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This contribution treats linear, steady-state dynamics for a metabolic network within a growing cell. Admissible steady-state reaction fluxes are assumed to form a pointed, convex, polyhedral, conical subset of the stoichiometric null-space. A solution of the problem is defined to consist of a linear basis for the stoichiometric null-space consisting of admissible fluxes called basic pathways. The algorithm used to construct the set of basic pathways scales as a polynomial of the system size in contrast to the NP-hard algorithms employed in the traditional notions of solution named extreme pathways, elementary flux modes, MEMos, and MinSpan, and that therefore suffer from the curse of dimensionality. The basic pathways approach is applied to a metabolic network consisting of a simplified version of the TCA cycle coupled to glycolysis highlighting that each basic pathway has a readily understood chemical interpretation. Generic admissible pathways are simply expressed in terms of basic pathways.

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Iron is an essential transition metal for all eukaryotic cells, and its trafficking throughout the cell is highly regulated. However, the overall cellular mechanism of regulation is poorly understood despite knowing many of the molecular players involved. Here, an ordinary-differential-equations (ODE) based kinetic model of iron trafficking within a growing yeast cell was developed that included autoregulation. The 9-reaction 8-component in-silico cell model was solved under both steady-state and time-dependent dynamical conditions. The ODE for each component included a dilution term due to cell growth. Conserved rate relationships were obtained from the null space of the stoichiometric matrix, and the reduced-row-echelon-form was used to distinguish independent from dependent rates. Independent rates were determined from experimentally estimated component concentrations, cell growth rates, and the literature. Simple rate-law expressions were assumed, allowing rate-constants for each reaction to be estimated. Continuous Heaviside logistical functions were used to regulate rate-constants. These functions acted like valves, opening or closing depending on component "sensor" concentrations. Two cellular regulatory mechanisms were selected from 134,217,728 possibilities using a novel approach involving 6 mathematically-defined filters. Three cellular states were analyzed including healthy wild-type cells, iron-deficient wild-type cells, and a frataxin-deficient strain of cells characterizing the disease Friedreich's Ataxia. The model was stable toward limited perturbations, as determined by the eigenvalues of Jacobian matrices. Autoregulation allowed healthy cells to transition to the diseased state when triggered by a mutation in frataxin, and to the iron-deficient state when cells are placed in iron-deficient growth medium. The in-silico phenotypes observed during these transitions were similar to those observed experimentally. The model also predicted the observed effects of hypoxia on the diseased condition. A similar approach could be used to solve ODE-based kinetic models associated with other biochemical processes operating within growing cells.

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A plausible biocontrol strategy for the eradication of invasive species involves augmenting wild populations with genetically modified supermales. Supermales contain double YY chromosomes. When they are augmented into a wild population, destabilization and eventual extinction occurs over time due to a strongly skewed gender ratio towards males. Here, we employ a mathematical model that considers an Allee effect, but we have discovered through simulation that the presence of supermales leads to an increase in the minimal number of females needed for survival at a value higher than the mathematically defined Allee effect. Using this effect, we focus our research on exploring the sensitivity of the optimized supply rate of supermale fish to the initial gender ratio and density of the wild populations. We find that the eradication strategy with optimized supply rate of supermales can be determined with knowledge of reproductive rate and survival fitness of supermale fish.

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This work presents a new force-based canonical approach that utilizes the average force rather than the pointwise force, on which previously developed canonical approaches were based. Advantageously, the average force based method only requires the evaluation of the potential function and not its derivative. The average force and the pointwise force based methods are applied to a variety of diatomic molecules, and their accuracy is compared. It is demonstrated that the average force based method gives an improved accuracy compared to the pointwise force based method. This improved accuracy is attributed to the fact that the average force based method eliminates the need to use the numerical approximation of the derivative of the potential function that, in practice, is only known at discrete points. In addition, an algorithm is developed to apply the average force based method as a practical tool for generating potential curves for pairwise interatomic interactions utilizing the classical Lennard-Jones potential as reference. Moreover, application of the average force based method leads to a new canonical approximation paradigm. In this new paradigm, only the coordinates of the equilibrium configuration (the bottom of the potential well) of a molecule are required for accurate generation of the potential function. Moreover, theoretical results are presented, demonstrating the effectiveness of the canonical transformation procedure in producing highly accurate potential approximations. In particular, it is proved that a certain general set of qualitative conditions on potential-like functions are sufficient for a given potential function to be in the same canonical transformation class as a (dimensionless) Lennard-Jones potential. For functions satisfying these assumptions, it is shown that they have canonical approximations with arbitrarily small approximation errors.

