Energy landscape reveals that the budding yeast cell cycle is a robust and adaptive multi-stage process

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.

An improved approach for calculating energy landscape of gene networks from moment equations

The energy landscape theory has widely been applied to study the stochastic dynamics of biological systems. Different methods have been developed to quantify the energy landscape for gene networks, e.g., using Gaussian approximation (GA) approach to calculate the landscape by solving the diffusion equation approximately from the first two moments. However, how high-order moments influence the landscape construction remains to be elucidated. Also, multistability exists extensively in biological networks. So, how to quantify the landscape for a multistable dynamical system accurately, is a paramount problem. In this work, we prove that the weighted summation from GA (WSGA), provides an effective way to calculate the landscape for multistable systems and limit cycle systems. Meanwhile, we proposed an extended Gaussian approximation (EGA) approach by considering the effects of the third moments, which provides a more accurate way to obtain probability distribution and corresponding landscape. By applying our generalized EGA approach to two specific biological systems: multistable genetic circuit and synthetic oscillatory network, we compared EGA with WSGA by calculating the KL divergence of the probability distribution between these two approaches and simulations, which demonstrated that the EGA provides a more accurate approach to calculate the energy landscape.

Dynamically modeling the effective range of IL-2 dosage in the treatment of systemic lupus erythematosus

Systemic lupus erythematosus (SLE) is a complex systemic autoimmune disease characterized by an overactive immune response to self-antigen. The overactivation of CD4+ Foxp3- conventional T cells (Tcons) and the inactivation of CD4+ CD25+ Foxp3+ regulatory T cells (Tregs) play important roles in the progression of SLE. Clinical trials showed that low-dose interleukin-2 (IL-2) is effective in treating SLE. Here, we developed a mathematical model involving Tcons, Tregs, natural killer (NK) cells, and IL-2 to simulate the dynamic processes involved in the treatment of SLE. We found an effective range of IL-2 dosage defined by the Tcon/Treg ratio in SLE treatment, termed the IL-2 dosage therapeutic window (IDTW). Our results showed that high levels of self-antigen result in a narrow IDTW and high post-treatment Tcon/Treg ratio. Furthermore, we proposed a classification method based on the ratio of pre-treatment Treg to CD4+ T cells to predict the treatment outcome of SLE patients.

A high-throughput microfluidic diploid yeast long-term culturing (DYLC) chip capable of bud reorientation and concerted daughter dissection for replicative lifespan determination

Background

Budding yeast, Saccharomyces cerevisiae, has been extensively favored as a model organism in aging and age-related studies, thanks to versatile microfluidic chips for cell dynamics assay and replicative lifespan (RLS) determination at single-cell resolution. However, previous microfluidic structures aiming to immobilize haploid yeast may impose excessive spatial constraint and mechanical stress on cells, especially for larger diploid cells that sprout in a bipolar pattern.

Results

We developed a high-throughput microfluidic chip for diploid yeast long-term culturing (DYLC), optical inspection and cell-aging analysis. The DYLC chip features 1100 “leaky bowl”-shaped traps formatted in an array to dock single cells under laminar-perfused medium and effectively remove daughter cells by hydraulic shear forces. The delicate microstructures of cell traps enable hydrodynamic rotation of newborn buds, so as to ensure bud reorientation towards downstream and concerted daughter dissection thereafter. The traps provide sufficient space for cell-volume enlargement during aging, and thus properly alleviate structural compression and external stress on budding yeast. Trapping efficiency and long-term maintenance of single cells were optimized according to computational fluid dynamics simulations and experimental characterization in terms of critical parameters of the trap and array geometries. Owing to the self-filling of daughter cells dissected from traps upstream, an initial trapping efficiency of about 70% can rapidly reach a high value of over 92% after 4-hour cell culturing. During yeast proliferation and aging, cellular processes of growth, budding and daughter dissection were continuously tracked for over 60 h by time-lapse imaging. Yeast RLS and budding time interval (BTI) were directly calculated by the sequential two-digit codes indicating the budding status in images. With the employed diploid yeast strain, we obtained an RLS of 24.29 ± 3.65 generations, and verified the extension of BTI in the first couple of generations after birth and the last several generations approaching death, as well as cell de-synchronization along diploid yeast aging.

