Single Cell Construction: Methods & Applications

• Single cell construction is the context-dependent process of building and analyzing isolated cells using computational, biological, and engineering frameworks.
• It employs tailored techniques such as levelwise CAD cell construction, variational optimization for biological dynamics, and cross-modal assembly in single-cell omics.
• The approach offers practical benefits in enhancing computational efficiency, understanding cellular behavior, and advancing biofabrication as well as battery engineering.

Searching arXiv for papers on “single cell construction” and related usages to ground the article in current literature. In the literature considered here, single cell construction is not a single standardized doctrine but a context-dependent technical expression. In symbolic computation, it denotes the construction of one cylindrical cell around a sample point instead of a full cylindrical algebraic decomposition (Nalbach et al., 2022). In theoretical biology, it denotes a derivation of optimal single-cell rules from a population-level utility and convex dissipation (Horiguchi et al., 2022). In single-cell omics, the phrase appears in descriptions of foundation-model, workflow, and resource construction, including joint cell-language pretraining, long-tail-aware annotation, and GPU-scalable analysis (Wang et al., 9 Jan 2026). In experimental engineering and biofabrication, it can refer to the physical construction of a single electrochemical cell or to 3D assembly with single-cell precision (Jenson et al., 2019, Ellison et al., 2022).

1. Terminological scope and disciplinary separation

The most formal use of the phrase occurs in CAD-derived reasoning for non-linear real arithmetic. There, the objective is to compute a single truth-invariant cell containing a sample point and making the relevant polynomials sign-invariant, because a full decomposition of $\mathbb{R}^n$ is often unnecessary for satisfiability checking or conflict explanation (Nalbach et al., 2022). This use is local, sample-centered, and algorithmic.

A distinct biological use appears in the gradient-flow formulation of multicellular dynamics. There, the “single-cell construction” is the closed-form construction of growth, immigration, and type-switching rules from a population-level utility $U_t(n)$ and a quadratic cost $C_n(g,m,v)$ (Horiguchi et al., 2022). The emphasis is not on geometric cells but on cell-level rules induced by a higher-level variational principle.

A broader omics-oriented use concerns the construction of single-cell models, paired corpora, annotations, and workflows. OKR-CELL explicitly presents “a full foundation-model recipe for single-cell construction,” consisting of richer text curation, robust alignment under noise, and scalable training (Wang et al., 9 Jan 2026). Related work uses “construction” for dataset assembly, GraphRAG-based annotation pipelines, organ-specific atlas pretraining, and GPU-backed analysis stacks (Han et al., 24 Apr 2025, Li et al., 30 Sep 2025, Dicks et al., 2 Mar 2026).

A common source of confusion is the proximity of this phrase to other “cell-construction” expressions. In condensed-matter theory, for example, the cell-construction method builds flat-band Hubbard models from overlapping complete graphs and proves unique ferromagnetic ground states under a connectivity condition; this is a different construction paradigm and should not be conflated with the CAD or single-cell biology usages (Tanaka, 2020).

2. Single cylindrical cell construction in CAD and SMT

In SMT solving over QF_NRA, polynomial constraints are Boolean combinations of relations of the form $p(x_1,\ldots,x_n)\sim 0$ with $\sim\in\{=,\neq,<,>,\leq,\geq\}$. Classical CAD decomposes real space into finitely many cylindrical cells on which relevant polynomials have invariant sign, but this is often more structure than the solver needs. The levelwise method of "Levelwise construction of a single cylindrical algebraic cell" instead constructs only one local cell containing the current sample, suitable for satisfiability checking or MCSAT/NLSAT-style conflict explanation (Nalbach et al., 2022).

The paper’s main methodological shift is from polynomial-by-polynomial refinement to levelwise construction. The cell is built variable by variable, with interval structure decided at level $i$ before only the necessary lower-level obligations are introduced. The formal presentation is cast as a proof system, separating correctness from heuristic choices. This allows the algorithm to vary the projection-polynomial reductions and root-ordering strategies without entangling those choices with the proof of soundness (Nalbach et al., 2022).

