Multistationarity, the basis of cell differentiation and memory. I. Structural conditions of multistationarity and other nontrivial behavior

Graph-based, dynamics-preserving reduction of (bio)chemical systems

Complex dynamical systems are often governed by equations containing many unknown parameters whose precise values may or may not be important for the system's dynamics. In particular, for chemical and biochemical systems, there may be some reactions or subsystems that are inessential to understanding the bifurcation structure and consequent behavior of a model, such as oscillations, multistationarity and patterning. Due to the size, complexity and parametric uncertainties of many (bio)chemical models, a dynamics-preserving reduction scheme that is able to isolate the necessary contributors to particular dynamical behaviors would be useful. In this contribution, we describe model reduction methods for mass-action (bio)chemical models based on the preservation of instability-generating subnetworks known as critical fragments. These methods focus on structural conditions for instabilities and so are parameter-independent. We apply these results to an existing model for the control of the synthesis of the NO-detoxifying enzyme Hmp in Escherichia coli that displays bistability.

Verifiable biology

The formalization of biological systems using computational modelling approaches as an alternative to mathematical-based methods has recently received much interest because computational models provide a deeper mechanistic understanding of biological systems. In particular, formal verification, complementary approach to standard computational techniques such as simulation, is used to validate the system correctness and obtain critical information about system behaviour. In this study, we survey the most frequently used computational modelling approaches and formal verification techniques for computational biology. We compare a number of verification tools and software suites used to analyse biological systems and biochemical networks, and to verify a wide range of biological properties. For users who have no expertise in formal verification, we present a novel methodology that allows them to easily apply formal verification techniques to analyse their biological or biochemical system of interest.

Effective Connectivity and Bias Entropy Improve Prediction of Dynamical Regime in Automata Networks

Biomolecular network dynamics are thought to operate near the critical boundary between ordered and disordered regimes, where large perturbations to a small set of elements neither die out nor spread on average. A biomolecular automaton (e.g., gene, protein) typically has high regulatory redundancy, where small subsets of regulators determine activation via collective canalization. Previous work has shown that effective connectivity, a measure of collective canalization, leads to improved dynamical regime prediction for homogeneous automata networks. We expand this by (i) studying random Boolean networks (RBNs) with heterogeneous in-degree distributions, (ii) considering additional experimentally validated automata network models of biomolecular processes, and (iii) considering new measures of heterogeneity in automata network logic. We found that effective connectivity improves dynamical regime prediction in the models considered; in RBNs, combining effective connectivity with bias entropy further improves the prediction. Our work yields a new understanding of criticality in biomolecular networks that accounts for collective canalization, redundancy, and heterogeneity in the connectivity and logic of their automata models. The strong link we demonstrate between criticality and regulatory redundancy provides a means to modulate the dynamical regime of biochemical networks.

Leveraging network structure in nonlinear control

Over the last twenty years, dynamic modeling of biomolecular networks has exploded in popularity. Many of the classical tools for understanding dynamical systems are unwieldy in the highly nonlinear, poorly constrained, high-dimensional systems that often arise from these modeling efforts. Understanding complex biological systems is greatly facilitated by purpose-built methods that leverage common features of such models, such as local monotonicity, interaction graph sparsity, and sigmoidal kinetics. Here, we review methods for controlling the systems of ordinary differential equations used to model biomolecular networks. We focus on methods that make use of the structure of the network of interactions to help inform, which variables to target for control, and highlight the computational and experimental advantages of such approaches. We also discuss the importance of nonperturbative methods in biomedical and experimental molecular biology applications, where finely tuned interventions can be difficult to implement. It is well known that feedback loops, and positive feedback loops in particular, play a major determining role in the dynamics of biomolecular networks. In many of the methods we cover here, control over system trajectories is realized by overriding the behavior of key feedback loops.

Multi-synchronization and other patterns of multi-rhythmicity in oscillatory biological systems

While experimental and theoretical studies have established the prevalence of rhythmic behaviour at all levels of biological organization, less common is the coexistence between multiple oscillatory regimes (multi-rhythmicity), which has been predicted by a variety of models for biological oscillators. The phenomenon of multi-rhythmicity involves, most commonly, the coexistence between two (birhythmicity) or three (trirhythmicity) distinct regimes of self-sustained oscillations. Birhythmicity has been observed experimentally in a few chemical reactions and in biological examples pertaining to cardiac cell physiology, neurobiology, human voice patterns and ecology. The present study consists of two parts. We first review the mechanisms underlying multi-rhythmicity in models for biochemical and cellular oscillations in which the phenomenon was investigated over the years. In the second part, we focus on the coupling of the cell cycle and the circadian clock and show how an additional source of multi-rhythmicity arises from the bidirectional coupling of these two cellular oscillators. Upon bidirectional coupling, the two oscillatory networks generally synchronize in a unique manner characterized by a single, common period. In some conditions, however, the two oscillators may synchronize in two or three different ways characterized by distinct waveforms and periods. We refer to this type of multi-rhythmicity as ‘multi-synchronization’.

Computational prediction of intracellular targets of wild-type or mutant vesicular stomatitis matrix protein

The matrix (M) protein of vesicular stomatitis virus (VSV) has a complex role in infection and immune evasion, particularly with respect to suppression of Type I interferon (IFN). Viral strains bearing the wild-type (wt) M protein are able to suppress Type I IFN responses. We recently reported that the 22–25 strain of VSV encodes a wt M protein, however its sister plaque isolate, strain 22–20, carries a M[MD52G] mutation that perturbs the ability of the M protein to block NFκB, but not M-mediated inhibition of host transcription. Therefore, although NFκB is activated in 22–20 infected murine L929 cells infected, no IFN mRNA or protein is produced. To investigate the impact of the M[D52G] mutation on immune evasion by VSV, we used transcriptomic data from L929 cells infected with wt, 22–25, or 22–20 to define parameters in a family of executable logical models with the aim of discovering direct targets of viruses encoding a wt or mutant M protein. After several generations of pruning or fixing hypothetical regulatory interactions, we identified specific predicted targets of each strain. We predict that wt and 22–25 VSV both have direct inhibitory actions on key elements of the NFκB signaling pathway, while 22–20 fails to inhibit this pathway.

