Reversible Enzyme Reaction Research Articles

The archaeal family 3 polyphosphate kinase reveals a function of polyphosphate as energy buffer under low energy charge.

Inorganic polyphosphate, a linear polymer of orthophosphate residues linked by phosphoanhydride bonds, occurs in all three domains of life and plays a diverse and prominent role in metabolism and cellular regulation. While the polyphosphate metabolism and its physiological significance have been well studied in bacteria and eukaryotes including human, there are only few studies in archaea available so far. In Crenarchaeota including members of Sulfolobaceae , the presence of polyphosphate and degradation via exopolyphosphatase has been reported and there is some evidence for a functional role in metal ion chelation, biofilm formation, adhesion and motility, however, the nature of the crenarchaeal polyphosphate kinase is still unknown. Here we used the crenarchaeal model organism Sulfolobus acidocaldarius to study the enzymes involved in polyphosphate synthesis. The two genes annotated as thymidylate kinase ( saci_2019 and saci_2020 ), localized downstream of the exopolyphosphatase, were identified as the missing polyphosphate kinase in S. acidocaldarius ( Sa PPK3). Thymidylate kinase activity was confirmed for Saci_0893. Notably Saci_2020 showed no polyphosphate kinase activity on its own but served as regulatory subunit (rPPK3) and was able to enhance polyphosphate kinase activity of the catalytically active subunit Saci_2019 (cPPK3). Heteromeric polyphosphate kinase activity is reversible and shows a clear preference for polyP-dependent nucleotide kinase activity, i.e. polyP-dependent formation of ATP from ADP (12.4 U/mg) and to a lower extent of GDP to GTP whereas AMP does not serve as substrate. PPK activity in the direction of ATP-dependent polyP synthesis is rather low (0.25 U/mg); GTP was not used as phosphoryl donor. A combined experimental modelling approach using quantitative 31 P NMR allowed to follow the reversible enzyme reaction for both ATP and polyP synthesis. PolyP synthesis was only observed when the ATP/ADP ratio was kept high, using an ATP recycling system. In absence of such a recycling system, all incubations with polyP and PPK would reach an equilibrium state with an ATP/ADP ratio between 3 and 4, independent of the initial conditions. Structural and sequence comparisons as well as phylogenetic analysis reveal that the S. acidocaldarius PPK is a member of a new PPK family, named PPK3, within the thymidylate kinase family of the P-loop kinase superfamily, clearly separated from PPK2. Our studies show that polyP, in addition to its function as phosphate storage, has a special importance for the energy homeostasis of S. acidocaldarius and due to its reversibility serves as energy buffer under low energy charge enabling a quick response to changes in cellular demand.

Forecasting and dynamical modeling of reversible enzymatic reactions with a hybrid proportional fractional derivative

Chemical kinetics is a branch of chemistry that investigates the rates of chemical reactions and has applications in cosmology, geology, and physiology. In this study, we develop a mathematical model for chemical reactions based on enzyme dynamics and kinetics, which is a two-step substrate–enzyme reversible reaction, applying chemical kinetics-based modeling of enzyme functions. The non-linear differential equations are transformed into fractional-order systems utilizing the constant proportional Caputo–Fabrizio (CPCF) and constant proportional Atangana–Baleanu–Caputo (CPABC) operators. The system of fractional differential equations is simulated using the Laplace–Adomian decomposition method at different fractional orders through simulations and numerical results. Both qualitative and quantitative analyses such as boundedness, positivity, unique solution, and feasible concentration for the proposed model with different hybrid operators are provided. The stability analysis of the proposed scheme is also verified using Picard’s stable condition through the fixed point theorem.

Apparent isotope effects of reversible enzymatic reactions

Accurately predicting apparent isotope effects of biogeochemical processes can aid the interpretation of laboratory batch experiments and field observations. It has been observed that the apparent instantaneous isotope fractionation factor changes with residual substrate concentration during a reaction process. The change can be attributed to the reversibility of a biogeochemical reaction, which is often overlooked for convenience. Here, we explore the influence of reverse reactions on high-dimensional isotope effects in the framework of Reverse Michaelis-Menten (RMM) kinetics. The apparent instantaneous isotope fractionation factors, as well as the isotope fractionation relationships between two elements of the same nonlabile compound in the RMM model, are functions of specific affinities of the product and substrate to an enzyme and the product-substrate ratios for both isotopes. In our cases, the influence of reaction reversibility is negligible when the substrate consumption rate is less than 63%. The analysis of the simultaneous stable isotope fractionation of two elements in a compound and of the relationship between the associated isotope fractionation factors can serve as a more effective indicator for characterizing the reaction mechanism.

