Design principles for robust biochemical reaction networks: what works, what cannot work, and what might almost work

Comparative analysis of kinetic realizations of insulin signaling

Several studies have developed dynamical models to understand the underlying mechanisms of insulin signaling, a signaling cascade that leads to the translocation of glucose, the human body's main source of energy. Fortunately, reaction network analysis allows us to extract properties of dynamical systems without depending on their model parameter values. This study focuses on the comparison of insulin signaling in healthy state (INSMS or INSulin Metabolic Signaling) and in type 2 diabetes (INRES or INsulin RESistance) using reaction network analysis. The analysis uses network decomposition to identify the different subsystems involved in insulin signaling (e.g., insulin receptor binding and recycling, GLUT4 translocation, and ERK signaling pathway, among others). Furthermore, results show that INSMS and INRES are similar with respect to some network, structo-kinetic, and kinetic properties. Their differences, however, provide insights into what happens when insulin resistance occurs. First, the variation in the number of species involved in INSMS and INRES suggests that when irregularities occur in the insulin signaling pathway, other complexes (and, hence, other processes) get involved, characterizing insulin resistance. Second, the loss of concordance exhibited by INRES suggests less restrictive interplay between the species involved in insulin signaling, leading to unusual activities in the signaling cascade. Lastly, GLUT4 losing its absolute concentration robustness in INRES may signify that the transporter has lost its reliability in shuttling glucose to the cell, inhibiting efficient cellular energy production. This study also suggests possible applications of the equilibria parametrization and network decomposition, resulting from the analysis, to potentially establish absolute concentration robustness in a species.

The effective deficiency of biochemical networks

The deficiency of a (bio)chemical reaction network can be conceptually interpreted as a measure of its ability to support exotic dynamical behavior and/or multistationarity. The classical definition of deficiency relates to the capacity of a network to permit variations of the complex formation rate vector at steady state, irrespective of the network kinetics. However, the deficiency is by definition completely insensitive to the fine details of the directionality of reactions as well as bounds on reaction fluxes. While the classical definition of deficiency can be readily applied in the analysis of unconstrained, weakly reversible networks, it only provides an upper bound in the cases where relevant constraints on reaction fluxes are imposed. Here we propose the concept of effective deficiency, which provides a more accurate assessment of the network's capacity to permit steady state variations at the complex level for constrained networks of any reversibility patterns. The effective deficiency relies on the concept of nonstoichiometric balanced complexes, which we have already shown to be present in real-world biochemical networks operating under flux constraints. Our results demonstrate that the effective deficiency of real-world biochemical networks is smaller than the classical deficiency, indicating the effects of reaction directionality and flux bounds on the variation of the complex formation rate vector at steady state.

A framework for deriving analytic steady states of biochemical reaction networks

The long-term behaviors of biochemical systems are often described by their steady states. Deriving these states directly for complex networks arising from real-world applications, however, is often challenging. Recent work has consequently focused on network-based approaches. Specifically, biochemical reaction networks are transformed into weakly reversible and deficiency zero generalized networks, which allows the derivation of their analytic steady states. Identifying this transformation, however, can be challenging for large and complex networks. In this paper, we address this difficulty by breaking the complex network into smaller independent subnetworks and then transforming the subnetworks to derive the analytic steady states of each subnetwork. We show that stitching these solutions together leads to the the analytic steady states of the original network. To facilitate this process, we develop a user-friendly and publicly available package, COMPILES (COMPutIng anaLytic stEady States). With COMPILES, we can easily test the presence of bistability of a CRISPRi toggle switch model, which was previously investigated via tremendous number of numerical simulations and within a limited range of parameters. Furthermore, COMPILES can be used to identify absolute concentration robustness (ACR), the property of a system that maintains the concentration of particular species at a steady state regardless of any initial concentrations. Specifically, our approach completely identifies all the species with and without ACR in a complex insulin model. Our method provides an effective approach to analyzing and understanding complex biochemical systems.

