Quantifying robustness and dissipation cost of yeast cell cycle network: the funneled energy landscape perspectives

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.

Programming Dissipation Systems by DNA Timer for Temporally Regulating Enzyme Catalysis and Nanostructure Assembly

Live cells precisely control their temporal pattern in energy dissipative processes such as catalysis and assembly. Here, we demonstrate a DNA-based artificial dissipative nonequilibrium system where the transient state is controlled by the processive digestion of λ-exonuclease (λ Exo). This enzyme reaction serves as an orthogonal and independent molecular timer allowing for the programmable regulation of the transient-state lifetime. This dissipation system is concatenated to enzyme catalysis and nanostructure assembly networks. Dynamic activation of enzyme catalysis and dynamic disassembly of DNA nanotubes (DNT) are realized, and the state lifetimes of these systems are accurately encoded by the DNA timer. This work demonstrates nontrivial dissipation systems with built-in molecular timers, which can be a useful tool for developing artificial reaction networks and nanostructures with enhanced complexities and intelligence.

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.

Noise-induced quasiperiod and period switching

We employ a typical genetic circuit model to explore how noise can influence dynamic structure. With the increase of a key interactive parameter, the model will deterministically go through two bifurcations and three dynamic structure regions. We find that a quasiperiodic component, which is not allowed by deterministic dynamics, will be generated by noise inducing in the first two regions, and this quasiperiod will be more and more stable along with the increase in noise. In particular, in the second region the quasiperiod will compete with a stable limit cycle and perform a new transient rhythm. Furthermore, we ascertain the entropy production rate and the heat dissipation rate, and discover a minimal value with theoretical elucidation. In the end, we unveil the mechanism of the formation of quasiperiods, and show a practical biological example. We expect this work to be helpful in solving some biological or ecological problems, such as the genetic origin of periodical cicadas and population dynamics with fluctuation.

Navigating Multi-Scale Cancer Systems Biology Towards Model-Driven Clinical Oncology and Its Applications in Personalized Therapeutics

Rapid advancements in high-throughput omics technologies and experimental protocols have led to the generation of vast amounts of scale-specific biomolecular data on cancer that now populates several online databases and resources. Cancer systems biology models built using this data have the potential to provide specific insights into complex multifactorial aberrations underpinning tumor initiation, development, and metastasis. Furthermore, the annotation of these single- and multi-scale models with patient data can additionally assist in designing personalized therapeutic interventions as well as aid in clinical decision-making. Here, we have systematically reviewed the emergence and evolution of (i) repositories with scale-specific and multi-scale biomolecular cancer data, (ii) systems biology models developed using this data, (iii) associated simulation software for the development of personalized cancer therapeutics, and (iv) translational attempts to pipeline multi-scale panomics data for data-driven in silico clinical oncology. The review concludes that the absence of a generic, zero-code, panomics-based multi-scale modeling pipeline and associated software framework, impedes the development and seamless deployment of personalized in silico multi-scale models in clinical settings.

Model-based robustness and bistability analysis for methylation-based, epigenetic memory systems

In recent years, epigenetic memory systems have been developed based on DNA methylation and positive feedback systems. Achieving a robust design for these systems is generally a challenging and multifactorial task. We developed and validated a novel mathematical model to describe methylation-based epigenetic memory systems that capture switching dynamics of methylation levels and methyltransferase amounts induced by different inputs. A bifurcation analysis shows that the system operates in the bistable range, but in its current setup is not robust to changes in parameters. An expansion of the model captures heterogeneity of cell populations by accounting for distributed cell division rates. Simulations predict that the system is highly sensitive to variations in temperature, which affects cell division and the efficiency of the zinc finger repressor. A moderate decrease in temperature leads to a highly heterogeneous response to input signals and bistability on a single-cell level. The predictions of our model were confirmed by flow cytometry experiments conducted in this study. Overall, the results of our study give insights into the functional mechanisms of methylation-based memory systems and demonstrate that the switching dynamics can be highly sensitive to experimental conditions.

Nonequilibrium Thermodynamics in Cell Biology: Extending Equilibrium Formalism to Cover Living Systems

We discuss new developments in the nonequilibrium dynamics and thermodynamics of living systems, giving a few examples to demonstrate the importance of nonequilibrium thermodynamics for understanding biological dynamics and functions. We study single-molecule enzyme dynamics, in which the nonequilibrium thermodynamic and dynamic driving forces of chemical potential and flux are crucial for the emergence of non-Michaelis-Menten kinetics. We explore single-gene expression dynamics, in which nonequilibrium dissipation can suppress fluctuations. We investigate the cell cycle and identify the nutrition supply as the energy input that sustains the stability, speed, and coherence of cell cycle oscillation, from which the different vital phases of the cell cycle emerge. We examine neural decision-making processes and find the trade-offs among speed, accuracy, and thermodynamic costs that are important for neural function. Lastly, we consider the thermodynamic cost for specificity in cellular signaling and adaptation.

Reverse engineering directed gene regulatory networks from transcriptomics and proteomics data of biomining bacterial communities with approximate Bayesian computation and steady-state signalling simulations

BACKGROUND

Network inference is an important aim of systems biology. It enables the transformation of OMICs datasets into biological knowledge. It consists of reverse engineering gene regulatory networks from OMICs data, such as RNAseq or mass spectrometry-based proteomics data, through computational methods. This approach allows to identify signalling pathways involved in specific biological functions. The ability to infer causality in gene regulatory networks, in addition to correlation, is crucial for several modelling approaches and allows targeted control in biotechnology applications.

METHODS

We performed simulations according to the approximate Bayesian computation method, where the core model consisted of a steady-state simulation algorithm used to study gene regulatory networks in systems for which a limited level of details is available. The simulations outcome was compared to experimentally measured transcriptomics and proteomics data through approximate Bayesian computation.

