Variable definitions

Firstly, we provide the meaning of abbreviations.

Expression of mRNA

mRNA

For mRNA, we have

\begin{equation}\frac{dM_F}{dt}=k_{M_F}-d_{M_F}M_F \end{equation}

According to equilibrium assumption

\begin{equation}\label{eq2}M_F=\frac{k_{M_F}}{d_{M_F}}\end{equation}

Similarly,we have

\begin{equation}\label{eq1}M_B=\frac{k_{M_B}}{d_{M_B}}\end{equation}

ccdA, ccdB, FNR

Take expression of mRNA, degrade of each protein and combining of ccdA and ccdB into consideration, we have

\begin{equation}\label{eq3}\frac{dC_{F}}{dt}=k_{C_F}M_F-d_{C_F}C_{F}-k_{iF}C_{F}\end{equation}

\begin{equation}\label{ep4}\frac{dC_{B}}{dt}=k_{C_B}M_B-d_{C_B}C_{B}-k_{com}C_{A}C_{B}\end{equation}

\begin{equation}\label{ep5}\frac{dC_{A}}{dt}=k_{C_A}M_A-d_{C_A}C_{A}-k_{com}C_{A}C_{B}\end{equation}

FNR protein

Deactivation

[Fe-S] structure consisted in FNR protein change to planar configuration which losses biological activity quickly under the catalysis of O2,

Considering the lack of multi-concentration gradient experiments in this part and the high sensitivity of FNR to detect oxygen, we consider the anaerobic state with all steric activity and the anoxic state with all planar deactivation, and consider the O2-catalyzed oxidation process by associating equations (1), (3) with multi-concentration gradient data.

FNR binding to HIP-1 promoter

Consider a simplified model, in which only the FNR that maintains the active configuration binds to DNA and starts the transcribe progress, turning inactive DNA to active DNA, we have

\begin{equation}\frac{dC_{aD}}{dt}=\frac{k_{2}C_{F}C_{iD}}{K_{1}+C_{iD}}-d_{aD}C_{aD}\end{equation}

Then the transcribe of activated DNA:

\begin{equation}\frac{dM_{A}}{dt}=\frac{V{max}C_{aD}}{K_{2}+C_{aD}}-d_{M_A}M_{A}\end{equation}

Ligands' cooperative behaviour

Two approaches can transform HIP-1 from inactivate states to translatable statue, and we assume that enzymes needed for translation are always abundant, so order of it is always 1.

In steady state, we have the following four equations:

\begin{equation}C_{HF}=K_{1}C_{H}C_{F}\end{equation}

\begin{equation}C_{RH}=K_{2}C_{H} \end{equation}

\begin{equation}C_{RHF}=K_{3}C_{HF}\end{equation}

\begin{equation}C_{HF}+C_{RH}+C_{RHF}+C_{H}=C_{total}\end{equation}

Then we have the ration translationable DNA takes in steady state

Therefore

$$\frac{dC_{ccdA}}{dt}=k_{mA}C_{MA}$$

Calculate to get the whole amount of $C_{ccdA}$ expressed in the process.

Now let's consider the ccdB

$$\frac{dC_{MB}}{dt}=\begin{cases}kC_{aDB}-k_{dmB}C_{MB} & t\le T_{B}\\-k_{dmB}C_{MB} & t>T_B\end{cases}$$

After we got the whole amount of ccdA and ccdB, consider their interation

which reach a constant state, which is established at a slower speed compred to

the expression of ccdA/B\\

We use $[A_2]$ to denote the concentration of free $ccdA ccdA$, $[B_2]$ to denote the concentration of free $ccdB ccdB$, $[A_2B_2A_2]$ to denote the concentration of free $ccdA_2 ccdB_2 ccdA_2$ $K_A$, $K_0$, $K_{0A}$, $K_B$ list the equations:

\begin{equation}A+A\rightleftharpoons A_2 \;\;\;\;\;\;\;\;\;\;\frac{[A_2]}{\;[A]^2}=K_{0A}\end{equation}

\begin{equation}B+B\rightleftharpoons B_2 \;\;\;\;\;\;\;\;\;\; \frac{[B_2]}{\;[B]^2}=K_{0B}\end{equation}

\begin{equation}A_2+B_2 \rightleftharpoons A_2B_2\;\;\;\;\;\;\;\;\;\; \frac{[A_2B_2]}{[A_2][B_2]}=K_0\end{equation}

\begin{equation}A_2B_2 + A_2 \rightleftharpoons A_2B_2A_2 \;\;\;\;\;\;\;\;\;\; \frac{[A_2B_2A_2]}{[A_2][A_2B_2]}=K_A\end{equation}

So we have

\begin{equation}[A_2B_2A_2]=K_A[A_2][A_2B_2]=K_AK_0[A_2]^2[B_2]\end{equation}

For the following two kinds of multi-complexes (each with $n$ repeating units), we have

$$\begin{equation}[A_2B_2A_2\cdots B_2] =K_A^{\frac{n}{2}-1}K_B^{\frac{n}{2}-1}K_0[A_2]^{\frac{n}{2}}[B_2]^{\frac{n}{2}}\end{equation}$$

$$\begin{equation}[A_2B_2A_2\cdots B_2A_2] =K_A^{\frac{n}{2}}K_B^{\frac{n}{2}-1}K_0[A_2]^{\frac{n}{2}+1}[B_2]^{\frac{n}{2}}\end{equation}$$

Add up all concentration of $A_2-B_2$ complex, and consider the total concentration of $A_2$ or $B_2$ is given through $[A_2]=K_{0A}[A]^2$

Simulation

We built a simbiology model to simulate the Suicide switch to confirm its feasibility of the suicide switch, we represent the effect of oxygen concentration by a function $\theta(O2,x)$, the combination of O$_2$ and FNR deactivates FNR, so the concentration of FNR is a function of $\theta$, and FNR activates ccdA expression, so the production rate of ccdA is a function of $\theta$. Since we consider two states, normoxia and hypoxia, and the combination of FNR and O$_2$ is very sensitive, we consider that $k_A(\theta)$ ranges from 0 to 1, with the hypoxic state close to 1 and the normoxic state close to 0.

However, in the real world, the concentration of O$_2$ is not binary, therefore, we need to evaluate how O$_2$ concentration affects ccdA. By numerical simulation, we derive the curve below.