Bistable Network Research Articles

Coupling between feedback loops in autoregulatory networks affects bistability range, open-loop gain and switching times

Biochemical regulatory networks governing diverse cellular processes such as stress-response, differentiation and cell cycle often contain coupled feedback loops. We aim at understanding how features of feedback architecture, such as the number of loops, the sign of the loops and the type of their coupling, affect network dynamical performance. Specifically, we investigate how bistability range, maximum open-loop gain and switching times of a network with transcriptional positive feedback are affected by additive or multiplicative coupling with another positive- or negative-feedback loop. We show that a network's bistability range is positively correlated with its maximum open-loop gain and that both quantities depend on the sign of the feedback loops and the type of feedback coupling. Moreover, we find that the addition of positive feedback could decrease the bistability range if we control the basal level in the signal-response curves of the two systems. Furthermore, the addition of negative feedback has the capacity to increase the bistability range if its dissociation constant is much lower than that of the positive feedback. We also find that the addition of a positive feedback to a bistable network increases the robustness of its bistability range, whereas the addition of a negative feedback decreases it. Finally, we show that the switching time for a transition from a high to a low steady state increases with the effective fold change in gene regulation. In summary, we show that the effect of coupled feedback loops on the bistability range and switching times depends on the underlying mechanistic details.

On the precision of quasi steady state assumptions in stochastic dynamics

Many biochemical networks have complex multidimensional dynamics and there is a long history of methods that have been used for dimensionality reduction for such reaction networks. Usually a deterministic mass action approach is used; however, in small volumes, there are significant fluctuations from the mean which the mass action approach cannot capture. In such cases stochastic simulation methods should be used. In this paper, we evaluate the applicability of one such dimensionality reduction method, the quasi-steady state approximation (QSSA) [L. Menten and M. Michaelis, "Die kinetik der invertinwirkung," Biochem. Z 49, 333369 (1913)] for dimensionality reduction in case of stochastic dynamics. First, the applicability of QSSA approach is evaluated for a canonical system of enzyme reactions. Application of QSSA to such a reaction system in a deterministic setting leads to Michaelis-Menten reduced kinetics which can be used to derive the equilibrium concentrations of the reaction species. In the case of stochastic simulations, however, the steady state is characterized by fluctuations around the mean equilibrium concentration. Our analysis shows that a QSSA based approach for dimensionality reduction captures well the mean of the distribution as obtained from a full dimensional simulation but fails to accurately capture the distribution around that mean. Moreover, the QSSA approximation is not unique. We have then extended the analysis to a simple bistable biochemical network model proposed to account for the stability of synaptic efficacies; the substrate of learning and memory [J. E. Lisman, "A mechanism of memory storage insensitive to molecular turnover: A bistable autophosphorylating kinase," Proc. Natl. Acad. Sci. U.S.A. 82, 3055-3057 (1985)]. Our analysis shows that a QSSA based dimensionality reduction method results in errors as big as two orders of magnitude in predicting the residence times in the two stable states.

Advances in Chemical Reaction Network Theory for the Identification of Kinetic Models

In this work, we illustrate the potential of Chemical Reaction Network Theory for model identification of kinetic models, setting up the basis for a novel method of inverse bifurcation analysis of bistable biochemical systems. The method exploits the structural properties of biochemical networks to infer the kinetic parameters from dose response curves. The methodology is applied to an example of a bistable enzymatic network.

Evolutionary Conservation and Network Structure Characterize Genes of Phenotypic Relevance for Mitosis in Human

The impact of gene silencing on cellular phenotypes is difficult to establish due to the complexity of interactions in the associated biological processes and pathways. A recent genome-wide RNA knock-down study both identified and phenotypically characterized a set of important genes for the cell cycle in HeLa cells. Here, we combine a molecular interaction network analysis, based on physical and functional protein interactions, in conjunction with evolutionary information, to elucidate the common biological and topological properties of these key genes. Our results show that these genes tend to be conserved with their corresponding protein interactions across several species and are key constituents of the evolutionary conserved molecular interaction network. Moreover, a group of bistable network motifs is found to be conserved within this network, which are likely to influence the network stability and therefore the robustness of cellular functioning. They form a cluster, which displays functional homogeneity and is significantly enriched in genes phenotypically relevant for mitosis. Additional results reveal a relationship between specific cellular processes and the phenotypic outcomes induced by gene silencing. This study introduces new ideas regarding the relationship between genotype and phenotype in the context of the cell cycle. We show that the analysis of molecular interaction networks can result in the identification of genes relevant to cellular processes, which is a promising avenue for future research.

