Beyond metaphor: quantitative reconstruction of Waddington landscape and exploration of cellular behavior

Abstract

Originally proposed as a conceptual metaphor, the Waddington landscape was used to illustrate the directional nature of embryonic development and the relative stability of distinct developmental states. While the Waddington landscape offers a valuable conceptual framework for understanding cellular dynamics, its quantitative reconstruction remains a significant challenge in systems biology and biophysics. Recent methodological advances in single-cell omics technologies, computational modeling approaches, and nonlinear dynamical systems theory have facilitated progress toward quantitative reconstruction of the Waddington landscape, thereby transforming this heuristic metaphor into a predictive theoretical framework. In this review, we summarize the theoretical foundations of the Waddington landscape, categorize current computational and mathematical approaches for the Waddington landscape reconstruction. Meanwhile, we highlight the potential applications and inherent limitations of these approaches in characterizing cellular behaviors, predicting cell fate decisions, and modulating developmental trajectories.

Introduction

The genome, serving as the fundamental code of life, harbors boundless potential and gives rise to a wide variety of cellular behaviors [1]. The exploration of the genome has profound parallels with the scholarly deciphering of oracle bone inscriptions [2]. These two require meticulous archeological reconstruction to elucidate: (i) primordial implications or functions, (ii) evolutionary trajectories, and (iii) mechanistic roles within specific environmental contexts of each individual element (gene or oracle bone inscription). Meanwhile, systematic analysis and research on the synergistic interactions of multiple individual elements (interaction rules of genes, grammar rules of oracle inscriptions) is essential, as the understanding of a single individual element will not automatically translate into a comprehensive grasp of the whole system containing multiple individual elements [3, 4]. Although contemporary researches have achieved groundbreaking advancements in elucidating the basic connotations and principles of the genome [5–7], these still do not constitute a comprehensive understanding of the complex cellular behaviors induced in organisms.

Conrad Hal Waddington conceived his seminal Waddington landscape as a conceptual framework to facilitate the comprehension of intricate cellular behaviors, particularly those governing cellular differentiation [8, 9]. The Waddington landscape ingeniously likens the dynamical process of cell differentiation to a ball rolling downhill, aptly capturing its series of carefully orchestrated, distinct, and gradual phenotypic transitions. The transitions commence from an initial state of totipotency or pluripotency and ultimately reach different final committed states with the specific location and function [10]. Notably, contemporary experimental evidence strongly supports that the reprogramming process can reverse the differentiation trajectory and facilitate the transition of cells from a differentiated state to a more potent or even totipotent state [11–13] through the careful regulation of a limited set of transcription factors. In addition, recent studies have revealed the potential to transfer a differentiated cell to another kind of differentiated cell type without reprogramming, which is defined as programmed transdifferentiation [14, 15]. All of these have led to the expansion of the Waddington landscape for elucidating the reprogramming process and programmed transdifferentiation [16]. In general, the Waddington landscape provides a powerful lens to explore the dynamical process governing cellular behavior [17, 18].

However, for many years following its proposition, the Waddington landscape persisted primarily as a vibrant metaphor or a metaphysical construct [9] due to the lack of corresponding quantitative methods. The maturation of the energy landscape [19, 20], coupled with revolutionary advances in sequencing technologies [21], has enabled the operational reconstruction of the Waddington landscape through quantitative and predictive paradigms. A growing number of computational and mathematical approaches have been proposed from various perspectives, which provides significant guiding value for gaining the deeper understanding of the molecular mechanisms underlying cellular behavior and elucidating biological issues, such as identifying key genes and regulations controlling cell behavior, inferring transition cells in the dynamical process of cellular behavior, mining biomarkers for different cell types, etc.

In this review, we commence by providing a comprehensive introduction to the conceptualization of the Waddington landscape. Afterwards, we provide a comprehensive overview for a large number of the Waddington landscape reconstructing approaches in a systematic manner, which can be mainly categorized into model-based and data-based approaches. Finally, we highlight key challenges and applications of these computational approaches, discuss the framework for leveraging them to build an effective system, and point out potential future directions.

The conceptual framework of the Waddington landscape

The fundamental concepts of the Waddington landscape, including state space and potential, stable states and attractors, trajectory and bifurcation, are outlined to elucidate the composition of the Waddington landscape, the qualitative relationships between stability and corresponding cell states, as well as the mechanistic connections between bifurcations and critical cell fate transitions.

State space and potential

The Waddington landscape serves as a metaphorical 3D topological surface that mechanistically encodes the dynamical transition process of cell phenotypic states (the dynamical process of cellular behavior) [22]. And there are three dimensions X, Y, and Z in Waddington landscape, where the 2D XY plane represents the cell phenotypic state space and the Z dimension represents the potential of the cell phenotypic state, which is shown in Fig. 1A.

Figure 1.

The illustrative diagram of the Waddington landscape. (A) The metaphorical 3D topological surface of the Waddington landscape; (B) the conceptualization of the Waddington landscape; (C) the reconstruction of the Waddington landscape; and (D) the application of the Waddington landscape.

To conceptualize this 3D topological surface, it is necessary to establish a foundational framework rooted in molecular biology for characterizing cell phenotypic states. Central to this paradigm lies the central dogma of molecular biology [23], wherein transcriptomic signatures emerge as cardinal determinants of phenotypic identity. Each cell phenotypic state establishes a biological observation uniquely associated with the gene expression pattern [24], formally represented as a state vector in -dimensional state space , where is the expression level of gene .

Subsequently, given that the original state space is high-dimensional, dimensionality reduction is required to visualize the Waddington landscape in a 3D configuration. Specifically, the N-dimensional state space must be transformed into a 2D state space that preserves the topological structure of cellular phenotypic states, such as the preservation of neighborhood relationships. In this reduced space , adjacent points (or states) signify similar gene expression patterns, thereby enabling a more intuitive understanding of the underlying cellular dynamics. Under this specified dimensionality reduction mapping, the cell phenotypic state is projected into a corresponding 2D state vector , where are the coordinates of the cell phenotypic state in the XY plane.

