Uncovering underlying physical principles and driving forces of cell differentiation and reprogramming from single-cell transcriptomics

Significance

Advances in single-cell technology enable transcriptome data at unprecedented scales. However, identifying the driving force of cell function from these data remains challenging. We learn cell state vector fields of cell differentiation/reprogramming from RNA velocity to quantify the global driving forces as Waddington landscape and flux from these data. Our methodology utilizes single-cell high-throughput experiments for validating the landscape and flux theory, with associated quantifications. We identify the deviation of the optimal path and transition state from the Waddington landscape alone as the driving force due to the presence of flux. We reveal the nucleation mechanism of cell fate decision-making with transition states as nucleation sites and pioneer genes as nucleation seeds. This provides a framework for uncovering underlying physical principles of biological processes via single-cell transcriptomics.

Abstract

Recent advances in single-cell sequencing technology have revolutionized our ability to acquire whole transcriptome data. However, uncovering the underlying transcriptional drivers and nonequilibrium driving forces of cell function directly from these data remains challenging. We address this by learning cell state vector fields from discrete single-cell RNA velocity to quantify the single-cell global nonequilibrium driving forces as landscape and flux. From single-cell data, we quantified the Waddington landscape, showing that optimal paths for differentiation and reprogramming deviate from the naively expected landscape gradient paths and may not pass through landscape saddles at finite fluctuations, challenging conventional transition state estimation of kinetic rate for cell fate decisions due to the presence of the flux. A key insight from our study is that stem/progenitor cells necessitate greater energy dissipation for rapid cell cycles and self-renewal, maintaining pluripotency. We predict optimal developmental pathways and elucidate the nucleation mechanism of cell fate decisions, with transition states as nucleation sites and pioneer genes as nucleation seeds. The concept of loop flux quantifies the contributions of each cycle flux to cell state transitions, facilitating the understanding of cell dynamics and thermodynamic cost, and providing insights into optimizing biological functions. We also infer cell–cell interactions and cell-type-specific gene regulatory networks, encompassing feedback mechanisms and interaction intensities, predicting genetic perturbation effects on cell fate decisions from single-cell omics data. Essentially, our methodology validates the landscape and flux theory, along with its associated quantifications, offering a framework for exploring the physical principles underlying cellular differentiation and reprogramming and broader biological processes through high-throughput single-cell sequencing experiments.

The cellular world, once perceived as a black box, has been illuminated in recent years by the explosion of technologies that allow for the study of multicellular organisms at single-cell resolution. This technological leap has revolutionized our understanding of cellular heterogeneity, developmental trajectories, and the intricate interplay of molecular networks. While many details are system-specific, common patterns emerge in the methodologies. However, despite these advancements, a formidable challenge persists: uncovering the underlying physical mechanisms of cell function, such as cell fate decision-making in differentiation and reprogramming, especially the quantification of global driving forces for the nonequilibrium dynamics. Integrating statistical mechanics with single-cell biology has unlocked novel opportunities to study cell development and its regulation (1). Statistical mechanics provides a robust framework for probing complex systems, molecular interactions, driving forces, and physical mechanisms behind cellular processes (2). On the other hand, single-cell biology delves into processes at the individual cell level, shedding light on population heterogeneity and cell differentiation mechanisms (3). This understanding is paramount, given that the single cell represents the fundamental unit of life.

Conventionally, researchers have focused on the trajectory dynamics or the local stability analyses around fixed points for nonlinear systems. However, these approaches often fall short of offering a global perspective. This has led to a quest for landscapes that characterize the whole system-wide stability. Yet, Waddington landscape alone, which is often used to quantify equilibrium system dynamics with detailed balance, cannot encapsulate the entirety of nonequilibrium system dynamics. An additional rotational flux, acting as a driving force, can become important for cell fate transitions (4). Transitions between gene expression states are determined by these driving forces, stemming from a nonequilibrium effective potential landscape shaped by steady-state probabilities and the rotational steady-state probability flux between system states (5). The steady-state probability flux, originating from energy input and resulting in thermodynamic dissipation, measures the degree of the nonequilibrium and the irreversible behavior of a system and serves as a dynamic driving force, linking the dynamical and thermodynamic aspects of the system (6). However, many previous studies, anchored in existing knowledge of gene regulatory networks, have often oversimplified network models, leading to significant deviations from genuine biological systems (7, 8). While low-throughput efforts have provided certain insights, they are often confined to only two or three genes (9–11). A holistic understanding necessitates a whole genomic approach to fully capture the landscape and flux as the global driving forces. With the advent of single-cell sequencing technology, we now have access to comprehensive transcriptomic data. Previous studies constructed the cell fate landscape from the state manifold and density of the single cell (12). MuTrans utilizes multiscale reduction to quantify attractors and their transition probabilities in snapshot data, as well as constructing a low-dimensional dynamical manifold (13). scEpath employs a statistical physics-based approach to quantify the energy landscape using “single-cell energy” and distance-based measures (14). PRESCIENT learns differentiation landscapes by modeling cell differentiation as diffusion (15). Waddington-Optimal Transport (OT) considers cells drawn from a probability distribution in gene expression space and uses OT to infer transport plans between two consecutive time points (16). However, many of these methods face difficulties in unraveling the underlying physical mechanisms governing cell signaling regulation. Thus, a pressing need arises to study the underlying mechanisms of cell function regulation directly from single-cell transcriptomic data, focusing on nonequilibrium dynamics and thermodynamics.

In this study, we elucidate the physical mechanisms of differentiation and reprogramming by harnessing nonequilibrium dynamics and thermodynamics, quantifying landscape and flux from single-cell transcriptomic data. Our approach involved assessing single-cell RNA velocity, reconstructing vector fields, dimensionality reduction, and mapping these onto lower-dimensional spaces to quantify landscape–flux and decipher interactions among signaling processes regulating cell differentiation and reprogramming. We revealed the dynamic flux driving cell fate transitions and the thermodynamic dissipation essential for maintaining stem/progenitor cell differentiation potency. These cells strive to preserve potency and rapid cell cycle oscillation for self-renewal by consuming more energy, leading to a nonequilibrium process where entropy production increases. Our methods, such as identifying the least action paths (LAPs), estimating mean first passage time (MFPT), and quantifying transition states, allowed us to reveal the nucleation mechanism and predict transcriptional drivers of optimal cell-fate switching. We highlighted how flux causes deviations in optimal paths and transition state kinetic rates from expected landscape-based predictions, identifying this as a driving force. Our findings also spotlight the role of dynamic loop-flux in cell fate decisions, driving continuous cell state transitions. By conducting in silico perturbations, we identified key genes for cell fate transitions and predicted the impacts of their genetic perturbations on cell development dynamics and thermodynamics. Finally, we inferred cell-type specific gene regulatory networks, estimated cell–cell interactions, and quantified cell-type specific pathway activity and biological function during cell development.