Modeling reaction noise with a desired accuracy by using the X level approach reaction noise estimator (XARNES) method

A novel computational method for modeling reaction noise characteristics has been suggested. The method can be classified as a moment closure method. The approach is based on the concept of correlation forms which are used for describing spatially extended many body problems where particle numbers change in space and time. In here, it was shown how the formalism of spatially extended correlation forms can be adapted to study well mixed reaction systems. Stochastic fluctuations in particle numbers are described by selectively capturing correlation effects up to the desired order, ξ. The method is referred to as the ξ-level Approximation Reaction Noise Estimator method (XARNES). For example, the ξ=1 description is equivalent to the mean field theory (first-order effects), the ξ=2 case corresponds to the previously developed PARNES method (pair effects), etc. The main idea is that inclusion of higher order correlation effects should lead to better (more accurate) results. Several models were used to test the method, two versions of a simple complex formation model, the Michaelis–Menten model of enzymatic kinetics, the smallest bistable reaction network, a gene expression network with negative feedback, and a random large network. It was explicitly demonstrated that increase in ξ indeed improves accuracy in all cases investigated. The approach has been implemented as automatic software using the Mathematica programming language. The user only needs to input reaction rates, stoichiometry coefficients, and the desired level of computation ξ.

Different Cell Fates from Cell-Cell Interactions: Core Architectures of Two-Cell Bistable Networks

The acquisition of different fates by cells that are initially in the same state is central to development. Here, we investigate the possible structures of bistable genetic networks that can allow two identical cells to acquire different fates through cell-cell interactions. Cell-autonomous bistable networks have been previously sampled using an evolutionary algorithm. We extend this evolutionary procedure to take into account interactions between cells. We obtain a variety of simple bistable networks that we classify into major subtypes. Some have long been proposed in the context of lateral inhibition through the Notch-Delta pathway, some have been more recently considered and others appear to be new and based on mechanisms not previously considered. The results highlight the role of posttranscriptional interactions and particularly of protein complexation and sequestration, which can replace cooperativity in transcriptional interactions. Some bistable networks are entirely based on posttranscriptional interactions and the simplest of these is found to lead, upon a single parameter change, to oscillations in the two cells with opposite phases. We provide qualitative explanations as well as mathematical analyses of the dynamical behaviors of various created networks. The results should help to identify and understand genetic structures implicated in cell-cell interactions and differentiation.

Mathematical model of GAL regulon dynamics in Saccharomyces cerevisiae

Genetic switches are prevalent in nature and provide cells with a strategy to adapt to changing environments. The GAL switch is an intriguing example which is not understood in all detail. The GAL switch allows organisms to metabolize galactose, and controls whether the machinery responsible for the galactose metabolism is turned on or off. Currently, it is not known exactly how the galactose signal is sensed by the transcriptional machinery. Here we utilize quantitative tools to understand the S. cerevisiae cell response to galactose challenge, and to analyze the plausible molecular mechanisms underlying its operation. We work at a population level to develop a dynamic model based on the interplay of the key regulatory proteins Gal4p, Gal80p, and Gal3p. To our knowledge, the model presented here is the first to reproduce qualitatively the bistable network behavior found experimentally. Given the current understanding of the GAL circuit induction (Wightman et al., 2008; Jiang et al., 2009), we propose that the most likely in vivo mechanism leading to the transcriptional activation of the GAL genes is the physical interaction between galactose-activated Gal3p and Gal80p, with the complex Gal3p–Gal80p remaining bound at the GAL promoters. Our mathematical model is in agreement with the flow cytometry profiles of wild type, gal3Δ and gal80Δ mutant strains from Acar et al. (2005), and involves a fraction of actively transcribing cells with the same qualitative features as in the data set collected by Acar et al. (2010). Furthermore, the computational modeling provides an explanation for the contradictory results obtained by independent laboratories when tackling experimentally the issue of binary versus graded response to galactose induction.

