Team:Tianjin/Model
Introduction
We developed a mathematical model to predict the influence of where to insert the centromere and the dynamics of cells affected. By doing so, we expect to estimate the efficiency with OD600 after using our chromosomal stabilization element group so as to modify our design.
During our experinment, Saccharomyces cerevisiae has been modified to different degrees. After the preliminary modification, S.cerevisiae was inserted with genes that produce lycopene. On this basis, we made two further modifications based on S.cerevisiae: one is to insert the centromere of Yarrowia lipolytica on its chromosome V, the other one is to replace the centromere of its chromosome V with that of Y.lipolytica. We named these three strains TJ01(natural condition S. cerevisiae),TJ02(centromere insertion of Yarrowia lipolytica) and TJ03(centromere replacement with Yarrowia lipolytica) respectively.
Processing
TJ01: Natural condition
When no centromere is inserted in Saccharomyces cerevisiae (assembled the lycopene gene), the population of cells is likely to follow a S-shaped growth curve (L.Ridenour curve), which can be formalized mathematically by logistic function
$$N(t)={1 \over 1+({L \over N_0})\times {\rm e}^{-b\times t}}$$The representations of the parameters in the equation are shown in the table below：
| t | Time, in hours. |
|---|---|
| N(t) | The number of bacteria at time t. |
| N0 | Initial number of bacteria. |
| L | Limit of bacterial population. |
| b | Correction factor. |
| m | Centromere insertion correction factor. |
After simulating the numerical solution (Figure 1), we can see that population of S.cerevisiae growing rapidly and reach to stationary state of 5.68 less than 22.5h. And the growth curve approximates:
$$y={5.68\over {1+95.85{\rm e}^{-0.4639t}}}$$This provides us a necessary comparison criteria.
Figure 1: The population growth curve of TJ01
TJ02: Insertion
After inserting centromere of Y.lipolytica on TJ01, it may increase the stability of non-homologous chromosome, but two centromeres may be simultaneously tugged by the spindle fiber during mitosis. If the two spindles pull in different directions, the chromosomes might be pulled apart, leading to the death of the bacteria, thus causing a decline in the population. Experiments shows just as expected: L is considerably smaller, 4.930, to be exact. Following are some basic properties and assumptions of our model.
Assumption
1. The population growth follows sigmoid growth curve, and has reached close to the stationary stage.
2. Dead cells are considered to be destructed in such a short period that it won’t affect the value of OD600.
3. Other factors that may affect the experiment are negligible.
Our model is constructed based on a correction factor m, modifying the correlation from
$${dN(t)\over dt}=bN_0\times {L-N_0\over L}$$to
$${dN(t)\over dt}=bN_0\times {L-N_0\over L}-m$$After simulating the numerical solution (Figure 2), we compute the numerical solution of m by MATLAB, which is 0.04017.
Figure 2: The population growth curve of centromere inserted S.cerevisiae
TJ03: Replacement
After discovering the negative effects of inserting centromere, we take another measure to avoid snapping the chromosome, that is, to replace the endogenous centromere with exogenous centromere instead of simply insert one.
By simulating the numerical solution (Figure 3), we can see that the growth curve approximates:
$$y={5.654\over {1+100.05{\rm e}^{-0.4851t}}}$$The population of S.cerevisiae hugely increased compared with that of TJ02 and has came very close to that of TJ01, showing that the centromere replacement could work well in S.cerevisiae.
Figure 3: The population growth curve of centromere-replaced S.cerevisiae
Results
The following curves (Figure 4) show dynamics of number change of each kinds of strains.
Figure 4: Combination figure
From the fitting parameters (see in table 2) we can see that simply inserting centromeres in chromosomes can lead to decreased population and replace endogenous centromere with exogenous centromere can successfully avoid that from happening.
| TJ01 | TJ02 | TJ03 | |
| L | 5.68 | 4.93 | 5.654 |
| N0 | 0.05865 | 0.06045 | 0.05586 |
| b | 0.4639 | 0.4237 | 0.4851 |
We specify the population from stationary stage, which stands for the quantity at equilibrium. By our improved model, we successfully predict the population dynamics of centromere-replaced S.cerevisiae. Estimate the key parameters L to be 5.7 (OD600) , can be considered as a good approximation through our estimation after referring to research papers and experimental data. This result is reasonable with our expectation, and it proved that our product indeed does no harm to the population.
Figure 5. The little programme that we made to provide aided support of gRNA sequence design
Figure 6. Programme running example