In summary, our study employs a unique combination of experimental high-throughput sequencing and theoretical landscape–flux approaches to understand the genetic drivers and dynamic driving forces for cell fate transition following cell development. By experimentally verifying the Waddington landscape and flux theory and providing its associated quantifications, we pave the way for transformative insights into cellular dynamics and their broader implications, with potential applications spanning regenerative medicine, disease modeling, and therapeutic interventions.

Results

Quantifying Cell Development Landscape and Flux from Single-Cell Transcriptomics Data.

Traditional systems biology studies, while insightful, often grapple with challenges stemming from network simplifications and parameter estimations, potentially leading to incomplete or skewed interpretations of complex biological systems (17). Although some studies have constructed the cell fate Waddington landscape based on the state manifold from single-cell transcriptomics data, they often face difficulties in obtaining the underlying physical mechanism related to cell signaling regulation (12, 14, 18). Addressing these challenges, we introduce an approach to study the underlying mechanism of cell development via nonequilibrium dynamics and thermodynamics by bypassing the prior known regulatory networks and quantify the cell development regulatory dynamics directly from the experimental data collected by single-cell high-throughput sequencing. Our quantification of the driving forces for cell differentiation and reprogramming as landscape and flux involved four primary steps, as illustrated in Fig. 1A. We began by processing transcriptomics data, capturing gene expression patterns and RNA-splicing dynamics over time. RNA velocity was then estimated to visualize the movements of cells throughout the cell development based on the unspliced and spliced counts, followed by the reconstruction of the cell development dynamics vector field. This captured the overall flow of cells and driving force field through the cell development. The final step involved quantifying the landscape and flux of global cell development dynamics and thermodynamics. Simply put, this method includes Uniform Manifold Approximation and Projection (UMAP)-based identification of cell-type clusters, estimation of RNA velocity, reconstruction of cell development dynamics’ vector field, and quantification of the cell development driving forces as the potential landscape and curl flux.

Fig. 1.

Quantifying Waddington landscape and flux of cell development from single-cell transcriptomics data. (A) Workflow of constructing cell development landscape–flux by single-cell transcriptomics data. (B) Color-coded UMAP based on cell-type identification of mouse retinal neuron development. (C) RNA velocity of cell development dynamics. (D) Reconstructed vector field of cell development dynamics. The color of digits reflects the type of fixed point: red, emitting fixed point; black, absorbing fixed point. The color of the numbered nodes corresponds to the confidence of the fixed points. (E) Quantified Waddington landscape of cell development dynamics in the UMAP space. Least action paths between different cell states in the 3D Waddington landscape, blue paths represent the optimal developmental paths, red paths are the reprogramming paths and the orange paths denote transdifferentiation paths. (F) The mean force (black arrow) of cell development dynamics. (G) The gradient force (white arrow) and curl flux (black arrow) of cell development dynamics.

To explore further into the intricacies of cellular dynamics and drivers during cell development, we employed a landscape–flux analysis using single-cell RNA sequencing (scRNA-seq) data from mouse retinal neuron (19) and single-cell metabolically labeled new RNA tagging sequencing (scNT-seq) data from human hematopoiesis development (20). Initially, we identified cell-type clusters of mouse retinal neurons and generated a low-dimensional representation of the cells using the UMAP algorithm after consideration of several dimensionality reduction techniques [including PCA (principal component analysis), t-SNE (t-distributed stochastic neighbor embedding), PHATE (Potential of Heat-diffusion for Affinity-based Trajectory Embedding), etc.], which effectively visualizes the distribution of the cell state manifold and the heterogeneity of the cell type following the cell developmental trajectories (Fig. 1B). Distinct cell type states exhibited varied gene expressions and regulatory patterns (SI Appendix, Fig. S1). Subsequently, we calculated the RNA velocity (21, 22) for each cell, capturing the direction and magnitude of gene expression changes (Fig. 1C). Dynamo (20) facilitated the reconstruction of the cell development dynamics vector field, enabling the derivation of the continuous and analytic velocity vector field from the sparse single-cell RNA velocity measurements (Fig. 1D). Concurrently, we undertook a differential geometry analysis of cell development based on this vector field (SI Appendix, Supporting Results). Using the reconstructed force field function, we quantified the potential landscape U ($\mathrm{U}=-ln{P}_{\mathit{ss}}$) of the global cell development dynamics in UMAP, representing the steady-state probability distribution ${\mathrm{P}}_{\mathit{ss}}$ of cells at different locations in different cell types (Fig. 1E). The different cell-type states correspond to different basins of attractions (colored balls) on the landscape. The basin depth quantifies the stability of the cell-type states, and the local barrier between these states or basins corresponds to the degree of difficulty for cell fate transition during the cell development (7). While the landscape attracts the cells down to the different valleys from the top ring valley under the gradient part of the driving force, the curl flux from the rotational part of the driving force helps the cell fate transition. So, this potential landscape also reflects the stability of different cell-type states in the cell development and highlights the key transition points. Additionally, we decompose the reconstructed force field into the curl flux and negative gradient forces of the nonequilibrium dynamic landscape of cell development, which clearly shows that the total driving forces of cell development are dictated by these two parts (Fig. 1 F and G). The curl flux indicates the amount of rotational flow in the cell development and the cell state transitions dynamics landscape that drives the stem/progenitor oscillations along the closed ring valley path, while the negative gradient force indicates the direction and magnitude of the forces stabilizing the cell state. In cell fate transition during cell differentiation and development, if the driving force from the flux is greater than the force from the negative potential gradient, this may help the cell state to switch to another cell state, the cell then will make a cell fate transition.

In a parallel analysis, we assessed the total RNA velocity from scNT-seq of the human hematopoiesis development and reconstructed the vector field for further quantifying the nonequilibrium landscape–flux (SI Appendix, Fig. S3). We also perform robustness analysis for fluctuation strength to select the diffusion constant value (D $=$ 0.2) that best matches the actual biology of human hematopoiesis (SI Appendix, Fig. S4). These results confirm the curl flux’s role as the driving force for cell fate transition, supplementing the gradient of the cell development landscape by experimental single-cell high-throughput sequencing data, providing deeper insights into the mechanisms behind the stability and behavior of intrinsic nonequilibrium cell development progression that cannot be discerned by the Waddington landscape quantification alone. What’s more, the creation of the cell development nonequilibrium landscape–flux through single-cell transcriptomics data presents a powerful tool for the study of stem cell development dynamics, allowing for insights into underlying mechanisms that drive cell development progression.

Metabolic Energy Dissipation Facilitates Rapid Cell Cycle to Preserve Cellular Self-Renewal and Pluripotency.

