LogicSR: prior-guided symbolic regression for gene regulatory network inference from single-cell transcriptomics data

Abstract

Deciphering gene regulatory mechanisms from high-dimensional biology data remains a central challenge in modern systems biology, despite the growing availability of single-cell datasets. The difficulty stems partly from the sparsity and noise inherent in single-cell data and partly from the complexity of dynamic combinatorial regulation mediated by transcription factors. In this work, we introduce LogicSR, a computational framework that reconstructs gene regulatory networks from single-cell gene expression data with high accuracy by integrating the mechanistic interpretability of Boolean logical models with the equation-discovery capabilities of symbolic regression. It incorporates prior knowledge into a multi-objective Monte Carlo tree search (MCTS) framework, leveraging it to ensure biological plausibility and accelerate the search for optimal governing equations. LogicSR outperforms existing methods on both synthetic and real-world benchmark datasets. When applied to a human embryonic stem cell dataset, it demonstrates superior performance in elucidating complex combinatorial TF–target gene regulations and identifying key regulators.

Introduction

A gene regulatory network (GRN) captures the complex interplay of genes, regulatory elements, and signaling molecules that orchestrate cellular behavior, driving processes such as tissue-specific differentiation, metabolic adaptation, and disease progression [1–3]. Deciphering GRNs is critical for elucidating the mechanistic basis of phenotypic plasticity in cancer [4–6], identifying master regulators of cell fate decisions [7–10], and designing synthetic circuits for therapeutic applications [11–13]. While bulk transcriptomics datasets have historically driven GRN inference, recent advances in single-cell RNA-sequencing (scRNA-seq) now enable resolution of regulatory dynamics at unprecedented cellular granularity [14, 15]. However, inferring causal interactions from such high-dimensional, sparse datasets remains a significant computational challenge, requiring models that balance scalability with biological interpretability [16, 17].

Transcription factors (TFs) frequently act synergistically to regulate gene expression by stabilizing each other’s binding or recruiting cofactors, enabling context-specific activation or repression of target genes (TGs) [18, 19]. Notably, the regulation of a single TG can hinge on intricate interactions among multiple TFs, underscoring the complexity of TF coordination. Despite its prevalence, the mechanistic underpinnings of this cooperative regulation remains incompletely understood, posing challenges for explaining how GRNs arise from such coordinated TF activity [20].

State-of-the-art (SOTA) GRN inference tools construct GRNs using a diverse computational approaches applied to gene expression data, including multi-omics integration [21], co-expression analysis [22], graphical models and dependency networks [23, 24], information-theoretic methods [25, 26], and ordinary differential equation models [27]. While these tools offer methodological strengths, they often neglect combinatorial cooperativity or antagonism among TFs, which is critical for context-specific regulation of TGs [28]. Deep learning frameworks, such as graph neural networks (GNNs) [29, 30] and attention-based architectures [31], have proven powerful for capturing complex, nonlinear regulatory relationships. However, their “black-box” nature limits mechanistic interpretability, restricting their utility for experimental validation [31]. Boolean network models (BNs), though conceptually simple, remain widely used for their ability to distill regulatory logic into discrete, interpretable rules [32]. By representing gene states as binary variables (active/inactive) and interactions as logical functions, BNs provide a flexible framework for modeling combinatorial TF regulations [12, 33–36]. Nevertheless, inferring logical functions is challenging due to the sparsity and noise of single-cell data, as well as the inherent complexity of Boolean rules discovery, which involves navigating an exponentially large search space [37, 38]. These limitations highlight the need for computational frameworks that integrate single-cell temporal data, prior biological knowledge, and adaptive rule-learning algorithms to infer GRNs with both high predictive accuracy and mechanistic interpretability.

