Mathematical Modeling is an essential component of our project for predictive purposes.
We utilized enzyme-kinetic equations as a predictive model to compare the substrate concentration [S] vs. enzyme concentration [E] for the reaction between alpha-amylase and 2-chloro-nitrophenyl (CNP-G3) and the reaction between alpha-amylase, CNP-G3, and the unmodified 0.19 inhibitor.
A final aspect of mathematically modeling was to help expand the opportunities for other iGEM teams to produce mathematical models. We created a massive open online course (MOOC), Mathematically Modeling Enzyme Kinetics, as an introductory course for high school students through the MOOC platform Udemy as well through the Youtube platform. Throughout this wiki page, we provide some of the lecture videos used in the MOOC for explanations of the Kinetic Laws we used throughout our project.
Modeling Assumptions
We modeled with the 0.19 protein interaction with alpha-amylase being competitive inhibition. While literature has stated both competitive and non-competitive inhibition, using competitive was more appropriate for modeling purposes.
The enzyme is pure.
The enzyme is fully active.
We also utilized the concept of chemical equilibrium throughout our modeling process. Here is our video explanation for chemical equilibrium:
Defining Kinetic Laws
In our mathematical modeling, we focused primarily on four Kinetic Laws: the Law of Mass Action, Henri-Michaelis-Menten, Noncompetitive Inhibition, and Competitive Inhibition.
Law of Mass-Action
The law of mass action is the principle that states the concentration of products produced from a reaction is directly proportional to the concentration of reactants used. The law of mass action is generally represented as an equation that shows the rate of the product produced in a chemical reaction is proportional to the concentration of the reactions.
The mass-action kinetic law was essential to understand as it forms the basis of the Henri-Michaelis-Menten, Noncompetitive inhibition, and competitive inhibition kinetic laws.
Henri-Michaelis-Menten
The Henri-Michaelis-Menten is an equation that utilizes mass action kinetics to model the patterns of enzyme catalysis. We focused on the reversible Henri-Michaelis-Menten modeling because this equation can be transformed to model the Noncompetitive Inhibition and the Competitive Inhibition binding.
Background to Henri-Michaelis-Menten
Derivation of Henri-Michaelis-Menten
Competitive Inhibition
Competitive inhibition occurs when the inhibitor directly blocks the active site on the enzyme to prevent the chemical reaction from being catalyzed. This inhibition type means the presence of enzyme, substrate, and inhibitor, there are two possible outcomes from the enzyme and its interaction with another substance. It will either form an enzyme-substrate complex or an enzyme-inhibitor complex. However, the enzyme-substrate complex is the only complex within this reaction to create the product.
Kinetic Parameters
The non-inhibitor Michaelis Menten equation, as discussed previously above, is as follows:
Our x-variable for our graph will be the [S] (substrate concentration). Thus, we will have to define the kinetic parameters: Vmax and m.
To determine the kinetic parameters for our model, we first found the Km (Michaelis Constant) value of the 2-Chloro-4-nitrophenyl-α-D-maltotrioside. (CNP-G3) and human salivary alpha-amylase as being 0.66 +- 0.04 (Lorentz 1999). We also found that when the concentration of the enzyme was at 50 mmol/L, the Vmax value was 4.0 mmol/S/L (Lorentz 1999).
The Vmax value is dependent on the concentration of the enzyme. Thus, we calculated the kcat (catalytic constant) by dividing the Vmax value by the enzyme concentration:
The kcat value will then allow us to determine the Vmax for any enzyme concentration between CNP-G3 and human salivary alpha-amylase. The Vmax for the between human salivary alpha-amylase and CNP-G3 would thus be as follows:
Our assay procedures called for 1,500 U/mg with 1.6U/microliter. We also needed to use the molecular weight of the non-glycosylated salivary alpha-amylase to calculate the concentration, which was 56 KDa (Fischer 2006). We thus did the below calculations to determine the concentration of the salivary alpha-amylase within our assay:
We thus used this value as our enzyme concentration value for our modeling. We then used the previously derived Vmax equation to determine the Vmax of the equation with the enzyme concentration. Our yield was thus:
With this information, we had enough data to determine the changes in product concentration based on the changes of substrate concentration, with the substrate concentration ranging from 0-30 mmol/L. The graph is shown below:
The equation to create this graph based on the previously discussed kinetic parameters is shown below:
Our next graph would be the predictive model of the CNP-G3 digestion with the presence of 0.19 protein. Thus, we need to follow the competitive inhibition Michaelis-Menten equation for our modeling, which is as follows:
Thus, our variables to create this graph would be as follows:
V max
K m
[I]
K i
We have already previously defined our Km and Vmax value. However, we must still determine our [I] value as well as our Ki value.
We then determined that we would like to model for when the inhibitor concentration is 0.05 mmol/L, due to this being near the concentration of the secretion tag 0.19 protein construct.
However, we encountered some gaps within research when determining the inhibition constant between human salivary alpha-amylase and 0.19 inhibitor. The only inhibition constant we could find of the interaction with an alpha-amylase and a 0.19 inhibitor was with porcine pancreatic alpha-amylase, which was determined to be 57.3 nM (Oneda 2004). Because the rest of our model is in micro units, we converted this value to become 0.0573 mM and used this as our inhibition constant.
The equation to create this graph based on the previously discussed kinetic parameters is as follows:
Python Code
Model Results
Limitations
Limitations with Predictive Model
Our predictive modeling has some clear limitations that must be addressed:
It is unreasonable to assume the inhibition constant of 0.19 protein and porcine pancreatic alpha-amylase would be the same value. However, this was the inhibition constant available for us for 0.19 protein.
We used two different papers (Oneda 2004 and Lorentz 1999) for our modeling parameters.
Limitations with Determination of Inhibition Constants
Our project had many limitations that ultimately led to the inability to use the data collected in the experiments to determine the inhibition constants of the 0.19 protein interacting with human salivary alpha-amylase as well as of our modified 0.19 protein interacting with human salivary alpha-amylase. These include:
No consistent access to 96 well plate reader for repeated measurements
Low 0.19 concentration expressed in constructs
Did not sample at different substrate concentrations (made the independent variable time instead of substrate concentration)
While we were able to clearly show that the enzymatic activity of α-amylase was weaker for our modified 0.19 through the quantitative measurements of the enzyme activity (as seen in the graph below), we were unsuccessful in implementing Michaelis-Menten equations to determine the inhibition constants.
Future Plans
It would be in our best interest to do an assay that would have the substrate concentration as the independent variable and the dependent variable being the reaction velocity. We would then be able to use the Dixon/Eadie-Hofstee/Hanes-Woolf/Hills Plot transformation to find the inhibitor constant of 0.19 and modified 0.19 with human salivary- alpha-amylase. If we were able to find the IC50 value, we could also use the Cheng-Prusoff relationship to determine the inhibition constant. This would then allow us another method to quantitatively determine that our modified 0.19 was a more effective alpha amylase inhibitor than the original 0.19 as well as contribute to the body of research regarding alpha-amylase inhibitors.
References:
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