Eddy current and coupled landscapes for nonadiabatic and nonequilibrium complex system dynamics

Irreversibility in bacterial regulatory networks

Irreversibility, in which a transient perturbation leaves a system in a new state, is an emergent property in systems of interacting entities. This property has well-established implications in statistical physics but remains underexplored in biological networks, especially for bacteria and other prokaryotes whose regulation of gene expression occurs predominantly at the transcriptional level. Focusing on the reconstructed regulatory network of Escherichia coli, we examine network responses to transient single-gene perturbations. We predict irreversibility in numerous cases and find that the incidence of irreversibility increases with the proximity of the perturbed gene to positive circuits in the network. Comparison with experimental data suggests a connection between the predicted irreversibility to transient perturbations and the evolutionary response to permanent perturbations.

Dynamical phase transition in models that couple chromatin folding with histone modifications

Genomic regions can acquire heritable epigenetic states through unique histone modifications, which lead to stable gene expression patterns without altering the underlying DNA sequence. However, the relationship between chromatin conformational dynamics and epigenetic stability is poorly understood. In this paper, we propose kinetic models to investigate the dynamic fluctuations of histone modifications and the spatial interactions between nucleosomes. Our model explicitly incorporates the influence of chemical modifications on the structural stability of chromatin and the contribution of chromatin contacts to the cooperative nature of chemical reactions. Through stochastic simulations and analytical theory, we have discovered distinct steady-state outcomes in different kinetic regimes, resembling a dynamical phase transition. Importantly, we have validated that the emergence of this transition, which occurs on biologically relevant timescales, is robust against variations in model design and parameters. Our findings suggest that the viscoelastic properties of chromatin and the timescale at which it transitions from a gel-like to a liquidlike state significantly impact dynamic processes that occur along the one-dimensional DNA sequence.

Quantifying nonequilibrium dynamics and thermodynamics of cell fate decision making in yeast under pheromone induction

Cellular responses to pheromone in yeast can range from gene expression to morphological and physiological changes. While signaling pathways are well studied, the cell fate decision-making during cellular polar growth is still unclear. Quantifying these cellular behaviors and revealing the underlying physical mechanism remain a significant challenge. Here, we employed a hidden Markov chain model to quantify the dynamics of cellular morphological systems based on our experimentally observed time series. The resulting statistics generated a stability landscape for state attractors. By quantifying rotational fluxes as the non-equilibrium driving force that tends to disrupt the current attractor state, the dynamical origin of non-equilibrium phase transition from four cell morphological fates to a single dominant fate was identified. We revealed that higher chemical voltage differences induced by a high dose of pheromone resulted in higher chemical currents, which will trigger a greater net input and, thus, more degrees of the detailed balance breaking. By quantifying the thermodynamic cost of maintaining morphological state stability, we demonstrated that the flux-related entropy production rate provides a thermodynamic origin for the phase transition in non-equilibrium morphologies. Furthermore, we confirmed that the time irreversibility in time series provides a practical way to predict the non-equilibrium phase transition.

Insights into the cell fate decision-making processes from chromosome structural reorganizations

The cell fate decision-making process, which provides the capability of a cell transition to a new cell type, involves the reorganizations of 3D genome structures. Currently, the high temporal resolution picture of how the chromosome structural rearrangements occur and further influence the gene activities during the cell-state transition is still challenging to acquire. Here, we study the chromosome structural reorganizations during the cell-state transitions among the pluripotent embryonic stem cell, the terminally differentiated normal cell, and the cancer cell using a nonequilibrium landscape-switching model implemented in the molecular dynamics simulation. We quantify the chromosome (de)compaction pathways during the cell-state transitions and find that the two pathways having the same destinations can merge prior to reaching the final states. The chromosomes at the merging states have similar structural geometries but can differ in long-range compartment segregation and spatial distribution of the chromosomal loci and genes, leading to cell-type-specific transition mechanisms. We identify the irreversible pathways of chromosome structural rearrangements during the forward and reverse transitions connecting the same pair of cell states, underscoring the critical roles of nonequilibrium dynamics in the cell-state transitions. Our results contribute to the understanding of the cell fate decision-making processes from the chromosome structural perspective.

Quantifying Chromosome Structural Reorganizations during Differentiation, Reprogramming, and Transdifferentiation

We developed a nonequilibrium model to study chromosome structural reorganizations within a simplified cell developmental system. From the chromosome structural perspective, we predicted that the neural progenitor cell is on the neural developmental path and very close to the transdifferentiation path from the fibroblast to the neuron cell. We identified an early bifurcation of stem cell differentiation processes and the cell-of-origin-specific reprogramming pathways. Our theoretical results are in good agreement with available experimental evidence, promoting future applications of our approach.