To address these gaps, we introduce LogicSR, a computational framework that models combinatorial TF coordination in regulating TGs and infers dynamic GRNs from scRNA-seq data. This is achieved this by encoding TF combinatorial mechanisms via logical rules and framing GRN inference as a rule discovery task. LogicSR employs a multi-objective symbolic regression framework that jointly optimizes predictive accuracy, biological interpretability, and model complexity. It then utilizes a Monte Carlo tree search (MCTS) algorithm, guided by prior biological knowledge, to efficiently explore regulatory rules, balancing exploration and exploitation rather than relying on random search in a vast solution space. Across synthetic and real single-cell datasets, and in comparisons with SOTA GRN inference methods, LogicSR consistently surpasses existing approaches in overall accuracy and TF–TG edge recovery while capturing combinatorial regulatory logic. Applied to a human embryonic stem (ES) cell dataset, LogicSR resolves dynamic TF cooperation, reveals interpretable rule-level mechanisms, and identified key regulators driving stage transitions. These results demonstrate LogicSR’s robustness to varying dropout rates and cell numbers, highlighting its utility for dissecting transcriptional regulation from scRNA-seq data.

Methods

The framework of LogicSR

LogicSR is a computational framework designed to infer GRNs from single-cell transcriptomics data. Using gene expression count matrices and TF–TF interaction data as input, it constructs dynamic GRNs and identifies key regulators (Fig. 1). Each GRN is represented as a logic model that explicitly captures combinatorial TF cooperation in regulating TGs. LogicSR stands out for its biological plausibility, mechanistic interpretability, and superior inference accuracy. To achieve this, we formalize these criteria as objectives in a multi-objective optimization problem and address them simultaneously via symbolic regression (SR). SR is a machine learning technique that searches the space of mathematical expressions to identify parsimonious equations mapping predictors to responses without assuming a predefined equation form.

Figure 1.

Schematic overview of LogicSR. (A) LogicSR accepts a single-cell gene expression matrix and a protein–protein interaction (PPI) network as inputs. (B) A symbolic tree is constructed by recursively inserting symbols from the root, expanding each operator node according to its arity and precedence, until all leaves are terminal symbols. (C) LogicSR infers dynamic GRNs through MCTS-guided multi-objective optimization, generating interpretable regulatory logic and enabling identification of key regulators.

We proposed a biologically informed MCTS search strategy within SR to identify governing regulatory functions. MCTS, which uses symbolic parse trees as search states, navigates the combinatorial space of regulatory logic by effectively balancing exploration and exploitation. “Biologically informed” refers to integrating biological prior knowledge into the inference process to enhance reliability, particularly under noisy data. Specifically, publicly available TF–TF interaction data were incorporated as a prior-consistency objective in our multi-objective framework, directing MCTS toward biologically plausible TF combinations and reducing redundant exploration. Critically, the prior network is soft—it guides MCTS without restricting it to known TF–TF combinations, thereby preserving the capacity to discover novel regulatory interactions.

LogicSR incorporates two complementary inputs (Fig. 1A). The first is a scRNA-seq matrix, encompassing either discrete time points or a pseudotime continuum. The second input is a TF-centered PPI prior network, constructed from integrated databases and restricted to expressed TF–TF interactions. LogicSR then generates candidate Boolean rules as parse trees: expressed TFs form the leaves, while logical operators (AND, OR, and NOT) occupy the internal nodes (Fig. 1B). Notably, prior to this, we conducted feature preselection with random forest. Subsequently, LogicSR uses iterative MCTS to explore the symbolic-tree space, evaluating each candidate rule via a multi-objective function that balances data fit, prior consistency, and parsimony. The highest-scoring rules are retained as interpretable Boolean expressions for each TG, which are mapped onto TF–TG edges to construct the final topological GRN. Using this topological GRN, we estimate TF activity through regulon-based enrichment scoring and prioritize key regulators by differential activity and trajectory association (Fig. 1C).

Gene regulatory network inference as an equation discovery task

Encoding gene regulation as Boolean equations provides an explicit, interpretable model of combinatorial control: a gene is activated only under specified TF-activity configurations (AND/OR/NOT). This approach approximates cis-regulatory module behavior, generating mechanistically testable hypotheses about synergistic and antagonistic TF interactions while maintaining high predictive accuracy for gene-expression dynamics.

Formally, let }, denote the set of n genes comprising the GRN. Each gene is represented as a binary-state node (OFF/ON), with its temporal evolution governed by a family of Boolean transition functions }. If gene is influenced by upstream genes, its update rule is defined as , where . The network is updated synchronously:

Continuous expression profiles measured at m + 1 consecutive time points yield the matrix, where . The corresponding binary data, , is obtained by binarizing using a specified discretization method.

