Not just a colourful metaphor: modelling the landscape of cellular development using Hopfield networks

[Nobel Prize in physics 2024 : John J. Hopfield and Geoffrey E. Hinton. From Hopfield and Hinton to AlphaFold: The 2024 Nobel Prize honors the pioneers of deep learning]

On October 8, 2024, the Nobel Prize in Physics was awarded to John J. Hopfield, professor at Princeton University, and Geoffrey E. Hinton, professor at the University of Toronto, for their "fundamental discoveries that made possible machine learning through artificial neural networks." According to the Nobel committee, John Hopfield designed an associative memory capable of storing and reconstructing images, while Geoffrey Hinton developed a method enabling tasks such as identifying specific elements within images. This article retraces the career paths of these two researchers and highlights their pioneering contributions.

Reconstructing Waddington Landscape from Cell Migration and Proliferation

The Waddington landscape was initially proposed to depict cell differentiation, and has been extended to explain phenomena such as reprogramming. The landscape serves as a concrete representation of cellular differentiation potential, yet the precise representation of this potential remains an unsolved problem, posing significant challenges to reconstructing the Waddington landscape. The characterization of cellular differentiation potential relies on transcriptomic signatures of known markers typically. Numerous computational models based on various energy indicators, such as Shannon entropy, have been proposed. While these models can effectively characterize cellular differentiation potential, most of them lack corresponding dynamical interpretations, which are crucial for enhancing our understanding of cell fate transitions. Therefore, from the perspective of cell migration and proliferation, a feasible framework was developed for calculating the dynamically interpretable energy indicator to reconstruct Waddington landscape based on sparse autoencoders and the reaction diffusion advection equation. Within this framework, typical cellular developmental processes, such as hematopoiesis and reprogramming processes, were dynamically simulated and their corresponding Waddington landscapes were reconstructed. Furthermore, dynamic simulation and reconstruction were also conducted for special developmental processes, such as embryogenesis and Epithelial-Mesenchymal Transition process. Ultimately, these diverse cell fate transitions were amalgamated into a unified Waddington landscape.

Chromatin-based memory as a self-stabilizing influence on cell identity

Cell types are traditionally thought to be specified and stabilized by gene regulatory networks. Here, we explore how chromatin memory contributes to the specification and stabilization of cell states. Through pervasive, local, feedback loops, chromatin memory enables cell states that were initially unstable to become stable. Deeper appreciation of this self-stabilizing role for chromatin broadens our perspective of Waddington's epigenetic landscape from a static surface with islands of stability shaped by evolution, to a plasticine surface molded by experience. With implications for the evolution of cell types, stabilization of resistant states in cancer, and the widespread plasticity of complex life.

TMELand: An End-to-End Pipeline for Quantification and Visualization of Waddington's Epigenetic Landscape Based on Gene Regulatory Network

Waddington's epigenetic landscape is a framework depicting the processes of cell differentiation and reprogramming under the control of a gene regulatory network (GRN). Traditional model-driven methods for landscape quantification focus on the Boolean network or differential equation-based models of GRN, which need sophisticated prior knowledge and hence hamper their practical applications. To resolve this problem, we combine data-driven methods for inferring GRNs from gene expression data with model-driven approach to the landscape mapping. Specifically, we build an end-to-end pipeline to link data-driven and model-driven methods and develop a software tool named TMELand for GRN inference, visualizing Waddington's epigenetic landscape, and calculating state transition paths between attractors to uncover the intrinsic mechanism of cellular transition dynamics. By integrating GRN inference from real transcriptomic data with landscape modeling, TMELand can facilitate studies of computational systems biology, such as predicting cellular states and visualizing the dynamical trends of cell fate determination and transition dynamics from single-cell transcriptomic data.

EnhancerNet: a predictive model of cell identity dynamics through enhancer selection

Understanding how cell identity is encoded by the genome and acquired during differentiation is a central challenge in cell biology. I have developed a theoretical framework called EnhancerNet, which models the regulation of cell identity through the lens of transcription factor-enhancer interactions. I demonstrate that autoregulation in these interactions imposes a constraint on the model, resulting in simplified dynamics that can be parameterized from observed cell identities. Despite its simplicity, EnhancerNet recapitulates a broad range of experimental observations on cell identity dynamics, including enhancer selection, cell fate induction, hierarchical differentiation through multipotent progenitor states and direct reprogramming by transcription factor overexpression. The model makes specific quantitative predictions, reproducing known reprogramming recipes and the complex haematopoietic differentiation hierarchy without fitting unobserved parameters. EnhancerNet provides insights into how new cell types could evolve and highlights the functional importance of distal regulatory elements with dynamic chromatin in multicellular evolution.

