Vector-field plots using the reduced 1D model with the same values as mentioned in the subset of C. (F) Vector-field plots using the reduced 1D model with the same values as mentioned in the subset of D. The XPPAUT files for the full and 2D models are given in the Supporting Information SI-2 and SI-1, respectively. The MATLAB code for the vector-field plots is given in (SI-3).

We comment here on the Markevich et al. model, and our proposed Michaelis–Menten-based PdP model. Both our models capture the distributive dual phosphorylation–dephosphorylation but differ in complexity and symmetry. Markevich et al. include ten reactions with multiple enzyme–substrate complexes and redistribution steps, especially in dephosphorylation via MKP3 (see Scheme 2, dephosphorylation steps in their paper), capturing biochemical realism through intermediate recycling and enzyme sequestration. In contrast, our model uses only eight reactions with symmetric, minimal MM stepseach phosphorylation and dephosphorylation step proceeds via one intermediate without feedback or rebinding. Despite its simplicity, we will show below that our model retains key nonlinearities and saturations to exhibit bistability and multiple bifurcations, revealing rich dynamics that can emerge from structurally minimal, symmetric, dual PdP cycles.

2.2. Reduction to the MM Kinetic Model Helps to Perform Experiments for Specific Dynamics

For biochemical regulatory modules such as phosphorylation–dephosphorylation (PdP) cycles, mathematical models can be formulated either using full mass-action kinetics or using Michaelis–Menten (MM) reduced kinetics. While the mass-action formulation provides the most mechanistically detailed description, the MM-based representation is often more practical and informative for biological interpretation and experimental design. In a full mass-action framework, each enzymatic reaction must explicitly account for enzyme–substrate binding and unbinding steps, catalytic conversion, and intermediate complex formation. As a result, a simple PdP cycle involving a kinase and a phosphatase typically requires several dynamical variables and a large number of kinetic parameters, including association rates, dissociation rates, catalytic rate constants, and total enzyme concentrations. This level of detail can significantly increase model dimensionality and parameter uncertainty, making it difficult to systematically analyze the dynamical behavior of the system or to relate model parameters directly to experimentally controllable quantities.

In contrast, the Michaelis–Menten formulation exploits the quasi-steady-state approximation for enzyme–substrate complexes and reduces the description of each enzymatic step to a small number of effective parameters, typically the maximum catalytic rate V _max and the Michaelis constant K _m . Consequently, the PdP module can be represented using a much smaller parameter set that directly captures the effective enzymatic activity and substrate affinity. For example, whereas the full mass-action representation of a kinase–phosphatase cycle may involve multiple rate constants for binding, dissociation, and catalysis by both enzymes, the MM formulation reduces these to a few composite parameters describing the effective phosphorylation and dephosphorylation fluxes. This reduction substantially simplifies the parameter space and makes it easier to perform systematic dynamical analyses such as steady-state mapping, sensitivity analysis, and bifurcation exploration.

From an experimental perspective, this reduction is particularly advantageous because the effective parameters in MM models are closely related to experimentally tunable variables. Parameters such as V _max depend on enzyme concentration and catalytic efficiency, while K _m reflects substrate affinity and can be modulated through mutations, competitive inhibitors, or substrate analogs. By using simulations of the reduced model, experimentalists can therefore identify which combinations of enzyme levels, substrate concentrations, or regulatory factors are most likely to shift the system into regimes of interest, such as ultrasensitive responses, biphasic dynamics, or potential switching behavior. In practice, this means that model-guided parameter sweeps can suggest targeted perturbationsfor example, increasing kinase abundance, reducing phosphatase activity, or modifying substrate availabilitythat are experimentally feasible. Thus, the MM-based representation serves as an effective bridge between theoretical modeling and experimental manipulation, enabling researchers to explore complex dynamical behaviors with a manageable number of biologically interpretable parameters and to guide the design of controlled perturbation experiments.

To further relate the present computational framework to experimentally characterized PdP systems, we note that the parameter ranges explored in this study are based on experimentally measured kinetic quantities reported for distributive dual-MAPK phosphorylation. Ferrell and Bhatt experimentally demonstrated that MAPK activation proceeds through a distributive dual-phosphorylation mechanism and reported Michaelis constants of approximately 300 nM and 46 nM for the first and second phosphorylation reactions, respectively, together with a phosphatase Michaelis constant of approximately 300 nM. The catalytic rate constants and enzyme concentration ranges were selected to be consistent with experimentally measured turnover rates and kinase/phosphatase concentrations reported by Witzel and Blüthgen. These experimentally determined values lie within or were used to define the sampling ranges adopted in the present ensemble analysis (Table ), indicating that the explored parameter space remains biologically plausible. We emphasize that these experimental studies validate the biochemical mechanism and physiologically relevant kinetic regime underlying the model rather than the bifurcation structures themselves, which are computational predictions arising from the systematic exploration of the experimentally admissible parameter space.

4. Parameter Ranges Used to Generate and Explore Different Bifurcation Regimes in the Dual Phosphorylation–Dephosphorylation (PdP) Model ^,

	Figures ,,,										Figure
	PF/SNTC		S-SN		Z-SN		common min–max		full variation		literature	wide range		S-SN		Z-SN		PF/SNTC
parameter	lower	upper	lower	upper	lower	upper	lower	upper	lower	upper		lower	upper	lower	upper	lower	upper	lower	upper
KTOT (nM)	100	100	100	100	100	100	1	100	1	500	100–1200 ,	100	1200	400	500	300	400	300	1000
PTOT (nM)	100	100	100	100	100	100	1	100	1	500	300–1000 a ,	300	1000	600	700	600	700	300	1000
K m1 (nM)	51	55	51	100	5	10	5	500	1	500	300	10	400	350	400	10	20	250	350
K m5 (nM)	51	55	5	10	100	500	5	100	1	500	300	10	400	15	20	300	400	250	350
K m3 (nM)	500	500	500	500	500	500	5	500	1	500	46	10	400	350	400	10	20	46	46
K m7 (nM)	500	500	500	500	500	500	5	500	1	500	300	10	400	30	40	100	200	300	300
k2 (s–1)	0.01	0.01	0.01	0.01	0.01	0.01	0.01	1	0.01	1	0.0747	0.01	0.10	0.06	0.08	0.03	0.06	0.03	0.08
k6 (s–1)	0.01	0.01	0.01	0.01	0.01	0.01	0.01	1	0.01	5	0.0585	0.01	0.10	0.01	0.02	0.03	0.06	0.03	0.08
k4 (s–1)	0.21	0.21	0.21	0.21	0.10	0.10	0.10	2	0.01	5	0.0357	0.01	0.10	0.05	0.055	0.06	0.08	0.03	0.08
k8 (s–1)	0.21	0.21	0.20	0.20	0.11	0.11	0.11	2	0.01	5	0.0315	0.01	0.10	0.03	0.06	0.045	0.05	0.03	0.08

^aThe columns grouped under Figures , , , and summarize the constrained parameter sets used to generate the PF/SNTC, S-SN, and Z-SN bifurcation regimes together with the expanded “Common Min–Max” and “Full Variation” sampling ranges used for ensemble bifurcation analysis. The columns grouped under Figure summarize the literature-derived reference values together with the parameter ranges used for the experimentally guided wide-range, S-enriched (LP only S-shaped), Z-enriched (LP only Z-shaped), and PF/SNTC-enriched (BP only/LP+BP) ensemble analyses. This table illustrates how targeted and progressively refined parameter exploration enables systematic mapping of bifurcation structures across the kinetic parameter space of the selected mathematical model. For the analyses presented in Figures , , , and , 500 parameter sets were generated for each sampling strategy. For Figure , 5000 parameter sets were generated for the wide-range ensemble, whereas 1000 parameter sets were generated for each of the S-enriched, Z-enriched, and PF/SNTC-enriched ensembles.

^bNote: reference Michaelis constants were obtained from the experimental study of distributive dual MAPK phosphorylation by Ferrell and Bhatt. The catalytic turnover rates (k ₂, k ₄, k ₆, and k ₈) and total enzyme concentrations (KTOT and PTOT) are taken from the mechanistic parametrization of Witzel and Blüthgen, which is based on experimentally measured kinetic data together with explicitly stated modeling assumptions. These reference values are provided to demonstrate that the parameter ranges explored in this study encompass the biologically plausible kinetic regimes. ^a Value assumed in the mechanistic model of Witzel and Blüthgen.

2.3. MATCONT–LHS Framework for Ensemble Bifurcation Analysis (EMBED)

Beyond identifying bifurcation points for individual parameter sets, MATCONT can be used through both a graphical user interface and a command-line interface (CLI). In the present MATCONT-LHS framework, we use the CLI to automate bifurcation analysis across ensembles of parameter sets, enabling the construction of ensemble bifurcation landscapes analogous to ensemble-based parameter estimation approaches widely used in systems biology. In the classical parameter estimation, large ensembles of parameter sets are generated to fit experimental time-series data or steady-state dose–response curves, yielding a family of models consistent with observations rather than a single best-fit solution. MATCONT–LHS extends this philosophy from trajectory fitting to dynamical-structure fitting. Latin Hypercube Sampling (LHS) is a statistical stratified sampling technique used to efficiently explore high-dimensional parameter spaces. Unlike simple random sampling, LHS divides each parameter range into equal-probability intervals and ensures that each interval is sampled exactly once. This guarantees more uniform space-filling coverage of the parameter space with significantly fewer samples. As a result, LHS reduces clustering and sampling bias, making it particularly suitable for nonlinear dynamical systems where bifurcation structures are sensitive to parameter variations. In systems biology, this becomes crucial because kinetic parameters are often uncertain, correlated, and experimentally heterogeneous. Therefore, LHS provides a computationally efficient and statistically robust way to generate diverse yet representative ensembles of parameter sets for bifurcation and uncertainty analysis.

