Law of Localization in Chemical Reaction Networks

BayesianSSA: a Bayesian statistical model based on structural sensitivity analysis for predicting responses to enzyme perturbations in metabolic networks

BACKGROUND

Chemical bioproduction has attracted attention as a key technology in a decarbonized society. In computational design for chemical bioproduction, it is necessary to predict changes in metabolic fluxes when up-/down-regulating enzymatic reactions, that is, responses of the system to enzyme perturbations. Structural sensitivity analysis (SSA) was previously developed as a method to predict qualitative responses to enzyme perturbations on the basis of the structural information of the reaction network. However, the network structural information can sometimes be insufficient to predict qualitative responses unambiguously, which is a practical issue in bioproduction applications. To address this, in this study, we propose BayesianSSA, a Bayesian statistical model based on SSA. BayesianSSA extracts environmental information from perturbation datasets collected in environments of interest and integrates it into SSA predictions.

RESULTS

We applied BayesianSSA to synthetic and real datasets of the central metabolic pathway of Escherichia coli. Our result demonstrates that BayesianSSA can successfully integrate environmental information extracted from perturbation data into SSA predictions. In addition, the posterior distribution estimated by BayesianSSA can be associated with the known pathway reported to enhance succinate export flux in previous studies.

CONCLUSIONS

We believe that BayesianSSA will accelerate the chemical bioproduction process and contribute to advancements in the field.

Historic obstacles and emerging opportunities in the field of developmental metabolism - lessons from Heidelberg

The field of developmental metabolism is experiencing a technological revolution that is opening entirely new fields of inquiry. Advances in metabolomics, small-molecule sensors, single-cell RNA sequencing and computational modeling present new opportunities for exploring cell-specific and tissue-specific metabolic networks, interorgan metabolic communication, and gene-by-metabolite interactions in time and space. Together, these advances not only present a means by which developmental biologists can tackle questions that have challenged the field for centuries, but also present young scientists with opportunities to define new areas of inquiry. These emerging frontiers of developmental metabolism were at the center of a highly interactive 2023 EMBO workshop 'Developmental metabolism: flows of energy, matter, and information'. Here, we summarize key discussions from this forum, emphasizing modern developmental biology's challenges and opportunities.

Information geometric bound on general chemical reaction networks

We investigate the convergence of chemical reaction networks (CRNs), aiming to establish an upper bound on their reaction rates. The nonlinear characteristics and discrete composition of CRNs pose significant challenges in this endeavor. To circumvent these complexities, we adopt an information geometric perspective, utilizing the natural gradient to formulate a nonlinear system. This system effectively determines an upper bound for the dynamics of CRNs. We corroborate our methodology through numerical simulations, which reveal that our constructed system converges more rapidly than CRNs within a particular class of reactions. This class is defined by the count of chemicals, the highest stoichiometric coefficients in the reactions, and the total number of reactions involved. Further, we juxtapose our approach with traditional methods, illustrating that the latter falls short in providing an upper bound for CRN reaction rates. Although our investigation centers on CRNs, the widespread presence of hypergraphs across various disciplines, ranging from natural sciences to engineering, indicates potential wider applications of our method, including in the realm of information science.

Patterns of change in regulatory modules of chemical reaction systems induced by network modification

Cellular functions are realized through the dynamics of chemical reaction networks formed by thousands of chemical reactions. Numerical studies have empirically demonstrated that small differences in network structures among species or tissues can cause substantial changes in dynamics. However, a general principle for behavior changes in response to network structure modifications is not known. The chemical reaction system possesses substructures called buffering structures, which are characterized by a certain topological index being zero. It was proven that the steady-state response to modulation of reaction parameters inside a buffering structure is localized in the buffering structure. In this study, we developed a method to systematically identify the loss or creation of buffering structures induced by the addition of a single degradation reaction from network structure alone. This makes it possible to predict the qualitative and macroscopic changes in regulation that will be caused by the network modification. This method was applied to two reaction systems: the central metabolic system and the mitogen-activated protein kinases signal transduction system. Our method enables identification of reactions that are important for biological functions in living systems.