Data mining and mathematical models in cancer prognosis and prediction

Cancer is a fetal and complex disease. Individual differences of the same cancer type or the same patient at different stages of cancer development may require distinct treatments. Pathological differences are reflected in tissues, cells and gene levels etc. The interactions between the cancer cells and nearby microenvironments can also influence the cancer progression and metastasis. It is a huge challenge to understand all of these mechanistically and quantitatively. Researchers applied pattern recognition algorithms such as machine learning or data mining to predict cancer types or classifications. With the rapidly growing and available computing powers, researchers begin to integrate huge data sets, multi-dimensional data types and information. The cells are controlled by the gene expressions determined by the promoter sequences and transcription regulators. For example, the changes in the gene expression through these underlying mechanisms can modify cell progressing in the cell-cycle. Such molecular activities can be governed by the gene regulations through the underlying gene regulatory networks, which are essential for cancer study when the information and gene regulations are clear and available. In this review, we briefly introduce several machine learning methods of cancer prediction and classification which include Artificial Neural Networks (ANNs), Decision Trees (DTs), Support Vector Machine (SVM) and naive Bayes. Then we describe a few typical models for building up gene regulatory networks such as Correlation, Regression and Bayes methods based on available data. These methods can help on cancer diagnosis such as susceptibility, recurrence, survival etc. At last, we summarize and compare the modeling methods to analyze the development and progression of cancer through gene regulatory networks. These models can provide possible physical strategies to analyze cancer progression in a systematic and quantitative way.

Dynamics and Pathways of Chromosome Structural Organizations during Cell Transdifferentiation

Direct conversion of one differentiated cell type into another is defined as cell transdifferentiation. In avoidance of forming pluripotency, cell transdifferentiation can reduce the potential risk of tumorigenicity, thus offering significant advantages over cell reprogramming in clinical applications. Until now, the mechanism of cell transdifferentiation is still largely unknown. It has been well recognized that cell transdifferentiation is determined by the underlying gene expression regulation, which relies on the accurate adaptation of the chromosome structure. To dissect the transdifferentiation at the molecular level, we develop a nonequilibrium landscape-switching model to investigate the chromosome structural dynamics during the state transitions between the human fibroblast and neuron cells. We uncover the high irreversibility of the transdifferentiation at the local chromosome structural ranges, where the topologically associating domains form. In contrast, the pathways in the two opposite directions of the transdifferentiation projected onto the chromosome compartment profiles are highly overlapped, indicating that the reversibility vanishes at the long-range chromosome structures. By calculating the contact strengths in the chromosome at the states along the paths, we observe strengthening contacts in compartment A concomitant with weakening contacts in compartment B at the early stages of the transdifferentiation. This further leads to adapting contacts toward the ones at the embryonic stem cell. In light of the intimate structure-function relationship at the chromosomal level, we suggest an increase of "stemness" during the transdifferentiation. In addition, we find that the neuron progenitor cell (NPC), a cell developmental state, is located on the transdifferentiation pathways projected onto the long-range chromosome contacts. The findings are consistent with the previous single-cell RNA sequencing experiment, where the NPC-like cell states were observed during the direct conversion of the fibroblast to neuron cells. Thus, we offer a promising microscopic and physical approach to study the cell transdifferentiation mechanism from the chromosome structural perspective.

Physics of biomolecular recognition and conformational dynamics

Biomolecular recognition usually leads to the formation of binding complexes, often accompanied by large-scale conformational changes. This process is fundamental to biological functions at the molecular and cellular levels. Uncovering the physical mechanisms of biomolecular recognition and quantifying the key biomolecular interactions are vital to understand these functions. The recently developed energy landscape theory has been successful in quantifying recognition processes and revealing the underlying mechanisms. Recent studies have shown that in addition to affinity, specificity is also crucial for biomolecular recognition. The proposed physical concept of intrinsic specificity based on the underlying energy landscape theory provides a practical way to quantify the specificity. Optimization of affinity and specificity can be adopted as a principle to guide the evolution and design of molecular recognition. This approach can also be used in practice for drug discovery using multidimensional screening to identify lead compounds. The energy landscape topography of molecular recognition is important for revealing the underlying flexible binding or binding-folding mechanisms. In this review, we first introduce the energy landscape theory for molecular recognition and then address four critical issues related to biomolecular recognition and conformational dynamics: (1) specificity quantification of molecular recognition; (2) evolution and design in molecular recognition; (3) flexible molecular recognition; (4) chromosome structural dynamics. The results described here and the discussions of the insights gained from the energy landscape topography can provide valuable guidance for further computational and experimental investigations of biomolecular recognition and conformational dynamics.