Measuring and Modeling Single-Cell Heterogeneity and Fate Decision in Mouse Embryos

Cellular heterogeneity is a property of any living system; however, its relationship with cellular fate decision remains an open question. Recent technological advances have enabled valuable insights, especially in complex systems such as the mouse embryo. In this review, we discuss recent studies that characterize cellular heterogeneity at different levels during mouse development, from the two-cell stage up to gastrulation. In addition to key experimental findings, we review mathematical modeling approaches that help researchers interpret these findings. Disentangling the role of heterogeneity in cell fate decision will likely rely on the refined integration of experiments, large-scale omics data, and mathematical modeling, complemented by the use of synthetic embryos and gastruloids as promising in vitro models.

Reconciling qualitative, abstract, and scalable modeling of biological networks

Predicting biological systems' behaviors requires taking into account many molecular and genetic elements for which limited information is available past a global knowledge of their pairwise interactions. Logical modeling, notably with Boolean Networks (BNs), is a well-established approach that enables reasoning on the qualitative dynamics of networks. Several dynamical interpretations of BNs have been proposed. The synchronous and (fully) asynchronous ones are the most prominent, where the value of either all or only one component can change at each step. Here we prove that, besides being costly to analyze, these usual interpretations can preclude the prediction of certain behaviors observed in quantitative systems. We introduce an execution paradigm, the Most Permissive Boolean Networks (MPBNs), which offers the formal guarantee not to miss any behavior achievable by a quantitative model following the same logic. Moreover, MPBNs significantly reduce the complexity of dynamical analysis, enabling to model genome-scale networks.

Bi-stability of the master gene regulatory network of the common dendritic precursor cell: Implications for cell differentiation

In cell lineage commitment decisions, a gene regulatory network (GRN) consisting of a limited number of transcription factors forms the regulatory pivot. Myeloid lineage dendritic cells or DCs are specialized cells having the antigen-presenting ability and are of immense importance in immune surveillance. In this report, we analyze the GRN that governs the lineage commitment of Common DC Progenitor (CDP) cells to conventional dendritic cells (cDC) and plasmacytoid dendritic cells (pDC). We have analyzed the quantitative behavior of the master regulatory circuit of CDP that governs the lineage commitment. Simulations showed that the GRN displays a bi-stable behavior within a range of parameter values. Several transcription factors, PU.1, IRF8, Flt3, and Stat3, whose concentrations vary significantly in the two steady states, appear to be the key players. We hypothesize that the two stable steady states are precursors of cDC and pDC, and the variation of concentration of these key transcription factors in the two states may be responsible for early events in lineage commitment.

Immune Network Modeling Predicts Specific Nasopharyngeal and Peripheral Immune Dysregulation in Otitis-Prone Children

Acute otitis media (AOM) pathogenesis involves nasopharyngeal colonization by potential otopathogens and a viral co-infection. Stringently-defined otitis prone (sOP) children show characteristic patterns of immune dysfunction. We hypothesized that otitis proneness is largely a result of altered signaling between immune components that are otherwise competent, resulting in increased susceptibility to infection by bacterial otopathogens. To test this, we constructed a regulatory immune network model linking immune cells and signaling elements known to be involved in AOM and/or dysregulated in sOP children. The alignment of immune response mechanisms with data from in vivo and in vitro experimental observations produced 82 putative immune network models, each describing variants of immune regulatory networks consistent with available observations. Analysis of these models suggested that new measurements of serum levels of IL-4 and CXCL8 could refine competing models and resulted in the elimination of 38 of the models. Further analysis of the remaining 44 models suggested specific deviations in the predicted regulation of nasopharyngeal and peripheral immunity during response to AOM. Specifically, immune responses active in sOP children during AOM were characterized by early and constitutive activation of pro-inflammatory signaling in the nasopharynx and a Th2- and Treg-dominated profile in the periphery. We conclude that sOP children have altered regulation of key immune mediators during both health and pathogenesis. This altered regulation may be amenable to therapeutic intervention.

Quantifying epigenetic stability with minimum action paths

Chromatin can adopt multiple stable, heritable states with distinct histone modifications and varying levels of gene expression. Insight on the stability and maintenance of such epigenetic states can be gained by mathematical modeling of stochastic reaction networks for histone modifications. Analytical results for the kinetic networks are particularly valuable. Compared to computationally demanding numerical simulations, they often are more convenient at evaluating the robustness of conclusions with respect to model parameters. In this communication, we developed a second-quantization-based approach that can be used to analyze discrete stochastic models with a fixed, finite number of particles using a representation of the SU(2) algebra. We applied the approach to a kinetic model of chromatin states that captures the feedback between nucleosomes and the enzymes conferring histone modifications. Using a path-integral expression for the transition probability, we computed the epigenetic landscape that helps to identify the emergence of bistability and the most probable path connecting the two steady states. We anticipate the generalizability of the approach will make it useful for studying more complicated models that couple epigenetic modifications with transcription factors and chromatin structure.

Dual dynamics of mitochondrial permeability transition pore opening

Mitochondria play an essential role in bioenergetics and cellular Ca\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{2+}$$\end{document}2+ handling. The mitochondrial permeability transition pore (mPTP) is a non-specific channel located in the inner mitochondrial membrane. Long-lasting openings of the pore allow the rapid passage of ions and large molecules, which can result in cell death. The mPTP also exhibits transient, low conductance openings that contribute to Ca\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{2+}$$\end{document}2+ homeostasis. Although many regulators of the pore have been identified, none of them uniquely governs the passage between the two operating modes, which thus probably relies on a still unidentified network of interactions. By developing a core computational model for mPTP opening under the control of mitochondrial voltage and Ca\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{2+}$$\end{document}2+, we uncovered the existence of a positive feedback loop leading to bistability. The characteristics of the two stable steady-states correspond to those of the two opening states. When inserted in a full model of Ca\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{2+}$$\end{document}2+ handling by mitochondria, our description of the pore reproduces observations in mitochondrial suspensions. Moreover, the model predicted the occurrence of hysteresis in the switching between the two modes, upon addition and removal of free Ca\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{2+}$$\end{document}2+ in the extra-mitochondrial medium. Stochastic simulations then confirmed that the pore can undergo transient openings resembling those observed in intact cells.