Localization of Noise in Biochemical Networks

Noise, or uncertainty in biochemical networks, has become an important aspect of many biological problems. Noise can arise and propagate from external factors and probabilistic chemical reactions occurring in small cellular compartments. For species survival, it is important to regulate such uncertainties in executing vital cell functions. Regulated noise can improve adaptability, whereas uncontrolled noise can cause diseases. Simulation can provide a detailed analysis of uncertainties, but parameters such as rate constants and initial conditions are usually unknown. A general understanding of noise dynamics from the perspective of network structure is highly desirable. In this study, we extended the previously developed law of localization for characterizing noise in terms of (co)variances and developed noise localization theory. With linear noise approximation, we can expand a biochemical network into an extended set of differential equations representing a fictitious network for pseudo-components consisting of variances and covariances, together with chemical species. Through localization analysis, perturbation responses at the steady state of pseudo-components can be summarized into a sensitivity matrix that only requires knowledge of network topology. Our work allows identification of buffering structures at the level of species, variances, and covariances and can provide insights into noise flow under non-steady-state conditions in the form of a pseudo-chemical reaction. We tested noise localization in various systems, and here we discuss its implications and potential applications. Results show that this theory is potentially applicable in discriminating models, scanning network topologies with interesting noise behavior, and designing and perturbing networks with the desired response.

Anticipating the occurrence and type of critical transitions

Critical transition can occur in many real-world systems. The ability to forecast the occurrence of transition is of major interest in a range of contexts. Various early warning signals (EWSs) have been developed to anticipate the coming critical transition or distinguish types of transition. However, no effective method allows to establish practical threshold indicating the condition when the critical transition is most likely to occur. Here, we introduce a powerful EWS, named dynamical eigenvalue (DEV), that is rooted in bifurcation theory of dynamical systems to estimate the dominant eigenvalue of the system. Theoretically, the absolute value of DEV approaches 1 when the system approaches bifurcation, while its position in the complex plane indicates the type of transition. We demonstrate the efficacy of the DEV approach in model systems with known bifurcation types and also test the DEV approach on various critical transitions in real-world systems.

A structural approach to understanding enzymatic regulation of chemical reaction networks

In living cells, chemical reactions are connected by sharing their products and substrates, and form complex systems, i.e. chemical reaction network. One of the largest missions in modern biology is to understand behaviors of such systems logically based on information of network structures. However, there are series of obstacles to study dynamical behaviors of complex network systems in biology. For example, network structure does not provide sufficient information to determine details of the dynamical behaviors. In this review, I will introduce a novel mathematical theory, structural sensitivity analysis, by which the responses of reaction systems upon the changes in enzyme activities/amounts are determined from network structure alone. The patterns of responses exhibit characteristic features, localization and hierarchy, depending on the topology of the network. The theory also shows that ranges of enzymatic regulations are governed by a mathematical law characterized by local topology of substructures. These findings imply that the network topology is one of the origins of biological robustness.

BioSANS: A software package for symbolic and numeric biological simulation

Modeling biochemical systems can provide insights into behaviors that are difficult to observe or understand. It requires software, programming, and understanding of the system to build a model and study it. Softwares exist for systems biology modeling, but most support only certain types of modeling tasks. Desirable features including ease in preparing input, symbolic or analytical computation, parameter estimation, graphical user interface, and systems biology markup language (SBML) support are not seen concurrently in one software package. In this study, we developed a python-based software that supports these features, with both deterministic and stochastic propagations. The software can be used by graphical user interface, command line, or as a python import. We also developed a semi-programmable and intuitively easy topology input method for the biochemical reactions. We tested the software with semantic and stochastic SBML test cases. Tests on symbolic solution and parameter estimation were also included. The software we developed is reliable, well performing, convenient to use, and compliant with most of the SBML tests. So far it is the only systems biology software that supports symbolic, deterministic, and stochastic modeling in one package that also features parameter estimation and SBML support. This work offers a comprehensive set of tools and allows for better availability and accessibility for studying kinetics and dynamics in biochemical systems.