Deciphering the molecular mechanism of the cancer formation by chromosome structural dynamics

Cancer reflects the dysregulation of the underlying gene network, which is strongly related to the 3D genome organization. Numerous efforts have been spent on experimental characterizations of the structural alterations in cancer genomes. However, there is still a lack of genomic structural-level understanding of the temporal dynamics for cancer initiation and progression. Here, we use a landscape-switching model to investigate the chromosome structural transition during the cancerization and reversion processes. We find that the chromosome undergoes a non-monotonic structural shape-changing pathway with initial expansion followed by compaction during both of these processes. Furthermore, our analysis reveals that the chromosome with a more expanding structure than those at both the normal and cancer cell during cancerization exhibits a sparse contact pattern, which shows significant structural similarity to the one at the embryonic stem cell in many aspects, including the trend of contact probability declining with the genomic distance, the global structural shape geometry and the spatial distribution of loci on the chromosome. In light of the intimate structure-function relationship at the chromosomal level, we further describe the cell state transition processes by the chromosome structural changes, suggesting an elevated cell stemness during the formation of the cancer cells. We show that cell cancerization and reversion are highly irreversible processes in terms of the chromosome structural transition pathways, spatial repositioning of chromosomal loci and hysteresis loop of contact evolution analysis. Our model draws a molecular-scale picture of cell cancerization from the chromosome structural perspective. The process contains initial reprogramming towards the stem cell followed by the differentiation towards the cancer cell, accompanied by an initial increase and subsequent decrease of the cell stemness.

Stochastic epigenetic dynamics of gene switching

Epigenetic modifications of histones crucially affect eukaryotic gene activity, while the epigenetic histone state is largely determined by the binding of specific factors such as the transcription factors (TFs) to DNA. Here, the way in which the TFs and the histone state are dynamically correlated is not obvious when the TF synthesis is regulated by the histone state. This type of feedback regulatory relation is ubiquitous in gene networks to determine cell fate in differentiation and other cell transformations. To gain insights into such dynamical feedback regulations, we theoretically analyze a model of epigenetic gene switching by extending the Doi-Peliti operator formalism of reaction kinetics to the problem of coupled molecular processes. Spin-1 and spin-1/2 coherent-state representations are introduced to describe stochastic reactions of histones and binding or unbinding of TFs in a unified way, which provides a concise view of the effects of timescale difference among these molecular processes; even in the case that binding or unbinding of TFs to or from DNA is adiabatically fast, the slow nonadiabatic histone dynamics gives rise to a distinct circular flow of the probability flux around basins in the landscape of the gene state distribution, which leads to hysteresis in gene switching. In contrast to the general belief that the change in the amount of TF precedes the histone state change, flux drives histones to be modified prior to the change in the amount of TF in self-regulating circuits. Flux-landscape analyses shed light on the nonlinear nonadiabatic mechanism of epigenetic cell fate decision making.

Quantifying the Stability of Coupled Genetic and Epigenetic Switches With Variational Methods

The Waddington landscape provides an intuitive metaphor to view development as a ball rolling down the hill, with distinct phenotypes as basins and differentiation pathways as valleys. Since, at a molecular level, cell differentiation arises from interactions among the genes, a mathematical definition for the Waddington landscape can, in principle, be obtained by studying the gene regulatory networks. For eukaryotes, gene regulation is inextricably and intimately linked to histone modifications. However, the impact of such modifications on both landscape topography and stability of attractor states is not fully understood. In this work, we introduced a minimal kinetic model for gene regulation that combines the impact of both histone modifications and transcription factors. We further developed an approximation scheme based on variational principles to solve the corresponding master equation in a second quantized framework. By analyzing the steady-state solutions at various parameter regimes, we found that histone modification kinetics can significantly alter the behavior of a genetic network, resulting in qualitative changes in gene expression profiles. The emerging epigenetic landscape captures the delicate interplay between transcription factors and histone modifications in driving cell-fate decisions.

Conformational state switching and pathways of chromosome dynamics in cell cycle

The cell cycle is a process and function of a cell with different phases essential for cell growth, proliferation, and replication. It depends on the structure and dynamics of the underlying DNA molecule, which underpins the genome function. A microscopic structural-level understanding of how a genome or its functional module chromosome performs the cell cycle in terms of large-scale conformational transformation between different phases, such as the interphase and the mitotic phase, is still challenging. Here, we develop a non-equilibrium, excitation-relaxation energy landscape-switching model to quantify the underlying chromosome conformational transitions through (de-)condensation for a complete microscopic understanding of the cell cycle. We show that the chromosome conformational transition mechanism from the interphase to the mitotic phase follows a two-stage scenario, in good agreement with the experiments. In contrast, the mitotic exit pathways show the existence of an over-expanded chromosome that recapitulates the chromosome in the experimentally identified intermediate state at the telophase. We find the conformational pathways are heterogeneous and irreversible as a result of the non-equilibrium dynamics of the cell cycle from both structural and kinetic perspectives. We suggest that the irreversibility is mainly due to the distinct participation of the ATP-dependent structural maintenance of chromosomal protein complexes during the cell cycle. Our findings provide crucial insights into the microscopic molecular structural and dynamical physical mechanism for the cell cycle beyond the previous more macroscopic descriptions. Our non-equilibrium landscape framework is general and applicable to study diverse non-equilibrium physical and biological processes such as active matter, differentiation/development, and cancer.