Stem and progenitor cells are characterized by their inherent ability for cellular self-renewal and pluripotency, which empower them to differentiate into a myriad of cell types and sustain tissue homeostasis (23). A defining attribute of these cells is their accelerated cell cycle, though the mechanisms propelling this remain largely uncharted. Our methodology embarks with the development of mouse retinal neurons and human hematopoiesis, revealing distinct phases of cell development characterized by evident phenotypic alterations (Fig. 2A and SI Appendix, Fig. S5A). These phase diagrams set the foundational understanding for comprehending how cell phenotype transition through various states during their developmental trajectory. Furthermore, we introduce a pseudotime concept, derived from RNA velocity of transcriptional dynamics (Fig. 2D and SI Appendix, Fig. S5D). This temporal perspective facilitates the visualization of the temporal evolution of gene expression changes during cell development and offers a dynamic window into the progression of developmental events. Subsequently, we probe the intricate dynamics of gene expression, with heatmaps highlighting the sequential gene expression patterns correlated with cell developmental pseudotime (Fig. 2C and SI Appendix, Fig. S5C). This paves the way for a deeper comprehension of cell differentiation dynamics and its determinants. Focusing on lineage-specific genes, we elucidate the retinal tree’s organizational structure, accentuated by pivotal marker genes (Fig. 2B and SI Appendix, Fig. S5B). This provides a spatial perspective to the differentiation process, enhancing our grasp of lineage determination and evolution (24). Specific marker genes facilitate the recognition of unique cell types, revealing their integral roles in developmental orchestration.

Fig. 2.

Methodological process for detecting cell differentiation in multicellular organisms. (A) The multiple phases of cell development are changes in cell phenotypes. (B) The retinal tree organization with key marker gene. (C) Heatmaps showing the dynamics of gene expression associated with cell developmental pseudotime orientation. Gene expression dynamics resolved along pseudotime shows a clear cascade of transcription in the top 300 likelihood-ranked genes. (D) The pseudotime based on RNA velocity of transcriptional dynamics. (E) Cell differentiational potency in 3D landscape. The cells differentiate from the state at a high value of potency to a lower value. (F) Inferred S and G2M phases based on cell cycle scores affirm the cell development identified by the dynamical model. The dynamical model accurately delineates the cycling population of retinal progenitors, their lineage commitment, cell cycle exit, and neuron differentiation. (G) The total metabolic capacity of single cell during cell development. (H) Progenitor cells undergoing rapid proliferation to maintain cell pluripotency adopt a glycolytic profile to meet high energy and biomass requirement; differentiated cells employ oxidative phosphorylation (OXPHOS).

We also in three-dimensional representation unravel the landscape of cell differentiation potency (Fig. 2E and SI Appendix, Fig. S5E). This visualization vividly portrays cells transitioning from heightened potency states to diminished ones, crucial for comprehending the cellular mechanisms underpinning self-renewal and pluripotency (25). By identifying S and G2M phases affirmed by cell cycle scores, our model captures the cycling population of retinal progenitors (Fig. 2F and SI Appendix, Fig. S5F), encompassing lineage determination, cell cycle exit, and eventual neuron differentiation (26, 27). Recognizing that cell development is a process that constantly consumes energy (28), we quantify individual cells’ metabolic capacities throughout development, examining energy production and consumption dynamics, elucidating the heterogeneity of metabolic capacity and type among single cells (Fig. 2G and SI Appendix, Fig. S5G). By comparing the metabolic differences of different cell types, stem/progenitor cells often have stronger metabolic capacity than differentiated cells in both mouse retinal neuronal development and human hematopoietic development, which indicates heightened energy demands in stem/progenitor cells compared to their differentiated cells (Fig. 2G and SI Appendix, Fig. S5H). For instance, the glycolytic capacity of progenitor cells is significantly higher than that of differentiated neuronal cells (Photoreceptor (PR), Amacrine cell/Horizontal cell (AC/HC)), with the exception of retinal ganglion cell (RGC). Key genes (such as Aldh1a1, Hk2, Pck2, etc.) related to glycolysis were significantly highly expressed in progenitor cells and low/no in differentiated cells (SI Appendix, Fig. S6 A–D). Conversely, key genes related to oxidative phosphorylation are underexpressed in progenitor cells and significantly highly expressed in differentiated cells (SI Appendix, Fig. S6 E–H). The same metabolic heterogeneity and gene expression also occur in human hematopoiesis (SI Appendix, Fig. S6 I–P). Pluripotent stem/progenitor cells have higher cell cycle scores, while differentiated cells have lower potency and lower cell cycle scores (Fig. 2 E and F and SI Appendix, Fig. S5 E and F). It is straightforward to sum up that stem/progenitor cells exhibit a unique metabolic profile characterized by enhanced glycolytic flux to meet the high energy and biomass requirement, which supports their rapidly proliferative state and maintains their pluripotency. However, the differentiated cells employ oxidative phosphorylation (OXPHOS) (Fig. 2H). We need to point out that the metabolism of cells is usually associated with thermodynamic energy dissipation, implying the role of thermodynamic dissipation in the cell fate determination (29). The transition of cells from a state of high potency to lower potency underscores the thermodynamic processes at play, with cells expending energy as they differentiate (30). In summary, these insights illuminate thermodynamic dissipation’s significance in accelerating the cell cycle, crucial for preserving cellular self-renewal and pluripotency, thereby enhancing our understanding of cells’ energy demands during differentiation and thermodynamic dissipation’s role in these processes.

Predicting Transcriptional Drivers and Dynamics of Optimal Cell-Fate Switching.

The potential to induce cellular state transitions is emerging as a pivotal strategy for disease modeling. Our quantified Waddington landscape serves as a tool to elucidate and characterize the biological pathways of developmental processes (4). Prior research has established that the dynamics of these biological paths on the landscape are governed by gradient and curl forces, rather than merely following the anticipated steepest descent path (17). To investigate cell development progression and cell fate transitions, we computed both the LAPs and MFPT between cell-type transitions. LAPs represent optimal paths in state space that minimize the action’s cost between states, while MFPT denotes the average transition time between states (31). Specifically, the optimal path between any two cell fate states is determined by iteratively modifying the continuous path connecting source and target states, minimizing its action (or maximizing the weight of the path), and updating the associated transition time. The resulting LAP has the highest transition probability and is correlated with a specific transition time. Initially, we employed the LAPs method to predict optimal cell state switching within the continuous vector field derived from scRNA-seq data of mouse retinal neuron development (Fig. 3A). The differential LAP of mouse retinal neuronal cell types from the progenitor cell to terminally differentiated neurons almost follow a curved flow specified by a vector field containing information about expression dynamics. For a given differentiation process, numerous paths closely aligned with the vector field’s streamlines exhibit near-zero action. In vivo, pluripotent undifferentiated cells typically exhibit stability (Fig. 1). The undifferentiated state remains stable at or near early developmental stages, with a minimal probability of escaping to a differentiated state due to fluctuations. Conversely, the action associated with the differentiation process is exceptionally low, corroborating the notion that in vitro cultured stem cells necessitate specific environmental factors to retain stemness; otherwise, they readily differentiate into somatic cells (32) (SI Appendix, Fig. S3). Cultured stem cells have become a cornerstone for regenerative medicine, developmental biology, and biophysical studies (33). Moreover, differentiation LAPs differ significantly from reprogramming LAPs, typically exhibiting shorter transition times and reduced action (Fig. 3C). Similarly, transdifferentiation LAPs, transitioning from one somatic cell type to another, also differ from reverse transdifferentiation, underscoring cellular irreversibility. However, vector field analysis alone insufficiently explains this irreversibility, which is well explained by the nonequilibrium landscape–flux theory we developed.

