Team:ITESMCEM/EnzymaticKinectics
From 2014.igem.org
ITESMCEM  Enzy7K me 


Modeling


 


OverviewDuring 2014, iGEM ITESM CEM Team worked on the development of a metabolic pathway for 7ketocholesterol degradation by enzyme therapy of human macrophages, using two hypothetic enzymes of the microorganism Rhodococcus jostii, active when the microbe grows using 7ketocholesterol as a sole source of carbon. These enzymes (oxoacyl reductase and 7dehydratase) were first described as potentially being used to catalyse these reactions by Mathieu (1); and were cloned and purified by iGEM ITESM CEM Team using E. coli and a diverse array of expression vectors. Figure 1 shows the general array of chemical reactions in the pathway, were red arrows indicate the direction proposed by iGEM ITESM CEM Team. Even though reactions can occur in almost any order, the most intuitive arrangement is that in which 7ketocholesterol is converted to 7βOHcholesterol, which is finally transformed to regular cholesterol; these two reactions are supposed to be catalysed by oxoacyl reductase and 7dehydratase respectively. Model DescriptionThe general MichaliesMenten model was used, which was stated in a differential form, as the following: Where Km, as usually, represents the substrate concentration needed to reach one half of the maximum reaction rate, a measurement of enzyme’s affinity for its substrate, and Rmax is the maximum reaction rate that can be achieved under the analysed pH, temperature, pressure and enzyme concentration conditions. In this case, three of these models were generated: each one describing the reaction rate of each component of the pathway (7ketocholesterol, 7βOHcholesterol, and cholesterol). In order to specifically focus on 7keto removal, no 7ketocholesterol synthesis or uptake was considered; and no cholesterol metabolism was included among the model. This means that an initial concentration of 7ketocholesterol is gradually degraded until it reaches zero, and that cholesterol slowly builds up until it reaches a limit concentration. As for 7 βOHcholesterol, its concentration varies according to the rate of reaction of both enzymes, firsts accumulating and then being degraded. The following system of differential equations was generated, where each compound has a particular abbreviation (7KC for 7ketocholesterol, 7βOHC for 7βOHCholesterol, and Ch for Cholesterol: This means that the concentration of 7ketocholesterol is always described by the MichaelisMenten reaction rate of the first enzyme, which tends to degrade it (a zero term stands for the nonexisting 7ketocholesterol synthesis); the concentration of 7 βOHcholesterol is described by both the first and the second enzyme MicahelisMenten kinetics, where the first one models its synthesis and the second one its degradation. Finally, cholesterol concentration is zero in the first stage of the reaction (when 7 βOHcholesterol has not been metabolized yet), and is modelled by the second enzyme kinetics in the second reaction stage; here, it is generated at a reaction rate of the same magnitude and opposite sign of that of 7βOHCholesterol degradation (a zero term stands for the nonexisting cholesterol metabolism). These iterative equations were programmed using Wolfram Mathematica; the results are presented in the following section. Model predictions The three coupled RungeKutta numerical approximation methods were performed between times 0 and 0.01, with varying parameter values. In order to perform a comparative assessment between the efficiency of the endogenous degradation of 7ketocholesterol, and our proposed synthetic pathway, some of the plots generated are included in figure 2. Because all of the enzymes cloned by iGEM ITESM CEM Team are supposed to be targeted to the lysosome via a signal peptide, they are assumed to work at low values of pH; this in turn promotes a high reaction rate for 7 βOHCholesterol degradation, because the hydroxyl group at the 7th position of the core rings becomes a strong nucleophile when surrounded by an acidic environment, and because usually this step of the pathway must be fast enough so that this intermediate, very reactive with lipid bilayers, does not accumulate. This is the reason why both the maximum rate (Rmax2) and the Michaelis constant (Km2) of the second enzyme were fixed to be very high and very low, respectively. Based of similar enzymes’ parameters reported in BRENDA, Rmax = 1100 and Km2 = 0.5 were used, this accounts for a fast reaction rate of reaction. The first plot might accurately represent the situation inside human macrophages, where 7ketocholesterol cannot be degraded at an appropriate rate, and gradually accumulates. If we further consider that this model does not include the endogenous synthesis and the uptake of 7ketocholesterol, we can easily see why this molecule’s concentration rapidly builds up in the bloodstream. However, as Km values for this reaction diminish, the plots tend to represent more efficient metabolic pathways: we expect our synthetic route, based of Rhodococcus jostii’s sterol metabolism, to be modelled by one of the last two plots in the figure; here, 7ketocholesterol is degraded faster, and regular cholesterol, which can readily be metabolized by human enzymes, is produced. ConclusionA mathematical model based on a system of three differential equations was built using MichaelisMenten enzyme kinetics in order to quantitatively describe the behaviour of a proposed metabolic pathway for 7ketocholesterol degradation and atherosclerosis prevention. The model was solved numerically using RungeKutta, fourth order approximations, and the results show that, the greater the affinity of the enzymes for 7ketocholesterol (which means lower Michaelis constant values), the larger will the efficiency of degradation be. When using this differential model, the proposed metabolic pathway, based on Rhodococcus jostii metabolism, is predicted to metabolize 7ketocholesterol way faster than regular human cells’ pathways. References 1. Mathieu JM. Strategies for the mitigation of oxysterolinduced cytotoxicity. Texas: Rice University; 2011.