Meanwhile, the potential function should be reconstructed to quantify the potential of cell phenotypic state as the Z-dimension in the landscape. The potential characterizes the stability of cell phenotypic state , which dominates the transition of cell phenotypic states during the dynamical process of cell behavior. For instance, during cell differentiation, cells will transition to a more stable phenotypic state, i.e. mature and differentiated cells demonstrate higher stability and greater homeostatic resilience, whereas undifferentiated cells retaining totipotent or pluripotent potential demonstrate heightened dynamical instability.

Thus, the transition of different cell phenotypic states on the 3D topological surface of the Waddington landscape is depicted by the potential of each cell phenotypic state at the given XY position, and serves dual mechanistic interpretations. On the one hand, the landscape elegantly models the dynamical process of cell differentiation as a ball rolling downhill for higher stability, commencing from an initial state of totipotency or pluripotency and ultimately reaching a final committed state with a specific location and function, which is shown as green trajectory in Fig. 1A. This downhill trajectory represents progressive restriction of differentiation potential. On the other hand, through the careful regulation of a limited set of transcription factors, the reprogramming process can reverse the differentiation trajectory and facilitate the transition of cells from a differentiated state to a more potent or even totipotent state, which is contrary to the downhill rolling ball symbolizing cell differentiation and can be manifested as a ball climbing uphill in the Waddington landscape, which is depicted as red trajectory in Fig. 1A. Additionally, programmed transdifferentiation differs from the aforementioned two processes in its manifestation within the Waddington landscape. It can be represented as a ball first climbing uphill and then rolling downhill, as is illustrated by the blue trajectory in Fig. 1A.

Stable states and attractors

The Waddington landscape employs the state vector as a unique identifier for cell phenotypic state, and potential is utilized to quantitatively characterize the stability of cell phenotypic state . Crucially, the concept of stable state in this landscape transcends mere temporal invariance (stationarity). Rather, it embodies Lyapunov stability wherein the system exhibits resilient maintenance of phenotypic identity despite exogenous perturbations [25].

Consequently, the states within this landscape are categorized into stationary and non-stationary. Stationary states are defined by their time-invariance, representing either a static or steady-state condition, such as and shown in Fig. 1B. Conversely, non-stationary states exhibit temporal variability, such as shown in Fig. 1B. Furthermore, it is important to note that not all stationary states can be qualified as stable. Only those stationary states that exhibit resilience are deemed to be stable, i.e. they can return to their original configurations following external perturbations, such as shown in Fig. 1B. This resilience ensures cells to be stable, which is determined by epigenetic modifications that lock specific gene expression patterns [26], such as DNA methylation and histone modifications.

Moreover, the stable state is defined as an attractor and its position in state space is a characteristic of the Waddington landscape. In the current paradigm of biological understanding, genes indirectly control development through the interaction of their corresponding biochemical products. In the landscape, the attractor will attract other states closer to it under the “force” inherent in this molecular interaction [27], and the movement of in state space toward the attractor in response to the molecular interaction. Then the stable state and nearby less-stable states together form a basin of attraction, similar to the potential energy well in classical physics [28]. Given that each cell type is uniquely specified by a distinct and stable gene expression configuration across the genome, attractors within the Waddington landscape serve as characteristic markers for differentiating various cell types in this context.

Experimental support for this concept is provided by the measurement of convergence in genome-wide transcriptional profile trajectories [29]. However, it is important to acknowledge that nominally defined cell types may, in reality, represent a heterogeneous assemblage of daughter cell types [30, 31]. These daughter cell types are correspondingly depicted by multiple and proximate small attractors that coalesce into a more confined and larger attractor within the landscape. In addition, the gene expression profiles of various cell types display distinct gene expression signatures, accompanied by lineage-specific molecular biomarkers, such as CD45+ for hematopoietic lineages [32], that facilitate the precise identification and functional annotation of corresponding attractors.

Trajectories and bifurcations

Under natural conditions, a coherent developmental change of cell phenotypic state is a movement toward a more stable state in the state space [33], equivalent to coordinated alterations in gene expression patterns across the genome, such as cell differentiation. Accordingly, the different molecular interactions cause the different transitions along the different trajectories. These trajectories are unidirectional in time dimension, and with respect to the developmental process of cellular differentiation, they exhibit characteristics that include multiple discrete termini from a single undifferentiated start and robust bifurcations [22], which is shown as green trajectory in Fig. 1A. Each independent, continuous trajectory signifies the temporal succession of gene expression patterns, thereby embodying a contiguous developmental pathway. Waddington termed these pathways “Chreods” [8], “chre” meaning “inevitable” in Greek and “hodos” meaning “pathway” in Greek. The combination of the two stands for the predetermined and stable developmental pathway during cellular behavior, representing the dynamical trajectory of cells toward a specific fate under epigenetic regulation. And Waddington introduced the concept of canalization to represent the ability of cells to stably develop into the same phenotypic state under genetic variation or environmental disturbance [8]. Its core mechanism is to restrict cellular behavior in a predetermined pathway through epigenetic constraints, which is defined as the process whereby developmental trajectory becomes increasingly resilient to perturbations. More specifically, canalization is visualized in the landscape as increased steepness of valley walls. The more canalized a trajectory, the lower the probability that external perturbations will drive development off its course.

In contrast, under special conditions (artificial use of certain transcription factors or specific genetic mutations), the transition of the cell phenotypic state is not necessarily a movement toward a more stable state in the state space, such as reprogramming process, which is shown as red trajectory in Fig. 1A. Correspondingly, its trajectory presented in the Waddington landscape may be in the opposite direction (compared with the developmental trajectories under the original natural conditions), and return to a less stable phenotypic state, such as pluripotent cells. Additionally, environmental factors (e.g. radiation and chemical exposure) can reshape the “topography” of the Waddington landscape by altering epigenetic modifications, thereby establishing new trajectories that drive cells toward a new phenotypic state, such as cancer cells [34].

