Based on experiments, we have designed sulforaphane production system, curcumin production system, and temperature-induced suicide system, as well as feasibility verification. However, quantified analysis is also needed, which can be realized by modelling. In order to analyze the dynamics properties of myrosinase and DCS-CURS, the inhibition effect to cancer cells of sulforaphane and curcumin, and the efficiency of temperature-induced suicide switch, we use Michaelis-Menten equation, Arrhenius equation, Hill equation, and Logistic equation to describe the system. By fitting, we derive dynamic properties of myrosinase and DCS-CURS in 37℃ and pH7.4. Numerical simulation via MATLAB shows that the suicide switch and inhibition system work well, whereas the inhibition rate is slow and not thorough.

Dynamic models

Introduction

This section is organized as follows.

1. Fitting of dynamic properties of myrosinase and DCS-CURS using Michaelis-Menten equations.

2. Description of TcI42 suicide switch using Hill equation and Arrhenius equation. This part also compute the generating velocity of sRRz.

3. Description of population growth vua Logistics equation. The lysis of engineering bacteria by sRRz and the inhibition of cancer cell growth by sulforaphane and curcumin are also considered.

4. Summarization of overall system ODEs.

Activity of enzymes

The Michaelis-Menten equation, which is widely used to describe the velocity of enzymatic reactions, is

$v=\frac{v_{\max}[S]}{K_m+[S]},$

where $v_{\max}$ is the maximum velocity for given enzyme concentration, $[S]$ is the concentration of substrate, $K_m$ is Michaelis-Menten constant, which is equal to the substrate concentration when the velocity reaches $v_{\max}/2$. According to experimental results, the dynamic properties for myrosinase and DCS-CURS in 37℃ and pH7.4 is shown below.

Table 1: Dynamic properties of myrosinase and DCS-CURS in 37℃ and pH7.4

Emzyme	$v_{\max}(\mu M/min)$	$K_m(\mu M)$
Myrosinase	15.51	84.15
DCS-CURS	0.2862	71.2

TcI42 and the production of sRRz

Temperature-inducible promoters are a type of sequences, which can adject transcriptional activity in a particular temperature range. They are widely used in bioengineering and synthetic biology to regulate temperature-sensitive gene expression. Temperature-inducible promoters work by changing the activity of promoters through changes in temperature, thereby regulating the transcription level of genes. At low temperatures, they are inhibited and the transcriptional activity of genes is low. At high temperatures, they are activated and the transcriptional activity of the gene increases. This temperature sensitivity is achieved by the interaction of specific regulatory elements and temperature-sensing factors in the promoter sequence.

For the activity of TcI42 versus temperature, as the principle of TcI42 can be simplified into the association of ligand[1], we can consider the dissociation reaction: $LR\rightleftarrows R+nL$, where $L$, $R$. and $LR$ represent the concentrations of ligand, TcI42, and associated TcI42, respectively, and $n$ represents Hill coefficient. According to Arrhenius equation, the velocity of positive and reversed reaction can be described as $v_1=A_1e^{-E_1/RT}[LR]$ and $v_2=A_2e^{-E_2/RT}[L]^n[R]$, where $E_1$ and $E_2$ represent the activation energy of positive and reversed reactions, respectively. When the reaction reaches equilibrium, according to $v_1=v_2$, the apparent dissociation constant is

$K_d=\frac{[L]^n[R]}{[LR]}=\frac{A_1}{A_2}e^{-(E_1-E_2)/RT}.$

Let $A=A_1/A_2$, the enthalpy change $\Delta H=E_1-E_2$, then the apparent dissociation constant is $K_d=Ae^{-\Delta H/RT}$. If the ratio of associated TcI42 is $\theta$, then according to Hill equation,