Conclusions

The DYLC chip offers a promising platform for reliable capture and culturing of diploid yeast cells and for life-long tracking of cell dynamics and replicative aging processes so that grasping comprehensive insights of aging mechanism in complex eukaryotic cells.

Graphical Abstract

Supplementary Information

The online version contains supplementary material available at 10.1186/s12951-022-01379-9.

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.

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...

Energy Landscape Analysis of the Epithelial-Mesenchymal Transition Network

The epithelial-mesenchymal transition (EMT) is a key developmental program that is often activated during the cancer invasion, metastasis, and drug resistance. However, it remains a critical question to elucidate the mechanisms of EMT. For example, how to quantify the global stability and stochastic transition dynamics of EMT under fluctuations is yet to be clarified. Here, we describe a framework and detailed steps for stochastic dynamics analysis of EMT. Starting from the underlying EMT gene regulatory network, we quantify the energy landscape of the EMT computationally. Multiple steady-state attractors are identified on the landscape surface, characterizing different cell phenotypes. The kinetic transition paths based on large deviation theory delineate the transition processes between different attractors quantitatively. The EMT or the reverse process, the mesenchymal-epithelial transition (MET), can be achieved by either a direct transition or a step-wise transition that goes through an intermediate state, depending on different extracellular environments. The landscape and transition paths presented in this chapter provide a new physical and quantitative picture to understand the underlying mechanisms of the EMT process. The approach for landscape and path analysis can be extended to other biological networks.

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.

Noise distorts the epigenetic landscape and shapes cell-fate decisions

The Waddington epigenetic landscape has become an iconic representation of the cellular differentiation process. Recent single-cell transcriptomic data provide new opportunities for quantifying this originally conceptual tool, offering insight into the gene regulatory networks underlying cellular development. While many methods for constructing the landscape have been proposed, by far the most commonly employed approach is based on computing the landscape as the negative logarithm of the steady-state probability distribution. Here, we use simple models to highlight the complexities and limitations that arise when reconstructing the potential landscape in the presence of stochastic fluctuations. We consider how the landscape changes in accordance with different stochastic systems and show that it is the subtle interplay between the deterministic and stochastic components of the system that ultimately shapes the landscape. We further discuss how the presence of noise has important implications for the identifiability of the regulatory dynamics from experimental data. A record of this paper's transparent peer review process is included in the supplemental information.

Optimal transition paths of phenotypic switching in a non-Markovian self-regulation gene expression

Gene expression is a complex biochemical process involving multiple reaction steps, creating molecular memory because the probability of waiting time between consecutive reaction steps no longer follows exponential distributions. What effect the molecular memory has on metastable states in gene expression remains not fully understood. Here, we study transition paths of switching between bistable states for a non-Markovian model of gene expression equipped with a self-regulation. Employing the large deviation theory for this model, we analyze the optimal transition paths of switching between bistable states in gene expression, interestingly finding that dynamic behaviors in gene expression along the optimal transition paths significantly depend on the molecular memory. Moreover, we discover that the molecular memory can prolong the time of switching between bistable states in gene expression along the optimal transition paths. Our results imply that the molecular memory may be an unneglectable factor to affect switching between metastable states in gene expression.

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.

Quantifying the Landscape and Transition Paths for Proliferation-Quiescence Fate Decisions

The cell cycle, essential for biological functions, experiences delicate spatiotemporal regulation. The transition between G1 and S phase, which is called the proliferation–quiescence decision, is critical to the cell cycle. However, the stability and underlying stochastic dynamical mechanisms of the proliferation–quiescence decision have not been fully understood. To quantify the process of the proliferation–quiescence decision, we constructed its underlying landscape based on the relevant gene regulatory network. We identified three attractors on the landscape corresponding to the G0, G1, and S phases, individually, which are supported by single-cell data. By calculating the transition path, which quantifies the potential barrier, we built expression profiles in temporal order for key regulators in different transitions. We propose that the two saddle points on the landscape characterize restriction point (RP) and G1/S checkpoint, respectively, which provides quantitative and physical explanations for the mechanisms of Rb governing the RP while p21 controlling the G1/S checkpoint. We found that Emi1 inhibits the transition from G0 to G1, while Emi1 in a suitable range facilitates the transition from G1 to S. These results are partially consistent with previous studies, which also suggested new roles of Emi1 in the cell cycle. By global sensitivity analysis, we identified some critical regulatory factors influencing the proliferation–quiescence decision. Our work provides a global view of the stochasticity and dynamics in the proliferation–quiescence decision of the cell cycle.