The unraveling of balanced complexes in metabolic networks

Balanced complexes in biochemical networks are at core of several theoretical and computational approaches that make statements about the properties of the steady states supported by the network. Recent computational approaches have employed balanced complexes to reduce metabolic networks, while ensuring preservation of particular steady-state properties; however, the underlying factors leading to the formation of balanced complexes have not been studied, yet. Here, we present a number of factorizations providing insights in mechanisms that lead to the origins of the corresponding balanced complexes. The proposed factorizations enable us to categorize balanced complexes into four distinct classes, each with specific origins and characteristics. They also provide the means to efficiently determine if a balanced complex in large-scale networks belongs to a particular class from the categorization. The results are obtained under very general conditions and irrespective of the network kinetics, rendering them broadly applicable across variety of network models. Application of the categorization shows that all classes of balanced complexes are present in large-scale metabolic models across all kingdoms of life, therefore paving the way to study their relevance with respect to different properties of steady states supported by these networks.

Network regulation meets substrate modification chemistry

Biochemical networks are at the heart of cellular information processing. These networks contain distinct facets: (i) processing of information from the environment via cascades/pathways along with network regulation and (ii) modification of substrates in different ways, to confer protein functionality, stability and processing. While many studies focus on these factors individually, how they interact and the consequences for cellular systems behaviour are poorly understood. We develop a systems framework for this purpose by examining the interplay of network regulation (canonical feedback and feed-forward circuits) and multisite modification, as an exemplar of substrate modification. Using computational, analytical and semi-analytical approaches, we reveal distinct and unexpected ways in which the substrate modification and network levels combine and the emergent behaviour arising therefrom. This has important consequences for dissecting the behaviour of specific signalling networks, tracing the origins of systems behaviour, inference of networks from data, robustness/evolvability and multi-level engineering of biomolecular networks. Overall, we repeatedly demonstrate how focusing on only one level (say network regulation) can lead to profoundly misleading conclusions about all these aspects, and reveal a number of important consequences for experimental/theoretical/data-driven interrogations of cellular signalling systems.

Foundations of static and dynamic absolute concentration robustness

Absolute Concentration Robustness (ACR) was introduced by Shinar and Feinberg (Science 327:1389-1391, 2010) as robustness of equilibrium species concentration in a mass action dynamical system. Their aim was to devise a mathematical condition that will ensure robustness in the function of the biological system being modeled. The robustness of function rests on what we refer to as empirical robustness-the concentration of a species remains unvarying, when measured in the long run, across arbitrary initial conditions. Even simple examples show that the ACR notion introduced in Shinar and Feinberg (Science 327:1389-1391, 2010) (here referred to as static ACR) is neither necessary nor sufficient for empirical robustness. To make a stronger connection with empirical robustness, we define dynamic ACR, a property related to long-term, global dynamics, rather than only to equilibrium behavior. We discuss general dynamical systems with dynamic ACR properties as well as parametrized families of dynamical systems related to reaction networks. We find necessary and sufficient conditions for dynamic ACR in complex balanced reaction networks, a class of networks that is central to the theory of reaction networks.

Reaction Network Analysis of Metabolic Insulin Signaling

Absolute concentration robustness (ACR) and concordance are novel concepts in the theory of robustness and stability within Chemical Reaction Network Theory. In this paper, we have extended Shinar and Feinberg's reaction network analysis approach to the insulin signaling system based on recent advances in decomposing reaction networks. We have shown that the network with 20 species, 35 complexes, and 35 reactions is concordant, implying at most one positive equilibrium in each of its stoichiometric compatibility class. We have obtained the system's finest independent decomposition consisting of 10 subnetworks, a coarsening of which reveals three subnetworks which are not only functionally but also structurally important. Utilizing the network's deficiency-oriented coarsening, we have developed a method to determine positive equilibria for the entire network. Our analysis has also shown that the system has ACR in 8 species all coming from a deficiency zero subnetwork. Interestingly, we have shown that, for a set of rate constants, the insulin-regulated glucose transporter GLUT4 (important in glucose energy metabolism), has stable ACR.

Maximization of information transmission influences selection of native phosphorelay architectures

Phosphorelays are signal transduction circuits that sense environmental changes and adjust cellular metabolism. Five different circuit architectures account for 99% of all phosphorelay operons annotated in over 9,000 fully sequenced genomes. Here we asked what biological design principles, if any, could explain selection among those architectures in nature. We began by studying kinetically well characterized phosphorelays (Spo0 of Bacillus subtilis and Sln1 of Saccharomyces cerevisiae). We find that natural circuit architecture maximizes information transmission in both cases. We use mathematical models to compare information transmission among the architectures for a realistic range of concentration and parameter values. Mapping experimentally determined phosphorelay protein concentrations onto that range reveals that the native architecture maximizes information transmission in sixteen out of seventeen analyzed phosphorelays. These results suggest that maximization of information transmission is important in the selection of native phosphorelay architectures, parameter values and protein concentrations.