Circular Causation, Circular Cognition: A Tour Around Some Common Confusions

Life and cognition are inherently circular dynamical processes, and people have difficulty understanding circular causation. I give case studies illustrating some resulting confusions, and propose that the problems may lie in failing to properly distinguish between similar concepts used to describe both local and global features of a system. I analyze how explanations in terms of circular causation work and how they rely on principles of normal settlement. Even though they typically will not explain the origins of phenomena (that is the province of linear causal explanation), circular explanations have predictive power for any persisting (i.e., stable or metastable) phenomena.

Leveraging Prior Knowledge of Endocrine Immune Regulation in the Therapeutically Relevant Phenotyping of Women With Chronic Fatigue Syndrome

PURPOSE

The complex and varied presentation of myalgic encephalomyelitis/chronic fatigue syndrome (ME/CFS) has made it difficult to diagnose, study, and treat. Its symptoms and likely etiology involve multiple components of endocrine and immune regulation, including the hypothalamic-pituitary-adrenal axis, the hypothalamic-pituitary-gonadal axis, and their interactive oversight of immune function. We propose that the persistence of ME/CFS may involve changes in the regulatory interactions across these physiological axes. We also propose that the robustness of this new pathogenic equilibrium may at least in part explain the limited success of conventional single-target therapies.

METHODS

A comprehensive model was constructed of female endocrine-immune signaling consisting of 28 markers linked by 214 documented regulatory interactions. This detailed model was then constrained to adhere to experimental measurements in a subset of 17 candidate immune markers measured in peripheral blood of patients with ME/CFS and healthy control subjects before, during, and after a maximal exercise challenge. A set of 26 competing numerical models satisfied these data to within 5% error.

FINDINGS

Mechanistically informed predictions of endocrine and immune markers that were either unmeasured or exhibited high subject-to-subject variability pointed to possible context-specific overexpression in ME/CFS at rest of corticotropin-releasing hormone, chemokine (C-X-C motif) ligand 8, estrogen, follicle-stimulating hormone (FSH), gonadotropin-releasing hormone 1, interleukin (IL)-23, and luteinizing hormone, and underexpression of adrenocorticotropic hormone, cortisol, interferon-γ, IL-10, IL-17, and IL-1α. Simulations of rintatolimod and rituximab treatment predicted a shift in the repertoire of available endocrine-immune regulatory regimens. Rintatolimod was predicted to make available substantial remission in a significant subset of subjects, in particular those with low levels of IL-1α, IL-17, and cortisol; intermediate levels of progesterone and FSH; and high estrogen levels. Rituximab treatment was predicted to support partial remission in a smaller subset of patients with ME/CFS, specifically those with low norepinephrine, IL-1α, chemokine (C-X-C motif) ligand 8, and cortisol levels; intermediate FSH and gonadotropin-releasing hormone 1 levels; and elevated expression of tumor necrosis factor-α, luteinizing hormone, IL-12, and B-cell activation.

IMPLICATIONS

Applying a rigorous filter of known signaling mechanisms to experimentally measured immune marker expression in ME/CFS has highlighted potential new context-specific markers of illness. These novel endocrine and immune markers may offer useful candidates in delineating new subtypes of ME/CFS and may inform on refinements to the inclusion criteria and instrumentation of new and ongoing trials involving rintatolimod and rituximab treatment protocols.

Multi-rhythmicity generated by coupling two cellular rhythms

The cell cycle and the circadian clock represent two major cellular rhythms, which are coupled because the circadian clock governs the synthesis of several proteins of the network that drives the mammalian cell cycle. Analysis of a detailed model for these coupled cellular rhythms previously showed that the cell cycle can be entrained at the circadian period of 24 h, or at a period of 48 h, depending on the autonomous period of the cell cycle and on the coupling strength. We show by means of numerical simulations that multiple stable periodic regimes, i.e. multi-rhythmicity, may originate from the coupling of the two cellular rhythms. In these conditions, the cell cycle can evolve to any one of two (birhythmicity) or three stable periodic regimes (trirhythmicity). When applied at the right phase, transient perturbations of appropriate duration and magnitude can induce the switch between the different oscillatory states. Such switching is characterized by final state sensitivity, which originates from the complex structure of the attraction basins. By providing a novel instance of multi-rhythmicity in a realistic model for the coupling of two major cellular rhythms, the results throw light on the conditions in which multiple stable periodic regimes may coexist in biological systems.

Computation of Robust Minimal Intervention Sets in Multi-Valued Biological Regulatory Networks

Enabled by rapid advances in computational sciences, in silico logical modeling of complex and large biological networks is more and more feasible making it an increasingly popular approach among biologists. Automated high-throughput, drug target identification is one of the primary goals of this in silico network biology. Targets identified in this way are then used to mine a library of drug chemical compounds in order to identify appropriate therapies. While identification of drug targets is exhaustively feasible on small networks, it remains computationally difficult on moderate and larger models. Moreover, there are several important constraints such as off-target effects, efficacy and safety that should be integrated into the identification of targets if the intention is translation to the clinical space. Here we introduce numerical constraints whereby efficacy is represented by efficiency in response and robustness of outcome. This paper introduces an algorithm that relies on a Constraint Satisfaction (CS) technique to efficiently compute the Minimal Intervention Sets (MIS) within a set of often complex clinical safety constraints with the aim of identifying the smallest least invasive set of targets pharmacologically accessible for therapy that most efficiently and reliably achieve the desired outcome.