Fig. 3.

LAPs and MFPT predict drivers and dynamics of optimal cell-fate switching. (A) Stable-steady points (black digits), unstable-steady point (red digit), and LAPs between different cell states in the velocity vector field of mouse retinal neuron development. The blue paths present the optimal developmental path; the red paths are the reprogramming LAPs; and the orange paths are transdifferentiation LAPs in the UMAP embedding. The color of the nodes along the paths indicates the LAP transition time from purple to yellow. (B) LAPs between different cell states in a 2D landscape. The white lines are the path alone the gradient of the potential landscape. The black arrows present the curl flux in the landscape. (C) The transition time of the LAPs between any two cell states. (D) The logarithm of MFPT between cell fate transitions (Cell differentiation from Progenitor to AC/HC cell, reprogramming from AC/HC back to Progenitor cell, and transdifferentiation from PR cell to AC/HC cell) versus the logarithm of Flux. (E) The correlation between the logarithm of EPR and the logarithm of MFPT when varying diffusion constant (D).

Regarding cellular irreversibility, we further analyzed LAPs within the nonequilibrium global dynamics landscape (Fig. 3B). Specifically, we demonstrate that developmental processes can be quantitatively described by quantifying biological pathways on the Waddington landscape that diverge from the steepest descent path (white path), typically with forward and reverse pathways bypassing saddle points on gradient pathways and controlled by a combination of gradient forces and curl forces (black arrows) on the landscape. The additional dynamic driving force excerpted from the curl flux causes the path to deviate from the anticipated steepest descending path based on potential U’s gradient. This quantitative representation differs from Waddington’s, where the development path is tumbled down by marble and follows a gradient of the underlying landscape. Instead, the true path of development is not simply descending along the gradient but is shaped by a curling flux force, resulting in a spiral motion, which is experimentally verifiable. We also observed that developmental processes, as described by the underlying landscape and associated biological pathways, are inherently stable and resilient to environmental perturbations. More quantitatively, we elucidated the correlation between MFPT and Flux (Fig. 3D), where an increased driving force (flux) facilitates faster transitions between cell states, revealing flux’s role in facilitating and accelerating cellular state transitions, corroborating our prior proposition that flux has the effect on accelerating the state switching while the flux does not change significantly (34). Additionally, the quantification of entropy production rate (EPR) and MFPT allows for a deeper exploration of cell state transition thermodynamics, we observed that differentiation progresses with less MFPT than reprogramming and transdifferentiation under the same EPR, indicating that differentiation processes consume less energy than reprogramming and transdifferentiation due to lower resistance to action (Fig. 3E). However, it is also observed that processes with longer MFPT consume more EPR (Fig. 3E). This may seem counterintuitive at first glance; the flux and EPR correlate with MFPT differently. But this is consistent with the thermodynamic understanding that systems farther away from the equilibrium (higher EPR) are more dissipative. It is important to note that EPR is a function of diffusion coefficient D, which can modulate the energy landscape, thus affecting the flux and EPR separately. For instance, in a noisy system, a higher EPR might not always coincide with a higher flux due to transient, nondirectional fluctuations that change the dissipation explicitly without significantly contributing to the flux for directed state transitions.

While we leverage LAPs and MFPT’s predictive capabilities to uncover the dynamics underpinning optimal cell-fate conversion, it’s imperative to recognize that the quantified Waddington landscape’s topology is contingent on the underlying gene regulatory circuits’ structure. Therefore, our ultimate objective is to devise a strategy that elucidates optimal paths, associated driving genes, and their corresponding expression dynamics. To identify the associated key regulators of specific cell fates, we concentrated on genes being used for transition, ranking them along the LAPs. Once the LAP is computed in PCA space, it can be projected back to the original gene expression space, predicting the entire transcriptome dynamics along the pathway (20). Analyzing the driving gene dynamics on LAPs and reverse LAPs between cell lineages further highlighted cellular irreversibility at the gene expression dynamic level (SI Appendix, Fig. S7). We demonstrated that differentiation is a multiday process wherein cells undergo a continuum of transcriptional changes with upregulation of differentiation genes preceding downregulation of typical stemness genes, which are in agreement with prior experimental findings (35). Conversely, in the case of cell reprogramming, where differentiated cells are induced to revert to a pluripotent state, the temporal sequence of gene expression changes suggests a complex and potentially variable sequence of events. In some instances, the upregulation of reprogramming factors appears to occur early in the process, effectively initiating the reprogramming event. This is sometimes followed by a gradual downregulation of differentiation-specific genes as the cells acquire a more stem-like transcriptional program (36). Yet, this sequence may not be uniform across all cell types or reprogramming methodologies. Detailed discussions in SI Appendix, Supporting Results. Upon quantifying the cellular development pathway, we further explored cellular irreversibility mechanisms from various aspects, such as nonequilibrium driver curl-flux and genetic driver gene expression pattern, enriching our understanding of the interplay between predictive dynamics, transcriptional regulation, and cell transition. These analyses underscore the LAP method’s potential in predicting the optimal pathway and transcription factor (TF) mixture of cell fate transitions with high accuracy, paving the way for rapid reprogramming between any cell type of interest in regenerative medicine applications (37).

Quantifying the Transition States Reveals the Underlying Nucleation Mechanism and Pioneer Genes.

Transition state is an intermediate state during cell fate decisions in which a cell exhibits a mixed identity between two or more states, which often represents the state of origin (that is, the initial state the cell is in) and that of destination (that is, the identity that the cell is adopting) (38). It has previously been analogized to an equilibrium molecular transition state and is hypothesized to be highly unstable and reversible (39). However, cell fate decision-making reveals a landscape far away from the equilibrium states traditionally analyzed in physics and chemistry. The cellular differentiation and reprogramming unfold in a realm dominated by nonequilibrium dynamics, where conventional models based on equilibrium dynamics are insufficient (4). The speed of cell fate decision-making extends beyond the barriers imposed by the putative transition states as understood in the context of chemical reactions (40). The transition state of the cell fate decision-making process is important in describing the underlying pioneer gene kinetics but still challenging.

