Stoichiometric Class Research Articles

Adding species to chemical reaction networks: Preserving rank preserves nondegenerate behaviours

We show that adding new chemical species into the reactions of a chemical reaction network (CRN) in such a way that the rank of the network remains unchanged preserves its capacity for multiple nondegenerate equilibria and/or periodic orbits. One consequence is that any bounded nondegenerate behaviours which can occur in a CRN can occur in a CRN with bounded stoichiometric classes. The main result adds to a family of theorems which tell us which enlargements of a CRN preserve its capacity for nontrivial dynamical behaviours. It generalises some earlier claims, and complements similar claims involving the addition of reactions into CRNs. The result gives us information on how ignoring some chemical species, as is common in biochemical modelling, might affect the allowed dynamics in differential equation models of CRNs. We demonstrate the scope and limitations of the main theorem via several examples. These illustrate how we can use the main theorem to predict multistationarity and oscillation in CRNs enlarged with additional species; but also how the enlargements can introduce new behaviours such as additional periodic orbits and new bifurcations.

Limit cycles in mass-conserving deficiency-one mass-action systems

We present some simple mass-action systems with limit cycles that fall under the scope of the Deficiency-One Theorem. All the constructed examples are mass-conserving and their stoichiometric subspace is two-dimensional. Using the continuation software MATCONT, we depict the limit cycles in all stoichiometric classes at once. The networks are trimolecular and tetramolecular, and some exhibit two or even three limit cycles. Finally, we show that the associated mass-action system of a bimolecular reaction network with two-dimensional stoichiometric subspace does not admit a limit cycle.

Profile of Polya's Problem-Solving Abilities Model Among Senior High School Students: Lesson From Stoichiometric Class

Based on several studies, it is known that there are students' difficulties in applying problem-solving as a problem-solving method, especially in stoichiometric material. Because solving stoichiometric' problems requires not only algorithmic skills but also a good understanding of chemical concepts. This study uses subjective test questions to determine the steps or methods students use to solve problem-solving test questions. The results showed that the students' problem-solving ability in the upper group was very high, the middle group was quite high. In contrast, the lower group was classified as low with mastery of Polya steps, respectively 85%, 68%, and 36%. There is a significant correlation between students' mastery of problem-solving steps. The correlation of mastery of problem-solving steps for students in the upper group was 0.786 (high), the middle group was 0.883 (very high), and the lower group had a correlation score of 0.803 (very high). REFERENCES Amalia, F.R. 2015. Identifikasi Kesulitan Siswa Kelas XI MIA SMA Negeri 1 Kauman Tulungagung dalam Menyelesaikan Soal-soal Stoikiometri Menggunakan Analisis Langkah Penyelesaian Soal . Skripsi tidak diterbitkan. Malang: FMIPA Universitas Negeri Malang. Bodner, George M. & Herron, J. Dudley. 2002. Problem Solving in Chemistry. Chemical Education: Towards Research-based Practice Kluwer , 235–266 Bodner, George M. & Domin, Daniels. 2000. Mental Models: The Role of Representations in Problem Solving in Chemistry. University Chemistry Education. 4: 24-30. Chang, Raymond. 2005. Kimia Dasar: Konsep-Konsep Inti Jilid 1/ Edisi ketiga. Bandung: Erlangga Chiu, M.H. 2001. Algorithmic Problem Solving and Conceptual Understanding of Chemistry by Student at a Local High School in Taiwan. Journal of Chemical Education International , 11(1): 20-38. Chiu, M.H. 2005. A National Survey of Students' Conceptions in Chemistry in Taiwan. Chemical Education International , 6 (1): 1-8. Effendy. 2006. A-Level Chemistry for Senior High Students Volume 2A . Malang: Bayumedia Publishing. Gabel, Dorothy L., Sherwood, Robert D., & Enochs, Larry. 1984. Problem-Solving Skills of High School Chemistry Students. Journal of Research in Science Teaching . 21 (2): 221-233 Hiebert, J. 1985. Conceptual and Procedural Knowledge : The Case of Mathematics . London: Lawrence Enibaum Associates. Nurjannah, Fitria. 2014. Analisis Langkah-Langkah Dalam Menyelesaikan Masalah Perhitungsn Pada Materi Stokiometri Siswa Kelas X SMA Negeri 1 Situbondo . Skripsi tidak diterbitkan. Malang: FMIPA Universitas Negeri Malang. Polya, G. 1957. How to solve it: a New Aspect of mathematical method. Princeton, USA: Princeton University Press Silberberg, Martin S. 2009. Chemistry The Molecular Nature of Matter Fifth Edition . New York: Mcgraw-hill Sugiyono, 2015. Memahami Penelitian Kualitatif . Bandung: Alfabeta. Zidny, Robby, Sopandi, Wahyu, Kusrijadi, Ali. 2013. Analisis Pemahaman Konsep Siswa SMA Kelas X pada Materi Persamaan Kimia dan Stoikiometri melalui Penggunaan Diagram Submikroskopik serta Hubungannya dengan Kemampuan Pemecaham Masalah. Jurnal Riset dan Praktik Pendidikan Kimia . 1(1)

