Team:HZAU-China/Model
We want to simulate the situation that tumor cells and Salmonella together in a liquid environment. We used the law of mass action to establish a model for the infection process of Salmonella, which is dimensionless.
$$N_{normal} + S_{almonella} \overset{Aw}{\rightarrow} N_{w} + S_{al\_normal}$$
$$N_{tumor} + S_{almonella} \overset{As}{\rightarrow} N_{s} + S_{al\_tumor}$$
$$S_{almonella} = S_{almonella0} - N_{normal\_cell} - N_{tumor}$$
$$\dfrac {dN_{w}} {d_{t}} = A_{w} S_{almonella} N_{w}$$
$$\dfrac {dN_{s}} {d_{t}} = A_{s} S_{almonella} N_{s}$$
$$\dfrac {dS_{almonella}} {d_{t}} = - \dfrac {dN_{w}} {d_{t}} - \dfrac {dN_{s}} {d_{t}}$$
\(N_{normal\_cell}\): The density of normal cells.
\(S_{almonella}\): The density of Salmonella in the liquid environment.
\(N_{tumor}\): The density of tumor cells.
\(N_w\): The number of Salmonella in the normal cells.
\(N_s\): The number of Salmonella in the tumor cells.
\(A_w\): The affinity between Salmonella and normal cells.
\(A_s\): he affinity between Salmonella and tumor cells.
Salmonella begins to replicate two hours after infection1 .
$$\dfrac {dN_{sal}} {d_{t}} = K_{break} N_{s} + N_{sal} 2^{\dfrac {t} {T}} \ln{2} \dfrac {1} {T} $$
According to our experimental results, we noticed that Salmonella follows Poisson distribution in normal cells, and an app was designed to judge the distribution of bacteria in the cells. We assume that the area less than 1 in the Poisson distribution is a part of cells which are not infected by Salmonella. According to our experimental results, cells which is infected by only one Salmonella can also die of pyroptosis. Based on this feature, we divide cells into uninfected and infected cells. When the average number of bacteria in the cell changes, which means that the λ of Possion distribution changes, the ratio of the two kind of cells will change. In summary, when the average number of Salmonella in cells changes, the proportion of dead cells will change.
Figure 1. Poisson distribution and Salmonella infection results. Figure 1a. Based on statistics on experimental results, we proved that the Salmonella follows Poisson distribution in normal cells. Figure 1b. We assume that the area less than 1 in the Possion distribution is a part of cells which are not infected by Salmonella. When the λ of Possion distribution changes, which means the average number of Salmonella in cells changes, the proportion of infected cells changes. Figure 1c. Cells which is infected by only one Salmonella can also die of pyroptosis.
We hope that the mathematical model can help the Salmonella infection experiment. In our final phenotypic experiment, the cell carries the GSDMD gene induced by atc, and we hope that the observed result is that the proportion of atc-induced cell death is more than which is not induced to prove the atc promoter is effective. In this experiment, the error may be big if the proportion of cells infected by Salmonella is different. What’s worse is that the experimental results we observed may be contrary to the actual situation.
Figure 2. Results caused by efficiency differences of infection. If the proportions of infections are different.
The experimental results can not prove atc promoter induced pyroptosis. The picture showed that the act promoter is induced and caused 90% cells’ death and 70% cells dead because of the promoter disclosure, but the difference of the propotion of infected cells is so big that the experimental results are contrary to the truth. However, we can solve this problem by improving the proportions of infected cells as much as possible.
We solve this problem by predicting the proportion of cells infected with bacteria over time.
Based on these, we designed an app with matlab (https://github.com/cccoolll/Pyroptosis.git) . In this app, different parameters got from the experiment can be input to predict the experimental results. Not only that, the app can povide guidance to our experiments.
Figure 3. The app we designed.
1: The change of the number of Salmonella in a single cell of tumor cell and normal cell infection
experiments, the red curve is the condition in the cancer cell, and the blue curve is the condition
within the normal cell;
2: Changes of the proportion of infected cells of tumor cell and normal
cell infection experiments, the blue curve is the condition in the cancer cell, and the red curve
is the condition in the normal cell.
