In 1864, Waage and Guldberg formulated the "law of mass action." Since that time, chemists, chemical engineers, physicists and mathematicians have amassed a great deal of knowledge on the topic. In our view, sufficient understanding has been acquired to warrant a formal mathematical consolidation. A major goal of this consolidation is to solidify the mathematical foundations of mass action chemistry -to provide precise definitions, elucidate what can now be proved, and indicate what is only conjectured. In addition, we believe that the law of mass action is of intrinsic mathematical interest and should be made available in a form that might transcend its application to chemistry alone. We present the law of mass action in the context of a dynamical theory of sets of binomials over the complex numbers.
BACKGROUND: The bioavailability of therapeutic proteins is improved through PEGylation. This chemical modification involves the production of isomers with different numbers and sites of attached PEG chains, which are difficult to separate efficiently. Their purification with chromatography requires an understanding of the operation and the evaluation of different operational conditions. The General Rate Model (GRM) was applied for modelling the linear salt gradient elution of mono-PEGylated and native lysozyme in Heparin Affinity Chromatography (HAC) considering mass balance equations for proteins in the bulk-fluid phase, in the particle phase and the kinetic adsorption. length (5, 13, 25 column volumes) and flow (0.8 and 1.2 mL min -1 ) with a relative error in retention times of less than 6% and correlation coefficients greater than 0.78. CONCLUSION: Simulation of the elution curves of PEGylated lysozyme in HAC was performed in this work and the diverse information generated by the model is explained through the physicochemical protein properties. This simulation represents a tool for optimization, prediction and future scale-up of PEGylated proteins purification, which would reduce the investment in time and resources to test several operating conditions. RESULTS: The model was able to simulate the individual proteins and the separation of these in a PEGylation reaction using as proof-of-concept a mono-PEGylated and native lysozyme mixture under changes of operational parameters such as the gradient
A continuous model of a metabolic network including gene regulation to simulate metabolic fluxes during batch cultivation of yeast Saccharomyces cerevisiae was developed. The metabolic network includes reactions of glycolysis, gluconeogenesis, glycerol and ethanol synthesis and consumption, the tricarboxylic acid cycle, and protein synthesis. Carbon sources considered were glucose and then ethanol synthesized during growth on glucose. The metabolic network has 39 fluxes, which represent the action of 50 enzymes and 64 genes and it is coupled with a gene regulation network which defines enzyme synthesis (activities) and incorporates regulation by glucose (enzyme induction and repression), modeled using ordinary differential equations. The model includes enzyme kinetics, equations that follow both mass‐action law and transport as well as inducible, repressible, and constitutive enzymes of metabolism. The model was able to simulate a fermentation of S. cerevisiae during the exponential growth phase on glucose and the exponential growth phase on ethanol using only one set of kinetic parameters. All fluxes in the continuous model followed the behavior shown by the metabolic flux analysis (MFA) obtained from experimental results. The differences obtained between the fluxes given by the model and the fluxes determined by the MFA do not exceed 25% in 75% of the cases during exponential growth on glucose, and 20% in 90% of the cases during exponential growth on ethanol. Furthermore, the adjustment of the fermentation profiles of biomass, glucose, and ethanol were 95%, 95%, and 79%, respectively. With these results the simulation was considered successful. A comparison between the simulation of the continuous model and the experimental data of the diauxic yeast fermentation for glucose, biomass, and ethanol, shows an extremely good match using the parameters found. The small discrepancies between the fluxes obtained through MFA and those predicted by the differential equations, as well as the good match between the profiles of glucose, biomass, and ethanol, and our simulation, show that this simple model, that does not rely on complex kinetic expressions, is able to capture the global behavior of the experimental data. Also, the determination of parameters using a straightforward minimization technique using data at only two points in time was sufficient to produce a relatively accurate model. Thus, even with a small amount of experimental data (rates and not concentrations) it was possible to estimate the parameters minimizing a simple objective function. The method proposed allows the obtention of reasonable parameters and concentrations in a system with a much larger number of unknowns than equations. Hence a contribution of this study is to present a convenient way to find in vivo rate parameters to model metabolic and genetic networks under different conditions. Biotechnol. Bioeng. 2012;109: 2325–2339. © 2012 Wiley Periodicals, Inc.
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