We first use the toggle model as an illustrative example to visually encapsulate the intricate pathways and transition states in cell fate decision-making (SI Appendix, Supporting Results). Then we translate this model to real cellular dynamics, particularly focusing on mouse retinal neuron development (SI Appendix, Fig. S8G). It traces the LAPs and saddle points, elucidating how cells navigate through developmental trajectories. The variation in action values and action integrals along the differentiation LAP and the reprogramming LAP underscores the dynamic nature of these transitions (SI Appendix, Figs. S8 H and S9 A, D, G, J, and M). On quantified nonequilibrium Waddington landscape, we identify the Hamilton–Jacobian LAPs of cell differentiation and reprogramming during mouse retinal neuron development, with red nodes on the path indicating the nonequilibrium transition states (Fig. 4A). Most of the transition states (the last global maximum point of the action value along the Hamilton–Jacobian LAP) are far away from saddle points of the landscape under finite noise (Fig. 4A, Inset and C and SI Appendix, Fig. S9 B, E, H, K, and N). This demonstrates the importance of flux as the driving force for the nonequilibrium dynamics. Typically, with only the landscape, the optimal path goes through the saddle point, where the barrier height at the saddle determines the kinetic rate. This indicates the nonequilibrium transition states, rather than the landscape’s saddle points, regulate the speed of cellular fate decisions.

Fig. 4.

Quantifying the transition states reveals the underlying nucleation mechanism and pioneer genes during cell fate decisions. (A) Quantified nonequilibrium Waddington landscape of mouse retinal neuron development with differentiation Hamilton–Jacobian LAPs (blue lines) and reprogramming Hamilton–Jacobian LAPs (red lines) between different lineages in UMAP. Red nodes on different Hamilton–Jacobian LAPs represent transition states. The blue node represents the saddle point on the gradient path (purple line). (B) The differentiation LAPs (blue lines) and reprogramming LAPs (red lines) between different lineages in the PCA velocity vector field of mouse retinal neuron development. Blue nodes represent cells closest to saddle points in the UMAP landscape, red nodes represent cells closest to transition states on different Hamilton–Jacobian LAPs. (C) The action value and action integral vary along the differentiation Hamilton–Jacobian LAP from Progenitor lineage to PR lineage in the quantified nonequilibrium Waddington landscape of mouse retinal neuron development. The red dash line and node represent the transition state on the Hamilton–Jacobian LAPs. (D) The pioneer genes expression trajectory along the differentiation LAP from Progenitor lineage to PR lineage. The blue dash line and node represent the position along LAP closest to the saddle points; the red dash line and node represent the position along the LAP closest to the transition states.

At the molecular level, a cell fate decision necessarily involves a change in the transcriptional state of a cell. When cells make a decision, they up-regulate the expression levels of the gene cohort of their chosen fate and down-regulate those of the alternative one. Physically speaking, cell fate switching can be viewed as a phase transition process. In the first-order phase transition process, the nucleation mechanism sets in the phase transition process. Typically, when studying biophysical phase nucleation mechanisms, one should determine where nucleation takes place and what the nucleation seeds are. During the cell fate decisions, the identification of transition states indicates where the process of nucleation—cell fate determination—takes place and which genes or regulations are crucial—they act as nucleation seeds that can initiate and direct the fate decision process, akin to catalysts in a chemical reaction. Usually, manipulating a small number of key genes can promote the transition of cells to a specific fate, such as reprogramming terminally differentiated somatic cells into induced pluripotent stem cells by expressing the transcription factors OCT4 (Octamer-Binding Transcription Factor 4), SOX2 (SRY-Box Transcription Factor 2), KLF4 (Krupple-like factor 4) and MYC (Myelocytomatosis) (41). They represent the molecular keystones of the nucleation mechanism, determining the propensity and direction of cellular differentiation and reprogramming. These factors are called pluripotency pioneer transcription factors. For the mouse retinal neuronal development system, we use the transition states (the nucleation sites) and the gene expression dynamics along optimal pathways to identify pioneer genes (the nucleation seeds) in cell fate decisions. We first determine the nearest neighbor point along the optimal path from the transition state in PCA space, which can be restored to the original gene expression space (Fig. 4B). Finally, pioneer genes were screened through gene expression dynamics along the optimal path, that is, gene expression is almost stable in the nonequilibrium transition state (Fig. 4D and SI Appendix, Fig. S9 C, F, I, L, and O). The identification of pioneer genes from nonequilibrium transition states, particularly those proximal to the target state, provides vital insights into the drivers of cell fate decisions. For example, during the differentiation process from progenitor cells to PR cells, genes such as Hspa5, Dkk3, Neurod4, Dpysl3, Slc2a1, and Arl6ip1 were selected as pioneer genes. Hspa5, Dkk3, Slc2a1, and Arl6ip1 are related to the regulation of the cell cycle and pluripotency (19), and are quickly silenced to very low levels during differentiation. This represents the loss of pluripotency of cells. At the same time, Neurod4 and Dpysl3, which are related to PR lineage functions (19), will be expressed quickly before other genes, and basically approach a very high and stable level in the transition state. In contrast, Pik3r1, Otx2, Crx, Dpysl3, Fos, and Cplx2 were identified as pioneer genes for PR lineage reprogramming. The Fos gene, as the progenitor cell marker gene, is quickly expressed to a very high level, and the PR cell marker genes, Pik3r1, Otx2, Crx, Dpysl3, and Cplx2 are quickly silenced (19). In essence, our quantification of landscape and flux advances beyond the traditional Waddington landscape both conceptually and mechanistically. Flux causes the nonequilibrium transition state on the optimal path to deviate from the potential landscape’s saddle point. This provides quantitative guidance for manipulating cell fate transitions in biological experiments and suggests targeting pioneer genes first to manipulate them to reach the transition state according to the gene expression trajectory might lead cells more effectively to the target state. These findings reinforce the idea that real-cell transition states are path-dependent and distinct from the landscape’s theoretical saddle points.

Dynamic Loop-Flux Drives Cell Development and Cell Fate Decision-Making.

