Team:HZAU-China/Comparison

In this part, we want to demonstrate some advantages of our design by making quantitative comparisons. These advantages include safety, energy efficiency and stability.

3.1 The post-transcriptional control ensures lower leakage

Firstly, we don't expect that our designed processing modules will alter its function by some noises of environment. So we design a coherent feedforward loop to filter noise. Here we use $x$ to represent a general input signal. As we mentioned before, the dynamic of the input module with post-transcriptional control can be described by \begin{equation} \begin{aligned} \frac{d[mRNA_{Cre}]}{dt}&=\beta_0+\beta_1\frac{[x]^n}{K^n+[x]^n}-K_{R}\cdot [mRNA_{Cre}]\\ \frac{d[taRNA]}{dt}&=\beta_0+\beta_1\frac{[x]^n}{K^n+[x]^n}-K_{R}\cdot [taRNA]\\ \frac{d[Cre]}{dt}&=K_{tl}\cdot [mRNA_{Cre}]\cdot\frac{[taRNA]}{K_m+[taRNA]}-K_{P}\cdot [Cre]. \end{aligned} \end{equation}

If there is no post-transcriptional control, this process can be described by \begin{equation} \begin{aligned} \frac{d[mRNA_{Cre}]}{dt}&=\beta_0+\beta_1\frac{[x]^n}{K^n+[x]^n}-K_{R}\cdot [mRNA_{Cre}]\\ \frac{d[Cre]}{dt}&=K_{tl}\cdot [mRNA_{Cre}]-K_{P}\cdot [Cre]. \end{aligned} \end{equation}

We compare the expression dynamic of Cre in either case at different level of input signal $x$. This comparison reveals that the riboregulator ensures lower leakage but doesn't affect the expression at a high input level. We also design some experiment to validate the model, the results are consistent with this model.

Figure 2: dynamic of the input module

(some data will be showed here)

3.2 The mutated Cre/loxP system determines the inversion direction

Secondly, once the engineered cells receive a sure signal, they will process information as we designed but not the other way around. So we must ensure the direction of the DNA invertion. In other words, the site-specific recombination can be regarded as a unidirectional one. To this end, we choose Cre recombinase and a pair of mutant lox sites, lox66 and lox71 to rearrange DNA sequence. Here, we explain why the forward reaction rate is higher than the reverse one. The mutant site will have a lower affinity for Cre. The binding event mainly depends on the rate of free diffusion, but the dissociation rate will be high if the binding strength is weak. According to the equation \begin{equation} r_c(2Cre\cdot lox)=\frac{k_8k_9\cdot [lox]\cdot [Cre]^2}{k_{-8}k_{-9}+k_9k_{-9}\cdot [Cre]}, \end{equation} we know that the second dissociation event is more significant than the first one. So mutant lox site like lox66 and lox71 can choose a wise way to benefit the formation of dimer. Here we assume that lox66 and lox71 will have higher $k_{-8}$. However, double mutant loxP site like lox72 will have high dissociation rate at both steps. For simplicity, we assume that the affected dissociation rate is $\epsilon$ times the original one. $\epsilon>1$.

After binding event, the two loxP-bound dimers associate to form a tetramer, and recombination proceeds via a Holiday Junction intermediate. Therefore, we can compare $r_c(2Cre\cdot loxP)\cdot r_c(2Cre\cdot lox72)$ with $r_c(2Cre\cdot lox66)\cdot r_c(2Cre\cdot lox71)$ to see what kind of synapsis is easy to form. \begin{equation} \begin{split} r_c(2Cre\cdot loxP)\cdot r_c(2Cre\cdot lox72)&=\frac{k_8^2k_9^2\cdot[lox]^2\cdot [Cre]^4}{({\epsilon}^2k_{-8}k_{-9}+\epsilon k_9k_{-9}[Cre])(k_{-8}k_{-9}+k_9k_{-9}[Cre])}\\ r_c(2Cre\cdot lox66)\cdot r_c(2Cre\cdot lox71)&=\frac{k_8^2k_9^2\cdot[lox]^2\cdot [Cre]^4}{({\epsilon}k_{-8}k_{-9}+k_9k_{-9}[Cre])^2} \end{split} \end{equation}

For $\epsilon>1$, \begin{equation} \begin{aligned} \frac{r_c(2Cre\cdot loxP)\cdot r_c(2Cre\cdot lox72)}{r_c(2Cre\cdot lox66)\cdot r_c(2Cre\cdot lox71)}&=\frac{({\epsilon}k_{-8}k_{-9}+k_9k_{-9}[Cre])^2}{({\epsilon}^2k_{-8}k_{-9}+\epsilon k_9k_{-9}[Cre])(k_{-8}k_{-9}+k_9k_{-9}[Cre])}\\ &=1-\frac{(\epsilon-1)(k_9k_{-9}[Cre]^2)+\epsilon(\epsilon-1)(k_{-8}k_9k_{-9}^2[Cre])}{({\epsilon}^2k_{-8}k_{-9}+\epsilon k_9k_{-9}[Cre])(k_{-8}k_{-9}+k_9k_{-9}[Cre])}\\ &<1 \end{aligned} \end{equation}

Under the condition that the $Cre\cdot lox$ is much more easy to get a Cre monomer rather than loss a Cre monomer, this proportion can be approximately equal to $\frac{1}{\epsilon}$, because \begin{equation} \begin{aligned} r_c(2Cre\cdot lox)&=\frac{k_8k_9\cdot [lox]\cdot [Cre]^2}{k_{-8}k_{-9}+k_9k_{-9}\cdot [Cre]} &\approx\frac{k_8\cdot [lox]\cdot [Cre]}{k_{-9}} \end{aligned} \end{equation}

Hence, $r_c($$2Cre\cdot loxP)$$\cdot r_c($$2Cre\cdot lox72)$$< r_c($$2Cre\cdot lox66)$$\cdot r_c($$2Cre\cdot lox71)$, which means the synapsis between lox66 and lox71 is easy to form.

This explanation is based on the kinetic model proposed in 1998. However, the real system may have additional processes. We introduce another explanation using a principle called kinetic proofreading that is widely employed to achieve high precision in diverse molecular recognition systems. Inspired by the DNA binding process described in available researches(Reardon and Sancar, 2004; Tlusty et al., 2004; Alon, 2007), we infer that the recombinase protein Cre undergoes a modification after binding to the half of lox site, and then it can recruit other proteins Cre to bind to another half of lox site. Such a modification can help the mutant lox site like lox66 and lox71 easily jump to the next state but prevent the incorrect binding of the double mutant lox site like lox72. This modification makes the second binding event more likely to happen and the corresponding dissociation event less likely to happen. So why $k_9>k_8$ and k_{-8}>$k_{-9}$ can also be explained by this modification.

In summary, we can explain Cre-mediated inversion using the mutated Cre/loxP system we used prefers the forward forward reaction by different mechanisms. No matter which theories we used to explain the phenomena, we emphasized that the two binding processes of one lox site are not identical. The first binding process contributes to the second one. Therefore, either the steady-state treatment of the first binding and dissociation or the kinetic proofreading by a modification is reasonable.