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In 2000, Elowitz and Leibler introduced the repressilator-a synthetic gene circuit with three genes that cyclically repress transcription of the next gene-as well as a corresponding mathematical model. Experimental data and model simulations exhibited oscillations in the protein concentrations across generations. Müller et al. (J Math Biol 53(6):905-937, 2006) generalized the model to an arbitrary number of genes and analyzed the resulting dynamics. Their new model arose from five key assumptions, two of which are restrictive given current biological knowledge. Accordingly, we propose a new repressilator system that allows for general functions to model transcription, degradation, and translation. We prove that, with an odd number of genes, the new model has a unique steady state and the system converges to this steady state or to a periodic orbit. We also give a necessary and sufficient condition for stability of steady states when the number of genes is even and conjecture a condition for stability for an odd number. Finally, we derive a new rate function describing transcription that arises under more reasonable biological assumptions than the widely used single-step binding assumption. With this new transcription-rate function, we compare the model's amplitude and period with that of a model with the conventional transcription-rate function. Taken together, our results enhance our understanding of genetic regulation by repression.

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A new force-based canonical approach for the accurate generation of multidimensional potential energy surfaces is demonstrated. Canonical transformations previously developed for diatomic molecules are used to construct accurate approximations to the 3-dimensional potential energy surface of the water molecule from judiciously chosen (adopting the right coordinate system) 1-dimensional planar slices that are shown to have the same canonical shape as the classical Lennard-Jones potential curve. Spline interpolation is then used to piece together the 1-dimensional canonical potential curves, to obtain the full 3-dimensional potential energy surface of a water molecule with a relative error less than 0.01. This work provides an approach to greatly reduce the computational cost of constructing potential energy surfaces in molecules from ab initio calculations. The canonical transformation techniques developed in this work illuminate a pathway to deepening our understanding of chemical bonding.

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The concept of chemical bonding is normally presented and simplified through two models: the covalent bond and the ionic bond. Expansion of the ideal covalent and ionic models leads chemists to the concepts of electronegativity and polarizability, and thus to the classification of polar and non-polar bonds. In addition, the intermolecular interactions are normally viewed as physical phenomena without direct correlation to the chemical bond in any simplistic model. Contrary to these traditional concepts of chemical bonding, recently developed canonical approaches demonstrate a unified perspective on the nature of binding in pairwise interatomic interactions. This new canonical model, which is a force-based approach with a basis in fundamental molecular quantum mechanics, confirms much earlier assertions that in fact there are no fundamental distinctions among covalent bonds, ionic bonds, and intermolecular interactions including the hydrogen bond, the halogen bond, and van der Waals interactions.

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Canonical approaches are applied to classic Morse, Lennard-Jones, and Kratzer potentials. Using the canonical transformation generated for the Morse potential as a reference, inverse transformations allow the accurate generation of the Born-Oppenheimer potential for the H2+ ion, neutral covalently bound H2, van der Waals bound Ar2, and the hydrogen bonded one-dimensional dissociative coordinate in a water dimer. Similar transformations are also generated using the Lennard-Jones and Kratzer potentials as references. Following application of inverse transformations, vibrational eigenvalues generated from the Born-Oppenheimer potentials give significantly improved quantitative comparison with values determined from the original accurately known potentials. In addition, an algorithmic strategy based upon a canonical transformation to the dimensionless form applied to the force distribution associated with a potential is presented. The resulting canonical force distribution is employed to construct an algorithm for deriving accurate estimates for the dissociation energy, the maximum attractive force, and the internuclear separations corresponding to the maximum attractive force and the potential well.

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Force-based canonical approaches have recently given a unified but different viewpoint on the nature of bonding in pairwise interatomic interactions. Differing molecular categories (covalent, ionic, van der Waals, hydrogen, and halogen bonding) of representative interatomic interactions with binding energies ranging from 1.01 to 1072.03 kJ/mol have been modeled canonically giving a rigorous semiempirical verification to high accuracy. However, the fundamental physical basis expected to provide the inherent characteristics of these canonical transformations has not yet been elucidated. Subsequently, it was shown through direct numerical differentiation of these potentials that their associated force curves have canonical shapes. However, this approach to analyzing force results in inherent loss of accuracy coming from numerical differentiation of the potentials. We now show that this serious obstruction can be avoided by directly demonstrating the canonical nature of force distributions from the perspective of the Hellmann-Feynman theorem. This requires only differentiation of explicitly known Coulombic potentials, and we discuss how this approach to canonical forces can be used to further explain the nature of chemical bonding in pairwise interatomic interactions. All parameter values used in the canonical transformation are determined through explicit physical based algorithms, and it does not require direct consideration of electron correlation effects.