Large deviations and fluctuation theorems for cycle currents defined in the loop-erased and spanning tree manners: A comparative study

The cycle current is a crucial quantity in stochastic thermodynamics. The absolute and net cycle currents of a Markovian system can be defined in the loop-erased (LE) or spanning tree (ST) manner. Here we make a comparative study between the large deviations and fluctuation theorems for LE and ST currents, i.e., cycle currents defined in the LE and ST manners. First, we derive the exact joint distribution and large deviation rate function for the LE currents of a system with a cyclic topology and also obtain the exact rate function for the ST currents of a general system. The relationship between the rate functions for LE and ST currents is clarified and the analytical results are applied to examine the fluctuations in the product rate of a three-step reversible enzyme reaction. Furthermore, we examine various types of fluctuation theorems satisfied by LE and ST currents and clarify their ranges of applicability. We show that both the absolute and net LE currents satisfy the strong form of all types of fluctuation theorems. In contrast, the absolute ST currents do not satisfy fluctuation theorems, while the net ST currents only satisfy the weak form of fluctuation theorems under the periodic boundary condition. Finally, the fluctuation theorems for cycle currents are applied to study the fluctuations in entropy production along a single stochastic trajectory.

Dynamics of two-step reversible enzymatic reaction under fractional derivative with Mittag-Leffler Kernel.

Chemical kinetics is a branch of chemistry that is founded on understanding chemical reaction rates. Chemical kinetics relates many aspects of cosmology, geology, and even in some cases of, psychology. There is a need for mathematical modelling of these chemical reactions. Therefore, the present research is based on chemical kinetics-based modelling and dynamics of enzyme processes. This research looks at the two-step substrate-enzyme reversible response. In the two step-reversible reactions, substrate combines with enzymes which is further converted into products with two steps. The model is displayed through the flow chart, which is then transformed into ODEs. The Atangana-Baleanu time-fractional operator and the Mittag-Leffler kernel are used to convert the original set of highly nonlinear coupled integer order ordinary differential equations into a fractional-order model. Additionally, it is shown that the solution to the investigated fractional model is unique, limited, and may be represented by its response velocity. A numerical scheme, also known as the Atangana-Toufik method, based on Newton polynomial interpolation technique via MATLAB software, is adopted to find the graphical results. The dynamics of reaction against different reaction rates are presented through various figures. It is observed that the forward reaction rates increase the reaction speed while backward reaction rates reduce it.

CH4 isotopic ordering records ultra-slow hydrocarbon biodegradation in the deep subsurface

Investigation of the biology and biochemistry of the deep subsurface provides invaluable information regarding the limit of life in extreme environments and its role in the global carbon cycle. It has been observed that subsurface microbial CH4 can form in apparent isotopic equilibrium, both with respect to methane clumped isotopic species and D/H fractionation with respect to coexisting water. This observation fostered the suggestion that methanogenic metabolisms in energy-starved environments can operate through slow and reversible enzymatic reactions. Here we present isotopic data including a vertical profile of clumped isotopic indices of methane from Paleozoic-aged pore waters in an aquiclude system from the Michigan Basin. We show evidence of both internal isotopic equilibrium of methane and intermolecular H-isotopic equilibrium between methane and co-occurring non-gaseous n-alkanes. Various mixing and microbial metabolic models were tested and allowed us to identify the possibility of the production of methane at thermodynamic equilibrium from the syntrophic degradation of sedimentary n-alkanes at ultra-slow rates. Significance StatementThe recent ability to measure the clumped isotopic composition of methane has fostered new ways of observing the deep subsurface biogeochemistry and has been proposed as a new independent geothermometer when methane is formed at or near isotopic equilibrium. In this manuscript, we present the first continuous Paleozoic sedimentary profile of clumped isotopes in pore methane (i.e. directly sampled from the pores of tight sedimentary rocks) from a paleo-bioreactor in the subsurface and show that a syntrophic metabolic network between fermentative bacteria and mainly acetotrophic methanogens can thrive in an energy limited environment.

