Reduction and Stability Analysis of a Transcription–Translation Model of RNA Polymerase

Abstract

The aim of this paper is to analyze the dynamical behavior of biological models of gene transcription and translation. We focus on a particular positive feedback loop governing the synthesis of RNA polymerase, needed for transcribing its own gene. We write a high-dimension model based on mass action laws and reduce it to a two-variable model (RNA polymerase and its mRNA) by means of monotone system theory and timescale arguments. We show that the reduced model has either a single globally stable trivial equilibrium in (0, 0), or an unstable zero equilibrium and a globally stable positive one. We give generalizations of this model, notably with a variable growth rate. The dynamical behavior of this system can be related to biological observations on the bacterium Escherichia coli.

Appendices

Appendix A: Monotone Systems

Monotone systems form an important class of dynamical systems. They are particularly well adapted to mathematical models in biology (Sontag 2004), because they are defined by conditions related to the signs of the Jacobian matrix. Such a sign for one element reflects the fact that some variable will contribute positively to the variation of some other variables, a kind of qualitative dependence frequently found in biological models. The reader may consult the reference Smith (1995) for a review and an exhaustive presentation of the theory of monotone systems.

In summary, if the system is cooperative, then the flow preserves the partial order of trajectories in $\mathfrak {R}^n$ (the flow is monotone). Consider an autonomous differential system:

$$\begin{aligned} \dot{x}=f(x) \end{aligned}$$

(25)

where, $x \in \mathfrak {R}^n$ and $f: \mathfrak {R}^n\rightarrow \mathfrak {R}^n$.

The system is monotone if $x_{01}\le x_{02}$ (this inequality must be understood coordinate by coordinate: i.e., $x_{01i} \le x_{02i}$, $\forall ~~ i \in [1,\ldots ,n]$ ), implies that $x(t,x_{01}) \le x(t,x_{02})~~\forall ~~t$ ($x(t,x_{0})$ corresponds to the evolution with respect to time starting from the initial condition $x_{0}$).

Cooperativity is easy to check by looking at the signs of the elements of the Jacobian matrix, which should verify

$$\begin{aligned} \displaystyle \frac{\partial f_i}{\partial x_j}(t,x) \ge 0\quad \forall i\ne j. \end{aligned}$$

These systems have a strong tendency to converge to the set of their equilibria (Smith 1995). It can be shown that almost any solution converges to the set of equilibria except a set of zero measure. In particular, there are no stable periodic solutions. For more precise theorems, see Smith (1995).

Appendix B: Tikhonov’s Theorem

This theorem applies to reduced systems of the form:

$$\begin{aligned} \begin{aligned} \dot{x}&=f(x,y,\varepsilon )\\ \dot{y}&= \frac{1}{\varepsilon } g(x,y,\varepsilon ).\\ \end{aligned} \end{aligned}$$

(26)

where $x\in \mathfrak {R}^n$, $y\in \mathfrak {R}^m$, and $0 <\varepsilon \ll 1$ ($\varepsilon$ a very small parameter), $x(0)=x_0, y(0)=y_0$. So, when $\varepsilon$ tends to 0 ($\dot{y}$ evolves very rapidly compared to $\dot{x}$), the system (26) is equivalent to the system:

$$\begin{aligned} \begin{aligned}&\dot{x}=f(x,y,0)\\&g(x,y,0)=0\\ \end{aligned} \end{aligned}$$

This is valid only if the fast subsystem $\dot{y}=g(x,y,0)$ satisfies some conditions which are given as follows:

• Existence and uniqueness of the steady state (there exists a unique solution, $y^*=\phi (x)$ of $g(x,y,0)=0$.

• Exponential stability of the steady state $y^*$ of the fast subsystem $\dot{y}=g(x,y,0)$ for fixed x.

