Bifurcations and multistability in inducible three-gene toggle switch networks

TL;DR

The study investigates inducible multistability in a three-gene toggle switch by embedding allosteric induction via the Monod-Wyman-Changeux model, linking external effector concentrations to active transcription-factor levels. It develops a baseline symmetric model, analyzes single- and multi-inducer regimes to map tristable, bistable, and monostable phases, and reveals how bifurcation thresholds depend on cooperativity and inducer geometry. By adding self-activation and exploring competitive versus non-exclusive binding, the work shows that the mechanistic interpretation of effector action—whether it controls activity or protein function—drastically alters the dynamical landscape and accessible stable states. These findings provide a framework for experimentally tuning inducible multistability, inform parameter inference from expression data, and guide design principles for synthetic gene circuits with robust, tunable fate decisions.

Abstract

Control of transcription presides over a vast array of biological processes, including those mediated by gene regulatory circuits that exhibit multistability. Within these circuits, two- and three-gene network motifs are particularly critical to the repertoire of metabolic and developmental pathways. Theoretical models of these circuits, however, often vary parameters such as dissociation constants, transcription rates, and degradation rates without specifying precisely how these parameters are controlled biologically. In this study, we examine the role of effector molecules, which can alter the concentrations of the active transcription factors that control regulation, and are ubiquitous in regulatory processes across many biological settings. We specifically consider allosteric regulation in the context of extending the standard bistable switch to three-gene networks, and explore the rich multistable dynamics exhibited in these architectures as a function of effector concentrations. We then analyze how the dynamics evolve under various interpretations of regulatory circuit mechanics, underlying inducer activity, and perturbations thereof. Notably, the biological mechanism by which we model effector control over dual-function proteins transforms not only the phenotypic trend of dynamic tuning but also the set of available dynamic regimes. In this way, we determine key parameters and regulatory features that drive phenotypic decisions, and offer an experimentally tunable structure for encoding inducible multistable behavior arising from both single and dual-function allosteric transcription factors.

Figures (23)

• Figure 1: Network of three mutually-repressing genes. Transcribing each gene $g_{i}$ produces a repressor with average concentration denoted by $R_{i}$.
• Figure 2: Expression of repressor 1 in the three-gene toggle switch, as described by a Hill function model. Equivalent definitions apply to expression of repressors 2 and 3. (A) Thermodynamic states, weights, and rates for expression of repressor 1 ($R_{1}$). The regulating repressors ($R_{2}$ and $R_{3}$) bind non-exclusively to the target promoter region to suppress gene transcription. Each repressor $R_{i}$ binds at the promoter region for repressor $R_{1}$ with affinity $K_{1i}$ and cooperativity $n$. (B) Thermodynamic states, weights, and rates for expression of repressor $R_{1}$ in the presence of inducers. Expression now depends on the active concentration of each repressor $R_{i}$, which is determined by a distinct inducer at concentration $c_{i}$, defining the probability of activity $p_{\text{act}}(c_{i})$.
• Figure 3: Repressor activity as a function of inducer concentration $c$, with $K_I < K_A$. (A) The probability of active repressor as a function of inducer concentration, with $m = 2$, $\Delta\epsilon = 4\:k_BT$, $K_A = 150$$\mu M$, and $K_I = 5$$\mu M$. The saturation and leakiness limits are denoted in blue and orange, respectively, and the $EC_{50}$, i.e., inducer concentration at which the probability is half maximal, is marked in purple. (B) Evolution of the probability for different values of $K_I$, with the curve in panel (A) shown again in green. The shift in probability spans the range of allowable $K_I$ values for the specified parameters up to the boundary at $K_I = 6.77\:\mu M$, derived from Appendix \ref{['app:MWC']}.
• Figure 4: Heatmaps tracking the bifurcation threshold beyond which the three-gene toggle switch can no longer have an $\bar{R}_1$-dominant steady state and be tristable. Note that we define the limit steady state case here as satisfying $\bar{R}_2^n$, $\bar{R}_3^n\rightarrow 0$, with $\bar{R}_2 = \bar{R}_3 \leq \varepsilon$. The results shown are specific to the choice $\varepsilon = \sqrt[n]{0.01}$. (A) The threshold probability of active repressor as a function of Hill coefficient $n$ and mRNA expression rate $\bar{a}$. Note that the beige-shaded region for smaller axis values denotes a regime in which the steady state is never possible at any inducer concentration. (B) The threshold specified in Fig. \ref{['fig:prob']}(A), now in terms of inducer concentration for the MWC model.
• Figure 5: Dynamics of the three-gene toggle switch with $n = 4$ and $\bar{a} = 2$ for increasing inducer concentration $c$. Allosteric regulation is defined by the probability curve shown in Fig. \ref{['fig:prob']}(A). (A)-(B) Bifurcation diagrams for $\bar{R}_1$, $\bar{R}_2$, and $\bar{R}_3$ steady state expression as a function of inducer concentration. (C) Fixed points for the three-gene toggle switch at a low inducer concentration. (D) Fixed points for the three-gene toggle switch at an intermediate inducer concentration. (E) Fixed points for the three-gene toggle switch at a high inducer concentration. In (C)-(E) the stability of each fixed point is color-coded as in panels (A) and (B), and each fixed point is labeled with its corresponding expression levels, colored as in Fig. \ref{['fig:3genetogglediagram']}.