Enhanced CO2 fixation in the biocatalytic carboxylation of resorcinol: Utilization of amines for amine scrubbing and in situ product precipitation

Abstract Reversible enzymatic decarboxylation reactions are known for their thermodynamic limitations regarding the carboxylation. Significant thermodynamic and kinetic improvements were achieved for the carboxylation of resorcinol by the 2,6-dihydroxybenzoic acid decarboxylase from Rhizobium sp. utilizing the amine scrubber triethanolamine for CO2 mediation. In a first approach, maximal yields of up to 50 % are achieved and are found to correlate with the CO2 loading capacity of triethanolamine. In a second approach, application of quaternary amines for in situ carboxylation product removal by precipitation improves the achievable yields depending on utilized amine concentration. This demonstrates the possibility for a reusable system. By variation of hydrated CO2 species ratio and concentration, CO2 and alternatively H2CO3 is identified acting as second substrate.

Behind Modeling Electrochemical Impedance Response of a Continuous Glucose Monitor

The continuous glucose monitor is used to measure the glucose concentration in interstitial fluids by means of a glucose oxidase enzyme that converts glucose into hydrogen peroxide, which can be detected electrochemically. The development of a model for the electrochemical impedance response of the sensor may be envisioned as a sequential series of steps. The electrochemical oxidation of hydrogen peroxide, in the absence of the enzyme, can be treated as a charge-transfer resistance coupled with a diffusion impedance. In the present case, however, the enzyme provides a homogeneous production of the hydrogen peroxide. The coupling of a homogeneous reaction with a heterogeneous electrochemical reaction was addressed, in the 1950s, by Gerischer, who assumed that homogeneous kinetics can be described by a linear equation, that all diffusion coefficients were equal, and that diffusion took place within a film. The resulting diffusion impedance had two loops, the high-frequency loop corresponding to the homogeneous reaction and the low-frequency loop corresponding to diffusion. While the simplified model represented by the electrochemical oxidation of hydrogen peroxide coupled with a Gerischer impedance yields insight into the impedance response of a glucose sensor, the chemistry of the sensor is much more complex. Three electrochemical reactions may take place in the potential range of interest. The enzymatic production of hydrogen peroxide involves more than one homogeneous reaction, and these reactions may not be modeled by linear equations. In addition, the diffusion coefficients of species associated with the coupled heterogeneous and homogeneous reactions are not equal. The present work describes the development of a mathematical model for the impedance response of glucose-oxidase based electrochemical biosensors [3]. The homogeneous reactions included anomerization between -D-glucose and -D-glucose, four reversible enzymatic catalytic reactions transforming -D-glucose and oxygen into gluconic acid and hydrogen peroxide, weak acid dissociation equilibrium, two reactions accounting of the pH-dependence of enzymatic activity, and a series of reactions associated with the buffer system. The electroactive hydrogen peroxide was considered to be oxidized or reduced at the electrode. In addition, reduction of oxygen was considered as a potential cathodic reaction. The heterogeneous reactions were coupled by the concentration of hydrogen ions, which appear as reactants for the cathodic reactions and as a product of the anodic reaction. Thus, the faradaic impedances associated with the three heterogeneous reactions could not be considered independent. The mathematical model was solved numerically by using the finite-difference method and Newman’s BAND algorithm [4]. The model demonstrates how the coupled non-linear homogeneous reactions affect the diffusion impedance, which has broadened the scope of the Gerischer impedance. The model can be used to explore the influence of various system parameters on limiting current, reaction profiles, and diffusion impedance. The system parameters, including interstitial glucose concentration, oxygen concentration, active enzyme concentration, diffusion coefficients, reaction rate constants and layer thickness, are related to various sensor working conditions such as body sugar level, inflammation, sensor degradation and sensor design. References H. Gerischer, “Wechselstrompolarisation Von Elektroden Mit Einem Potentialbes- timmenden Schritt Beim Gleichgewichtspotential,” Zeitschrift fur Physikalische Chemie, 198 (1951) 286-313.M. E. Orazem and B. Tribollet, Electrochemical Impedance Spectroscopy (John Wiley & Sons, Hoboken, NJ, 2017), 2nd edition, p. 279-293.M. Gao, M. S. Hazelbaker, R. Kong, and M. E. Orazem, “Mathematical Model for the Electrochemical Impedance Response of a Continuous Glucose Monitor,” Electrochimica Acta, 275 (2018), 119-132J. S. Newman and K. E. Thomas-Alyea, Electrochemical Systems (John Wiley & Sons, Hoboken, NJ, 2004), 3rd edition. Acknowledgement The support of Medtronic Diabetes (Northridge, CA) and Andrea Varsavsky, program monitor, is gratefully acknowledged.