RESULTS

The structure of small gene regulatory networks responsible for the regulation of biological functions involved in biomining were inferred from multi OMICs data of mixed bacterial cultures. Several causal inter- and intraspecies interactions were inferred between genes coding for proteins involved in the biomining process, such as heavy metal transport, DNA damage, replication and repair, and membrane biogenesis. The method also provided indications for the role of several uncharacterized proteins by the inferred connection in their network context.

CONCLUSIONS

The combination of fast algorithms with high-performance computing allowed the simulation of a multitude of gene regulatory networks and their comparison to experimentally measured OMICs data through approximate Bayesian computation, enabling the probabilistic inference of causality in gene regulatory networks of a multispecies bacterial system involved in biomining without need of single-cell or multiple perturbation experiments. This information can be used to influence biological functions and control specific processes in biotechnology applications.

Determining Relative Dynamic Stability of Cell States Using Boolean Network Model

Cell state transition is at the core of biological processes in metazoan, which includes cell differentiation, epithelial-to-mesenchymal transition (EMT) and cell reprogramming. In these cases, it is important to understand the molecular mechanism of cellular stability and how the transitions happen between different cell states, which is controlled by a gene regulatory network (GRN) hard-wired in the genome. Here we use Boolean modeling of GRN to study the cell state transition of EMT and systematically compare four available methods to calculate the cellular stability of three cell states in EMT in both normal and genetically mutated cases. The results produced from four methods generally agree but do not totally agree with each other. We show that distribution of one-degree neighborhood of cell states, which are the nearest states by Hamming distance, causes the difference among the methods. From that, we propose a new method based on one-degree neighborhood, which is the simplest one and agrees with other methods to estimate the cellular stability in all scenarios of our EMT model. This new method will help the researchers in the field of cell differentiation and cell reprogramming to calculate cellular stability using Boolean model, and then rationally design their experimental protocols to manipulate the cell state transition.

How Criticality of Gene Regulatory Networks Affects the Resulting Morphogenesis under Genetic Perturbations

Whereas the relationship between criticality of gene regulatory networks (GRNs) and dynamics of GRNs at a single-cell level has been vigorously studied, the relationship between the criticality of GRNs and system properties at a higher level has not been fully explored. Here we aim at revealing a potential role of criticality of GRNs in morphogenesis, which is hard to uncover through the single-cell-level studies, especially from an evolutionary viewpoint. Our model simulated the growth of a cell population from a single seed cell. All the cells were assumed to have identical intracellular GRNs. We induced genetic perturbations to the GRN of the seed cell by adding, deleting, or switching a regulatory link between a pair of genes. From numerical simulations, we found that the criticality of GRNs facilitated the formation of nontrivial morphologies when the GRNs were critical in the presence of the evolutionary perturbations. Moreover, the criticality of GRNs produced topologically homogeneous cell clusters by adjusting the spatial arrangements of cells, which led to the formation of nontrivial morphogenetic patterns. Our findings correspond to an epigenetic viewpoint that heterogeneous and complex features emerge from homogeneous and less complex components through the interactions among them. Thus, our results imply that highly structured tissues or organs in morphogenesis of multicellular organisms might stem from the criticality of GRNs.

ATLANTIS - Attractor Landscape Analysis Toolbox for Cell Fate Discovery and Reprogramming

Boolean modelling of biological networks is a well-established technique for abstracting dynamical biomolecular regulation in cells. Specifically, decoding linkages between salient regulatory network states and corresponding cell fate outcomes can help uncover pathological foundations of diseases such as cancer. Attractor landscape analysis is one such methodology which converts complex network behavior into a landscape of network states wherein each state is represented by propensity of its occurrence. Towards undertaking attractor landscape analysis of Boolean networks, we propose an Attractor Landscape Analysis Toolbox (ATLANTIS) for cell fate discovery, from biomolecular networks, and reprogramming upon network perturbation. ATLANTIS can be employed to perform both deterministic and probabilistic analyses. It has been validated by successfully reconstructing attractor landscapes from several published case studies followed by reprogramming of cell fates upon therapeutic treatment of network. Additionally, the biomolecular network of HCT-116 colorectal cancer cell line has been screened for therapeutic evaluation of drug-targets. Our results show agreement between therapeutic efficacies reported by ATLANTIS and the published literature. These case studies sufficiently highlight the in silico cell fate prediction and therapeutic screening potential of the toolbox. Lastly, ATLANTIS can also help guide single or combinatorial therapy responses towards reprogramming biomolecular networks to recover cell fates.

Systems Level Modeling of the Cell Cycle Using Budding Yeast

Proteins involved in the regulation of the cell cycle are highly conserved across all eukaryotes, and so a relatively simple eukaryote such as yeast can provide insight into a variety of cell cycle perturbations including those that occur in human cancer. To date, the budding yeast Saccharomyces cerevisiae has provided the largest amount of experimental and modeling data on the progression of the cell cycle, making it a logical choice for in-depth studies of this process. Moreover, the advent of methods for collection of high-throughput genome, transcriptome, and proteome data has provided a means to collect and precisely quantify simultaneous cell cycle gene transcript and protein levels, permitting modeling of the cell cycle on the systems level. With the appropriate mathematical framework and sufficient and accurate data on cell cycle components, it should be possible to create a model of the cell cycle that not only effectively describes its operation, but can also predict responses to perturbations such as variation in protein levels and responses to external stimuli including targeted inhibition by drugs. In this review, we summarize existing data on the yeast cell cycle, proteomics technologies for quantifying cell cycle proteins, and the mathematical frameworks that can integrate this data into representative and effective models. Systems level modeling of the cell cycle will require the integration of high-quality data with the appropriate mathematical framework, which can currently be attained through the combination of dynamic modeling based on proteomics data and using yeast as a model organism.

