arXiv subjects

Bryan S. Hernandez

Publications and source records attributed to Bryan S. Hernandez.

17 recordsLinked to original sources

arXiv2026

CRITERIA: A network decomposition and elementary flux mode translation-based tool for computing equilibria of biochemical systems

Understanding how biochemical systems settle into stable states, such as how protein concentrations reach equilibrium, is central to explaining cellular behavior and designing synthetic biological circuits. However, existing analytical tools for computing these equilibria, such as COMPILES, are limited by computational bottlenecks and can only be applied to a restricted class of reaction networks. In this work, we introduce CRITERIA (Computing paRametrized posITive EquilibRIA), a new computational framework that makes equilibrium analysis more efficient and broadly applicable. CRITERIA uses a graph-based approach built on elementary flux modes to streamline key steps in the computation. It also changes how the problem is solved by combining subnetworks into a single system before computing equilibria, which avoids complicated symbolic calculations required in previous methods. We demonstrate the usefulness of CRITERIA by studying biologically important systems, including the EnvZ-OmpR signaling pathway and a synthetic CRISPRi circuit. Our approach enables faster and more scalable analysis, allowing researchers to better understand how complex biochemical networks behave over time.

math.DS↗

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.

q-bio.MN↗

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.

math.DS↗

arXiv2024

Positive steady states of a class of power law systems with independent decompositions

Power law systems have been studied extensively due to their wide-ranging applications, particularly in chemistry. In this work, we focus on power law systems that can be decomposed into stoichiometrically independent subsystems. We show that for such systems where the ranks of the augmented matrices containing the kinetic order vectors of the underlying subnetworks sum up to the rank of the augmented matrix containing the kinetic order vectors of the entire network, then the existence of the positive steady states of each stoichiometrically independent subsystem is a necessary and sufficient condition for the existence of the positive steady states of the given power law system. We demonstrate the result through illustrative examples. One of which is a network of a carbon cycle model that satisfies the assumption, while the other network fails to meet the assumption. Finally, using the aforementioned result, we present a systematic method for deriving positive steady state parametrizations for the mentioned subclass of power law systems, which is a generalization of our recent method for mass action systems.

math.DS↗

arXiv2024

A decomposition-based approach for deriving positive steady states of a class of chemical reaction networks with non-mass-action kinetics

Steady states are frequently used to investigate the long-term behaviors of (bio)-chemical systems. Recently, there has been a growing interest in network-based approaches due to their efficiency in deriving parametrizations of positive steady states in systems with mass-action kinetics. In this study, we extend this approach to derive positive steady states in networks under non-mass-action kinetics, specifically mixed kinetics. In a system with mixed kinetics, some reactions {may follow} mass-action kinetics, while others in the same network follow different rate laws, such as quotient rate laws. An example of such complexity is evident in a mathematical model of the insulin signaling pathway in type 2 diabetes. To compute its positive {steady states}, we adapt our existing network decomposition approach, originally designed for mass-action kinetics, to handle networks with non-mass-action kinetics. This approach involves breaking down a given network into smaller, independent subnetworks to derive the positive steady states of each subnetwork separately. These individual steady states are then combined to obtain the positive steady states of the entire network. This strategy makes computations more manageable for complex and large networks. More importantly, this method could separate reactions with purely mass-action kinetics into certain subnetworks from those that follow different rate laws. We also present an illustrative example that provides insights into methods for transforming networks with mixed kinetics into their associated mass-action systems.

math.DS↗

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.

q-bio.MN↗

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.

math.DS↗

arXiv2022

Computational translation framework identifies biochemical reaction networks with special topologies and their long-term dynamics

Long-term behaviors of biochemical systems are described by steady states in deterministic models and stationary distributions in stochastic models. Obtaining their analytic solutions can be done for limited cases, such as linear or finite-state systems, as it generally requires solving many coupled equations. Interestingly, analytic solutions can be easily obtained when underlying networks have special topologies, called weak reversibility (WR) and zero deficiency (ZD), and the kinetic law follows a generalized form of mass-action kinetics. However, such desired topological conditions do not hold for the majority of cases. Thus, translating networks to have WR and ZD while preserving the original dynamics was proposed. Yet, this approach is limited because manually obtaining the desired network translation among the large number of candidates is challenging. Here, we prove necessary conditions for having WR and ZD after translation, and based on these conditions, we develop a user-friendly computational package, TOWARDZ, that automatically and efficiently identifies translated networks with WR and ZD. This allows us to quantitatively examine how likely it is to obtain WR and ZD after translation depending on the number of species and reactions. Importantly, we also describe how our package can be used to analytically derive steady states of deterministic models and stationary distributions of stochastic models. TOWARDZ provides an effective tool to analyze biochemical systems.

