Modeling Starts from the minor
Introduction
Our modeling consists of the following three parts:
- To make the project applicable in real life, we designed a device with modified engineering bacteria inside, which can be placed in soil to attract and kill nematodes.
- Assuming there is a farmland, we took advantage of gas diffusion and nematodes’ movement analogue simulation to find the best position for the device to be placed.
- We established a database to broaden the scope of our applications, combined with our methods, to kill more different pests. We hope this new environment-friendly method, based on principles of synthetic biology, could be shared with and improved by the researchers all over the world.
Design
Device 1.0
In order to enable our project to be applied in real environment, we designed and made Device 1.0. Because the productivity of attracting substance by E.coli is limited and the price of man-made attracting substance is quite high. We choose low-cost CO2 as our assistant attracting substance since it was demonstrated that carbon dioxide has a function of attracting nematodes[1-3].
Our device has four areas. The first area is CO2 generating area. We produce CO2 by mixing limestone and dilute hydrochloric acid together, which is widely used in industry. The second area is E.coli culturing area. It includes a medium inside the device - to culture modified engineering bacteria. The third area is light controlling area, which includes a LED light. When it is turned on, red emission will activate the promoter and bacterial cells will express attracting substance; while the LED light is turned off, toxalbumin will be produced. The fourth area is made up of a cuboid outer shell, which can support the device.
The figure above shows our product – Device 1.0. We have simulated our device in the lab and our results show that our system is able to work under real-world conditions.
There are 8 steps to apply our device in farmland as shown in the video above. And the procedures below were conducted in a fume hood.
- Step 1. We gathered a box of soil from our farmland.
- Step 2. We applied culture medium onto slide glasses.
- Step 3. We gathered several small stones used as CaCO3 and put them into the test tube.
- Step 4. We added HCl into the separating funnel.
- Step 5. We opened the faucet of the separating funnel in order to let HCl flow into the test tube under the atmospheric pressure. Then small stones reacted with HCl to liberate CO2. We also demonstrated how to use red LED light in the video.
- Step 6. We put the device into the soil.
- Step 7. After 3 hours of incubation, we took out the device and removed the slide glasses.
- Step 8. Finally, we tested the results using a microscope. In the video, we showed the movement of a nematode that we separated from soil from Hebei Province, China.
Device 2.0
After discussion among team members about device 1.0, we found several deficiencies. First of all, the size of our device is so limited that the reaction substrate (limestone and diluted hydrochloric acid) is not enough to generate CO2 constantly. Replenishing the reaction substrate frequently will greatly increase the cost. Secondly, the space utilization percentage is low on the slide medium; as a result, the production of both attractant and toxalbumin is low. Lastly, LED tube is sizable and needed to be powered constantly, thus not suitable for farmland. In device 2.0, we improved device 1.0 in those three aspects mentioned above. Indeed, we need to further improve our device.
Firstly, we assume that engineering bacteria can produce ideal concentration of attractant and toxalbumin (the ideal concentration is in reasonable range). Then, considering the problem of space utilization rate and the fact that E.coli can only grow on the surface of the medium, we changed the medium’s shape to sphericity to reach the highest space utilization percentage. Meanwhile, our device has been shaped to sphericity too, and the LED light has been moved to the center so that the whole surface can get the radiation evenly. A design like this enable us to use mini LED light bulb, and use solar energy instead of electricity as our power resource. This change fulfills our aim to save money and energy, which is also more economical and enviromentally friendly.
As shown above, device 2.0 has two shells – an outer shell and an inner shell. There are tiny holes on the outer shell, which allow nematodes to enter the device. The holes are biosynthetically designed to prevent other soil organisms from entering. The red LED light can be supplied with the power from solar power pane, which will be put on the surface of the farmland. And we can achieve remote control by using radio technology.