Within this framework, parameter ensembles are generated not only to reproduce observed system outputs but also to realize specific qualitative dynamics, such as bistability, oscillations, or excitability. This ensemble-based approach reveals that the same biological network can support multiple bifurcation mechanisms across parameter space. For example, transitions traditionally attributed to a pitchfork bifurcation may, for different parameter ensembles, arise via saddle-node or saddle-node–transcritical bifurcations.

Importantly, this perspective allows one to generate parameter ensembles conditioned on bifurcation features, in direct analogy to generating parameter ensembles conditioned on experimental data. Instead of asking whether a given parameter set exhibits a bifurcation, the question is inverted: which regions of parameter space give rise to a desired bifurcation topology or dynamical phenotype? In this sense, bifurcation diagrams are treated as targets of inference, not merely diagnostic tools. By aggregating bifurcation information across the LHS ensemble, MATCONT–LHS constructs probabilistic maps of bifurcation types and stability regimes, effectively producing an ensemble bifurcation diagram. These diagrams capture both the diversity and prevalence of dynamical behaviors, uncovering previously hidden bifurcation regimes and revealing how qualitative system behavior reorganizes under parametric uncertainty. Overall, MATCONT–LHS establishes a conceptual parallel between parameter estimation and bifurcation discovery, positioning bifurcation analysis as a statistical, ensemble-driven process rather than a single-model exercise. This shift is particularly relevant for biological systems, where variability, uncertainty, and context dependence are intrinsic features rather than modeling artifacts.

The computational workflow underlying the MATCONT–LHS framework is organized into a sequence of modular steps, including parameter initialization, sampling bounds definitions, Latin Hypercube Sampling (LHS), steady-state computation, numerical continuation, branch extension, dynamical classification, and visualization of bifurcation structures. These steps are executed through the MATCONT command-line interface (CLI), allowing the bifurcation analysis to be automated across the sampled parameter ensemble. For each sampled parameter set, the ODE system is first integrated to steady state, followed by equilibrium continuation using MATCONT to identify bifurcation points such as limit points and branch points. When branch points are detected, additional continuation is performed to capture disconnected solution branches. The resulting bifurcation structures are then categorized and aggregated across the ensemble to construct statistical summaries and visual representations. The algorithm corresponding to this workflow is provided below. Full implementation details, including code blocks and parameter configurations, are provided in Section S4 of the Supporting Information SI-7.

Although the present study demonstrates EMBED using a Michaelis–Menten formulation of the dual phosphorylation–dephosphorylation cycle, the computational framework itself is not restricted to this kinetic representation. EMBED can be applied to any ODE-based biochemical model compatible with MATCONT, provided that the corresponding MATLAB function describing the governing equations is supplied to the CLI-based workflow. While the automated workflow remains unchanged, the resulting bifurcation landscapes and distributions of dynamical regimes are expected to depend on the underlying mathematical model and its associated nonlinearities.

For all parameter sets, numerical continuation was performed using a common set of MATCONT continuation settings with adaptive step-size control, automatic detection of limit points (LPs) and branch points (BPs), and branch switching at detected BP points. Parameter sets for which numerical continuation failed or terminated prematurely were automatically excluded from the ensemble statistics. The remaining parameter sets were classified according to their bifurcation topology, while the corresponding steady-state response curves were independently categorized as graded, ultrasensitive, or biphasic based on their qualitative dose–response characteristics. A complete description of the continuation settings, classification criteria, failure-handling procedure, and MATLAB implementation is provided in Section S4 of the Supporting Information S1.7.

3. Results and Discussions

3.1. Dynamics from a Single Kinetic Parameter Set

We first analyze the bifurcation structure of the dual PdP model using XPPAUT and MATCONT, which reliably detect limit points (LPs) and branch points (BPs) for a single, specific parameter set. However, BP points cannot be unambiguously classified as transcritical or pitchfork bifurcations using these tools alone. To resolve this ambiguity and rigorously characterize the nature of the bifurcations, we employ Sotomayor’s theorem. , Details of MATCONT, XPPAUT, and Sotomayor’s theorem are provided in Section (S2) of the Supporting Information SI-7.

In Section (S3) of the Supporting Information (SI-7), we apply the conditions of Sotomayor’s theorem to generic one-dimensional models of saddle-node–transcritical (SNTC) and saddle-node–pitchfork (SNPF) bifurcations to verify consistency. The conditions of the theorem are satisfied for the observed dynamics, as shown in supplementary Tables S3 and S4. The corresponding dynamical behaviors are illustrated in Supplementary Figure S1.

To perform numerical bifurcation analysis, we choose the parameters that is close to the Markevich et al. model. The parameters are provided in Supporting Information SI-7, Table S2. We fix all parameters and use to the total substrate ATOT as the bifurcation parameter. The codimension-1 bifurcation is shown in Figure A. We would like to comment here on the choice of bifurcation parameter which plays a critical role in shaping the qualitative dynamics of dual phosphorylation–dephosphorylation (PdP) systems. In our model, the total substrate concentration (ATOT) serves as the bifurcation parameter, while kinase and phosphatase levels remain fixed. This allows the system to preserve structural and mathematical symmetry, enabling the emergence of transcritical (TC), saddle-node (SN), and pitchfork (PF) bifurcations through parameter variations alone. As ATOT increases, enzyme saturation and substrate competition drive nonlinearities that shift the system from graded responses (via TC), to bistable switches (via SN), and, in symmetric cases, to PF bifurcations when two symmetric steady states lose stability simultaneously. In contrast, the Markevich model uses MAPKK_TOT (the kinase) as the bifurcation parameter, holding total MAPK and phosphatase constant. This setup introduces structural asymmetry due to complex enzyme–substrate interactions and enzyme recycling, particularly within phosphatase-mediated dephosphorylation steps. As a result, the Markevich model may not exhibit TC or PF bifurcations but may generate bistability exclusively through SN bifurcations arising from enzyme sequestration and rebinding. Thus, our model’s bifurcation behavior stems from intrinsic mass-action and Michaelis–Menten nonlinearities, while the Markevich model relies on complex enzymatic redistribution mechanisms. The use of ATOT emphasizes substrate-driven ultrasensitivity and allows spontaneous symmetry breaking, while MAPKK_TOT as a parameter reflects kinase-limited regimes and introduces indirect feedback-like behavior. Ultimately, the choice of bifurcation parameter governs the system’s symmetry, feedback structure, and nonlinearity; and therefore the types and conditions of bifurcations that can emerge. We will use the rate-balance plot, and 1D-vector-field plot in the next section to explain how a small change in the kinetic constant k ₄ results in the appearance of the SNT bifurcation. And how many steady states are created/annihilated/exchange stabilities between them.

3.2. Bifurcation Analysis, Classification, and Model Reduction

3.2.1. Numerical Bifurcation Analysis

To characterize the nonlinear dynamics of the dual phosphorylation–dephosphorylation (PdP) system, we performed numerical continuation using a hierarchy of mathematically equivalent models derived from the same biochemical reaction network. The full mechanistic model is formulated using mass-action (MA) kinetics, in which every elementary enzyme–substrate binding, dissociation, and catalytic reaction is explicitly represented. Applying the quasi-steady-state approximation (QSSA) reduces the full model to a two-dimensional (2D) Michaelis–Menten formulation by eliminating the rapidly equilibrating enzyme–substrate complexes. In this reduced formulation, the elementary kinetic rate constants are incorporated into the corresponding Michaelis constants (for example, K _m1 = (k _–1 + k ₂)/k ₁) and the maximum rates V _max1 = k ₂ KTOT. A further reduction of the 2D model yields a one-dimensional (1D) formulation from which the phase-line diagrams are constructed, providing a simple graphical interpretation of the steady states, their stability, and bifurcation points (see Supporting Information SI-7, Section S1.2 for the reduction of the model from the full MA model).

Importantly, these three formulations represent the same biochemical mechanism and are linked through systematic model reduction, rather than independent assumptions. The kinetic parameter sets used to generate the various bifurcation behaviors in the full mass-action model are consistently mapped to the corresponding parameters in the reduced models through QSSA relationships. Consequently, every parameter regime investigated produces identical steady-state branches (but not identical but similar steady states), stability changes, and bifurcation structures in the full, 2D, and 1D formulations. This agreement demonstrates that the successive reductions preserve the essential nonlinear dynamics of the PdP system and neither introduce nor eliminate the bifurcation behavior.

Using these equivalent formulations, we systematically explored the parameter space by varying selected kinetic parameters while keeping the biochemical reaction topology and governing equations unchanged in the 2D-ODE model. Consequently, Figures , , , and do not represent different mathematical models but rather different dynamical regimes arising from distinct kinetic parameter sets within the same biochemical network. Each figure follows the same analysis workflow. Panel (A) presents the codimension-1 bifurcation diagram obtained from the full mass-action model, whereas Panel (B) shows the corresponding diagram for the reduced 2D QSSA model. Panels (C) and (D) present the corresponding 1D phase-line diagrams, which provide an intuitive graphical interpretation of the steady states and their stability. Finally, Panels (E) and (F) present representative codimension-2 bifurcation diagrams illustrating the sensitivity of the observed dynamics to selected kinetic parameters. Together, these complementary analyses demonstrate that the observed saddle-node (SN), transcritical (TC), pitchfork (PF), and saddle-node–transcritical (SNTC) bifurcations arise solely from different kinetic parameter regimes rather than changes in either the reaction topology or the mathematical formulation. Below, explain in detail the various dynamics exhibited by the PdP model.

3.2.2. Saddle-Node Bifurcation (Figure )

Figure A presents the codimension-1 bifurcation diagram obtained from the full mass-action model, exhibiting classical bistability generated by two saddle-node (SN) bifurcations. Figure B shows the corresponding bifurcation diagram of the reduced 2D QSSA model, which reproduces the identical bifurcation structure, confirming that the reduction preserves the nonlinear dynamics.

Figure C,D presents the corresponding 1D phase-line diagrams derived from the reduced model. These diagrams provide a graphical interpretation of the steady states and their stabilities. As the bifurcation parameter is varied, the phase-line diagrams illustrate the transition from a monostable regime to a bistable regime and subsequently back to a monostable regime, with the tangential condition identifying the saddle-node bifurcation points.

