Cell lineage tracing has evolved into a rigorous quantitative discipline, enabling the inference of complex cellular dynamics from static snapshots of genealogical trees. This Review examines representative mathematical and statistical frameworks that underpin this field, including selected recent contributions from our own work. We categorize the current inference landscape into three hierarchical scales: (1) Population Dynamics, where we discuss the use of Markov branching processes and coalescent theory—including our work on quantifying division and death rates—to decode progenitor pool behaviors; (2) Cell Fate Dynamics, focusing on the integration of transcriptomic information with lineage topology, highlighting our development of velocity-based models for reconstructing continuous state transitions; and (3) Gene Regulatory Dynamics, exploring how lineage structures serve as indispensable priors for inferring directed regulatory networks. By addressing the fundamental challenge of identifiability, this review synthesizes how these multi-scale frameworks allow researchers to move beyond descriptive mapping toward a quantitative understanding of tissue morphogenesis and tumor evolution.

Deciphering stochastic gene regulatory dynamics and their governing network architectures remains a central challenge in systems biology. While single-cell sequencing advancements have significantly enhanced our understanding of large-scale gene regulatory networks, these snapshot measurements inherently lack temporal resolution, thereby limiting our ability to capture underlying dynamical processes. Current dynamical reconstruction approaches face complementary limitations: data-driven methods usually lack interpretation of molecular mechanisms, whereas model-driven strategies usually lack integration of quantitative experimental data. Here, we introduce RESIDE, a unified single-cell framework that integrates model- and data-driven methods for stochastic dynamics reconstruction, coupled with concurrent network structure inference and quantification of steady-state distributions, vector fields, and underlying landscapes. Across the tested in silico systems, RESIDE showed favorable performance relative to the compared methods. Applied to single-cell data from mouse preimplantation development, it recovered differentiation features consistent with the observed cell-state organization and predicted directionally asymmetric transdifferentiation paths. Overall, RESIDE provides a mechanistically interpretable framework for jointly inferring network interactions and effective stochastic dynamics from single-cell snapshot data, with potential value for hypothesis generation and experimental investigation.

We study a chemotactic Ross–Lotka malaria model on a compact metric graph, where spatial movement is described by graph diffusion together with mosquito chemotaxis toward infectious humans. Using the Crandall-Rabinowitz theorem together with a higher-order expansion, we derive a second-order Lyapunov-type coefficient, which determines the local stability of the bifurcating branch. Near the bifurcation point, the stable nonconstant steady state is locally unique and dominated by the critical Laplacian eigenmode. Numerical simulations are presented on a diamond-shaped graph, which illustrate that the chemotaxis can induce the occurrence of Turing patterns, and the emerging pattern is selected by the first nonconstant Laplacian mode.

We study traveling wave solutions in a nonlinear reaction-diffusion model incorporating a weak spatiotemporal distributed memory kernel. The model describes a single-species population whose movement is influenced by both random diffusion and memory-based dispersal, with the latter expressed as a convolution term involving a temporal weighting function and a spatial Green’s function. This framework captures the gradual decay of spatial memory and its effect on dispersal dynamics. Using perturbation expansions, operator theory, and the Banach fixed-point theorem, we establish the existence of traveling wavefronts connecting equilibrium states in two parameter regimes: (i) a small memory-based diffusion coefficient and (ii) a large wave speed. The analysis addresses significant challenges arising from the nonlocal, nonlinear memory term by employing integral equation representations and precise estimates. Numerical simulations illustrate how the memory diffusion coefficient and mean delay influence wave speed, front shape, and population distribution. The results provide a rigorous characterization of wave propagation in systems with weak distributed memory, offering a unified approach applicable to models in population dynamics, biological invasion, and other spatiotemporal processes with memory effects.

Evolutionary stability (or evolutionarily stable strategy) is the core concept of evolutionary game theory. Although evolutionary game theory has achieved great success in evolutionary biology, economics and social science, etc. over the past forty years, how to understand the impact of environmental stochastic fluctuations on the evolutionary stability is still one of the most challenging issues in evolutionary game theory. This paper mainly introduces the recent advances in the study of stochastic evolutionary stability and stochastic Nash equilibrium within the framework of stochastic evolutionary game dynamics. These studies may provide a possible way to better understand the complexity of evolutionary game dynamics in real world situations.