These conditions are given by the Tikhonov’s theorem (see Khalil 2002 for a complete description), which ensures that y will converge rapidly to a quasi-steady state ($y=\phi (x)$, depending only on x). Therefore, the reduced system using the Tikhonov’s Theorem is:

$$\begin{aligned} \dot{x}=f(x,\phi (x),0),\quad x(0)=x_0. \end{aligned}$$

There are also extensions for infinite time (Hoppensteadt 1966; Khalil 2002; Sakamoto 1990).

Appendix C: Rescaling the System

To make the time scales more obvious and verify that the evolution of z and q is slow, we scale the variables $y, y^1,\ldots , y^{L-1}$ with respect to a scaling factor $\alpha$, and the variables $x, x^1,\ldots , x^{H-1}$ with respect to a scaling factor $\beta$. Consider $\overline{y}= \alpha y, \overline{y}^i= \alpha y^i$ and $\overline{x}= \beta x, \overline{x^i}= \beta x^i$, this gives:

$$\begin{aligned} \begin{aligned} \dot{c}&=k_+\,p\,(d_0-c)-k_-\,c-k_c\,c- k^{'}_p\,c\\ \dot{p}&=-k_+\,p\,(d_0-c)+\frac{k_t}{\alpha }\,\overline{y}^{L-1}+k_-\,c+\frac{k^{'}_t}{\beta }\,\overline{x}^{H-1}-k^{'}_p\,p\\ \overline{\dot{y}}&= \alpha k_c\,c-k_t\,\overline{y}-k^{'}_p\,\overline{y}\\ \overline{\dot{y}}^1&=k_t\,\overline{y}-k_t\,\overline{y}^1-k^{'}_p\,\overline{y}^1\\ \overline{\dot{y}}^2&=k_t\,\overline{y}^1-k_t\,\overline{y}^2-k^{'}_p\,\overline{y}^2\\ \vdots \\ \overline{\dot{y}}^{L-1}&=k_t\,\overline{y}^{L-2}-k_t\,\overline{y}^{L-1}-k^{'}_p\,\overline{y}^{L-1}\\ \dot{w}&=k^{'}_{+}\,r\,m-k^{'}_{-}\,w-k_w\,w-k^{'}_m\,w\\ \dot{m}&=-k^{'}_{+}\,r\,m+k^{'}_{-}\,w+k_w\,w+ \frac{k_t}{\alpha } \,\overline{y}^{L-1}-k^{'}_m\,m\\ \dot{r}&=-k^{'}_{+}\,r\,m+k^{'}_{-}\,w+\frac{k^{'}_t}{\beta }\,\overline{x}^{H-1}+k^{'}_m\,w \\ \overline{\dot{x}}&=\beta k_w\,w-k^{'}_t\,\overline{x}\\ \overline{\dot{x}}^1&=k^{'}_t\,\overline{x}-k^{'}_t\,\overline{x}^1\\ \vdots \\ \overline{\dot{x}}^{H-1}&=k^{'}_t\,\overline{x}^{H-2}-k^{'}_t\,\overline{x}^{H-1}\\ \end{aligned} \end{aligned}$$

(27)

where $\frac{k_t}{\alpha }$, $\frac{k^{'}_t}{\beta }$ are small compared to $k_t$ and $k^{'}_t$, but where the first one is bigger than the second one $\left( \frac{k_t}{\alpha }=200~\text {min}^{-1}~\hbox {and}~\frac{k^{'}_t}{\beta }=10~ \text {min}^{-1}\right)$.⁷

Finally, the slow evolution part is given by the equation: $z = c+p+\frac{1}{\alpha }(\overline{y}+\overline{y}^1+\cdots +\overline{y}^{L-1})$, which gives $\dot{z} = \frac{k^{'}_t}{\beta }\, \overline{x}^{H-1} - k^{'}_p\,z$. Similarly, $q=m+w$ and therefore $\dot{q} = \frac{k_t}{\alpha }\,\overline{y}^{L-1} - k_m'\,q$. Having introduced the two new variables z and q we return, for simplicity, to the scale of the original system (see system (4)).