The resulting cell is represented levelwise by symbolic intervals built from indexed root expressions. The method uses CAD notions such as delineability, sign-invariance, order-invariance, analytic submanifold structure, and connectedness. It also introduces several heuristics for shaping the cell and controlling projection complexity, including the equational constraint heuristic, the biggest cell heuristic, the lowest degree barriers heuristic, the chain heuristic, and the full heuristic. The direct sensitivity to input-polynomial ordering that characterized earlier refinement-based methods is thereby reduced, although the final cell still depends on boundary and ordering choices (Nalbach et al., 2022).

The method is sound but not complete in general because it is formulated in a McCallum-style projection framework and may return FAIL in the presence of nullification. The paper explicitly notes fallback to a complete method such as NLSAT with Collins projection, and it also remarks that a future Lazard-based version could be complete (Nalbach et al., 2022). This is a central limitation: single-cell construction in this sense is a targeted local explanation mechanism, not a replacement for full CAD in every regime.

3. Projective delineability as a refinement of the single-cell paradigm

"Projective Delineability for Single Cell Construction" weakens the classical delineability requirements by allowing roots to be tracked on the real projective line $P=\mathbb{R}\cup\{\infty\}$ rather than only on $\mathbb{R}$ (Nalbach et al., 1 Aug 2025). The point of the modification is precise: some leading coefficients can be omitted from the projection set because roots are allowed to pass through infinity projectively. The paper adapts the single-cell-construction algorithm accordingly.

The projective variant preserves the overall goal—compute a locally cylindrical cell containing a sample point on which all polynomials in $P\subseteq\mathbb{Q}[x_1,\ldots,x_n]$ are sign-invariant—but changes the middle layer of the lifting argument. Ordinary delineability is retained for the boundary-defining polynomials, while projective delineability is used for polynomials that only need to remain away from the interior of the selected cell. The paper’s caveat is explicit: this optimization must be applied carefully, especially for unbounded cells, because leading coefficients may still be needed to prevent roots from crossing $\infty$ in a way that matters for the chosen interval (Nalbach et al., 1 Aug 2025).

The implementation in SMT-RAT compared BC, LDB, BC-PD, and LDB-PD on 12,154 SMT-LIB QF_NRA instances with a 60 second timeout and 4 GB memory limit. The aggregate conclusion is conservative: projective delineability does not greatly affect which problems are solvable within the time limit, and the runtime curves show gains and losses that are roughly balanced overall (Nalbach et al., 1 Aug 2025).

The micro-level statistics are more revealing. For the instances solved by BC-PD, the leading coefficient could be omitted for 307,822 polynomials; the optimization could not be applied for 826,795 polynomials because the cell was unbounded in some direction; and for 4,089 polynomials it could not be applied because the polynomial did not have a root on both sides of the bounds or no suitable resultant was available. Among applicable cases, the optimization was used in about 37% of cases. The paper also reports that about 55% of algebraic computation time was spent on discriminants, about 5% on resultants, and almost none on coefficients; among the highest-total-degree polynomials encountered, 15% were discriminants, 30% were resultants, and only 3% were coefficients (Nalbach et al., 1 Aug 2025). A plausible implication is that omitting leading coefficients is mathematically clean and sometimes useful, but often not the dominant lever on runtime because discriminants and resultants remain the main cost centers.

4. Population-level utility and the biological single-cell construction

In "Cellular gradient flow structure connects single-cell-level rules and population-level dynamics," the heterogeneous population state is

$U_t(n)$0

with dynamics generated by growth, type switching, and immigration (Horiguchi et al., 2022). The mechanistic description is

$U_t(n)$1

The key construction is variational: rates are chosen to maximize instantaneous net utility gain,

$U_t(n)$2

where $U_t(n)$3 is a population-level utility and $U_t(n)$4 is a quadratic dissipation or cost functional (Horiguchi et al., 2022).

