The bacteria grow in solution with limited resources, both in the lab and in the real-life soil environment. The cell grows, replicates, ages, and dies in a dynamic way as in the classical logistic model.

However, when we talk about cell growth, one of the less noticeable parts is controlled cell death or called cell suicide. We focus on cell suicide for many different concerns below:

• One is about the distribution of herbicides.
  

  AA, as far as we know, cannot be secreted by E. coli automatically. So, we take autolysis as the way to bring the molecule into the environment. We choose a thermo-switch, to interrupt the process of herbs' metabolisms at their maximum.

• The other concern is biosafety.
  

  One of the codes of synthetic biology is to prevent the leakage of GM materials. Our modified genome may pollute the genetic background of wild-type E. coli, and by the process of lysis, the modified genome shall be degraded naturally.

• Also, cell suicide controls the gross amount of herbicide, which reduces the risk of causing great select pressure for mutation against the herbicide.We use negative control on the suicide switch due to safety concerns.

The logistic equation (sometimes called the Verhulst model or logistic growth curve) is a model of population growth first published by Pierre Verhulst (1845, 1847)[1]. The model is continuous in time, but a modification of the continuous equation to a discrete quadratic recurrence equation known as the logistic map is also widely used.

• The growth of cells follows the classical logistic model, which means:
  • No lagging, the amount of capacity immediately responds
  • Not considering other factors, such as competition, toxic exoenzyme and change of environment
  • Not considering spores, as they do not have metabolism and do not produce our product
• Cell suicides independently. One cell may use some information molecule to induce others’ suicide (quorum sensing), but as no papers prove it on our circuit, we ignore it.
• Cell suicide circuit has a stable efficiency. Once activated, cell death will be triggered at a constant rate, which brings a linear relation to us. As no more complex dynamics were detected, we take this hypothesis as acceptable.
• Lysate will be degraded into nutrition immediately. It corresponds with the former growth model and has been proven acceptable before.

Name	Value	Description	Reference
K	6.08OD	Environmental capacity	[2]
r	1.28h−1	Birth rate of E. coli.	[2]
N0	1OD	E. coli population initial values	[2]

Let N denotes the population density of E. coli. Wi th the Logistic equation, we obtain:

where r and K are the growth rate and the environmental capacity of E. coli respectively. Whenever the temperature control switch is activated, the bacterial The lysis gene is expressed and causes the cells to die. Usually, when we control the apoptosis of bacteria to 5% of the original, we turn off the temperature control switch. This process can be described using the following equations:

N = 0.05 × N

The bacterial population then resumes proliferation, a process we call a growth cycle.

Fig(1). Temperature control switch is on

Fig(2). Temperature control switch is off

With cell growth, they produce two critical products for our project.

Herbs are one of the major threats against crops, it competes with crops in sunlight, nutrition, and space, which causes the depression of productivity and further food crisis. However, traditional chemicals damage the biosphere, while they are more able to select mutation against it. Bioherbicide puts an effect on key metabolic processes, meaning less chance to be naturally selected

The bacteria produce aspterric acid (known as AA) with the induction of blue light. AA is a natural, non-toxic, environment-friendly efficient biobased herbicide, produced by a cluster of genes. AA takes effect by depressing DCHD, the key enzyme in the biosynthesis of amino acids.

On the other side, we also produce EPS. The erosion threat grows globally nowadays, rich soil, nutrition, and microbes are flown away by flood or even average rainfall when soil particles do not join together. EPS works as a glue here, joining bacteria, roots, soil animals, soil microbe, and soil itself into an organic structure. A joint structure means a higher effect of our herbicide, less erosion, and higher ecological value.

The schematic of this process is shown below:

Fig(1). Synthesis mechanism of AA

Fig(2). Synthesis mechanism of EPS

Our goal is to simulate the process of gene expression, and the situation of products, which makes us possible to evaluate the actual effect, possible further improvement, and further commercial value.

In an attempt to simplify the model, we take these assumptions:

• As the cycle of growth, autolysis, and re-growth continues, we simply take one cycle representing the whole process. Therefore, all products shall be distributed into the lysate in our model.
• Bluelight promoter as well as lac promoter responses without lagging，but with a certain level of leak expressionwhich means a low-level transcription without blue light.
• The biobrick is relatively independent of background metabolism and expression, such as the energy level, nutrition level, and possible interaction.
• Upstream and downstream genes in one cluster are abstracted as one gene to simplify modeling.
• All herbicides in the lysate are in their activated form. We do not find research about the inclusion body of AA molecule, also, AA is not a protein, and cannot be folded wrongly. So our assumption stands.
• ‘Modeling for herbicide’s blue-light protein promoter’ equals ‘modeling for blue-light protein promoter only, with a theoretical product’ owing to the hardship of directly detecting the herbicide molecular.