Flux-augmented bifurcation analysis in chemical reaction network systems

The dynamics of biochemical reaction networks are considered to be responsible for biological functions in living systems. Since real networks are immense and complicated, it is difficult to determine which reactions can cause a significant change of dynamical behaviors, namely, bifurcations. Also to what extent numerical results of network systems depend on the chosen kinetic rate parameters is not known. In this paper, an analytical setting that splits the information of the dynamics into the network structure and reaction kinetics is introduced. This setting possesses a factorization structure for some class of network systems which allows one to determine which subnetworks are responsible for the occurrence of a bifurcation. Subsequently, the bifurcation criteria are reformulated in a manner that allows the efficient determination of relevant reactions for bifurcations.

Field theory of reaction-diffusion: Law of mass action with an energetic variational approach

We extend the energetic variational approach so it can be applied to a chemical reaction system with general mass action kinetics. Our approach starts with an energy-dissipation law. We show that the chemical equilibrium is determined by the choice of the free energy and the dynamics of the chemical reaction is determined by the choice of the dissipation. This approach enables us to couple chemical reactions with other effects, such as diffusion and drift in an electric field. As an illustration, we apply our approach to a nonequilibrium reaction-diffusion system in a specific but canonical setup. We show by numerical simulations that the input-output relation of such a system depends on the choice of the dissipation.

Pyrophosphate inhibits gluconeogenesis by restricting UDP-glucose formation in vivo

Pyrophosphate (PPi) is produced by anabolic reactions and serves as an energy donor in the cytosol of plant cells; however, its accumulation to toxic levels disrupts several common biosynthetic pathways and is lethal. Before acquiring photosynthetic capacity, young seedlings must endure a short but critical heterotrophic period, during which they are nourished solely by sugar produced from seed reserves by the anabolic process of gluconeogenesis. Previously, we reported that excess PPi in H⁺-PPase-knockout fugu5 mutants of Arabidopsis thaliana severely compromised gluconeogenesis. However, the precise metabolic target of PPi inhibition in vivo remained elusive. Here, CE-TOF MS analyses of major metabolites characteristic of gluconeogenesis from seed lipids showed that the Glc6P;Fru6P level significantly increased and that Glc1P level was consistently somewhat higher in fugu5 compared to wild type. In contrast, the UDP-Glc level decreased significantly in the mutants. Importantly, specific removal of PPi in fugu5, and thus in AVP1_pro:IPP1 transgenic lines, restored the Glc1P and the Glc6P;Fru6P levels, increased the UDP-Glc level ~2.0-fold, and subsequently increased sucrose synthesis. Given the reversible nature of the Glc1P/UDP-Glc reaction, our results indicate that UGP-Glc pyrophosphorylase is the major target when excess PPi exerts inhibitory effects in vivo. To validate our findings, we analyzed metabolite responses using a mathematical theory called structural sensitivity analysis (SSA), in which the responses of concentrations in reaction systems to perturbations in enzyme activity are determined from the structure of the network alone. A comparison of our experimental data with the results of pure structural theory predicted the existence of unknown reactions as the necessary condition for the above metabolic profiles, and confirmed the above results. Our data support the notion that H⁺-PPase plays a pivotal role in cytosolic PPi homeostasis in plant cells. We propose that the combination of metabolomics and SSA is powerful when seeking to identify and predict metabolic targets in living cells.

Structural bifurcation analysis in chemical reaction networks

In living cells, chemical reactions form complex networks. Dynamics arising from such networks are the origins of biological functions. We propose a mathematical method to analyze bifurcation behaviors of network systems using their structures alone. Specifically, a whole network is decomposed into subnetworks, and for each of them the bifurcation condition can be studied independently. Further, parameters inducing bifurcations and chemicals exhibiting bifurcations can be determined on the network. We illustrate our theory using hypothetical and real networks.

Sensitivity and network topology in chemical reaction systems

In living cells, biochemical reactions are catalyzed by specific enzymes and connect to one another by sharing substrates and products, forming complex networks. In our previous studies, we established a framework determining the responses to enzyme perturbations only from network topology, and then proved a theorem, called the law of localization, explaining response patterns in terms of network topology. In this paper, we generalize these results to reaction networks with conserved concentrations, which allows us to study any reaction system. We also propose network characteristics quantifying robustness. We compare E. coli metabolic network with randomly rewired networks, and find that the robustness of the E. coli network is significantly higher than that of the random networks.