arXiv subjects

Eduardo R. Mendoza

Publications and source records attributed to Eduardo R. Mendoza.

At least 19 recordsLinked to original sources

arXiv2026

The finest decompositions' architecture of a reaction network

Biochemical and environmental modeling typically relies on reaction networks to represent complex transformations. While the Linkage Class Decomposition (LCD) partitions networks based on visual standard connectivity, it often misaligns with the algebraic properties governing long-term dynamics. This work establishes the Finest Decompositions' Architecture (FDA) framework by analyzing hierarchical relationships between the LCD and two algebraic structures: the Finest Independent Decomposition (FID) and the Finest Incidence-Independent Decomposition (FIID). These algebraic decompositions serve as the respective building blocks for characterizing general equilibria and complex-balanced equilibria of a reaction network. Under the partial order of "coarsens to," we categorize reaction networks into six architectures, distinguishing three subclasses of Independent Linkage Classes (ILC) from three subclasses of Dependent Linkage Classes (DLC). To facilitate the classification, we introduce the Deficiency Difference (Delta), measuring the discrepancy between total and subnetwork deficiencies, and the Common Complexes Cardinality CC of the FID. Results show that Delta uniquely identifies all the ILC classes and one DLC subclass, while CC distinguishes the remaining DLC subclasses. A number of results on mass action systems such as the Deficiency One Theorem as well as on power law systems essentially rely on the ILC property of the underlying networks. These suggest that the FDA classification of ILC and DLC networks signify a certain alignment of both structural and kinetic attributes. This work opens up direction for the study of the structure and equilibria analysis of reaction networks across diverse decomposition architectures.

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arXiv2025

The Long-Term Impact of Direct Capture Approaches to Carbon Dioxide Removal

Understanding the similarities and differences of the long term impact of different carbon dioxide removal (CDR) techniques is essential in determining the most effective and sustainable strategies to mitigate climate change. In particular, direct ocean capture (DOC) has emerged as a promising approach. In contrast to direct air capture (DAC) which separates carbon dioxide from the atmosphere, DOC performs the separation directly from seawater before storing it in geological reservoirs. In this study, we construct and analyze a kinetic system for CDR via DOC using chemical reaction network theory. Our analysis reveals the necessary conditions for the existence of positive steady states and highlights the potential for multistationarity, where the carbon cycle may admit multiple positive steady states, emphasizing the critical importance of addressing tipping points, thresholds beyond which the system could undergo irreversible changes. Furthermore, we examine conditions under which certain carbon pools exhibit absolute concentration robustness, remaining resistant to change regardless of initial conditions. We also determine the conditions for the carbon reduction capability of the model with the DOC intervention. Importantly, a comparative analysis is then presented, where we compare the DOC model with the well-established DAC model by Fortun et al., and explore an integrated DOC-DAC approach for CDR. This comparison is important given that DAC is already being implemented in large-scale projects, while DOC remains in its early stages with limited trials and is geographically constrained to oceanic vicinity. Our comparative modeling framework provides valuable insights into the long-term impacts and complementary roles of DOC, DAC, and their integration into broader CDR strategies for climate mitigation.

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arXiv2025

Parameter-minimal analysis of carbon dioxide removal through direct air capture

The potential for multistationarity, or the existence of steady-state multiplicity, in the Earth System raises concerns that the planet could reach a climatic `tipping point,' rapidly transitioning to a warmer steady-state from which recovery may be practically unattainable. In detailed Earth models that require extensive computation time, it is difficult to make an a priori prediction of the possibility of multistationarity. In this study, we demonstrate Chemical Reaction Network Theory (CRNT) analysis of a simple heuristic box model of the Earth System carbon cycle with the human intervention of Direct Air Capture. CRNT leverages parameter-minimal analysis, relying primarily on the graphical and kinetic structure of the reaction network system, to identify necessary conditions for steady-state multiplicity. The analysis reveals necessary conditions for the combination of system parameters where steady-state multiplicity may exist. With this method, other negative emissions technologies (NET) may be screened in a relatively simple manner to aid in the priority setting by policymakers. Beyond multistationarity, the analysis provides insights into key system properties, such as absolute concentration robustness and some conditions for atmospheric carbon reduction.

