Team:UCAS-China/Model for the Design


Model for the the biosafety module

This module is designed to trigger suicide of engineered bacteria if and only if they leave human bodies. We take temperature as the detection indicator to determine whether the bacterium is in vivo. The module can be divided into two parts. CI434 and TEV makes the control part and the Doc gene is the toxin part. For the toxin part, only the equilibrium is of our interest, while for the control part the dynamics are more important.

All parameters involved in this page are listed here

Variable Meaning
$$C_C$$ Concentration of CI434
$$C_T$$ Concentration of TEV
$$C_D$$ Concentration of Doc
$$V_C$$ Max production speed of CI434
$$V_T$$ Max production speed of TEV
$$V_D$$ Max production speed of Doc
$$\gamma_C$$ Degradation rate of CI434
$$\gamma_T$$ Degradation rate of TEV
$$\gamma_D$$ Degradation rate of Doc
$$V_m$$ Max production rate
$$K_m$$ Michaelis-Menten constant
$$K_C$$ Parameter of Hill equilibrium
$$n_C$$ Hill coefficient
$$\theta$$ Function of temperature

Part 1. the Control Part.

The design of the control part is a simplified version of T. Gardner’s toggle switch. Our design can also achieve a system with temperature depended bistability. The variables and parameters involved in the model are listed. It is known that the functioning of TEV follows the Michaelis-Menten kinetics, while the repressing effect of CI434 can be well described by Hill’s equation. We further assume that degeneration is always a first order process and CI434 has a constant production rate. Here we get the equation

$$ \begin{array}{c} \frac{d C_{C}}{d t}=V_{C}-\gamma_{C} C_{C}-\frac{V_{m} C_{T}}{K_{m}+C_{C}} C_{C} \\ \frac{d C_{T}}{d t}=\frac{V_{T}}{1+\left(\theta \frac{C_{C}}{K_{C}}\right)^{n_{C}}}-\gamma_{T} C_{T} \end{array} $$

where \(\theta\) is a dimensionless increasing function of temperature. Note that \(\theta\) is bijective with temperature, it is convenient to use \(\theta\) to represent temperature. With dimensionless parameters listed, we nondimensionalize the equations

Variable Definition Value
$$v_C$$ $$V_C/\gamma_TK_C$$ 50
$$\eta$$ $$\gamma_C/\gamma_T$$ 1
$$n_C$$ 2
$$v_m$$ $$V_mV_T/\gamma_C\gamma_TK_C$$ 200
$$k_m$$ $$K_m/K_C$$ 2
$$c_C$$ $$C_C/K_C$$
$$c_T$$ $$C_T\gamma_T/V_T$$
$$\tau$$ $$\gamma_Tt$$

$$ \begin{array}{c} \frac{d c_{C}}{d \tau}=v_{C}-\left(1+\frac{v_{m} c_{T}}{k_{m}+c_{c}}\right) \eta c_{C} \\ \frac{d c_{T}}{d \tau}=\frac{1}{1+\left(\theta c_{C}\right)^{n_{C}}}-c_{T} \end{array} $$

Let \(\frac{d c_{C}}{d \tau}=\frac{d c_{T}}{d \tau}=0\), the static state solution is obtained. The figure clearly shows that between \([0.096,0.612]\) the system possesses a bistability. Bacterium is safe in state with low \(c_T\) and high \(c_C\), while for the state with high \(c_T\) and low \(c_C\), the toxin part would be triggered and the bacterium would die. As long as we adjust the \(\theta\) for body temperature between \([0.096,0.612]\) and \(\theta\) for room temperature lower than \(0.096\), our design is successful.

Figure 1. static state solution

Here we give a simple explanation of why bistability can play the role of a switch. Set the system initially in safe state and \(\theta\) the body temperature. Let \(\theta\) slowly decrease, the system point would also shift left together with \(\theta\). However, when \(\theta\) come across the critical value, system cannot follow \(\theta\) if it keeps in the safe state, therefore system would jump to the suicide state. Now even if \(\theta\) increase again, system would keep its current state—suicide state and the bacterium would soon die.

Form the process description given above, the state transition is irreversible. With such property, bacterium would have less chance to survive were it left human body, even if it soon finds a worm place (e.g. eaten by animal). However, such property brings new problems—would fluctuation trigger the switch? What kind of fluctuation would trigger the switch? Can patients eat ice-cream? To answer these questions, we do an analysis we call “ice-cream analysis”. We take a square pulse of duration \(dt\) and amplitude \(\Delta \theta \) over the base line value \(\theta_{0}\). Through simulation, we figure out the boundary surface between i-state and p-state in \(\theta_{0}-\Delta \theta-d t\) space. The result may help to design a special diet that could minimize the risk of Hp infection.

Part 2. the Toxin Part

For the toxin part, we can similarly write down its equation (Note that the repressing effect is only related to CI434)

$$ \frac{d C_{D}}{d t}=\frac{V_{D}}{1+\left(\theta \frac{C_{C}}{K_{C}}\right)^{n_{C}}}-\gamma_{D} C_{D} $$

For static state, we have

$$ \frac{V_{D}}{1+\left(\theta \frac{C_{C}}{K_{C}}\right)^{n_{C}}}-\gamma_{D} C_{D}=0 \Rightarrow C_{D}=\frac{\frac{V_{D}}{\gamma_{D}}}{1+\left(\theta \frac{C_{C}}{K_{C}}\right)^{n_{C}}}=\frac{\gamma_{T} V_{D}}{\gamma_{D} V_{T}} C_{T} $$

Thus the ratio between take “+” the subscribe for the suicide state and “-” for the safe sate, we have

$$ \frac{C_{D+}}{C_{D-}}=\frac{C_{T+}}{C_{T-}} \approx 600 $$

Moreover, if Doc must be cut by TEV to be effective, as we have improved the design, we get

$$ \frac{d C_{D}^{*}}{d t}=\frac{V_{m} C_{T} C_{D}}{K_{m}+C_{D}}-\gamma_{D} C_{D}^{*} $$

For static state, since by last part it is expect to have \(K_{m} \sim C_{T} \sim C_{D}\), here we have

$$ \frac{C_{D+}^{*}}{C_{D-}^{*}}=\frac{C_{T+} C_{D+}}{C_{T-} C_{D-}} \frac{K_{m}+C_{D-}^{*}}{K_{m}+C_{D+}^{*}} \approx \frac{C_{T+} C_{D+}}{C_{T-} C_{D-}}=\left(\frac{C_{T+}}{C_{T-}}\right)^{2} \approx 10^{5} $$

We expect leakage of bacteria (suicide in body temperature or keep alive in room temperature in unit time) of similar order of magnitude. This proofs that our design can ensure the biosafety.