General kinetic model

The modeling of the trap-RNA system is based on two general reaction schemes \begin{align} {\color{blue}m} \mathop{\rightleftharpoons}_{k_{-1,s}}^{k_{1,s}

\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 \ref{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 \ref{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{eqnarray} \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{eqnarray} 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 ${\bf f}({\bf x}) = (f_1({\bf x}), f_2({\bf x}), f_3({\bf x}))$.

Catalytic trap-RNA

Assuming $p_s = 0$ and $p_r = 0$ the steady state solution of equations \eqref{system1}, \eqref{system2} and \eqref{system3} is derivable and unique. It is furthermore stable because the eigenvalues of the \emph{Jacobian} given by $J_{ij} = \frac{\partial f_i ({\bf x})}{\partial x_j}$ all have negative real parts. [{\color{red} ref, cite systems biology book}]. The solution specifies the steady-state of m \begin{equation} \label{ss} m^* = \frac{\alpha_m (\beta_r \beta_s + \alpha_r k_r)}{\beta_r \alpha_s k_s + \beta_m \beta_r \beta_s + \beta_m \alpha_r k_r} \end{equation} To eliminate the dependence on $\alpha_m$ the steady-state is scaled with respect to the maximum amount of expression $m^*_{max} = \frac{\alpha_m}{\beta_m}$. This maximum value of $m$ expression is achieved by not having any sRNA which is equivalent to $\alpha_s = 0$. A new measure of steady-state $\frac{m^*_{max}}{m^*}$ is interpreted as the \emph{fold repression} caused by the system. To analyze how this fold repression depends on the variables and parameters, equation \eqref{ss} is scaled and restated into its simpler form \begin{equation} \label{phi1} \phi = \frac{m^*_{max}}{m^*} = 1 + \frac{\frac{k_s \alpha_s}{\beta_m \beta_s}}{1 + \frac{k_r \alpha_r}{\beta_s \beta_r}} \end{equation} where $\phi$ is a measure of the fold repression.

The dynamic range is described by the function \begin{equation} \phi(x) = 1 + \frac{a}{1+b x} \label{dyn} \end{equation} where $x = \alpha_r$. The form of $\phi(x)$ depends on the two effective parameters $a = \frac{k_s \alpha_s}{\beta_m \beta_s}$ and $b = \frac{k_r}{\beta_s \beta_r}$. The parameter $a$ governs the fold repression at $x = 0$ as seen by rewriting \eqref{dyn} into $\phi(0) = 1 + a$. The other parameter $b$ governs the sensitivity with respect to $x$. In a biological perspective the governing parameters of $a$ and $b$ are flexible because the underlying parameters are changeable. Especially $\alpha_s$ are changeable by simply altering the induction of sRNA but also $k_r$ might be changeable by altering the binding affinity of sRNA and trap-RNA. Thus the catalytic trap-RNA model suggests a highly modular dynamic range.

Partly stoichiometric trap-RNA

Assuming $p_s = 0$ and $0 < p_r \leq 1$ the steady state solution of equations \eqref{system1}, \eqref{system2} and \eqref{system3} with respect to m is [{\color{red}ref}] \begin{equation} m^* = \frac{2}{\beta_m} \bigg( \frac{p_r \alpha_m \lambda_s}{\alpha_s p_r - \alpha_r -\lambda_r + 2 p_r \lambda_s + \sqrt{(\alpha_s p_r - \alpha_r - \lambda_r)^2 + 4 p_r \alpha_s \lambda_r}} \bigg) \end{equation} where $\lambda_r = \frac{\beta_s \beta_r}{k_r}$ and $\lambda_s = \frac{\beta_s \beta_m}{k_s}$. The fold repression is again defined removing the dependence on $\alpha_m$ and achieving more generality \begin{equation} \label{phi2} \phi = \frac{m^*_{max}}{m^*} = \frac{\alpha_s p_r - \alpha_r -\lambda_r + 2 p_r \lambda_s + \sqrt{(\alpha_s p_r - \alpha_r - \lambda_r)^2 + 4 p_r \alpha_s \lambda_r}}{2 p_r \lambda_s} \end{equation} Comparing the two expressions for fold repression \eqref{phi1} and \eqref{phi2} are reassuringly equivalent when $\alpha_r = 0$, which corresponds to ignoring reaction \eqref{eq:r2}. This makes sense since \eqref{eq:r1} are identical for both dual degradation models. This result leads to arguments concerning the design of the dynamic range.

Design of dynamic range

When $\alpha_r = 0$ both the catalytic expression \eqref{phi1} and the partly stoichiometric expression \eqref{phi2} reduces to \begin{equation} \label{lin} \phi = 1 + \frac{k_s \alpha_s}{\beta_m \beta_s} \end{equation} Thus the effect of the parameter $a = \frac{k_s \alpha_s}{\beta_m \beta_s}$ on the maximum fold repression is valid for both proposed models. Because $a$ is easily changeable by altering the induction of sRNA the starting point for the dynamic range is also easily changeable. The dynamic range can be expression as a function of $\alpha_r = x$, called $\phi(x)$. Because the induction of trap-RNA, $x$, is limited to some interval of biological plausible induction levels $[0;x_{max}]$, the value of $\phi(x_{max})$ must be sufficiently small to give rise to a functional dynamic range. For the catalytic model this value depends on $a$ and $b$ whereas for the partly stoichiometric model the dependence on parameters is more complicated. But for both models the tradeoff for high fold repression

Steepness...