Studies on the reversible enzyme reaction of rabbit muscle glycogen phosphorylase b using isothermal titration calorimetry

Glycogen phosphorylase enzymes (GP) catalyse reversible reactions; the glucose transfer from glycogen to inorganic phosphate (Pi, phosphorolysis) or the reverse glucose transfer from glucose-1-phosphate (G-1-P) to glycogen (synthesis). Rabbit muscle GPb (rmGPb) was used as a model enzyme to study the reversible enzyme reaction. To follow both directions of this reversible reaction, we have developed a novel isothermal titration calorimetry (ITC) method for the determination of the direct reaction rate. The preference of forward or reverse reaction was ensured by the 0.1 or 10 concentration ratios of G-1-P/Pi, respectively. Substrate specificity was studied using different maltooligosaccharides and glycogen. Based on the KM values, glycogen and 2-chloro-4-nitrophenyl maltoheptaoside (CNP-G7) were found to be analogous substrates, which allowed to optimize the method by taking advantage of the CNP chromophore being detectable in HPLC. In case of CNP-G7, substrate inhibition was observed and characterised by Ki of 23 ± 7 mM. Inhibition of human GP is a promising strategy for the treatment of diabetes. Our ITC measurements have confirmed that caffeine and glucopyranosylidene-spiro-thiohydantoin (GTH), as known GPb inhibitors, inhibit the rmGPb-catalysed reversible reaction in both directions. Ki values obtained in the direction of synthesis (1.92 ± 0.14 mM for caffeine and 11.5 ± 2.0 μM for GTH) have been shown to be in good agreement with the Ki values obtained in the direction of phosphorolysis (4.05 ± 0.26 mM for caffeine and 13.8 ± 1.6 μM for GTH). The higher difference between the inhibition constants of caffeine was explained by the non-competitive mechanism.The described ITC method using the developed experimental design and reaction conditions is suitable for activity measurements of different phosphorylase enzymes on various substrates and is applicable for inhibition studies as well.

Dynamic control of hydrogel crosslinking via sortase-mediated reversible transpeptidation

Cell-laden hydrogels whose crosslinking density can be dynamically and reversibly tuned are highly sought-after for studying pathophysiological cellular fate processes, including embryogenesis, fibrosis, and tumorigenesis. Special efforts have focused on controlling network crosslinking in poly(ethylene glycol) (PEG) based hydrogels to evaluate the impact of matrix mechanics on cell proliferation, morphogenesis, and differentiation. In this study, we sought to design dynamic PEG-peptide hydrogels that permit cyclic/reversible stiffening and softening. This was achieved by utilizing reversible enzymatic reactions that afford specificity, biorthogonality, and predictable reaction kinetics. To that end, we prepared PEG-peptide conjugates to enable sortase A (SrtA) induced tunable hydrogel crosslinking independent of macromer contents. Uniquely, these hydrogels can be completely degraded by the same enzymatic reactions and the degradation rate can be tuned from hours to days. We further synthesized SrtA-sensitive peptide linker (i.e., KCLPRTGCK) for crosslinking with 8-arm PEG-norbornene (PEG8NB) via thiol-norbornene photocrosslinking. These hydrogels afford diverse softening paradigms through control of network structures during crosslinking or by adjusting enzymatic parameters during on-demand softening. Importantly, user-controlled hydrogel softening promoted spreading of human mesenchymal stem cells (hMSCs) in 3D. Finally, we designed a bis-cysteine-bearing linear peptide flanked with SrtA substrates at the peptide’s N- and C-termini (i.e., NH2-GGGCKGGGKCLPRTG-CONH2) to enable cyclic/reversible hydrogel stiffening/softening. We show that matrix stiffening and softening play a crucial role in growth and chemoresistance in pancreatic cancer cells. These results represent the first dynamic hydrogel platform that affords cyclic gel stiffening/softening based on reversible enzymatic reactions. More importantly, the chemical motifs that affords such reversible crosslinking were built-in on the linear peptide crosslinker without any post-synthesis modification. Statement of SignificanceCell-laden ‘dynamic’ hydrogels are typically designed to enable externally stimulated stiffening or softening of the hydrogel network. However, no enzymatic reaction has been used to reversibly control matrix crosslinking. The application of SrtA-mediated transpeptidation in crosslinking and post-gelation modification of biomimetic hydrogels is innovative because of the specificity of the reaction and reversible tunability of crosslinking kinetics. While SrtA has been previously used to crosslink and fully degrade hydrogels, matrix softening and reversible stiffening of cell-laden hydrogels has not been reported. By designing simple peptide substrates, this unique enzymatic reaction can be employed to form a primary network, to gradually soften hydrogels, or to reversibly stiffen hydrogels. As a result, this dynamic hydrogel platform can be used to answer important matrix-related biological questions that are otherwise difficult to address.

