Team:BIT-China/Model
Model
Modeling is to use mathematical tools to simulate and predict the system. The experiment is restricted by time and cost, which may not be comprehensive. Modeling can solve this problem by establishing mathematical system and improve the whole project. Also, the results of modeling can guide the experiment in turn, and provide clues for the selection of each material in the experiment. Last but not least, modeling can also optimize the production process when satisfying some assumptions. In our design, the overall process will be divided into two parts: cell growth part and cell production part. Our main purpose is to obtain the changes of variables concerned in this process, such as the interaction of different promoter intensity and the factors that affect the final yield. Through the detailed thinking of the whole process, we divide the whole model into the following four models:
Promoter intensity relationship model. The model considers how the intensity of promoter which inducing LuxR affects the production during the growth process. We use the fitting method to get the relationship curve between the promoter intensity and the fluorescence intensity, and then decide which promoter should be used in the actual production.
Cell growth simulation model. In this model, we mainly consider the relationship between cell density and signal factor concentration during cell growth. In this model, we simulate the LuxR-AHL system. And because the system are related to the number of cells, we add the Logistics growth model of cells, and finally describe the relationship between the components in the growth process.
Promoter sequencing model. The purpose of this model is to adjust the sequencing of the three promoters to maximize the final yield. In this model we simplify the model to a certain extent for too many parameters. By considering the solution of the whole process in the equilibrium state, the recurrence relation of each component is obtained, and the optimal solution is achieved by enumerating.
The optimization model of the balance between production and growth. The model is used to consider the transformation of cells in growth and production condition by periodic regulation. The aim is to explore whether it is necessary to regulate and decide the regulation time when required.
According to these four models, we has basically full consideration to the project. The establishment of the accurate numerical relationship between these quantities is great help to the experiment and even the subsequent production.
Purpose
In the gene pathway, we linked the gene of GFP fluorescent protein after pLuxL promoter. By giving different promoter which inducing LuxR expression and then measured the fluorescence intensity. We obtained the relationship curve between the promoter intensity and the fluorescence intensity, then the best promoter was found.
Model Construction:
The promoter data of this model were derived from iGEM2006_Berkeley, where we used the fluorescence intensity in their program to characterize the promoter intensity. In this model, we choose the promoter and their intensities are as followed.
We obtained the curve of fluorescence intensity changing with time under different promoters which inducing LuxR expression according to experiments and get the figure below. In which we averaged the data of the last 6 hours of each promoter and defined it as the corresponding average production intensity AP. We made the following scatter map of the average production intensity AP and the intensity of the promoter.
In this figure, we can find the promoter intensity and the fluorescence intensity have an approximate linear relationship. When the promoter is stronger, and there may be more possible to have a better production. We made a linear fit curve of AP and promoter intensity as the prediction of this small interval. The following equation is obtained.
Where [P]intensity is the promoter intensity.
Results：
In this model, we got the trend of overall change between the AP and promoter intensity and made a linear fit curve to predict it. Subject to the experiment, we may not be able to get a more accurate result, but we have a promoter selection criteria: the intensity of the promoter should be higher than J23102. This is of great significance for practical production and follow-up experiments, and a further step forward to the goal of maximizing production.
Purpose:
To simulate the variation of different components with time, and discover the relationship between these components and the number of cells in the process of cell growth. This model lays the foundation for the whole process.
The process of the quorum sensing:
First of all, in order to describe the intracellular signaling factors accurately, we added LuxR-AHL system to the model. The mode of operation is shown in the following figure. AHL is an important signal factor connecting the two systems, in which the LuxR-AHL system can produce AHL by enzyme reaction. LuxI connects the gene fragment of the production part and is responsible for starting the lycopene production.
Model Hypothesis:
(1) The concentration of signal factors in all cells is the same.
(2) The cell growth environment is not affected by other conditions
The ode model:
(1)the model of signal factor concentration in cell:
When considering the change of signal factor concentration in a single cell, we
refer to the model of Maike
In the model, the RA is AHL-LuxR complex and C is the dimerized complex. The value of the parameters is given when the population density reaches 5×108.The meaning and values of the parameters in the model are as follows:
(2)Cell growth model：
Because the growth of cells in the process of growth is only limited by environmental space, and nutritional factors. We use the Logistic growth model:
The parameters in the model are fitted by the experimental data, and the fitting results are as follows:
Results:
We combine the above models and solve them with software. We can get the concentration curve of all substances, and the results are shown in the figure below:
In this result, we can clearly see that AHL showed very low growth rate in the early stage. At a certain time, when the population density reached a certain level, AHL showed a sharp increase trend, indicating that quorum sensing has occurred. The growth rate of LuxI also began to increase suddenly, and a large number of products were produced. This result shows that our model is very consistent with the phenomenon of quorum sensing, and can be used as a mathematical description of the cell growth state in the experiment. This model can be great significance to the experimental guidance.
