Modeling results

Gene Expression

Here we show our models to predict gene expression from our main circuits using roxA and lcp. We also show a detailed study of the kill switch mechanism we have used.

The main process we want to model in this section is the expression and translation of the genes involved in our circuits. Qualitatively, this whole process is triggered by the attachment of an enzyme that synthesizes RNA, known as RNA polymerase (RNAP), to the DNA near a gene. Promoters contain specific DNA sequences such as response elements that provide a secure initial binding site for RNA polymerase and for proteins called transcription factors that recruit RNAP. Then, mRNA associated with that gene is generated which can then be translated into a protein by ribosomes. This second process (translation) is triggered when a ribosome binds at a special region, called Ribosomal Binding Site (RBS). At the end of the day, a protein is produced.

At this point, the protein might be in its function conformal structure or not. We will stop at this point for now and model this whole process from rather coarser level of details: we will simulate populations of mRNAs and proteins and predict their concentration over time. As a sneaky-peek of what is to come, this is the basic prediction of our model:

Notice that when Rhamnose concentration (black line) drops, TetR also drops and HokD starts to be produced, killing the cell.

Main objective

Before we start dwelling with our mathematical modeling, let's state what are our main objectives and goals with this model. We want to ...

predict protein/enzyme concentration
take into account Rhamnose concentration
...

Modeling with Differential Equations

As mentioned above, we want to model the gene expression. Fundamentally, RNA polymerase (RNAP) binds to a transcription factor and triggers the transcription of a gene into mRNA; subsequently, ribosomes starts the translation of that mRNA into a protein/enzyme. The schematics below summarizes it.

Each process is based on the combination of two elements: first, a RNAP and the gene; then, ribosome and mRNA. The simplest way to model such combinations and their products is to use a Dynamical Systems approach [1,2], similar to techniques employed in Population Dynamics [3]. For a introductory text on this subject, click here.

RoxA and Lcp Production

Particularly, we want to study . Let's define $P_{rham}(t)$ as the Rhamnose concentration as function of time. This will promote the expression of our circuit. The concentration of RoxA and Lcp proteins are $P_{r}(t)$ and $P_{\ell}(t)$, respectively. The Kill Switch is promoted with TetR concentration, denoted by $P_{t}(t)$. The kill switch should then express HokD, $P_{h}(t)$, to activate a self-destruction mechanism.

Let's start defining the differential equation for the production of RoxA and Lcp. Their concentration depend only on the concentration of Rhamnose and their own degradation rate. Let's define $\delta_r$ and $\delta_{\ell}$ as the degradation rate of RoxA and Lcp respectively. Additionally, to set the promotion levels of Rhamnose, we define the $\beta$ parameter. Then, the differential equations become:

\[ \frac{dP_{r}(t)}{dt} = - \delta_r P_{r}(t) + \beta \frac{P^n_{Rham}(t)}{K^n + P^n_{Rham}(t)} \] \[ \frac{dP_{\ell}(t)}{dt} = - \delta_{\ell} P_{\ell}(t) + \beta \frac{P^n_{Rham}(t)}{K^n + P^n_{Rham}(t)} \]

Notice that we have used a Hill function for the Rhamnose promotion. Jointly, tetR will be also expressed and its differential equation can be written as follows:

\[ \frac{dP_{t}(t)}{dt} = - \delta_{t} P_{t}(t) + \beta \frac{P^n_{Rham}(t)}{K^n + P^n_{Rham}(t)} \]

Kill Switch: hokD

The Kill Switch mechanism triggers a self-destruction mechanism based on a toxic enzyme, HokD. expression will then depend on $P_t (t)$,

\[ \frac{dP_{h}(t)}{dt} = - \delta_{h} P_{h}(t) + \beta_0 + \beta_t \frac{P^n_{t}(t)}{K^n + P^n_{t}(t)} , \]

where $\beta_0$ is the leakeage term [2] and $\beta_t$ defines the level of promotion of the tetR promoter.

Fitting the parameters

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