Figure E,F presents representative codimension-2 bifurcation diagrams illustrating the dependence of the bistable region on the kinetic parameters k ₄ and K _m5, together with the bifurcation parameter ATOT. These diagrams demonstrate that both the existence and extent of bistability are highly sensitive to variations in these kinetic parameters.

3.2.3. Saddle-Node–Transcritical Interaction (Figure )

Figure A presents the codimension-1 bifurcation diagram of the full mass-action model, revealing the coexistence of a branch point (BP) and a limit point (LP), characteristic of a saddle-node–transcritical (SNTC) interaction. Figure B shows that the reduced 2D QSSA model reproduces the identical bifurcation structure, demonstrating that the SNTC dynamics are preserved after model reduction.

Figure C,D presents the corresponding 1D phase-line diagrams, illustrating the qualitative reorganization of the steady states as the bifurcation parameter varies. Unlike the pure saddle-node case, the phase-line diagrams reveal how the steady-state branches exchange through the transcritical interaction, thereby explaining the coexistence of the branch point and the limit point.

Figure E,F presents representative codimension-2 bifurcation diagrams illustrating the sensitivity of the SNTC regime to variations in the selected kinetic parameters. The positions of both the branch point and the limit point vary substantially with parameter changes, highlighting the strong dependence of this bifurcation structure on the underlying biochemical kinetics.

3.2.4. Pitchfork Bifurcation (Figure )

Figure A presents the codimension-1 bifurcation diagram of the full mass-action model exhibiting a pitchfork (PF) bifurcation. Figure B demonstrates that the reduced 2D QSSA model reproduces the identical symmetric bifurcation structure, confirming that model reduction preserves the symmetry of the underlying dynamics.

Figure C,D presents the corresponding 1D phase-line diagrams, which provide an intuitive visualization of the symmetry breaking process. As the bifurcation parameter crosses its critical value, the single stable steady state loses stability and gives rise to two symmetric stable steady states, illustrating the defining characteristic of a pitchfork bifurcation.

Figure E,F presents representative codimension-2 bifurcation diagrams illustrating the dependence of the pitchfork bifurcation on the selected kinetic parameters. These diagrams identify the regions of the parameter space over which the symmetric bifurcation structure is maintained.

3.2.5. Unfolding of the Pitchfork into an SNTC Interaction (Figure )

Figure A presents the codimension-1 bifurcation diagram obtained after perturbing the symmetric kinetic parameter set. Figure B shows that the reduced 2D QSSA model reproduces the identical bifurcation structure, confirming that the unfolding is an intrinsic property of the biochemical system rather than an artifact of model reduction.

Figure C,D presents the corresponding 1D phase-line diagrams illustrating how perturbation of the kinetic parameters breaks the underlying symmetry. The phase-line diagrams clearly show the transition from the symmetric pitchfork structure to the asymmetric saddle-node–transcritical interaction.

Figure E,F presents representative codimension-2 bifurcation diagrams illustrating the sensitivity of the unfolded bifurcation structure to the selected kinetic parameters. These diagrams show that the locations of both the limit point and the branch point vary considerably with parameter changes, demonstrating that unfolding is governed entirely by the kinetic parameter regime.

Collectively, Figures ,,, establish two important findings. First, the full mass-action model, the reduced 2D QSSA model, and the 1D phase-line formulation produce identical bifurcation structures for the same kinetic parameter sets, confirming that the successive model reductions preserve the essential nonlinear dynamics of the PdP system. Second, the emergence of saddle-node (SN), transcritical (TC), pitchfork (PF), and saddle-node–transcritical (SNTC) behaviors is governed entirely by the kinetic parameter regime while the underlying biochemical reaction topology and governing equations remain unchanged.

While numerical continuation using MATCONT and XPPAUT robustly detects singularities such as limit points (LPs) and branch points (BPs), these tools identify both transcritical and pitchfork bifurcations simply as branch points and therefore do not explicitly distinguish between these bifurcation types. Furthermore, coupled bifurcation interactions, such as the saddle-node–transcritical (SNTC) bifurcation, cannot be rigorously classified from numerical continuation alone. Consequently, although continuation accurately locates the bifurcation points and constructs the global bifurcation diagrams, a rigorous analytical framework is required to determine the precise mathematical nature of each bifurcation. This motivates the application of Sotomayor’s theorem in the following section.

3.2.6. Bifurcation Classification Using Sotomayor’s Theorem

To rigorously classify the bifurcation points identified numerically, we apply Sotomayor’s theorem, which provides necessary and sufficient conditions for codimension-1 bifurcations based on eigenvalue degeneracy and higher-order derivatives.

Before applying Sotomayor’s theorem to the biochemical PdP model, we first validated our application of the theorem using canonical one-dimensional normal forms whose bifurcation structures are analytically well established. Specifically, we considered the normal forms for saddle-node–transcritical (SNTC; $ẋ=Sx\pm a{x}^2\pm b{x}^3$ , where S, a, and b are parameters) and saddle-node–pitchfork (SNPF; $ẋ=Sx\pm a{x}^3\pm b{x}^5$ , where S, a, and b are parameters) bifurcations. These benchmark systems enabled us to gain confidence in the application of Sotomayor’s theorem before analyzing the considerably more complex biochemical PdP model. For these one-dimensional systems, we first performed the conventional linear stability analysis by explicitly determining the steady states and their stability. The bifurcation classifications obtained analytically were then independently verified using numerical methods implemented in Mathematica, confirming the expected bifurcation behavior and the corresponding nondegeneracy conditions prescribed by Sotomayor’s theorem. The complete derivations, linear stability analysis, and numerical verification are provided in the Supporting Information SI-7 (Sections S3.1 and S3.2).

Having established confidence in the methodology, we then applied the same analytical framework directly to the PdP reaction network. Unlike the canonical normal forms, where different governing equations describe different bifurcation types, the PdP system retains the same reaction topology and governing equations throughout the analysis. The distinct bifurcation behaviors (SN, TC, PF, and SNTC) arise solely from different kinetic parameter regimes. Owing to the complexity of the biochemical model, conventional analytical linear stability analysis is generally intractable. Therefore, the bifurcation classification based on Sotomayor’s theorem was verified numerically using Mathematica for representative kinetic parameter sets corresponding to each bifurcation type before interpreting the results.

The steady states used for the application of Sotomayor’s theorem were first computed independently in Mathematica for the corresponding kinetic parameter sets and bifurcation parameter values associated with each codimension-1 bifurcation. These equilibrium solutions were then verified against the steady states obtained from the numerical continuation results generated using both MATCONT and XPPAUT to ensure consistency across the three computational approaches. The validated steady states at the corresponding limit points (LPs) and branch points (BPs) were subsequently used to evaluate the nondegeneracy conditions prescribed by Sotomayor’s theorem. For completeness, the steady-state values obtained for all bifurcation cases, together with the corresponding verification, are provided in tabular form in the Supporting Information SI-7 (Table S3)

For the bistable regime (Figure ), the limit points satisfy the nondegeneracy conditions for saddle-node bifurcations (Table ), confirming that bistability arises from SN mechanisms. In the SNTC case (Figure ), the branch point satisfies the conditions for a transcritical (TC) bifurcation, while the corresponding limit point satisfies the saddle-node conditions (Table ), establishing the coexistence of SN and TC dynamics. In the pitchfork case (Figure ), the branch point satisfies higher-order degeneracy conditions, including vanishing first- and second-order terms and nonzero higher-order derivatives (Table ), distinguishing it from transcritical bifurcations. This highlights a key difference: while both PF and TC appear as branch points in numerical continuation, PF requires additional symmetry-induced degeneracy that cannot be inferred from numerical continuation tools alone.

1. Verification of Sotomayor’s Theorem Conditions for Identifying Saddle-Node (SN) Bifurcations at Representative Parameter Values for the 2D-ODE Model of PdP .

conditions	SN	point	condition values	satisfied
W T F p (APP ss, ATOT)	≠ 0	ATOT = 60.96	0.00273042 ≠ 0	yes
		ATOT = 56.455	0.000826113 ≠ 0	yes
W T [D 2 F(APP ss,ATOT)(V,V)]	≠ 0	ATOT = 60.96	0.00405055 ≠ 0	yes
		ATOT = 56.455	–0.00113208 ≠ 0	yes

^aThe table reports the evaluation of key nondegeneracy conditions at two bifurcation points (ATOT = 60.96 and 56.455), where APP _ss denotes the steady-state solution. The vectors V and W represent the right and left eigenvectors, respectively, associated with the zero eigenvalue of the Jacobian at the bifurcation point, and F _p denotes the partial derivative with respect to the bifurcation parameter (ATOT). The nonzero values of W ^T F _p and W ^T [D ² F(V,V)] confirm that the necessary conditions for a saddle-node bifurcation are satisfied at both points, thereby ruling out transcritical and pitchfork bifurcations.

2. Verification of Sotomayor’s Theorem Conditions for Identifying a Saddle-Node–Transcritical (SNTC) Interaction at Representative Parameter Values for the 2D-ODE Model of PdP .

Conditions	SN	TC	point	condition values	satisfied
W T F p (APP ss, ATOT)	≠ 0	= 0	ATOT = 84.84	–3.48 × 10–11 ≃ 0	yes (TC)
			ATOT = 60.1	0.01 ≠ 0	yes (SN)
W T [D 2 F(APP ss,ATOT)(V,V)]	≠ 0	≠ 0	ATOT = 84.84	0.00759767 ≠ 0	yes
			ATOT = 60.1	–0.00118 ≠ 0	yes
W T [DF p (x ss,pB)V]	-	≠ 0	ATOT = 84.84	0.000186368 ≠ 0	yes (TC)
			ATOT = 60.1	-	-

^aThe table reports the evaluation of conditions at two nearby bifurcation points (ATOT = 84.84 and 60.1), where APP _ss denotes the steady-state solution. The vectors V and W correspond to the right and left eigenvectors associated with the zero eigenvalue of the Jacobian, respectively, and F _p denotes differentiation with respect to the bifurcation parameter (ATOT). At ATOT = 84.84, the conditions W ^T F _p ≃ 0 and W ^T [DF _p V] ≠ 0 confirm a transcritical (TC) bifurcation, whereas at ATOT = 60.1, W ^T F _p ≠ 0 establishes a saddle-node (SN) bifurcation. The coexistence of these conditions indicates an SNTC interaction. Unlike a pitchfork (PF) bifurcation, which requires higher-order degeneracy (W ^T [D ² F(V, V)] = 0), the SNTC interaction involves coupling between SN and TC without such symmetry-induced degeneracy.

