The core problem the team set out to solve is one of biological interpretation. Standard trajectory inference methods frequently produce trees with multifurcations, meaning nodes where three, four, or more branches radiate from a single point. Mathematically, such high-degree branching is often the cheapest way to connect all the cells in a dataset, which is why minimum spanning tree approaches favor it. Biologically, however, a node with five simultaneous outgoing branches is hard to defend. Cell fate decisions, as conceptualized in Waddington’s famous epigenetic landscape, are typically depicted as a ball rolling down a valley that splits into two channels, not five. A binary structure, where each decision point offers at most two alternatives, aligns far more naturally with this classical picture of progressive restriction of cell potential.

To impose this binary discipline, the researchers turned to integer programming, a branch of discrete optimization in which decisions are encoded as variables restricted to whole-number values. In their formulation, the choice of which edges to include in the spanning tree becomes a set of binary variables, each indicating whether a particular connection between two cell cluster representatives is selected. The objective function minimizes the total edge weight, preserving the spirit of the minimum spanning tree, while additional constraints enforce that no node in the resulting tree exceeds a degree of two in the branching sense appropriate for trajectory topology. This transforms a problem that classical greedy algorithms such as Kruskal’s or Prim’s can solve efficiently into a harder combinatorial optimization problem, but one whose solutions carry direct biological meaning.

The degree restriction is not merely cosmetic. When an unconstrained minimum spanning tree is built over cluster centroids in a high-dimensional gene expression space, the algorithm has no incentive to respect the sequential nature of cell differentiation. It may connect a mature cell type directly to a primitive progenitor population through a shortcut that skips intermediate states, or it may hang several distinct lineages off a single hub. By capping the degree, the integer programming formulation forces the tree to route connections through intermediate nodes, producing a hierarchy in which each branching event represents a discrete, interpretable fate choice. The result is a topology that reads like a family tree of cell states rather than a tangled web of shortest connections.

A key practical strength of the new method is its modularity. Rather than building a complete trajectory pipeline from scratch, the authors implemented their degree-restricted tree construction as a plug-in module that can be inserted into existing graph-based and tree-based trajectory inference frameworks. To demonstrate this flexibility, they integrated the module into Slingshot, one of the most widely used pseudotime inference tools in the single-cell genomics community. Slingshot traditionally constructs a minimum spanning tree over cluster centroids and then fits principal curves along its branches to order cells in pseudotime. By swapping the unconstrained tree for the degree-restricted one, the researchers showed that the downstream pseudotime machinery can operate on a topology that is simultaneously cost-efficient and biologically constrained.

Validation of the constrained topologies required an independent reference against which the reconstructed trees could be compared. The team turned to CoSpar, a method that infers cell fate hierarchies using lineage information, often derived from experimental lineage tracing. By evaluating the consistency of their degree-restricted trees against a CoSpar-derived reference hierarchy, the authors could assess whether the binary constraint, imposed purely on the basis of expression data and optimization, recovered relationships that an orthogonal lineage-aware approach also supports. This cross-method consistency check is important because trajectory inference is notoriously sensitive to the assumptions baked into any single algorithm, and agreement between structurally different approaches lends credibility to the inferred hierarchy.

The researchers applied their framework to three biologically rich datasets: bone marrow mononuclear cells, human fetal immune cells, and a mouse hematopoiesis dataset. Hematopoiesis, the process by which blood cells are generated, is a canonical testing ground for trajectory methods because its branching structure, from stem cells through progenitors to mature lineages, has been mapped in detail over decades of experimental work. Across these datasets, the degree-restricted trees produced trajectories whose branching patterns were more interpretable than those generated by unconstrained minimum spanning tree baselines. The comparison with the unconstrained baselines was descriptive rather than a claim of universal superiority, but it illustrated concretely how the binary constraint reshapes the inferred lineage structure into forms that match developmental expectations.

Behind the biological narrative lies a substantial computational challenge. Minimum spanning trees can be computed in near-linear time for the number of edges considered, but degree-constrained spanning tree problems belong to a family of combinatorial optimization problems that are computationally hard in general. Integer programming solvers handle such problems by exploring the space of feasible integer solutions with techniques such as branch and bound, cutting planes, and linear programming relaxations. The feasibility of the approach at the scale of single-cell datasets, where cluster-level graphs may contain dozens to hundreds of nodes after preprocessing and clustering, depends on careful formulation that keeps the integer program tractable. The authors’ decision to operate at the level of cluster representatives rather than individual cells is central to this tractability, since it dramatically reduces the number of nodes over which the constrained tree must be built.

The conceptual appeal of the method extends beyond hematopoiesis. Waddington’s epigenetic landscape, sketched in the 1950s as a metaphor for how cells navigate a landscape of developmental potential, has become a guiding intuition for the entire field of pseudotime analysis. Yet most computational implementations of trajectory inference treat topology as an emergent property of the data rather than an explicit modeling choice. By encoding the binary branching structure directly into the optimization objective and constraints, the new approach makes the Waddingtonian assumption a formal part of the inference procedure. This explicitness is a virtue: it allows users to see exactly what biological prior is being imposed, and it opens the door to future variants in which different degree restrictions or additional structural constraints could encode other hypotheses about how cell states relate.

The work, led by Jiaying Zhao and Takuma Iwaki, who contributed equally, together with Tomoya Mori, Wai-Ki Ching, and Tatsuya Akutsu, was supported by the Hong Kong Research Grants Council and released as open access, with source code publicly available on GitHub for researchers who wish to integrate the degree-restricted module into their own pipelines. As single-cell RNA sequencing continues to generate atlases of ever greater scale and complexity, the demand for trajectory inference methods that are both computationally principled and biologically legible will only grow. This study demonstrates that ideas from discrete optimization, a field seemingly distant from developmental biology, can supply exactly the kind of structural discipline that turns a tangle of gene expression similarities into a readable map of cellular destiny.

Subject of Research: Degree-restricted minimum spanning tree algorithms for binary cell trajectory inference in single-cell RNA-seq data

Article Title: Degree-restricted minimum spanning trees for binary cell trajectory inference in single-cell RNA-seq data

Article References: Zhao, J., Iwaki, T., Mori, T., Ching, W.-K., & Akutsu, T. (2026). Degree-restricted minimum spanning trees for binary cell trajectory inference in single-cell RNA-seq data. BMC Bioinformatics. https://doi.org/10.1186/s12859-026-06674-y

Image Credits: AI Generated

DOI: 10.1186/s12859-026-06674-y

Keywords: single-cell RNA-seq, trajectory inference, pseudotime, minimum spanning tree, integer programming, Waddington's epigenetic landscape, hematopoiesis, Slingshot, CoSpar, cell fate decisions, computational biology, discrete optimization