The Basal Level of Gene Expression Associated with Chromatin Loosening Shapes Waddington Landscapes and Controls Cell Differentiation

The baseline level of transcription, which is variable and difficult to quantify, seriously complicates the normalization of comparative transcriptomic data, but its biological importance remains unappreciated. We show that this currently neglected ingredient is essential for controlling gene network multistability and therefore cellular differentiation. Basal expression is correlated to the degree of chromatin loosening measured by DNA accessibility and systematically leads to cellular dedifferentiation as assessed by transcriptomic signatures, irrespective of the molecular and cellular tools used. Modeling gene network motifs formally involved in developmental bifurcations reveals that the epigenetic landscapes of Waddington are restructured by the level of nonspecific expression, such that the attractors of progenitor and differentiated cells can be mutually exclusive. This mechanism is universal and holds beyond the particular nature of the genes involved, provided the multistable circuits are correctly described with autonomous basal expression. These results explain the relationships long established between gene expression noise, chromatin decondensation and cellular dedifferentiation, and highlight how heterochromatin maintenance is essential for preventing pathological cellular reprogramming, age-related diseases, and cancer.

Quantifying the flux as the driving force for nonequilibrium dynamics and thermodynamics in non-Michaelis-Menten enzyme kinetics

The driving force for active physical and biological systems is determined by both the underlying landscape and nonequilibrium curl flux. While landscape can be experimentally quantified from the histograms of the collected real-time trajectories of the observables, quantifying the experimental flux remains challenging. In this work, we studied the single-molecule enzyme dynamics of horseradish peroxidase with dihydrorhodamine 123 and hydrogen peroxide (H2O2) as substrates. Surprisingly, significant deviations in the kinetics from the conventional Michaelis-Menten reaction rate were observed. Instead of a linear relationship between the inverse of the enzyme kinetic rate and the inverse of substrate concentration, a nonlinear relationship between the two emerged. We identified nonequilibrium flux as the origin of such non-Michaelis-Menten enzyme rate behavior. Furthermore, we quantified the nonequilibrium flux from experimentally obtained fluorescence correlation spectroscopy data and showed this flux to led to the deviations from the Michaelis-Menten kinetics. We also identified and quantified the nonequilibrium thermodynamic driving forces as the chemical potential and entropy production for such non-Michaelis-Menten kinetics. Moreover, through isothermal titration calorimetry measurements, we identified and quantified the origin of both nonequilibrium dynamic and thermodynamic driving forces as the heat absorbed (energy input) into the enzyme reaction system. Furthermore, we showed that the nonequilibrium driving forces led to time irreversibility through the difference between the forward and backward directions in time and high-order correlations were associated with the deviations from Michaelis-Menten kinetics. This study provided a general framework for experimentally quantifying the dynamic and thermodynamic driving forces for nonequilibrium systems.

Landscape perspectives of tumor, EMT, and development

A tumor is rarely fatal until becoming metastatic. Recent discoveries suggest that epithelial mesenchymal transition(EMT) is an important process which contributes to not only cancer metastasis but also increased stemness. Cancer cells with stem cell characteristics are called cancer stem cells (CSCs). We review recent efforts to quantify and delineate the relationship among EMT, CSC and tumor development. When the gene regulatory network is tightly regulated through the effectively fast regulatory binding, Cancer, Premalignant, Normal, CSC, stem cell (SC), Lesion and Hyperplasia states emerged. The corresponding landscape topography for all of these states can be quantified to a global way for uncovering the relationship among the tumor, metastasis, and development. On the other hand, phenotypic and functional heterogeneity is regarded as one of the greatest challenge in cancer treatment. Cancer and CSCs are heterogeneous and give rise to tumorigenic and non-tumorigenic cells during self-renewal, differentiation and epigenetic diversification. Further, if the gene regulatory network is weakly regulated through the effective slow regulatory binding (by DNA methylation or histone modification etc), multiple meta-stable states can emerge. This model can provide an epigenetic and physical rather than genetic and fixed origin of heterogeneity. Elucidating the origin of and dynamic nature of tumor cells will likely help better understand the cellular basis of therapeutic response, resistance, and relapse.

