Team:BroadRun-Baltimore/Model

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Models

Having a kinematic model that mimics the behavior of process water in the ceiling tile plant is a valuable design and analysis tool to investigate potential solution strategies such as the proposed synthetic biology solution of engineered yeast. As a first step towards this goal of a simulated design and testing environment, basic mathematical models are developed here to characterize both the kinetics of yeast cell growth and the enzymatic reaction of starch degradation by alpha-amylase. Such a kinetic model for microbial growth and product formation must account for the different phases of growth, most importantly an exponential growth phase during cell seeding,a stationary phase relating to steady-state operation, and a death phase associated with nutrient deficiency or toxic products.

Several models can be found in the literature [1-4] to describe cell growth, substrate utilization, and product formation. The most widely used models are simpler unstructured models that also use the Monod equation to represent the logistic growth characteristics typical of microbials. This model is developed mathematically with the following differential equations.

Future Modeling

A model to include the bacteria prevalent in the ceiling tile process water and wastewater will be developed. To reflect both organisms, bacteria and yeast, the present model will be extended to incorporate the bacteria model and mathematical simulations will be run to gather insight on biomass, starch, and butyric acid kinematics under varying plant conditions of DO, temperature, pH, and starch input starch fed through the recycling process.

References

1. Shuler, M. L., & Kargi, F. (2002). Bioprocess engineering: Basic concepts. Chapter 6. Upper Saddle River, NJ: Prentice Hall.

2. Zangirolami, T.C., Carlsen, M., Nielsen, J., & Jørgensen, S.B.. (2002). Growth and enzyme production during continuous cultures of a high amylase-producing variant of Aspergillus Oryzae. Brazilian Journal of Chemical Engineering, 19(1), 55-68

3. Wang, L., D. Ridgway, T. Gu and M.Y. Murray, 2009. Kinetic modeling of cell growth and product formation in submerged culture of recombinant Aspergillus niger. Chem. Eng. Com., 196:481-490.

4. Akpa J (2012) Modeling of a Bioreactor for the Fermentation of Palm wine by Saccaharomyce cerevisiae (yeast) and lactobacillus (bacteria). Bioresource Technology 3: 231-240.

Inspiration

Above Math expressions from https://2015.igem.org/Team:Oxford/Modeling: \[(Arab:AraC)\overset{K}{\rightarrow}mRNA\overset{\alpha}{\rightarrow}P\] \[\dfrac{d[mRNA]}{dt}=K_{max}\dfrac{[Arab:AraC]^{n}}{K_{half}^{n}+[Arab:AraC]^{n}}-\gamma_{1}[mRNA]\] \[\dfrac{d\left[P\right]}{dt}=\alpha\left[mRNA\right]-\gamma_{2}\left[P\right]\]

Michaelis-Menten Kinetics

It is possible to formulate a rate law that describes enzyme-catalysed reactions, this is known as Michaelis-Menten kinetics. In order to derive this rate law we will look at a generalised system of chemical events detailed below:

\[S + E\mathop{\rightleftharpoons}C_{1}\mathop{\rightleftharpoons}C_{2}\mathop{\rightleftharpoons}P + E\]

In this system \(S\) represents our substrate, \(E\) our enzyme, \(C_{1}\) our enzyme-substrate complex, \(C_{2}\) our enzyme-product complex, and \(P\) our product.

Initially we will make two simplifications. Firstly, we will combine \(C_{1}\) and C2 into a single complex, \(C\), as we will assume that the time-scale of the conversion \(C_{1}\mathop{\rightleftharpoons}C_{2}\) is much faster than that of the association and dissociation events. Secondly, we will assume that the product never binds with the free enzyme. These two assumptions lead to the simplified network:

\[S + E\mathrel{\mathop{\rightleftharpoons}^{k_{1}}_{k_{-1}}}C\mathop{\rightarrow}^{k_{2}}P + E\]

Using the laws of mass action (detailed above) we arrive at the following differential equation model:

\[\dfrac{d}{dt}s(t)=-k_{1}s(t)e(t) + k_{-1}c(t)\] \[\dfrac{d}{dt}e(t)=k_{-1}c(t) - k_{1}s(t)e(t) + k_{2}c(t)\] \[\dfrac{d}{dt}c(t)=-k_{-1}c(t) + k_{1}s(t)e(t) - k_{2}c(t)\] \[\dfrac{d}{dt}p(t) = k_{2}c(t)\]

Concentrations are denoted as the lowercase letter eg the concentration of \(S\) is given by \(s\).

You may have spotted that the enzyme is not consumed in this series of reactions. Therefore the total amount of enzyme remains constant. We reflect this in the expression \(e_{T}=e+c\). We can now use this expression to eliminate \(e(t)\) from our model, leaving:

\[\dfrac{d}{dt}s(t)=-k_{1}s(t)(e_{T}-c(t)) + k_{-1}c(t)\] \[\dfrac{d}{dt}c(t)=-k_{-1}c(t) + k_{1}s(t)(e_{T}-c(t)) - k_{2}c(t)\] \[\dfrac{d}{dt}p(t) = k_{2}c(t)\]

We have one further simplification to make. This is the rapid equilibrium approximation, by which we assume that equilibrium between \(s+e\) and \(c\) on a much faster time scale than the reaction of \(c\) to \(p\). With this approximation we can write the following equation:

\[0=-k_{-1}c(t) + k_{1}s(t)(e_{T}-c(t)) - k_{2}c(t)\]

Leading to:

\[c^{*}(t)= \dfrac{k_{1}s(t)e_{T}}{k_{-1} + k_{2} + k_{1}s(t)}\]

Now we can see that \(c\) is no longer an independent variable, but instead tracks the other variables. This can be referred to as \(c\) being in quasi-steady state. By including this new expression into our model we are left with:

\[\dfrac{d}{dt}s(t) = -\dfrac{k_{2}k_{1}s(t)e_{T}}{k_{-1} + k_{2} + k_{1}s(t)}\] \[\dfrac{d}{dt}p(t) = \dfrac{k_{2}k_{1}s(t)e_{T}}{k_{-1} + k_{2} + k_{1}s(t)}\]

With this we have an expression describing our enzyme catalysed system in the form of a single reaction. The rate of this reaction is known as a Michaelis-Menten rate law. By defining \(K_{max}=k_{2}e_{T}\) and \(K_{half}=\dfrac{k_{-1}+k_{2}}{k_{1}}\) we can express in the more familiar form:

\[rate \: of \: S \mathop{\rightarrow} P = K_{max}\dfrac{s}{K_{half} + s}\]

It is worth noting that there are a number of ways to derive this formula, each involving different approximations. These methods may lead to the constants \(K_{max}\) and \(K_{half}\) being defined differently.