Emergence of Switch-Like Behavior in a Large Family of Simple Biochemical Networks

Bistability plays a central role in the gene regulatory networks (GRNs) controlling many essential biological functions, including cellular differentiation and cell cycle control. However, establishing the network topologies that can exhibit bistability remains a challenge, in part due to the exceedingly large variety of GRNs that exist for even a small number of components. We begin to address this problem by employing chemical reaction network theory in a comprehensive in silico survey to determine the capacity for bistability of more than 40,000 simple networks that can be formed by two transcription factor-coding genes and their associated proteins (assuming only the most elementary biochemical processes). We find that there exist reaction rate constants leading to bistability in ∼90% of these GRN models, including several circuits that do not contain any of the TF cooperativity commonly associated with bistable systems, and the majority of which could only be identified as bistable through an original subnetwork-based analysis. A topological sorting of the two-gene family of networks based on the presence or absence of biochemical reactions reveals eleven minimal bistable networks (i.e., bistable networks that do not contain within them a smaller bistable subnetwork). The large number of previously unknown bistable network topologies suggests that the capacity for switch-like behavior in GRNs arises with relative ease and is not easily lost through network evolution. To highlight the relevance of the systematic application of CRNT to bistable network identification in real biological systems, we integrated publicly available protein-protein interaction, protein-DNA interaction, and gene expression data from Saccharomyces cerevisiae, and identified several GRNs predicted to behave in a bistable fashion.

Novel visualisation and analysis of natural and complex systems using systemic computation

The study, analysis and understanding of natural processes are difficult tasks considering the complex nature of such processes. In this respect, the visual analysis of such systems can be of great help in the understanding of their behaviour. The increasing power of modern computers enables novel possible uses of computer graphics for such tasks. Previous work introduced systemic computation, a new model of computation and corresponding computer architecture aiming at enabling a clear formalism of natural and complex systems and providing tools for their analysis. Here, we present an online visualisation of dynamic systems based on this novel paradigm. The observation is done at a high level of abstraction, focussing on information flow, interactions and emergent behaviour, and enabling the identification of similarities and differences between models of complex systems. This visualisation framework is then applied to two biological networks: a bistable gene network and a MAPK signalling cascade.

Novel Visualisation and Analysis of Natural and Complex Systems Using Systemic Computation

The study, analysis and understanding of natural processes are difficult tasks considering the complex nature of such processes. In this respect, the visual analysis of such systems can be of great help in the understanding of their behaviour. The increasing power of modern computers enables novel possible uses of computer graphics for such tasks. Previous work introduced systemic computation, a new model of computation and corresponding computer architecture aiming at enabling a clear formalism of natural and complex systems and providing tools for their analysis. Here, we present an online visualisation of dynamic systems based on this novel paradigm. The observation is done at a high level of abstraction, focussing on information flow, interactions and emergent behaviour, and enabling the identification of similarities and differences between models of complex systems. This visualisation framework is then applied to two biological networks: a bistable gene network and a MAPK signalling cascade.

Phenotypic switching of Escherichia coli cells containing cyclic digenic systems with negative feedback upon changes in cultivation conditions

One of the mechanisms for the epigenetic control of cell phenotypes is based on switching the functioning regimes of bistable gene networks, which can maintain the two alternative levels of gene expression under the same conditions. Cyclic digenic systems with negative feedback represent an example of a simple bistable gene network. Cells carrying artificial cyclic digenic systems on plasmids inherit each alternative phenotype upon exponential growth on rich medium during several cell generations. The action of specific inducers is necessary for switching. In this work, the impact of changes in cell cultivation conditions on the phenotypic composition of the clonal Escherichia coli cell population containing artificial cyclic digenic systems with negative feedback was studied. Phenotypes differ with respect to the expression level of marker proteins: beta-galactosidase and GFP. Slow growth on a medium containing little-available carbon sources was shown to cause the transition from the phenotype Lac- to Lac+ in the absence of inducers. Phenotypic switching cannot be explained by transcriptional activation of the lactose operon, because 80 +/- 15% of cells inherit the acquired phenotype after replating bacteria on rich medium. Inheritance of the phenotype Lac- in batch culture depends on the medium and duration of cultivation. Dynamics of changes in the activity of beta-galactosidase and culture fluorescence suggests that a decrease in the level of metabolism resulted in the switch of these cyclic systems from bistable to monostable functioning regime, which corresponds to the Lac+ phenotype with respect to the ratio of regulatory proteins. Thus, the instability of growth conditions may cause phenotypic heterogeneity in the clonal population of cells containing bistable gene networks.