Funneled potential and flux landscapes dictate the stabilities of both the states and the flow: Fission yeast cell cycle

Using fission yeast cell cycle as an example, we uncovered that the non-equilibrium network dynamics and global properties are determined by two essential features: the potential landscape and the flux landscape. These two landscapes can be quantified through the decomposition of the dynamics into the detailed balance preserving part and detailed balance breaking non-equilibrium part. While the funneled potential landscape is often crucial for the stability of the single attractor networks, we have uncovered that the funneled flux landscape is crucial for the emergence and maintenance of the stable limit cycle oscillation flow. This provides a new interpretation of the origin for the limit cycle oscillations: There are many cycles and loops existed flowing through the state space and forming the flux landscapes, each cycle with a probability flux going through the loop. The limit cycle emerges when a loop stands out and carries significantly more probability flux than other loops. We explore how robustness ratio (RR) as the gap or steepness versus averaged variations or roughness of the landscape, quantifying the degrees of the funneling of the underlying potential and flux landscapes. We state that these two landscapes complement each other with one crucial for stabilities of states on the cycle and the other crucial for the stability of the flow along the cycle. The flux is directly related to the speed of the cell cycle. This allows us to identify the key factors and structure elements of the networks in determining the stability, speed and robustness of the fission yeast cell cycle oscillations. We see that the non-equilibriumness characterized by the degree of detailed balance breaking from the energy pump quantified by the flux is the cause of the energy dissipation for initiating and sustaining the replications essential for the origin and evolution of life. Regulating the cell cycle speed is crucial for designing the prevention and curing strategy of cancer.

We have uncovered that the non-equilibrium network dynamics and global properties are determined by two essential features: the potential landscape and the flux landscape. We have found that the funneled potential landscape is crucial for the stability of the states on the cell cycle, however, the stabilities of the oscillation states cannot guarantee the stable directional flows. We have uncovered that the funneled flux landscape is important for the emergence and maintenance of the stable limit cycle oscillation flow. This work will allow us to identify the key factors and structure elements of the networks in determining the stability, speed and robustness of the fission yeast cell cycle oscillations. We see that the non-equilibriumness characterized by the degree of detailed balance breaking from the energy pump quantified by the flux is the cause of the energy dissipation for initiating and sustaining the replications essential for the origin and evolution of life. Regulating the cell cycle speed is crucial for designing the prevention and curing strategy of cancer.

Modeling the epigenetic attractors landscape: toward a post-genomic mechanistic understanding of development

Robust temporal and spatial patterns of cell types emerge in the course of normal development in multicellular organisms. The onset of degenerative diseases may result from altered cell fate decisions that give rise to pathological phenotypes. Complex networks of genetic and non-genetic components underlie such normal and altered morphogenetic patterns. Here we focus on the networks of regulatory interactions involved in cell-fate decisions. Such networks modeled as dynamical non-linear systems attain particular stable configurations on gene activity that have been interpreted as cell-fate states. The network structure also restricts the most probable transition patterns among such states. The so-called Epigenetic Landscape (EL), originally proposed by C. H. Waddington, was an early attempt to conceptually explain the emergence of developmental choices as the result of intrinsic constraints (regulatory interactions) shaped during evolution. Thanks to the wealth of molecular genetic and genomic studies, we are now able to postulate gene regulatory networks (GRN) grounded on experimental data, and to derive EL models for specific cases. This, in turn, has motivated several mathematical and computational modeling approaches inspired by the EL concept, that may be useful tools to understand and predict cell-fate decisions and emerging patterns. In order to distinguish between the classical metaphorical EL proposal of Waddington, we refer to the Epigenetic Attractors Landscape (EAL), a proposal that is formally framed in the context of GRNs and dynamical systems theory. In this review we discuss recent EAL modeling strategies, their conceptual basis and their application in studying the emergence of both normal and pathological developmental processes. In addition, we discuss how model predictions can shed light into rational strategies for cell fate regulation, and we point to challenges ahead.

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.

The potential and flux landscape theory of ecology

The species in ecosystems are mutually interacting and self sustainable stable for a certain period. Stability and dynamics are crucial for understanding the structure and the function of ecosystems. We developed a potential and flux landscape theory of ecosystems to address these issues. We show that the driving force of the ecological dynamics can be decomposed to the gradient of the potential landscape and the curl probability flux measuring the degree of the breaking down of the detailed balance (due to in or out flow of the energy to the ecosystems). We found that the underlying intrinsic potential landscape is a global Lyapunov function monotonically going down in time and the topology of the landscape provides a quantitative measure for the global stability of the ecosystems. We also quantified the intrinsic energy, the entropy, the free energy and constructed the non-equilibrium thermodynamics for the ecosystems. We studied several typical and important ecological systems: the predation, competition, mutualism and a realistic lynx-snowshoe hare model. Single attractor, multiple attractors and limit cycle attractors emerge from these studies. We studied the stability and robustness of the ecosystems against the perturbations in parameters and the environmental fluctuations. We also found that the kinetic paths between the multiple attractors do not follow the gradient paths of the underlying landscape and are irreversible because of the non-zero flux. This theory provides a novel way for exploring the global stability, function and the robustness of ecosystems.

Therapeutic hints from analyzing the attractor landscape of the p53 regulatory circuit

Genes are interconnected in the cell to form a genetic network that regulates cell fate. Targeting multiple genes is expected to be more effective in developing therapeutics than targeting single genes. A recent study demonstrated the possibility of systematically searching for such combinatorial treatments by characterizing the attractor landscape of the p53 regulatory circuit.