The objective of GRN inference is to recover the rule set that most faithfully reproduces the observed state transitions encoded in , thereby framing network reconstruction as an equation discovery task.

Discovering governing regulatory equations via symbolic regression

SR provides a powerful framework for inferring the underlying mathematical structure of complex systems directly from observational data [39–41]. Unlike traditional regression techniques, which optimize parameters within a predefined model form, SR performs a combinatorial search over a space of mathematical expressions (composed of operators, functions, and state variables) to identify both the functional form and the coefficients of governing equations that best describe the observed dynamics.

Formally, let   denote a dataset with samples, where represents -dimensional input variables and is a scalar output. The objective of SR is to find an optimal symbolic expression that minimizes the empirical loss:

where denotes the hypothesis space (i.e. the set of all possible symbolic expressions), and is a loss function quantifying prediction error. Given that is exponentially large, identifying is computationally intractable without specialized optimization strategies.

The BN inference problem can be readily decomposed into n subproblems using a divide-and-conquer strategy. In each subproblem, we determine the Boolean equation for gene by applying SR to its observed state transitions. Formally, for gene with potential regulators, we construct a target vector representing its state across time steps, and a predictor matrix encoding the states of its candidate regulators. The Boolean governing equation is then inferred via:

where denotes the space of Boolean functions defined over the basis , and denotes the loss function. Crucially, this formulation enables the discovery of interpretable logical rules directly from noisy biological data, with the inferred Boolean function explicitly revealing canonical regulatory interactions (e.g. ).

Representing equations as symbolic trees

A mathematical expression can be formalized as a parse tree generated by a context-free grammar (CFG). Let , where is the finite set of non-terminal symbols, the finite set of terminal symbols, the production rules, and the start symbol. In our formulation, the start symbol denotes the entire equation. Leaf nodes (terminals) correspond to independent variables (e.g, , while interior nodes (elements of act as structural markers distinct from both the root and the leaves (Fig. 1B). Production rules encode standard Boolean operators: unary rules map a single non-terminal to one child (e.g. , whereas binary rules map a non-terminal to two children (e.g. . A well-formed expression is obtained via a pre-order expansion that begins at , and terminates once every frontier node is a terminal symbol. Hence each valid formula corresponds uniquely to the traversal-induced sequence of production rules.

Monte Carlo tree search for symbolic regression

MCTS is a heuristic method that expands a search tree via Monte-Carlo sampling to approximate optimal decisions in high-dimensional combinatorial space [42–44]. When applied to logic rule learning, MCTS treats expression synthesis as a sequential decision process: grammar-constrained operators {AND, OR, NOT}, and regulator literals (TFs), and structural delimiters form the action set, so every root-to-leaf path encodes a candidate formula (e.g. ). At each iteration the algorithm alternates the following four phases (Fig. 2).

Figure 2.

Monte Carlo tree search workflow. The search tree expands iteratively through four phases: (A) Selection. Starting at the root, repeatedly select the child with the highest UCB score until reaching a leaf node. (B) Expansion. Apply one production rule to the selected leaf to generate a new child node and add it to the tree. (C) Simulation. For each new child, perform stochastic roll-outs: replace all remaining nonterminal symbols using the grammar to form a complete expression, then fit any free constants to the gene expression data to compute a reward. (D) Backpropagation. Update the reward and visit count along the selected path, adjusting the statistics of each node up to the root.

Selection

Starting from the root, the algorithm repeatedly chooses a child that maximizes an optimism-in-the-face-of-uncertainty score—typically the Upper Confidence Bound (UCB):

where is the average multi-objective reward obtained so far after taking action in state (refer to the next section for multi-objective criteria), is the visit count of state and is the number of times action has been selected at . The logarithmic term grows slowly, so the second summand favors actions that have been tried less often, ensuring continued exploration; the constant tunes the exploration–exploitation balance and is typically set empirically for a given problem domain.

