# Team:HUST-China/Modelling/MCOS

### Multi Cells Oscillation Simulation

We notice that a single oscillator cell would not excrete enough propionate to lower blood pressure - we need a group of them. Standing on the basis that a single cell would generate periodical signal, we wanted to know if a group of these cells would oscillate as well. So we built a computer model to simulate this situation.

#### Methods

1.Establish behaviors of cells according to life circle of E.coli;
2.Investigate reasonable parameters set from previous researches;
3.Use Q-Q plot to find the correspodent time intervals in experiment data and simulation data; comparing them after normalization using Kolmogorov-Smirnov test.

#### Background

Life cycle of E.coli is approximately 60 minutes. They would take 40 minutes preparing for cell fission and during that time they will express proteins, which in our case is mRFP. Then they would start to divide themselves and cease to express protein in a 20 minutes interval. In terms of death of cells, according to logistic model of cell population, death rate is linear to population itself. Based on this, we set that a certain amount of cells, which is proportion to squares of population, would be killed due to limited food. Those cells are picked randomly.
In terms of population of cells, we take real-life situation in to consideration. For safety(Frequently Asked Question), we planned to wrap our engineered cells into microencapsulation to avoid direct contact with human intestine. Such drug deliver system is called OCDDS. Bacteria concentration in microencapsulation can reach $10^{10}cfu/mL$ and diameter of microencapsulation can reach 433 67μm. For convenience, we assumed that each monocolony originated from a single bacteria. Based on that, the number of bacteria are approximately 81713~208333. We set the population to 100,000.
Moreover, we assumed that the quantity of AraC are linear with plasmid copies. Given a specific environment, the number of initial plasmid copies are a constant. The replication of plasmids can be thought to be completed instantly (within 0.05min). When cells started to divide, the plasmids are allocated into two filial cells evenly. During the lifespan of a single cell, the number of replicated plasmids obey Poisson distribution.

#### Assumptions

(1)The expression interval of cells $x\sim N(40, 2^2).$
(2)When cells exit expression interval, they start to divide and cease to express mRFP.
(3)When reenter expression interval, AraC's concentration would be reset to 0 and restart oscillation.
(4)The division interval of cells $x\sim N(20, 1^2).$
(5)Certain amount of cells, which is proportion to square of population, is 'sentenced' to dead in every round. They are picked randomly.
(6)AraC's concentration is proportion to plasmid copies.
(7)When cells are dividing, the plasmid copies would increase by y, and y~Pois(50), then split evenly into two filial cells when division is ended.
(8)Two filial cells would experience different division intervals.
(9)All the cell generate the same AraC's curve, whose period is 44.8 minutes and is a numeric solve of DDEs mentioned in the last section.
(10)AraC output of each cell are not synchronized in the beginning.

#### Results and Fitting

Fig 1.Population of bacteria against time. Average = 105180, largest difference Δ = 5667

The population of bacteria is fluctuating within a small range (5667/105180 = 0.0539) and generally steady, showing that the logistic model is feasible and this model is successfully simulating the population within microencapsulation.

Fig 2.AraC concentration of oscillating cells within microencapsulation.

The multi cells oscillation simulation suggests that even with that amount of cells, the oscillation will still exist just like a single one. We then fit this curve with our wet-lab result, which is a curve of a oscillating cells' fluorescence intensity.
We first draw the Q-Q plot between two groups of data for a explicit visual comparison to see whether further analysis is needed for determination:
In this figure, we can see that these two groups of data are quite similar. Then time dimension should be taken into consideration.
Since data from simuation and data from experiment have different unit and different period, we choose data of two periods and then have them normalized using their correspondent maximum difference. Then we get the figure that:
In this figure, we can see that curve of experimet data and curve of simulation data is quite close, and residue is almost symmetrically disrtributed. Thus, we can conclude that our data, either experiment one or simulated one, confirms the other.
Furthermore, we use K-S test in these two groups to confirm ourselves. P-value is calculated: P= 0.2827; in other words, we fail to reject the null hypothesis, namely there is lack of evidence that these two groups of data are different.

#### References

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Catherine Tomaro-Duchesneau, et al., 2013, Microencapsulation for the Therapeutic Delivery of Drugs, Live Mammalian and Bacterial Cells, and Other Biopharmaceutics: Current Status and Future Directions., J. Pharm., Article ID 103527
Chandra Iyer, et al., 2005, Studies on co-encapsulation of probiotics and prebiotics and its efficacy in survival, delivery, release and immunomodulatory activity in the host intestine., Thesis (Ph.D.)--University of Western Sydney, 2005.
Nordström K., 2006, Plasmid R1--replication and its control., Plasmid., 55(1):1-26
Shiru Jia, 1988, Dynamic Model of Plasmid Replication and Expression, Journal of Tianjin Institute of Light Industry
Gustafsson P, Nordström K, Perram JW., 1978, Selection and timing of replication of plasmids R1drd-19 and F'lac in Escherichia coli., Plasmid., 1(2):187-203.