Estimating the stochastic bifurcation structure of cellular networks.

High throughput measurement of gene expression at single-cell resolution, combined with systematic perturbation of environmental or cellular variables, provides information that can be used to generate novel insight into the properties of gene regulatory networks by linking cellular responses to external parameters. In dynamical systems theory, this information is the subject of bifurcation analysis, which establishes how system-level behaviour changes as a function of parameter values within a given deterministic mathematical model. Since cellular networks are inherently noisy, we generalize the traditional bifurcation diagram of deterministic systems theory to stochastic dynamical systems. We demonstrate how statistical methods for density estimation, in particular, mixture density and conditional mixture density estimators, can be employed to establish empirical bifurcation diagrams describing the bistable genetic switch network controlling galactose utilization in yeast Saccharomyces cerevisiae. These approaches allow us to make novel qualitative and quantitative observations about the switching behavior of the galactose network, and provide a framework that might be useful to extract information needed for the development of quantitative network models.

Towards a dynamical network view of brain ischemia and reperfusion. Part I: background and preliminaries

The general failure of neuroprotectants in clinical trials of ischemic stroke points to the possibility of a fundamental blind spot in the current conception of ischemic brain injury, the "ischemic cascade". This is the first in a series of four papers whose purpose is to work towards a revision of the concept of brain ischemia by applying network concepts to develop a bistable model of brain ischemia. This first paper sets the stage for developing the bistable model of brain ischemia. Necessary background in network theory is introduced using examples from developmental biology which, perhaps surprisingly, can be adapted to brain ischemia with only minor modification. Then, to move towards a network model, we extract two core generalizations about brain ischemia from the mass of empirical data. First, we conclude that all changes induced in the brain by ischemia can be classified as either damage mechanisms that contribute to cell death, or stress responses that contribute to cell survival. Second, we move towards formalizing the idea of the "amount of ischemia", I, as a continuous, nonnegative, monotonically increasing quantity. These two generalizations are necessary precursors to reformulating brain ischemia as a bistable network.

Towards a dynamical network view of brain ischemia and reperfusion. Part I: background and preliminaries

The general failure of neuroprotectants in clinical trials of ischemic stroke points to the possibility of a fundamen-tal blind spot in the current conception of ischemic brain injury, the “ischemic cascade”. This is the first in a se-ries of four papers whose purpose is to work towards a revision of the concept of brain ischemia by applying network concepts to develop a bistable model of brain ischemia. This first paper sets the stage for developing the bistable model of brain ischemia. Necessary background in network theory is introduced using examples from developmental biology which, perhaps surprisingly, can be adapted to brain ischemia with only minor mod-ification. Then, to move towards a network model, we extract two core generalizations about brain ischemia from the mass of empirical data. First, we conclude that all changes induced in the brain by ischemia can be classified as either damage mechanisms that contribute to cell death, or stress responses that contribute to cell survival. Second, we move towards formalizing the idea of the “amount of ischemia”, I, as a continuous, non-negative, monotonically increasing quantity. These two generalizations are necessary precursors to reformulat-ing brain ischemia as a bistable network.