Expansion

When traversal reaches a non-terminal, partially explored node —i.e. one whose action set still contains unvisited options—the algorithm appends exactly one new child. Concretely, it samples an admissible action (often uniformly or via a light heuristic), generates the successor state , and inserts into the tree with initial statistics and . Adding a single child at a time keeps the tree balanced and ensures that every new node immediately undergoes simulation and backpropagation.

Simulation

Starting from , one or more playouts are executed using a default policy—most often uniform random selection of admissible actions—until a valid mathematical expression is generated. The expression is then evaluated on the input data to obtain its fitness value, which acts as a reward for subsequent computation.

Backpropagation

After the simulation, we update the maximum rewards and visit counts of nodes along the path from the current node to the root. The search algorithm repeatedly cycles through aforementioned steps until the stopping criterion is met. Each gene’s rule is taken from the root’s best-scoring, most-visited children; combining these rules produces a BN.

Multi-objective reward function

Multi-objective optimization provides a rigorous theoretical framework for integrating the competing design objectives inherent to GRN reconstruction. A candidate rule set must (i) accurately capture empirical state-transition trajectories, (ii) ensure topological simplicity to preserve interpretability, and (iii) adhere to established mechanistic priors within empirical constraints. Simultaneous optimization of these criteria generates regulatory models that exhibit both predictive accuracy and biological plausibility. We ensure this by considering the following criteria:

(1) Inconsistency Error (IE): predictive fidelity is quantified by counting the mismatches between model predictions and observed gene-expression transitions:

where is the measured state of at , is the indicator function that maps a true or false value to 1 or 0, respectively, and is the state forecast by from the configuration at . A lower value indicates that the inferred logic recapitulates the experimental trajectory more accurately.

(2) Function Complexity (FC): Each input variable in the Boolean function of corresponds to a regulator of . Given a Boolean function , we evaluate its complexity as the number of input variables, termed as :

(3) Prior information consistency (PlC): To prioritize TFs with strong functional associations and topological importance in PPI networks, we design an optimal objective that integrates edge weights and node degrees:

Where is the interaction weight between and , is candidate TF sets in Boolean function, and denotes the degree of in the PPI database. The degree normalization between candidate TFs, acts as a normalization factor to mitigate hub bias. This objective function encourages that candidate TFs with strong interactions are more likely to be selected together rather than in isolation, thereby enhancing the biological plausibility of the inferred GRN.

All three criteria are incorporated as objectives in a unified multi-objective optimization framework. To approximate the Pareto front, we minimize a weighted linear aggregate of these objectives with respect to , and these three objectives form the multi-objective optimization problem as follows:

(1)

The sparsity parameter is selected via cross-validation to balance model simplicity and predictive accuracy. The prior information consistency parameter is user-specified to control the influence of prior TF–TF interactions on coefficient smoothing.

Reward function

During the simulation stage of MCTS, each candidate expression tree is scored by a scalar reward derived from the aggregate loss (Eq. 1) and a length-based penalty:

where is the number of nodes visited in the tree’s pre-order traversal and denotes the penalty constant. The numerator exponentially penalizes overly large expressions, thereby favoring concise and interpretable rules, while the denominator rewards trees that achieve lower multi-objective loss values. Hence, higher rewards are assigned to compact expressions that realize superior trade-offs among predictive accuracy, sparsity, and prior consistency, effectively guiding the search toward biologically plausible and computationally efficient solutions.

Benchmark datasets

We generated simulated datasets with GeneNetWeaver (GNW) [45], which performs stochastic simulations of gene-expression dynamics based on thermodynamics-based transcription–translation models on network topologies derived from empirical connectivity patterns in Escherichia coli and Saccharomyces cerevisiae. To emulate single-cell data, we simulated thousands of time-series experiments per network under default settings and sampled mRNA at the 21 default time points (0–1000; step 50). The benchmark comprised 30 networks derived from E. coli and S. cerevisiae transcriptional networks—10 networks for each size (30, 60, and 100 genes), with five per species per size. Across all networks, we sampled 50 cells at each time point (1050 cells per network) and then created four replicas by applying random dropout events (each gene has a 50% probability of being set to 0) at 5%, 10%, 20%, and 30%. We obtained four different scRNA-seq datasets, two from mouse, one from yeast and one from human (Supplementary Table S1). These datasets encompass a total of four cell types. Each dataset was preprocessed following the procedure described in the corresponding publication. Following prior work [46, 47], we complied multiple experimentally validated regulatory interaction networks from public databases and primary literature to serve as gold-standard references for evaluating our network-inference algorithms.