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arXiv2024

Comparison of reaction networks of insulin signaling

Understanding the insulin signaling cascade provides insights on the underlying mechanisms of biological phenomena such as insulin resistance, diabetes, Alzheimer's disease, and cancer. For this reason, previous studies utilized chemical reaction network theory to perform comparative analyses of reaction networks of insulin signaling in healthy (INSMS: INSulin Metabolic Signaling) and diabetic cells (INRES: INsulin RESistance). This study extends these analyses using various methods which give further insights regarding insulin signaling. Using embedded networks, we discuss evidence of the presence of a structural "bifurcation" in the signaling process between INSMS and INRES. Concordance profiles of INSMS and INRES show that both have a high propensity to remain monostationary. Moreover, the concordance properties allow us to present heuristic evidence that INRES has a higher level of stability beyond its monostationarity. Finally, we discuss a new way of analyzing reaction networks through network translation. This method gives rise to three new insights: (i) each stoichiometric class of INSMS and INRES contains a unique positive equilibrium; (ii) any positive equilibrium of INSMS is exponentially stable and is a global attractor in its stoichiometric class; and (iii) any positive equilibrium of INRES is locally asymptotically stable. These results open up opportunities for collaboration with experimental biologists to understand insulin signaling better.

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arXiv2024

Embedding-based comparison of reaction networks of Wnt signaling

This work introduces a new method for comparing two reaction networks of the same or closely related systems through their embedded networks in terms of the shared set of species. Hence, we call this method the Common Species Embedded Networks (CSEN) analysis. Using this approach, we conduct a comparison of existing reaction networks associated with Wnt signaling models (Lee, Schmitz, MacLean, and Feinberg) that we have identified. The analysis yields three important results for these Wnt models. First, the CSEN analysis of the Lee (mono-stationary) and Feinberg (multi-stationary) shows a strong similarity, justifying the study of the Feinberg model, which was a modified Lee model constructed to study an important network property called "concordance". It also challenge the absoluteness of discrimination of the models into mono-stationarity versus multi-stationarity, which is a main result of Maclean et al. (PNAS USA 2015). Second, the CSEN analysis provides evidence supporting a strong similarity between the Schmitz and MacLean models, as indicated by the "proximate equivalence" that we have identified. Third, the analysis underscores the absence of a comparable relationship between the Feinberg and MacLean models, highlighting distinctive differences between the two. Thus, our approach could be a useful tool to compare mathematical models of the same or closely related systems.

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arXiv2024

Equilibria decomposition-based comparison of reaction networks of Wnt signaling

The Wnt signaling pathway plays a critical role in various biochemical processes, including embryonic development, tissue homeostasis, and cancer progression. In this paper, we conduct a comparative analysis of $β$-catenin-dependent Wnt signaling reaction networks, which we refer to as the Feinberg, Schmitz, and MacLean models, based on the previous study by MacLean et al. (PNAS USA 2015). Our analysis is based on the (unique) finest independent decomposition (FID) of each reaction network and our comparative techniques include equilibria parametrizations (EP) and the newly developed methods of Common Reactions Equilibria (CORE) analysis and Concordance Profile (CP) analysis. Our investigation yields three interesting results concerning the equilibria sets of these models. Firstly, we explore the concept of absolute concentration robustness (ACR), wherein a system exhibits ACR in a specific species if the equilibrium value for that species is the same for any positive equilibrium. Through ACR analysis employing FID and EP, we observe that both the Schmitz and MacLean models lack ACR, whereas the Feinberg model demonstrates ACR in a single species. Second, our analyses using FID and CORE reveal important relationships within the equilibria sets of the augmented Schmitz and MacLean models. Furthermore, FID and CORE identify the lack of a substantial relationship between the equilibria sets of the Feinberg and MacLean models. Hence, these methods detect subtle differences between the Feinberg and MacLean models and also between the Schmitz and MacLean models, which are not evident in the standard reaction network analysis. Finally, based on the concordance levels, CP analysis indicates that the MacLean and Schmitz models are more similar than the MacLean and Feinberg models.

