A Dimension Reduction Approach for Energy Landscape: Identifying Intermediate States in Metabolism-EMT Network

Energy Landscape Reveals the Underlying Mechanism of Cancer-Adipose Conversion in Gene Network Models

Cancer is a systemic heterogeneous disease involving complex molecular networks. Tumor formation involves an epithelial-mesenchymal transition (EMT), which promotes both metastasis and plasticity of cancer cells. Recent experiments have proposed that cancer cells can be transformed into adipocytes via a combination of drugs. However, the underlying mechanisms for how these drugs work, from a molecular network perspective, remain elusive. To reveal the mechanism of cancer-adipose conversion (CAC), this study adopts a systems biology approach by combing mathematical modeling and molecular experiments, based on underlying molecular regulatory networks. Four types of attractors are identified, corresponding to epithelial (E), mesenchymal (M), adipose (A) and partial/intermediate EMT (P) cell states on the CAC landscape. Landscape and transition path results illustrate that intermediate states play critical roles in the cancer to adipose transition. Through a landscape control approach, two new therapeutic strategies for drug combinations are identified, that promote CAC. These predictions are verified by molecular experiments in different cell lines. The combined computational and experimental approach provides a powerful tool to explore molecular mechanisms for cell fate transitions in cancer networks. The results reveal underlying mechanisms of intermediate cell states that govern the CAC, and identified new potential drug combinations to induce cancer adipogenesis.

EPR-Net: constructing a non-equilibrium potential landscape via a variational force projection formulation

We present EPR-Net, a novel and effective deep learning approach that tackles a crucial challenge in biophysics: constructing potential landscapes for high-dimensional non-equilibrium steady-state systems. EPR-Net leverages a nice mathematical fact that the desired negative potential gradient is simply the orthogonal projection of the driving force of the underlying dynamics in a weighted inner-product space. Remarkably, our loss function has an intimate connection with the steady entropy production rate (EPR), enabling simultaneous landscape construction and EPR estimation. We introduce an enhanced learning strategy for systems with small noise, and extend our framework to include dimensionality reduction and the state-dependent diffusion coefficient case in a unified fashion. Comparative evaluations on benchmark problems demonstrate the superior accuracy, effectiveness and robustness of EPR-Net compared to existing methods. We apply our approach to challenging biophysical problems, such as an eight-dimensional (8D) limit cycle and a 52D multi-stability problem, which provide accurate solutions and interesting insights on constructed landscapes. With its versatility and power, EPR-Net offers a promising solution for diverse landscape construction problems in biophysics.

Perspectives on computational modeling of biological systems and the significance of the SysMod community

Motivation

In recent years, applying computational modeling to systems biology has caused a substantial surge in both discovery and practical applications and a significant shift in our understanding of the complexity inherent in biological systems.

Results

In this perspective article, we briefly overview computational modeling in biology, highlighting recent advancements such as multi-scale modeling due to the omics revolution, single-cell technology, and integration of artificial intelligence and machine learning approaches. We also discuss the primary challenges faced: integration, standardization, model complexity, scalability, and interdisciplinary collaboration. Lastly, we highlight the contribution made by the Computational Modeling of Biological Systems (SysMod) Community of Special Interest (COSI) associated with the International Society of Computational Biology (ISCB) in driving progress within this rapidly evolving field through community engagement (via both in person and virtual meetings, social media interactions), webinars, and conferences.

Availability and implementation

Additional information about SysMod is available at https://sysmod.info.