$\theta=\frac{[L]^n}{K_d+[L]^n}=\frac{[L]^n}{Ae^{-\Delta H/RT}+[L]^n}.$

As $[L]$ and $n$ do not change when temperature changes, we have

$\theta=\frac{1}{1+C_1e^{C_2/T}},$

where $C_1$ and $C_2$ are two constants. Denote $T_0$ as the temperature when half of TcI42 is associated. As $C_2=T_0\ln(2+1/C_1)$, and for temperature-induced switch, $C_1$ is usually a real number which is much smaller than 1, then

$\theta=\frac{1}{1+e^{CT_0/T-C}}.$

In suicide switch, the transcription and translation process can be described by

\frac{d[mRNA]}{dt}=\hat\alpha_1+\hat\alpha_2\theta-\hat\delta_1[mRNA], \\ \frac{d[sRRz]}{dt}=\hat\alpha_3[mRNA]-\hat\delta_2[P].

where $\alpha_1$ represents the transcription rate when all TcI42 is not associated. By assuming that the concentration of mRNA stablizes instantly, we have

$\frac{d[sRRz]}{dt}=\alpha+\beta\theta-\delta[sRRz].$

For zero initial value of $[sRRz]$, integration result is

$[sRRz]=\left(A+\frac{B}{1+e^{CT_0/T-C}}\right)(1-e^{-\delta t}),$

where $A=\alpha/\delta$ and $B=\beta/\delta$.

Growth

In an environment where the space is finite, the growth rate is restricted by population. For an environment, if the growth rate without restriction is $r$, environmental capacity is $N_{\max}$, then the change of population follows Logistic model.

$\frac{dN}{dt}=rN\left(1-\frac{N}{N_{\max}}\right).$

For engineering bacteria, temperature-induced suicide system can synthesize sRRz, the concentration of which affects the lysis rate of bacteria. That is, the number of engineering bacteria $N_1$ satisfies

$\frac{dN_1}{dt}=r_1N_1\left(1-\frac{N_1}{N_{\max,1}}\right)-\lambda N[sRRz].$

By substituting the result from previous section, we have

$\frac{dN_1}{dt}=r_1N_1\left(1-\frac{N_1}{N_{\max,1}}\right)-N_1\left(\mu_1+\frac{\mu_2}{1+e^{CT_0/T-C}}\right)(1-e^{-\delta t}),$

where $\mu_1=A\lambda$ and $\mu_2=B\lambda$. For cancer cells, if we use $N_2$ to represent the population, $[P_1]$ and $[P_2]$ to represent the concentration of sulforaphane and curcumin, respectively, then

$\frac{dN_2}{dt}=(r_2-\lambda_1[P_1]-\lambda_2[P_2])N_2\left(1-\frac{N_2}{N_{\max,2}}\right),$

where $\lambda_1$ and $\lambda_2$ represent inhibition intensity of sulforaphane and curcumin, respectively.

Production of sulforaphane and curcumin

We assume that the concentrations of enzymes reach stability instantly, then for sulforaphane and curcumin, there is

$\frac{d[P_i]}{dt}=\frac{N_1v_{\max,i}[S_i]}{K_{m,i}+[S_i]}-\delta_i[P_i],\quad i = 1, 2,$

where $[S_1]$ and $[S_2]$ represent the concentrations of substates synthesizing sulforaphane and curcumin (the synthesis of curcumin is simplified to a one-substrate-one-enzyme reaction), respectively，$v_{\max,i}$ and $K_{m,i}$ represent the maximum speed and Michaelis-Menten constants when the number of engineering bacteria reaches environmental capacity, respectively, and $\delta_1$ and $\delta_2$ represent the degradation rate of sulforaphane and curcumin, respectively.