In addition to trajectories, bifurcations play an equally pivotal role, which refers to the branching points of developmental pathways corresponding to cellular behavior. They not only characterize cell fate decisions but also provide insight into the underlying mechanisms governing cell fate transitions [35]. This enables cells to select and stabilize within a potential fate state when confronted with a multitude of possible fates. However, it is important to note that not all fate transitions are exclusively governed by bifurcations. Occasionally, specific intrinsic noise or, more frequently, external perturbations may facilitate a cell’s “escape” from its current basin of attraction, allowing it to traverse a relatively low energetic barrier and settle into a neighboring attractor (e.g. cellular programmed transformation [36]).

The quantification and reconstruction of the Waddington landscape

As the conceptual framework of the Waddington landscape continues to mature, the challenge of quantifying and reconstructing this landscape from both theoretical models and empirical data has emerged as a focal research area within the domain of analyzing cellular states and dynamics. There are mainly two kinds of approaches to constructing the Waddington landscape: (i) model-based Waddington landscape reconstructing approaches, and (ii) data-based Waddington landscape reconstructing approaches, which is shown in Fig. 1C. The key aspects of these approaches are briefly summarized and discussed, which are given in the next sections.

Furthermore, since the Waddington landscape can effectively model the cell behavior in a low-dimensional space, it can serve as an effective basic tool and develop corresponding algorithms to understand the molecular mechanisms of cell behavior and clarify related biological issues (see Fig. 1D for schematic illustration). For instance, the identification of key regulatory genes governing cell fate decisions can be achieved by quantifying energy barrier fluctuations between adjacent attractor basins; cell type specific biomarker discovery is facilitated through comparative analysis of attractor state configurations, wherein distinct molecular signatures emerge as characteristic features of differential attractor topologies; pathological mechanisms may be elucidated by reconstructing developmental trajectory deviations. Through these applications, the Waddington landscape undergoes comprehensive multidimensional analyses that substantially augment both its methodological value and theoretical implications in cellular biology.

Model-based Waddington landscape reconstructing approaches

Model-based Waddington landscape reconstructing approaches aim to derive predictions about cell behavior, from mathematical descriptions of interacting molecules. In general, model-based approaches typically follow a three-step methodological framework: (i) mechanistic abstraction; (ii) formal mathematical encoding; and (iii) computational landscape reconstruction.

• Mechanistic abstraction involves the simplification of biological knowledge or mechanistic descriptions into a interaction network (i.e. gene regulatory networks [37]), as shown in Fig. 2A. This process typically entails distilling complex interactions into topological interaction networks, where nodes represent molecules such as genes, and edges denote regulatory relationships. It is crucial to acknowledge that some details of the molecular mechanism is omitted in this abstraction, including cellular structures, spatial organization, and molecular structure. Despite this, such abstraction preserves the core logical structure of the system, ensuring clarity and precision in the representation of underlying mechanisms while eliminating potential ambiguities or vagueness associated.

• Formal mathematical encoding involves the translation of the interaction network into a formal mathematical framework, as shown in Fig. 2B. Since the interaction network is a representation of network structure, it is natural to think of using Boolean networks and Bayesian networks [38] for the translation. However, these formal frameworks are infrequently employed for landscape representation owing to their inherent properties, such as conditionality or discreteness. Conversely, dynamic system modeling utilizing ordinary differential equations proves highly effective in simulating network dynamics of cellular system, and is more prevalently utilized for this translation. Moreover, due to the random fluctuations in the dynamics of cell behavior, such as oxygen concentration, the interaction network is usually translated into a stochastic dynamics model.

• Computational landscape reconstruction involves the utilization of numerical solutions to deduce the landscape, as shown in Fig. 2C. Within the framework of modeling stochastic dynamical systems and inspired by Boltzmann’s distribution law in equilibrium statistical mechanics, numerical methods are employed to solve for the steady-state distribution of stochastic differential equations, and then enables the reconstruction of landscape representation.

Figure 2.

An overview of model-based approaches for the reconstruction of the Waddington landscape.

Stochastic dynamics model without proliferation

Utilizing the interaction network as a foundation, cellular behavior is frequently modeled in a simplified manner as cell migration. The cell migration is constituted by the general chemical reaction kinetic model [39] and is mathematically expressed as follows:

(1)

where represents the cell phenotypic state over time; represents the biological interactions between molecular elements in the interaction network; is a parameter standing for the noise amplitude; is a standard Brownian motion with independent components, which encapsulates the diverse and numerous fluctuations observed within the intracellular environment, including but not limited to DNA damage, defective signal transduction, alterations in protein concentration, pH variations, and oxygen consumption [40–42]; is the initial cell phenotypic state. In a practical application, is usually modeled using either activated or inhibited Hill functions, according to the considered interaction network.

The probability distribution of the above stochastic dynamics model is predictable, and denoted as . It can be computed as the weighted expectation of all cells, and follows the well-known Fokker–Planck equation [43, 44]:

The steady-state distribution can be obtained by solving , and the potential of the Waddington landscape is defined by landscape and flux theory [45] as:

In practical applications, there are primarily two methodologies employed for numerically computing the steady-state probability distribution . The most direct method involves the utilization of deterministic numerical techniques, which inherently require the imposition of appropriate boundary conditions. However, the computational expense associated with this strategy escalates exponentially, rendering it infeasible even for systems with dimensions [46]. The other method is the approximation, and can be obtained by the mean field approximation [47] or Monte Carlo simulation [48, 49] in the high-dimensional scenarios.