Benchmark methods

We evaluated recent regulatory network inference algorithms using diverse approaches—including Boolean model, probabilistic models, and information theory—for both bulk and scRNA-seq data. Each algorithm is briefly summarized below (Supplementary Table S5).

SCNS [48]

It reconstructs executable BNs from binarized single-cell time-course data. Under a single-gene-flip adjacency assumption, it constructs a state-transition graph and formulates per-gene rule synthesis as a SAT problem using a predefined Boolean functional form.

BPN [49]

BPN firstly reduces the search space by preselecting candidate regulators via partial correlation. It embeds cells using diffusion maps and applies Dijkstra’s algorithm to identify the shortest path from initial to terminal cell; this trajectory constrains the Boolean optimizer to states on or near the path, reducing computational cost. The hypothesis space is further restricted by a fixed Boolean rule.

GENIE3 [50]

GENIE3 decomposes GRN inference into multiple supervised learning tasks: for each TG, an ensemble of regression trees (Random Forest or Extra-Trees) is trained using the remaining genes as predictors, and variable-importance scores define edge weights. Aggregating these scores across all targets yields a directed, weighted network, from which edges are selected via thresholding or top-k ranking.

SINCERITIES [50]

SINCERITIES infers GRNs from timestamped single-cell expression profiles. It models temporal evolution of gene expression distributions using regularized linear regression, estimating directed influences under the assumption that transcription-factor changes drive subsequent target-gene expression changes.

PIDC [25]

PIDC infers GRNs from single-cell data using information theory. For each regulator–target pair, it computes mutual information and, across gene triplets, applied partial information decomposition to retain the regulator’s unique contribution while discounting redundancy mediated by other genes. Summing these contributions across all contexts generates an edge score; edges are ranked to form the network.

Results

LogicSR shows state-of-art performance in extensive benchmarks

To systematically evaluate the performance of our approach on GRN inference, we benchmarked it against several SOTA methods selected based on prior benchmarking studies. The evaluated methods include GENIE3 [51], SINCERITIES [50], PIDC [25], SCNS [48], and BPN [49]. Among them, GENIE3 is one of the most widely used tools in GRN reconstruction, while SCNS and BPN are logic-based approaches.

For robust and fair evaluation, we utilized both synthetic and real benchmark datasets. The synthetic datasets were generated using GNW [45], a widely accepted benchmark in GRN research. The real datasets included published scRNA-seq datasets from human, mouse, and yeast, covering diverse biological contexts and network sizes [47].

Given the inherent sparsity of GRNs, we adopted evaluation metrics more appropriate than simple classification accuracy. In particular, we used the Area Under the Receiver Operating Characteristic Curve (AUROC), and Area Under the Precision-Recall Curve (AUPRC) which are better suited for imbalanced settings where true regulatory links are rare compared to non-interactions.

Results for synthetic dataset

We first evaluated the accuracy of LogicSR using simulated datasets with known ground-truth networks. These datasets were generated by simulating stochastic differential equations encoding predefined regulatory structures, enabling principled performance assessment. Specifically, we used data produced by GNW, a widely adopted simulator for benchmarking gene expression and GRN inference. The benchmark included 30 networks derived from E. coli and S. cerevisiae transcriptional networks, with 10 networks for each network size (30, 60, and 100 genes), comprising five networks per species per size. LogicSR consistently demonstrated superior performance across all benchmark networks. In terms of AUROC, it ranks among the top-performing methods, achieving results comparable to the best baseline algorithms. However, its advantage is more pronounced in AUPRC, where it consistently outperformed all competitors. On average, LogicSR surpassed the strongest baseline, GENIE3, by ~0.16 in AUPRC, underscoring its effectiveness in recovering true regulatory interactions under class imbalance (Fig. 3A).

Figure 3.