3. Verification of Sotomayor’s Theorem Conditions for Identifying a Pitchfork (PF) Bifurcation at the Representative Parameter Value ATOT = 132.7 .

Conditions	PF	point	condition values	satisfied
W T F p (APP ss, ATOT)	= 0	ATOT = 132.7	–5.69 × 10–9 ≃ 0	yes
W T [D 2 F(APP ss,ATOT)(V,V)]	= 0	ATOT = 132.7	–2.1 × 10–7 ≃ 0	yes
W T [DF p (x ss,pB)V]	≠ 0	ATOT = 132.7	0.000067 ≠ 0	yes
W T [D 3 F(x ss,pB)(V,V,V)]	≠ 0	ATOT = 132.7	–7.58 × 10–6 ≠ 0	yes

^aThe table reports the evaluation of the necessary degeneracy and nondegeneracy conditions at the bifurcation point, where APP _ss denotes the steady-state solution. The vectors V and W represent the right and left eigenvectors associated with the zero eigenvalue of the Jacobian at the bifurcation point, respectively, and F _p denotes the partial derivative with respect to the bifurcation parameter (ATOT). The near-zero values of W ^T F _p and W ^T [D ² F(V, V)] satisfy the degeneracy conditions, while the nonzero values of W ^T [DF _p V] and W ^T [D ³ F(V, V, V)] confirm the required nondegeneracy conditions for a pitchfork bifurcation. Together, these results establish the presence of a PF bifurcation and rule out saddle-node and transcritical bifurcations.

Thus, Sotomayor’s theorem plays a critical role by providing a rigorous, parameter-based classification of bifurcations. It resolves ambiguities inherent in numerical detection, enables discrimination between closely related bifurcation types, and confirms higher-order interactions such as SNTC. Without this analytical validation, branch points identified numerically could be misclassified, leading to incorrect interpretation of system dynamics.

Finally, we emphasize that the global organization of dynamical regimes reported here refers to the complete bifurcation landscape of the PdP system within the adopted mechanistic framework. Our conclusions are not inferred solely from the reduced Michaelis–Menten formulation but are consistently supported by the full mass-action model, the reduced two-dimensional (2D) QSSA model, and the corresponding one-dimensional (1D) phase-line formulation, all of which reproduce identical bifurcation structures for the same kinetic parameter sets. Thus, the observed organization of monostable, bistable, symmetric, and asymmetric dynamical regimes is an intrinsic property of the underlying PdP reaction mechanism and is preserved across successive model reductions. While any mathematical model represents an approximation of biological reality, the agreement among these independent yet mathematically equivalent formulations provides strong evidence that the identified bifurcation landscape reflects the inherent nonlinear organization of the adopted biochemical mechanism rather than an artifact of model simplification.

3.2.7. Consistency across Model Reductions and Rate-Balance Analysis

To facilitate both analysis and visualization, we considered three mathematically equivalent representations of the PdP system: (i) the full mechanistic model, (ii) a reduced two-dimensional (2D) model obtained using the quasi-steady-state approximation (QSSA), and (iii) a one-dimensional (1D) model derived from the corresponding rate-balance equation.

Across all kinetic parameter regimes examined, the full model, the reduced 2D model, and the 1D rate-balance formulation produce identical steady-state branches, stability changes, and bifurcation structures, respectively. In particular, the SN, TC, PF, and SNTC bifurcation diagrams shown in Figures ,,,, respectively, are reproduced consistently by all three formulations. This agreement demonstrates that the model reductions preserve the essential nonlinear dynamics of the biochemical system while substantially simplifying the mathematical analysis.

The QSSA-based reduction decreases the dimensionality of the governing equations while retaining the fundamental biochemical mechanisms, including enzyme saturation, substrate competition, and conservation constraints. The further reduction to a one-dimensional rate-balance representation provides an intuitive geometric interpretation of nonlinear dynamics.

The rate-balance plots are constructed by plotting the net reaction rate, $\frac{d[AP]}{dt}$ , as a function of the substrate concentration [APP]. Steady states correspond to the zero crossings of the curve, while the local slope determines their stability. Single, double, and triple intersections correspond to mono-, bi-, and multistable regimes, respectively, whereas tangential intersections identify bifurcation points. Consequently, the rate-balance representation provides a simple graphical interpretation of how steady states are created, annihilated, and exchanged stability as kinetic parameters are varied.

The complete agreement among the full mechanistic model, the reduced 2D model, and the 1D rate-balance formulation confirms that the observed bifurcation behaviors are intrinsic properties of the underlying PdP mechanism rather than artifacts introduced by model reduction. The reduced formulations therefore preserve dynamic fidelity while substantially improving analytical tractability, enabling efficient exploration, visualization, and rigorous classification of the complete bifurcation landscape.

3.2.8. Codimension-2 Bifurcations: Small Region of Important Dynamics

The 2D model consists of 11 parameters. Among these parameters, changes in k ₄ and K _m5 give rise to different bifurcations such as saddle-node, transcritical, and pitchfork. The parameter k ₄ is the kinetic rate constant for the release of kinase at the end of the second phosphorylation of the two-step dual phosphorylation. Meanwhile, K _m5 is the Michaelis–Menten constant for the first dephosphorylation reaction of the two-step dual dephosphorylation. To show the ranges of k ₄ and K _m5 for each bifurcation, we simulate the two-parameter bifurcation of k ₄ and K _m5 against ATOT, respectively. While these two-parameter diagrams provide a structured view of how bifurcation regimes are organized locally, they do not capture how frequently these regimes occur across the full parameter space. In particular, the narrowness of the regions observed here raises an important question: do these bifurcation regimes remain equally restricted when explored across large parameter ensembles? We address this question in the subsequent section using the EMBED ensemble bifurcation framework, where we systematically examine whether large-scale parameter sampling leads to similar conclusions regarding the limited extent of these dynamical regimes.

Figure E,F shows the two-parameter bifurcation with k ₄ and K _m5 against ATOT, respectively, using the parameter set for saddle-node bifurcation. Two-parameter bifurcation between k ₄ and ATOT is divided into three regions: “I” and “III” have one steady state and “I”’ has three steady states. The two-parameter bifurcation between K _m5 and ATOT is divided into three regions, where “IV” has one steady state and “I”–“III” have three steady states. The vector field at each region is shown in the subset figures of both the two-parameter bifurcations. Notably, the region corresponding to bistability (three steady states) is confined to a very narrow band in parameter space, indicating that saddle-node-driven switching requires fine-tuned kinetic balance. Even small variations in k ₄, K _m5, or ATOT readily shift the system out of this regime.

Similarly, Figure E and F are generated using the parameter set for saddle-node–transcritical (SNTC) bifurcation. The two-parameter bifurcation between k ₄ and ATOT in Figure E is the same as in Figure E, as the only differing parameter between SN and SNTC is k ₄. The two-parameter bifurcation between K _m5 and ATOT (Figure F) is divided into four regions, where “I”–“III” have three steady states and “VI” has one steady state. The vector field at each region is shown in the subset figures of both the two-parameter bifurcations. Importantly, the parameter region supporting SNTC behavior is even more restricted than that of the SN case, appearing as a thin and highly localized region in the two-parameter plane. This indicates that SNTC represents a structurally sensitive regime, where slight perturbations in parameters readily shift the system away from this behavior.

Figure E and F corresponds to the pitchfork bifurcation. The two-parameter bifurcation between K _m5 and ATOT in Figure F is the same as in Figure F, as the only differing parameter between SNTC and PF is K _m5. The two-parameter bifurcation between k ₄ and ATOT is divided into three regions: “I” and “III” have one steady state and “II” has three steady states. As in the SNTC case, the region corresponding to PF-like behavior is extremely narrow and requires near-perfect symmetry conditions in kinetic parameters.

Overall, these two-parameter bifurcation analyses show that regions corresponding to bistability and higher-order bifurcations such as SNTC and PF occupy exceedingly small and finely tuned regions of the parameter space. Consequently, even minor variations in kinetic parameters arising from intrinsic noise, experimental uncertainty, or environmental fluctuations can shift the system away from the desired dynamical regimes. This suggests that although such parameter regimes can be identified computationally, they are difficult to achieve and maintain experimentally. Whether this intrinsic narrowness persists at the level of large-scale parameter exploration is investigated in the subsequent section by using ensemble bifurcation analysis.

3.3. Bridging the Design Gap in DBTL through Ensemble Bifurcation Analysis

The Design–Build–Test–Learn (DBTL) cycle provides a structured approach for engineering biological systems, where computational models guide experimental implementation and iterative refinement. , Despite its widespread adoption, a persistent challenge lies in the design phase, where models are typically used to identify specific parameter sets that produce desired behaviors such as bistability, oscillations, or ultrasensitivity. While such approaches demonstrate theoretical feasibility, they often fail to account for the global organization of the underlying parameter space. Consequently, designs based on single or narrowly tuned parameter sets are highly sensitive to perturbations and frequently fail during experimental realization.