3: the concentration of added Salmonella;
4: Output value for
the result of the optimal infection time.
5:The density of tumor cells;
6: Rate constant of
Salmonella infecting tumor cells.
7: The density of normal cells;
8: Rate constant of Salmonella
infecting normal cells;
9: Changes of the proportion of infected tumor cells of tumor cell
infection experiments.
Figure 4. According to our experimental protocol, the MOI is 100 and the we correspond the MOI to concentration of cells. The result showed that the infection time is at least 2 hours to eliminate unnecessary variables.
Figure 5. In order to reflect the affinity of Salmonella between tumor cells and normal cells, we hope that the difference between experimental results of tumor cells and normal cells infecting experiment is obvious. However, the number of bacteria in different cells is difficult to count, we can only obvious experimental results by counting the number of infected cells and calculate the proportion of infected cells. Therefore, we need to predict the time when the difference of experimental results are the best.
The Tet repressor protein (tetR) regulates transcription of tetracyclines resistance protein, tetA. The expression of tetA must be strictly regulated since tetA is a membrane-spanning H+-[Tc-Mg]2+ antiporter which means it can lower the pH environment of cytoplasm. As a result, the natural circuit of tetracyclines regulation is a negative-feedback circuit2. Tc is the inducer, which shows high affinity to tetR protein. The tetR protein binds to tetO sequence on DNA specifically, thus inhibits the expression of Tet promoter. When Tc, or other substantial molecule like ATc (anhydrotetracycline) diffuse into bacteria, it will bind to tetR protein and unleash the tetR protein from DNA, thus release the inhibit and start the expression of Tet promoter.
In our project, we choose ATc (anhydrotetracyclines) as the inducer. ATc is less harmful to bacteria than Tc and about 100-fold higher affinity to tetR than Tc2.
The ATc model aims to predict and solve two problems: Firstly, how fast dose the circuit react to ATc; Secondly, how much target gene will express in the bacteria community under a certain concentration of ATc.
There are two tetO sites on the Tet Promoter and both can bind to tetR protein randomly and inhibit the promoter’s expression independently. To make the condition simple, we consider the two tetO sites into one as we just want to explain the relationship of the inhibition of the promoter and the tetR protein.
In our project, the ATc concentration in our incubation environment is uniform, and the diffusion rate of anhydrotetracycline can be ignored3 . In spite of this, the degradation rate of ATc under 37℃ must be taken into account as reported4.
Based on these consideration and truth, we give out these hypotheses:
1. Regard two tetO operon as one equivalently.
2. Ignore the diffusion of ATc
3. The reaction time between ATc and tetR, tetR and DNA is much faster than transcription and
transformation.
$$tetR + [tetR - ATc_2] = tetR_{total}$$
$$tetR + 2 \times ATc = [tetR - ATc_2]$$
$$P_{tet} + [tetR_2 - P_{tet}] = [P_{tet}]_{total}$$
$$2 \times tetR + P_{tet} = [tetR_2 - P_{tet}]$$
$$$$
Based on Hill function, we can determine the amount of activated tetR, tetRact:
$$ tetR_{act} + n \times S(t) \rightarrow [tetR \times S(t)_n] $$
$$ K_X = \dfrac {tetR_{act} \times S^n (t)} {[tetR \times S(t)_n]} $$
$$ tetR = tetR_{act} + [tetR \times S(t)_n] $$
$$ tetR_{act} = \dfrac {tetR} {1 + \dfrac {S^n (t)} {K_{X}}} $$