Considering the differentiation, reprogramming, and transdifferentiation between cells, if the cell-type state is treated as a network node, and the interchangeable cell states are interconnected to form a bidirectional edge, the cell development system can be conceptualized as a complex network within state space (Fig. 5A and SI Appendix, Fig. S10A). For analyzing nonequilibrium dynamics in such networks, researchers have proposed a way of decomposing them into circular diagrams (42). When the system reaches a steady state, the flux across the cellular states of this cell developmental system in state space can form closed loops and are coordinated by multiple circle maps (43). The concept of loop flux, derived from graph theory, allows for the precise quantification of each cycle map’s contribution to cell state transitions (44). By ranking the loop flux, the corresponding spatial closure trajectory that dominates the transition of a cell state can be easily found from many possible processes, that is, the dominant cycle (45). Analyzing the dominant cycle’s behavior can elucidate the dynamic and thermodynamic transport characteristics of the cell state transition, providing insights for optimizing biological manipulations. In this study, we validated this analytical approach by exploring mouse retinal neuronal generation and human hematopoietic development systems. By leveraging the reconstructed vector field, we computed the transition rate between any two cell-type transitions, subsequently characterizing the Markov dynamics of cell fate transitions using the master equation for probabilistic evolution (Fig. 5B, Left and SI Appendix, Fig. S10 B, Left). This facilitated the derivation of a stable probability distribution of cell fates during development and quantified the probability flux between any two cell-type transitions (Fig. 5B, Right and SI Appendix, Fig. S10 B, Right). Notably, a positive correlation exists between the cell state transition rate and the probability flux, indicating the latter’s role in facilitating cell state transitions. In the landscape context, the probability flux’s role becomes even more pronounced, as barriers impede cell state transitions, necessitating the probability fluxes, potentially sourced from environmental factors, to facilitate cell fate transformations (46).

Fig. 5.

Dynamic loop-flux drives the cell-type state transition. (A) The network of cell differentiation, reprogramming, and transdifferentiation with cell-type state transition. (B) The transition rate of the LAPs between any two cell states versus the probability flux. (C) Schematic of the loop-flux decomposition. (D) The loop fluxes drive the cell-type state transition. The width of arrows is the strength of loop-flux. (E) The main loop fluxes drive the cell-type state transition in the potential landscape.

In fact, discrete probabilistic fluxes are abstract concepts, and in real biochemical processes, fluxes are interconnected on various loops to form loop flux. Given the multiple cell-type states in the cell developmental system, numerous loops connect different cell states (Fig. 5C and SI Appendix, Fig. S10C). For the mouse retinal neuronal development system we analyzed, with four cell-type states (excluding intermediate states such as Neuroblast), five cycles are feasible. The human hematopoietic development system, with six cellular states [excluding intermediate states like MEP (Megakaryocyte-erythroid progenitor) and GMP (Granulocyte-monocyte progenitor)], presents 42 potential loop pathways. Biologists aim for the most time-saving and labor-saving way to manipulate cell fate, so our objective was to identify the dominant loops beneficial for biochemical cell fate manipulation. Loop-flux decomposition on the two cell development systems (SI Appendix, Supporting Materials and Methods) yielded 3 and 10 loop fluxes, respectively (Fig. 5D and SI Appendix, Fig. S10D). Quantitative ranking revealed the dominant loop-fluxes governing cell state transitions, we found that Progenitor cell state, AC/HC cell state, and RGC cell state are involved in the most dominant loop-flux during the mouse retinal neuronal development, and the loop-flux connecting hematopoietic stem cell (HSC) cell state, Ery cell state, and Mon cell state are most dominant in the human hematopoietic development system. Surprisingly, the most dominant loop-flux connects the three cell-type states with the lowest probability distribution in the cell development system. On the landscape, these low-probability cell-type states possess a low basin barrier, indicating a relatively shallow attraction domain. This can be metaphorically understood as shallow puddles in a stream interacting more frequently, while deeper ones remain isolated (47). Finally, by projecting the dominant loop-fluxes onto the landscape, it becomes evident that the previously decomposed curl fluxes can form cohesive loops to facilitate cell fate transitions (Fig. 5E and SI Appendix, Fig. S10E). It provides evidence that the dominant loop-fluxes are not merely a feature of the system’s architecture but are also pivotal in the temporal progression of the cell development system. These dominant loops signify more than the facilitation of cell fate transitions. They may indeed act as the initial pathways in the differentiation process. The prominence of these loops suggests a hierarchical organization within the differentiation landscape, where certain pathways are “primed” to be traversed before others. This priming could be attributed to lower energy barriers or higher transition probabilities within these dominant loops, effectively guiding the cell’s developmental journey.

Genetic Perturbations in Cell Development: Insights from In Silico Analysis.

Perturbation, characterized by the modification of a biological system’s function due to factors as environmental stimuli, drug inhibition, or gene knockdown (48), offers insights into gene interactions and their influence on cellular processes and phenotypes (49). Within the realm of cell development, genetic perturbations can introduce significant changes in the dynamics governing cellular behavior. To investigate the effects of genetic perturbations on cell development through nonequilibrium dynamics and thermodynamics, we performed in silico perturbations of different genes in a controlled and systematic manner. Similar to Fig. 1A, we first estimated the RNA velocity of cell development diverse genetic backgrounds through in silico perturbation (SI Appendix, Fig. S11). Subsequently, we reconstructed distinct vector fields based on the perturbed RNA velocity (SI Appendix, Fig. S12). Our analysis culminated in quantifying the global cell development dynamic landscapes under varied genetic perturbations, incorporating noise in the dynamic simulation (SI Appendix, Figs. S13A and S14A). Distinct gene perturbations led to specific alterations in the cell development landscape, cell fate conversion, and variations in EPR and flux (SI Appendix, Figs. S13 B and C and S14 B and C). Detailed discussions in SI Appendix, Supporting Results.

Inferring Cell-Type-Specific Gene Regulatory Networks and Cell–Cell Interactions of Cell Differentiation.

Biological molecular networks determine cell identity, gene function, and phenotypic traits. During development, dynamically evolving gene regulatory networks (GRNs) are sculpted by an array of cell-type-specific TFs (50). A deep understanding of these networks and their dynamic interactions is essential for a holistic view of cellular development. We began by inferring and illustrating the cell-type specific gene regulatory networks related to mouse retinal neuron development (Fig. 6A). By quantifying the Jacobian of gene–gene interaction matrices for different cell types, we can provide a quantitative perspective on the regulatory relationships between genes (SI Appendix, Fig. S15). The positive Jacobian represents activation while negative for inhibition (20). In these GRNs, the directional arrows, differentiated by color and width, offer a clear understanding of the nature (activation or inhibition) and strength of interactions between genes. Our analysis revealed that gene regulatory networks exhibit a variety of cell-type specificity: First, the expression level of genes shows cell-type specificity, such as Cenpf is highly expressed in progenitor cells but low in differentiated neuronal cells, and conversely, Meis2, Tfap2b, and Pou6f2 are only relatively highly expressed in PR, AC/HC, and RGC cells and low in other cell lines. Second, the interaction intensity between genes also showed the specificity of cell type, such as Gngt2 has a strong promotion effect on Dmd in progenitor cells, and the promotion intensity of Gngt2 to Dmd in RGC cells is weakened. Third, the type of interactions between genes presents the specificity of cell type, such as Cyr61 promotes Dmd in PR cells but inhibits Dmd in AC/HC cells. These specificities were also evident in the human hematopoietic development system (SI Appendix, Fig. S19 A and B). Such insights into cell-type specific gene regulatory networks can greatly benefit targeted medical research, especially in areas as cell synthesis, regeneration, and cell fate transformation (51).