Attractor landscape analysis reveals feedback loops in the p53 network that control the cellular response to DNA damage

The protein p53 functions as a tumor suppressor and can trigger either cell cycle arrest or apoptosis in response to DNA damage. We used Boolean network modeling and attractor landscape analysis to analyze the state transition dynamics of a simplified p53 network for which particular combinations of activation states of the molecules corresponded to specific cellular outcomes. Our results identified five critical interactions in the network that determined the cellular response to DNA damage, and simulations lacking any of these interactions produced states associated with sustained p53 activity, which corresponded to a cell death response. Attractor landscape analysis of the cellular response to DNA damage of the breast cancer cell line MCF7 and the effect of the Mdm2 (murine double minute 2) inhibitor nutlin-3 indicated that nutlin-3 would exhibit limited efficacy in triggering cell death, because the cell death state was not induced to a large extent by simulations with nutlin-3 and instead produced a state consistent with oscillatory p53 dynamics and cell cycle arrest. Attractor landscape analysis also suggested that combining nutlin-3 with inhibition of Wip1 would synergize to stimulate a sustained increase in p53 activity and promote p53-mediated cell death. We validated this synergistic effect in stimulating p53 activity and triggering cell death with single-cell imaging of a fluorescent p53 reporter in MCF7 cells. Thus, attractor landscape analysis of p53 network dynamics and its regulation can identify potential therapeutic strategies for treating cancer.

Potential flux landscapes determine the global stability of a Lorenz chaotic attractor under intrinsic fluctuations

We developed a potential flux landscape theory to investigate the dynamics and the global stability of a chemical Lorenz chaotic strange attractor under intrinsic fluctuations. Landscape was uncovered to have a butterfly shape. For chaotic systems, both landscape and probabilistic flux are crucial to the dynamics of chaotic oscillations. Landscape attracts the system down to the chaotic attractor, while flux drives the coherent motions along the chaotic attractors. Barrier heights from the landscape topography provide a quantitative measure for the robustness of chaotic attractor. We also found that the entropy production rate and phase coherence increase as the molecular numbers increase. Power spectrum analysis of autocorrelation function provides another way to quantify the global stability of chaotic attractor. We further found that limit cycle requires more flux and energy to sustain than the chaotic strange attractor. Finally, by detailed analysis we found that the curl probabilistic flux may provide the origin of the chaotic attractor.

The energy pump and the origin of the non-equilibrium flux of the dynamical systems and the networks

The global stability of dynamical systems and networks is still challenging to study. We developed a landscape and flux framework to explore the global stability. The potential landscape is directly linked to the steady state probability distribution of the non-equilibrium dynamical systems which can be used to study the global stability. The steady state probability flux together with the landscape gradient determines the dynamics of the system. The non-zero probability flux implies the breaking down of the detailed balance which is a quantitative signature of the systems being in non-equilibrium states. We investigated the dynamics of several systems from monostability to limit cycle and explored the microscopic origin of the probability flux. We discovered that the origin of the probability flux is due to the non-equilibrium conditions on the concentrations resulting energy input acting like non-equilibrium pump or battery to the system. Another interesting behavior we uncovered is that the probabilistic flux is closely related to the steady state deterministic chemical flux. For the monostable model of the kinetic cycle, the analytical expression of the probabilistic flux is directly related to the deterministic flux, and the later is directly generated by the chemical potential difference from the adenosine triphosphate (ATP) hydrolysis. For the limit cycle of the reversible Schnakenberg model, we also show that the probabilistic flux is correlated to the chemical driving force, as well as the deterministic effective flux. Furthermore, we study the phase coherence of the stochastic oscillation against the energy pump, and argue that larger non-equilibrium pump results faster flux and higher coherence. This leads to higher robustness of the biological oscillations. We also uncovered how fluctuations influence the coherence of the oscillations in two steps: (1) The mild fluctuations influence the coherence of the system mainly through the probability flux while maintaining the regular landscape topography. (2) The larger fluctuations lead to flat landscape and the complete loss of the stability of the whole system.

Landscape topography determines global stability and robustness of a metabolic network

Metabolic networks have gained broad attention in recent years as a result of their important roles in biological systems. However, how to quantify the global stability of the metabolic networks is still challenging. We develop a probabilistic landscape approach to investigate the global natures of the metabolic system under external fluctuations. As an example, we choose a model of the carbohydrate metabolism and the anaplerotic synthesis of oxalacetate in Aspergillus niger under conditions of citric acid accumulation to explore landscape topography. The landscape has a funnel shape, which guarantees the robustness of system under fluctuations and perturbations. Robustness ratio (RR), defined as the ratio of gap between lowest potential and average potential versus roughness measured by the dispersion or square root of variations of potentials, can be used to quantitatively evaluate the global stability of metabolic networks, and the larger the RR value, the more stable the system. Results of the entropy production rate imply that nature might evolve such that the network is robust against perturbations from environment or network wirings and performs specific biological functions with less dissipation cost. We also carried out a sensitivity analysis of parameters and uncovered some key network structure factors such as kinetic rates or wirings connecting the protein species nodes, which influence the global natures of the system. We found there is a strong correlation between the landscape topography and the input-output response. The more stable and robust the metabolic network is, the sharper the response is.

Systematic search for recipes to generate induced pluripotent stem cells

Generation of induced pluripotent stem cells (iPSCs) opens a new avenue in regenerative medicine. One of the major hurdles for therapeutic applications is to improve the efficiency of generating iPSCs and also to avoid the tumorigenicity, which requires searching for new reprogramming recipes. We present a systems biology approach to efficiently evaluate a large number of possible recipes and find those that are most effective at generating iPSCs. We not only recovered several experimentally confirmed recipes but we also suggested new ones that may improve reprogramming efficiency and quality. In addition, our approach allows one to estimate the cell-state landscape, monitor the progress of reprogramming, identify important regulatory transition states, and ultimately understand the mechanisms of iPSC generation.