Random Parametric Perturbations of Gene Regulatory Circuit Uncover State Transitions in Cell Cycle

Many biological processes involve precise cellular state transitions controlled by complex gene regulation. Here, we use budding yeast cell cycle as a model system and explore how a gene regulatory circuit encodes essential information of state transitions. We present a generalized random circuit perturbation method for circuits containing heterogeneous regulation types and its usage to analyze both steady and oscillatory states from an ensemble of circuit models with random kinetic parameters. The stable steady states form robust clusters with a circular structure that are associated with cell cycle phases. This circular structure in the clusters is consistent with single-cell RNA sequencing data. The oscillatory states specify the irreversible state transitions along cell cycle progression. Furthermore, we identify possible mechanisms to understand the irreversible state transitions from the steady states. We expect this approach to be robust and generally applicable to unbiasedly predict dynamical transitions of a gene regulatory circuit.

Critical slowing down and attractive manifold: A mechanism for dynamic robustness in the yeast cell-cycle process

Biological processes that execute complex multiple functions, such as the cell cycle, must ensure the order of sequential events and maintain dynamic robustness against various fluctuations. Here, we examine the mechanisms and fundamental structure that achieve these properties in the cell cycle of the budding yeast Saccharomyces cerevisiae. We show that this process behaves like an excitable system containing three well-decoupled saddle-node bifurcations to execute DNA replication and mitosis events. The yeast cell-cycle regulatory network can be divided into three modules-the G1/S phase, early M phase, and late M phase-wherein both positive feedback loops in each module and interactions among modules play important roles. Specifically, when the cell-cycle process operates near the critical points of the saddle-node bifurcations, a critical slowing down effect takes place. Such interregnum then allows for an attractive manifold and sufficient duration for cell-cycle events, within which to assess the completion of DNA replication and mitosis, e.g., spindle assembly. Moreover, such arrangement ensures that any fluctuation in an early module or event will not transmit to a later module or event. Thus, our results suggest a possible dynamical mechanism of the cell-cycle process to ensure event order and dynamic robustness and give insight into the evolution of eukaryotic cell-cycle processes.

Landscape inferred from gene expression data governs pluripotency in embryonic stem cells

Embryonic stem cells (ESCs) can differentiate into diverse cell types and have the ability of self-renewal. Therefore, the study of cell fate decisions on embryonic stem cells has far-reaching significance for regenerative medicine and other biomedical fields. Mathematical models have been used to study emryonic stem cell differentiation. However, the underlying mechanisms of cell differentiation and lineage reprogramming remain to be elucidated. Especially, how to integrate the computational models with quantitative experimental data is still challenging. In this work, we developed a data-constrained modelling approach, and established a model of mouse embryonic stem cells. We used the truncated moment equations (TME) method to quantify the potential landscape of the ESC network. We identified two attractors on the landscape, which represent the embryonic stem cell (ESC) state and differentiated cell (DC) state, respectively, and quantified high dimensional biological paths for differentiation and reprogramming process. Through identifying the optimal combinations of gene targets based on a landscape control strategy, we offered some predictions about the key regulatory factors that govern the differentiation and reprogramming in ESCs.