Symmetry breaking meets multisite modification

Multisite modification is a basic way of conferring functionality to proteins and a key component of post-translational modification networks. Additional interest in multisite modification stems from its capability of acting as complex information processors. In this paper, we connect two seemingly disparate themes: symmetry and multisite modification. We examine different classes of random modification networks of substrates involving separate or common enzymes. We demonstrate that under different instances of symmetry of the modification network (invoked explicitly or implicitly and discussed in the literature), the biochemistry of multisite modification can lead to the symmetry being broken. This is shown computationally and consolidated analytically, revealing parameter regions where this can (and in fact does) happen, and characteristics of the symmetry-broken state. We discuss the relevance of these results in situations where exact symmetry is not present. Overall, through our study we show how symmetry breaking (i) can confer new capabilities to protein networks, including concentration robustness of different combinations of species (in conjunction with multiple steady states); (ii) could have been the basis for ordering of multisite modification, which is widely observed in cells; (iii) can significantly impact information processing in multisite modification and in cell signalling networks/pathways where multisite modification is present; and (iv) can be a fruitful new angle for engineering in synthetic biology and chemistry. All in all, the emerging conceptual synthesis provides a new vantage point for the elucidation and the engineering of molecular systems at the junction of chemical and biological systems.

Proteins help our cells perform the chemical reactions necessary for life. Once proteins are made, they can also be modified in different ways. This can simply change their activity, or otherwise make them better suited for their specific jobs within the cell. Biological ‘catalysts’ called enzymes carry out protein modifications by reversibly adding (or removing) chemical groups, such as phosphate groups.

‘Multisite modifications’ occur when a protein has two or more modifications in different areas, which can be added randomly or in a specific sequence. The combination of all the modifications attached to a protein acts like a chemical barcode and confers a specific function to the protein.

Modification networks add levels of complexity above individual proteins. These encompass not only the proteins in a cell or tissue, but also the different enzymes that can modify them, and how they all interact with each other. Although our knowledge of these networks is substantial, basic aspects, such as how the ordering of multisite modification systems emerges, is still not well understood.

Using a simple set of multisite modifications, Ramesh and Krishnan set out to study the potential mechanisms allowing the creation of order in this context. Symmetry is a pervasive theme across the sciences. In biology, symmetry and how it may be broken, is important to understand, for example, how organism develop. Ramesh and Krishnan used the perspective of symmetry in protein networks to uncover the origins of ordering.

First, mathematical models of simple modification networks were created based on their basic descriptions. This system centred on proteins that could have phosphate modifications at two possible sites. The network was ‘symmetric’, meaning that the rate of different sets of chemical reactions was identical, as were the amounts of all the enzymes involved.

Dissecting the simulated network using a variety of mathematical approaches showed that its initial symmetry could break, giving rise to sets of ordered multisite modifications. Breaking symmetry did not require any additional features or factors; the basic chemical ‘ingredients’ of protein modification were all that was needed. The prism of symmetry also revealed other aspects of these multisite modification networks, such as robustness and oscillations.

This study sheds new light on the mechanism behind ordering of protein modifications. In the future, Ramesh and Krishnan hope that this approach can be applied to the study of not just proteins but also a wider range of biochemical networks.

Energy-Based Modeling of the Feedback Control of Biomolecular Systems With Cyclic Flow Modulation

Energy-based modelling brings engineering insight to the understanding of biomolecular systems. It is shown how well-established control engineering concepts, such as loop-gain, arise from energy feedback loops and are therefore amenable to control engineering insight. In particular, a novel method is introduced to allow the transfer function based approach of classical linear control to be utilised in the analysis of feedback systems modelled by network thermodynamics and thus amalgamate energy-based modelling with control systems analysis. The approach is illustrated using a class of metabolic cycles with activation and inhibition leading to the concept of Cyclic Flow Modulation.