This limitation is particularly critical in nonlinear biochemical systems, where distinct dynamical regimes are separated by bifurcations that occupy narrow and fragile regions of parameter space. Classical studies in signaling and gene regulatory networks have demonstrated bistability, oscillations, and switch-like behavior under carefully chosen parameter conditions. ,− However, these studies typically focus on specific parameter regimes and do not quantify how prevalent or robust these behaviors are across the full parameter space. As a result, rare but mathematically valid dynamical regimes may be overrepresented in modeling studies, while more robust and experimentally accessible behaviors remain underexplored. By systematically exploring the full parameter space, as in the EMBED framework, it becomes possible to identify experimentally feasible parameter sets that not only realize these rare dynamics but also uncover parameter regimes where such behaviors persist over wider regions of parameter space.

Several studies have highlighted the role of parameter sloppiness and uncertainty in limiting predictive accuracy in systems biology models. , In such systems, large variations in parameters can produce similar outputs, while small changes in critical directions can lead to qualitative shifts in the dynamics. This anisotropic structure of the parameter space implies that desirable behaviors such as bistability or symmetry-driven bifurcations may exist only in finely tuned regions that are difficult to achieve experimentally. Indeed, experimental studies of synthetic circuits often report deviations from predicted behaviors due to context dependence, stochasticity, and unmodeled interactions. ,

To address these limitations, ensemble-based approaches have been proposed to explore parameter uncertainty and robustness in biological systems. , However, these methods primarily focus on fitting experimental data or exploring steady-state variability and do not explicitly characterize the global organization of dynamical regimes across the kinetic parameter space of a given mathematical model. In the present work, “global organization” refers to the distribution and connectivity of dynamical regimes across the admissible kinetic parameter space of the selected mathematical model. It does not imply a complete characterization of the underlying biological system, whose behavior ultimately depends on the chosen modeling framework and associated assumptions. In contrast, the framework introduced here extends ensemble concepts to dynamical systems analysis by integrating Latin Hypercube Sampling (LHS) with numerical continuation, as summarized in Figure . This enables systematic exploration of high-dimensional parameter space of the selected mathematical model and classification of the resulting dynamical behaviors based on their underlying bifurcation structure.

Within the DBTL paradigm, this approach fundamentally enhances the design phase by shifting the focus from identifying single-parameter solutions to mapping the distribution, prevalence, and robustness of dynamical regimes. Instead of asking whether a system can exhibit a particular behavior, the framework quantifies how likely it is for that behavior to occur under realistic parameter variability. This distinction is crucial for experimental design as it prioritizes parameter regimes that are both functionally desirable and robust to perturbations.

By constructing ensemble bifurcation landscapes for the selected mathematical model, the proposed EMBED framework reveals that classical bifurcation-driven behaviors such as saddle-node, pitchfork, and saddle-node–transcritical bifurcations are confined to narrow regions of parameter space, whereas robust nonsingular dynamics such as ultrasensitive and biphasic responses dominate. This observation provides a mechanistic explanation for the frequent discrepancy between theoretical predictions and experimental observations in synthetic biology.

The parameter ranges explored in this study are consistent with those of experimentally characterized distributive phosphorylation systems. Ferrell and Bhatt experimentally demonstrated that MAPK activation proceeds through a distributive dual-phosphorylation mechanism and reported Michaelis constants of approximately 300 nM and 46 nM for the first and second phosphorylation reactions, respectively, together with a phosphatase Michaelis constant of approximately 300 nM. These experimentally determined values fall within the sampling ranges considered in the present ensemble analysis (Table ), indicating that the investigated parameter space encompasses biologically plausible kinetic regimes. Although the experimental study focused on ultrasensitive steady-state responses rather than bifurcation phenomena, it provides experimental support for the biochemical mechanisms and kinetic assumptions underlying the EMBED framework.

Thus, EMBED bridges a critical gap in DBTL by providing a global structure-based mapping from parameters to dynamics. It enables the rational selection of control parameters, guides experimental perturbations, and improves the likelihood of successful implementation in the build and test phases. More broadly, this framework establishes ensemble bifurcation analysis as a key component of the design stage in DBTL, complementing existing approaches in systems and synthetic biology.

4. Mapping Dynamical Regimes and Structural Constraints via Ensemble Bifurcation Analysis

The 2D dual PdP model exhibits a diverse spectrum of dynamical behaviors arising from variations in kinetic parameters, spanning both singular (bifurcation-driven) and nonsingular responses. Using the EMBED–PdP framework, we systematically uncover and classify these behaviors across high-dimensional parameter space, revealing not only the accessible dynamics but also the structural constraints that govern their realization. The dominant codimension-1 bifurcations observed are saddle-node (SN) and pitchfork (PF), along with their coupled form, the saddle-node–transcritical (SNTC) interaction, while nonsingular behaviors include ultrasensitive, biphasic, and graded responses. Importantly, comparison between single-parameter bifurcation results and ensemble analysis reveals that while individual parameter sets suggest well-defined bifurcation structures, these represent only isolated realizations; ensemble analysis exposes how these structures persist, shift, or vanish under parameter variability, thereby identifying which kinetic parameters must be tightly constrained and which can be varied to achieve desired dynamics.

4.1. PF, SNTC, and DBTL Organizing Principle

A central organizing principle governing the dynamic landscape of the dual PdP system is the presence or absence of symmetry. Ensemble bifurcation analysis reveals that symmetry is the primary structural constraint controlling the accessibility of higher-order bifurcation phenomena. In particular, pitchfork (PF) bifurcations arise only when the phosphorylation and dephosphorylation modules possess complete kinetic symmetry, corresponding to identical enzyme abundances, Michaelis constants, and catalytic rate constants in the opposing reaction pathways. Under these conditions, the governing equations exhibit a ${\mathbb{Z}}_2$ symmetry, whereby interchanging the phosphorylation and dephosphorylation variables leaves the system invariant, producing the characteristic supercritical PF bifurcation shown in Figure I.

7.

Pitchfork (PF) bifurcation and associated dynamical regimes obtained by varying kinetic parameters. (I) A supercritical pitchfork bifurcation arises when symmetry constraints are imposed on the kinetic parameters, namely, KTOT = PTOT, K _m1 = K _m5, K _m3 = K _m7, k ₂ = k ₆, and k ₄ = k ₈. The parameter values used are indicated in the figure and is given in Table (PF/SNTC). In this panel, only K _m1 and K _m5 are varied. (II) Biphasic dynamics is observed when the parameters KTOT, K _m7, k ₆, k ₄, and k ₈ are varied while all other parameters remain fixed as in panel (I). (III) Ultrasensitive/graded dynamics emerge when the parameters PTOT, K _m3, and k ₂ are varied while keeping the remaining parameters the same as in panel (I). (IV) Increasing the variation range of K _m5, while keeping all other parameters identical to panel (I), leads to the coexistence of pitchfork and saddle-node–transcritical (SNTC) bifurcations with LP < BP. (V) Increasing the variation range of KM1 under the same baseline parameter set produces both pitchfork and SNTC bifurcations with LP > BP. The network diagram in the center highlights the parameters varied relative to the baseline set in panel (I). Parameters are given in colored boxes according to the dynamical behavior induced by their variation: yellow indicates SNTC with LP > BP, red indicates SNTC with LP < BP, blue denotes graded/ultrasensitive responses, and green corresponds to biphasic dynamics. For each case, 500 samples were generated; however, for clarity of representation, only 5–10 samples are shown. Increasing the number of samples did not alter the relative proportions of the observed dynamical behaviors. Notably, variations in most parameters predominantly give rise to biphasic and ultrasensitive dynamics. The MATLAB code EMBED is given in the supplementary (SI-4).

The symmetric operating point requires

$$\mathrm{K}\mathrm{T}\mathrm{O}\mathrm{T}=\mathrm{P}\mathrm{T}\mathrm{O}\mathrm{T},K_{m1}=K_{m5},K_{m3}=K_{m7},k_2=k_6,k_4=k_8$$

with the corresponding parameter values listed in Table (PF/SNTC). These constraints define a lower-dimensional symmetry manifold within the kinetic parameter space, indicating that PF bifurcations are highly specialized rather than generic dynamical behaviors.

Figure examines the consequences of progressively breaking this symmetry. Panel I represents the symmetric reference system, in which every sampled parameter set preserves the PF bifurcation. Panels II–V systematically relax individual symmetry constraints while maintaining the remaining baseline parameters. The central network schematic identifies the perturbed kinetic parameters, where green denotes biphasic responses, blue indicates ultrasensitive or graded responses, red corresponds to saddle-node–transcritical (SNTC) bifurcations with LP < BP, and yellow represents SNTC bifurcations with LP > BP. Each perturbation experiment consists of an ensemble of 500 independently sampled parameter sets, with only representative realizations shown for clarity.

Most symmetry breaking perturbations do not preserve the PF bifurcation. Instead, perturbations of KTOT, K _m7, k ₆, k ₄, and k ₈ predominantly generate biphasic responses (Figure II), whereas perturbations of PTOT, K _m3, and k ₂ primarily produce ultrasensitive or graded dynamics (Figure III). These nonsingular responses occupy substantially larger regions of parameter space than the symmetric PF regime, demonstrating that exact symmetry is highly sensitive to kinetic perturbations and that smooth nonlinear responses constitute the generic operating mode of the PdP network.

The Michaelis constants K _m1 and K _m5 play a unique structural role in controlling symmetry breaking. Expanding the admissible variation of K _m5 while maintaining the remaining symmetry conditions generates the coexistence of PF and SNTC bifurcations with LP < BP (Figure IV), whereas increasing the variation of K _m1 produces PF and SNTC bifurcations with LP > BP (Figure V). These results demonstrate that substrate saturation governs the unfolding of the symmetric PF bifurcation into the asymmetric SNTC structures. From the perspective of singularity theory, the PF bifurcation exists only on the symmetry manifold, whereas symmetry breaking separates the degenerate bifurcation into interacting saddle-node (SN) and transcritical (TC) bifurcations. Consequently, the PF bifurcation represents a limiting symmetric case of the broader SNTC family rather than an independently robust dynamic regime.