Phase separation provides a mechanism to drive phenotype switching

Phenotypic switching plays a crucial role in cell fate determination across various organisms. Recent experimental findings highlight the significance of protein compartmentalization via liquid-liquid phase separation in influencing such decisions. However, the precise mechanism through which phase separation regulates phenotypic switching remains elusive. To investigate this, we established a mathematical model that couples a phase separation process and a gene expression process with feedback. We used the chemical master equation theory and mean-field approximation to study the effects of phase separation on the gene expression products. We found that phase separation can cause bistability and bimodality. Furthermore, phase separation can control the bistable properties of the system, such as bifurcation points and bistable ranges. On the other hand, in stochastic dynamics, the droplet phase exhibits double peaks within a more extensive phase separation threshold range than the dilute phase, indicating the pivotal role of the droplet phase in cell fate decisions. These findings propose an alternative mechanism that influences cell fate decisions through the phase separation process. As phase separation is increasingly discovered in gene regulatory networks, related modeling research can help build biomolecular systems with desired properties and offer insights into explaining cell fate decisions.

Transcriptome free energy can serve as a dynamic patient-specific biomarker in acute myeloid leukemia

Acute myeloid leukemia (AML) is prevalent in both adult and pediatric patients. Despite advances in patient categorization, the heterogeneity of AML remains a challenge. Recent studies have explored the use of gene expression data to enhance AML diagnosis and prognosis, however, alternative approaches rooted in physics and chemistry may provide another level of insight into AML transformation. Utilizing publicly available databases, we analyze 884 human and mouse blood and bone marrow samples. We employ a personalized medicine strategy, combining state-transition theory and surprisal analysis, to assess the RNA transcriptome of individual patients. The transcriptome is transformed into physical parameters that represent each sample's steady state and the free energy change (FEC) from that steady state, which is the state with the lowest free energy.We found the transcriptome steady state was invariant across normal and AML samples. FEC, representing active molecular processes, varied significantly between samples and was used to create patient-specific barcodes to characterize the biology of the disease. We discovered that AML samples that were in a transition state had the highest FEC. This disease state may be characterized as the most unstable and hence the most therapeutically targetable since a change in free energy is a thermodynamic requirement for disease progression. We also found that distinct sets of ongoing processes may be at the root of otherwise similar clinical phenotypes, implying that our integrated analysis of transcriptome profiles may facilitate a personalized medicine approach to cure AML and restore a steady state in each patient.

Data-driven energy landscape reveals critical genes in cancer progression

The evolution of cancer is a complex process characterized by stable states and transitions among them. Studying the dynamic evolution of cancer and revealing the mechanisms of cancer progression based on experimental data is an important topic. In this study, we aim to employ a data-driven energy landscape approach to analyze the dynamic evolution of cancer. We take Kidney renal clear cell carcinoma (KIRC) as an example. From the energy landscape, we introduce two quantitative indicators (transition probability and barrier height) to study critical shifts in KIRC cancer evolution, including cancer onset and progression, and identify critical genes involved in these transitions. Our results successfully identify crucial genes that either promote or inhibit these transition processes in KIRC. We also conduct a comprehensive biological function analysis on these genes, validating the accuracy and reliability of our predictions. This work has implications for discovering new biomarkers, drug targets, and cancer treatment strategies in KIRC.

Landscape quantifies the intermediate state and transition dynamics in ecological networks

Understanding the ecological mechanisms associated with the collapse and restoration is especially critical in promoting harmonious coexistence between humans and nature. So far, it remains challenging to elucidate the mechanisms of stochastic dynamical transitions for ecological systems. Using an example of plant-pollinator network, we quantified the energy landscape of ecological system. The landscape displays multiple attractors characterizing the high, low and intermediate abundance stable states. Interestingly, we detected the intermediate states under pollinator decline, and demonstrated the indispensable role of the intermediate state in state transitions. From the landscape, we define the barrier height (BH) as a global quantity to evaluate the transition feasibility. We propose that the BH can serve as a new early-warning signal (EWS) for upcoming catastrophic breakdown, which provides an earlier and more accurate warning signal than traditional metrics based on time series. Our results promote developing better management strategies to achieve environmental sustainability.