A network property necessary for concentration robustness

Maintenance of functionality of complex cellular networks and entire organisms exposed to environmental perturbations often depends on concentration robustness of the underlying components. Yet, the reasons and consequences of concentration robustness in large-scale cellular networks remain largely unknown. Here, we derive a necessary condition for concentration robustness based only on the structure of networks endowed with mass action kinetics. The structural condition can be used to design targeted experiments to study concentration robustness. We show that metabolites satisfying the necessary condition are present in metabolic networks from diverse species, suggesting prevalence of this property across kingdoms of life. We also demonstrate that our predictions about concentration robustness of energy-related metabolites are in line with experimental evidence from Escherichia coli. The necessary condition is applicable to mass action biological systems of arbitrary size, and will enable understanding the implications of concentration robustness in genetic engineering strategies and medical applications.

Absolute concentration robustness (ACR), independence of the steady-state concentration of a molecule from the environment, is difficult to predict. Here, the authors derive a network structure-based necessary condition for ACR, and suggest that metabolites satisfying the condition are prevalent.

Can biological complexity be reverse engineered?

Concerns with the use of engineering approaches in biology have recently been raised. I examine two related challenges to biological research that I call the synchronic and diachronic underdetermination problem. The former refers to challenges associated with the inference of design principles underlying system capacities when the synchronic relations between lower-level processes and higher-level systems capacities are degenerate (many-to-many). The diachronic underdetermination problem regards the problem of reverse engineering a system where the non-linear relations between system capacities and lower-level mechanisms are changing over time. Braun and Marom argue that recent insights to biological complexity leave the aim of reverse engineering hopeless - in principle as well as in practice. While I support their call for systemic approaches to capture the dynamic nature of living systems, I take issue with the conflation of reverse engineering with naïve reductionism. I clarify how the notion of design principles can be more broadly conceived and argue that reverse engineering is compatible with a dynamic view of organisms. It may even help to facilitate an integrated account that bridges the gap between mechanistic and systems approaches.

Invariants reveal multiple forms of robustness in bifunctional enzyme systems

Experimental and theoretical studies have suggested that bifunctional enzymes catalyzing opposing modification and demodification reactions can confer steady-state concentration robustness to their substrates. However, the types of robustness and the biochemical basis for them have remained elusive. Here we report a systematic study of the most general biochemical reaction network for a bifunctional enzyme acting on a substrate with one modification site, along with eleven sub-networks with more specialized biochemical assumptions. We exploit ideas from computational algebraic geometry, introduced in previous work, to find a polynomial expression (an invariant) between the steady state concentrations of the modified and unmodified substrate for each network. We use these invariants to identify five classes of robust behavior: robust upper bounds on concentration, robust two-sided bounds on concentration ratio, hybrid robustness, absolute concentration robustness (ACR), and robust concentration ratio. This analysis demonstrates that robustness can take a variety of forms and that the type of robustness is sensitive to many biochemical details, with small changes in biochemistry leading to very different steady-state behaviors. In particular, we find that the widely-studied ACR requires highly specialized assumptions in addition to bifunctionality. An unexpected result is that the robust bounds derived from invariants are strictly tighter than those derived by ad hoc manipulation of the underlying differential equations, confirming the value of invariants as a tool to gain insight into biochemical reaction networks. Furthermore, invariants yield multiple experimentally testable predictions and illuminate new strategies for inferring enzymatic mechanisms from steady-state measurements.

Understanding bistability in yeast glycolysis using general properties of metabolic pathways

Glycolysis is the central pathway in energy metabolism in the majority of organisms. In a recent paper, van Heerden et al. showed experimentally and computationally that glycolysis can exist in two states, a global steady state and a so-called imbalanced state. In the imbalanced state, intermediary metabolites accumulate at low levels of ATP and inorganic phosphate. It was shown that Baker's yeast uses a peculiar regulatory mechanism--via trehalose metabolism--to ensure that most yeast cells reach the steady state and not the imbalanced state.

RESULTS

Here we explore the apparent bistable behaviour in a core model of glycolysis that is based on a well-established detailed model, and study in great detail the bifurcation behaviour of solutions, without using any numerical information on parameter values.

CONCLUSION

We uncover a rich suite of solutions, including so-called imbalanced states, bistability, and oscillatory behaviour. The techniques employed are generic, directly suitable for a wide class of biochemical pathways, and could lead to better analytical treatments of more detailed models.