From the Design–Build–Test–Learn (DBTL) perspective, Figure provides a quantitative design map identifying the kinetic parameters that govern structural topology and those that primarily tune response thresholds. In particular, KM1 and K _m5 act as structural control parameters whose perturbation fundamentally reorganizes the steady-state manifold, whereas several other kinetic parameters mainly modulate quantitative operating characteristics while preserving the qualitative dynamics. Thus, EMBED extends conventional continuation analysis by identifying not only mathematically admissible bifurcations but also the parameter regions in which they remain experimentally accessible and robust under biological variability, providing practical design guidance for engineering biochemical networks.

4.2. SN Bifurcation and DBTL Perspective

Unlike the PF bifurcation, which requires strict global symmetry of the phosphorylation and dephosphorylation modules, saddle-node (SN) bifurcations arise under conditions of partial symmetry in which the nonlinear balance between phosphorylation and dephosphorylation fluxes generates multiple steady states. Geometrically, the SN bifurcation corresponds to the formation of a folded steady-state manifold in which a stable and an unstable equilibrium coalesce at a limit point (LP), producing the characteristic S-shaped bistability response. Although SN bifurcations are generally more accessible than PF bifurcations, the present ensemble analysis demonstrates that their occurrence remains strongly constrained by the underlying kinetic parameter space.

Figure systematically investigates the robustness of the S-shaped SN bifurcation through controlled perturbations around a baseline bistable parameter set. Panel I shows the reference system, obtained by allowing the Michaelis constants K _m1 and K _m5 to vary over a wider range while maintaining the remaining parameters at the values listed in Table (S-SN). The resulting folded steady-state manifold generates the classical S-shaped bifurcation diagram, with two stable branches separated by an unstable intermediate branch. The remaining panels progressively perturb selected kinetic parameters, as summarized by the central network diagram, where purple denotes S-shaped SN bistability, blue indicates ultrasensitive or graded responses, and green represents biphasic dynamics. Each perturbation experiment consists of an ensemble of 500 independently sampled parameter sets with representative realizations shown for clarity.

8.

Saddle-node (S-shaped) bifurcation and different dynamical regimes obtained by varying specific kinetic parameters in the network. (I) An S-shaped saddle-node (SN) bifurcation is observed when the parameters K _m1 and K _m5 are varied over a wider range while keeping the remaining parameters fixed at the values indicated in the panel and is given in Table (S-SN). (II) Biphasic response dynamics arises when the parameters K _m5, K _m7, k ₆, and k ₈ are varied while the remaining parameters are kept the same as in panel (I). (III) Ultrasensitive/graded (US) dynamics is obtained by varying PTOT, K _m3, k ₂, and k ₄ while the other parameters remain fixed as in panel (I). (IV) When the parameter KTOT is varied over a broader range, the system exhibits a combination of S-shaped saddle-node bifurcation and biphasic dynamics. The schematic network in the center highlights the parameters that are varied relative to the baseline parameter set. The colored boxes indicate the type of dynamical behavior produced when those parameters are varied: purple corresponds to S-shaped saddle-node/bistable dynamics, blue indicates ultrasensitive (US) responses, and green represents biphasic (BI) behavior. For each case, 500 samples were generated; however, for clarity of representation, only 5–10 samples are shown. Increasing the number of samples did not alter the relative proportions of the observed dynamical behaviors. Notably, variations in most parameters predominantly give rise to biphasic and ultrasensitive dynamics. The MATLAB code EMBED is given in the Supporting Information SI-4.

The ensemble analysis reveals that the bistable SN structure is considerably more fragile than that suggested by an individual continuation diagram. Perturbations of K _m5, K _m7, k ₆, and k ₈ predominantly collapse the folded steady-state manifold into biphasic responses (Figure II), whereas variations in PTOT, K _m3, k ₂, and k ₄ primarily generate ultrasensitive or graded dynamics (Figure III). In both cases, the disappearance of the limit points eliminates bistability while preserving the underlying nonlinear response, indicating that ultrasensitive and biphasic behaviors represent the dominant operating modes, once the balance required for bistability is disrupted.

An interesting exception is observed when the admissible range of the total kinase concentration KTOT is expanded (Figure IV). In this case, both classical S-shaped SN bifurcations and biphasic responses coexist, demonstrating that enzyme abundance influences the accessibility of bistability without completely destroying the folded manifold. This distinction highlights the difference between structural robustness, which governs the existence of the bistable manifold, and threshold robustness, which determines the location of the switching thresholds while preserving their qualitative topology.

Mechanistically, these results identify substrate saturation as the primary driver of the bistability in dual PdP systems. The Michaelis constants K _m1 and K _m5 act as structural control parameters that determine the existence of the folded manifold, whereas several other kinetic parameters mainly regulate quantitative response characteristics, such as activation thresholds or response steepness, without altering the overall topology of the steady-state manifold.

From the Design–Build–Test–Learn (DBTL) perspective, Figure provides quantitative guidance for engineering robust biochemical switches. Rather than identifying isolated bistable parameter sets, EMBED distinguishes the kinetic parameters that must be tightly controlled to preserve bistability from those that primarily tune its operating characteristics. This landscape-based perspective enables rational prioritization of experimental design by focusing on parameter regions that maximize both the robustness and experimental accessibility of bistable behavior.

4.2.1. Z-Shaped

Although the S-shaped SN bifurcation shown in Figure represents the canonical realization of bistability in dual PdP systems, ensemble bifurcation analysis further reveals an alternative manifestation of the same underlying nonlinear equilibrium manifold, namely, the Z-shaped saddle-node (Z-SN) bifurcation (Figure ). Although the S-shaped and Z-shaped bifurcation diagrams differ in their visual appearance, both originate from the same fundamental mechanism: the formation of a folded steady-state manifold through the nonlinear balance between the phosphorylation and dephosphorylation fluxes. Their differences primarily reflect the geometric arrangement of the equilibrium branches and the ordering of the associated limit points rather than distinct biochemical mechanisms.

9.

Z-shaped saddle-node (Z-SN) bifurcation and associated dynamical regimes obtained by varying specific kinetic parameters in the network. (I) A Z-shaped saddle-node bifurcation is observed when the parameters K _m1 and K _m5 are varied over the indicated ranges while the remaining parameters are fixed as shown in the panel and is given in Table (Z-SN). (II) Biphasic (BI) dynamics emerges when the parameter k ₆ is varied while keeping the remaining parameters the same as in panel (I). (III) Ultrasensitive/graded (US) dynamics is obtained by varying the parameters k ₂ and k ₄, with all other parameters fixed as in panel (I). (IV) Increasing the variation range of KTOT, K _m1, K _m7, and k ₄ leads to the coexistence of Z-shaped saddle-node bifurcation and biphasic response. (V) Varying PTOT, K _m3, and k ₆ produces a transition between Z-SN and graded/ultrasensitive regimes, depending on the parameter ranges considered. The schematic network in the center highlights the parameters that are varied relative to the baseline parameter set used in panel (I). Parameters are given in colored boxes according to the type of dynamical response obtained upon variation: brown denotes Z-SN/US behavior, purple corresponds to Z-SN/BI regimes, blue indicates ultrasensitive (US) responses, and green represents biphasic (BI) dynamics. For each case, 500 samples were generated; however, for clarity of representation, only 5–10 samples are shown. Increasing the number of samples did not alter the relative proportions of the observed dynamical behaviors. Notably, variations in most parameters predominantly give rise to biphasic and ultrasensitive dynamics. The MATLAB code EMBED is given in the supplementary (SI-4).

Figure examines the robustness of the Z-shaped bifurcation by perturbing selected kinetic parameters around a baseline parameter set supporting the folded manifold (Figure I). As in the S-shaped analysis, the Z-SN bifurcation is generated primarily by varying the Michaelis constants K _m1 and K _m5, while the remaining parameters are maintained at the values listed in Table (Z-SN). The reference solution served as the organizing center for the subsequent ensemble perturbation analysis.

The perturbation experiments reveal behavior closely parallel to that of the S-shaped bifurcation. Perturbation of the catalytic constant k ₆ predominantly transforms the bistable response into biphasic dynamics (Figure II), whereas variations in k ₂ and k ₄ eliminate the folded manifold and generate ultrasensitive or graded responses (Figure III). Thus, regardless of the orientation of the folded equilibrium surface, disruption of the kinetic balance redirects the system toward smooth nonlinear-response regimes.

A notable feature of the Z-shaped manifold is its coexistence with nonsingular responses over extended parameter ranges. Expanding the admissible ranges of KTOT, K _m1, K _m7, and k ₄ generates ensembles containing both Z-SN and biphasic dynamics (Figure IV), whereas perturbations of PTOT, K _m3, and k ₆ produce transitions between Z-SN and ultrasensitive or graded responses (Figure V). The central network schematic summarizes these perturbations, with brown boxes denoting Z-SN/US transitions, purple indicating Z-SN/BI coexistence, blue representing ultrasensitive responses, and green corresponding to biphasic behavior. Each perturbation experiment consists of an ensemble of 500 independently sampled parameter sets with representative realizations shown for clarity.

Comparison of Figures and highlights a common principle governing bistability in PdP systems. Despite their different geometries, both S-shaped and Z-shaped bifurcations require the formation of a folded steady-state manifold generated through balanced enzyme saturation in opposing phosphorylation and dephosphorylation pathways. Consequently, both depend critically on the Michaelis constants controlling substrate saturation and exhibit a similar sensitivity to perturbations of enzyme abundance and catalytic activity.

From the Design–Build–Test–Learn perspective, Figure complements the S-shaped analysis by demonstrating that different manifestations of bistability are governed by the same structural constraints. Rather than treating S-shaped and Z-shaped bifurcations as independent design objectives, EMBED identifies the common kinetic parameters responsible for both behaviors and quantifies their accessibility across the global parameter space. Together, Figures and show that the primary challenge is not establishing the mathematical existence of bistability but identifying experimentally accessible parameter regions capable of sustaining bistable behavior under realistic biological variability.