Reference
[1] Melke P, Sahlin P, Levchenko A, et al. A cell-based model for quorum sensing in heterogeneous bacterial colonies[J]. PLoS computational biology, 2010, 6(6): e1000819.
[2] Marc, W., & Javier, B. (2013). Dynamics of the quorum sensing switch: stochastic and non-stationary effects. BMC Systems Biology,7,1(2013-01-16), 7(1), 6-6.
[3] James, S., Nilsson, P., James, G., Kjelleberg, S., & Fagerström, T. (2000). Luminescence control in the marine bacterium vibrio fischeri: an analysis of the dynamics of lux regulation. Journal of Molecular Biology, 296(4), 1127-1137.
Purpose
In the production procedure, adjusting different arrangement of the promoter PσB,PσF,PT7 will lead to differences to the final production. It will change the concentration of crtE, crtB and crtI, thus influent the concentration of intermediate product, and finally affect the quantity of Lycopene. This model is aimed to find the best arrangement of the three kinds of promoter which can maximize the Lycopene output.
Model analysis and construction
Since σB,σF,T7 are simultaneously transcribed in genes, we can assume that the amount of these three proteins is the same, so the quantities of these three enzymes crtE, crtB and crtI is only related to their promoter intensities. The ratios of these three promoter strengths were measured by fluorescence experiments. [P]intensity means the intensity of the promoter P, and the calculated data is as followed:
The ratio of three promoter intensities can be assumed to be the ratio of the quantity of the three enzymes crtE, crtB and crtI attached to these promoters.
From the process above we can obtain the following equation. The reaction was assumed to be reached an equilibrium state and that the rate of FPP generated from the previous reaction is assumed to be constant. We express the reaction process in the following equations:
Where kEA[crtE] is the maximum speed of enzyme reaction when formation of GGPP catalyzed by crtE, due to the maximum speed of enzyme reaction is proportional to amount of enzyme. Other equations are the same.
From the above equation, we can get
Our goal is to reach max[Lycopene]. Through the above formula, it can be found that the size between each component is only related to the ratio of the amount of the three enzymes crtE, crtB and crtI. Because the total amount of calculation is very small, the optimal results can be obtained through the enumeration method. The parameters in this model are given by ourselves.
Simulation results
Given the value of following parameters:
Based on the premise above, we obtained the optimization result: max[Lycopene]=8.74.The enzymes and their attached promoters are: crtE with PσF, crtB with PσB, crtI with PT7, which corresponds to the results used in the experiment.
purpose
In industrial production, it could be possible to use the cycle regulation and makes the system constantly transform between growth and production. It may be more effective to use a period of growth time to get more cells to produce than only to keep the cell production. Therefore, our following model is to discuss such an occasion and explore the periodic adjustment scheme in the quantitative case.
Model analysis and construction
First we take a look on the Logistic growth model:
The special solution of this equation under a given initial value N0 when t=t0 is the function as below:
Generally speaking, the regulated growth and production process may be shown in the following figure.
We analyze it only in one cycle and add some analytical variables to this.
In a circular sub process, it can actually be divided into two processes. The growth process from 0 to t1 and the decreasing process from t1 to t2. Two differential equations are satisfied in two processes
The special solution is obtained when the initial value x1，x2 is given separately.
Let t=t1 in the (1) function，then N1 (t1 ) equal to x1. We can get the relationship between x2 and x1.
Let t=T=t1+t2 in the (2) function，then N2(t1+t2) equal to x1. We can get the relationship between t2 and t1.
We can begin to produce only in the period from t1 to T in one cycle. We assume that the production per unit time ΔE(t) is proportional to the population, that is,
Then the production in one cycle is
Assume that the amount spent on each cycle adjustment is m, so our optimization goal is
x2，t2 can be expressed by x1，t1，So the problem is to optimize x1 and t1. The whole optimization problem is
Result
In this model, we give the following parameters
Because of the complexity of this function, the common optimization algorithm can not solve the equation well, so we use the interval enumerating method to do it. We enumerate x1 with every unit from 1000 to 2000, and enumerate t1 every unit from 0 to 100. The optimization answer is
Finally, it is concluded that if the biological properties of the population can not be adjusted, then the adjustment is meaningless. If the adjustment can affect k1, then this adjustment is useful. Such as if k1 turn from 0.01 to 0.001 when the system transform from growth to production, then the answer is
This two results could be used to explain the problems of adjustment in actual production. And this model plays a very positive guiding role in real industrial production.