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Canonical approaches are applied for investigation of the extraordinarily accurate electronic ground state potentials of H2(+), H2, HeH(+), and LiH using the virial theorem. These approaches will be dependent on previous investigations involving the canonical nature of E(R), the Born-Oppenheimer potential, and F(R), the associated force of E(R), that have been demonstrated to be individually canonical to high accuracy in the case of the systems investigated. Now, the canonical nature of the remaining functions in the virial theorem [the electronic kinetic energy T(R), the electrostatic potential energy V(R), and the function W(R) = RF(R)] are investigated and applied to H2, HeH(+), and LiH with H2(+) chosen as reference. The results will be discussed in the context of a different perspective of molecular bonding that goes beyond previous direct applications of the virial theorem.

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The Trojan Y-Chromosome (TYC) strategy, an autocidal genetic biocontrol method, has been proposed to eliminate invasive alien species. In this work, we develop a Markov jump process model for this strategy, and we verify that there is a positive probability for wild-type females going extinct within a finite time. Moreover, when sex-reversed Trojan females are introduced at a constant population size, we formulate a stochastic differential equation (SDE) model as an approximation to the proposed Markov jump process model. Using the SDE model, we investigate the probability distribution and expectation of the extinction time of wild-type females by solving Kolmogorov equations associated with these statistics. The results indicate how the probability distribution and expectation of the extinction time are shaped by the initial conditions and the model parameters.

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A generalized formulation of canonical transformations and spectra are used to investigate the concept of a canonical potential strictly within the Born-Oppenheimer approximation. Data for the most accurate available ground electronic state pairwise intermolecular potentials in H2, HD, D2, HeH(+), and LiH are used to rigorously evaluate such transformations. The corresponding potentials are generated explicitly using parameters calculated with algebraic functions from that of the single canonical potential of the simplest molecule, H2(+). The efficacy of this approach is further tested by direct comparison of the predicted eigenvalues of all vibrational states in the selected molecular systems considered with the corresponding most accurately known Born-Oppenheimer eigenvalues currently available. Deviations are demonstrated to be less than 2 cm(-1) for all vibrational states in H2, HD, D2, HeH(+), and LiH, with an average standard deviation of 0.27 cm(-1) for the 87 states considered. The implications of these results for molecular quantum chemistry are discussed.

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A generalized formulation of explicit force-based transformations is introduced to investigate the concept of a canonical potential in both fundamental chemical and intermolecular bonding. Different classes of representative ground electronic state pairwise interatomic interactions are referenced to a chosen canonical potential illustrating application of such transformations. Specifically, accurately determined potentials of the diatomic molecules H2, H2(+), HF, LiH, argon dimer, and one-dimensional dissociative coordinates in Ar-HBr, OC-HF, and OC-Cl2 are investigated throughout their bound potentials. Advantages of the current formulation for accurately evaluating equilibrium dissociation energies and a fundamentally different unified perspective on nature of intermolecular interactions will be emphasized. In particular, this canonical approach has significance to previous assertions that there is no very fundamental distinction between van der Waals bonding and covalent bonding or for that matter hydrogen and halogen bonds.

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The Trojan Y-Chromosome (TYC) strategy, an autocidal genetic biocontrol method, has been proposed to eliminate invasive alien species. In this work, we analyze the dynamical system model of the TYC strategy, with the aim of studying the viability of the TYC eradication and control strategy of an invasive species. In particular, because the constant introduction of sex-reversed trojan females for all time is not possible in practice, there arises the question: What happens if this injection is stopped after some time? Can the invasive species recover? To answer that question, we perform a rigorous bifurcation analysis and study the basin of attraction of the recovery state and the extinction state in both the full model and a certain reduced model. In particular, we find a theoretical condition for the eradication strategy to work. Additionally, the consideration of an Allee effect and the possibility of a Turing instability are also studied in this work. Our results show that: (1) with the inclusion of an Allee effect, the number of the invasive females is not required to be very low when the introduction of the sex-reversed trojan females is stopped, and the remaining Trojan Y-Chromosome population is sufficient to induce extinction of the invasive females; (2) incorporating diffusive spatial spread does not produce a Turing instability, which would have suggested that the TYC eradication strategy might be only partially effective, leaving a patchy distribution of the invasive species.

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This paper analyzes the stability of a coexistence equilibrium point of a model for competition between two stage-structured populations. In this model, for each population, competition for resources may affect any one of the following population parameters: reproduction, juvenile survival, maturation rate, or adult survival. The results show that the competitive strength of a population is affected by (1) the ratio of the population parameter influenced by competition under no resource limitation (maximum compensatory capacity) over the same parameter under a resource limitation due to competition (equilibrium rate) and (2) the ratio of interspecific competition over intraspecific competition; this ratio was previously shown to depend on resource-use overlap. The former ratio, which we define as fitness, can be equalized by adjusting organisms' life history strategies, thereby promoting coexistence. We conclude that in addition to niche differentiation among populations, the life history strategies of organisms play an important role in coexistence.

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