Simulation
Modeling in farmland
With these improvements to our device, we want to figure out how we can put our technology into practice in farmland, and whether our device would be more efficient and economical than the methods used currently used in killing nematodes (crop-dusting mostly). In order to answer these two questions, especially the second one, we need to conduct a simulation of the movement of nematodes to determine the most suitable place in the farmland to place our device. Our modeling process is given below:
Modeling assumptions
- Nematodes can be attracted if the attraction concentration of limonene is higher than the lowest attractant concentration and then can move towards the direction of the highest odor concentration after attracted by the limonene smell[4].
- Engineering bacteria in our device are able to produce enough limonene as we need.
- The creeping speed of nematode is fixed.
- Nematodes move on a 2D plane. Because the movement range of nematodes is within 10cm and the distance between two devices is much longer than 10cm, we can ignore the depth.
- When the gas diffuses in space, the concentration is not affected by temperature, aerodynamics or other factors.
- The device emits gas continuously and the concentration is uniform in space. And after the gas diffusion, the attractant concentration will stabilize.
Explanation of symbols
Table 1. Explanation of symbols
| Symbol | Meaning | Symbol | Meaning |
|---|---|---|---|
| V | Nematodes’ movement velocity | T | Attraction time |
| N | Distance between two equipment | S | Distance between two devices |
| C0 | The lowest concentration | C | The concentration of every device |
Data
Table 2. Data we use in the simulation
| Constant | Value | |
|---|---|---|
| Nematodes’ movement velocity | \(290 \mu\)m/s[5] | |
| Nematodes’ density | \(1.89 \times {10^6 m^3}\)[6] | |
| The lowest concentration | \(99.3\space g/m^3\)[7] | |
| Diffusion coefficient of limonene | \(2.46\)[8] | |
Modeling approach
Model solution
Considering that we need to control the density of nematodes in a proper range, we hope to do this by killing all the mature nematodes within certain period of time under the identical condition. Because different species of nematodes have different life cycles, our attraction times vary. And we found that the Meloidogyne needs 8.5d-10.5d to develop into second instar larvae from monoplast egg according to an exiting literature[9]. We define d as the half time of which nematodes grow from eggs to larvae. We plan to apply our “attract – and – kill” process in two rounds. We will attract nematodes into our device in time d and kill them in time d. We found that all nematodes are killed after two “slaughters”. Additionally, it takes time for nematodes to reach the device, so we can determine the distance between the devices we place according to the three-day-period and the crawl speed of nematodes.
$$L = V \times {T}$$
Let L be the longest distance between nematodes and the equipment, V is nematodes’ crawl speed, and T is time nematodes need to develop into adult.
Firstly, we considered a square farmland, to minimize the number of equipment we use, the arrangement mode of equipment is as follows (firstly, we consider the situation with four devices arranged together).
As shown above, the white circles represent the devices of colon bacillus, while the bigger circles represent diffusion, with the same odorousness in a circle. What’s more, squares ABCD are the simplified farmland. It’s obvious that in this farmland (do not consider points on the sides of the square), point F is the farthest point away from the four equipment. Based on the longest distance between nematodes and the equipment which has been acquired in the first question, we obtain:
$$AB = {2\sqrt{2}EF} = N$$
N is the distance between two equipment.
Table.3 The results of the relationship between attraction time and distance
| Attraction time(Day) | Distance between two divices(m) |
|---|---|
| 0.5 | 17.7137 |
| 1 | 35.4345 |
| 1.5 | 53.1518 |
| 2 | 70.8691 |
| 2.5 | 88.5863 |
| 3 | 106.3036 |
Considering that odorousness will be reduced with the increase of distance away from the device, we should only consider about the superposed odorousness at point F(considering four equipment now) to reach the lowest attractant concentration. Thus we obtain:
$$C_1 = C_0/4$$
Where C1 is the superposed odorousness diffused from the device, C0 is the lowest attractant concentration.
According to Fick's Law, We obtain a 3-D gas diffusion model. Since the three dimensions have no difference between each other, we simplify the question into a 1-D situation according to isotropy.