A Mathematical Model for Electrochemical Impedance Response of a Continuous Glucose Monitor

Enzymatic biosensors commonly used for management of diabetes must be maintained in good working condition to provide reliable real-time data. Electrochemical impedance spectroscopy has potential applications in in-vivo sensor diagnostics and sensor calibration. In the present work, a mathematical model for the impedance response of glucose oxidase electrochemical biosensors has been developed [1,2]. The coupling between the homogeneous reactions and heterogeneous reactions considered in the model included anomerization between α-D-glucose and β-D-glucose and four reversible enzymatic catalytic reactions transforming β-D-glucose and oxygen into gluconic acid and hydrogen peroxide. The electroactive hydrogen peroxide was considered to be reversibly oxidized or reduced at the electrode. In the model, the transport of the species were accounted for and the homogeneous enzymatic reaction followed ping-pong kinetics and the laws of mass action. The electrochemical system was modeled mathematically as a one-dimensional boundary value problem and solved using Newman’s BAND algorithm [3]. This model accounts for two layers, a layer adjacent to the electrode where the glucose oxidase is immobilized and the glucose-limiting membrane (GLM), which controls the amount of glucose participating in the enzymatic reaction. The steady-state concentrations of reacting species were calculated, including the concentrations of α- and β-D-glucose, oxygen, gluconic acid, hydrogen peroxide, oxidized and reduced enzyme, and enzymatic-complex intermediates. The solution of the nonlinear coupled set of differential equations for the steady-state was used as input for the set of coupled linear differential equations for the frequency-domain calculations. The corresponding impedance was calculated for each specified frequency [4]. The model was used to explore the influence of system parameters on limiting current, reaction profiles, and diffusion impedance. The system parameters, including interstitial glucose concentration, oxygen concentration, active enzyme concentration, reaction rate constants and layer thickness, are related to various sensor working conditions such as body sugar level, inflammation, sensor degradation and sensor design. A condition was found in which a reduced activity sensor exposed to a high glucose concentration yielded the same current as would be generated by an intact sensor exposed to a lower glucose concentration. The simulation showed measurement of impedance to be a potential tool for differentiating these cases. The value of diffusion resistance and capacitance can be extracted from the impedance at the potential on the mass-transfer limited plateau, which may have some discriminatory value. At a potential below the plateau, the impedance can contain more information. The present mathematical model can guide sensor design and can be used for diagnosing potential sensor failure mechanisms. References M. S. Harding, Mathematical Models for Impedance Spectroscopy, PhD dissertation, University of Florida, 2017.M. Gao, M. S. Hazelbaker, R. Kong, and M. E. Orazem, “Mathematical Model for the Electrochemical Impedance Response of a Continuous Glucose Monitor,” Electrochimica Acta, submitted, 2018.J. S. Newman and K. E. Thomas-Alyea, Electrochemical Systems (John Wiley & Sons, Hoboken, NJ, 2004), 3rd edition.M. E. Orazem and B. Tribollet, Electrochemical Impedance Spectroscopy (John Wiley & Sons, Hoboken, NJ, 2017), 2nd edition, p. 279-293. Acknowledgement The support of Medtronic Diabetes (Northridge, CA) and Andrea Varsavsky, program monitor, is gratefully acknowledged.

