Modeling

1. Model Assumptions

To establish a more reasonable mathematical model, the following assumptions are made:

1. The timescale for binding and transcription reactions is much faster than translation
2. The growth of bacteria conforms to the assumptions of the logistics equations
3. The enzymatic reaction process conforms to the Michaelis–Menten equation
4. The binding reaction of the transcription factor to the environmental signal and the binding reaction of the transcription factor to the promoter will reach equilibrium in a short period of time and can be regarded as a balancing process in subsequent analysis

2. Model Building

2.1 Bacterial growth and lysis models

We developed an ordinary differential equation model to simulate the case of the time curve of the number of bacteria at a certain concentration of L-rhamnose.

We denote$r$ as the growth rate of E. coli flora, $N$ as the number of E. coli flora. If we assume that $r$ doesn't change over time, then the simplest model is that: $\frac{dN}{dt}=rN$. In practice, for nutrients and living space in the bacterial culture environment are limited, there will be a maximum upper limit of $N_{max}$ in the number of flora, and at the same time, as the number of flora increases, the growth rate of the flora should decrease, so $r$ is related to $N$. Under the assumption that the natural growth rate is $r_0$, we can obtain: $r\left(N\right)=aN+r_0$.

When$N=N_{max}$, the number of flora reaches its maximum, at which point the growth rate of the flora should be 0, so it can be obtained:$r\left(N_{max}\right)=a{N}_{max}+r_0=0$.

Therefore, we have:$a=-\frac{r_0}{N_{max}}$;

$a=-\frac{r_0}{N_{max}}$$\frac{dN}{dt}=r_0\left(1-\frac{N}{N_{max}}\right)N$.

Since the experiment stops adding L-rhamnose at $t=9h$, causing the bacteria to initiate the lysis gene and make the bacteria lysis. Therefore, we note that $\gamma \left(t\right)$ is the mortality rate of the bacteria under the action of the lysis gene at the moment, so the total equation is as follows[2]:

Where $\gamma \left(t\right)$is defined as follows:

If we denote ${\mathrm{N}}_{\mathrm{i}}^{\mathrm{j}}$ as the experimental data of the $\text{ i }$th parallel group at $\text{ j }$th sampling time, and $N$ is the number of data samples in each parallel group, then the data sampling time can be denoted as ${\left\{{\mathrm{t}}_{\mathrm{j}}\right\}}_{\mathrm{j}=1}^{\mathrm{N}}$. Since our experiment was conducted in triplicate control group, $\mathrm{i}\in \{1,2,3\}$. Our goal is to select the above parameters reasonably so as to minimize the error between the curve of the formula and the experimental data.

Geometrically, our model curve needs to be the closest to the experimental data curve, mathematically, we need to find the parameter ${\mathrm{r}}_0,{\mathrm{N}}_{max},\mathrm{a},\mathrm{b}$ that minimizes the error between the model and the experimental data and satisfies the following conditions:

The essence of this problem is an optimization problem. So, for the target function: $\sum _{j=1}^{\mathrm{N}}{({\tilde{\mathrm{N}}}_{\mathrm{j}}-{\overline{\mathrm{N}}}_{\mathrm{j}})}^2$, we apply the particle swarm optimization algorithm(PSO)[3],To find the global optimal parameter${\mathrm{r}}_0,{\mathrm{N}}_{max},\mathrm{a},\mathrm{b}$ , the following is the iterative solution procedure of the particle swarm optimization algorithm. The iteration process is shown in the following figure

After that, we selected the average $O{D}_{600}$ concentration of three parallel groups as the data point, plotted the test data curve that generated the change of $O{D}_{600}$ concentration over time, and plotted the model curve and the test data curve at the same time, as shown in the following figure:

Parameters	Values	Meaning
$r_0$	0.1707	The natural growth rate of engineered strains
${\mathbf{N}}_{max}$	1.1147	The Maximum ${\mathrm{O}\mathrm{D}}_{600}$ value of engineered strains
a	0.5457	The lysis rate of engineered strains

2.2 Models of optimal temperature and pH for cell production

From the data and theoretical knowledge, the reaction rate of the engineered strains we constructed showed a bell-shaped curve with pH value and temperature[3], and there was an optimal pH value and an optimal temperature.

To find the optimal pH and temperature model for the reaction of engineered strains, we try to use a two-dimensional Gaussian distribution model to find the optimal pH and temperature of the reaction, as well as the theoretical optimal reaction yield.

Let $x,y$ be two random variables, and if they satisfy a two-dimensional Gaussian distribution, the probability density function $\rho (x,y)$ is[5]:

$\rho (x,y)=\frac{1}{2}{e}^{-\frac{x^2+y^2}{2}}$

In general, if we set the reaction rate of engineering strains to be $\gamma (\mathrm{x},\mathrm{y})$, and the temperature and pH of the reaction system are $x,y$, then the optimal model we propose can be expressed as:

$\gamma (\mathrm{x},\mathrm{y})={\mathrm{a}\mathrm{e}}^{-\mathrm{b}{(\mathrm{x}-{\mathrm{x}}_0)}^2-\mathrm{c}{(\mathrm{y}-{\mathrm{y}}_0)}^2}$

We fit the resulting data to minimize the error between the data and the model, and finally get the optimization parameters as follows:

Parameters	Values	Meaning
a	11.7	Theoretical maximum yield of engineered strains
b	0.2033	The amount of extrusion to the x-axis
c	0.001264	The amount of extrusion to the y-axis
${\mathrm{x}}_0$	6.999	The best pH value of engineered strains
${\mathrm{y}}_0$	39.03	The best temperature value of engineered strains

From the above table, we can get: The theoretical maximum yield of engineered strains is 11.7g/l in 24h, The best pH value of engineered strains is 6.999, The best temperature value of engineered strains is 39.03. Finally, we plot the fitted 2D Gaussian surface in the same spatial coordinate system as the actual data curve, and the result is shown in the following figure:

3. Reference

1. Gang, X. U. , H. Wen , and W. U. Kun . "Primal chaos data system and feedback control research of Logistic population increase model." Journal of Natural Science of Heilongjiang University (2003).
2. Alon, U. An Introduction to Systems Biology: Design Principles of Biological Circuits 3–19 (Chapman & Hall/CRC, 2007).
3. Clerc, Maurice . Particle Swarm Optimization. Ashgate, 2006.
4. Srinivasan, Bharath. "A guide to the Michaelis–Menten equation: steady state and beyond." The FEBS journal (2021).
5. Silvestre, M.P.C., Carreira, R.L., Silva, M.R. et al. Effect of pH and Temperature on the Activity of Enzymatic Extracts from Pineapple Peel. Food Bioprocess Technol 5, 1824–1831 (2012).
6. Bodin, N. A. , and V. A. Zalgaller . "Concavity of certain functions connected with the two-dimensional normal distribution. " Litovsk.mat.sb (1967):389-393.