Team:SYSU-Medicine/M1 Cultivation
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Modeling Part 1
M1 Cultivation
The Curve of Titer of M1 with Time
In the process of the culture in vitro of oncolytic virus M1, there were uninfected vero cells, infected vero cells and oncolytic virus M1 in the medium. And the change of the microorganism numbers are related to the number at the current moment of it.
We used ordinary differential equation model to describe the change of the number of the three kinds of microorganisms with time.
The relationship among the three kings of microorganism is shown in the figure below:
According to the mutual transformation relationship between the microorganism, the ordinary differential equations are established:
$$\left\{ \begin{aligned}
\frac{dU(t)}{dt}&=rU(t)(1-\frac{U(t)+I(t)}{K})-\beta U(t)M(t)\\ \frac{dI(t)}{dt}&=\beta
U(t)M(t)-\alpha I(t)\\ \frac{dM(t)}{dt}&=\sigma I(t)-uM(t) \end{aligned}
\right.$$
According to the experimental data and references, we obtained the values of parameters of M1 as follows [1]:
r = 0.95, K = 600, β = 0.27, σ = 1, u = 0.6
By using ode45 function in MATLAB to solve the system of ordinary differential equations, we obtained the number change curve of M1, infected vero cells and uninfected vero cells with time:
This is consistent with the growth rule of virus in vitro culture, which proves that our ordinary differential equations are reasonable.
We use the solution to calculate the time-varying growth rate of M1, and find that 67.4701h is the time for the highest growth rate.
M1 RNA Point Mutation Model
We first assumed that during the process, each site of the RNA mutates independently. That is, the mutation probability of each site is not affected by the mutation of other sites. At this time, the number of mutated sites of M1 obey Poisson Distribution within a certain time.
$$P(N(s,s+t]=k) = \frac{(\lambda t)^k}{k!}e^{-\lambda t}, k=0, 1,
2\cdots$$
The derivation with respect to t:
$$\frac{\partial \frac{(\lambda t)^k}{k!}e^{-\lambda t}}{\partial t} =
\frac{(\lambda)^k}{k!}e^{\lambda t}t^{k-1}(k-\lambda t) = 0$$
We have:
$$\hat{t} = \frac{k}{\lambda} = \frac{48k}{11000\alpha}$$
During the experiment, the mutation rate of the RNA of M1 is 0.01%, so we can get the curves describing the best time to collect M1 with specified number of mutation sites.
The best collection times are listed as below:
| k | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Best Time | 0h | 43.636h | 87.273h | 130.909h | 174.545h |
We can see that the probability of 2 point mutation is the highest at 87.273h. Combined with the best
time for the growth rate of M1, the best interval to collect M1 who has a high growth rate and a
high probability of 2 point mutation is
[67.4701h, 87.273h].
However, the growth
rate curve seems to decent more sharply, so the best time is ought to get closer to 67.4701h. In a
sense, 72h is a good option. Luckily, 72h is actually the best time for collection, confirmed by
experiment. In a sense, our model is quite suitable for our experiment.
Additionally, In order to make the model more scientific and reasonable, we consider that RNA sites of M1 may mutate interactively. Therefore, we adopt the Pearson Correlation Test to calculate the correlation of the RNA sites. [2] In fact, mutation rate 0.01% can be used under the assumption that the mutation is independent between sites. 0.01% is measured under experiment, which contains the situation that one site will be highly related to another site. Because in RNA, the relationship between sites is complex. If we consider the relationship of sites and estimate which site will mutate, the model will become very complex. In addition, the mutation data is hard to collect, and the relationship between sites is unknown. So there is significant for us to choose the Poisson Process, which can be accurate and effective.
Renference
[1]Xiangming Zhang, Zhihua Liu. Bifurcation Analysis of an Age Structured HIV Infection Model with Both Virus-to-Cell and Cell-to-Cell Transmissions. International Journal of Bifurcation and Chaos 2018 28:09
[2]Ren Yonghui. Study on several tumor treatment models [D]. Shanxi normal university,2012.