Based on the above framework, the Waddington landscape is reconstructed from the steady-state distribution of the biological systems and adapted to the analysis of many basic cellular behaviors, e.g. stem cell differentiation [50], budding yeast cell cycle [47], and calcium oscillation [51]. Furthermore, Waddington landscapes of complex diseases have also been constructed to analyze the action mechanisms of specific molecules and disease pathogenesis, e.g. pan-cancer [52], breast cancer [53], and gastric cancer [54].

Stochastic dynamics model with proliferation

From a modeling perspective, cellular behavior is usually characterized as a non-equilibrium process, which is attributed to the dynamical processes of cell birth and death. Consequently, cell behavior should be represented as a coupled system that integrates both cell migration and cell proliferation. Thus, the birth and death term is introduced into the stochastic dynamics (Equation (1)),

where represents the weight of the cell phenotypic state at time . And it also can be transformed into a weighted stochastic dynamics which follows a generalized form of the well-known Fokker–Planck equation:

(2)

Subsequently, the original potential can be disaggregated into two components: potential and by the energy landscape decomposition theory [55]:

where is the steady-state distribution of Equation (2) with the known , and is the steady-state distribution of Equation (2) with . The potential retains the same semantic content as the original potential , where its stable states correspond to distinct cell types. Conversely, potential originates from the proliferative effect, whose value implicates pluripotency and indicates the migratory direction. Similarly, numerous recent studies have employed this framework to investigate various phenomena. Specifically, low-dimensional instances, exemplified by the two landscapes of T-cell differentiation, have been reconstructed to examine the progression through different stages of T-cell development [55]. In parallel, the two landscapes of a high-dimensional case for lung adenocarcinoma have also been reconstructed to delve into the physical mechanisms underlying the composition and development of lung adenocarcinoma [56].

In conclusion, the model-based Waddington landscape reconstructing approaches represent a significant methodological advancement by transitioning from traditional qualitative descriptions to a robust quantitative analytical framework through the implementation of mathematical modeling. This paradigm shift establishes a computationally tractable theoretical foundation for the Waddington landscape, thereby substantially enhancing the precision, reproducibility, and generalizability of developmental biology research. Furthermore, this approach offers novel mechanistic insights into the underlying mechanism governing cellular development, while simultaneously providing a predictive theoretical framework to guide experimental design and hypothesis testing, ultimately accelerating the pace of discovery in related fields.

Data-based Waddington landscape reconstructing approaches

With advancements in single-cell omics technologies [57–59] and time-lapse microscopy [60–62], there exists a significant opportunity to reconstruct and quantitatively assess the Waddington landscape, thereby enabling a thorough analysis of cellular dynamics. Data-based approaches for landscape quantification and reconstruction can be systematically categorized based on the type of data utilized (e.g. single-cell sequencing data, spatial transcriptome data, and time-lapse microscopy data) and the analytical perspective employed (e.g. probability distributions, differential equations, or geometric manifolds).

Nevertheless, regardless of the specific categorization, the core challenge in landscape quantification or reconstruction typically involves either parameterizing the potential function (or its gradient) that selects optimal parameters to capture the observed temporal evolution of cell distribution patterns, or developing an energy metric that accurately captures the observed evolution of cell differentiation potential under the potential . Thus, the data-based Waddington landscape reconstructing approaches can be divided into three main categories: (i) the parameterization for the potential function of the Waddington landscape; (ii) the parameterization for the dynamical trajectory of the Waddington landscape; and (iii) the non-parameterization for the differentiation potential of the Waddington landscape. The key aspects of these approaches are briefly summarized and discussed, which are given in the next subsections.

The parameterization for the potential function of the Waddington landscape

As elucidated in the preceding section, cellular behavior is governed by an intricate interaction network comprising transcription factors and intercellular signals that be frequently conceptualized as a complex stochastic dynamical system. In this context, the Waddington landscape can be quantitatively characterized and reconstructed through the parameterization of potential function or its gradient [63] to establish the corresponding stochastic dynamical system. This parameterization process can be computationally implemented using neural network architectures, wherein the network weights serve as the fundamental parameters. For instance, Hashimoto et al. [64] have demonstrated the application of this approach by modeling cell differentiation processes through diffusive recurrent neural networks. Zhao et al. [65] presented EPR-Net to construct potential landscapes for the high-dimensional non-equilibrium steady-state systems with either moderate or small noise. Nevertheless, it is crucial to recognize that the underlying dynamics regulating cellular behavior are inherently high-dimensional [66], rendering the potential function or its gradient parameterization process computationally intensive.

Fortunately, typical perturbations typically merely alter the proportions of known cell types without giving rise to novel ones, which aligns well with the concepts of generality and structural stability within the theoretical framework of dynamical systems [67]. Consequently, it is permissible to enumerate the qualitative possibilities of dynamical behaviors and to recover the potential function of the Waddington landscape directly with a limited number of parameters (as shown in Fig. 3A), in particular, the bifurcation in the Waddington landscape. Specifically, this involves focusing not on gene expression profiles in the high-dimensional concrete state space, but rather on the geometric properties of the manifold within an abstract low-dimensional space. This type of reconstructing approaches of the Waddington landscape is referred to as geometrical models, which have been used to fit experimental data on cell fate and describe how spatial interactions between cells can be understood geometrically. Such a model is constructed within an abstract mathematical space, where the coordinate axes are arbitrarily defined and solely represent the fate decisions associated with cellular behavior.

Figure 3.

An overview of data-based approaches for the reconstruction of the Waddington landscape.

Geometrical models generally use time-lapse microscopy data in systems that are being perturbed by a couple of external signal controls, which is available in the form of proportions of end point fates as the signals are perturbed. These experimental data are used to identify the attractors and the geometric form of the dynamical landscape. Based on the experimentally validated geometric form, appropriate generic parameterized landscapes (e.g. dual cusp, fold bifurcation, heteroclinic flip, etc.) were systematically selected to construct a comprehensive global landscape model [68–70]. The parameterized landscape uses polynomials with parameters as prototypes, e.g. dual cusp is prototyped using the following polynomial [69]:

where is the parameter. In other words, the qualitative structure of the landscape can be described by a relatively small corpus of universal normal forms [71]. The priori parameters in parameterized landscapes are associated in linear combinations or otherwise with biological identities that influence cell fate decisions, such as effective levels of signals, the strength of lateral inhibition, or coefficient of activation for a transcription factor [35], e.g.

where is the weight of the linear combinations, is the effective level of signal or the strength of transcription factor .