Because the cost is quadratic and convex, the optimization has a unique closed-form solution:

$U_t(n)$5

$U_t(n)$6

$U_t(n)$7

with $U_t(n)$8 and $U_t(n)$9 (Horiguchi et al., 2022). These formulas are the paper’s “single-cell construction”: growth is proportional to the local utility derivative, immigration occurs only into types with positive utility advantage, and switching from $C_n(g,m,v)$0 to $C_n(g,m,v)$1 occurs only if $C_n(g,m,v)$2 is better than $C_n(g,m,v)$3 in the utility landscape.

Under these optimal rates, the utility evolves according to

$C_n(g,m,v)$4

so for time-independent utility the population evolves uphill in utility and the utility is nondecreasing (Horiguchi et al., 2022). This is why the system is described as a generalized gradient flow. The functional derivative $C_n(g,m,v)$5 acts as a single-cell-level landscape, and because switching is only allowed toward higher utility derivative, the induced switching graph is automatically acyclic. The paper therefore derives unidirectional differentiation and hierarchical order in types as consequences of the gradient structure rather than as externally imposed assumptions (Horiguchi et al., 2022).

The framework also couples behaviors that are often modeled separately. In the constant-weight case,

$C_n(g,m,v)$6

and

$C_n(g,m,v)$7

Thus immigration and proliferation are aligned, and switching is coordinated with growth-rate ordering (Horiguchi et al., 2022). The three-state T-cell example with naive $C_n(g,m,v)$8, activated effector $C_n(g,m,v)$9, and memory $p(x_1,\ldots,x_n)\sim 0$0 shows how a time-dependent utility reproduces the expansion–contraction–memory program and why only transitions compatible with the utility ordering remain accessible.

5. Construction in single-cell omics: corpora, models, and workflows

Within single-cell omics, “construction” frequently refers to the assembly of large corpora, cross-modal representations, and scalable workflows. OKR-CELL is exemplary. It is pretrained on SCxGEN-32M, a dataset of 32 million cell-text pairs collected from CELLxGENE, standardizes metadata with the Cell Ontology and OBO Foundry, enriches text through an LLM + retrieval-augmented generation (RAG) workflow over a biomedical knowledge base encoded with BioBERT, generates descriptions with DeepSeek-V3, and filters hallucination-prone augmentations by a reliability-screening step based on Clinical-Longformer embeddings and cosine similarity (Wang et al., 9 Jan 2026). Architecturally it uses scGPT as the cell encoder, Clinical-Longformer as the text encoder, and a cross-modal projector; the training objective is

$p(x_1,\ldots,x_n)\sim 0$1

where the Cross-modal Robust Alignment (CRA) objective combines Progressive Sample Weighting, Coupled Momentum-updated Memory Bank, and Cross-modal Complementary Alignment (Wang et al., 9 Jan 2026). The model is evaluated on six tasks—cell clustering, cell-type annotation, batch-effect correction, few-shot annotation, zero-shot annotation, and bidirectional cell-text retrieval—and the paper frames the result as a full recipe for single-cell construction (Wang et al., 9 Jan 2026).

Related systems extend the same construction logic toward generation and annotation. Cell2Text turns scRNA-seq profiles into structured natural-language descriptions using Geneformer, a two-layer feedforward adapter, and instruction-tuned LLM decoders, with a multimodal dataset derived from CELLxGENE Census containing 1,000,000 cells, 7,331 donors, 783 cell types, 347 tissue types, and 128 disease conditions (Kharouiche et al., 29 Sep 2025). ReCellTy reconstructs CellMarker 2.0 into a graph-structured feature-marker database with 18,850 biological entity nodes, 48,944 relationships, 7 node types, and 7 relationship types, then uses Neo4j, Cypher, and LangChain’s GraphCypherQAChain in a five-task annotation pipeline designed to mimic manual annotation logic (Han et al., 24 Apr 2025).