Quantifying cancer cell plasticity with gene regulatory networks and single-cell dynamics

Phenotypic plasticity of cancer cells can lead to complex cell state dynamics during tumor progression and acquired resistance. Highly plastic stem-like states may be inherently drug-resistant. Moreover, cell state dynamics in response to therapy allow a tumor to evade treatment. In both scenarios, quantifying plasticity is essential for identifying high-plasticity states or elucidating transition paths between states. Currently, methods to quantify plasticity tend to focus on 1) quantification of quasi-potential based on the underlying gene regulatory network dynamics of the system; or 2) inference of cell potency based on trajectory inference or lineage tracing in single-cell dynamics. Here, we explore both of these approaches and associated computational tools. We then discuss implications of each approach to plasticity metrics, and relevance to cancer treatment strategies.

Controlling brain dynamics: Landscape and transition path for working memory

Understanding the underlying dynamical mechanisms of the brain and controlling it is a crucial issue in brain science. The energy landscape and transition path approach provides a possible route to address these challenges. Here, taking working memory as an example, we quantified its landscape based on a large-scale macaque model. The working memory function is governed by the change of landscape and brain-wide state switching in response to the task demands. The kinetic transition path reveals that information flow follows the direction of hierarchical structure. Importantly, we propose a landscape control approach to manipulate brain state transition by modulating external stimulation or inter-areal connectivity, demonstrating the crucial roles of associative areas, especially prefrontal and parietal cortical areas in working memory performance. Our findings provide new insights into the dynamical mechanism of cognitive function, and the landscape control approach helps to develop therapeutic strategies for brain disorders.

Structure and function in artificial, zebrafish and human neural networks

Network science provides a set of tools for the characterization of the structure and functional behavior of complex systems. Yet a major problem is to quantify how the structural domain is related to the dynamical one. In other words, how the diversity of dynamical states of a system can be predicted from the static network structure? Or the reverse problem: starting from a set of signals derived from experimental recordings, how can one discover the network connections or the causal relations behind the observed dynamics? Despite the advances achieved over the last two decades, many challenges remain concerning the study of the structure-dynamics interplay of complex systems. In neuroscience, progress is typically constrained by the low spatio-temporal resolution of experiments and by the lack of a universal inferring framework for empirical systems. To address these issues, applications of network science and artificial intelligence to neural data have been rapidly growing. In this article, we review important recent applications of methods from those fields to the study of the interplay between structure and functional dynamics of human and zebrafish brain. We cover the selection of topological features for the characterization of brain networks, inference of functional connections, dynamical modeling, and close with applications to both the human and zebrafish brain. This review is intended to neuroscientists who want to become acquainted with techniques from network science, as well as to researchers from the latter field who are interested in exploring novel application scenarios in neuroscience.

circCAPRIN1 interacts with STAT2 to promote tumor progression and lipid synthesis via upregulating ACC1 expression in colorectal cancer

BACKGROUND

Circular RNAs (circRNAs) generated by back-splicing of precursor mRNAs (pre-mRNAs) are often aberrantly expressed in cancer cells. Accumulating evidence has revealed that circRNAs play a critical role in the progression of several cancers, including colorectal cancer (CRC). However, the current understandings of the emerging functions of circRNAs in CRC lipid metabolism and the underlying molecular mechanisms are still limited. Here, we aimed to explore the role of circCAPRIN1 in regulating CRC lipid metabolism and tumorigenesis.

METHODS

circRNA microarray was performed with three pairs of tumor and non-tumor tissues from CRC patients. The expression of circRNAs were determined by quantitative PCR (qPCR) and in situ hybridization (ISH). The endogenous levels of circRNAs in CRC cells were manipulated by transfection with lentiviruses overexpressing or silencing circRNAs. The regulatory roles of circRNAs in the occurrence of CRC were investigated both in vitro and in vivo using gene expression array, RNA pull-down/mass spectrometry, RNA immunoprecipitation assay, luciferase reporter assay, chromatin immunoprecipitation analysis, and fluorescence in situ hybridization (FISH).