Single-cell transcriptional uncertainty landscape of cell differentiation

Background: Single-cell studies have demonstrated the presence of significant cell-to-cell heterogeneity in gene expression. Whether such heterogeneity is only a bystander or has a functional role in the cell differentiation process is still hotly debated. Methods: In this study, we quantified and followed single-cell transcriptional uncertainty - a measure of gene transcriptional stochasticity in single cells - in 10 cell differentiation systems of varying cell lineage progressions, from single to multi-branching trajectories, using the stochastic two-state gene transcription model. Results: By visualizing the transcriptional uncertainty as a landscape over a two-dimensional representation of the single-cell gene expression data, we observed universal features in the cell differentiation trajectories that include: (i) a peak in single-cell uncertainty during transition states, and in systems with bifurcating differentiation trajectories, each branching point represents a state of high transcriptional uncertainty; (ii) a positive correlation of transcriptional uncertainty with transcriptional burst size and frequency; (iii) an increase in RNA velocity preceding the increase in the cell transcriptional uncertainty. Conclusions: Our findings suggest a possible universal mechanism during the cell differentiation process, in which stem cells engage stochastic exploratory dynamics of gene expression at the start of the cell differentiation by increasing gene transcriptional bursts, and disengage such dynamics once cells have decided on a particular terminal cell identity. Notably, the peak of single-cell transcriptional uncertainty signifies the decision-making point in the cell differentiation process.

Characterizing Cellular Differentiation Potency and Waddington Landscape via Energy Indicator

The precise characterization of cellular differentiation potency remains an open question, which is fundamentally important for deciphering the dynamics mechanism related to cell fate transition. We quantitatively evaluated the differentiation potency of different stem cells based on the Hopfield neural network (HNN). The results emphasized that cellular differentiation potency can be approximated by Hopfield energy values. We then profiled the Waddington energy landscape of embryogenesis and cell reprogramming processes. The energy landscape at single-cell resolution further confirmed that cell fate decision is progressively specified in a continuous process. Moreover, the transition of cells from one steady state to another in embryogenesis and cell reprogramming processes was dynamically simulated on the energy ladder. These two processes can be metaphorized as the motion of descending and ascending ladders, respectively. We further deciphered the dynamics of the gene regulatory network (GRN) for driving cell fate transition. Our study proposes a new energy indicator to quantitatively characterize cellular differentiation potency without prior knowledge, facilitating the further exploration of the potential mechanism of cellular plasticity.

Intrinsically Disordered Proteins: Critical Components of the Wetware

Despite the wealth of knowledge gained about intrinsically disordered proteins (IDPs) since their discovery, there are several aspects that remain unexplored and, hence, poorly understood. A living cell is a complex adaptive system that can be described as a wetware─a metaphor used to describe the cell as a computer comprising both hardware and software and attuned to logic gates─capable of "making" decisions. In this focused Review, we discuss how IDPs, as critical components of the wetware, influence cell-fate decisions by wiring protein interaction networks to keep them minimally frustrated. Because IDPs lie between order and chaos, we explore the possibility that they can be modeled as attractors. Further, we discuss how the conformational dynamics of IDPs manifests itself as conformational noise, which can potentially amplify transcriptional noise to stochastically switch cellular phenotypes. Finally, we explore the potential role of IDPs in prebiotic evolution, in forming proteinaceous membrane-less organelles, in the origin of multicellularity, and in protein conformation-based transgenerational inheritance of acquired characteristics. Together, these ideas provide a new conceptual framework to discern how IDPs may perform critical biological functions despite their lack of structure.

Statistical mechanics meets single-cell biology

Single-cell omics is transforming our understanding of cell biology and disease, yet the systems-level analysis and interpretation of single-cell data faces many challenges. In this Perspective, we describe the impact that fundamental concepts from statistical mechanics, notably entropy, stochastic processes and critical phenomena, are having on single-cell data analysis. We further advocate the need for more bottom-up modelling of single-cell data and to embrace a statistical mechanics analysis paradigm to help attain a deeper understanding of single-cell systems biology.