Mathematical model for the electrochemical impedance response of a continuous glucose monitor

A mathematical model is developed for the impedance response of immobilized glucose oxidase electrochemical biosensors. The coupling between the homogeneous reactions and heterogeneous reactions considered in the model included anomerization between α-d-glucose and β-d-glucose and four reversible enzymatic catalytic reactions transforming β-d-glucose and oxygen into gluconic acid and hydrogen peroxide. The electroactive hydrogen peroxide was considered to be reversibly oxidized or reduced at the electrode. The electrochemical system was modeled mathematically as a one-dimensional boundary-value problem and solved by use of Newman's BAND algorithm. The corresponding impedance was calculated for each specified frequency. The resulted limiting current, reaction profiles, and impedance response provide insights into the influence of system parameters such as interstitial glucose concentration and enzymatic rate constants. This model has a potential application in predicting sensor design and diagnosing sensor failure mechanisms.

Convergence to equilibrium of renormalised solutions to nonlinear chemical reaction\u2013diffusion systems

The convergence to equilibrium for renormalised solutions to nonlinear reaction–diffusion systems is studied. The considered reaction–diffusion systems arise from chemical reaction networks with mass action kinetics and satisfy the complex balanced condition. By applying the so-called entropy method, we show that if the system does not have boundary equilibria, i.e. equilibrium states lying on the boundary of {mathbb {R}}_+^N, then any renormalised solution converges exponentially to the complex balanced equilibrium with a rate, which can be computed explicitly up to a finite-dimensional inequality. This inequality is proven via a contradiction argument and thus not explicitly. An explicit method of proof, however, is provided for a specific application modelling a reversible enzyme reaction by exploiting the specific structure of the conservation laws. Our approach is also useful to study the trend to equilibrium for systems possessing boundary equilibria. More precisely, to show the convergence to equilibrium for systems with boundary equilibria, we establish a sufficient condition in terms of a modified finite-dimensional inequality along trajectories of the system. By assuming this condition, which roughly means that the system produces too much entropy to stay close to a boundary equilibrium for infinite time, the entropy method shows exponential convergence to equilibrium for renormalised solutions to complex balanced systems with boundary equilibria.

Trend to Equilibrium for a Reaction-Diffusion System Modelling Reversible Enzyme Reaction.

A spatio-temporal evolution of chemicals appearing in a reversible enzyme reaction and modelled by a four-component reaction-diffusion system with the reaction terms obtained by the law of mass action is considered. The large time behaviour of the system is studied by means of entropy methods.

Towards in vivo estimation of reaction kinetics using high-throughput metabolomics data: a maximum likelihood approach.

BackgroundHigh-throughput assays such as mass spectrometry have opened up the possibility for large-scale in vivo measurements of the metabolome. This data could potentially be used to estimate kinetic parameters for many metabolic reactions. However, high-throughput in vivo measurements have special properties that are not taken into account in existing methods for estimating kinetic parameters, including significant relative errors in measurements of metabolite concentrations and reaction rates, and reactions with multiple substrates and products, which are sometimes reversible. A new method is needed to estimate kinetic parameters taking into account these factors.ResultsA new method, InVEst (In Vivo Estimation), is described for estimating reaction kinetic parameters, which addresses the specific challenges of in vivo data. InVEst uses maximum likelihood estimation based on a model where all measurements have relative errors. Simulations show that InVEst produces accurate estimates for a reversible enzymatic reaction with multiple reactants and products, that estimated parameters can be used to predict the effects of genetic variants, and that InVEst is more accurate than general least squares and graphic methods on data with relative errors. InVEst uses the bootstrap method to evaluate the accuracy of its estimates.ConclusionsInVEst addresses several challenges of in vivo data, which are not taken into account by existing methods. When data have relative errors, InVEst produces more accurate and robust estimates. InVEst also provides useful information about estimation accuracy using bootstrapping. It has potential applications of quantifying the effects of genetic variants, inference of the target of a mutation or drug treatment and improving flux estimation.Electronic supplementary materialThe online version of this article (doi:10.1186/s12918-015-0214-7) contains supplementary material, which is available to authorized users.

Global Stability of Reversible Enzymatic Metabolic Chains

We consider metabolic networks with reversible enzymatic reactions. The model is written as a system of ordinary differential equations, possibly with inputs and outputs. We prove the global stability of the equilibrium (if it exists), using techniques of monotone systems and compartmental matrices. We show that the equilibrium does not always exist. Finally, we consider a metabolic system coupled with a genetic network, and we study the dependence of the metabolic equilibrium (if it exists) with respect to concentrations of enzymes. We give some conclusions concerning the dynamical behavior of coupled genetic/metabolic systems.