Performance comparison of GRN inference methods on synthetic datasets. (A) Box plots showing AUROC and AUPRC scores of six GRN inference methods. (B) Robustness evaluation under increasing levels of dropout. LogicSR consistently outperformed other methods, particularly at higher noise levels. (C) AUPRC performance across varying sample sizes. LogicSR and GENIE3 demonstrated greater improvement with increasing sample size. (D) Dot plot summarizing AUPRC values for six datasets simulated from E. coli and S. cerevisiae networks of different sizes (30, 60, and 100 genes).

We further assessed LogicSR’s robustness under increasing noise levels (5%–30%). As shown in Fig. 3B, it maintained the highest AUPRC across all conditions, with minimal performance degradation even at 30% noise. In contrast, other methods, including GENIE3 and PIDC, exhibited substantial declines. LogicSR also demonstrated strong data efficiency, achieving high AUPRC with relatively few samples and rapidly converging as sample size increased (Fig. 3C). Moreover, it maintained stable and high performance across networks of varying sizes (30, 60, and 100 genes) and species backgrounds (E. coli and S. cerevisiae), highlighting its scalability and generalizability (Fig. 3D). To evaluate the distinct impact of TF-TF prior data, we developed a variant called “LogicSR-np” that functions identically to the complete LogicSR framework but omits the TF-TF prior. As demonstrated in Figs S3–S5 available online at http://bib.oxfordjournals.org/, LogicSR incorporating the prior consistently surpassed the prior-free version across all evaluation metrics.

Although SCNS and BPN also adopt logic-based frameworks, they consistently underperformed across all scenarios. In contrast, LogicSR—despite being logic-based—substantially outperformed both, indicating that the success of logical models critically depends on the underlying inference strategy. Its SR–driven framework enables more accurate and robust reconstruction of gene regulatory logic.

These results collectively highlight LogicSR’s strengths in accuracy, robustness, and adaptability. Beyond predictive performance, LogicSR offers structural interpretability by producing explicit Boolean rules. Unlike probabilistic methods that require threshold tuning to infer network edges, LogicSR directly outputs interpretable binary regulatory links, enabling more transparent and causally grounded network reconstruction.

Results for real dataset

We further evaluated LogicSR on four experimentally derived gene expression datasets, including yeast [52], mESC [53], mDC [54], and hESC [55], to assess its ability to reconstruct biologically meaningful GRNs. To comprehensively evaluate GRN inference performance, we adopted a dual assessment framework capturing both local edge-level accuracy (Fig. 4A and C) and global topological structure (Fig. 4B). For edge-level evaluation, we compared predicted regulatory interactions against reference datasets using standard metrics, including AUPRC and the AUROC. Following prior work [46]., we employed three types of ground-truth datasets to accommodate different aspects of biological prior knowledge: Cell-type-specific ChIP–seq (Cs-ChIP), Nonspecific ChIP–seq (Ns-ChIP) and STRING functional networks. To assess global topological fidelity, we examined hub-related structural features—specifically betweenness and degree centrality, which reflect the presence of regulatory masters—and quantified their concordance with reference networks via the Jaccard coefficient.

Figure 4.

Benchmarking on real gene expression datasets. (A) Relative ranks (1 = best) for AUROC and AUPRC across four datasets and three reference sets. (B) Topological concordance with references, measured as the Jaccard similarity of betweenness and degree-centrality profiles. (C) AUPRC as a function of network size (100, 300, 500, and 700 genes); LogicSR consistently outperforms all baselines across scales, while BPN and SCNS are reported only for 100-gene networks due to computational constraints.

Figure 4A presents algorithmic rankings (with 1 denoting the top-performing method) derived from independent sorting of AUROC and AUPRC values, enabling scale-independent comparisons across datasets and prior knowledge sources. LogicSR achieves ranks 1 or 2 in AUPRC for 11 of 12 dataset-reference combinations and never falls below rank 3 in AUROC, demonstrating clear dominance over baseline approaches. Due to computational constraints, BPN and SCNS were restricted to 100-gene networks but still significantly underperformed compared to LogicSR, further highlighting our method’s practical computational efficacy. Subsampling analysis (Fig. 4C) reveals exceptional data efficiency: with only 100 genes, LogicSR exceeds 0.52 AUPRC in yeast and hESC, while the closest competitor remains below 0.40; although the absolute gap narrows as gene count expands to 700, LogicSR maintains the first place, underscoring rapid convergence and scalable performance. To assess higher-order structural fidelity, we quantified betweenness and degree centralities of inferred networks and computed Jaccard similarities to reference architectures (Fig. 4B); LogicSR again establishes supremacy, outperforming the second-best method by an average of 20%, affirming its superior preservation of hub configurations and master-regulatory frameworks. Collectively, these results establish LogicSR as the most precise, and topologically faithful GRN inference method.