Digital Cell Sorter (DCS): a cell type identification, anomaly detection, and Hopfield landscapes toolkit for single-cell transcriptomics

MOTIVATION

Analysis of singe cell RNA sequencing (scRNA-seq) typically consists of different steps including quality control, batch correction, clustering, cell identification and characterization, and visualization. The amount of scRNA-seq data is growing extremely fast, and novel algorithmic approaches improving these steps are key to extract more biological information. Here, we introduce: (i) two methods for automatic cell type identification (i.e., without expert curator) based on a voting algorithm and a Hopfield classifier, (ii) a method for cell anomaly quantification based on isolation forest, and (iii) a tool for the visualization of cell phenotypic landscapes based on Hopfield energy-like functions. These new approaches are integrated in a software platform that includes many other state-of-the-art methodologies and provides a self-contained toolkit for scRNA-seq analysis.

RESULTS

We present a suite of software elements for the analysis of scRNA-seq data. This Python-based open source software, Digital Cell Sorter (DCS), consists in an extensive toolkit of methods for scRNA-seq analysis. We illustrate the capability of the software using data from large datasets of peripheral blood mononuclear cells (PBMC), as well as plasma cells of bone marrow samples from healthy donors and multiple myeloma patients. We test the novel algorithms by evaluating their ability to deconvolve cell mixtures and detect small numbers of anomalous cells in PBMC data.

AVAILABILITY

The DCS toolkit is available for download and installation through the Python Package Index (PyPI). The software can be deployed using the Python import function following installation. Source code is also available for download on Zenodo: DOI 10.5281/zenodo.2533377.

SUPPLEMENTARY INFORMATION

Supplemental Materials are available at PeerJ online.

Master regulators as order parameters of gene expression states

Cell type-specific gene expression patterns are represented as memory states of a Hopfield neural network model. It is shown that order parameters of this model can be interpreted as concentrations of master transcription regulators that form concurrent positive feedback loops with a large number of downstream regulated genes. The order parameter free energy then defines an epigenetic landscape in which local minima correspond to stable cell states. The model is applied to gene expression data in the context of hematopoiesis.

Modeling Basins of Attraction for Breast Cancer Using Hopfield Networks

Cancer is a genetic disease for which traditional treatments cause harmful side effects. After two decades of genomics technological breakthroughs, personalized medicine is being used to improve treatment outcomes and mitigate side effects. In mathematical modeling, it has been proposed that cancer matches an attractor in Waddington's epigenetic landscape. The use of Hopfield networks is an attractive modeling approach because it requires neither previous biological knowledge about protein-protein interactions nor kinetic parameters. In this report, Hopfield network modeling was used to analyze bulk RNA-Seq data of paired breast tumor and control samples from 70 patients. We characterized the control and tumor attractors with respect to their size and potential energy and correlated the Euclidean distances between the tumor samples and the control attractor with their corresponding clinical data. In addition, we developed a protocol that outlines the key genes involved in tumor state stability. We found that the tumor basin of attraction is larger than that of the control and that tumor samples are associated with a more substantial negative energy than control samples, which is in agreement with previous reports. Moreover, we found a negative correlation between the Euclidean distances from tumor samples to the control attractor and patient overall survival. The ascending order of each node's density in the weight matrix and the descending order of the number of patients that have the target active only in the tumor sample were the parameters that withdrew more tumor samples from the tumor basin of attraction with fewer gene inhibitions. The combinations of therapeutic targets were specific to each patient. We performed an initial validation through simulation of trastuzumab treatment effects in HER2+ breast cancer samples. For that, we built an energy landscape composed of single-cell and bulk RNA-Seq data from trastuzumab-treated and non-treated HER2+ samples. The trajectory from the non-treated bulk sample toward the treated bulk sample was inferred through the perturbation of differentially expressed genes between these samples. Among them, we characterized key genes involved in the trastuzumab response according to the literature.

Development and utilization of human decidualization reporter cell line uncovers new modulators of female fertility

Failure of embryo implantation accounts for a significant percentage of female infertility. Exquisitely coordinated molecular programs govern the interaction between the competent blastocyst and the receptive uterus. Decidualization, the rapid proliferation and differentiation of endometrial stromal cells into decidual cells, is required for implantation. Decidualization defects can cause poor placentation, intrauterine growth restriction, and early parturition leading to preterm birth. Decidualization has not yet been systematically studied at the genetic level due to the lack of a suitable high-throughput screening tool. Herein we describe the generation of an immortalized human endometrial stromal cell line that uses yellow fluorescent protein under the control of the prolactin promoter as a quantifiable visual readout of the decidualization response (hESC-PRLY cells). Using this cell line, we performed a genome-wide siRNA library screen, as well as a screen of 910 small molecules, to identify more than 4,000 previously unrecognized genetic and chemical modulators of decidualization. Ontology analysis revealed several groups of decidualization modulators, including many previously unappreciated transcription factors, sensory receptors, growth factors, and kinases. Expression studies of hits revealed that the majority of decidualization modulators are acutely sensitive to ovarian hormone exposure. Gradient treatment of exogenous factors was used to identify EC₅₀ values of small-molecule hits, as well as verify several growth factor hits identified by the siRNA screen. The high-throughput decidualization reporter cell line and the findings described herein will aid in the development of patient-specific treatments for decidualization-based recurrent pregnancy loss, subfertility, and infertility.