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arXiv2023

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.

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arXiv2023

Absolute Concentration Robustness in Rank-One Kinetic Systems

A kinetic system has an absolute concentration robustness (ACR) for a molecular species if its concentration remains the same in every positive steady state of the system. Just recently, a condition that sufficiently guarantees the existence of an ACR in a rank-one mass-action kinetic system was found. In this paper, it will be shown that this ACR criterion does not extend in general to power-law kinetic systems. Moreover, we also discussed in this paper a necessary condition for ACR in multistationary rank-one kinetic system which can be used in ACR analysis. Finally, a concept of equilibria variation for kinetic systems which are based on the number of the system's ACR species will be introduced here.

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arXiv2022

Comparative analysis of carbon cycle models via kinetic representations

The pre-industrial state of the global carbon cycle is a significant aspect of studies related to climate change. In this paper, we recall the power law kinetic representations of the pre-industrial models of Schmitz (2002) and Anderies et al. (2013) from our earlier work. The power law kinetic representations, as uniform formalism, allow for a more extensive analysis and comparison of the different models for the same system. Using the mathematical theories of chemical reaction networks (with power-law kinetics), this work extends the analysis of the kinetic representations of the two models and assesses the similarities and differences in their structural and dynamic properties in relation to model construction assumptions. The analysis includes but is not limited to the coincidence of kinetic and stoichiometric spaces of the networks, capacity for equilibria multiplicity and co-multiplicity, and absolute concentration robustness in some species. Moreover, we bring together previously published results about the power law kinetic representations of the two models and consolidate them with new observations here. We also illustrate how the pre-industrial model of Anderies et al. may serve as a building block in the analysis of a kinetic representation of a global carbon cycle with carbon dioxide removal intervention.

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arXiv2022

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.

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arXiv2022

Positive equilibria of power law kinetics on networks with independent linkage classes

Studies about the set of positive equilibria ($E_+$) of kinetic systems have been focused on mass action, and not that much on power law kinetic (PLK) systems, even for PL-RDK systems (PLK systems where two reactions with identical reactant complexes have the same kinetic order vectors). For mass action, reactions with different reactants have different kinetic order rows. A PL-RDK system satisfying this property is called factor span surjective (PL-FSK). In this work, we show that a cycle terminal PL-FSK system with $E_+\ne \varnothing$ and has independent linkage classes (ILC) is a poly-PLP system, i.e., $E_+$ is the disjoint union of log-parametrized sets. The key insight for the extension is that factor span surjectivity induces an isomorphic digraph structure on the kinetic complexes. The result also completes, for ILC networks, the structural analysis of the original complex balanced generalized mass action systems (GMAS) by Müller and Regensburger. We also identify a large set of PL-RDK systems where non-emptiness of $E_+$ is a necessary and sufficient condition for non-emptiness of each set of positive equilibria for each linkage class. These results extend those of Boros on mass action systems with ILC. We conclude this paper with two applications of our results. Firstly, we consider absolute complex balancing (ACB), i.e., the property that each positive equilibrium is complex balanced, in poly-PLP systems. Finally, we use the new results to study absolute concentration robustness (ACR) in these systems. In particular, we obtain a species hyperplane containment criterion to determine ACR in the system species.