4.3. Overall Observations without Constraining the Parameters: Implications in Designing Experiments and DBTL Cycle

While Figures ,, elucidate the mechanisms underlying individual bifurcation phenomena, they do not address how frequently these behaviors occur across the complete kinetic parameter space. EMBED addresses this question by performing large-scale ensemble continuation over independently sampled parameter sets, thereby quantifying the prevalence of competing dynamical regimes.

The ensemble statistics in Figure reveal a clear and consistent trend. Across all sampling strategies, codimension-1 bifurcations occupy only a small fraction of the accessible parameter space, whereas nonsingular responses overwhelmingly dominate. Under constrained sampling (Figure A,B), only 4 of the 500 sampled parameter sets exhibit saddle-node bifurcations, while the remaining 496 generate ultrasensitive, graded, or biphasic responses. Notably, no pitchfork or saddle-node–transcritical bifurcations are observed despite the parameter ranges encompassing the symmetric configurations identified in Figures ,,. When the parameter space is expanded to broader unbiased sampling (Figure C,D), the number of saddle-node bifurcations decreases further to only 2 among 500 parameter sets, whereas nonsingular responses increase to 498. The consistency of these observations across both constrained and unconstrained sampling indicates that the rarity of bifurcation-driven dynamics is an intrinsic property of the PdP network rather than an artifact of the sampling strategy.

10.

Prevalence of ultrasensitive and biphasic dynamics across parameter ensembles. Two sampling strategies were used to explore the dynamical landscape of the PdP model. (A,B) correspond to the constrained sampling scenario in which minimum and maximum values across the previous configurations were taken as given in Table (Common Min–Max). (A) Representative bifurcation diagrams obtained from the sampled parameter sets. Only a small fraction of the parameter sets exhibit saddle-node behavior, while the majority display graded/ultrasensitive or biphasic steady-state responses. (B) Statistical summary of the ensemble classification showing the number of parameter sets exhibiting LP-only (SN), BP-only, LP + BP, or no singularities. The results indicate that ultrasensitive and biphasic responses (496) dominate the parameter space, with only a few parameter sets producing saddle-node bifurcations (4) and no occurrences of pitchfork (PF) or saddle-node–transcritical (SNTC) bifurcations. (C,D) correspond to the unbiased sampling scenario in which all model parameters were varied over broader ranges as given in Table (full variation). (C) Ensemble bifurcation diagrams obtained under this expanded parameter exploration, again showing occasional limit points corresponding to SN bifurcations. (D) Statistical distribution of dynamical behaviors across the ensemble. Compared to the constrained sampling case, the occurrence of saddle-node bifurcations (2) decreases further as the parameter ranges increase, while ultrasensitive and biphasic responses (498) remain dominant. In both the cases, 500 parameter sets were generated. The MATLAB code EMBED is given in the Supporting Information SI-4.

5. Bridging Experimental Kinetics and Predicted Dynamical Regimes

One of the primary motivations of the proposed EMBED framework is to bridge the computational bifurcation analysis with experimentally realizable biochemical systems. While the previous sections characterized the global dynamic landscape of the PdP network, it is equally important to determine whether these predicted behaviors remain consistent with experimentally measured phosphorylation kinetics. To address this question, experimentally reported kinetic parameters were incorporated into the EMBED workflow and progressively expanded to investigate how experimentally accessible perturbations reorganized the underlying dynamical landscape.

Experimentally measured kinetic parameters for distributive MAPK phosphorylation were compiled from Ferrell and Bhatt and Witzel and Blüthgen. Ferrell and Bhatt reported Michaelis constants of approximately 300 nM for the first phosphorylation reaction, 46 nM for the second phosphorylation reaction, and approximately 300 nM for phosphatase-mediated dephosphorylation, while Witzel and Blüthgen provided experimentally grounded enzyme concentrations and catalytic rate constants for a mechanistic MAPK model. Together, these values define biologically realistic constraints for the reduced Michaelis–Menten model used in this study. All literature-derived parameters lie within the broader sampling ranges explored by EMBED (Table ).

While Figures ,, establish the mechanistic origins of the different bifurcation classes, Figure quantifies their prevalence across the global kinetic parameter space using large-scale ensemble sampling. The analysis reveals that the PdP network is intrinsically dominated by graded, ultrasensitive, and biphasic responses, whereas saddle-node, pitchfork, and saddle-node–transcritical (SNTC) bifurcations occupy only small regions of parameter space. Importantly, this statistical organization is consistently recovered under both constrained and broad parameter sampling, indicating that the rarity of higher-order bifurcations is an inherent property of the biochemical network rather than a consequence of the sampling strategy. Figure provides an experimental validation of these theoretical predictions by applying the EMBED framework to experimentally measured kinetic parameters.

11.

Experimentally guided parameter exploration and emergence of distinct bifurcation regimes in the dual phosphorylation–dephosphorylation (PdP) model using EMBED. Each row corresponds to a different parameter sampling strategy and consists of three components: (left) parameter sampling bounds, (middle) representative bifurcation diagrams generated from at most 10 representative parameter sets, and (right) the distribution of dynamical classes obtained from the complete ensemble. In the parameter-bound panels, the red stars denote literature-derived reference values, , while the horizontal bars indicate the sampling ranges used for each ensemble. Gray bars represent the initial wide-range sampling intervals, whereas blue, yellow, and pink bars denote the progressively refined sampling ranges used for the S-shaped, Z-shaped, and PF/SNTC-enriched ensembles, respectively. For visualization, the horizontal extent of each parameter row is normalized to the corresponding wide-range interval for that parameter. Consequently, bar lengths should be interpreted relative to the admissible range of each individual kinetic parameter rather than compared directly across different parameters. (A) Experimental parameter set. Literature-derived values of KTOT, PTOT, K _m1, K _m3, K _m5, K _m7, k ₂, k ₄, k ₆, and k ₈ compiled from distributive MAPK phosphorylation studies , were used directly. The resulting ensemble exhibits only graded, ultrasensitive, and biphasic responses. (B) Wide parameter exploration. Literature-derived reference values were embedded within broader biologically plausible sampling ranges to perform an unbiased ensemble exploration. Rare S-shaped and Z-shaped saddle-node bifurcations emerge in addition to the dominant ultrasensitive and biphasic responses, whereas PF and SNTC bifurcations remain absent. (C) S-enriched sampling. Parameter ranges were narrowed around representative S-shaped solutions identified from the wide exploration, increasing the occurrence of S-shaped saddle-node bifurcations while preserving the remaining dynamical behaviors. (D) Z-enriched sampling. Parameter ranges were similarly centered around representative Z-shaped solutions obtained from the wide exploration, enriching the occurrence of Z-shaped saddle-node bifurcations. (E) PF/SNTC-enriched sampling. Symmetry-constrained parameter sampling was performed around representative parameter regions supporting pitchfork and saddle-node–transcritical bifurcations, resulting in enrichment of these higher-order bifurcation structures. The parameter-bound panels illustrate the progressive refinement of the sampling ranges relative to the initial wide-range parameter space, beginning from literature-derived reference values and culminating in targeted sampling strategies that enrich specific bifurcation regimes. The bifurcation panels show representative continuation curves for each ensemble, while the histograms summarize the frequencies of all seven dynamical classes across the complete parameter ensemble (1000 parameter sets for Panels A, C, D, and E; 5000 parameter sets for Panel B). Collectively, the figure demonstrates that experimentally supported kinetic parameters predominantly occupy robust ultrasensitive and biphasic regimes, whereas progressively richer bistable and codimension-2 bifurcation structures emerge only through systematic perturbation of experimentally accessible kinetic parameters using the EMBED framework. The parameter values are given in Table , and the MATLAB code EMBED is given in the Supporting Information SI-4.

Figure A shows the ensemble obtained using only experimentally reported kinetic parameters. Under these experimentally constrained conditions, all sampled parameter sets exhibit graded, ultrasensitive, or biphasic responses, with no saddle-node, pitchfork, or saddle-node–transcritical (SNTC) bifurcations. These observations agree well with the experimental studies of distributive MAPK phosphorylation, which reported robust ultrasensitive steady-state responses under physiologically relevant enzyme concentrations rather than bistable or higher-order bifurcation phenomena. ,

To investigate whether additional dynamic behaviors become accessible within experimentally plausible kinetic regimes, the sampling intervals were expanded while retaining the experimentally reported values within the sampled ranges. As shown in Figure B, ultrasensitive and biphasic responses continue to dominate the ensemble, while only a small number of S-shaped and Z-shaped saddle-node bifurcations emerge. Consistent with the global ensemble statistics in Figure , neither pitchfork nor SNTC bifurcations are observed, indicating that these higher-order bifurcations occupy substantially smaller regions of the admissible parameter space.

The rare S-shaped and Z-shaped parameter sets identified during the wide-range exploration were subsequently used as seed regions for targeted local sampling. Figure C,D progressively enriches the S-shaped and Z-shaped bistable regimes by locally expanding the parameter ranges around representative solutions. These targeted searches demonstrate that once a rare dynamical region is identified, EMBED can efficiently characterize the surrounding parameter landscape and identify neighboring parameter combinations that preserve desired qualitative dynamics.

Finally, Figure E investigates parameter regions supporting pitchfork and saddle-node–transcritical (SNTC) bifurcations. Symmetry-constrained sampling was performed by imposing KTOT = PTOT, K _m3 = K _m7, k ₂ = k ₆, and k ₄ = k ₈, while allowing K _m1 and K _m5 to vary independently. This broken-symmetry condition enriches the occurrence of SNTC bifurcations. Parameter sets that additionally satisfy the symmetry condition K _m1 = K _m5 recover the fully symmetric case and consequently exhibit pitchfork bifurcations.

Collectively, Figure reveals a hierarchical organization of the PdP parameter space. Experimentally characterized parameter values occupy regions dominated by robust-graded, ultrasensitive, and biphasic responses. Broad exploration of biologically plausible parameter ranges uncovers rare bistable regimes, while targeted local sampling progressively enriches S-, Z-, SNTC-, and pitchfork bifurcations. These results demonstrate that higher-order bifurcations are not excluded by the underlying biochemical mechanism but occupy increasingly restricted regions of the admissible kinetic parameter space.

