Team:USP-Brazil/Model:Deterministic

Deterministic model

Introduction

We used a deterministic approach in order to develop a simple mathematical model to relate the expression of a report protein RFP with different concentrations of ethanol and methanol. By making some hypothesis, we were able to end up with a very simplified model that can help us to understand about the biological mechanism of the activation of the promoter P_AOX. This activation is mediated by the transcript factor Mxr1 that in the presence of methanol, activates the expression of the promoter. Nevertheless, the Mxr1 repress the expression of the promoter in the presence of other sources of carbon, like ethanol. Because of that, in our experimental design we decided to use a modified Mxr1 transcript factor that instead of repressing the expression of the promoter in the presence of ethanol, it activates the expression.

The model

We first model the reaction of the “activation” of the transcript factor, referred as (X), by ethanol or methanol. Initially, the transcript factor is free, X_f. Then, after interacting with the ethanol or methanol, it becomes X_et or X_met. This reaction can be described as follows:

\begin{equation} X_f + et \leftrightarrows X_{et} \end{equation} \begin{equation} X_f + met \leftrightarrows X_{met} \end{equation}

Our first hypothesis is that this reaction occurs much faster than any protein production. Because of that, we can assume that this reaction reaches quickly the equilibrium. Then at the equilibrium point we have:

\begin{equation} \frac{d[X_{et}]}{dt} = \beta^{+}_{et}[et][X_f] - \beta^{-}_{et}[X_{et}] = 0 \end{equation} \begin{equation} \frac{d[X_{met}]}{dt} = \beta^{+}_{met}[met][X_f] - \beta^{-}_{met}[X_{met}] = 0 \end{equation}

and so,

\begin{equation} [X_{et}]_{eq} = \beta_{et}[et][X_f]_{eq} \end{equation} \begin{equation} [X_{met}]_{eq} = \beta_{met}[met][X_f]_{eq} \end{equation}

where

$$\beta = \frac{\beta^+}{\beta^-}$$

A second hypothesis we will assume is that in the equilibrium, the concentration of X_f is much lower than the transcript factor modified by ethanol or methanol. In mathematical formulation:

\begin{equation} [X_{f}]_{eq} << ( [X_{et}]_{eq}+[X_{met}]_{eq} ) \end{equation}

This hypothesis seems quite reasonable since the concentration of methanol or ethanol is expected to be orders of magnitude higher than the concentration of the transcript factor. Then it is expected that the majority of the transcript factor have interacted with those metabolites. Because of that, we can also assume that the concentration of the metabolites do not change in time.

Now we can use the following conservation rule:

\begin{equation} [X_{f}]_{0} = [X_f]_{eq} + [X_{et}]_{eq} + [X_{met}]_{eq} \end{equation}

then we have:

\begin{equation} [X_{et}]_{eq} = [X_f]_{0}\frac{\beta_{et}[et]}{1 + \beta_{et}[et] + \beta_{met}[met]} \end{equation} \begin{equation} [X_{met}]_{eq} = [X_f]_{0}\frac{\beta_{met}[met]}{1 + \beta_{et}[et] + \beta_{met}[met]} \end{equation} and using equation 7: \begin{equation} [X_{et}]_{eq} \approx [X_f]_{0}\frac{\beta_{et}[et]}{\beta_{et}[et] + \beta_{met}[met]} \end{equation} \begin{equation} [X_{met}]_{eq} \approx [X_f]_{0}\frac{\beta_{met}[met]}{\beta_{et}[et] + \beta_{met}[met]} \end{equation}

Now, lets describe the model for the report protein (R) production. After interacting with ethanol or methanol the transcript factor activates the promoter. However, it transcript factor X_et or X_met activates with different rates k_et or k_met, respectively. Then, the protein production can be described by the following equation:

\begin{equation} \frac{d[R]}{dt} = k_{et}\theta_{et} + k_{met}\theta_{met} - \gamma [R] \end{equation}

where θ_et represents the fraction of the time the promoter is activated by X_et. Since the activation of the promoter occurs at the same scale of the protein production promoter P, that can be represent as:

\begin{equation} P_f + X_{et} \leftrightarrows P_{et} \end{equation} \begin{equation} P_f + X_{met} \leftrightarrows P_{met} \end{equation}

from this system:

\begin{equation} \frac{d[P_{et}]}{dt} = \alpha^{+}_{et}[X_{et}][P_f] - \alpha^{-}_{et}[P_{et}] \end{equation} \begin{equation} \frac{d[P_{met}]}{dt} = \alpha^{+}_{met}[X_{met}][P_f] - \alpha^{-}_{met}[P_{met}] \end{equation}

and at equilibrium:

\begin{equation} [P_{et}]_{eq} = \alpha_{et}[X_{et}]_{eq}[P_f]_{eq} \end{equation} \begin{equation} [P_{met}]_{eq} = \alpha_{met}[X_{met}]_{eq}[P_f]_{eq} \end{equation}

we can obtain the fraction of time the promoter is bound by each type of transcriptor factor.

\begin{equation} \theta_{et} = \frac{\alpha_{et}[X_{et}]}{\alpha_{et}[X_{et}] + \alpha_{met}[X_{met}]} = \frac{1}{1 + \lambda \frac{[X_{met}]}{[X_{et}]}} \end{equation} \begin{equation} \theta_{met} = \frac{\alpha_{met}[X_{met}]}{\alpha_{et}[X_{et}] + \alpha_{met}[X_{met}]} = \frac{1}{1 + \lambda^{-1} \frac{[X_{et}]}{[X_{met}]}} \end{equation}

where

$$\lambda = \frac{\alpha_{met} \beta_{met}}{\alpha_{et} \beta_{et}}$$

Here, is important to note that we also assumed that the promoter is bound the majority of the time.

Then, at the equilibrium we have:

\begin{equation} [R]_{eq} = \frac{1}{\gamma} \left[ k_{et}\frac{1}{1 + \lambda \frac{[met]}{[et]}} + k_{met}\frac{1}{1 + \lambda^{-1} \frac{[et]}{[met]}}\right] \end{equation}

or

\begin{equation} [R]_{eq} = \hat{k}_{et}\frac{1}{1 + \lambda \frac{[met]}{[et]}} + \hat{k}_{met}\frac{1}{1 + \lambda^{-1} \frac{[et]}{[met]}} \end{equation}

where

$\hat{k} = \frac{k}{\gamma}$

Now, we ended up with a simplified model, that depends only on three variables, k_et, k_met and λ.

References

[1]

RFP Visibility | Deterministic Model | Stochastic Model