Adaptive therapy (AT) is designed to postpone the emergence of drug resistance by exploiting evolutionary competition among tumor subclones. Most mathematical models of AT assume a binary population structure of drug-sensitive and drug-resistant cells, which neglects the continuous nature of phenotypic plasticity. In this study, we propose a mathematical model that integrates a continuous drug susceptibility index with a probabilistic inheritance function to describe clonal dynamics under therapy. The resulting integro-differential system generalizes traditional two-type competition models and captures both heterogeneity and plasticity of tumor cells. Analytical and numerical studies show that (i) continuous therapy drives rapid expansion of resistant clones, (ii) adaptive therapy maintains long-term tumor control by dynamically regulating sensitive populations, and (iii) high phenotypic plasticity accelerates phenotype switching, leading to earlier tumor relapse following continuous therapy. These results identify critical parameter regimes where adaptive therapy outperforms fixed regimens and highlight the essential role of plasticity in shaping treatment outcomes. The proposed framework provides a more realistic mathematical foundation for the design of clinically relevant adaptive therapy strategies.

Adaptive locomotion of living organisms contributes to enhancing their competitive abilities and helps to maintain their fitness in diverse environments. To date, our understanding of searching behavior and its ultimate cause remains limited, partly because integrative experimental–theoretical studies are still scarce in ecology and evolutionary biology. Here, we investigate how motion patterns of biofilm-inhabiting marine raphid diatom (Navicula arenaria var. rostellata) optimize nutrient search efficiency in two-dimensional space. We report that individual Navicula cells display a universal “circular run-and-reversal” motion behavior at different concentrations of dissolved silicic acid. We then show that the gliding motions of diatom cells can be elucidated accurately by behavioral optimization of the rotational diffusivity, reversal rate, and angular velocity coupled to a Langevin model. We hypothesize that the optimal search efficiency is achieved near a critical region of specific behavioral parameters. Using the maximized diffusivity as the search-efficiency metric, our theoretical results demonstrate that these optimal parameters coincide with the experimental observations on diatom cells undergoing natural selection.

In this paper, we focus on efficiently and flexibly simulating the Fokker-Planck equation associated with the nonlinear noisy leaky integrate-and-fire model, which reflects the dynamic behavior of neuron networks. We apply the Galerkin spectral method to discretize the spatial domain by constructing a variational formulation that satisfies complex boundary conditions. Moreover, the boundary conditions in the variational formulation include only zeroth-order terms, with first-order conditions being naturally incorporated. This allows the numerical scheme to be further extended to an excitatory-inhibitory population model with synaptic delays and refractory states. Additionally, we establish the consistency of the numerical scheme. Experimental results, including accuracy tests, blow-up events, and periodic oscillations, validate the properties of our proposed method.

Mass spectrometry-based single-cell proteomics (MS-SCP) enables the quantification of protein abundance in individual cells, offering a molecular perspective on post-transcriptional regulation and heterogeneity that cannot be inferred from transcriptomic data alone. However, MS-SCP data exhibit high rates of missing values, batch effects, and low-input noise, which require tailored computational models. In this review, we examine computational developments across the MS-SCP pipeline, covering protein identification and quantification, public repositories, data enhancement, and downstream analysis. We emphasize algorithms that integrate statistical modeling and deep learning for identification, quantification, and the joint correction of missingness and batch effects. We highlight the unique role of deep learning in modeling non-linear batch-dependent effects and learning robust protein representations from sparse, high-dimensional MS-SCP data. Finally, we outline future directions for method developers, including the incorporation of biological priors, the construction of abundance-level foundation models, the curation of single-cell perturbation datasets, and the integration of proteomic information with spatial and multimodal data.

Imaging-based spatial transcriptomics provides single-cell resolution but is restricted to targeted gene panels, leaving most transcriptional variation unexplored. Existing imputation approaches often rely on latent-space alignment, which can distort biological structure and capture limited within-type heterogeneity. We present BRIDGE, an interpretable framework that expands panel-limited spatial transcriptomics to transcriptome-level resolution through anchor-guided linear calibration and multi-scale neighborhood-based prediction. BRIDGE learns a global calibration matrix that directly adjusts shared-gene expression between technologies and provides explicit gene-level interpretability. Using the calibrated reference, BRIDGE predicts unmeasured genes by integrating multi-scale local neighborhoods to achieve both accuracy and robustness. Across seven datasets from CosMx, MERFISH, Xenium and multiple tissues and species, BRIDGE exceeds or matches existing methods on gene-level and cell-level accuracy. The calibration matrix offers clear biological interpretation, and the completed transcriptomes recover fine-scale spatial patterns such as cortical lamination and support refined characterization of cell-state heterogeneity, including B-cell states and Stromal subtypes in human breast cancer. BRIDGE provides a robust and interpretable solution for extending imaging-based spatial transcriptomics to transcriptome-scale analysis and enables deeper investigation of microenvironment-dependent cellular programs.