Iterative approximate solutions of kinetic equations for reversible enzyme reactions

We study kinetic models of reversible enzyme reactions and compare two techniques for analytic approximate solutions of the model. Analytic approximate solutions of non-linear reaction equations for reversible enzyme reactions are calculated using the Homotopy Perturbation Method (HPM) and the Simple Iteration Method (SIM). The results of the approximations are similar. The Matlab programs are included in appendices.

The reversibility of dissimilatory sulphate reduction and the cell-internal multi-step reduction of sulphite to sulphide: insights from the oxygen isotope composition of sulphate

Dissimilatory sulphate reduction (DSR) leads to an overprint of the oxygen isotope composition of sulphate by the oxygen isotope composition of water. This overprint is assumed to occur via cell-internally formed sulphuroxy intermediates in the sulphate reduction pathway. Unlike sulphate, the sulphuroxy intermediates can readily exchange oxygen isotopes with water. Subsequent to the oxygen isotope exchange, these intermediates, e.g. sulphite, are re-oxidised by reversible enzymatic reactions to sulphate, thereby incorporating the oxygen used for the re-oxidation of the sulphur intermediates. Consequently, the rate and expression of DSR-mediated oxygen isotope exchange between sulphate and water depend not only on the oxygen isotope exchange between sulphuroxy intermediates and water, but also on cell-internal forward and backward reactions. The latter are the very same processes that control the extent of sulphur isotope fractionation expressed by DSR. Recently, the measurement of multiple sulphur isotope fractionation has successfully been applied to obtain information on the reversibility of individual enzymatically catalysed steps in DSR. Similarly, the oxygen isotope signature of sulphate has the potential to reveal complementary information on the reversibility of DSR. The aim of this work is to assess this potential. We derived a mathematical model that links sulphur and oxygen isotope effects by DSR, assuming that oxygen isotope effects observed in the oxygen isotopic composition of ambient sulphate are controlled by the oxygen isotope exchange between sulphite and water and the successive cell-internal oxidation of sulphite back to sulphate. Our model predicts rapid DSR-mediated oxygen isotope exchange for cases where the sulphur isotope fractionation is large and slow exchange for cases where the sulphur isotope fractionation is small. Our model also demonstrates that different DSR-mediated oxygen isotope equilibrium values are observed, depending on the importance of oxygen isotope exchange between sulphite and water relative to the re-oxidation of sulphite. Comparison of model results to experimental data further leads to the conclusion that sulphur isotope fractionation in the reduction of sulphite to sulphide is not a single-step process.

Modeling of uncertainties in biochemical reactions

Mathematical modeling is an indispensable tool for research and development in biotechnology and bioengineering. The formulation of kinetic models of biochemical networks depends on knowledge of the kinetic properties of the enzymes of the individual reactions. However, kinetic data acquired from experimental observations bring along uncertainties due to various experimental conditions and measurement methods. In this contribution, we propose a novel way to model the uncertainty in the enzyme kinetics and to predict quantitatively the responses of metabolic reactions to the changes in enzyme activities under uncertainty. The proposed methodology accounts explicitly for mechanistic properties of enzymes and physico-chemical and thermodynamic constraints, and is based on formalism from systems theory and metabolic control analysis. We achieve this by observing that kinetic responses of metabolic reactions depend: (i) on the distribution of the enzymes among their free form and all reactive states; (ii) on the equilibrium displacements of the overall reaction and that of the individual enzymatic steps; and (iii) on the net fluxes through the enzyme. Relying on this observation, we develop a novel, efficient Monte Carlo sampling procedure to generate all states within a metabolic reaction that satisfy imposed constrains. Thus, we derive the statistics of the expected responses of the metabolic reactions to changes in enzyme levels and activities, in the levels of metabolites, and in the values of the kinetic parameters. We present aspects of the proposed framework through an example of the fundamental three-step reversible enzymatic reaction mechanism. We demonstrate that the equilibrium displacements of the individual enzymatic steps have an important influence on kinetic responses of the enzyme. Furthermore, we derive the conditions that must be satisfied by a reversible three-step enzymatic reaction operating far away from the equilibrium in order to respond to changes in metabolite levels according to the irreversible Michelis-Menten kinetics. The efficient sampling procedure allows easy, scalable, implementation of this methodology to modeling of large-scale biochemical networks.

Helping themselves

By using reversible enzyme reactions involving short peptides, molecular synthesis can be controlled by the self-assembly of the resulting products.