Reverse engineering and identification in systems biology: strategies, perspectives and challenges

The interplay of mathematical modelling with experiments is one of the central elements in systems biology. The aim of reverse engineering is to infer, analyse and understand, through this interplay, the functional and regulatory mechanisms of biological systems. Reverse engineering is not exclusive of systems biology and has been studied in different areas, such as inverse problem theory, machine learning, nonlinear physics, (bio)chemical kinetics, control theory and optimization, among others. However, it seems that many of these areas have been relatively closed to outsiders. In this contribution, we aim to compare and highlight the different perspectives and contributions from these fields, with emphasis on two key questions: (i) why are reverse engineering problems so hard to solve, and (ii) what methods are available for the particular problems arising from systems biology?

Dual regulation of gene expression mediated by extended MAPK activation and salicylic acid contributes to robust innate immunity in Arabidopsis thaliana

Network robustness is a crucial property of the plant immune signaling network because pathogens are under a strong selection pressure to perturb plant network components to dampen plant immune responses. Nevertheless, modulation of network robustness is an area of network biology that has rarely been explored. While two modes of plant immunity, Effector-Triggered Immunity (ETI) and Pattern-Triggered Immunity (PTI), extensively share signaling machinery, the network output is much more robust against perturbations during ETI than PTI, suggesting modulation of network robustness. Here, we report a molecular mechanism underlying the modulation of the network robustness in Arabidopsis thaliana. The salicylic acid (SA) signaling sector regulates a major portion of the plant immune response and is important in immunity against biotrophic and hemibiotrophic pathogens. In Arabidopsis, SA signaling was required for the proper regulation of the vast majority of SA-responsive genes during PTI. However, during ETI, regulation of most SA-responsive genes, including the canonical SA marker gene PR1, could be controlled by SA-independent mechanisms as well as by SA. The activation of the two immune-related MAPKs, MPK3 and MPK6, persisted for several hours during ETI but less than one hour during PTI. Sustained MAPK activation was sufficient to confer SA-independent regulation of most SA-responsive genes. Furthermore, the MPK3 and SA signaling sectors were compensatory to each other for inhibition of bacterial growth as well as for PR1 expression during ETI. These results indicate that the duration of the MAPK activation is a critical determinant for modulation of robustness of the immune signaling network. Our findings with the plant immune signaling network imply that the robustness level of a biological network can be modulated by the activities of network components.

Robustness of a network is defined by how consistently it performs upon removal of some of its components. It is a common strategy for plant pathogens to attack components of the plant immune signaling network in an attempt to dampen plant immunity. Therefore, it is crucial for the plant immune signaling network to have a high level of robustness. We previously reported that the robustness level of the plant immune signaling network is higher during Effector-Triggered Immunity (ETI) than Pattern-Triggered Immunity (PTI). Here we discovered a molecular switch that determines two robustness levels during ETI and PTI. Salicylic acid (SA) is a major plant immune signal molecule that regulates many immune-related genes. SA-independent alternative mechanisms also regulated the majority of SA-responsive genes during ETI but not PTI. One of the SA-independent mechanisms was mediated by prolonged activation of MAP kinases (MAPKs). MAPK activation was prolonged during ETI but transient during PTI. Thus, the duration of MAPK activation switches the robustness level of the plant immune signaling network. Our findings imply that the robustness level of a biological network can be modulated by activities of its components.

Concordant chemical reaction networks and the Species-Reaction Graph

In a recent paper it was shown that, for chemical reaction networks possessing a subtle structural property called concordance, dynamical behavior of a very circumscribed (and largely stable) kind is enforced, so long as the kinetics lies within the very broad and natural weakly monotonic class. In particular, multiple equilibria are precluded, as are degenerate positive equilibria. Moreover, under certain circumstances, also related to concordance, all real eigenvalues associated with a positive equilibrium are negative. Although concordance of a reaction network can be decided by readily available computational means, we show here that, when a nondegenerate network's Species-Reaction Graph satisfies certain mild conditions, concordance and its dynamical consequences are ensured. These conditions are weaker than earlier ones invoked to establish kinetic system injectivity, which, in turn, is just one ramification of network concordance. Because the Species-Reaction Graph resembles pathway depictions often drawn by biochemists, results here expand the possibility of inferring significant dynamical information directly from standard biochemical reaction diagrams.