According to Fick's Law, points out that during the process of unsteady diffusion, at a distance of x Since the three dimension has no difference between each other, we simplify the question into 1-D situation according to isotropy.
A brief introduction to the Fick's Law: It points out that during the process of unsteady diffusion, at a distance of X, the change rate of concentration to time is equal to the negative value of the change rate of diffusion flux to distance. Its mathematical expression is as follows:
$$\frac{\partial C(t,x)}{\partial t} = D_{A}\frac{\partial ^2 C(t,x)}{\partial x^2}$$
Boundary value condition is: C（t,0）== C2
C(x,t) is the concentration at the distance of X from gas source in t, N0 is the concentration of gas source, D is the diffusion coefficient.
According to the model above, we obtain the gas diffusion model in 3D. And according to the principle of isotropy, we simplify the problem in 1D.
We then obtain the solution
$$c(x,t) = \frac{C_{0}}{2}(1-erf\frac{x}{2\sqrt{Dt}})$$
Adding that
$$erf(x) = \frac{2}{\sqrt{\pi}}\int^{x}_{0}e^{-\lambda^{2}}d\lambda$$
$$erf(x) = \frac{2}{\sqrt{\pi}}\int^{x}_{0}e^{-\lambda^{2}}d\lambda$$
And we apply the following two fomulae to get the approximate value
$$erf(x) = \frac{2}{\sqrt{\pi}}$$
$$\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{n!(2n+1)}$$
We build a gas diffusion model to calculate the gas concentration at each point in matrices. In matrix A, we assume that the device is put on row i and column j (i, j). The distance between any point (m, n) in any matrix and point (i, j) is:
$$x = \sqrt{(m-1)^2 + (n-j)^2}$$
The concentration at the point at time t is:
\begin{equation} \begin{array}{l} c(x,t) = \frac{C_{0}}{2}(1-erf\frac{x}{2\sqrt{Dt}})\\ =\frac{c_0}{2}(1-\sum_{i=0}^{18}(-1)^i\frac{x^{2i+1}}{i!(2i+1)(2\sqrt{Dt})^{2i+1}}) \end{array} \end{equation}
This is the schematic diagram of our device in a 300m × 300m farmland according to results of simulation. We set the attraction time as 3 days and the distance between two devices is 100 m. As the evenly distribution of nematodes is from 5 cm to 15 cm underground, we put the device into the place which is 10 cm underground. In order to meet the lowest attractant concentration, the concentration of limonene produced by every device should be \(24.8 g/cm^3\).
The explanations of simulation: In the video above, we built up a simulation model of the movement of nematodes using the cellular automaton in MATLAB. It shows how nematode moves in a real farmland with the device. We built up a 301×301 matrix in MATLAB, in place of a 20m×20m piece of farmland. The proportion of virtual nematode and real nematode is 0.0084.
The nematodes is distributed randomly at the beginning. The rules of the movement are as follows:
- The nematodes move towards 4 different directions: front, behind, right and left;
- Every time the nematode choose one of the 4 directions randomly and the probability of these 4 directions are the same;
- If the nematode move to the place where the gas concentration is lower than the previous position, then the nematode will move back to the previous position.
We did the different simulation experiments with one device in the farmland and four devices in the farmland separately.
Database
In order to apply our hypothesis and methodology to more agricultural pests control, we established a database. We hope to take a better use of this new environmental-friendly method from the context synthetic biology and share it with researchers all over the world. Our database contains 3 daughter databases: attractant, pests and toxalbumins bases. We will update relevant information about the engineering bacteria we designed to kill the pests in our database. Currently, our database contains the biobrick we established in this project and the one established by team ZJU-CHINA.
All the users have the permission to edit and add new contents, and we welcome everyone to use and enrich our database!
Our link address is: http://fwpz.github.io/BNU-Database/
All the users have the permission to edit and add new contents, and we welcome everyone to use and enrich our database!
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