Name	Value	Descriptions	Reference
αVVD	5.214 * 10−5M * h−1	Promoter activity for VVD	[1]
αEPS	6.840 * 101M * h−1	Promoter activity for EPS	[1]
βVVD	0.136mol * L−1 * h−1	Ribosome binding efficiency to VVD mRNA	[1]
βEPS	2.160 * 104mol * L−1 * h−1	Ribosome binding efficiency to EPS mRNA	[1]
γrVVD	4.667 * 10−5M * h−1	Degradation rate of VVD mRNA	[1]
γpVVD	0.79mol * L−1 * h−1	Degradation rate of blue-induced proteins expressed by VVD genes	[1]
γrEPS	4.680M * h−1	Degradation rate of EPS mRNA	[1]
γpEPS	4.896 * 102mol * L−1 * h−1	Degradation rate of EPS protein	[1]

Based on above parameters, let N_rx denotes the concentration of mRNA transcribed from the x gene, where x can be VVD or EPS gene. Then we use 4 ODEs to describe the whole process and solve them in MatLab.

$\frac{dN_{rVVD}}{dt}=\alpha_{VVD}-\gamma_{rVVD}N_{rVVD}\tag{1}$

$\frac{dN_{pVVD}}{dt}=\beta_{VVD}N_{rVVD}-\gamma_{pVVD}N_{pVVD}\tag{2}$

$\frac{dN_{rEPS}}{dt}=\alpha_{EPS}-\gamma_{rEPS}N_{rEPS}\tag{3}$

$\frac{dN_{pEPS}}{dt}=\beta_{EPS}N_{rEPS}-\gamma_{pEPS}N_{pEPS}\tag{4}$

Noting that the object of the above equation is a single individual in the population, let S_x denotes the total amount of x protein produced by the E. coli population, where x can be VVD or lacO. Note that in the population growth model we get the total population as N, then we obtain:

S_VVD = N × N_pVVD

S_EPS = N × N_pEPS

Next, we will make adjustments on the underlying model according to the different characteristics of blue light-inducing protein and EPS protein, so this article will have two branches next to describe our model solving process for each of both.

An intriguing topic in the environment is microbial expansion. However, it is too complicated to study accurate motion because of the complexity of a realistic system. therefore, theoretical models are needed in that area.

The cellular automata model is proposed by Von Neumann in the 1950s to study the phenomenon of cell self-reproduction. It is a modeling method assuming that after setting the initial state, each cell is constantly updated according to certain evolution rules. CA is a system that is discrete in time, space, and state. Its basic elements include cellular, cellular space, cellular neighbor, and evolution rules.

• Cellular: Cellular is the basic unit of cellular automata. A cell can be one-dimensional or two-dimensional, and its state is determined by a set of states at each discrete time point.
• Cellular space: Cellular space is the space occupied by a distribution of cells. The appropriate shape can be selected according to the interaction relationship between cells in practical problems. The most common is the grid-type regular arrangement, including triangle, quadrilateral and hexagon.
• Cellular neighbor: Cellular neighbors are a set of cells which have an effect on the change of cellular state. Generally, in the cellular automata model, the state of a cell is affected by the state of itself and surrounding cells, such as competitive nutrients or mutually beneficial symbiosis. In a numerical simulation, the range of neighbors can be determined by controlling the retrieval radius.
• Evolution rules: The evolution rules are set according to a practical problem. But generally, the state of the cell at time t + 1 is a function of the state of the cell neighbor at time t s_i^{t + 1} = f(s_i^t,s_N^t)

where

• t denotes time and is dicrete.
• s_i^t denotes the state of the cell i at time t.
• s_N^t denotes the state of neighbors of the cell i

We adopt Fick’s first law, a classic equation to describe particles’ diffusion in solution. Under the assumption that the diffusion flux (J) goes from regions of high solute concentration to regions of low concentration, with a magnitude proportional to concentration (ϕ) gradient, we can write: J = − P∇ϕ
In 2-D space, that is:

where P is the diffusion coefficient, an intrinsic property of solute and the medium. A large D value means higher diffusivity. is a 2D vector that can be decomposed into several directions. Therefore, we can calculate J in any direction separately.
In this discrete model, we may use $\frac{\triangle\phi}{\triangle x}$ rather than the first derivative. △ϕ is the difference of the number (of cells, EPS or nutrients) between neighbors. Under our unit system, △x = 1 . Thus, we can represent it as: where d denotes the direction for migration and (u,v) denotes the change of direction:

• d = 1 denote left, which means (u,v) = (−1,0)
• d = 2 denote up, which means (u,v) = (0,1)
• d = 3 denote right, which means (u,v) = (1,0)
• d = 4 denote down, which means (u,v) = (0,−1)

Calculate Number of Migration

Based on the above theory, we present the solution of a mathematical model of E. coli migration on a plate. For each passing minute, for each grid of E. coli populations, we calculate the total number of individuals whose migration will occur and simulate the process of their random movement by generating random numbers. The change in the number of individuals in the population between the meshes during migration satisfies the following equation:

Then the popularity model and the protein expression model will be used to get the next time step data. Finally, we get the results of th model of the spread of cells and proteins on a plate

For the protein synthesized by E. coli (because the VVD protein is close to the expression process characteristics of the EPS protein, the VVD protein is used as an example), because there is a delay in E. coli synthesis of proteins, the spread rate of the protein on the plate is lower than that of E. coli, which is common.

Fig(3). The VVD protein expand in the center around the plate

Fig(4). The VVD protein expand in four points around the plate