Exposing the Underlying Relationship of Cancer Metastasis to Metabolism and Epithelial-Mesenchymal Transitions

Cancer is a disease governed by the underlying gene regulatory networks. The hallmarks of cancer have been proposed to characterize the cancerization, e.g., abnormal metabolism, epithelial to mesenchymal transition (EMT), and cancer metastasis. We constructed a metabolism-EMT-metastasis regulatory network and quantified its underlying landscape. We identified four attractors, characterizing epithelial, abnormal metabolic, mesenchymal, and metastatic cell states, respectively. Importantly, we identified an abnormal metabolic state. Based on the transition path theory, we quantified the kinetic transition paths among these different cell states. Our results for landscape and paths indicate that metastasis is a sequential process: cells tend to first change their metabolism, then activate the EMT and eventually reach the metastatic state. This demonstrates the importance of the temporal order for different gene circuits switching on or off during metastatic progression of cancer cells and underlines the cascading regulation of metastasis through an abnormal metabolic intermediate state.

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.

A landscape view on the interplay between EMT and cancer metastasis

The epithelial-mesenchymal transition (EMT) is a basic developmental process that converts epithelial cells to mesenchymal cells. Although EMT might promote cancer metastasis, the molecular mechanisms for it remain to be fully clarified. To address this issue, we constructed an EMT-metastasis gene regulatory network model and quantified the potential landscape of cancer metastasis-promoting system computationally. We identified four steady-state attractors on the landscape, which separately characterize anti-metastatic (A), metastatic (M), and two other intermediate (I1 and I2) cell states. The tetrastable landscape and the existence of intermediate states are consistent with recent single-cell measurements. We identified one of the two intermediate states I1 as the EMT state. From a MAP approach, we found that for metastatic progression cells need to first undergo EMT (enter the I1 state), and then become metastatic (switch from the I1 state to the M state). Specifically, for metastatic progression, EMT genes (such as ZEB) should be activated before metastasis genes (such as BACH1). This suggests that temporal order is important for the activation of cellular programs in biological systems, and provides a possible mechanism of EMT-promoting cancer metastasis. To identify possible therapeutic targets from this landscape view, we performed sensitivity analysis for individual molecular factors, and identified optimal interventions for landscape control. We found that minimizing transition actions more effectively identifies optimal combinations of targets that induce transitions between attractors than single-factor sensitivity analysis. Overall, the landscape view not only suggests that intermediate states increase plasticity during cell fate decisions, providing a possible source for tumor heterogeneity that is critically important in metastatic progress, but also provides a way to identify therapeutic targets for preventing cancer progression.

Parameter sensitivity analysis for a stochastic model of mitochondrial apoptosis pathway

Understanding how gene alterations induce oncogenesis plays an important role in cancer research and may be instructive for cancer prevention and treatment. We conducted a parameter sensitivity analysis to the mitochondrial apoptosis model. Both a nonlinear bifurcation analysis of the deterministic dynamics and energy barrier analysis of the corresponding stochastic models were performed. We found that the parameter sensitivity ranking according to the change of the bifurcation-point locations in deterministic models and the change of the barrier heights from a living to death state of the cell in stochastic models are highly correlated. For the model we considered, in combination with previous knowledge that the parameters significantly affecting the system’s bifurcation point are strongly associated with frequently mutated oncogenic genes, we conclude that the energy barrier height can be used as indicator of oncogenesis as well as bifurcation point. We provide a possible mechanism that may help elucidate the logic of cancer initiation from the view of stochastic dynamics and energy landscape. And we show the equivalence of energy barrier height and bifurcation-point location in determining the parameter sensitivity spectrum for the first time.

Nanog induced intermediate state in regulating stem cell differentiation and reprogramming

Background

Heterogeneous gene expressions of cells are widely observed in self-renewing pluripotent stem cells, suggesting possible coexistence of multiple cellular states with distinct characteristics. Though the elements regulating cellular states have been identified, the underlying dynamic mechanisms and the significance of such cellular heterogeneity remain elusive.

Results

We present a gene regulatory network model to investigate the bimodal Nanog distribution in stem cells. Our model reveals a novel role of dynamic conversion between the cellular states of high and low Nanog levels. Model simulations demonstrate that the low-Nanog state benefits cell differentiation through serving as an intermediate state to reduce the barrier of transition. Interestingly, the existence of low-Nanog state dynamically slows down the reprogramming process, and additional Nanog activation is found to be essential to quickly attaining the fully reprogrammed cell state.