Identification of key TFs in hESC differentiation

To demonstrate LogicSR’s capacity to recover master regulators, we applied LogicSR to a human ES cell dataset, which contains cells developing from human ES cells to induced definitive endoderm (DE) cells at 0, 12, 24, 36, 72, and 96 h of differentiation [56]. LogicSR first inferred an interpretable TF–target Boolean regulatory network, from which we defined TF-specific regulons. We then quantified TF activity on a per-cell basis using AUCell [24], which scores the enrichment of each regulon within a cell’s ranked expression profile, with higher values indicating greater regulon activation in that cell.

For each TF, we extracted its regulon as all predicted targets in the high-confidence GRN reconstructed at each time point and recorded its size; network connectivity density was then summarized as the mean regulon size across TFs per time point. As shown in Fig. 5B, connectivity density increases from 0 to 24 h, peaks at 36 h, and subsequently declines at 72 and 96 h to intermediate levels. This transient peak indicates a short-lived intensification of regulatory connectivity and suggests pronounced network rewiring around the 36-h checkpoint during hESC to DE differentiation. Consistent with this pattern, prior studies have reported marked upregulation of mesodermal and endodermal markers within 36 h of induced differentiation in hESCs, supporting 36 h as a critical inflection in lineage commitment [57]. Figure 5C and F depict the temporal trajectories of TF activity during the hESC to DE transition. Activities of pluripotency factors (POU5F1, SOX2, and NANOG) decline sharply over the first 24 h, consistent with loss of stem-cell identity [58]. Mesoderm-associated TFs (TBXT, FGF4, and CDX1) show a transient surge peaking at 24–36 h, followed by attenuation. Endoderm specification factors display delayed activation: EOMES rises at 24–36 h and precedes robust increases in SOX17, FOXA2, GATA6 that culminate at 72–96 h [59]. Figure 5D presents a two-dimensional UMAP projection constructed from the activity matrix of the top LogicSR-derived TF regulons. Cells segregate according to sampling time, forming six well-defined clusters that trace the progression from pluripotency (0 h) through intermediate stages (12, 24, and 36 h) to late DE (72 and 96 h). Notably, the 72 h and 96 h populations overlap extensively, indicating that by 72 h the cells have largely converged on a common transcriptional profile that remains stable at 96 h. These activity trajectories recapitulate the canonical ordering of lineage programs and align with expression-based trends for the same markers, demonstrating that LogicSR-based TF activity estimation effectively identifies critical time points in the hESC differentiation process while also pinpointing key regulators driving this differentiation.

Figure 5.

LogicSR identifies driver regulators during cellular differentiation. (A) Experimental timeline of hESC to DE differentiation and schematic of GRN-based TF–target regulons. (B) Temporal profile of mean regulon size, expressed as the average number of target genes per TF across differentiation time points. (C) TF activity and mean gene expression of canonical markers for each developmental stage (D) UMAP plot based on the TF activity matrix. (E) GATA6 regulon at 96 h, showing extensive endoderm-specific targeting. (F) Single-cell TF activity maps of representative TFs.

Discussion

Inferring dynamic gene regulatory processes deepens our understanding of cellular regulation in development and disease and enables the discovery of key regulators driving differentiation. Single-cell technologies have made it possible to reconstruct GRNs with unprecedented cellular granularity, yet inferring dynamic GRNs at single-cell resolution remains challenging. The first challenge is the sparsity and noise in the data, which complicates the accurate capture of underlying regulatory relationships. Second, inference algorithms often treat TFs in isolation, typically relying on correlation-based methods, without considering the combinatorial nature of their regulatory effects. Finally, the search space for even moderate-sized GRNs is NP-hard, and as a result, current methods frequently perform only marginally better than random predictors.