Identifying (un)controllable dynamical behavior in complex networks

We present a technique applicable in any dynamical framework to identify control-robust subsets of an interacting system. These robust subsystems, which we call stable modules, are characterized by constraints on the variables that make up the subsystem. They are robust in the sense that if the defining constraints are satisfied at a given time, they remain satisfied for all later times, regardless of what happens in the rest of the system, and can only be broken if the constrained variables are externally manipulated. We identify stable modules as graph structures in an expanded network, which represents causal links between variable constraints. A stable module represents a system "decision point", or trap subspace. Using the expanded network, small stable modules can be composed sequentially to form larger stable modules that describe dynamics on the system level. Collections of large, mutually exclusive stable modules describe the system's repertoire of long-term behaviors. We implement this technique in a broad class of dynamical systems and illustrate its practical utility via examples and algorithmic analysis of two published biological network models. In the segment polarity gene network of Drosophila melanogaster, we obtain a state-space visualization that reproduces by novel means the four possible cell fates and predicts the outcome of cell transplant experiments. In the T-cell signaling network, we identify six signaling elements that determine the high-signal response and show that control of an element connected to them cannot disrupt this response.

Gene Regulatory Network Modeling of Macrophage Differentiation Corroborates the Continuum Hypothesis of Polarization States

Macrophages derived from monocyte precursors undergo specific polarization processes which are influenced by the local tissue environment: classically activated (M1) macrophages, with a pro-inflammatory activity and a role of effector cells in Th1 cellular immune responses, and alternatively activated (M2) macrophages, with anti-inflammatory functions and involved in immunosuppression and tissue repair. At least three different subsets of M2 macrophages, namely, M2a, M2b, and M2c, are characterized in the literature based on their eliciting signals. The activation and polarization of macrophages is achieved through many, often intertwined, signaling pathways. To describe the logical relationships among the genes involved in macrophage polarization, we used a computational modeling methodology, namely, logical (Boolean) modeling of gene regulation. We integrated experimental data and knowledge available in the literature to construct a logical network model for the gene regulation driving macrophage polarization to the M1, M2a, M2b, and M2c phenotypes. Using the software GINsim and BoolNet, we analyzed the network dynamics under different conditions and perturbations to understand how they affect cell polarization. Dynamic simulations of the network model, enacting the most relevant biological conditions, showed coherence with the observed behavior of in vivo macrophages. The model could correctly reproduce the polarization toward the four main phenotypes as well as to several hybrid phenotypes, which are known to be experimentally associated to physiological and pathological conditions. We surmise that shifts among different phenotypes in the model mimic the hypothetical continuum of macrophage polarization, with M1 and M2 being the extremes of an uninterrupted sequence of states. Furthermore, model simulations suggest that anti-inflammatory macrophages are resilient to shift back to the pro-inflammatory phenotype.

Revisiting a skeleton model for the mammalian cell cycle: From bistability to Cdk oscillations and cellular heterogeneity

A network of cyclin-dependent kinases (Cdks) regulated by multiple negative and positive feedback loops controls progression in the mammalian cell cycle. We previously proposed a detailed computational model for this network, which consists of four coupled Cdk modules. Both this detailed model and a reduced, skeleton version show that the Cdk network is capable of temporal self-organization in the form of sustained Cdk oscillations, which correspond to the orderly progression along the different cell cycle phases G1, S (DNA replication), G2 and M (mitosis). We use the skeleton model to revisit the role of positive feedback (PF) loops on the dynamics of the mammalian cell cycle by showing that the multiplicity of PF loops extends the range of bistability in the isolated Cdk modules controlling the G1/S and G2/M transitions. Resorting to stochastic simulations we show that, through their effect on the range of bistability, multiple PF loops enhance the robustness of Cdk oscillations with respect to molecular noise. The model predicts that a rise in the total level of Cdk1 also enlarges the domain of bistability in the isolated Cdk modules as well as the range of oscillations in the full Cdk network. Surprisingly, stochastic simulations indicate that Cdk1 overexpression reduces the robustness of Cdk oscillations towards molecular noise; this result is due to the increased distance between the two branches of the bistable switch at higher levels of Cdk1. At intermediate levels of growth factor stochastic simulations show that cells may randomly switch between cell cycle arrest and cell proliferation, as a consequence of fluctuations. In the presence of Cdk1 overexpression, these transitions occur even at low levels of growth factor. Extending stochastic simulations from single cells to cell populations suggests that stochastic switches between cell cycle arrest and proliferation may provide a source of heterogeneity in a cell population, as observed in cancer cells characterized by Cdk1 overexpression.

A mathematical model of the biochemical network underlying left-right asymmetry establishment in mammals

The expression of the TGF-β protein Nodal on the left side of vertebrate embryos is a determining event in the development of internal-organ asymmetry. We present a mathematical model for the control of the expression of Nodal and its antagonist Lefty consisting entirely of realistic elementary reactions. We analyze the model in the absence of Lefty and find a wide range of parameters over which bistability (two stable steady states) is observed, with one stable steady state a low-Nodal state corresponding to the right-hand developmental fate, and the other a high-Nodal state corresponding to the left. We find that bistability requires a transcription factor containing two molecules of phosphorylated Smad2. A numerical survey of the full model, including Lefty, shows the effects of Lefty on the potential for bistability, and on the conditions that lead to the system reaching one or the other steady state.

Self-sustaining positive feedback loops in discrete and continuous systems

We consider a dynamic framework frequently used to model gene regulatory and signal transduction networks: monotonic ODEs that are composed of Hill functions. We derive conditions under which activity or inactivity in one system variable induces and sustains activity or inactivity in another. Cycles of such influences correspond to positive feedback loops that are self-sustaining and control-robust, in the sense that these feedback loops "trap" the system in a region of state space from which it cannot exit, even if the other system variables are externally controlled. To demonstrate the utility of this result, we consider prototypical examples of bistability and hysteresis in gene regulatory networks, and analyze a T-cell signal transduction ODE model from the literature.