q-bio.MN↗

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.

math.DS↗

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).

math.DS↗

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.

math.DS↗

arXiv2021

Independent Decompositions of Chemical Reaction Networks

A chemical reaction network (CRN) is composed of reactions that can be seen as interactions among entities called species, which exist within the system. Endowed with kinetics, CRN has a corresponding set of ordinary differential equations (ODEs). In Chemical Reaction Network Theory, we are interested with connections between the structure of the CRN and qualitative properties of the corresponding ODEs. One of the results in Decomposition Theory of CRNs is that the intersection of the sets of positive steady states of the subsystems is equal to the set of positive steady states of the whole system, if the decomposition is independent. Hence, computational approach using independent decompositions can be used as an efficient tool in studying large systems. In this work, we provide a necessary and sufficient condition for the existence of a nontrivial independent decomposition of a CRN, which leads to a novel step-by-step method to obtain such decomposition, if it exists. We also illustrate these results using real-life examples. In particular, we show that a CRN of a popular model of anaerobic yeast fermentation pathway has a nontrivial independent decomposition, while a particular biological system, which is a metabolic network with one positive feedforward and a negative feedback has none. Finally, we analyze properties of steady states of reaction networks of specific influenza virus models.

math.DS↗

arXiv2020

Positive Equilibria of Hill-Type Kinetic Systems

This work introduces a novel approach to study properties of positive equilibria of a chemical reaction network $\mathscr{N}$ endowed with Hill-type kinetics $K$, called a Hill-type kinetic (HTK) system $\left(\mathscr{N},K\right)$, including their multiplicity and concentration robustness in a species. We associate a unique positive linear combination of power-law kinetic systems called poly-PL kinetic (PYK) system $\left( {\mathscr{N},{K_\text{PY}}} \right)$ to the given HTK system. The associated system has the key property that its equilibria sets coincide with those of the Hill-type system, i.e., ${E_ + }\left( {\mathscr{N},K} \right) = {E_ + }\left( {\mathscr{N},{K_\text{PY}}} \right)$ and ${Z_ + }\left( {\mathscr{N},K} \right) = {Z_ + }\left( {\mathscr{N},{K_\text{PY}}} \right)$. This allows us to identify two novel subsets of the Hill-type kinetics, called PL-equilibrated and PL-complex balanced kinetics, to which recent results on absolute concentration robustness (ACR) of species and complex balancing at positive equilibria of power-law (PL) kinetic systems can be applied. Our main results also include the Shinar-Feinberg ACR Theorem for PL-equilibrated HT-RDK systems (i.e., subset of complex factorizable HTK systems), which establishes a foundation for the analysis of ACR in HTK systems, and the extension of the results of Müller and Regensburger on generalized mass action systems to PL-complex balanced HT-RDK systems. In addition, we derive the theory of balanced concentration robustness (BCR) in an analogous manner to ACR for PL-equilibrated systems. Finally, we provide further extensions of our results to a more general class of kinetics, which includes quotients of poly-PL functions.

math.DS↗

arXiv2020

A Computational Approach to Multistationarity in Poly-PL Kinetic Systems

One important question that interests those who work in chemical reaction network theory (CRNT) is this: Does the system obtained from a reaction network admit a positive equilibrium and if it does, can there be more than one within a stoichiometric class? The higher deficiency algorithm (HDA) of Ji and Feinberg provided a method of determining the multistationarity capacity of a CRN with mass action kinetics (MAK). An extension of this, called Multistationarity Algorithm (MSA), recently came into the scene tackling CRNs with power law kinetics (PLK), a kinetic system which is more general (having MAK systems as a special case). For this paper, we provide a computational approach to study the multistationarity feature of reaction networks endowed with kinetics which are non-negative linear combinations of power law functions called poly-PL kinetics (PYK). The idea is to use MSA and combine it with a transformation called STAR-MSC (i.e., $S$-invariant Termwise Addition of Reactions via Maximal Stoichiometric Coefficients) producing PLKs that are dynamically equivalent to PYKs. This leads us to being able to determinine the multistationarity capacity of a much larger class of kinetic systems. We show that if the transformed dynamically equivalent PLK system is multistationary for a stoichiometric class for a set of particular rate constants, then so is its original corresponding PYK system. Moreover, the monostationarity property of the transformed PLK system also implies the monostationarity property of the original PYK system.