RESULTS

Among circRNAs, circCAPRIN1 was most significantly upregulated in CRC tissue specimens. circCAPRIN1 expression was positively correlated with the clinical stage and unfavorable prognosis of CRC patients. Downregulation of circCAPRIN1 suppressed proliferation, migration, and epithelial-mesenchymal transition of CRC cells, while circCAPRIN1 overexpression had opposite effects. RNA sequencing and gene ontology analysis indicated that circCAPRIN1 upregulated the expressions of genes involved in CRC lipid metabolism. Moreover, circCAPRIN1 promoted lipid synthesis by enhancing Acetyl-CoA carboxylase 1 (ACC1) expression. Further mechanistic assays demonstrated that circCAPRIN1 directly bound signal transducer and activator of transcription 2 (STAT2) to activate ACC1 transcription, thus regulating lipid metabolism and facilitating CRC tumorigenesis.

CONCLUSIONS

These findings revealed the oncogenic role and mechanism of circCAPRIN1 in CRC. circCAPRIN1 interacted with STAT2 to promote CRC tumor progression and lipid synthesis by enhancing the expression of ACC1. circCAPRIN1 may be considered as a novel potential diagnostic and therapeutic target for CRC patients.

Inferring structural and dynamical properties of gene networks from data with deep learning

The reconstruction of gene regulatory networks (GRNs) from data is vital in systems biology. Although different approaches have been proposed to infer causality from data, some challenges remain, such as how to accurately infer the direction and type of interactions, how to deal with complex network involving multiple feedbacks, as well as how to infer causality between variables from real-world data, especially single cell data. Here, we tackle these problems by deep neural networks (DNNs). The underlying regulatory network for different systems (gene regulations, ecology, diseases, development) can be successfully reconstructed from trained DNN models. We show that DNN is superior to existing approaches including Boolean network, Random Forest and partial cross mapping for network inference. Further, by interrogating the ensemble DNN model trained from single cell data from dynamical system perspective, we are able to unravel complex cell fate dynamics during preimplantation development. We also propose a data-driven approach to quantify the energy landscape for gene regulatory systems, by combining DNN with the partial self-consistent mean field approximation (PSCA) approach. We anticipate the proposed method can be applied to other fields to decipher the underlying dynamical mechanisms of systems from data.

Perspectives on the landscape and flux theory for describing emergent behaviors of the biological systems

We give a review on the landscape theory of the equilibrium biological systems and landscape-flux theory of the nonequilibrium biological systems as the global driving force. The emergences of the behaviors, the associated thermodynamics in terms of the entropy and free energy and dynamics in terms of the rate and paths have been quantitatively demonstrated. The hierarchical organization structures have been discussed. The biological applications ranging from protein folding, biomolecular recognition, specificity, biomolecular evolution and design for equilibrium systems as well as cell cycle, differentiation and development, cancer, neural networks and brain function, and evolution for nonequilibrium systems, cross-scale studies of genome structural dynamics and experimental quantifications/verifications of the landscape and flux are illustrated. Together, this gives an overall global physical and quantitative picture in terms of the landscape and flux for the behaviors, dynamics and functions of biological systems.

Network modeling-based identification of the switching targets between pyroptosis and secondary pyroptosis

The newly identified cell death type, pyroptosis plays crucial roles in various diseases. Most recently, mounting evidence accumulates that pyroptotic signaling is highly correlated with coronavirus disease 2019 (COVID-19). Thus, understanding the induction of the pyroptotic signaling and dissecting the detail molecular control mechanisms are urgently needed. Based on recent experimental studies, a core regulatory model of the pyroptotic signaling is constructed to investigate the intricate crosstalk dynamics between the two cell death types, i.e., pyroptosis and secondary pyroptosis. The model well reproduces the experimental observations under different conditions. Sensitivity analysis determines that only the expression level of caspase-1 or GSDMD has the potential to individually change death modes. The decrease of caspase-1 or GSDMD level switches cell death from pyroptosis to secondary pyroptosis. Besides, eight biochemical reactions are identified that can efficiently switch death modes. While from the viewpoint of bifurcation analysis, the expression level of caspase-3 is further identified and twelve biochemical reactions are obtained. The coexistence of pyroptosis and secondary pyroptosis is predicted to be observed not only within the bistable range, but also within proper monostable range, presenting two potential different control mechanisms. Combined with the landscape theory, we further explore the stochastic dynamic and global stability of the pyroptotic system, accurately quantifying how each component mediates the individual occurrence probability of pyroptosis and secondary pyroptosis. Overall, this study sheds new light on the intricate crosstalk of the pyroptotic signaling and uncovers the regulatory mechanisms of various stable state transitions, providing potential clues to guide the development for prevention and treatment of pyroptosis-related diseases.