Dissipative structures in biological systems: bistability, oscillations, spatial patterns and waves

The goal of this review article is to assess how relevant is the concept of dissipative structure for understanding the dynamical bases of non-equilibrium self-organization in biological systems, and to see where it has been applied in the five decades since it was initially proposed by Ilya Prigogine. Dissipative structures can be classified into four types, which will be considered, in turn, and illustrated by biological examples: (i) multistability, in the form of bistability and tristability, which involve the coexistence of two or three stable steady states, or in the form of birhythmicity, which involves the coexistence between two stable rhythms; (ii) temporal dissipative structures in the form of sustained oscillations, illustrated by biological rhythms; (iii) spatial dissipative structures, known as Turing patterns; and (iv) spatio-temporal structures in the form of propagating waves. Rhythms occur with widely different periods at all levels of biological organization, from neural, cardiac and metabolic oscillations to circadian clocks and the cell cycle; they play key roles in physiology and in many disorders. New rhythms are being uncovered while artificial ones are produced by synthetic biology. Rhythms provide the richest source of examples of dissipative structures in biological systems. Bistability has been observed experimentally, but has primarily been investigated in theoretical models in an increasingly wide range of biological contexts, from the genetic to the cell and animal population levels, both in physiological conditions and in disease. Bistable transitions have been implicated in the progression between the different phases of the cell cycle and, more generally, in the process of cell fate specification in the developing embryo. Turing patterns are exemplified by the formation of some periodic structures in the course of development and by skin stripe patterns in animals. Spatio-temporal patterns in the form of propagating waves are observed within cells as well as in intercellular communication. This review illustrates how dissipative structures of all sorts abound in biological systems.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)'.

High-fidelity discrete modeling of the HPA axis: a study of regulatory plasticity in biology

BACKGROUND

The hypothalamic-pituitary-adrenal (HPA) axis is a central regulator of stress response and its dysfunction has been associated with a broad range of complex illnesses including Gulf War Illness (GWI) and Chronic Fatigue Syndrome (CFS). Though classical mathematical approaches have been used to model HPA function in isolation, its broad regulatory interactions with immune and central nervous function are such that the biological fidelity of simulations is undermined by the limited availability of reliable parameter estimates.

METHOD

Here we introduce and apply a generalized discrete formalism to recover multiple stable regulatory programs of the HPA axis using little more than connectivity between physiological components. This simple discrete model captures cyclic attractors such as the circadian rhythm by applying generic constraints to a minimal parameter set; this is distinct from Ordinary Differential Equation (ODE) models, which require broad and precise parameter sets. Parameter tuning is accomplished by decomposition of the overall regulatory network into isolated sub-networks that support cyclic attractors. Network behavior is simulated using a novel asynchronous updating scheme that enforces priority with memory within and between physiological compartments.

RESULTS

Consistent with much more complex conventional models of the HPA axis, this parsimonious framework supports two cyclic attractors, governed by higher and lower levels of cortisol respectively. Importantly, results suggest that stress may remodel the stability landscape of this system, favoring migration from one stable circadian cycle to the other. Access to each regime is dependent on HPA axis tone, captured here by the tunable parameters of the multi-valued logic. Likewise, an idealized glucocorticoid receptor blocker alters the regulatory topology such that maintenance of persistently low cortisol levels is rendered unstable, favoring a return to normal circadian oscillation in both cortisol and glucocorticoid receptor expression.

CONCLUSION

These results emphasize the significance of regulatory connectivity alone and how regulatory plasticity may be explored using simple discrete logic and minimal data compared to conventional methods.

Complexity-aware simple modeling

Mathematical models continue to be essential for deepening our understanding of biology. On one extreme, simple or small-scale models help delineate general biological principles. However, the parsimony of detail in these models as well as their assumption of modularity and insulation make them inaccurate for describing quantitative features. On the other extreme, large-scale and detailed models can quantitatively recapitulate a phenotype of interest, but have to rely on many unknown parameters, making them often difficult to parse mechanistically and to use for extracting general principles. We discuss some examples of a new approach-complexity-aware simple modeling-that can bridge the gap between the small-scale and large-scale approaches.

Identifying parameter regions for multistationarity

Mathematical modelling has become an established tool for studying the dynamics of biological systems. Current applications range from building models that reproduce quantitative data to identifying systems with predefined qualitative features, such as switching behaviour, bistability or oscillations. Mathematically, the latter question amounts to identifying parameter values associated with a given qualitative feature. We introduce a procedure to partition the parameter space of a parameterized system of ordinary differential equations into regions for which the system has a unique or multiple equilibria. The procedure is based on the computation of the Brouwer degree, and it creates a multivariate polynomial with parameter depending coefficients. The signs of the coefficients determine parameter regions with and without multistationarity. A particular strength of the procedure is the avoidance of numerical analysis and parameter sampling. The procedure consists of a number of steps. Each of these steps might be addressed algorithmically using various computer programs and available software, or manually. We demonstrate our procedure on several models of gene transcription and cell signalling, and show that in many cases we obtain a complete partitioning of the parameter space with respect to multistationarity.

Dissipative structures and biological rhythms

Sustained oscillations abound in biological systems. They occur at all levels of biological organization over a wide range of periods, from a fraction of a second to years, and with a variety of underlying mechanisms. They control major physiological functions, and their dysfunction is associated with a variety of physiological disorders. The goal of this review is (i) to give an overview of the main rhythms observed at the cellular and supracellular levels, (ii) to briefly describe how the study of biological rhythms unfolded in the course of time, in parallel with studies on chemical oscillations, (iii) to present the major roles of biological rhythms in the control of physiological functions, and (iv) the pathologies associated with the alteration, disappearance, or spurious occurrence of biological rhythms. Two tables present the main examples of cellular and supracellular rhythms ordered according to their period, and their role in physiology and pathophysiology. Among the rhythms discussed are neural and cardiac rhythms, metabolic oscillations such as those occurring in glycolysis in yeast, intracellular Ca⁺⁺ oscillations, cyclic AMP oscillations in Dictyostelium amoebae, the segmentation clock that controls somitogenesis, pulsatile hormone secretion, circadian rhythms which occur in all eukaryotes and some bacteria with a period close to 24 h, the oscillatory dynamics of the enzymatic network driving the cell cycle, and oscillations in transcription factors such as NF-ΚB and tumor suppressors such as p53. Ilya Prigogine's concept of dissipative structures applies to temporal oscillations and allows us to unify within a common framework the various rhythms observed at different levels of biological organization, regardless of their period and underlying mechanism.