Towards a dynamical network view of brain ischemia and reperfusion. Part II: a post-ischemic neuronal state space

The general failure of neuroprotectants in clinical trials of ischemic stroke points to the possibility of a fundamen-tal blind spot in the current conception of ischemic brain injury, the “ischemic cascade”. This is the second in a series of four papers whose purpose is to work towards a revision of the concept of brain ischemia by applying network concepts to develop a bistable model of brain ischemia. We here build the bistable network model of brain ischemia. The central concept is that of a post-ischemic state space. Ischemia, as a quantitative pertur-bation, is envisioned to push the brain through a series of four phenotypes as a function of the amount of ischemia: the homeostatic, preconditioned, delayed neuronal death and necrotic phenotypes. The phenotypes are meta-stable attractors in the landscape of the post-ischemic state space. The sequence of the phenotypes derives from the mutual antagonism between damage mechanisms and stress responses, each conceived as aggregate ensemble variables. The competition between damage mechanisms and stress responses is posited to have the form of a bistability. Application of bistability to brain ischemia is grounded in the incontrovertible fact that post-ischemic neurons face the mutually exclusive decision to either live or die.

Towards a dynamical network view of brain ischemia and reperfusion. Part II: a post-ischemic neuronal state space

The general failure of neuroprotectants in clinical trials of ischemic stroke points to the possibility of a fundamental blind spot in the current conception of ischemic brain injury, the "ischemic cascade". This is the second in a series of four papers whose purpose is to work towards a revision of the concept of brain ischemia by applying network concepts to develop a bistable model of brain ischemia. We here build the bistable network model of brain ischemia. The central concept is that of a post-ischemic state space. Ischemia, as a quantitative perturbation, is envisioned to push the brain through a series of four phenotypes as a function of the amount of ischemia: the homeostatic, preconditioned, delayed neuronal death and necrotic phenotypes. The phenotypes are meta-stable attractors in the landscape of the post-ischemic state space. The sequence of the phenotypes derives from the mutual antagonism between damage mechanisms and stress responses, each conceived as aggregate ensemble variables. The competition between damage mechanisms and stress responses is posited to have the form of a bistability. Application of bistability to brain ischemia is grounded in the incontrovertible fact that post-ischemic neurons face the mutually exclusive decision to either live or die.

Additional comments on the bistable network model of brain ischemia

Dr. Hossmann provides a lucid and concise summary and a sympathetic critique of the bistable model and its potential role in cerebral ischemia research. He rightly gets to the heart of the matter in recognizing that the bistable model as currently presented is inspired primarily by envisioning transient global ischemia. Such an insult is characteristic of commonly used laboratory models of global ischemia including cardiac arrest, bilateral carotid artery occlusion (with or without hypotension) or four vessel occlusion.

Towards a dynamical network view of brain ischemia and reperfusion. Part IV: additional considerations

The general failure of neuroprotectants in clinical trials of ischemic stroke points to the possibility of a fundamen-tal blind spot in the current conception of ischemic brain injury, the “ischemic cascade”. This is the fourth in a series of four papers whose purpose is to work towards a revision of the concept of brain ischemia by applying network concepts to develop a bistable model of brain ischemia. Here we consider additional issues to round out and close out this initial presentation of the bistable network view of brain ischemia. Initial considerations of the network architecture underlying the post-ischemic state space are discussed. Network and differential equa-tion models of brain ischemia are compared. We offer a first look at applying the bistable model to focal cere-bral ischemia. The limitations of the present formulation of the bistable model are discussed. This work con-cludes with a series of questions by which to direct future efforts.

Making Cellular Memories

The induction of a protracted response to a brief stimulus is a form of cellular memory. Here we describe the role of transcriptional regulation in both natural and synthetic memory networks and discuss the potential applications of engineered memory networks in medicine and industrial biotechnology.

Towards a dynamical network view of brain ischemia and reperfusion. Part IV: additional considerations

The general failure of neuroprotectants in clinical trials of ischemic stroke points to the possibility of a fundamental blind spot in the current conception of ischemic brain injury, the "ischemic cascade". This is the fourth in a series of four papers whose purpose is to work towards a revision of the concept of brain ischemia by applying network concepts to develop a bistable model of brain ischemia. Here we consider additional issues to round out and close out this initial presentation of the bistable network view of brain ischemia. Initial considerations of the network architecture underlying the post-ischemic state space are discussed. Network and differential equation models of brain ischemia are compared. We offer a first look at applying the bistable model to focal cerebral ischemia. The limitations of the present formulation of the bistable model are discussed. This work concludes with a series of questions by which to direct future efforts.