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arXiv2022

Weakly Reversible CF-Decompositions of Chemical Kinetic Systems

This paper studies chemical kinetic systems which decompose into weakly reversible complex factorizable (CF) systems. Among power law kinetic systems, CF systems (denoted as PL-RDK systems) are those where branching reactions of a reactant complex have identical rows in the kinetic order matrix. Mass action and generalized mass action systems (GMAS) are well-known examples. Schmitz's global carbon cycle model is a previously studied non-complex factorizable (NF) power law system (denoted as PL-NDK). We derive novel conditions for the existence of weakly reversible CF-decompositions and present an algorithm for verifying these conditions. We discuss methods for identifying independent decompositions, i.e., those where the stoichiometric subspaces of the subnetworks form a direct sum, as such decompositions relate positive equilibria sets of the subnetworks to that of the whole network. We then use the results to determine the positive equilibria sets of PL-NDK systems which admit an independent weakly reversible decomposition into PL-RDK systems of PLP type, i.e., the positive equilibria are log-parametrized, which is a broad generalization of a Deficiency Zero Theorem of Fortun et al. (2019).

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arXiv2021

Absolutely complex balanced kinetic systems

A complex balanced kinetic system is absolutely complex balanced (ACB) if every positive equilibrium is complex balanced. Two results on absolute complex balancing were foundational for modern chemical reaction network theory (CRNT): in 1972, M. Feinberg proved that any deficiency zero complex balanced system is absolutely complex balanced. In the same year, F. Horn and R. Jackson showed that the (full) converse of the result is not true: any complex balanced mass action system, regardless of its deficiency, is absolutely complex balanced. In this paper, we present initial results on the extension of the Horn and Jackson ACB Theorem. In particular, we focus on other kinetic systems with positive deficiency where complex balancing implies absolute complex balancing. While doing so, we found out that complex balanced power law reactant determined kinetic systems (PL-RDK) systems are not ACB. In our search for necessary and sufficient conditions for complex balanced systems to be absolutely complex balanced, we came across the so-called CLP systems (complex balanced systems with a desired "log parametrization" property). It is shown that complex balanced systems with bi-LP property are absolutely complex balanced. For non-CLP systems, we discuss novel methods for finding sufficient conditions for ACB in kinetic systems containing non-CLP systems: decompositions, the Positive Function Factor (PFF) and the Coset Intersection Count (CIC) and their application to poly-PL and Hill-type systems.

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arXiv2021

Complex balanced equilibria of weakly reversible poly-PL systems: existence, stability, and robustness

Poly-PL kinetic systems (PYK) are kinetic systems consisting of nonnegative linear combinations of power law functions. In this contribution, we analyze these kinetic systems using two main approaches: (1) we define a canonical power law representation of a poly-PL system, and (2) we transform a poly-PL system into a dynamically equivalent power law kinetic system that preserves the stoichiometric subspace of the system. These approaches led us to establish results that concern important dynamical properties of poly-PL systems that extend known results for generalized mass actions systems (GMAS) such as existence, uniqueness and parametrization of complex balanced steady states, and linear stability of complex balanced equilibria. Furthermore, the paper discusses subsets of poly-PL systems that exhibit two types of concentration robustness in some species namely absolute concentration robustness and balanced concentration robustness.

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arXiv2021

Concentration Robustness in LP Kinetic Systems

For a reaction network with species set $\mathscr{S}$, a log-parametrized (LP) set is a non-empty set of the form $E(P, x^*) = \{x \in \mathbb{R}^\mathscr{S}_> \mid \log x - \log x^* \in P^\perp\}$ where $P$ (called the LP set's flux subspace) is a subspace of $\mathbb{R}^\mathscr{S}$, $x^*$ (called the LP set's reference point) is a given element of $\mathbb{R}^\mathscr{S}_>$, and $P^\perp$ (called the LP set's parameter subspace) is the orthogonal complement of $P$. A network with kinetics $K$ is a positive equilibria LP (PLP) system if its set of positive equilibria is an LP set. Analogously, it is a complex balanced equilibria LP (CLP) system if its set of complex balanced equilibria is an LP set. An LP kinetic system is a PLP or CLP system. This paper studies concentration robustness of a species on subsets of equilibria. We present the "species hyperplane criterion", a necessary and sufficient condition for absolute concentration robustness (ACR) for a species of a PLP system. An analogous criterion holds for balanced concentration robustness (BCR) for species of a CLP system. These criteria also lead to interesting necessary properties of LP systems with concentration robustness. Furthermore, we show that PLP and CLP power law systems with Shinar-Feinberg reaction pairs in species $X$ in a linkage class have ACR and BCR in $X$, respectively. This leads to a broadening of the "low deficiency building blocks" framework to include LP systems of Shinar-Feinberg type with arbitrary deficiency. Finally, we apply our results to species concentration robustness in LP systems with poly-PL kinetics.