Association of rs1801157 single nucleotide polymorphism of CXCL12 gene in breast cancer in Pakistan and in-silico expression analysis of CXCL12-CXCR4 associated biological regulatory network

Background

C-X-C chemokine ligand 12 (CXCL12) has important implications in breast cancer (BC) pathogenesis. It is selectively expressed on B and T lymphocytes and is involved in hematopoiesis, thymocyte trafficking, stem cell motility, neovascularization, and tumorigenesis. The single nucleotide polymorphism (SNP) rs1801157 of CXCL12 gene has been found to be associated with higher risk of BC.

Methods

Our study focuses on the genotypic and allelic distribution of SNP (rs1801157; G/A) in Pakistani population as well as its association with the clinico-pathological features. The association between rs1801157 genotypes (G/A) and BC risks was assessed by a multivariate logistic regression (MLR) analysis. Genotyping was performed in both healthy individuals and patients of BC using PCR-restriction fragment length polymorphism (PCR-RFLP) method. Furthermore, in-silico approaches were adapted to investigate the association of CXCL12 and its receptor CXCR4 with genes/proteins involved in BC signalling.

Results

Significant differences in allelic and genotypic distribution between BC patients and healthy individuals of genotype (G/G) and (A/G) (p < 0.05) were observed. The frequency of the allele G in the BC group (77%) was significantly higher as compared to control group (61%) (p = 0.01). The association of genotype GG with clinico-pathological features including age, stages of cancer and organ (lung, liver, bones and brain) metastasis (p > 0.05) was assessed. In a MLR analysis, a number of variables including age, weight of an individual, affected lymph nodes, hormonal status (estrogen and progesterone receptor), alcohol consumption and family history associated with the GG genotype (GG:AA, odds ratio (OR) = 1.30, 95% CI [1.06–1.60]) were found to be independent risk factors for BC. Our in-vitro results suggest that genotype GG is possibly increasing the risk of BC in Pakistani cohorts. in-silico analysis finds that CXCL12–CXCR4 is associated with an increased expression of PDZK1, PI3k and Akt which lead the breast tumor towards metastasis.

Conclusion

Multiple targets such as CXCL12, CXCR4, PDZK1, PI3k and Akt can be inhibited in combined strategies to treat BC metastasis.

Random positioning of nucleosomes enhances heritable bistability

Chromosomal regions are often dynamically modified by histones, leading to the uncertainty of nucleosome positions. Experiments have provided evidence for this randomness, but it is unclear how it impacts epigenetic heritability. Here, by analyzing a mechanic model at the molecular level, which considers three representative types of nucleosomes (unmodified, methylated, and acetylated) and dynamic nucleosome modifications, we find that in contrast to the equidistance partition of nucleosomes, random partition can significantly enhance heritable bistability. Moreover, the more "chaotic" the nucleosome positions are, the better the heritable bistability is, in contrast to the previous view. In both cases of nucleosome positioning, heritable bistability occurs only when the total nucleosome number is beyond a threshold, and it depends strongly on the allocation rate that enzymes regulate transitions between different nucleosome types. Thus, we conclude that random positioning of nucleosomes is an unneglectable factor impacting heritable bistability. A point worth mentioning is that our model established on a master equation can easily be extended to include other more complex processes underlying dynamic nucleosome modifications.

Expected Number of Fixed Points in Boolean Networks with Arbitrary Topology

Boolean network models describe genetic, neural, and social dynamics in complex networks, where the dynamics depend generally on network topology. Fixed points in a genetic regulatory network are typically considered to correspond to cell types in an organism. We prove that the expected number of fixed points in a Boolean network, with Boolean functions drawn from probability distributions that are not required to be uniform or identical, is one, and is independent of network topology if only a feedback arc set satisfies a stochastic neutrality condition. We also demonstrate that the expected number is increased by the predominance of positive feedback in a cycle.

Formal reasoning about systems biology using theorem proving

System biology provides the basis to understand the behavioral properties of complex biological organisms at different levels of abstraction. Traditionally, analysing systems biology based models of various diseases have been carried out by paper-and-pencil based proofs and simulations. However, these methods cannot provide an accurate analysis, which is a serious drawback for the safety-critical domain of human medicine. In order to overcome these limitations, we propose a framework to formally analyze biological networks and pathways. In particular, we formalize the notion of reaction kinetics in higher-order logic and formally verify some of the commonly used reaction based models of biological networks using the HOL Light theorem prover. Furthermore, we have ported our earlier formalization of Zsyntax, i.e., a deductive language for reasoning about biological networks and pathways, from HOL4 to the HOL Light theorem prover to make it compatible with the above-mentioned formalization of reaction kinetics. To illustrate the usefulness of the proposed framework, we present the formal analysis of three case studies, i.e., the pathway leading to TP53 Phosphorylation, the pathway leading to the death of cancer stem cells and the tumor growth based on cancer stem cells, which is used for the prognosis and future drug designs to treat cancer patients.

Dynamical predictors of an imminent phenotypic switch in bacteria

Single cells can stochastically switch across thresholds imposed by regulatory networks. Such thresholds can act as a tipping point, drastically changing global phenotypic states. In ecology and economics, imminent transitions across such tipping points can be predicted using dynamical early warning indicators. A typical example is 'flickering' of a fast variable, predicting a longer-lasting switch from a low to a high state or vice versa. Considering the different timescales between metabolite and protein fluctuations in bacteria, we hypothesized that metabolic early warning indicators predict imminent transitions across a network threshold caused by enzyme saturation. We used stochastic simulations to determine if flickering predicts phenotypic transitions, accounting for a variety of molecular physiological parameters, including enzyme affinity, burstiness of enzyme gene expression, homeostatic feedback, and rates of metabolic precursor influx. In most cases, we found that metabolic flickering rates are robustly peaked near the enzyme saturation threshold. The degree of fluctuation was amplified by product inhibition of the enzyme. We conclude that sensitivity to flickering in fast variables may be a possible natural or synthetic strategy to prepare physiological states for an imminent transition.