We developed LogicSR, a SR framework designed to uncover how combinations of TFs regulate genes. It achieves this by modeling their complex interactions using explicit and intuitive Boolean rules. We employ a Boolean model, which uses binary representation for gene expression, as numerous studies have shown that gene expression exhibits bimodality distributed at the single-cell level [60]. Synchronous models update all genes simultaneously—this is computationally straightforward but less biologically realistic, as genes do not change at uniform rates. Asynchronous models update genes one by one—this is more biologically plausible but poses greater challenges in design and analysis. For large networks, there is currently no efficient method to enumerate all attractors under either approach, underscoring the trade-off between realism and computational feasibility [37]. Beyond its current use in inferring regulatory rules, the BN framework demonstrates significant power and is extensively applied in systems biology for in-depth network analysis, including attractor dynamics [38], and network controllability [36]. To efficiently identify the most relevant rules, LogicSR employs a MCTS guided by existing biological knowledge (soft priors), which accelerates the process without preventing the discovery of novel interactions. The framework simultaneously optimizes for model accuracy, simplicity (parsimony), and consistency with prior knowledge, ultimately producing human-readable rules that offer clear and direct insights into the underlying biological mechanisms. Our extensive benchmarking reveals that LogicSR is more robust to noise inherent in single-cell data, recovers global network topology more faithfully, and achieves higher AUROC/AUPRC scores than current probabilistic and correlation-based methods. Moreover, its utility is demonstrated in a case study of hESC to DE differentiation, where LogicSR-derived rules enabled precise, temporally-resolved identification of master regulators. Collectively, these features position LogicSR as a practical and biologically transparent tool for dissecting dynamic regulatory circuits at the single-cell level.

While LogicSR is a powerful tool, its current design has several limitations that motivate future development. The method’s reliance on prior biological knowledge, while beneficial, can introduce biases and restrict its application to well-studied organisms or pathways. Furthermore, although LogicSR is designed to handle dynamic processes, accurately capturing temporal dynamics remains challenging, especially when data lacks time resolution or has suboptimal sampling. Combining chromatin accessibility data and CRISPR perturbation screens enhances accuracy by constraining candidate regulators through context-specific regulatory element activity and TF motif occupancy, and by offering causal, signed perturbation evidence to confirm edges and refine inferred Boolean functions. Future work integrating multi-omics information into our model would enhance performance and provide deeper insights into biological processes of interest. Inspired by DeepRIG’s [30] utilization of global network structure, we plan to incorporate graph-based priors in the future to improve regulator selection and direct the MCTS search. By addressing these challenges and further extending the framework, LogicSR has the potential to become an even more powerful tool, enabling more accurate and comprehensive insights into dynamic gene regulation across diverse biological contexts.

Key Points

• A novel prior-guided symbolic regression model, LogicSR, is proposed for inferring gene regulatory networks from scRNA-seq data collected across multiple time points.

• LogicSR encodes combinatorial TF coordination in target-gene regulation as a logic model and solves rule discovery via MCTS-based multi-objective optimization.

• We demonstrated the superior performance of LogicSR on synthetic and real scRNA-seq benchmark datasets, comparing with existing methods.

• We also showcased LogicSR’s predictive power in effectively identifying key transcription factors during hESC differentiation.

Conflict of interest

None declared.

Funding

This work was partially supported by the Noncommunicable Chronic Diseases-National Science and Technology Major Project [No. 2024ZD0531902]; the National Key Research and Development Program of China [No. 2020YFA0712402]; the National Natural Science Foundation of China [Nos. 62373216, 92374107]; Shandong Provincial Natural Science Foundation [No. ZR2024MF015].

Data availability

All datasets used in this paper are publicly available. The scRNA-seq expression matrix is available under the following accession numbers: GSE114952(mESC), GSE48968(mDC), GSE107552(hESC), GSE125162(yeast). AUCell computations were performed using the AUCell package https://github.com/aertslab/AUCell. Monte Carlo Tree Search implementation was adapted from https://github.com/suragnair/alpha-zero-general.