Some consequences of thermodynamic feasibility for chemical reaction networks

Power law dynamics is used to describe the stability behavior in metabolic networks such as chemical reaction networks (CRN’s). These systems allow multiple steady states within a single stoichiometric class. On the other side thermodynamic constraints such as loop-less fluxes represented by the Gorban theorem of alternatives applied to these networks reveal considerable restrictions to their dynamics by eliminating multistability of CRN’s in general. Thermodynamic feasible CRN’s are contained in the class of injective CRN’s. We can give an alternative proof of the detailed balance with Brewer’s Fixed Point Theorem. Furthermore we can derive by the loop-less principle the extended detailed balance. This paper establishes a link between recent research in CRN theory and thermodynamic basics. The result has also consequences for the picture of multiple steady states as assumed for cell differentiation and regulation. CRN’s provide from their perspective not enough means to maintain multistability without regulation or external control.

Convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria

In this paper we study the rate of convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria. We first analyze a three-species system with boundary equilibria in some stoichiometric classes, and whose right hand side is bounded above by a quadratic nonlinearity in the positive orthant. We prove similar results on the convergence to the positive equilibrium for a fairly general two-species reversible reaction-diffusion network with boundary equilibria.

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.

Complex-balanced equilibria of generalized mass-action systems: necessary conditions for linear stability.

It is well known that, for mass-action systems, complex-balanced equilibria are asymptotically stable. For generalized mass-action systems, even if there exists a unique complex-balanced equilibrium (in every stoichiometric class and for all rate constants), it need not be stable. We first discuss several notions of matrix stability (on a linear subspace) such as D-stability and diagonal stability, and then we apply abstract results on matrix stability to complex-balanced equilibria of generalized mass-action systems. In particular, we show that linear stability (on the stoichiometric subspace and for all rate constants) implies uniqueness. For cyclic networks, we characterize linear stability (in terms of D-stability of the Jacobian matrix); and for weakly reversible networks, we give necessary conditions for linear stability (in terms of D-semistability of the Jacobian matrices of all cycles in the network). Moreover, we show that, for classical mass-action systems, complex-balanced equilibria are not just asymptotically stable, but even diagonally stable (and hence linearly stable). Finally, we recall and extend characterizations of D-stability and diagonal stability for matrices of dimension up to three, and we illustrate our results by examples of irreversible cycles (of dimension up to three) and of reversible chains and S-systems (of arbitrary dimension).

Existence of Positive Steady States for Weakly Reversible Mass-Action Systems

We prove the following. For each weakly reversible mass-action system, there exists a positive steady state in each positive stoichiometric class.

Regulated, greenhouse gas, and particulate emissions from lean-burn and stoichiometric natural gas heavy-duty vehicles on different fuel compositions

The gaseous and particulate matter (PM) emissions from three heavy-duty natural gas vehicles, including a lean-burn bus with an oxidation catalyst and two stoichiometric Class 8 trucks with three-way catalysts were evaluated. Testing was performed on a range of three to seven different test fuels with varying Wobbe and methane numbers. The lean-burn vehicle showed general trends of higher emissions of nitrogen oxides (NOx) and non-methane hydrocarbons (NMHC), and lower emissions of total hydrocarbons (THC), methane (CH4), and formaldehyde, and improved fuel economy for the fuels with low methane numbers. The stoichiometric trucks showed some trends toward lower THC, CH4, and NOx emissions with the low methane number fuels, whereas some increases in NMHC, carbon monoxide (CO), and ammonia (NH3) emissions were also observed. Results of the particle size distributions revealed bimodal size distribution profiles for all three vehicles, with a predominant nucleation mode close to 10nm for the lean-burn bus and one of the stoichiometric trucks.