Resource construction also targets scale and imbalance. Celler contributes Celler-75, a dataset of 40 million cells distributed across 80 human tissues and 75 specific diseases, and couples it with Gaussian Inflation (GInf) Loss and Hard Data Mining (HDM) to improve long-tailed disease annotation (Zhao et al., 28 Mar 2025). Nephrobase Cell+ is a kidney-specific foundation model pretrained on ~100 billion tokens from ~39.5 million single-cell and single-nucleus profiles across 4,319 samples and 4 mammalian species, with a transformer encoder-decoder, gene-token cross-attention, and a Mixture-of-Experts module (Li et al., 30 Sep 2025). These works make explicit that, in current omics usage, construction includes not only model fitting but also atlas assembly, ontology alignment, multimodal harmonization, and class-imbalance-aware objective design.

A further shift concerns the construction of the latent manifold itself. CellRefine inserts a post-pretraining stage between initial pretraining and downstream fine-tuning, using marker-gene prototypes, lineage-aware separation, and a Gaussian Mixture Variational Encoder; the paper reports gains up to 15% (Weerasekara et al., 8 May 2026). Cell-JEPA moves from reconstructing sparse counts to latent-space prediction and reports 0.72 AvgBIO in zero-shot transfer versus 0.53 for scGPT, a 36% relative improvement (ElSheikh et al., 2 Feb 2026). On the infrastructure side, rapids-singlecell makes single-cell workflow construction GPU-native while preserving AnnData and the scverse contract; on a 1 million-cell workflow it reports execution in 26 seconds on a single NVIDIA DGX B200 GPU versus over 52 minutes on a 32-core CPU workstation, and it analyzes the Tahoe 100M dataset in less than 20 minutes using Dask and multi-GPU execution (Dicks et al., 2 Mar 2026). This suggests that omics “construction” has become a layered enterprise spanning corpus curation, representation learning, and workflow execution.

6. Physical construction at the level of an individual cell

In electrochemistry, "Hydrogel Leclanché Cell: Construction and Characterization" uses the term in the literal engineering sense of constructing a single cell. The device preserves traditional Leclanché chemistry—zinc anode, manganese dioxide cathode, conductive carbon as acetylene black or graphite, and an aqueous electrolyte hydrogel containing NH4Cl and ZnCl2—but suspends the components in agar hydrogel, which serves as support and separator (Jenson et al., 2019). The paper describes both a 20 mL vial configuration and a refillable 11 mL glass chamber with purge/fill ports and current collection ports. Under a 10 kΩ load, the proof-of-concept graphite-based cells showed average open circuit voltage at construction of ~1.4 V and initial closed circuit voltage of ~0.75 V, while acetylene black generally outperformed graphite; the closed chamber demonstrated purge-and-refill behavior, with potential peaking at 0.35 V at about 125000 s and a second peak of about 0.28 V after refill (Jenson et al., 2019).

In biofabrication, "Cellular Micromasonry" constructs three-dimensional cellular structures with subcellular-to-single-cell precision by combining a jammed microgel support medium, micromanipulation with a glass microcapillary, and confocal microscopy (Ellison et al., 2022). The medium is optimized at 5–6 wt%, with elastic shear modulus about 10–20 Pa and yield stress about 1.3–2.4 Pa. With capillary translation at about 0.5 mm/s, the estimated maximum Reynolds number is about $p(x_1,\ldots,x_n)\sim 0$2, and the field-wide RMS displacement is about 4 μm, less than a cell diameter (Ellison et al., 2022). The method was used to build alternating-color lines, triangles, rings, checkerboards, multilayer structures, a hollow spherical shell approximating an acinus, and nearly 100-cell “UF” initials; the authors state that no imperfection exceeds one cell diameter (Ellison et al., 2022).

The two physical uses are distinct in substrate and purpose, but both are literal constructions of bounded cellular units. The hydrogel Leclanché work constructs an electrochemical cell whose discharged suspension can be replaced structurally rather than electrically recharged (Jenson et al., 2019). Cellular Micromasonry constructs biological microarchitectures in which post-assembly function is directly tested, including gap junction communication by calcein transfer over about 5 hours and acini development over about 10 days in modified microgel medium (Ellison et al., 2022). In this engineering sense, single cell construction denotes fabrication rather than inference or representation learning.