Converting somatic cells back to the stem cell state (called induced pluripotent stem cells or iPSCs) exemplifies the recent advancement of cellular reprogramming that holds great promise for developing regenerative medicine. Generation of iPSCs is often achieved by overexpressing three to four genes in somatic cells that are critical for regulating pluripotency. Developing optimal reprogramming recipe is a non-trivial task that requires significant effort. We present here a computational method that can facilitate discovery of effective recipes to generate iPSCs with high efficiency and better quality. In addition, our approach provides a new way to estimate the landscape in the cell-state space and monitor the trajectory of cellular reprogramming from a differentiated cell to an iPS cell. This work provides not only practical recipes for iPSC generation but also theoretical understanding of the reprogramming process.

NONLINEAR DYNAMICS OF CELL CYCLES WITH STOCHASTIC MATHEMATICAL MODELS

Cell cycles are fundamental components of all living organisms and their systematic studies extend our knowledge about the interconnection between regulatory, metabolic, and signaling networks, and therefore open new opportunities for our ultimate efficient control of cellular processes for disease treatments, as well as for a wide variety of biomedical and biotechnological applications. In the study of cell cycles, nonlinear phenomena play a paramount role, in particular in those cases where the cellular dynamics is in the focus of attention. Quantification of this dynamics is a challenging task due to a wide range of parameters that require estimations and the presence of many stochastic effects. Based on the originally deterministic model, in this paper we develop a hierarchy of models that allow us to describe the nonlinear dynamics accounting for special events of cell cycles. First, we develop a model that takes into account fluctuations of relative concentrations of proteins during special events of cell cycles. Such fluctuations are induced by varying rates of relative concentrations of proteins and/or by relative concentrations of proteins themselves. As such fluctuations may be responsible for qualitative changes in the cell, we develop a new model that accounts for the effect of cellular dynamics on the cell cycle. Finally, we analyze numerically nonlinear effects in the cell cycle by constructing phase portraits based on the newly developed model and carry out a parametric sensitivity analysis in order to identify parameters for an efficient cell cycle control. The results of computational experiments demonstrate that the metabolic events in gene regulatory networks can qualitatively influence the dynamics of the cell cycle.

Landscape, flux, correlation, resonance, coherence, stability, and key network wirings of stochastic circadian oscillation

Circadian rhythms with a period of ~24 h, are natural timing machines. They are broadly distributed in living organisms, such as Neurospora, Drosophila, and mammals. The underlying natures of the rhythmic behavior have been explored by experimental and theoretical approaches. However, the global and physical natures of the oscillation under fluctuations are still not very clear. We developed a landscape and flux framework to explore the global stability and robustness of a circadian oscillation system. The potential landscape of the network is uncovered and has a global Mexican-hat shape. The height of the Mexican-hat provides a quantitative measure to evaluate the robustness and coherence of the oscillation. We found that in nonequilibrium dynamic systems, not only the potential landscape but also the probability flux are important to the dynamics of the system under intrinsic noise. Landscape attracts the systems down to the oscillation ring while flux drives the coherent oscillation on the ring. We also investigated the phase coherence and the entropy production rate of the system at different fluctuations and found that dissipations are less and the coherence is higher for larger number of molecules. We also found that the power spectrum of autocorrelation functions show resonance peak at the frequency of coherent oscillations. The peak is less prominent for smaller number of molecules and less barrier height and therefore can be used as another measure of stability of oscillations. As a consequence of nonzero probability flux, we show that the three-point correlations from the time traces show irreversibility, providing a possible way to explore the flux from the observations. Furthermore, we explored the escape time from the oscillation ring to outside at different molecular number. We found that when barrier height is higher, escape time is longer and phase coherence of oscillation is higher. Finally, we performed the global sensitivity analysis of the underlying parameters to find the key network wirings responsible for the stability of the oscillation system.

Recipes and mechanisms of cellular reprogramming: a case study on budding yeast Saccharomyces cerevisiae

Background

Generation of induced pluripotent stem cells (iPSCs) and converting one cell type to another (transdifferentiation) by manipulating the expression of a small number of genes highlight the progress of cellular reprogramming, which holds great promise for regenerative medicine. A key challenge is to find the recipes of perturbing genes to achieve successful reprogramming such that the reprogrammed cells function in the same way as the natural cells.

Results

We present here a systems biology approach that allows systematic search for effective reprogramming recipes and monitoring the reprogramming progress to uncover the underlying mechanisms. Using budding yeast as a model system, we have curated a genetic network regulating cell cycle and sporulation. Phenotypic consequences of perturbations can be predicted from the network without any prior knowledge, which makes it possible to computationally reprogram cell fate. As the heterogeneity of natural cells is important in many biological processes, we find that the extent of this heterogeneity restored by the reprogrammed cells varies significantly upon reprogramming recipes. The heterogeneity difference between the reprogrammed and natural cells may have functional consequences.

Conclusions

Our study reveals that cellular reprogramming can be achieved by many different perturbations and the reprogrammability of a cell depends on the heterogeneity of the original cell state. We provide a general framework that can help discover new recipes for cellular reprogramming in human.

Potential landscape and probabilistic flux of a predator prey network

Predator-prey system, as an essential element of ecological dynamics, has been recently studied experimentally with synthetic biology. We developed a global probabilistic landscape and flux framework to explore a synthetic predator-prey network constructed with two Escherichia coli populations. We developed a self consistent mean field method to solve multidimensional problem and uncovered the potential landscape with Mexican hat ring valley shape for predator-prey oscillations. The landscape attracts the system down to the closed oscillation ring. The probability flux drives the coherent oscillations on the ring. Both the landscape and flux are essential for the stable and coherent oscillations. The landscape topography characterized by the barrier height from the top of Mexican hat to the closed ring valley provides a quantitative measure of global stability of system. The entropy production rate for the energy dissipation is less for smaller environmental fluctuations or perturbations. The global sensitivity analysis based on the landscape topography gives specific predictions for the effects of parameters on the stability and function of the system. This may provide some clues for the global stability, robustness, function and synthetic network design.