Fig. 6.

Cell-type-specific gene regulatory networks and cell–cell interactions. (A) The cell-type-specific gene regulatory networks of mouse retinal neuron development. The node size is scaled based on gene expression in the specific cell type and the color of nodes represents the node centrality in the network, the black arrows represent the activation and the red arrows represent the inhibition, and the width of connection arrows is scaled based on the strength of gene interactions. (B) The cell–cell interaction analysis.

Cell development is not solely a single-cell process, especially in vivo, it frequently requires the coordinated interaction of multiple cells. For instance, during in vivo embryonic development, cells communicate through secreted factors, guiding other cells toward specific functional states (52). Yet, the characterization of cell development at the level of the macroscopic patterns resulting from cell-to-cell interactions remains largely qualitative. Here, we highlight the interactions between different cell types, emphasizing the collaborative nature of cellular development (Fig. 6B). Our findings indicated that, during mouse retinal neuron development, RGC cells predominantly interacted with progenitor and AC/HC cells. In human hematopoietic development, Neu and Meg cells exhibited more interactions with other cells, especially Neu and Bas cells communicate most with each other (SI Appendix, Fig. S19C). These cell–cell interactions are facilitated through surface ligand–receptor interactions (53), and we further detailed specific ligand–receptor interactions in both mouse retinal neuronal and human hematopoietic development, which provides insights into the molecular communication that might be directing cell differentiation and development (SI Appendix, Figs. S16 and S20). For instance, RGC cells express higher levels of genes that encode teneurin transmembrane protein (TENM) family members and the gene family that encodes a member of the latrophilin subfamily of G-protein coupled receptors (ADGRLs), which triggers massive exocytosis from the neurons and neuroendocrine cells and is the receptor for TENM2 that mediates heterophilic synaptic cell–cell contact and postsynaptic specialization (SI Appendix, Fig. S16). TENMs interact with ADGRLs in embryonic cortical neurons which have previously been described (54). We pinpoint this series of family interactions specifically to the RGC cells. Similarly, SI Appendix, Fig. S20 highlights the underlying mechanism of the pronounced communication between the different cells in the human hematopoietic system, underlining the significance of these interactions in guiding cell fate decisions. For example, neutrophils have a high expression level of FPR1 (Formyl Peptide Receptor 1), the ligand of which (ANXA1, Annexin A1) is expressed by basophils. This interaction pattern is particularly evident between Neu and Bas, underscoring a specialized communication route that may be critical for differentiation and immune responses within the hematopoietic niche (55). Moreover, we studied the differences in pathway activity across cell types or species, observed both conservation and divergence in pathway activity between mouse and human cellular systems, providing insights into the active signaling pathways during cell development (SI Appendix, Supporting Results).

Discussion

Over the past decade, remarkable advancements in experimental techniques, such as single-cell dynamics in microscopy and high-throughput data acquisition, have provided an extensive dataset on cell dynamics, genetic regulation, and organismal development. This wealth of experimental data has spurred the evolution of tools and concepts to elucidate the physical mechanisms underpinning biological processes (1). To harness this data effectively, an analytical framework is imperative, especially for exploring the nonequilibrium dynamics and thermodynamic behavior of cells (56). Single-cell transcriptomics has emerged as a transformative tool in this endeavor. It not only provides a detailed snapshot of cellular states but also offers a unique opportunity to quantify the driving forces for cellular network dynamics. By leveraging this technology, our study has been able to experimentally validate the landscape and flux theory, bridging the theoretical constructs directly with high-throughput experimental data. This direct connection underscores the potential of single-cell transcriptomics in offering a holistic view of cellular dynamics and their underlying driving forces. The core of our methodology is the reconstruction of a continuous vector field based on single-cell RNA velocity data to directly quantify the underlying landscape–flux as the driving force for the nonequilibrium dynamics and thermodynamics, providing insights into the underlying physical mechanisms of the cell function. For the RNA velocity estimation, although many methods have been developed, different approaches are suitable for different types of data and biological systems (20–22, 57) (SI Appendix, Supporting Discussion). While our study leverages a computational framework to reconstruct cell state vector fields and quantify the landscape–flux in single-cell transcriptomics, the reconstructed force field is an average stationarity field that includes all cells together, ignoring the heterogeneity between cells, which cannot capture temporally evolving dynamics, while the cell proliferation and death effects are not considered in the current study. TrajectoryNet is the first method to consider growth/death by incorporating it as a separate discrete static unbalanced OT model in the continuous setting (58). TIGON (Trajectory Inference with Growth via Optimal transport and Neural network) reconstructs dynamic trajectories and population growth simultaneously as well as the underlying gene regulatory network from multiple snapshots with unbalanced optimal transport algorithm (59). Incorporating insights from these studies, we acknowledge that our model could be extended to explicitly consider the effects of cell proliferation and death (60).

Conventionally, cell development dynamics have been depicted using the Waddington landscape, where development is viewed as a ball rolling through a valley, providing a visual metaphor for differentiation (61). Another view posits that cell fate commitment arises from stochastic fluctuations, both intrinsic from limited molecule numbers and extrinsic from environmental factors (62), with experimental evidence supporting both the destabilization of the stem/progenitor undifferentiated state and fluctuation-induced transitions (63). Our study proposes a different cytogenetic landscape, indicating that cell development is influenced not only by traditional landscape gradient forces but also by a rotational force, the curl flux. This emergence of curl force accounts for the irreversibility of cell fate switching and the deviation of optimal paths for differentiation and reprogramming from the expected gradient paths, without necessarily crossing the landscape’s saddle point. Our previous works can experimentally quantify landscape–flux and transition time with low throughput of several genes (9–11). Combining the single-cell transcriptome data, we have further elucidated the irreversibility or deviation of the pathways from the expected landscape gradient, i.e., the irreversibility of gene expression dynamics along LAPs. This provides a better way to manipulate the fate of cells.