As parsimonious parametric frameworks that recapitulate cell behavioral dynamics, geometric models have been used to calibrate against empirical observations of cell fate determination and describe how spatial cell interactions can be understood geometrically. For instance, Corson and Siggia [72] established a minimal geometric model to elucidate vulval development in Caenorhabditis elegans. This parsimonious model incorporated three distinct cell fate specifications governed by the interplay between epidermal growth factor (EGF) and Notch signaling pathways, which can fit available end-point fate data on perturbations of EGF and levels of Notch ligands and further predict fate outcomes for specific timed perturbations to these signals. Sáez et al. [35] established a geometrical model by the linear combination of binary choice and binary flip, and approximate Bayesian computation to formulate a quantitative dynamical landscape that accurately predicts cell fate outcomes of pluripotent stem cells exposed to different combinations of signaling factors. Raju and Siggia [73] extended flip bifurcation for a single cell to the entire inner cell mass by means of a self-consistently defined time-dependent FGF signal in the mouse blastocyst, which is a very plausible model for any situation where the embryo needs control over the relative proportions of two fates by a morphogen feedback.

Overall, geometric models take a more abstract but mathematically rigorous approach that can reveal the fundamentals of development and can provide immediate quantitative predictions that are difficult to obtain through model-based approaches. Despite geometric models effectively address the issue of parameter redundancy in biological system modeling through the identification of minimal models that are consistent with empirical observations, they confront intrinsic limitations due to the non-convex nature of their parameter spaces. This fundamental characteristic makes it difficult to establish rigorous theoretical guarantees for this approach of directly fitting the landscape [74]. A particularly challenging manifestation of this limitation is the existence of multiple local minima that exhibit comparable goodness-of-fit to experimental data yet correspond to distinct biological interpretations, thereby compromising the reliability of model-based inferences.

The parameterization for the dynamical trajectory of the Waddington landscape

Intriguingly, for certain classes of problems, a strategic reformulation of the optimization objective can yield significant computational advantages. Specifically, while the direct reconstruction of the potential function constitutes a fundamentally non-convex optimization challenge, the alternative convex approaches can be formulated to infer the dynamical trajectories induced by the potential function (as shown in Fig. 3B). These inferred trajectories are the important parts of the Waddington landscape, and provide the valuable insights into the underlying biological mechanisms, thereby maintaining the analytical utility while circumventing the computational limitations associated with non-convex optimization. These convex approaches leverage the mathematical principles of optimal transport (OT) theory, which provides a rigorous foundation for the mass distribution transfer between the probability measures [75] and has been successfully adapted for the analysis of time-series single-cell RNA sequencing (scRNA-seq) data. The emergence of scRNA-seq technology basically solves the problem of how to discover cell classes in a population [76], providing a data basis for tracking the development of each cell class based on optimal transmission.

The methodological evolution in this domain commenced with the introduction of Dynamic OT [77], which extends traditional OT theory by incorporating a temporal dimension, provides an alternative interpretation with notable connections to fluid dynamics, and remarkably results in a convex optimization formulation. However, despite its pioneering approach to temporal data analysis, Dynamic OT exhibited limitations in capturing intricate branching structures, primarily due to its dependence on pairwise comparisons between discrete temporal points. A paradigm shift occurred with the development of Waddington-OT [78], which systematically integrates the concept of the Waddington landscape into the OT framework. This methodology operates on cellular populations represented as probability distributions within gene expression space, employing OT to infer transport mechanisms between consecutive temporal points. To address the inherent non-convexity of the optimization problem, Waddington-OT implements entropic regularization, thereby facilitating smoother transitions between cellular states. Furthermore, it incorporates growth hallmark gene expression to approximate growth as a temporal coupling term, enabling the reconstruction of developmental trajectories within the Waddington landscape. However, the approximation of growth is based on the growth marker genes, and it demonstrates a significant dependency on the selection of biological knowledge databases. This dependency arises because different knowledge bases provide distinct sets of gene annotations and associated biological pathways, such as the Kyoto Encyclopedia of Genes and Genomes [79] and the Gene Ontology [80]. Consequently, the growth marker genes’ lists derived from these databases may exhibit substantial variability, which influences the approximation of growth. Despite this, Waddington-OT represents a substantial progression in the quantitative analysis of developmental processes, offering researchers a sophisticated yet computationally tractable approach to investigate the complex dynamics of cellular differentiation through the Waddington landscape perspective.

Subsequent to these developments, TrajectoryNet [81] emerged as a significant methodological advancement by integrating neural network architectures into the OT framework, which addresses the non-convexity challenge through deep learning-based approximation, provides a smoother trajectory reconstruction and interpolates well to intermediate time points between measured ones. Notably, TrajectoryNet represents the first methodology to explicitly incorporate cellular growth by incorporating it as a separate discrete static unbalanced OT model in the continuous setting. In contrast to Waddington-OT, TrajectoryNet approximates growth by learning the relationships between gene expressions at different time points, identifying genes whose expression changes were associated with cell growth and using their collective expression patterns to infer growth. This innovation bridges dynamical OT and continuous normalizing flows to infer continuous trajectories of cellular dynamics.