Concordant chemical reaction networks

We describe a large class of chemical reaction networks, those endowed with a subtle structural property called concordance. We show that the class of concordant networks coincides precisely with the class of networks which, when taken with any weakly monotonic kinetics, invariably give rise to kinetic systems that are injective - a quality that, among other things, precludes the possibility of switch-like transitions between distinct positive steady states. We also provide persistence characteristics of concordant networks, instability implications of discordance, and consequences of stronger variants of concordance. Some of our results are in the spirit of recent ones by Banaji and Craciun, but here we do not require that every species suffer a degradation reaction. This is especially important in studying biochemical networks, for which it is rare to have all species degrade.

Dimerization and bifunctionality confer robustness to the isocitrate dehydrogenase regulatory system in Escherichia coli

An important goal of systems biology is to develop quantitative models that explain how specific molecular features give rise to systems-level properties. Metabolic and regulatory pathways that contain multifunctional proteins are especially interesting to study from this perspective because they have frequently been observed to exhibit robustness: the ability for a system to perform its proper function even as levels of its components change. In this study, we use extensive biochemical data and algebraic modeling to develop and analyze a model that shows how robust behavior arises in the isocitrate dehydrogenase (IDH) regulatory system of Escherichia coli, which was shown in 1985 to experimentally exhibit robustness. E. coli IDH is regulated by reversible phosphorylation catalyzed by the bifunctional isocitrate dehydrogenase kinase/phosphatase (IDHKP), and the level of IDH activity determines whether carbon flux is directed through the glyoxylate bypass (for growth on two-carbon substrates) or the full tricarboxylic acid cycle. Our model, which incorporates recent structural data on IDHKP, identifies several specific biochemical features of the system (including homodimerization of IDH and bifunctionality of IDHKP) that provide a potential explanation for robustness. Using algebraic techniques, we derive an invariant that summarizes the steady-state relationship between the phospho-forms of IDH. We use the invariant in combination with kinetic data on IDHKP to calculate IDH activity at a range of total IDH levels and find that our model predicts robustness. Our work unifies much of the known biochemistry of the IDH regulatory system into a single quantitative framework and highlights the importance of constructing biochemically realistic models in systems biology.

Characterization of the reconstituted UTase/UR-PII-NRII-NRI bicyclic signal transduction system that controls the transcription of nitrogen-regulated (Ntr) genes in Escherichia coli

A reconstituted UTase/UR-PII-NRII-NRI bicyclic cascade regulated PII uridylylation and NRI phosphorylation in response to glutamine. We examined the sensitivity and robustness of the responses of the individual cycles and of the bicyclic system. The sensitivity of the glutamine response of the upstream UTase/UR-PII monocycle depended upon the PII concentration, and we show that PII exerted substrate inhibition of the UTase activity of UTase/UR, potentially contributing to this dependence of sensitivity on PII. In the downstream NRII-NRI monocycle, PII controlled NRI phosphorylation state, and the response to PII was hyperbolic at both saturating and unsaturating NRI concentration. As expected from theory, the level of NRI∼P produced by the NRII-NRI monocycle was robust to changes in the NRII or NRI concentrations when NRI was in excess over NRII, as long as the NRII concentration was above a threshold value, an example of absolute concentration robustness (ACR). Because of the parameters of the system, at physiological protein levels and ratios of NRI to NRII, the level of NRI∼P depended upon both protein concentrations. In bicyclic UTase/UR-PII-NRII-NRI systems, the NRI phosphorylation state response to glutamine was always hyperbolic, regardless of the PII concentration or sensitivity of the upstream UTase/UR-PII cycle. In these bicyclic systems, NRI phosphorylation state was only robust to variation in the PII/NRII ratio within a narrow range; when PII was in excess NRI∼P was low, and when NRII was in excess NRI phosphorylation was elevated, throughout the physiological range of glutamine concentrations. Our results show that the bicyclic system produced a graded response of NRI phosphorylation to glutamine under a range of conditions, and that under most conditions the response of NRI phosphorylation state to glutamine levels depended on the concentrations of NRI, NRII, and PII.