Adiabatic and non-adiabatic non-equilibrium stochastic dynamics of single regulating genes

We explore the stochastic dynamics of self-regulative genes from fluctuations of molecular numbers and of on and off switching of gene states due to regulatory protein binding/unbinding to the genes. We found when the binding/unbinding is relatively fast (slow) compared with the synthesis/degradation of proteins in adiabatic (nonadiabatic) case the self-regulators can exhibit one or two peak (two peak) distributions in protein concentrations. This phenomena can also be quantified through Fano factors. This shows that even with the same architecture (topology of wiring) networks can have quite different functions (phenotypes), consistent with recent single molecule single gene experiments. We further found the inhibition and activation curves to be consistent with previous results (monomer binding) in adiabatic regime, but, in nonadiabatic regimes, show significantly different behaviors with previous predictions (monomer binding). Such difference is due to the slow (nonadiabatic) dimer binding/unbinding effect, and it has never been reported before. We derived the nonequilibrium phase diagrams of monostability and bistability in adiabatic and nonadiabatic regimes. We studied the dynamical trajectories of the self-regulating genes on the underlying landscapes from nonadiabatic to adiabatic limit, and we provide a global picture of understanding and show an analogy to the electron transfer problem. We studied the stability and robustness of the systems through mean first passage time (MFPT) from one peak (basin of attraction) to another and found both monotonic and nonmonotonic turnover behavior from adiabatic to nonadiabatic regimes. For the first time, we explore global dissipation by entropy production and the relation with binding/unbinding processes. Our theoretical predictions for steady state peaks, fano factos, inhibition/activation curves, and MFPT can be probed and tested from experiments.

The chemical master equation approach to nonequilibrium steady-state of open biochemical systems: linear single-molecule enzyme kinetics and nonlinear biochemical reaction networks

We develop the stochastic, chemical master equation as a unifying approach to the dynamics of biochemical reaction systems in a mesoscopic volume under a living environment. A living environment provides a continuous chemical energy input that sustains the reaction system in a nonequilibrium steady state with concentration fluctuations. We discuss the linear, unimolecular single-molecule enzyme kinetics, phosphorylation-dephosphorylation cycle (PdPC) with bistability, and network exhibiting oscillations. Emphasis is paid to the comparison between the stochastic dynamics and the prediction based on the traditional approach based on the Law of Mass Action. We introduce the difference between nonlinear bistability and stochastic bistability, the latter has no deterministic counterpart. For systems with nonlinear bistability, there are three different time scales: (a) individual biochemical reactions, (b) nonlinear network dynamics approaching to attractors, and (c) cellular evolution. For mesoscopic systems with size of a living cell, dynamics in (a) and (c) are stochastic while that with (b) is dominantly deterministic. Both (b) and (c) are emergent properties of a dynamic biochemical network; We suggest that the (c) is most relevant to major cellular biochemical processes such as epi-genetic regulation, apoptosis, and cancer immunoediting. The cellular evolution proceeds with transitions among the attractors of (b) in a "punctuated equilibrium" manner.

Potential and flux landscapes quantify the stability and robustness of budding yeast cell cycle network

Studying the cell cycle process is crucial for understanding cell growth, proliferation, development, and death. We uncovered some key factors in determining the global robustness and function of the budding yeast cell cycle by exploring the underlying landscape and flux of this nonequilibrium network. The dynamics of the system is determined by both the landscape which attracts the system down to the oscillation orbit and the curl flux which drives the periodic motion on the ring. This global structure of landscape is crucial for the coherent cell cycle dynamics and function. The topography of the underlying landscape, specifically the barrier height separating basins of attractions, characterizes the capability of changing from one part of the system to another. This quantifies the stability and robustness of the system. We studied how barrier height is influenced by environmental fluctuations and perturbations on specific wirings of the cell cycle network. When the fluctuations increase, the barrier height decreases and the period and amplitude of cell cycle oscillation is more dispersed and less coherent. The corresponding dissipation of the system quantitatively measured by the entropy production rate increases. This implies that the system is less stable under fluctuations. We identified some key structural elements for wirings of the cell cycle network responsible for the change of the barrier height and therefore the global stability of the system through the sensitivity analysis. The results are in agreement with recent experiments and also provide new predictions.

Communications: Hamiltonian regulated cell signaling network

Cell signaling is fundamental to cell survival and disease progression. Traditional approaches to study these networks have focused largely on probabilistic approaches, with a large number of ad hoc assumptions. In this paper, we develop a linear Hamiltonian model to study the integrin signaling network. The integrin signaling network is central to cell adhesion, migration, and differentiation, but has not been studied in the same detail as other cell cycle networks. In this study, the integrin signaling network with 16 nodes in thermal fluctuations is analyzed through ensemble averages on the linear Hamiltonian model. This new and analytically rigorous approach offers a quick method to find out the dominant nodes in the complex network, which operate in the thermal noise regime. The robust on/off transitions due to the different initial inputs also reflect the inherent structure in the network, providing new insights into structure and function of the network.

Robustness and coherence of a three-protein circadian oscillator: landscape and flux perspectives

Three-protein circadian oscillations in cyanobacteria sustain for weeks. To understand how cellular oscillations function robustly in stochastic fluctuating environments, we used a stochastic model to uncover two natures of circadian oscillation: the potential landscape related to steady-state probability distribution of protein concentrations; and the corresponding flux related to speed of concentration changes which drive the oscillations. The barrier height of escaping from the oscillation attractor on the landscape provides a quantitative measure of the robustness and coherence for oscillations against intrinsic and external fluctuations. The difference between the locations of the zero total driving force and the extremal of the potential provides a possible experimental probe and quantification of the force from curl flux. These results, correlated with experiments, can help in the design of robust oscillatory networks.

