Team:Rice/Model/ORibosomes
Orthogonal Expression
Overview
Our model incorporates the transcription of orthogonal 16S rRNAs that bind to ribosomal proteins to form orthogonal ribosomes. These orthogonal ribosomes can be restricted to translate only mRNAs from circuit genes. We have the equations for the orthogonal T7 RNA polymerases on the polymerase addition page.
Explanation
Note: if the equations below seem messed up, please follow this link to a simplified version of this page.
Orthogonal ribosomes are composed of two parts: host ribosomal proteins ($p_R$) and orthogonal 16S rRNA ($r_O$). The orthogonal rRNA is transcribed just like the host rRNA, with a maximum transcription rate of $w_O$ and a transcription energy threshold of $o_O$. These rRNAs degrade at the same rate as all other RNAs, with a rate constant of $d_r$. The orthogonal rRNAs bind to the host ribosomal proteins in a reversible fashion, effectively functioning as competitive inhibitors of the binding of host rRNAs. The orthogonal ribosomes formed only bind to circuit mRNAs ($m_C$) upon which they form a translation complex $c_C$ and generate circuit proteins. Because the orthogonal rRNAs bind to host ribosomal proteins $p_R$, we need to also modify the differential equation governing $p_R$.
Equations
\begin{equation*} \begin{aligned} \frac{dr_O}{dt} &= w_O \frac{E_c}{E_c+o_O} - b_r p_R r_O + u_r R_O - (d_r + \lambda) r_O \\ \frac{dR_O}{dt} &= b_r p_R r_O - u_r R_O +\gamma \frac{c_C}{n_C} + u_r c_C - b_r R_O m_C - \lambda R_O \\ \frac{dp_R}{dt} &= \gamma \frac{c_R}{n_R} - b_r p_R r_R - b_r p_R r_O + u_r R_h + u_r R_O - \lambda p_R \\ \end{aligned} \end{equation*}