The subsequent development of MIOFlow [82] represented a paradigm shift in addressing non-convex trajectory inference challenges through its manifold interpolation. The method reformulates trajectory reconstruction as a manifold-constrained interpolation problem, where developmental dynamics are modeled via neural ordinary differential equations [83] trained to interpolate static cellular population snapshots. Critically, these solutions of the ordinary differential equations are regularized by an OT objective incorporating a manifold-adapted ground metric, ensuring that the inferred flow trajectories conform to the intrinsic geometry of the single-cell data manifold. The architecture operates within a geodesic autoencoder derived latent space, where latent representations are explicitly constrained to preserve multiscale topological relationships observed in the original high-dimensional transcriptomic space. Specifically, MIOFlow introduces a multiscale geodesic regularization that enforces consistency between pairwise latent space distances and a novel data-based geodesic metric quantifying developmental proximity across cell states. For approximating growth, MIOFlow analyzed the gene expression changes across different cell populations in the latent space and implicitly capture developmental dynamics without requiring explicit growth markers. It identified genes that were differentially expressed along the inferred trajectories and used these to build a model of growth, taking into account the geometric relationships between different cell states. This method advances computational cell biology by providing a mathematically coherent framework for lineage inference that respects both the geometric organization of cellular states and the inherent noise structure of scRNA-seq data, which can significantly improve the accuracy of trajectory inference by preserving the geometric structure of the single-cell data manifold.

TIGON, the latest methodological innovation in this domain, synthesizes the most effective attributes of its antecedent approaches [84]. Specifically, TIGON constitutes a deep learning framework designed for the extraction of dynamical patterns and gene regulatory mechanisms from time-resolved single-cell transcriptomic datasets. This methodology enables the simultaneous modeling of gene expression velocity at the single-cell level and population growth dynamics. Technically, TIGON represents a dynamical, unbalanced OT model grounded in the Wasserstein–Fisher–Rao (WFR) metric [85, 86], thereby extending traditional OT theory to accommodate measures with varying mass distributions. This innovative application of WFR-based dynamical unbalanced OT demonstrates significant potential for the integration of temporal datasets while effectively accounting for critical cellular processes, including cell division and apoptosis.

Collectively, these methods represent a progressive refinement of OT-based approaches to reconstruct the Waddington landscape. Each iteration has built upon the limitations of its predecessors, introducing innovative solutions to the challenges of non-convexity and the approximation of cellular growth. The evolution from Dynamic OT to TIGON underscores the field’s ongoing efforts to bridge the gap between theoretical models and the complex reality of cellular development, ultimately providing deeper insights into the mechanisms governing cell fate determination.

Notably, in addition to OT-based methods, several trajectory inference algorithms based on RNA velocity are widely used. For instance, scVelo predicts dynamic cell state changes by inferring gene-specific RNA velocity from complementary unspliced and spliced transcript abundance matrices, enabling trajectory inference [87]. Moreover, CellRank integrates multimodal data including RNA velocity, gene expression similarity, and pseudo-time to construct a directed Markov chain model for predicting cell fate probabilities and developmental trajectories [88]. However, both approaches critically depend on high-quality intronic data, as excessive noise in the input may degrade RNA velocity estimation accuracy.

The non-parameterization for the differentiation potential of the Waddington landscape

The reconstruction of both the continuous potential function and the associated dynamical trajectory fundamentally constitutes the landscape’s reconstruction from a continuous dynamics perspective based on the parameterization. When abstracted from continuous cellular dynamics, the Waddington landscape essentially represents the potential energy of various cell states based on the non-parameterization, which corresponds to their relative stability. Precise characterization of this relative stability is essential for reconstructing the Waddington landscape, albeit through a discrete approach. Consequently, numerous studies have employed methodologies that construct energy metrics to quantify the potential energy of cell states, thereby reconstructing the discrete Waddington landscape (as shown in Fig. 3C). While this approach does not precisely capture the dynamical behavior of cell states, it offers distinct advantages in analyzing the quantitative characteristics of cell differentiation potential and state transitions. Within the framework of the Waddington landscape, the biological interpretation of this potential energy is predominantly associated with the differentiation potential, which refers to the capacity of a cell to develop into specialized cell phenotypic states in the process of cell differentiation. Thus, the following discussion will primarily focus on the energy metric that quantifies the differentiation potential of cell phenotypic states.

Conventionally, transcriptomic signatures of known gene markers are relied on to characterize the natural cellular differentiation potential [89]. For instance, the cellular differentiation potential of totipotent cells is evaluated by the expression of totipotency molecular features, e.g. MERVL, Dux, etc. [90]. Similarly, the cellular differentiation potential of pluripotent cells is evaluated by the expression of pluripotent molecular features, including Nanog, Zfp42, and Pou5f1. [91]. However, on the one hand, characterizing the potential of various cell types in this manner necessitates unique transcriptomic signatures of markers, especially for the identification of novel cell populations [92]. Conversely, relying solely on a restricted set of specific gene markers is inadequate for precisely portraying and understanding the differentiation potential of cells [93].

Therefore, more comprehensive methodologies are required to assess the cellular transcriptome [93, 94]. And various approaches have been proposed from the theory of graph, network, entropy, and dynamical systems. Given that graphs and networks are inherently discrete mathematical structures, they provide a robust framework for modeling complex molecular interactions within cells as well as intercellular relationships. This discrete nature enables the precise representation of biological systems, where nodes and edges can, respectively, denote specific molecular entities (e.g. genes, proteins, and cells) and their interactions (e.g. regulatory relationships, signaling pathways, and cell–cell communication). Consequently, graphs and networks hold significant potential for the development of energy-based metrics, which can serve as quantitative indicators to characterize the differentiation potential of cells. Such metrics offer the systematic and scalable approaches to assessing cellular plasticity and fate determination in a data-based manner, which is derived from the topological and dynamical properties of these discrete structures. One of the pioneering works comes from Weinreb et al. [95], who proposed population balance analysis, an elegant approach with provable guarantees using spectral graph theory. In this method, the potential of each cell is calculated by the inner product of the Laplace matrix’s pseudo-inverse, and the Laplace matrix is corresponded to the K-nearest neighbor graph which is constructed from the sampled cells.