Conclusions

Nanog has been recognized as a critical pluripotency gene in stem cell regulation. Our modeling results quantitatively show a dual role of Nanog during stem cell differentiation and reprogramming, and the importance of the intermediate state during cell state transitions. Our approach offers a general method for analyzing key regulatory factors controlling cell differentiation and reprogramming.

Electronic supplementary material

The online version of this article (10.1186/s12918-018-0552-3) contains supplementary material, which is available to authorized users.

Identifying the optimal anticancer targets from the landscape of a cancer-immunity interaction network

Cancer immunotherapy, an approach of targeting immune cells to attack tumor cells, has been suggested to be a promising way for cancer treatment recently. However, the successful application of this approach warrants a deeper understanding of the intricate interplay between cancer cells and the immune system. Especially, the mechanisms of immunotherapy remain elusive. In this work, we constructed a cancer-immunity interplay network by incorporating interactions among cancer cells and some representative immune cells, and uncovered the potential landscape of the cancer-immunity network. Three attractors emerge on the landscape, representing the cancer state, the immune state, and the hybrid state, which can correspond to escape, elimination, and equilibrium phases in the immunoediting theory, respectively. We quantified the transition processes between the cancer state and the immune state by calculating transition actions and identifying the corresponding minimum action paths (MAPs) between these two attractors. The transition actions, directly calculated from the high dimensional system, are correlated with the barrier heights from the landscape, but provide a more precise description of the dynamics of a system. By optimizing the transition actions from the cancer state to the immune state, we identified some optimal combinations of anticancer targets. Our combined approach of the landscape and optimization of transition actions offers a framework to study the stochastic dynamics and identify the optimal combination of targets for the cancer-immunity interplay, and can be applied to other cell communication networks or gene regulatory networks.

Response to "Comment on 'Construction of the landscape for multi-stable systems: Potential landscape, quasi-potential, A-type integral and beyond"' [J. Chem. Phys. 145, 147104 (2016)]

The uniqueness issue of SDE decomposition theory proposed by Ao and his co-workers has recently been discussed. A comprehensive study to investigate connections among different landscape theories [J. Chem. Phys. 144, 094109 (2016)] has pointed out that the decomposition is generally not unique, while Ao et al. recently argue that such conclusions are "incorrect" because the uniqueness of the decomposition for Ornstein-Uhlenbeck (O-U) process has been claimed before. In this response, we will demonstrate that the claimed "uniqueness" of the O-U process decomposition is invalid to serve as a counterexample according to the original definition of SDE decomposition. The absence of effective and concrete boundary conditions in previous SDE decomposition papers will be pointed out, and some other issues in the comment will also be responded.

Mesoscopic kinetic basis of macroscopic chemical thermodynamics: A mathematical theory

Gibbs' macroscopic chemical thermodynamics is one of the most important theories in chemistry. Generalizing it to mesoscaled nonequilibrium systems is essential to biophysics. The nonequilibrium stochastic thermodynamics of chemical reaction kinetics suggested a free energy balance equation dF^{(meso)}/dt=E_{in}-e_{p} in which the free energy input rate E_{in} and dissipation rate e_{p} are both non-negative, and E_{in}≤e_{p}. We prove that in the macroscopic limit by merely allowing the molecular numbers to be infinite, the generalized mesoscopic free energy F^{(meso)} converges to φ^{ss}, the large deviation rate function for the stationary distributions. This generalized macroscopic free energy φ^{ss} now satisfies a balance equation dφ^{ss}(x)/dt=cmf(x)-σ(x), in which x represents chemical concentration. The chemical motive force cmf(x) and entropy production rate σ(x) are both non-negative, and cmf(x)≤σ(x). The balance equation is valid generally in isothermal driven systems and is different from mechanical energy conservation and the first law; it is actually an unknown form of the second law. Consequences of the emergent thermodynamic quantities and equalities are further discussed. The emergent "law" is independent of underlying kinetic details. Our theory provides an example showing how a macroscopic law emerges from a level below.

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.