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arXiv2021

Common Complexes of Decompositions and Complex Balanced Equilibria of Chemical Reaction Networks

A decomposition of a chemical reaction network (CRN) is produced by partitioning its set of reactions. The partition induces networks, called subnetworks, that are "smaller" than the given CRN which, at this point, can be called parent network. A complex is called a common complex if it occurs in at least two subnetworks in a decomposition. A decomposition is said to be incidence independent if the image of the incidence map of the parent network is the direct sum of the images of the subnetworks' incidence maps. It has been recently discovered that the complex balanced equilibria of the parent network and its subnetworks are fundamentally connected in an incidence independent decomposition. In this paper, we utilized the set of common complexes and a developed criterion to investigate decomposition's incidence independence properties. A framework was also developed to analyze decomposition classes with similar structure and incidence independence properties. We identified decomposition classes that can be characterized by their sets of common complexes and studied their incidence independence. Some of these decomposition classes occur in some biological and chemical models. Finally, a sufficient condition was obtained for the complex balancing of some power law kinetic (PLK) systems with incidence independent and complex balanced decompositions. This condition led to a generalization of the Defficiency Zero Theorem for some PLK systems.

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arXiv2021

Independent, Incidence Independent and Weakly Reversible Decompositions of Chemical Reaction Networks

Chemical reaction networks (CRNs) are directed graphs with reactant or product complexes as vertices, and reactions as arcs. A CRN is weakly reversible if each of its connected components is strongly connected. Weakly reversible networks can be considered as the most important class of reaction networks. Now, the stoichiometric subspace of a network is the linear span of the reaction vectors (i.e., difference between the product and the reactant complexes). A decomposition of a CRN is independent (incidence independent) if the direct sum of the stoichiometric subspaces (incidence maps) of the subnetworks equals the stoichiometric subspace (incidence map) of the whole network. Decompositions can be used to study relationships between steady states of the whole system (induced from partitioning the reaction set of the underlying network) and those of its subsystems. In this work, we revisit our novel method of finding independent decomposition, and use it to expand applicability on (vector) components of steady states. We also explore CRNs with embedded deficiency zero independent subnetworks. In addition, we establish a method for finding incidence independent decomposition of a CRN. We determine all the forms of independent and incidence independent decompositions of a network, and provide the number of such decompositions. Lastly, for weakly reversible networks, we determine that incidence independence is a sufficient condition for weak reversibility of a decomposition, and we identify subclasses of weakly reversible networks where any independent decomposition is weakly reversible.

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arXiv2021

Complex balanced equilibria of weakly reversible poly-PL systems: multiplicity, robustness and stability

Poly-PL kinetic systems are kinetic systems consisting of nonnegative linear combinations of power law functions. In this contribution, we analyze these kinetic systems using two main approaches: (1) we define a canonical power law representation of a poly-PL system, and (2) we transform a poly-PL system into a dynamically equivalent power law kinetic system that preserves the stoichiometric subspace of the system. These approaches led us to establish results that concern important dynamical properties of poly-PL systems such as existence and parametrization of complex balanced steady states, capacity for multiple complex balanced equilibria in a stoichiometric compatibility class, concentration robustness of some species, and linear stability of complex balanced equilibria.

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