Steady states are the fixed points of a network’s dynamics, the configurations of gene activity that, once reached, persist indefinitely. In the language of cell biology, these steady states are interpreted as cellular phenotypes, the distinct identities a cell can adopt. A stem cell deciding between self-renewal and differentiation, an immune cell settling into an activated or exhausted state, or a cancer cell sliding into an invasive mode all correspond, in this framework, to different attractors of the underlying regulatory circuitry. The first step in understanding any network’s dynamics is therefore describing the full collection of steady states it can support under different conditions. But there is a catch: for any given gene regulatory network, there are usually many possible Boolean models consistent with its known interactions, and different models can disagree about which steady states exist.

The research team confronted this ambiguity head-on. Rather than picking a single Boolean model and analyzing it, they considered the collection of all monotone Boolean function models compatible with a given gene regulatory network. Monotone means that each regulatory interaction has a consistent sign: an activator never becomes an inhibitor and vice versa, which matches the biological reality of signed interactions in gene networks. Within this vast family of compatible models, the authors asked a deceptively simple question: which steady states are supported by most of the models? A steady state endorsed by nearly every compatible model is far more likely to reflect genuine cellular behavior than one that appears in only a handful of exotic parameterizations.

The key insight that unlocks the problem comes from an unexpected corner of mathematics: the algebraic structure of the lattices of monotone Boolean functions. The authors proved that the set of monotone Boolean models supporting a given equilibrium can be decomposed as a product of prime ideals and prime filters of these lattices. In practical terms, this structural theorem converts a seemingly intractable counting problem, enumerating and weighing enormous numbers of Boolean models, into an explicit computation. Instead of simulating thousands of network variants and tallying outcomes, researchers can now characterize, with precision, exactly which models support a given fixed point and how large that supporting set is relative to the whole space of compatible models.

The theorem delivers its most striking payoff for balanced networks, a class of regulatory networks whose signed interaction structure satisfies a consistency condition around every feedback loop. In such networks, the authors found an explicit hierarchy in the prevalence of individual steady states, as well as in the prevalence of bistability and multistability. Bistability refers to a network’s capacity to settle into two alternative stable states, a property widely believed to underlie biological switch-like decisions, while multistability extends this to three or more coexisting stable outcomes. The new results mean that for balanced networks, one can rank steady states and multistable configurations by how commonly they arise across all compatible models, without ever committing to a specific parameter choice.

This hierarchy has immediate biological consequences, because it turns model uncertainty from a nuisance into a source of information. When most models compatible with a network agree that a particular pair of states is bistable, that bistability is robust to the unknown details of the regulatory functions. Conversely, a steady state that only rare models support can be flagged as fragile, a prediction that experiments could test by perturbing the network and observing whether the state appears. The framework thus provides a principled way to distinguish between the behaviors a network reliably produces and the artifacts of particular modeling assumptions, a distinction that has often been blurred in the computational biology literature.

To demonstrate the power of the approach, the authors applied it to a network of direct clinical relevance: the epithelial-mesenchymal transition network, a regulatory circuit implicated in cancer metastasis. During this transition, epithelial cells, which are typically stationary and tightly bound to their neighbors, lose their adhesions and acquire the motile, invasive characteristics of mesenchymal cells. The process is a hallmark of cancer progression, enabling tumor cells to detach from the primary tumor, migrate through surrounding tissue, and seed metastases at distant sites. Understanding which states this network can stably maintain, and how easily it flips between them, is therefore a central goal of computational oncology.

The analysis of the EMT network produced an elegant result. Across all monotone Boolean models compatible with the network’s structure, the most common equilibria correspond precisely to the fully epithelial state and the fully mesenchymal state, the two biologically recognized extremes of the spectrum. Moreover, the bistability between these two states emerged as the most common bistability among all network-compatible monotone Boolean models. In other words, the mathematical hierarchy of model prevalence aligns with experimental observations of the transition: cells genuinely occupy these two principal phenotypes, and the switch between them is the network’s dominant binary decision. The agreement between the abstract counting of models and the concrete biology of metastasis suggests that the structure of a gene network alone encodes a surprising amount of its functional behavior.

Why should this alignment exist? One interpretation is that evolution has wired regulatory networks so that their most robust dynamical behaviors, those supported by the widest range of mechanistic implementations, are exactly the behaviors the organism needs. A phenotype that survives variation in the fine details of gene regulation is a phenotype that can be reliably produced across genetic backgrounds and environmental conditions. The prevalence hierarchy computed by the new method can be read as a measure of this robustness, offering evolutionary theorists a quantitative handle on why certain cellular decisions, like the epithelial-mesenchymal switch, appear so consistently across tissues and species while others remain rare and context-dependent.

The broader implications reach beyond cancer biology. Because the method applies to any monotone gene regulatory network, and yields explicit results for balanced networks, it provides a general toolkit for interrogating the multistability of developmental circuits, immune regulatory modules, and synthetic gene networks designed in the laboratory. Engineers building synthetic circuits could use the hierarchy to choose wiring diagrams whose desired states are maximally prevalent across models, maximizing the odds that the circuit behaves as intended despite uncertainty in its molecular details. For experimentalists, the framework generates testable predictions about which phenotypes should be common, which should be rare, and which transitions should be easy or hard to trigger. What began as an abstract question about Boolean functions and their ideals has matured into a practical lens on one of biology’s deepest puzzles: how the same wiring diagram can give rise to the reliable, switch-like decisions that cells make every day.

Subject of Research: Characterization of steady states and multistability in monotone Boolean models of gene regulatory networks

Article Title: Characterization of monotone Boolean models supporting fixed points and multistability in balanced networks

Article References: Adigwe, S., BV, H., Jolly, M. K., & Gedeon, T. (2026). Characterization of monotone Boolean models supporting fixed points and multistability in balanced networks. PLOS Complex Systems, 3(5), e0000103. https://doi.org/10.1371/journal.pcsy.0000103

Image Credits: AI Generated

DOI: 10.1371/journal.pcsy.0000103

Keywords: gene regulatory networks, Boolean models, steady states, multistability, bistability, balanced networks, monotone functions, epithelial-mesenchymal transition, cancer metastasis, systems biology, fixed points, cellular phenotypes