Robustness of integrin signaling network

Integrin signaling network is responsible for regulating a wide variety of fundamental biological processes ranging from cell survival to cell death. While individual components of the network have been studied through experimental and computational methods, the network robustness and the flow of information through the network have not been characterized in a quantitative framework. Using a probability based model implemented through GRID computing, we approach the reduced signaling network and show that the network is highly robust and the final stable steady state is independent of the initial configurations. However, the path from the initial and the final state is intrinsically dependent on the state of the input nodes. Our results demonstrate a rugged funnel-like landscape for the signaling network where the final state is unique, but the paths are dependent on initial conditions.

Computational systems biology of the cell cycle

One of the early success stories of computational systems biology was the work done on cell-cycle regulation. The earliest mathematical descriptions of cell-cycle control evolved into very complex, detailed computational models that describe the regulation of cell division in many different cell types. On the way these models predicted several dynamical properties and unknown components of the system that were later experimentally verified/identified. Still, research on this field is far from over. We need to understand how the core cell-cycle machinery is controlled by internal and external signals, also in yeast cells and in the more complex regulatory networks of higher eukaryotes. Furthermore, there are many computational challenges what we face as new types of data appear thanks to continuing advances in experimental techniques. We have to deal with cell-to-cell variations, revealed by single cell measurements, as well as the tremendous amount of data flowing from high throughput machines. We need new computational concepts and tools to handle these data and develop more detailed, more precise models of cell-cycle regulation in various organisms. Here we review past and present of computational modeling of cell-cycle regulation, and discuss possible future directions of the field.

Robustness and topology of the yeast cell cycle Boolean network

Yeast cell cycle Boolean network was used as a case study of robustness to protein noise. Robustness was interpreted as involving stability of G1 steady state and sequence of gene expression from cell cycle START to stationary G1. A robustness measure to evaluate robustness strength of a network was proposed. Robust putative networks corresponding to the same steady state and sequence of gene expression of wild-type network were sampled. Architecture of wild-type yeast cell cycle network can be revealed by average topology profile of sampled robust putative networks.

Robustness, dissipations and coherence of the oscillation of circadian clock: potential landscape and flux perspectives

Finding the global probabilistic nature of a non-equilibrium circadian clock is essential for addressing important issues of robustness and function. We have uncovered the underlying potential energy landscape of a simple cyanobacteria biochemical network, and the corresponding flux which is the driving force for the oscillation. We found that the underlying potential landscape for the oscillation in the presence of small statistical fluctuations is like an explicit ring valley or doughnut shape in the three dimensional protein concentration space. We found that the barrier height separating the oscillation ring and other area is a quantitative measure of the oscillation robustness and decreases when the fluctuations increase. We also found that the entropy production rate characterizing the dissipation or heat loss decreases as the fluctuations decrease. In addition, we found that, as the fluctuations increase, the period and the amplitude of the oscillations is more dispersed, and the phase coherence decreases. We also found that the properties from exploring the effects of the inherent chemical rate parameters on the robustness. Our approach is quite general and can be applied to other oscillatory cellular network.

PACS Codes: 87.18.-h, 87.18.Vf, 87.18.Yt

Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: the Schlögl model revisited

Schlögl's model is the canonical example of a chemical reaction system that exhibits bistability. Because the biological examples of bistability and switching behaviour are increasingly numerous, this paper presents an integrated deterministic, stochastic and thermodynamic analysis of the model. After a brief review of the deterministic and stochastic modelling frameworks, the concepts of chemical and mathematical detailed balances are discussed and non-equilibrium conditions are shown to be necessary for bistability. Thermodynamic quantities such as the flux, chemical potential and entropy production rate are defined and compared across the two models. In the bistable region, the stochastic model exhibits an exchange of the global stability between the two stable states under changes in the pump parameters and volume size. The stochastic entropy production rate shows a sharp transition that mirrors this exchange. A new hybrid model that includes continuous diffusion and discrete jumps is suggested to deal with the multiscale dynamics of the bistable system. Accurate approximations of the exponentially small eigenvalue associated with the time scale of this switching and the full time-dependent solution are calculated using Matlab. A breakdown of previously known asymptotic approximations on small volume scales is observed through comparison with these and Monte Carlo results. Finally, in the appendix section is an illustration of how the diffusion approximation of the chemical master equation can fail to represent correctly the mesoscopically interesting steady-state behaviour of the system.

Floral morphogenesis: stochastic explorations of a gene network epigenetic landscape

In contrast to the classical view of development as a preprogrammed and deterministic process, recent studies have demonstrated that stochastic perturbations of highly non-linear systems may underlie the emergence and stability of biological patterns. Herein, we address the question of whether noise contributes to the generation of the stereotypical temporal pattern in gene expression during flower development. We modeled the regulatory network of organ identity genes in the Arabidopsis thaliana flower as a stochastic system. This network has previously been shown to converge to ten fixed-point attractors, each with gene expression arrays that characterize inflorescence cells and primordial cells of sepals, petals, stamens, and carpels. The network used is binary, and the logical rules that govern its dynamics are grounded in experimental evidence. We introduced different levels of uncertainty in the updating rules of the network. Interestingly, for a level of noise of around 0.5–10%, the system exhibited a sequence of transitions among attractors that mimics the sequence of gene activation configurations observed in real flowers. We also implemented the gene regulatory network as a continuous system using the Glass model of differential equations, that can be considered as a first approximation of kinetic-reaction equations, but which are not necessarily equivalent to the Boolean model. Interestingly, the Glass dynamics recover a temporal sequence of attractors, that is qualitatively similar, although not identical, to that obtained using the Boolean model. Thus, time ordering in the emergence of cell-fate patterns is not an artifact of synchronous updating in the Boolean model. Therefore, our model provides a novel explanation for the emergence and robustness of the ubiquitous temporal pattern of floral organ specification. It also constitutes a new approach to understanding morphogenesis, providing predictions on the population dynamics of cells with different genetic configurations during development.