Importantly, the perturbed parameters investigated here correspond to experimentally controllable biochemical quantities rather than abstract mathematical variables. The Michaelis constant (KM) reflects the effective affinity between an enzyme and its substrate and can be modified experimentally through site-directed mutagenesis, engineering of docking interactions, competitive inhibitors, substrate analogs, scaffold proteins, or post-translational modifications that alter enzyme–substrate binding affinity. ,, Likewise, total kinase and phosphatase concentrations (KTOT and PTOT) can be systematically varied using inducible promoters, gene overexpression, CRISPR-mediated transcriptional regulation, RNA interference, or targeted protein degradation strategies. , Consequently, the parameter perturbations explored by EMBED correspond directly to experimentally accessible biochemical manipulations.

Taken together, Figure demonstrates how experimentally measured kinetic parameters provide biologically realistic starting points for systematic exploration of biochemical parameter space. Using experimentally validated kinetic values, EMBED progressively identifies broader admissible regions, isolates rare dynamical behaviors, and enriches parameter regimes supporting higher-order bifurcations. Rather than relying on exhaustive experimental trial-and-error, the framework identifies which experimentally controllable kinetic parameters should be perturbed, the direction of perturbation, and the approximate parameter ranges required to access specific dynamical regimes. EMBED helps identify experimentally relevant parameter regimes for designing phosphorylation–dephosphorylation systems within the Design–Build–Test–Learn (DBTL) paradigm.

6. Discussion and Conclusions

In this work, we developed EMBED, an ensemble bifurcation framework that transforms conventional MATCONT-based analysis into a scalable, high-throughput approach for mapping the global dynamical landscape of dual phosphorylation–dephosphorylation (PdP) systems. By integrating large-scale parameter sampling with automated continuation, EMBED overcomes the inherent limitations of single-parameter bifurcation analysis, which captures only a narrow slice of system behavior. In contrast, ensemble analysis reveals how bifurcation structures distribute across parameter uncertainty, showing that features appearing sharp and well defined in local analyses are often sparse and fragile in the full parameter space. Importantly, despite the high dimensionality and large ensemble sizes, the computational cost remains remarkably low, enabling rapid, on-the-fly exploration of parameter space without compromising the rigor of numerical continuation. Sotomayor’s theorem provides the theoretical basis for interpreting these results by classifying nonhyperbolic steady states and distinguishing transcritical (TC), pitchfork (PF), and saddle-node–transcritical (SNTC) bifurcations. Our analysis shows that many apparent PF-like behaviors arise from structurally sensitive interactions between SN and TC bifurcations and readily unfold under parameter perturbations. Thus, these bifurcations act as organizing centers rather than robust dynamical regimes.

A key finding is that bifurcation-driven behaviors in PdP systems occupy extremely narrow and finely tuned regions of parameter space. Two-parameter bifurcation analysis reveals thin regime boundaries, while ensemble analysis further demonstrates that these regions have low measure and are statistically rare. Consequently, small parameter variationsarising from biological variability or experimental uncertaintyreadily shift the system away from these regimes. In contrast, nonsingular dynamics such as ultrasensitive, biphasic, and graded responses dominate across biologically relevant conditions, highlighting their intrinsic robustness.

This provides a mechanistic explanation for the long-standing discrepancy between the model predictions and experimental observations. Although classical models predict SN, TC, and PF bifurcations, these behaviors are difficult to realize experimentally because of parameter sloppiness, structural model approximations, and the limited volume of bifurcation-supporting regions. EMBED quantitatively resolves this gap by linking theoretical possibilities to the experimentally accessible dynamics.

It is important to note that the dynamical landscape reported here is interpreted within the adopted mechanistic PdP framework rather than as a universal property of all phosphorylation–dephosphorylation systems. As discussed in Section , the reduced two-dimensional Michaelis–Menten formulation was deliberately chosen because its effective kinetic parameters, namely, V _max and K _M , are experimentally measurable and directly amenable to biochemical perturbation, whereas many elementary rate constants in the full mass-action formulation are difficult to estimate or manipulate experimentally. Importantly, we demonstrated that the reduced model reproduces the same bifurcation structures as the full mass-action model, thereby preserving the essential nonlinear dynamics while providing a more experimentally tractable parametrization. Consequently, the ensemble bifurcation landscapes obtained here identify experimentally feasible kinetic regimes within the adopted PdP model and provide quantitative guidance for realizing specific dynamical behaviors, while recognizing that the precise locations and extents of these regimes may vary across alternative mechanistic models, parameter priors, or biological systems.

Beyond the specific PdP application, the principal contribution of EMBED is methodological. Unlike conventional ensemble modeling and global sensitivity analysis, which primarily quantify parameter sensitivity, uncertainty propagation, or statistical variability in model outputs, and parameter-space mapping approaches that identify regions associated with particular responses, EMBED explicitly characterizes the underlying bifurcation structure across an ensemble of kinetic parameter sets. By integrating large-scale parameter sampling, automated numerical continuation, systematic model reduction, codimension-1 and codimension-2 bifurcation analyses, and rigorous analytical classification using Sotomayor’s theorem, the framework transforms classical bifurcation analysis from a local, single-parameter diagnostic tool into a global, ensemble-based methodology. This enables not only the identification of distinct dynamical regimes but also the quantification of their prevalence, accessibility, and robustness within the admissible kinetic parameter space. Consequently, EMBED provides a practical framework for identifying experimentally feasible kinetic regimes that realize desired dynamical behaviors, thereby establishing a direct link between nonlinear dynamical systems analysis and model-guided experimental design within the DBTL paradigm.

Recent advances in agentic artificial intelligence (AI) further highlight the potential of mechanistic frameworks such as EMBED for autonomous scientific discovery. Unlike conventional AI approaches that primarily perform prediction or optimization, agentic AI systems are designed to iteratively reason, formulate hypotheses, select informative computational or experimental actions, and continuously refine their decisions based on newly acquired evidence. In this context, the ensemble bifurcation landscapes generated by EMBED provide substantially richer information than individual continuation diagrams by quantifying the prevalence, robustness, and accessibility of competing dynamical regimes across the admissible parameter space. Such mechanistic landscape information could serve as an interpretable knowledge base for future autonomous Design–Build–Test–Learn (DBTL) workflows, enabling intelligent agents to prioritize experimentally accessible and robust operating regimes rather than searching blindly through high-dimensional parameter spaces. Although the present work focuses on the development of the EMBED framework itself, integrating ensemble bifurcation analysis with emerging agentic AI systems represents a promising direction toward more interpretable and autonomous model-guided biological design.

From a DBTL perspective, the EMBED framework bridges the gap between model-based design and experimental implementation by enabling global exploration of parameter space and identification of robust dynamical regimes. Rather than focusing on isolated parameter sets, the framework facilitates the selection of parameter regimes that are both functionally relevant and robust to variability. By revealing that bifurcation-driven behaviors are intrinsically rare and confined to narrow regions, while nonsingular dynamics dominate, this work provides a mechanistic explanation for the persistent discrepancy between computational predictions and experimental outcomes. Furthermore, the framework highlights how the choice of control parameterssuch as ATOT, KTOT, or PTOTconstrains accessible dynamics, establishing control parameter selection as a key design axis. The ability to perform rapid, ensemble-scale bifurcation analysis enables efficient in silico iteration of design–test cycles, reducing reliance on finely tuned parameter choices and accelerating the overall DBTL workflow. As such, EMBED offers a scalable and generalizable strategy for the rational design of biochemical networks across signaling and gene regulatory systems. Overall, EMBED establishes ensemble bifurcation analysis as a necessary extension of classical bifurcation theory, converting it from a local diagnostic tool into a global, design-oriented framework. While demonstrated for PdP systems, the approach is general and applicable to broader signaling and gene regulatory networks. Future work will incorporate intrinsic stochasticity, where noise is expected to further blur bifurcation boundaries and reinforce the dominance of robust dynamical regimes.

1.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.6c05467.

• Mathematical models, source codes, analytical calculations, and Supporting Information used in this study; (SI-1) DualPdP2D.txt XPPAUT ODE file for the reduced two-dimensional Michaelis–Menten PdP model; (SI-2) DualPdPFull.txt XPPAUT ODE file for the full mechanistic mass-action PdP model; (SI-3) DualPdPVF.m MATLAB script used to generate the vector-field plots of the reduced one-dimensional PdP model (Figures 3C,D, 4C,D, 5C,D, and 6E,F); (SI-4) Ensemble2D.m MATLAB implementation of the EMBED framework for Latin Hypercube Sampling (LHS), MATCONT-based numerical continuation, automatic bifurcation detection, and ensemble bifurcation analysis (Figures 7–11); (SI-5) SotomayersConditions.nb Mathematica notebook implementing Sotomayor’s theorem for evaluating and classifying saddle-node, transcritical, and pitchfork bifurcations; (SI-6) DualPdPJacobiansigns.nb Mathematica notebook used to derive the Jacobian matrix and evaluate the Jacobian sign structure of the reduced two-dimensional PdP model; (SI-7) Supplementary.pdf Supplementary document containing the full mechanistic model, derivation of the reduced two-dimensional model, generic one-dimensional SNTC and SNPF models, Sotomayor conditions, EMBED workflow, Jacobian calculations, additional tables, and supplementary figures; (SI-8) Generic1DS1.m MATLAB script used to generate the generic one-dimensional normal-form bifurcation diagrams and vector-field plots presented in Supplementary Figure S1; (SI-9) MAPK2D.m MATLAB function implementing the reduced two-dimensional PdP model used by Ensemble2D.m for MATCONT continuation; and (SI-10) (PDF)

• SupportingInformation2.zip (ZIP)

• README file describing the contents of all supporting source files and providing usage instructions (TXT)

The authors declare no competing financial interest.