Functional architecture and global properties of the Corynebacterium glutamicum regulatory network: Novel insights from a dataset with a high genomic coverage

Corynebacterium glutamicum is a Gram-positive, anaerobic, rod-shaped soil bacterium able to grow on a diversity of carbon sources like sugars and organic acids. It is a biotechnological relevant organism because of its highly efficient ability to biosynthesize amino acids, such as l-glutamic acid and l-lysine. Here, we reconstructed the most complete C. glutamicum regulatory network to date and comprehensively analyzed its global organizational properties, systems-level features and functional architecture. Our analyses show the tremendous power of Abasy Atlas to study the functional organization of regulatory networks. We created two models of the C. glutamicum regulatory network: all-evidences (containing both weak and strong supported interactions, genomic coverage=73%) and strongly-supported (only accounting for strongly supported evidences, genomic coverage=71%). Using state-of-the-art methodologies, we prove that power-law behaviors truly govern the connectivity and clustering coefficient distributions. We found a non-previously reported circuit motif that we named complex feed-forward motif. We highlighted the importance of feedback loops for the functional architecture, beyond whether they are statistically over-represented or not in the network. We show that the previously reported top-down approach is inadequate to infer the hierarchy governing a regulatory network because feedback bridges different hierarchical layers, and the top-down approach disregards the presence of intermodular genes shaping the integration layer. Our findings all together further support a diamond-shaped, three-layered hierarchy exhibiting some feedback between processing and coordination layers, which is shaped by four classes of systems-level elements: global regulators, locally autonomous modules, basal machinery and intermodular genes.

Formal modeling and analysis of ER-α associated Biological Regulatory Network in breast cancer

BACKGROUND

Breast cancer (BC) is one of the leading cause of death among females worldwide. The increasing incidence of BC is due to various genetic and environmental changes which lead to the disruption of cellular signaling network(s). It is a complex disease in which several interlinking signaling cascades play a crucial role in establishing a complex regulatory network. The logical modeling approach of René Thomas has been applied to analyze the behavior of estrogen receptor-alpha (ER-α) associated Biological Regulatory Network (BRN) for a small part of complex events that leads to BC metastasis.

METHODS

A discrete model was constructed using the kinetic logic formalism and its set of logical parameters were obtained using the model checking technique implemented in the SMBioNet software which is consistent with biological observations. The discrete model was further enriched with continuous dynamics by converting it into an equivalent Petri Net (PN) to analyze the logical parameters of the involved entities.

RESULTS

In-silico based discrete and continuous modeling of ER-α associated signaling network involved in BC provides information about behaviors and gene-gene interaction in detail. The dynamics of discrete model revealed, imperative behaviors represented as cyclic paths and trajectories leading to pathogenic states such as metastasis. Results suggest that the increased expressions of receptors ER-α, IGF-1R and EGFR slow down the activity of tumor suppressor genes (TSGs) such as BRCA1, p53 and Mdm2 which can lead to metastasis. Therefore, IGF-1R and EGFR are considered as important inhibitory targets to control the metastasis in BC.

CONCLUSION

The in-silico approaches allow us to increase our understanding of the functional properties of living organisms. It opens new avenues of investigations of multiple inhibitory targets (ER-α, IGF-1R and EGFR) for wet lab experiments as well as provided valuable insights in the treatment of cancers such as BC.

A Qualitative Model of the Differentiation Network in Chondrocyte Maturation: A Holistic View of Chondrocyte Hypertrophy

Differentiation of chondrocytes towards hypertrophy is a natural process whose control is essential in endochondral bone formation. It is additionally thought to play a role in several pathophysiological processes, with osteoarthritis being a prominent example. We perform a dynamic analysis of a qualitative mathematical model of the regulatory network that directs this phenotypic switch to investigate the influence of the individual factors holistically. To estimate the stability of a SOX9 positive state (associated with resting/proliferation chondrocytes) versus a RUNX2 positive one (associated with hypertrophy) we employ two measures. The robustness of the state in canalisation (size of the attractor basin) is assessed by a Monte Carlo analysis and the sensitivity to perturbations is assessed by a perturbational analysis of the attractor. Through qualitative predictions, these measures allow for an in silico screening of the effect of the modelled factors on chondrocyte maintenance and hypertrophy. We show how discrepancies between experimental data and the model’s results can be resolved by evaluating the dynamic plausibility of alternative network topologies. The findings are further supported by a literature study of proposed therapeutic targets in the case of osteoarthritis.

A Method to Identify and Analyze Biological Programs through Automated Reasoning

Predictive biology is elusive because rigorous, data-constrained, mechanistic models of complex biological systems are difficult to derive and validate. Current approaches tend to construct and examine static interaction network models, which are descriptively rich, but often lack explanatory and predictive power, or dynamic models that can be simulated to reproduce known behavior. However, in such approaches implicit assumptions are introduced as typically only one mechanism is considered, and exhaustively investigating all scenarios is impractical using simulation. To address these limitations, we present a methodology based on automated formal reasoning, which permits the synthesis and analysis of the complete set of logical models consistent with experimental observations. We test hypotheses against all candidate models, and remove the need for simulation by characterizing and simultaneously analyzing all mechanistic explanations of observed behavior. Our methodology transforms knowledge of complex biological processes from sets of possible interactions and experimental observations to precise, predictive biological programs governing cell function.

Realization of tristability in a multiplicatively coupled dual-loop genetic network

Multistability is a crucial recurring theme in cell signaling. Multistability is attributed to the presence of positive feedback loops, but the general condition and essential mechanism for realizing multistability remain unclear. Here, we build a generic circuit model comprising two transcription factors and a microRNA, representing a kind of core architecture in gene regulatory networks. The circuit can be decomposed into two positive feedback loops (PFLs) or one PFL and one negative feedback loop (NFL), which are multiplicatively coupled. Bifurcation analyses of the model reveal that the circuit can achieve tristability through four kinds of bifurcation scenarios when parameter values are varied in a wide range. We formulate the general requirement for tristability in terms of logarithmic gain of the circuit. The parameter ranges for tristability and possible transition routes among steady states are determined by the combination of gain features of individual feedback loops. Coupling two PFLs with bistability or one NFL with a bistable PFL is most likely to generate tristability, but the underlying mechanisms are largely different. We also interpret published results and make testable predictions. This work sheds new light on interlinking feedback loops to realize tristability. The proposed theoretical framework can be of wide applicability.

