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 two general reaction schemes %Include intro figure of system. \begin{align} \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{align} In the first reaction ${\color{blue} m}$RNA binds to a ${\color{red}s}$RNA forming a complex called $c_{ms}$. The RNAs of the duplex is then irreversibly degraded 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}]. 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 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.

Due to the unknown mechanism of reaction \eqref{eq:r2} multiple models are proposed. All models are based on ordinary differential equations (ODEs) set up by applying law of mass action to reaction \eqref{eq:r1} and \eqref{eq:r2} and the background production and degradation \eqref{proddeg}. The ODEs are then simplified by assuming pseudo-steady state of the RNA complexes $c_{ms} $ and $ c_{sr}$. For the general reaction scheme

\begin{align} \label{system1} \frac{dm}{dt} &= \alpha_m - \beta_m m - k_s ms \\ \label{system2} \frac{ds}{dt} &= \alpha_s - \beta_s s - p_s k_s ms - k_r sr \\ \label{system3} \frac{dr}{dt} &= \alpha_r - \beta_r r - p_r k_r sr \end{align} where the kinetic constants of reaction \eqref{eq:r1} and \eqref{eq:r2} have been lumped into the overall kinetic constants $k_s = \frac{k_{1,s} k_{2,s}}{k_{-1,s} + k_{2,s}}$ and $k_r = \frac{k_{1,r} k_{2,r}}{k_{-1,r} + k_{2,r}}$. It is seen from the equations that setting either $p_s = 0$ or $p_r = 0$ simplifies the expression. Strong evidence exists that $p_s = 0$ [{\color{red}cite}] whereas the value of $p_r$ is unknown. Hence the general model is split up into a catalytic or partly stoichiometric trap-RNA mechanism corresponding to $p_r = 0$ or $0 < p_r \leq 1$ respectively.

To simplify the notation in the following section on steady-state analysis we define a vector of variables ${\bf x} = (m, s, r)$ and a hovering dot as ${\bf \dot{x}} = \frac{d \bf x}{dt}$. Thus the steady-state problem can be compactly specified as ${\bf \dot{x}} = 0$ with some solution $\bf x^*$ satisfying the condition. In this notation the system of differential equations are given by $\dot{\bf x} = {\bf f}({\bf x})$ where ${\bf f}({\bf x}) = (f_1({\bf x}), f_2({\bf x}), f_3({\bf x}))$.