Landscape of gene networks for random parameter perturbation

Landscape approaches have been exploited to study the stochastic dynamics of gene networks. However, how to calculate the landscape with a wide range of parameter variations and how to investigate the influence of the network topology on the global properties of gene networks remain to be elucidated. Here, I developed an approach for the landscape of random parameter perturbation (LRPP) to address this issue. Based on a self-consistent approximation approach, by making perturbations to parameters in a given range, I obtained the landscape for gene network systems. I applied this approach to two biological models, one for the mutual repression model and the other for the embryonic stem (ES) cell differentiation network. For the mutual repression model, my results confirm quantitatively that positive feedback promotes the robustness of multistability. For the ES cell differentiation model, I identify three cell states, representing the ES cell, the differentiation cell, and the intermediate state cell, respectively. I propose that the intermediate states and the wide range of parameter values coming from inhomogeneous cellular environments provide possible explanations for the heterogeneity observed in single cell experiments. I also offer a counterintuitive result that noise could reduce heterogeneity and promote the stability of cell states. These results support that the network topology determines the operating principles of the genetic networks, reflected by the representative landscapes from LRPP. This work provides a new route to obtain the potential landscape for a gene network system given a wide range of parameter values and study the influences of the network topology on the global properties of the system.

Analysis of dynamically stable patterns in a maze-like corridor using the Wasserstein metric

The two-dimensional optimal velocity (2d-OV) model represents a dissipative system with asymmetric interactions, thus being suitable to reproduce behaviours such as pedestrian dynamics and the collective motion of living organisms. In this study, we found that particles in the 2d-OV model form optimal patterns in a maze-like corridor. Then, we estimated the stability of such patterns using the Wasserstein metric. Furthermore, we mapped these patterns into the Wasserstein metric space and represented them as points in a plane. As a result, we discovered that the stability of the dynamical patterns is strongly affected by the model sensitivity, which controls the motion of each particle. In addition, we verified the existence of two stable macroscopic patterns which were cohesive, stable, and appeared regularly over the time evolution of the model.

Exploring the Underlying Mechanisms of the Xenopus laevis Embryonic Cell Cycle

The cell cycle is an indispensable process in proliferation and development. Despite significant efforts, global quantification and physical understanding are still challenging. In this study, we explored the mechanisms of the Xenopus laevis embryonic cell cycle by quantifying the underlying landscape and flux. We uncovered the Mexican hat landscape of the Xenopus laevis embryonic cell cycle with several local basins and barriers on the oscillation path. The local basins characterize the different phases of the Xenopus laevis embryonic cell cycle, and the local barriers represent the checkpoints. The checkpoint mechanism of the cell cycle is revealed by the landscape basins and barriers. While landscape shape determines the stabilities of the states on the oscillation path, the curl flux force determines the stability of the cell cycle flow. Replication is fundamental for biology of living cells. We quantify the input energy (through the entropy production) as the thermodynamic requirement for initiation and sustainability of single cell life (cell cycle). Furthermore, we also quantify curl flux originated from the input energy as the dynamical requirement for the emergence of a new stable phase (cell cycle). This can provide a new quantitative insight for the origin of single cell life. In fact, the curl flux originated from the energy input or nutrition supply determines the speed and guarantees the progression of the cell cycle. The speed of the cell cycle is a hallmark of cancer. We characterized the quality of the cell cycle by the coherence time and found it is supported by the flux and energy cost. We are also able to quantify the degree of time irreversibility by the cross correlation function forward and backward in time from the stochastic traces in the simulation or experiments, providing a way for the quantification of the time irreversibility and the flux. Through global sensitivity analysis upon landscape and flux, we can identify the key elements for controlling the cell cycle speed. This can help to design an effective strategy for drug discovery against cancer.

Genomic Energy Landscapes

Energy landscape theory, developed in the context of protein folding, provides, to our knowledge, a new perspective on chromosome architecture. We review what has been learned concerning the topology and structure of both the interphase and mitotic chromosomes from effective energy landscapes constructed using Hi-C data. Energy landscape thinking raises new questions about the nonequilibrium dynamics of the chromosome and gene regulation.

Multiple coupled landscapes and non-adiabatic dynamics with applications to self-activating genes