In addition to the spectral properties of graphs or networks, their topological structures and weight matrices significantly influence the number and stability of attractors. Consequently, these properties can be utilized to characterize the differentiation potential of cells within the context of the Waddington landscape. The Hopfield energy , a concept derived from the Hopfield neural network, is defined based on the interactions between nodes and serves as a robust indicator of network stability [96, 97], where is the interaction weight matrix of the Hopfield neural network, denotes the feature vector representing the input sample to the Hopfield network. Furthermore, the Hopfield energy has been employed as a quantitative metric to assess the differentiation potential of individual cells across various stages of the cellular behavior. For instance, based on the expression profiles of highly variable genes, Fard et al. [98] constructed a Hopfield neural network using gene co-expression relationships, and the interaction weight matrix is quantified through Pearson correlation coefficients. Utilizing the Hopfield energy derived from this network, the differentiation potential of cells during processes such as stem cell differentiation [98], the progression of complex diseases [99], etc., were calculated, and the Waddington landscape corresponding to the corresponding process was reconstructed. Furthermore, Li et al. employed the outer product method to construct the corresponding Hopfield neural network for each sampled cell, utilizing the expression profiles of highly variable genes as input feature vectors. And then quantitatively characterized the cell differentiation potential without prior knowledge, which is helpful to further explore the potential mechanism of cell plasticity [92].

As a fundamental concept in statistical physics, entropy serves as a quantitative measure of the degree of disorder within a system. In the context of cellular biology, the process of cell differentiation fundamentally represents the transition of gene expression patterns from a disordered state to an ordered state. Furthermore, entropy functions as an unbiased metric capable of capturing dynamical changes during cell differentiation, thereby providing a robust quantitative framework for understanding cell fate determination. Consequently, entropy has been extensively adopted as a key indicator to quantify the differentiation potential of cells based on graph and network modeling. For instance, the SLICE model grounded in Shannon entropy, calculated the entropy of single cells based on their probability distributions of functional activation, which was developed to quantitatively assess cellular differentiation states and reconstruct cell differentiation trajectories by characterizing the uncertainty of a biological system [93]. Subsequently, a generalized computational framework termed SCENT was introduced to estimate cellular plasticity by calculating signaling entropy [94]. Unlike methods requiring feature selection, SCENT calculates signaling entropy for individual cells by integrating gene expression data across a comprehensive gene regulatory network. This approach inherently incorporates systems-level information, enabling a genome-wide assessment of cellular plasticity. By leveraging the entire gene network, SCENT provides a robust and unbiased quantification of cellular differentiation potential at the single-cell level. Additionally, the Markov Chain Entropy model [100] was proposed to quantify cell differentiation potency by leveraging scRNA-seq data in conjunction with protein–protein interaction networks, thereby providing a comprehensive measure of cellular differentiation potential.

Most of the above methods are based on statistical indicators that are inherently sample-dependent. Alternatively, most of them rely on approximations of equilibrium processes and ignore kinetic perspectives that are essential to elucidate the complex mechanisms behind cell behaviors. Therefore, defining an energy metric, i.e. based on continuous cellular dynamics is crucial for deciphering the dynamical mechanisms underlying cellular behaviors and fate transitions. For instance, Shi et al. [101] proposed the Landscape of Differentiation Dynamics method, which calculates cellular differentiation potential by a continuous birth-death process from scRNA-seq data. Inspired by the energy decomposition theory [55], Han et al. [102] proposed a feasible framework for reconstructing the Waddington landscape based on the sparse autoencoder and the reaction–diffusion advection equation, which is from the perspective of cell migration and proliferation to calculate the dynamically interpretable energy metric.

In conclusion, the energy metrics offer a robust framework for precisely characterizing cellular differentiation potential and state transition dynamics within the Waddington landscape. These metrics, developed from multiple methodological perspectives, demonstrate complementary advantages in terms of computational accuracy and biological interpretability. Collectively, they significantly enhance both the theoretical understanding and practical application of the Waddington landscape model in cellular biology research.

Ultimate challenges and future directions

The reconstruction of the Waddington landscape has evolved through diverse computational frameworks, including data-based and model-based approaches. However, each paradigm exhibits intrinsic limitations that constrain its applicability, with several fundamental challenges remaining unresolved.

Model-based approaches necessitate precise a priori biological knowledge to parameterize the system. However, owing to the inherent computational intractability of high-dimensional system modeling, these approaches are constrained to generate simplified stochastic dynamical systems that only provide coarse approximations of true cellular behavior. This constraint fundamentally limits their capacity to investigate complex cellular dynamics. Notably, while both gene regulatory networks and epigenetic modifications critically govern cell fate decisions, they operate on markedly distinct timescales: gene interactions typically occur on the order of milliseconds, whereas epigenetic modifications evolve over hours to days. The current modeling frameworks, coupling of epigenetic modifications and gene interactions, inadequately account for this temporal divergence, thereby failing to capture the emergent cross-scale interactions between epigenetic regulation and transcriptional dynamics that collectively drive cellular behaviors.

Conversely, data-based approaches face critical challenges in addressing data sparsity and measurement noise, both of which are essential to accurately quantify differentiation potential and reconstruct dynamical trajectory of the complex cellular behaviors. While nonlinear dimensionality reduction techniques (e.g. t-SNE [103] and UMAP [104]) are commonly employed to address these issues, such manifold learning methods may distort the true topological relationships between cell states, leading to artifactual landscape structures. Furthermore, although single-cell sequencing technologies enable large-scale sampling, investigations of dynamical process involving transient states (e.g. early embryogenesis) remain challenged by the inherent sparsity of rare cell population observations. It results in model overfitting due to insufficient sampling of transitional states, and reduces generalizability from inadequate representation of low-abundance cell subsets.