Summary

By summarizing the above results, we obtain that the entire process satisfies the following differential equations:

$\begin{aligned} \frac{dN_1}{dt}&=r_1N_1\left(1-\frac{N_1}{N_{\max,1}}\right)-N_1\left(\mu_1+\frac{\mu_2}{1+e^{CT_0/T-C}}\right)(1-e^{-\delta t})\\ \frac{d[P_1]}{dt}&=\frac{N_1v_{\max,1}[S_1]}{N_{\max,1}(K_{m,1}+[S_1])}-\delta_1[P_1]\\ \frac{d[P_2]}{dt}&=\frac{N_1v_{\max,2}[S_2]}{N_{\max,1}(K_{m,2}+[S_2])}-\delta_2[P_2]\\ \frac{dN_2}{dt}&=(r_2-\lambda_1[P_1]-\lambda_2[P_2])N_2\left(1-\frac{N_2}{N_{\max,2}}\right) \end{aligned}$

Numerical simulation

We set the concentration of substrates synthesizing sulforaphane and curcumin to the corresponding Michaelis-Menten constants in order to synthesize sulforaphane and curcumin at half of the maximum velocity. Initially, we set the number of engineering bacteria to a low level, and the number of cancer cells to half of the environmental capacity. For the environment, we set the temperature to 37℃ for the first 7 hours, and after 7 hours the temperature is set to 42℃. We use MATLAB to simulate. The parameter list and the simulation result is shown below.

Table 2: Parameter list

Symbol	Meaning	Unit	Value
$N_1$	Number of engineering bacteria	None	Initial 10
$N_2$	Number of cancer cells	None	Initial 50000
$[P_1]$	Concentration of sulforaphane	$\mu$M	Initial 0
$[P_2]$	Concentration of curcumin	$\mu$M	Initial 0
$r_1$	Growth rate of engineering bacteria	min$^{-1}$	0.0083
$r_2$	Growth rate of cancer cells	min$^{-1}$	0.01
$N_{\max,1}$	Environmental capacity of engineering bacteria	None	10000
$N_{\max,2}$	Environmental capacity of cancer cells	None	100000
$\mu_1$	Maximum lysis rate when TcI42 is incativated	min$^{-1}$	0.0017
$\mu_2$	Maximum lysis rate caused by activation of TcI42	min$^{-1}$	0.1
$C$	Activation constant	None	300
$T_0$	Temperature when half of TcI42 is associated	K	315
$T$	Environmental temperature	K	300$\rightarrow$315
$\delta$	Degradation rate of sRRz	min$^{-1}$	0.5
$\delta_1$	Degradation rate of myrosinase	min$^{-1}$	0.0033
$\delta_2$	Degradation rate of DCS-CURS	min$^{-1}$	0.0008
$t$	Time	min	Variable
$v_{\max,1}$	Maximum velocity of myrosinase when the number of engineering bacteria reaches environmental capacity	$\mu$M/min	15.51
$v_{\max,2}$	Maximum velocity of DCS-CURS when the number of engineering bacteria reaches environmental capacity	$\mu$M/min	0.2862
$K_{m,1}$	Michaelis-Menten constant of myrosinase	$\mu$M	84.15
$K_{m,2}$	Michaelis-Menten constant of DCS-CURS	$\mu$M	71.2
$[S_1]$	Concentration of substrate synthesizing sulforaphane	$\mu$M	84.15
$[S_2]$	Concentration of substrate synthesizing curcumin	$\mu$M	71.2
$\lambda_1$	Inhibition intensity of sulforaphane	($\mu$M$\cdot$min)$^{-1}$	0.0025
$\lambda_2$	Inhibition intensity of curcumin	($\mu$M$\cdot$min)$^{-1}$	0.0025

When the temperature increases to 42℃, the suicide switch turns on, leading to fast death of engineering bacteria. As the degradation of sulforaphane and curcumin needs time, the number of cancer cells will still decrease for some time after increase of temperature. However, after 14h, the concentration of sulforaphane and curcumin will be not low enough to inhibit the growth of cancer cells due to degradation. The simulation result demonstates that the suicide switch and inhibition system work well, whereas the inhibition rate is slow and not thorough.

References

[1] Abedi, M. H., Yao, M. S., Mittelstein, D. R., Bar-Zion, A., Swift, M. B., Lee-Gosselin, A., Barturen-Larrea, P., Buss, M. T., & Shapiro, M. G. (2022). Ultrasound-controllable engineered bacteria for cancer immunotherapy. Nature communications, 13(1), 1585.