Many physical, chemical and biochemical systems (e.g. electronic dynamics and gene regulatory networks) are governed by continuous stochastic processes (e.g. electron dynamics on a particular electronic energy surface and protein (gene product) synthesis) coupled with discrete processes (e.g. hopping among different electronic energy surfaces and on and off switching of genes). One can also think of the underlying dynamics as the continuous motion on a particular landscape and discrete hoppings among different landscapes. The main difference of such systems from the intra-landscape dynamics alone is the emergence of the timescale involved in transitions among different landscapes in addition to the timescale involved in a particular landscape. The adiabatic limit when inter-landscape hoppings are fast compared to continuous intra-landscape dynamics has been studied both analytically and numerically, but the analytical treatment of the non-adiabatic regime where the inter-landscape hoppings are slow or comparable to continuous intra-landscape dynamics remains challenging. In this study, we show that there exists mathematical mapping of the dynamics on 2(N) discretely coupled N continuous dimensional landscapes onto one single landscape in 2N dimensional extended continuous space. On this 2N dimensional landscape, eddy current emerges as a sign of non-equilibrium non-adiabatic dynamics and plays an important role in system evolution. Many interesting physical effects such as the enhancement of fluctuations, irreversibility, dissipation and optimal kinetics emerge due to non-adiabaticity manifested by the eddy current illustrated for an N = 1 self-activator. We further generalize our theory to the N-gene network with multiple binding sites and multiple synthesis rates for discretely coupled non-equilibrium stochastic physical and biological systems.

Effects of Collective Histone State Dynamics on Epigenetic Landscape and Kinetics of Cell Reprogramming

Cell reprogramming is a process of transitions from differentiated to pluripotent cell states via transient intermediate states. Within the epigenetic landscape framework, such a process is regarded as a sequence of transitions among basins on the landscape; therefore, theoretical construction of a model landscape which exhibits experimentally consistent dynamics can provide clues to understanding epigenetic mechanism of reprogramming. We propose a minimal gene-network model of the landscape, in which each gene is regulated by an integrated mechanism of transcription-factor binding/unbinding and the collective chemical modification of histones. We show that the slow collective variation of many histones around each gene locus alters topology of the landscape and significantly affects transition dynamics between basins. Differentiation and reprogramming follow different transition pathways on the calculated landscape, which should be verified experimentally via single-cell pursuit of the reprogramming process. Effects of modulation in collective histone state kinetics on transition dynamics and pathway are examined in search for an efficient protocol of reprogramming.

DNA-Binding Kinetics Determines the Mechanism of Noise-Induced Switching in Gene Networks

Gene regulatory networks are multistable dynamical systems in which attractor states represent cell phenotypes. Spontaneous, noise-induced transitions between these states are thought to underlie critical cellular processes, including cell developmental fate decisions, phenotypic plasticity in fluctuating environments, and carcinogenesis. As such, there is increasing interest in the development of theoretical and computational approaches that can shed light on the dynamics of these stochastic state transitions in multistable gene networks. We applied a numerical rare-event sampling algorithm to study transition paths of spontaneous noise-induced switching for a ubiquitous gene regulatory network motif, the bistable toggle switch, in which two mutually repressive genes compete for dominant expression. We find that the method can efficiently uncover detailed switching mechanisms that involve fluctuations both in occupancies of DNA regulatory sites and copy numbers of protein products. In addition, we show that the rate parameters governing binding and unbinding of regulatory proteins to DNA strongly influence the switching mechanism. In a regime of slow DNA-binding/unbinding kinetics, spontaneous switching occurs relatively frequently and is driven primarily by fluctuations in DNA-site occupancies. In contrast, in a regime of fast DNA-binding/unbinding kinetics, switching occurs rarely and is driven by fluctuations in levels of expressed protein. Our results demonstrate how spontaneous cell phenotype transitions involve collective behavior of both regulatory proteins and DNA. Computational approaches capable of simulating dynamics over many system variables are thus well suited to exploring dynamic mechanisms in gene networks.

The Regulatory Landscape of Lineage Differentiation in a Metazoan Embryo

Elucidating the mechanism of cell lineage differentiation is critical for our understanding of development and fate manipulation. Here we combined systematic perturbation and direct lineaging to map the regulatory landscape of lineage differentiation in early C. elegans embryogenesis. High-dimensional phenotypic analysis of 204 essential genes in 1,368 embryos revealed that cell lineage differentiation follows a canalized landscape with barriers shaped by lineage distance and genetic robustness. We assigned function to 201 genes in regulating lineage differentiation, including 175 switches of binary fate choices. We generated a multiscale model that connects gene networks and cells to the experimentally mapped landscape. Simulations showed that the landscape topology determines the propensity of differentiation and regulatory complexity. Furthermore, the model allowed us to identify the chromatin assembly complex CAF-1 as a context-specific repressor of Notch signaling. Our study presents a systematic survey of the regulatory landscape of lineage differentiation of a metazoan embryo.

Distinguishing the rates of gene activation from phenotypic variations

Background

Stochastic genetic switching driven by intrinsic noise is an important process in gene expression. When the rates of gene activation/inactivation are relatively slow, fast, or medium compared with the synthesis/degradation rates of mRNAs and proteins, the variability of protein and mRNA levels may exhibit very different dynamical patterns. It is desirable to provide a systematic approach to identify their key dynamical features in different regimes, aiming at distinguishing which regime a considered gene regulatory network is in from their phenotypic variations.