math.DS↗

arXiv2020

Fundamental Decompositions and Multistationarity of Power-Law Kinetic Systems

The fundamental decomposition of a chemical reaction network (also called its "$\mathscr{F}$-decomposition") is the set of subnetworks generated by the partition of its set of reactions into the "fundamental classes" introduced by Ji and Feinberg in 2011 as the basis of their "higher deficiency algorithm" for mass action systems. The first part of this paper studies the properties of the $\mathscr{F}$-decomposition, in particular, its independence (i.e., the network's stoichiometric subspace is the direct sum of the subnetworks' stoichiometric subspaces) and its incidence-independence (i.e., the image of the network's incidence map is the direct sum of the incidence maps' images of the subnetworks). We derive necessary and sufficient conditions for these properties and identify network classes where the $\mathscr{F}$-decomposition coincides with other known decompositions. The second part of the paper applies the above-mentioned results to improve the Multistationarity Algorithm for power-law kinetic systems (MSA), a general computational approach that we introduced in previous work. We show that for systems with non-reactant determined interactions but with an independent $\mathscr{F}$-decomposition, the transformation to a dynamically equivalent system with reactant-determined interactions -- required in the original MSA -- is not necessary. We illustrate this improvement with the subnetwork of Schmitz's carbon cycle model recently analyzed by Fortun et al.

math.DS↗

arXiv2020

A Computational Approach to Multistationarity of Power-Law Kinetic Systems

This paper presents a computational solution to determine if a chemical reaction network endowed with power-law kinetics (PLK system) has the capacity for multistationarity, i.e., whether there exist positive rate constants such that the corresponding differential equations admit multiple positive steady states within a stoichiometric class. The approach, which is called the "Multistationarity Algorithm for PLK systems" (MSA), combines (i) the extension of the "higher deficiency algorithm" of Ji and Feinberg for mass action to PLK systems with reactant-determined interactions, and (ii) a method that transforms any PLK system to a dynamically equivalent one with reactant-determined interactions. Using this algorithm, we obtain two new results: the monostationarity of a popular model of anaerobic yeast fermentation pathway, and the multistationarity of a global carbon cycle model with climate engineering, both in the generalized mass action format of biochemical systems theory. We also provide examples of the broader scope of our approach for deficiency one PLK systems in comparison to the extension of Feinberg's "deficiency one algorithm" to such systems.

math.DS↗

arXiv2020

On the Independence of Fundamental Decompositions of Power-Law Kinetic Systems

The fundamental decomposition of a chemical reaction network (CRN) is induced by partitioning the reaction set into "fundamental classes". It was the basis of the Higher Deficiency Algorithm for mass action systems of Ji and Feinberg, and the Multistationarity Algorithm for power-law kinetic systems of Hernandez et al. In addition to our previous work, we provide important properties of the independence (i.e., the network's stoichiometric subspace is the direct sum of the subnetworks' stoichiometric subspaces) and the incidence-independence (i.e., the image of the network's incidence map is the direct sum of the incidence maps' images of the subnetworks) of these decompositions. Feinberg established the essential relationship between independent decompositions and the set of positive equilibria of a network, which we call the Feinberg Decomposition Theorem (FDT). Moreover, Farinas et al. recently documented its version for incidence-independence. Fundamental decomposition divides the network into subnetworks of deficiency either 0 or 1 only. Hence, available results for lower deficiency networks, such as the Deficiency Zero Theorem (DZT), can be used. These justify the study of independent fundamental decompositions. A MATLAB program which (i) computes the subnetworks of a CRN under the fundamental decomposition and (ii) is useful for determining whether the decomposition is independent and incidence-independent is also created. Finally, we provide the following solution for determining multistationarity of CRNs with the following steps: (1) the use of the program, (2) the application of available results for CRNs with deficiency 0 or 1 (e.g., DZT), and (3) the use of FDT. We illustrate the solution by showing that the generalization of a subnetwork of Schmitz's carbon cycle model by Hernandez et al., endowed with mass action kinetics, has no capacity for multistationarity.

math.DS↗