Potential landscape and flux framework of nonequilibrium networks: robustness, dissipation, and coherence of biochemical oscillations

We established a theoretical framework for studying nonequilibrium networks with two distinct natures essential for characterizing the global probabilistic dynamics: the underlying potential landscape and the corresponding curl flux. We applied the idea to a biochemical oscillation network and found that the underlying potential landscape for the oscillation limit cycle has a distinct closed ring valley (Mexican hat-like) shape when the fluctuations are small. This global landscape structure leads to attractions of the system to the ring valley. On the ring, we found that the nonequilibrium flux is the driving force for oscillations. Therefore, both structured landscape and flux are needed to guarantee a robust oscillating network. The barrier height separating the oscillation ring and other areas derived from the landscape topography is shown to be correlated with the escaping time from the limit cycle attractor and provides a quantitative measure of the robustness for the network. The landscape becomes shallower and the closed ring valley shape structure becomes weaker (lower barrier height) with larger fluctuations. We observe that the period and the amplitude of the oscillations are more dispersed and oscillations become less coherent when the fluctuations increase. We also found that the entropy production of the whole network, characterizing the dissipation costs from the combined effects of both landscapes and fluxes, decreases when the fluctuations decrease. Therefore, less dissipation leads to more robust networks. Our approach is quite general and applicable to other networks, dynamical systems, and biological evolution. It can help in designing robust networks.

A molecular noise generator

Recent studies have demonstrated that intracellular variations in the rate of gene expression are of fundamental importance to cellular function and development. While such 'noise' is often considered detrimental in the context of perturbing genetic systems, it can be beneficial in processes such as species diversification and facilitation of evolution. A major difficulty in exploring such effects is that the magnitude and spectral properties of the induced variations arise from some intrinsic cellular process that is difficult to manipulate. Here, we present two designs of a molecular noise generator that allow for the flexible modulation of the noise profile of a target gene. The first design uses a dual-signal mechanism that enables independent tuning of the mean and variability of an output protein. This is achieved through the combinatorial control of two signals that regulate transcription and translation separately. We then extend the design to allow for DNA copy-number regulation, which leads to a wider tuning spectrum for the output molecule. To gain a deeper understanding of the circuit's functionality in a realistic environment, we introduce variability in the input signals in order to ascertain the degree of noise induced by the control process itself. We conclude by illustrating potential applications of the noise generator, demonstrating how it could be used to ascertain the robust or fragile properties of a genetic circuit.

Intrinsic noise, dissipation cost, and robustness of cellular networks: the underlying energy landscape of MAPK signal transduction

We develop a probabilistic method for analyzing global features of a cellular network under intrinsic statistical fluctuations, which is important when there are finite numbers of molecules. By making a self-consistent mean field approximation of splitting the variables in order to reduce the large number of degrees of freedom, which is reasonable for a not very strongly interacting network, we discovered that the underlying energy landscape of the mitogen-activated protein kinases (MAPKs) signal transduction network (with experimentally measured or inferred parameters such as chemical reaction rate coefficients in the network) is funneled toward a global minimum characterized by the nonequilibrium steady-state fixed point of the system at the end of the signal transduction process. For this system, we also show that the energy landscape is robust against intrinsic fluctuations and random perturbation to the inherent chemical reaction rates. The ratio of the slope versus the roughness of the energy landscape becomes a quantitative measure of robustness and stability of the network. Furthermore, we quantify the dissipation cost of this nonequilibrium system through entropy production, caused by the nonequilibrium flux in the system. We found that a lower dissipation cost corresponds to a more robust network. This least dissipation property might provide a design principle for robust and functional networks. Finally, we find the possibility of bistable and oscillatory-like solutions, which are important for cell fate decisions, upon perturbations. The method described here can be used in a variety of biological networks.

Exact probabilistic solution of spatial-dependent stochastics and associated spatial potential landscape for the bicoid protein

We investigated the spatial-dependent stochastic effects originating from the finite number of bicoid proteins in Drosophila melanogaster, which are crucial to cell development. We obtained an exact solution to the spatial-dependent stochastic chemical master equation and recovered the usual reaction-diffusion solution for the average of the bicoid concentration, valid in the bulk. We also used the steady state probability to get the spatial potential landscape. The stochastic effects are captured by the Poisson distribution; so, as the average of the bicoid concentration decreases from the anterior (A) to the posterior (P) of the embryo, the statistical fluctuations also decrease. An alternative way of interpreting this is that the shape of the spatial potential landscape shrinks from A to P. While the mathematical result is known, we offer a simple approach to understanding why it is what it is and give associated physical intuitions. The approach can be generalized and applied to any problem with a particle that diffuses, decays, and has a stochastic source.

Least dissipation cost as a design principle for robustness and function of cellular networks

From a study of the budding yeast cell cycle, we found that the cellular network evolves to have the least cost for realizing its biological function. We quantify the cost in terms of the dissipation or heat loss characterized through the steady-state properties: the underlying landscape and the associated flux. We found that the dissipation cost is intimately related to the stability and robustness of the network. With the least dissipation cost, the network becomes most stable and robust under mutations and perturbations on the sharpness of the response from input to output as well as self-degradations. The least dissipation cost may provide a general design principle for the cellular network to survive from the evolution and realize the biological function.