On the dependence of the existence of the positive steady states on the rate coefficients for deficiency-one mass action systems: single linkage class

The Deficiency-One Theorem states that there exists a unique positive steady state in each positive stoichiometric class for weakly reversible deficiency-one mass action systems with one linkage class (regardless of the values of the rate coefficients). The non-emptiness of the set of positive steady states does not remain valid if we omit the weak reversibility. A recently published paper provided an equivalent condition to the existence of a positive steady state for deficiency-one mass action systems that are not weakly reversible, but still has only one linkage class. Based on that result, we characterise in this paper those of these mass action systems for which the non-emptiness of the set of positive steady states holds regardless of the values of the rate coefficients. Also, we provide an equivalent condition to the existence of rate coefficients such that the set of positive steady states is nonempty for the resulting mass action system.

On the existence of the positive steady states of weakly reversible deficiency-one mass action systems

In this paper we prove that there exists a positive steady state in each positive stoichiometric class for every weakly reversible deficiency-one mass action system. This qualitative property does not depend on the values of the rate coefficients. Since not all weakly reversible deficiency-one mass action systems satisfy the assumptions of the Deficiency-One Theorem, our result contributes to the theory of deficiency-one mass action systems.

Power-Law Kinetics and Determinant Criteria for the Preclusion of Multistationarity in Networks of Interacting Species

We present determinant criteria for the preclusion of nondegenerate multiple steady states in net- works of interacting species. A network is modeled as a system of ordinary differential equations in which the form of the species formation rate function is restricted by the reactions of the network and how the species influence each reaction. We characterize families of so-called power-law kinet- ics for which the associated species formation rate function is injective within each stoichiometric class and thus the network cannot exhibit multistationarity. The criterion for power-law kinetics is derived from the determinant of the Jacobian of the species formation rate function. Using this characterization, we further derive similar determinant criteria applicable to general sets of kinet- ics. The criteria are conceptually simple, computationally tractable, and easily implemented. Our approach embraces and extends previous work on multistationarity, such as work in relation to chemical reaction networks with dynamics defined by mass-action or noncatalytic kinetics, and also work based on graphical analysis of the interaction graph associated with the system. Further, we interpret the criteria in terms of circuits in the so-called DSR-graph.

Preclusion of switch behavior in networks with mass-action kinetics

We study networks taken with mass-action kinetics and provide a Jacobian criterion that applies to an arbitrary network to preclude the existence of multiple positive steady states within any stoichiometric class for any choice of rate constants. We are concerned with the characterization of injective networks, that is, networks for which the species formation rate function is injective in the interior of the positive orthant within each stoichiometric class. We show that a network is injective if and only if the determinant of the Jacobian of a certain function does not vanish. The function consists of components of the species formation rate function and a maximal set of independent conservation laws. The determinant of the function is a polynomial in the species concentrations and the rate constants (linear in the latter) and its coefficients are fully determined. The criterion also precludes the existence of degenerate steady states. Further, we relate injectivity of a network to that of the network obtained by adding outflow, or degradation, reactions for all species.

Stoichiometric measurements of 30S and 50S ribosomal proteins from Escherichia coli.

Stoichiometric investigations were made for all 21 proteins of the 30S subunit and for 31 proteins of the 50S subunit. While the results for the 30S proteins strongly suggest the existence of two different stoichiometric classes, the so-called “fractional” (F) and “unit” (U) proteins with average molar amounts of 0.1–0.5 and 0.8–1.2 per mole 30S respectively, at least one further group is present in isolated 50S particles. Therefore 50S proteins may be subdivided in the following way: a) 9–10 proteins with average molar amounts of 0.2–0.6 (F-proteins), b) 15–16 proteins with 0.8–1.2 copies per 50S particle (U-proteins), c) at least one protein with an average value of 1.8–2.2 copies, the “repeated” (R) proteins, d) a group of 7 proteins with 1.4–1.7 copies per particle; they can be named “fractional repeated” (FR) proteins. These findings indicate that 50S ribosomal particles show a more differentiated degree of structural heterogeneity at least in vitro than has been shown for 30S subunits.