Team:BroadRun-Baltimore/Model

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Models

Having a kinematic model that simulates the plant is a valuable platform to investigate potential solution strategies such as modified yeast in a simulated testing environment. Towards that goal, basic mathematical models for

\[(Arab:AraC)\overset{K}{\rightarrow}mRNA\overset{\alpha}{\rightarrow}P\] \[\dfrac{d[mRNA]}{dt}=K_{max}\dfrac{[Arab:AraC]^{n}}{K_{half}^{n}+[Arab:AraC]^{n}}-\gamma_{1}[mRNA]\] \[\dfrac{d\left[P\right]}{dt}=\alpha\left[mRNA\right]-\gamma_{2}\left[P\right]\]

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.