Experimental analysis and parameters

In order to use the expressions for steady-state \eqref{phi1} and \eqref{phi2} the fold repression, $\phi$, must somehow be measurable. To arrive at some expression relating $\phi$ to empirical data, the steady-state of the expressed target gene is assumed to be proportional to the steady-state of the target mRNA. Using translational or transcriptional fusion of a reporter gene to the target gene the fold repression can be approximated by \begin{equation} \phi \approx \frac{[A_{max}]}{[A]} \end{equation} where $[A]$ is some measure proportional to the amount of reporter gene and $[A_{max}]$ is some measure at maximum expression. The measures of reporter gene could be either fluorescence from GFP or absorbance of X-gal product due to LacZ activity.

The parameters of $\lambda_s = \frac{\beta_m \beta_s}{k_s}$ and $\lambda_r = \frac{\beta_s \beta_r}{k_r}$ which arises in both steady-state solutions \eqref{phi1} and \eqref{phi2} can be interpreted as \emph{leakage rates}. Or in other words the ratio between the background degradation rate and the mediated specific degradation rate. The leakage rate $\lambda_s$ can be determined by fitting \eqref{lin} to a plot of $\phi$ against $\alpha_s$. Similarly $\lambda_s$ and $\lambda_r$ can be determined by fitting empirically determined $\phi$ to values of $\alpha_s$ and $\alpha_r$ using equation \eqref{phi1} or \eqref{phi2}. The goodness of fit should provide evidence discerning the two alternative hypotheses of catalytic or partly stoichiometric trap-RNA.

The production terms $\alpha_m$, $\alpha_s$ and $\alpha_r$ are either constant when constitutive promoters are used or dependable on inducer concentration when inducible promoters are used. Inducible promoter activity can be approximated by a linear function \begin{equation} \alpha = c [I] \end{equation} where $[I]$ is the inducer concentration and $c$ is a constant of proportionality. Alternatively inducer promoter activity can be approximated by Hill binding \begin{equation} \alpha = \frac{\alpha_{max} [I]^n}{K_d + [I]^n} \end{equation} where $[I]$ is the inducer concentration, $n$ is the Hill coefficient, $\alpha_{max}$ is the maximum production rate at full saturation and $K_d$ is the binding coefficient for the inducer-promoter complex.

The background degradation rates can be modeled as first order decay processes and determined by fitting to \begin{equation} N(t) = N_0e^{-\beta t} \end{equation} where $N(t)$ is the amount of RNA and intracellular production of RNA is stopped at $t=0$. Experiments indicate (Overgaard, 2009) that $\beta_s = \beta_m = 0.0257 min ^{-1}$.

\begin{table}[htbp] \centering \begin{tabular}{lll} \hline Parameter & Meaning & Unit \\ \hline $\alpha_m$ & Transcription rate of target mRNA & [amount/time] \\ $\alpha_s$ & Transcription rate of sRNA (s) & [amount/time] \\ $\alpha_r$ & Transcription rate of sRNA (r) & [amount/time] \\ $\beta_m$ & Degradation rate of free mRNA & [1/time] \\ $\beta_s$ & Degradation rate of free sRNA (s)& [1/time] \\ $\beta_r$ & Degradation rate of free sRNA (r) & [1/time] \\ $p_s$ & Probability that s is codegraded with m & [-] \\ $p_r$ & Probability that r is codegraded with s & [-] \\ $k_s$ & Kinetic constant of sRNA mediated degradation of mRNA & [1/(time*amount)] \\ $k_r$ & Kinetic constant of trap-RNA mediated degradation of sRNA & [1/(time*amount)] \\ $\lambda_s$ & Leakage rate of sRNA mediated degradation of mRNA & [amount/time] \\ $\lambda_r$ & Leakage rate of trap-RNA mediated degradation of sRNA & [amount/time] \\ \hline \end{tabular} \caption{Interpretation of model parameters} \label{tab:Def_par} \end{table}

The role of Hfq

In terms of reaction mechanism the second order kinetic parameters $k_s$ and $k_r$ has the following interpretation \begin{equation} k = \frac{k_1 k_2}{k_{-1} + k_2} \end{equation} The dissociation constant is defined by $K_d = \frac{k_{-1}}{k_1}$. To relate these quantities we assume that the on rate $k_1$ is constant with respect to $K_d$ and that any change in $K_d$ only affects the off-rate $k_{-1}$. By inserting $k_{-1} = k_1 K_d$ we get the following linear relationship. \begin{equation} \frac{1}{k} = \frac{1}{k_1} + \frac{1}{k_2} K_d \end{equation} Using experimentally determined values of $k$ and $K_d$ prediction from secondary RNA structure predictors we should be able to investigate the relationship.

One might speculate on the role of Hfq in RNA-RNA hybridization. A proposed mechanism is, that Hfq increases the on rate by some factor. If Hfq works by increasing the local concentration of RNA the factor would describe the increased probability of RNA collision caused by the presence of Hfq. Incorporating this idea we define the following altered on rate $k_{1} = \alpha_H k_{1}^{\circ}$ where $\alpha_H \geq 1$ and get \begin{equation} \frac{1}{k} = \frac{1}{\alpha_H k_1^{\circ}} + \frac{1}{k_2} K_d \end{equation}