Unraveling the stochastic transition mechanism between oscillation states by landscape and minimum action path theory

Cell fate transitions have been studied from various perspectives, such as the transition between stable states, or the transition between stable states and oscillation states. However, there is a lack...

Dynamic inference of cell developmental complex energy landscape from time series single-cell transcriptomic data

Time series single-cell RNA sequencing (scRNA-seq) data are emerging. However, dynamic inference of an evolving cell population from time series scRNA-seq data is challenging owing to the stochasticity and nonlinearity of the underlying biological processes. This calls for the development of mathematical models and methods capable of reconstructing cellular dynamic transition processes and uncovering the nonlinear cell-cell interactions. In this study, we present GraphFP, a nonlinear Fokker-Planck equation on graph based model and dynamic inference framework, with the aim of reconstructing the cell state-transition complex potential energy landscape from time series single-cell transcriptomic data. The free energy of our model explicitly takes into account of the cell-cell interactions in a nonlinear quadratic term. We then recast the model inference problem in the form of a dynamic optimal transport framework and solve it efficiently with the adjoint method of optimal control. We evaluated GraphFP on the time series scRNA-seq data set of embryonic murine cerebral cortex development. We illustrated that it 1) reconstructs cell state potential energy, which is a measure of cellular differentiation potency, 2) faithfully charts the probability flows between paired cell states over the dynamic processes of cell differentiation, and 3) accurately quantifies the stochastic dynamics of cell type frequencies on probability simplex in continuous time. We also illustrated that GraphFP is robust in terms of cluster labelling with different resolutions, as well as parameter choices. Meanwhile, GraphFP provides a model-based approach to delineate the cell-cell interactions that drive cell differentiation. GraphFP software is available at https://github.com/QiJiang-QJ/GraphFP.

Dynamic inference of cell development processes from time series scRNA-seq data is a major challenge. Here, we present GraphFP, a coherent computational framework that simultaneously reconstructs the cell state-transition complex potential energy landscape and infers cell-cell interactions from time series single-cell transcriptomic data. Based on the mathematical framework of nonlinear Fokker-Planck equation on graph, GraphFP models the stochastic dynamics of the cell state/type frequencies on probability simplex in continuous time, where the free energy with a nonlinear quadratic interaction term is employed to characterize cell-cell interactions. We formulate the model inference problem in the form of a dynamic optimal transport framework and solve it efficiently with the celebrated adjoint method. GraphFP allows for 1) reconstructing cell state potential energy, which is a measure of cellular differentiation potency, 2) charting the probability flows between paired cell states over dynamic processes, 3) quantifying the stochastic dynamics of cell type frequencies on probability simplex in continuous time, and 4) delineating cell-cell interactions that drive cell differentiation. We show how GraphFP can be used to faithfully reveal and accurately quantify the cell development processes using the embryonic murine cerebral cortex development time series scRNA-seq dataset.