Results

We studied a gene expression model with positive feedbacks when genetic switching rates vary over a wide range. With the goal of providing a method to distinguish the regime of the switching rates, we first focus on understanding the essential dynamics of gene expression system in different cases. In the regime of slow switching rates, we found that the effective dynamics can be reduced to independent evolutions on two separate layers corresponding to gene activation and inactivation states, and the transitions between two layers are rare events, after which the system goes mainly along deterministic ODE trajectories on a particular layer to reach new steady states. The energy landscape in this regime can be well approximated by using Gaussian mixture model. In the regime of intermediate switching rates, we analyzed the mean switching time to investigate the stability of the system in different parameter ranges. We also discussed the case of fast switching rates from the viewpoint of transition state theory. Based on the obtained results, we made a proposal to distinguish these three regimes in a simulation experiment. We identified the intermediate regime from the fact that the strength of cellular memory is lower than the other two cases, and the fast and slow regimes can be distinguished by their different perturbation-response behavior with respect to the switching rates perturbations.

Conclusions

We proposed a simulation experiment to distinguish the slow, intermediate and fast regimes, which is the main point of our paper. In order to achieve this goal, we systematically studied the essential dynamics of gene expression system when the switching rates are in different regimes. Our theoretical understanding provides new insights on the gene expression experiments.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-015-0172-0) contains supplementary material, which is available to authorized users.

Landscape and flux reveal a new global view and physical quantification of mammalian cell cycle

Cell cycles, essential for biological function, have been investigated extensively. However, enabling a global understanding and defining a physical quantification of the stability and function of the cell cycle remains challenging. Based upon a mammalian cell cycle gene network, we uncovered the underlying Mexican hat landscape of the cell cycle. We found the emergence of three local basins of attraction and two major potential barriers along the cell cycle trajectory. The three local basins of attraction characterize the G1, S/G2, and M phases. The barriers characterize the G1 and S/G2 checkpoints, respectively, of the cell cycle, thus providing an explanation of the checkpoint mechanism for the cell cycle from the physical perspective. We found that the progression of a cell cycle is determined by two driving forces: curl flux for acceleration and potential barriers for deceleration along the cycle path. Therefore, the cell cycle can be promoted (suppressed), either by enhancing (suppressing) the flux (representing the energy input) or by lowering (increasing) the barrier along the cell cycle path. We found that both the entropy production rate and energy per cell cycle increase as the growth factor increases. This reflects that cell growth and division are driven by energy or nutrition supply. More energy input increases flux and decreases barrier along the cell cycle path, leading to faster oscillations. We also identified certain key genes and regulations for stability and progression of the cell cycle. Some of these findings were evidenced from experiments whereas others lead to predictions and potential anticancer strategies.

Exploring the mechanisms of differentiation, dedifferentiation, reprogramming and transdifferentiation

We explored the underlying mechanisms of differentiation, dedifferentiation, reprogramming and transdifferentiation (cell type switchings) from landscape and flux perspectives. Lineage reprogramming is a new regenerative method to convert a matured cell into another cell including direct transdifferentiation without undergoing a pluripotent cell state and indirect transdifferentiation with an initial dedifferentiation-reversion (reprogramming) to a pluripotent cell state. Each cell type is quantified by a distinct valley on the potential landscape with higher probability. We investigated three driving forces for cell fate decision making: stochastic fluctuations, gene regulation and induction, which can lead to cell type switchings. We showed that under the driving forces the direct transdifferentiation process proceeds from a differentiated cell valley to another differentiated cell valley through either a distinct stable intermediate state or a certain series of unstable indeterminate states. The dedifferentiation process proceeds through a pluripotent cell state. Barrier height and the corresponding escape time from the valley on the landscape can be used to quantify the stability and efficiency of cell type switchings. We also uncovered the mechanisms of the underlying processes by quantifying the dominant biological paths of cell type switchings on the potential landscape. The dynamics of cell type switchings are determined by both landscape gradient and flux. The flux can lead to the deviations of the dominant biological paths for cell type switchings from the naively expected landscape gradient path. As a result, the corresponding dominant paths of cell type switchings are irreversible. We also classified the mechanisms of cell fate development from our landscape theory: super-critical pitchfork bifurcation, sub-critical pitchfork bifurcation, sub-critical pitchfork with two saddle-node bifurcation, and saddle-node bifurcation. Our model showed good agreements with the experiments. It provides a general framework to explore the mechanisms of differentiation, dedifferentiation, reprogramming and transdifferentiation.