Hope4Genes: a Hopfield-like class prediction algorithm for transcriptomic data

After its introduction in 1982, the Hopfield model has been extensively applied for classification and pattern recognition. Recently, its great potential in gene expression patterns retrieval has also been shown. Following this line, we develop Hope4Genes a single-sample class prediction algorithm based on a Hopfield-like model. Differently from previous works, we here tested the performances of the algorithm for class prediction, a task of fundamental importance for precision medicine and therapeutic decision-making. Hope4Genes proved better performances than the state-of-art methodologies in the field independently of the size of the input dataset, its profiling platform, the number of classes and the typical class-imbalance present in biological data. Our results provide encoraging evidence that the Hopfield model, together with the use of its energy for the estimation of the false discoveries, is a particularly promising tool for precision medicine.

Quantitative Modelling of the Waddington Epigenetic Landscape

C.H. Waddington introduced the epigenetic landscape as a metaphor to represent cellular decision-making during development. Like a population of balls rolling down a rough hillside, developing cells follow specific trajectories (valleys) and eventually come to rest in one or another low-energy state that represents a mature cell type. Waddington depicted the topography of this landscape as determined by interactions among gene products, thereby connecting genotype to phenotype. In modern terms, each point on the landscape represents a state of the underlying genetic regulatory network, which in turn is described by a gene expression profile. In this chapter we demonstrate how the mathematical formalism of Hopfield networks can be used to model this epigenetic landscape. Hopfield networks are auto-associative artificial neural networks; input patterns are stored as attractors of the network and can be recalled from noisy or incomplete inputs. The resulting models capture the temporal dynamics of a gene regulatory network, yielding quantitative insight into cellular development and phenotype.

Towards a Quantitative Understanding of Cell Identity

Cells have traditionally been characterized using expression levels of a few proteins that are thought to specify phenotype. This requires a priori selection of proteins, which can introduce descriptor bias, and neglects the wealth of additional molecular information nested within each cell in a population, which often makes these sparse descriptors qualitative. Recently, more unbiased and quantitative cell characterization has been made possible by new high-throughput, information-dense experimental approaches and data-driven computational methods. This review discusses such quantitative descriptors in the context of three central concepts of cell identity: definition, creation, and stability. Collectively, these concepts are essential for constructing quantitative phenotypic landscapes, which will enhance our understanding of cell biology and facilitate cell engineering for research and clinical applications.

HopLand: single-cell pseudotime recovery using continuous Hopfield network-based modeling of Waddington's epigenetic landscape

Motivation

The interpretation of transcriptional dynamics in single-cell data, especially pseudotime estimation, could help understand the transition of gene expression profiles. The recovery of pseudotime increases the temporal resolution of single-cell transcriptional data, but is challenging due to the high variability in gene expression between individual cells. Here, we introduce HopLand, a pseudotime recovery method using continuous Hopfield network to map cells to a Waddington’s epigenetic landscape. It reveals from the single-cell data the combinatorial regulatory interactions among genes that control the dynamic progression through successive cell states.

Results

We applied HopLand to different types of single-cell transcriptomic data. It achieved high accuracies of pseudotime prediction compared with existing methods. Moreover, a kinetic model can be extracted from each dataset. Through the analysis of such a model, we identified key genes and regulatory interactions driving the transition of cell states. Therefore, our method has the potential to generate fundamental insights into cell fate regulation.

Availability and implementation

The MATLAB implementation of HopLand is available at https://github.com/NetLand-NTU/HopLand.

NetLand: quantitative modeling and visualization of Waddington's epigenetic landscape using probabilistic potential

Summary

Waddington’s epigenetic landscape is a powerful metaphor for cellular dynamics driven by gene regulatory networks (GRNs). Its quantitative modeling and visualization, however, remains a challenge, especially when there are more than two genes in the network. A software tool for Waddington’s landscape has not been available in the literature. We present NetLand, an open-source software tool for modeling and simulating the kinetic dynamics of GRNs, and visualizing the corresponding Waddington’s epigenetic landscape in three dimensions without restriction on the number of genes in a GRN. With an interactive and graphical user interface, NetLand can facilitate the knowledge discovery and experimental design in the study of cell fate regulation (e.g. stem cell differentiation and reprogramming).