In addition to the optimal pathway, biologists are usually more concerned about the transition state reached during the cell fate switching. Traditionally, the transition states were considered reversible in Waddington’s epigenetic landscape. In this study, the transition states in our quantified nonequilibrium landscape are shown to be irreversible. The key to understanding this process lies in recognizing that the transition states in cellular systems do not always conform to the classical saddle points described in the energy landscapes of chemical reactions. Instead, these states manifest as path-dependent intermediates, with each path from undifferentiated to differentiated states (and vice versa in reprogramming) possessing its unique transition state and barrier. This revelation is crucial, as it implies that the new transition state representing the highest barriers on the optimal pathway does not necessarily coincide with the saddle barrier on the Waddington landscape in cell fate decision-making. The actual transition states, as our analysis indicates, emerge from a combination of the quantified Waddington landscape surface and the rotational curl flux. This latter element, a measure of deviation from equilibrium, reshapes the optimal paths, guiding the transition states away from the expected saddle points on the landscape under finite fluctuations.

Our research also delves into the physical and molecular aspects of cell fate determination, namely the nucleation mechanism of cell state phase transitions, specifically when nucleation takes place—the transition states and the nucleation seeds—pioneer genes. The nucleation mechanism akin to first-order phase transitions in physical systems takes place through bubble nucleation—a process where small clusters of a new phase emerge within the old phase. In cellular systems, this translates to discrete groups of cells or molecular interactions that initiate the switching from one cell fate to another. Transition states determined by the action of optimal paths can help reveal the nucleation mechanisms of cell differentiation, where the dynamic interplay of gene regulation, signaling pathways, and epigenetic factors combine for actions. The rationale for considering pioneer genes as nucleation seeds lies in their capacity to modulate the cell fate decision-making process. These genes do not merely influence the direction of differentiation but can significantly accelerate it. By altering the regulatory landscape at critical junctures, pioneer genes effectively facilitate a more rapid and directed switching of cell states. By identifying the pioneer genes that act as nucleation seeds within these states, we gain the potential to influence and direct the cell fate decision process. By elucidating these nucleation mechanisms, identifying the corresponding pioneer genes and understanding these complex dynamics can lead to more effective strategies for cell manipulation and regenerative medicine.

Our earlier proposal that curl flux acts as a dynamic origin of bifurcation suggests that flux enhances cell fate diversity during development (64). This concept stemmed from force decomposition, leading to our introduction of the loop-flux framework, which robustly quantifies how each cycle flux contributes to cell state transitions. In fact, discrete curl fluxes in the cell development landscape are connected to these distinct loop-fluxes. During cell development, several dominant loop-fluxes between different cell states, drive the transformation of cell fate and the diversity of cell fate. The implication of this finding is profound: it suggests that the differentiation landscape is sculpted not only by the final cell fates but also by the optimal paths cells take to achieve these fates. While our analysis has focused on several dominant loop-fluxes, which drive cell fate transformation and diversity, we acknowledge that not only the most prominent but also the secondary or tertiary loop-fluxes can have substantial impacts on the system’s dynamics (45). Our detailed analysis uncovers that, although the dominant loops capture the most direct pathways of cell state transitions, the subdominant loops often embody alternative, yet significant pathways that could provide resilience and adaptability to the cell fate landscape. These subdominant loops represent potential paths that cells might take under different conditions or perturbations, offering a “plan B” that ensures the robustness of the differentiation process. For instance, the second and third-ranking loops in SI Appendix, Fig. S10, while not the primary drivers of state transitions, may become increasingly relevant under certain biological contexts or stress conditions. These loops may facilitate transitions that bypass blocked or less efficient primary routes, thus maintaining the system’s fluidity and responsiveness. By understanding the hierarchy and interconnections of these loop-fluxes, we can more accurately predict cellular behavior and engineer conditions that favor desired cell fate outcomes. The implications of our findings are important for the field of cellular biology, suggesting that the differentiation landscape is sculpted by a complex interplay of multiple pathways, each contributing to the overall dynamics of the system. Subdominant loop-fluxes may enhance the flexibility of cell fate decisions, offering alternative routes for differentiation that could be exploited in regenerative medicine and disease modeling.

Additionally, the maintenance of adult tissues depends on the sustained activity of resident stem cells, yet the mechanisms governing their self-renewal during homeostasis remain largely unknown. Thermodynamically, cell development systems function as open, nonequilibrium systems that necessitate material input and energy consumption to sustain their steady-state and organizational order (29, 31). So, we shed light on the thermodynamic processes that underlie the preservation of cellular differentiation potency. A central revelation from our study is the heightened energy requirements of stem/progenitor cells, which is pivotal for maintaining rapid cell cycles and ensuring pluripotency in cell differentiation. The differentiated cells employ OXPHOS and efficiently catabolize carbon chains. Cells undergoing rapid growth and proliferation must adopt a glycolytic profile to provide anabolism with both carbon chains and ATP (Adenosine 5’-triphosphate) at high rates (65). This insight challenges traditional perspectives on cellular energy dynamics and paves the way for further investigations into the metabolic demands of different cell types and their implications in developmental biology.

Despite significant advancements, GRN inference remains challenging, including dynamic rewiring, causal inference, feedback loop modeling, and context specificity (51). Some existing cell-specific network inference methods vary in capability; some only infer undirected edges, failing to indicate the direction of information flow, while others can infer direction but not interaction intensity. Recently, the popular algorithms based on Jacobian function inference, such as spliceJAC and Dynamo, can infer the direction and intensity of the interactions. However, Dynamo’s Jacobian calculations are based on the entire average vector field, and although the Jacobian values are distributed differently across different cells and cell type-specific interaction networks can be inferred, this Jacobian based on the average field seems unreasonable (20). SpliceJAC is a good way to circumvent this shortcoming, and it calculates Jacobian based on RNA velocity for different cell types. However, spliceJAC does not use all genes to calculate Jacobian, but selects the top genes for calculation (66). locaTE, employs the localized transfer entropy to infer cell-specific GRNs with several choices of kernels (velocity kernel, diffusion pseudotime kernel, optimal transport kernel, and population balance analysis kernel) is a dynamical inference method that models single cell dynamics using Markov processes, which leads to an information-theoretic approach (67). These GRNs elucidate the cell-type specificity of gene expression and regulation, providing assistance for in vitro culture or synthetic engineering of cells (33).

In conclusion, the findings presented in this study underscore the transformative potential of integrating advanced single-cell sequencing technology with landscape and flux theory to decode the intricate dynamics of cell development. Our research not only bridges the gap between transcriptomic data and the underlying transcriptional drivers but also offers a comprehensive understanding of the nonequilibrium dynamical and thermodynamic forces that shape cellular functions. As we progress, integrating our landscape–flux theory and methodologies with advanced techniques and experimental validations will undoubtedly lead to further discoveries in single-cell biology, disease diagnostics, and therapeutic interventions.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

All study data are included in the article and/or SI Appendix. Previously published data were used for this work [The scRNA-seq raw data of the early-born retinal neurons are available at GEO with accession GSE12682016 (68). The scNT-seq raw data of human hematopoiesis are available at GEO with accession number GSE19351717 (69)].