Moreover, the emergence of multi-modal measurement technologies presents both opportunities and challenges. On the one hand, current technology has supported the simultaneous measurement of multiple modes [105], and on the other hand, different measurement modes are able to supplement key information on different dimensions in cellular behaviors. These multi-modal measurements open a high-dimensional window of development and present unique, novel opportunities. For instance, the integration of spatial transcriptomics with live-cell imaging techniques holds promise for the systematic reconstruction of the Waddington landscape with single-cell spatiotemporal resolution. Similarly, integrating ATAC-seq (Assay for Transposase-Accessible Chromatin with high-throughput sequencing) with spatial transcriptomics would allow simultaneous mapping of epigenetic regulation and gene expression across tissue architectures, helping to resolve spatial heterogeneity in cell fate decisions. Furthermore, coupling ATAC-seq with scRNA-seq can integrate the regulatory potential and transcriptional output, which allows quantitative modeling of chromatin openness gradients correlate within the Waddington landscape. However, it is difficult to effectively combine these different types of data, such as the destructive sampling of single-cell sequencing and continuous observation data of live-cell imaging. Also, current methodologies lack robust frameworks for reconciling these disparate data modalities, representing a critical area for future development.

In addition to the unique limitations and challenges, the different Waddington landscape reconstructing approaches also face the common challenge. Most critically, the field lacks formal theoretical frameworks for systematically validating the biological plausibility of reconstructed landscape. That is to say, we need a rigorous theoretical understanding to judge whether the reconstructed Waddington landscape is plausible. While recent studies have begun to establish preliminary theoretical guarantees for certain reconstructing approaches [95, 106], rigorous theoretical analysis of the performance characteristics of these approaches is still severely lacking. This theoretical gap impedes objective assessment of whether computational models faithfully recapitulate the multi-scale, dynamical processes for exploring the cellular behaviors. And the development of such frameworks represents an essential prerequisite for advancing the field.

Discussion

In this review, we provide a brief elucidation of the fundamental concepts underlying the Waddington landscape, including potential, attractor, trajectory, bifurcation, etc. These core concepts establish a robust theoretical foundation for understanding this powerful metaphorical framework in developmental biology. Based on methodological distinctions, we categorize the Waddington landscape reconstructing approaches into two primary categories: model-based and data-based.

Within the model-based framework, we summarize a structured overview of the fundamental principles and implementation procedures employed in the reconstruction of the Waddington landscape. Furthermore, we highlight recent applications of model-based approaches in elucidating cellular development and fate determination processes, demonstrating their utility in advancing our understanding of developmental dynamics. Regarding data-based approaches, we further divide them into three main categories based on parameterization and non-parameterization. We critically evaluate recent advancements within each methodological category, and conduct a comparative analysis of their respective advantages and limitations, which is shown in Table 1. This analysis aims to guide researchers in selecting the appropriate reconstructing strategies, and stimulate further developments in the quantitative approaches to developmental biology.

Table 1.

The comparison of the Waddington landscape reconstructing approaches

Methodology	Category	Classical method/Model	Advantages	Limitations
Landscape and flux theory	Model-based	Stochastic dynamical model without proliferation	Mechanistic modeling of interaction networks	Requires prior biological knowledge
			Capable of simulating molecular-level dynamics	High computational complexity in high-dimensional system
				Difficult to calibrate noise and heterogeneity in real data
Energy landscape decomposition theory	Model-based	Stochastic dynamical model with proliferation	Quantifies non-equilibrium effects on landscape topology	Requires experimental parameter calibration
			Incorporates key biological processes (e.g. cell proliferation)	Limited adaptability to complex tissue dynamics
The parameterization for the potential function	Data-based	Geometric models	Rigorous mathematical framework for bifurcation analysis	Requires predefined landscape topology
			The intervention prediction of parameterized models	Non-convex parameter space prone to local optima
The parameterization for the dynamical trajectory	Data-based	OT-based methods	Dynamical model-free analysis of high-dimensional data	Sensitive to data sampling density
			Captures complex branching trajectories	Dependent on the selection of growth marker genes
The non-parameterization for the differentiation potential	Data-based	Energy metric-based methods	Computational efficiency for large-scale data	Lacks continuous dynamics modeling
			Multi-omics integration capability	Requires efficient feature engineering

In general, the ongoing interplay between model-based and data-based approaches is progressively transforming the Waddington landscape from a conceptual metaphor to a quantifiable framework, and such quantitative reconstruction of the landscape paradigm enables more precise interrogation of developmental mechanisms at both single-cell and population levels. This synergistic advancement of the Waddington landscape reconstructing approaches provides crucial mechanistic insights into fundamental biological processes, including the molecular regulation of cell fate determination, the dynamical balance between cellular state stability and plasticity, and the spatiotemporal control of gene expression patterns during the cellular behaviors.

Key Points

• The fundamental concepts of the Waddington landscape, such as state space, etc., are outlined to clarify its composition, stability-cell state qualitative relationships, and bifurcation-critical cell fate transition mechanistic connections.

• Dividing the reconstructing approaches of the Waddington landscape into model-based and data-based categories helps readers clarify the quantification and reconstruction of the Waddington landscape from a methodological perspective.

• The significant insights into the quantitative reconstruction of the Waddington landscape, such as ultimate challenges, etc., are offered to clarify the future directions.

Author contributions

B.C. initialized this study. Y.H. and B.C. discussed many times to finalized the work plan. X.S. and B.C. gave suggestions many times to modify this study. Y.H. drafted the manuscript. Everyone read the manuscript and revised it, and agreed with the final version.

Conflict of interest: None declared.

Funding

This work was supported the National Natural Science Foundation of China under Grant No. 62433016 and the National Key R&D Program of China under Grant No. 2021YFA1000402.

Data availability

This review article does not generate any new original data. All data discussed in the manuscript are derived from previously published studies, which have been cited appropriately in the text and listed in the reference section. The original data can be accessed by referring to the corresponding cited publications.