Tristability in mitochondrial permeability transition pore opening

Mitochondrial permeability transition pore (PTP), a key regulator of cell life and death processes, is triggered by calcium ions (Ca²⁺) and potentiated by reactive oxygen species (ROS). Although the two modes of PTP opening, i.e., transient and persistent, have been identified for a long time, its dynamical mechanism is still not fully understood. To test a proposed hypothesis that PTP opening acts as a tristable switch, which is characterized by low, medium, and high open probability, we develop a three-variable model that focused on PTP opening caused by Ca²⁺ and ROS. For the system reduced to two differential equations for Ca²⁺ and ROS, both the stability analysis and the potential landscape feature that it exhibits tristability under standard parameters. For the full system, the bifurcation analysis suggests that it can achieve tristability over a wide range of input parameters. Furthermore, parameter sensitivity analysis demonstrates that the existence of tristability is a robust property. In addition, we show how the deterministic tristable property can be understood within a stochastic framework, which also explains the PTP dynamics at the level of a single channel. Overall, this study may yield valuable insights into the intricate regulatory mechanism of PTP opening.

Quantifying the Landscape of Decision Making From Spiking Neural Networks

The decision making function is governed by the complex coupled neural circuit in the brain. The underlying energy landscape provides a global picture for the dynamics of the neural decision making system and has been described extensively in the literature, but often as illustrations. In this work, we explicitly quantified the landscape for perceptual decision making based on biophysically-realistic cortical network with spiking neurons to mimic a two-alternative visual motion discrimination task. Under certain parameter regions, the underlying landscape displays bistable or tristable attractor states, which quantify the transition dynamics between different decision states. We identified two intermediate states: the spontaneous state which increases the plasticity and robustness of changes of minds and the "double-up" state which facilitates the state transitions. The irreversibility of the bistable and tristable switches due to the probabilistic curl flux demonstrates the inherent non-equilibrium characteristics of the neural decision system. The results of global stability of decision-making quantified by barrier height inferred from landscape topography and mean first passage time are in line with experimental observations. These results advance our understanding of the stochastic and dynamical transition mechanism of decision-making function, and the landscape and kinetic path approach can be applied to other cognitive function related problems (such as working memory) in brain networks.

Landscape and kinetic path quantify critical transitions in epithelial-mesenchymal transition

Epithelial-mesenchymal transition (EMT), a basic developmental process that might promote cancer metastasis, has been studied from various perspectives. Recently, the early warning theory has been used to anticipate critical transitions in EMT from mathematical modeling. However, the underlying mechanisms of EMT involving complex molecular networks remain to be clarified. Especially, how to quantify the global stability and stochastic transition dynamics of EMT and what the underlying mechanism for early warning theory in EMT is remain to be fully clarified. To address these issues, we constructed a comprehensive gene regulatory network model for EMT and quantified the corresponding potential landscape. The landscape for EMT displays multiple stable attractors, which correspond to E, M, and some other intermediate states. Based on the path-integral approach, we identified the most probable transition paths of EMT, which are supported by experimental data. Correspondingly, the results of transition actions demonstrated that intermediate states can accelerate EMT, consistent with recent studies. By integrating the landscape and path with early warning concept, we identified the potential barrier height from the landscape as a global and more accurate measure for early warning signals to predict critical transitions in EMT. The landscape results also provide an intuitive and quantitative explanation for the early warning theory. Overall, the landscape and path results advance our mechanistic understanding of dynamical transitions and roles of intermediate states in EMT, and the potential barrier height provides a new, to our knowledge, measure for critical transitions and quantitative explanations for the early warning theory.

Partial EMT in Squamous Cell Carcinoma: A Snapshot

In the process of cancer EMT, some subgroups of cancer cells simultaneously exhibit both mesenchymal and epithelial characteristics, a phenomenon termed partial EMT (pEMT). pEMT is a plastic state in which cells coexpress epithelial and mesenchymal markers. In squamous cell carcinoma (SCC), pEMT is regulated, and the phenotype is maintained via the HIPPO pathway, NOTCH pathway and TGF-β pathways and by microRNAs, lncRNAs and the cancer microenvironment (CME); thus, SCC exhibits aggressive tumorigenic properties and high stemness, which leads collective migration and therapy resistance. Few studies have reported therapeutic interventions to address cells that have undergone pEMT, and this approach may be an effective way to inhibit the plasticity, drug resistance and metastatic potential of SCC.