Purpose of modeling

In this project we developed a model of a trap-RNA system with the dual purpose of both developing hypotheses about the system and providing synthetic biologists the framework to incorporate the trap-RNA systems into advanced modeling of their designs. Furthermore we provide the ability to predict the caused gene repression based on key parameters some of which are changeable by altering the primary sequences involved...

biologist who wish to employ a trap-RNA system with the means of how to assess the tunability of gene repression by altering...

General kinetic model

The modeling of the trap-RNA system is based on reaction kinetics and multiple models are proposed based on two general reaction schemes

\begin{eqnarray} \leftrightharpoons_k^l \label{eq:r1} &\ce{\textit{\color{blue}m} + \textit{\color{red}s} <=>[k_{1,s}][k_{-1,s}] $c_{ms}$ ->[k_{2,s}]} (1 - p_s) {\color{red}s} \\ \label{eq:r2} &\ce{\textit{\color{red}s} + \textit{\color{green}r} <=>[k_{1,r}][k_{-1,r}] $c_{sr}$ ->[k_{2,r}]} (1-p_r) {\color{green}r} \end{eqnarray}

In the first reaction ${\color{blue} m}$RNA binds to a ${\color{red}s}$RNA forming a RNA:RNA complex called $c_{ms}$. The RNAs of the duplex is then irreversibly degraded in an RNAse-E dependent reaction with stoichiometries defined by $p_s$, which denotes the probability that $s$ is codegraded in the reaction. With this definition $(1 - p_s) {\color{red}s}$ represents sRNA that is not codegraded or released following the degradation of mRNA. The majority of investigated small regulatory RNA (srRNA) acts stoichiometrically, they are degraded 1:1 with their target mRNA corresponding to $p_s = 1$. Interestingly studies indicate that the trap-RNA system acts catalytically with $p_s$ close to zero[{\color{red}citation}], leading to different regulatory properties some of which we try to explore mathematically. For the second reaction the main experimental observation is that the t{\color{green}r}ap RNA inhibits the activity of the ${\color{red}s}$RNA but the mechanism is largely unknown. The general scheme allows different hypotheses; either trap-RNA works by competing for the sRNA binding site or it could mediate RNAse-E dependent degradation of the sRNA either catalytically or stoichiometrically.

\emph{In vivo} genes end their RNAs are constantly being expressed and degraded giving rise to finite lifetimes of RNA molecules. To model the trap-RNA system in context of living cells the expression of each RNA is described by a production term $\alpha_i$ and the degradation is described by a first order degradation term $\beta_i[RNA]_i$. \begin{equation} \label{proddeg} \ce{->[production] [RNA]_i ->[degradation]} \end{equation} Eventually the amount of RNA will settle into a steady state where production equals degradation providing stability for mathematical analysis.

\begin{eqnarray} \label{test} y & = & ax^2 + bx + c \\ f(x) & = & x^2 + 2xy + y^2 \end{eqnarray} \eqref{test}