Architecture of chaotic attractors for flows in the absence of any singular point

Some chaotic attractors produced by three-dimensional dynamical systems without any singular point have now been identified, but explaining how they are structured in the state space remains an open question. We here want to explain-in the particular case of the Wei system-such a structure, using one-dimensional sets obtained by vanishing two of the three derivatives of the flow. The neighborhoods of these sets are made of points which are characterized by the eigenvalues of a 2 × 2 matrix describing the stability of flow in a subspace transverse to it. We will show that the attractor is spiralling and twisted in the neighborhood of one-dimensional sets where points are characterized by a pair of complex conjugated eigenvalues. We then show that such one-dimensional sets are also useful in explaining the structure of attractors produced by systems with singular points, by considering the case of the Lorenz system.

Coherent organization in gene regulation: a study on six networks

Structural and dynamical fingerprints of evolutionary optimization in biological networks are still unclear. Here we analyze the dynamics of genetic regulatory networks responsible for the regulation of cell cycle and cell differentiation in three organisms or cell types each, and show that they follow a version of Hebb's rule which we have termed coherence. More precisely, we find that simultaneously expressed genes with a common target are less likely to act antagonistically at the attractors of the regulatory dynamics. We then investigate the dependence of coherence on structural parameters, such as the mean number of inputs per node and the activatory/repressory interaction ratio, as well as on dynamically determined quantities, such as the basin size and the number of expressed genes.

Noise Reduction in Complex Biological Switches

Cells operate in noisy molecular environments via complex regulatory networks. It is possible to understand how molecular counts are related to noise in specific networks, but it is not generally clear how noise relates to network complexity, because different levels of complexity also imply different overall number of molecules. For a fixed function, does increased network complexity reduce noise, beyond the mere increase of overall molecular counts? If so, complexity could provide an advantage counteracting the costs involved in maintaining larger networks. For that purpose, we investigate how noise affects multistable systems, where a small amount of noise could lead to very different outcomes; thus we turn to biochemical switches. Our method for comparing networks of different structure and complexity is to place them in conditions where they produce exactly the same deterministic function. We are then in a good position to compare their noise characteristics relatively to their identical deterministic traces. We show that more complex networks are better at coping with both intrinsic and extrinsic noise. Intrinsic noise tends to decrease with complexity, and extrinsic noise tends to have less impact. Our findings suggest a new role for increased complexity in biological networks, at parity of function.

Computational Modeling, Formal Analysis, and Tools for Systems Biology

As the amount of biological data in the public domain grows, so does the range of modeling and analysis techniques employed in systems biology. In recent years, a number of theoretical computer science developments have enabled modeling methodology to keep pace. The growing interest in systems biology in executable models and their analysis has necessitated the borrowing of terms and methods from computer science, such as formal analysis, model checking, static analysis, and runtime verification. Here, we discuss the most important and exciting computational methods and tools currently available to systems biologists. We believe that a deeper understanding of the concepts and theory highlighted in this review will produce better software practice, improved investigation of complex biological processes, and even new ideas and better feedback into computer science.

From Systems Understanding to Personalized Medicine: Lessons and Recommendations Based on a Multidisciplinary and Translational Analysis of COPD

Systems medicine, using and adapting methods and approaches as developed within systems biology, promises to be essential in ongoing efforts of realizing and implementing personalized medicine in clinical practice and research. Here we review and critically assess these opportunities and challenges using our work on COPD as a case study. We find that there are significant unresolved biomedical challenges in how to unravel complex multifactorial components in disease initiation and progression producing different clinical phenotypes. Yet, while such a systems understanding of COPD is necessary, there are other auxiliary challenges that need to be addressed in concert with a systems analysis of COPD. These include information and communication technology (ICT)-related issues such as data harmonization, systematic handling of knowledge, computational modeling, and importantly their translation and support of clinical practice. For example, clinical decision-support systems need a seamless integration with new models and knowledge as systems analysis of COPD continues to develop. Our experience with clinical implementation of systems medicine targeting COPD highlights the need for a change of management including design of appropriate business models and adoption of ICT providing and supporting organizational interoperability among professional teams across healthcare tiers, working around the patient. In conclusion, in our hands the scope and efforts of systems medicine need to concurrently consider these aspects of clinical implementation, which inherently drives the selection of the most relevant and urgent issues and methods that need further development in a systems analysis of disease.

Cooperative stabilization of the SIR complex provides robust epigenetic memory in a model of SIR silencing in Saccharomyces cerevisiae

How alternative chromatin-based regulatory states can be made stable and heritable in order to provide robust epigenetic memory is poorly understood. Here, we develop a stochastic model of the silencing system in Saccharomyces cerevisiae that incorporates cooperative binding of the repressive SIR complex and antisilencing histone modifications, in addition to positive feedback in Sir2 recruitment. The model was able to reproduce key features of SIR regulation of an HM locus, including heritable bistability, dependence on the silencer elements, and sensitivity to SIR dosage. We found that antisilencing methylation of H3K79 by Dot1 was not needed to generate these features, but acted to reduce spreading of SIR binding, consistent with its proposed role in containment of silencing. In contrast, cooperative inter-nucleosome interactions mediated by the SIR complex were critical for concentrating SIR binding around the silencers in the absence of barriers, and for providing bistability in SIR binding. SIR-SIR interactions magnify the cooperativity in the Sir2-histone deacetylation positive feedback reaction and complete a double-negative feedback circuit involving antisilencing modifications. Thus, our modeling underscores the potential importance of cooperative interactions between nucleosome-bound complexes both in the SIR system and in other chromatin-based complexes in epigenetic regulation.