Relating biomarkers and phenotypes using dynamical trap spaces

TL;DR

The paper defines dynamical phenotypes as complete trap spaces with a fixed set of phenotype-determining nodes and free inputs, providing a principled, scalable link between Boolean network structure and experimentally observable cell states. It introduces a binary decision diagram–based symbolic method to identify these spaces without enumerating all attractors, and demonstrates how to construct input-to-phenotype maps efficiently. Through four case studies, including a 70-node T cell differentiation model, the approach recapitulates known cell types and reveals additional phenotypes not captured by traditional attractor analyses. A validation suite using attractor clustering and mutual information shows that LDOI-based PDN sets yield complementary, sometimes more specific, phenotype descriptions, enabling principled biomarker selection and environment–phenotype predictions. The work suggests extensions to multi-level or continuous models and highlights practical implications for model-guided experimental design and phenotype control.

Abstract

Connecting the dynamics of biomolecular networks to experimentally measurable cell phenotypes remains a central challenge in systems biology. Here we introduce a model-based definition of phenotype as a partial steady state that is committed to a certain dynamical outcome while otherwise being minimally constrained. We focus on Boolean models and define \emph{dynamical phenotypes} as complete trap spaces that maximally specify a chosen set of phenotype-determining nodes that correspond to biomarkers while keeping external inputs unconstrained. We show that dynamical phenotypes can be efficiently identified without full attractor enumeration. Using four published models, including a 70-node Boolean model of T cell differentiation, we show that dynamical phenotypes recover known cell types and activation states, and indicate the environmental conditions ensuring their existence. We also propose a method to identify informative phenotype-determining nodes based on the canalization of the Boolean functions. This method reveals biologically relevant cell state information that is complementary to the phenotypes manually defined by model creators and is validated by two attractor-based approaches. Our results demonstrate that dynamical phenotypes provide a scalable framework for linking model structure, external inputs, and phenotypic outcomes, and offer a principled tool for model-guided biomarker selection.

Figures (9)

• Figure 1: Characterization of the simple Boolean model of Example \ref{['example:toy-network']}. (a) Influence graph showing positive influences as green edges that end in triangle arrows, and negative influences as red edges that end in flat arrows. (b) Asynchronous state transition graph, showing two fixed points ($011$ and $111$) and a complex attractor ($100 \leftrightarrow 101$). The rectangular outlines indicate the model's repertoire of trap spaces, with continuous outlines corresponding to complete trap spaces. (c) Succession diagram consisting of all complete trap spaces; the directed edges depict subspace inclusion. The non-minimal complete trap space ${\star}11$, wherein the network is committed to $\texttt{Out}=1$, is highlighted using a blue gradient.
• Figure 2: The influence graph of the 70-node Boolean version of the T cell differentiation model of Naldi et al. with 12 constant-value nodes removed. Regular arrows represent positive influences (activation), and round arrows represent negative influences (inhibition). The thicker edges indicate canalizing regulators (either sufficient or necessary to elicit the activation of the target node). Red border indicates 7 biologically-motivated PDNs, blue border indicates 3 PDNs chosen based on regulatory logic, and purple border indicates 2 shared PDNs. Green borders indicate input nodes.
• Figure 3: Two subgraphs of the influence graph of the T cell differentiation model, illustrating the canalizing regulation of PDNs and the canalizing influences incident on them. Regular arrows represent positive influences (activation), and round arrows represent negative influences (inhibition). Panel A focuses on the 7 nodes that are exclusive to the biologically-motivated PDNs. The canalizing influences of these PDNs tend to remain within the set. Panel B focuses on the 3 PDNs exclusive to the LDOI-based PDN set. The canalizing influences of these PDNs extend to 14 nodes, including 4 of the 9 biologically-motivated PDNs. The LDOI-based PDNs are connected to 22 nodes by canalizing paths. The nodes with green outlines are external inputs.
• Figure 4: Mapping between the biologically-motivated dynamical phenotypes (rows, see Table \ref{['Bio_phenotypes']} for the meaning of the notations) and the LDOI-based dynamical phenotypes (columns). The LDOI phenotypes are grouped according to the values of NFAT, TBET, proliferation, IL4R_b1 and STAT1, as explained by the figure legend. The coloring is based on the values of NFAT and TBET (also see legend). Finally, the numbers and color intensity indicate the attractor count in thousands ($1 \equiv [1, 1000]$, $2 \equiv [1000, 1999]$, etc.). The matrix positions with a colored outline and grey background indicate intersections of biological and LDOI-based dynamical phenotypes that could but do not contain any attractors.
• Figure 5: Mapping between the 32 LDOI-based phenotypes (rows) and the 31 attractor clusters (columns). The LDOI phenotypes are denoted by a binary string indicating the state of IL4R_b1, NFAT, STAT1, TBET, and proliferation, respectively. The numbers and color intensity indicate the attractor count in thousands ($1 \equiv [1, 1000]$, $2 \equiv [1000, 1999]$, etc.).

Theorems & Definitions (4)

• Example 1
• Example 2
• Definition 1: dynamical phenotype
• Example 3