Stem cell differentiation as a many-body problem

Stem cell differentiation has been viewed as coming from transitions between attractors on an epigenetic landscape that governs the dynamics of a regulatory network involving many genes. Rigorous definition of such a landscape is made possible by the realization that gene regulation is stochastic, owing to the small copy number of the transcription factors that regulate gene expression and because of the single-molecule nature of the gene itself. We develop an approximation that allows the quantitative construction of the epigenetic landscape for large realistic model networks. Applying this approach to the network for embryonic stem cell development explains many experimental observations, including the heterogeneous distribution of the transcription factor Nanog and its role in safeguarding the stem cell pluripotency, which can be understood by finding stable steady-state attractors and the most probable transition paths between those attractors. We also demonstrate that the switching rate between attractors can be significantly influenced by the gene expression noise arising from the fluctuations of DNA occupancy when binding to a specific DNA site is slow.

Epigenetic state network approach for describing cell phenotypic transitions

Recent breakthroughs of cell phenotype reprogramming impose theoretical challenges on unravelling the complexity of large circuits maintaining cell phenotypes coupled at many different epigenetic and gene regulation levels, and quantitatively describing the phenotypic transition dynamics. A popular picture proposed by Waddington views cell differentiation as a ball sliding down a landscape with valleys corresponding to different cell types separated by ridges. Based on theories of dynamical systems, we establish a novel 'epigenetic state network' framework that captures the global architecture of cell phenotypes, which allows us to translate the metaphorical low-dimensional Waddington epigenetic landscape concept into a simple-yet-predictive rigorous mathematical framework of cell phenotypic transitions. Specifically, we simplify a high-dimensional epigenetic landscape into a collection of discrete states corresponding to stable cell phenotypes connected by optimal transition pathways among them. We then apply the approach to the phenotypic transition processes among fibroblasts (FBs), pluripotent stem cells (PSCs) and cardiomyocytes (CMs). The epigenetic state network for this case predicts three major transition pathways connecting FBs and CMs. One goes by way of PSCs. The other two pathways involve transdifferentiation either indirectly through cardiac progenitor cells or directly from FB to CM. The predicted pathways and multiple intermediate states are supported by existing microarray data and other experiments. Our approach provides a theoretical framework for studying cell phenotypic transitions. Future studies at single-cell levels can directly test the model predictions.

Time scales in epigenetic dynamics and phenotypic heterogeneity of embryonic stem cells

A remarkable feature of the self-renewing population of embryonic stem cells (ESCs) is their phenotypic heterogeneity: Nanog and other marker proteins of ESCs show large cell-to-cell variation in their expression level, which should significantly influence the differentiation process of individual cells. The molecular mechanism and biological implication of this heterogeneity, however, still remain elusive. We address this problem by constructing a model of the core gene-network of mouse ESCs. The model takes account of processes of binding/unbinding of transcription factors, formation/dissolution of transcription apparatus, and modification of histone code at each locus of genes in the network. These processes are hierarchically interrelated to each other forming the dynamical feedback loops. By simulating stochastic dynamics of this model, we show that the phenotypic heterogeneity of ESCs can be explained when the chromatin at the Nanog locus undergoes the large scale reorganization in formation/dissolution of transcription apparatus, which should have the timescale similar to the cell cycle period. With this slow transcriptional switching of Nanog, the simulated ESCs fluctuate among multiple transient states, which can trigger the differentiation into the lineage-specific cell states. From the simulated transitions among cell states, the epigenetic landscape underlying transitions is calculated. The slow Nanog switching gives rise to the wide basin of ESC states in the landscape. The bimodal Nanog distribution arising from the kinetic flow running through this ESC basin prevents transdifferentiation and promotes the definite decision of the cell fate. These results show that the distribution of timescales of the regulatory processes is decisively important to characterize the fluctuation of cells and their differentiation process. The analyses through the epigenetic landscape and the kinetic flow on the landscape should provide a guideline to engineer cell differentiation.

Embryonic stem cells (ESCs) can proliferate indefinitely by keeping pluripotency, i.e., the ability to differentiate into any cell-lineage. ESCs, therefore, have been the focus of intense biological and medical interests. A remarkable feature of ESCs is their phenotypic heterogeneity: ESCs show large cell-to-cell fluctuation in the expression level of Nanog, which is a key factor to maintain pluripotency. Since Nanog regulates many genes in ESCs, this fluctuation should seriously affect individual cells when they start differentiation. In this paper we analyze this phenotypic fluctuation by simulating the stochastic dynamics of gene network in ESCs. The model takes account of the mutually interrelated processes of gene regulation such as binding/unbinding of transcription factors, formation/dissolution of transcription apparatus, and histone-code modification. We show the distribution of timescales of these processes is decisively important to characterize the dynamical behavior of the gene network, and that the slow formation/dissolution of transcription apparatus at the Nanog locus explains the observed large fluctuation of ESCs. The epigenetic landscapes are calculated based on the stochastic simulation, and the role of the phenotypic fluctuation in the differentiation process is analyzed through the landscape picture.