Availability and Implementation

NetLand can run under operating systems including Windows, Linux and OS X. The executive files and source code of NetLand as well as a user manual, example models etc. can be downloaded from http://netland-ntu.github.io/NetLand/.

Supplementary information

Supplementary data are available at Bioinformatics online.

A stochastic and dynamical view of pluripotency in mouse embryonic stem cells

Pluripotent embryonic stem cells are of paramount importance for biomedical sciences because of their innate ability for self-renewal and differentiation into all major cell lines. The fateful decision to exit or remain in the pluripotent state is regulated by complex genetic regulatory networks. The rapid growth of single-cell sequencing data has greatly stimulated applications of statistical and machine learning methods for inferring topologies of pluripotency regulating genetic networks. The inferred network topologies, however, often only encode Boolean information while remaining silent about the roles of dynamics and molecular stochasticity inherent in gene expression. Herein we develop a framework for systematically extending Boolean-level network topologies into higher resolution models of networks which explicitly account for the promoter architectures and gene state switching dynamics. We show the framework to be useful for disentangling the various contributions that gene switching, external signaling, and network topology make to the global heterogeneity and dynamics of transcription factor populations. We find the pluripotent state of the network to be a steady state which is robust to global variations of gene switching rates which we argue are a good proxy for epigenetic states of individual promoters. The temporal dynamics of exiting the pluripotent state, on the other hand, is significantly influenced by the rates of genetic switching which makes cells more responsive to changes in extracellular signals.

Modeling the Attractor Landscape of Disease Progression: a Network-Based Approach

Genome-wide regulatory networks enable cells to function, develop, and survive. Perturbation of these networks can lead to appearance of a disease phenotype. Inspired by Conrad Waddington's epigenetic landscape of cell development, we use a Hopfield network formalism to construct an attractor landscape model of disease progression based on protein- or gene-correlation networks of Parkinson's disease, glioma, and colorectal cancer. Attractors in this landscape correspond to normal and disease states of the cell. We introduce approaches to estimate the size and robustness of these attractors, and take a network-based approach to study their biological features such as the key genes and their functions associated with the attractors. Our results show that the attractor of cancer cells is wider than the attractor of normal cells, suggesting a heterogeneous nature of cancer. Perturbation analysis shows that robustness depends on characteristics of the input data (number of samples per time-point, and the fraction which converge to an attractor). We identify unique gene interactions at each stage, which reflect the temporal rewiring of the gene regulatory network (GRN) with disease progression. Our model of the attractor landscape, constructed from large-scale gene expression profiles of individual patients, captures snapshots of disease progression and identifies gene interactions specific to different stages, opening the way for development of stage-specific therapeutic strategies.

Dynamics of embryonic stem cell differentiation inferred from single-cell transcriptomics show a series of transitions through discrete cell states

The complexity of gene regulatory networks that lead multipotent cells to acquire different cell fates makes a quantitative understanding of differentiation challenging. Using a statistical framework to analyze single-cell transcriptomics data, we infer the gene expression dynamics of early mouse embryonic stem (mES) cell differentiation, uncovering discrete transitions across nine cell states. We validate the predicted transitions across discrete states using flow cytometry. Moreover, using live-cell microscopy, we show that individual cells undergo abrupt transitions from a naïve to primed pluripotent state. Using the inferred discrete cell states to build a probabilistic model for the underlying gene regulatory network, we further predict and experimentally verify that these states have unique response to perturbations, thus defining them functionally. Our study provides a framework to infer the dynamics of differentiation from single cell transcriptomics data and to build predictive models of the gene regulatory networks that drive the sequence of cell fate decisions during development.

DOI:http://dx.doi.org/10.7554/eLife.20487.001

Respectful Modeling: Addressing Uncertainty in Dynamic System Models for Molecular Biology

Although there is still some skepticism in the biological community regarding the value and significance of quantitative computational modeling, important steps are continually being taken to enhance its accessibility and predictive power. We view these developments as essential components of an emerging 'respectful modeling' framework which has two key aims: (i) respecting the models themselves and facilitating the reproduction and update of modeling results by other scientists, and (ii) respecting the predictions of the models and rigorously quantifying the confidence associated with the modeling results. This respectful attitude will guide the design of higher-quality models and facilitate the use of models in modern applications such as engineering and manipulating microbial metabolism by synthetic biology.