Based on Hill function, we can determine the amount of activated promoter, with which we can calculate the total transcription speed of all promoters per cell:
$$ P_{tet\_act} + n\cdot tetR_{act} \rightarrow [P_{tet} \cdot (tetR_{act})_n] $$
$$ P_{tet\_copy} = P_{tet\_act} + [P_{tet} \cdot (tetR_{act})_n] $$
$$ K_d = \dfrac {P_{tet\_copy} \times tetR^n_{act} } {[P_{tet} \cdot (tetR_{act})_n]} $$
$$ A_{mRNA} = P_{tet\_act} \times beta $$
$$ A_{mRNA} = \dfrac {P_{tet\_copy} \times beta } { 1 + \dfrac {tetR^n_{act}} {K_{d}}} $$
Kinetic equations of transcription and transformation:
$$ \dfrac {dmRNA} {dt} = A_{mRNA} - K_{deg\_mRNA} \times mRNA $$
$$ \dfrac {dtetR} {dt} = K_{trans\_tetR} \times mRNA - K_{deg\_tetR} \times tetR $$
$$ \dfrac {dGSDMD} {dt} = K_{trans\_GSDMD} \times - K_{deg\_GSDMD} \times GSDMD $$
Degradation function of ATc by time3:
$$ \dfrac {dS_x(t)} {d_t} = -K_{deg\_ATc} \times S_x(t) $$
$$ \ln(S_x(t)) = \ln(S_x(0)) - K_{deg\_ATc} \times t $$
Growth curve of bacteria based on logistics model by P. F. Verhulst:
$$ N(t) = \dfrac {K_{max}} {1 + C \cdot e^{-rt}} $$
Total GSDMD expressed in bacteria community:
$$ GSDMD_{total} = N(t) \cdot GSDMD $$
\(S_x(t)\): concentration of ATc, as a function of time
\(tetR_{act}\): concentration of activated tetR
\(tetR \): concentration of total tetR
\(GSDMD \): concentration of GSDMD
\(A_{mRNA} \): transcription rate constant of the promoter
\(P_{tet\_copy} \): plasmid copy number
\(K_X \): disassociation rate constant of tetR and ATc
\(K_d \): disassociation rate constant of tetR and DNA
\(beta \): original transcription rate constant of the promoter
\(K_{deg\_mRNA} \): degradation rate constant of mRNA
\(K_{deg\_tetR} \): degradation rate constant of tetR
\(K_{trans\_tetR} \): translation rate constant of tetR
\(mRNA \): concentration of mRNA
\(K_{deg\_GSDMD} \): degradation rate constant of GSDMD
\(K_{trans\_GSDMD} \): transcription rate constant of GSDMD
\(K_{deg\_ATc} \): degradation rate constant of ATc
\(n \): Hill coefficient
\(N(t) \): initial OD600 value of the bacteria
\(r \): growth rate of the bacteria
\(K_{max} \): Maximum OD of the bacteria in cultivation
As is hard to obtain the initial parameters in the equations above on our own without any experiments, the only way to obtain these parameters is to look up in former research or other teams work. Fortunately we got a copy of these parameters from team William and Mary iGEM 20166. These parameters include \(K_X = 0.36 \), \(K_d = 0.1 \), \(beta = 0.0023 \), \(K_{deg\_mRNA} = 0.009 \), \(K_{deg\_tetR} = 0.631 \), \(K_{trans\_tetR} = 235.5 \) (All unites are combined of nM and s). Considering that both Salmonella and E. coli are in Enterobacteriaceae, we assumed that in Salmonella these parameters are the same with those in E. coli since we just wanted to figure out an abstract instruction to wet lab.
To gain the parameters in bacteria growth curve, we carried out an experiment to measure the growth of Salmonella. Then we fit the data obtained into the logistics. By doing these we figure out that \(r = 60min^{-1} \), \(K_{max} = 0.9997 \) and \(C = 7.2319 \). Results and diagram are shown below (Figure 5):
Figure 5. Bacteria growth curve
After complete the works above, we used MATLABTM to solve the equations above and acquired a series of diagrams which visually demonstrated the relationships, which would help the wet lab group get an abstract view of how ATc influence on the expression of GSDMD. We assumed that \(P_{tet\_copy} = 4 \), \(K_{deg\_GSDMD} = 0.8 \), \(K_{trans\_GSDMD} = 200 \), \(K_{deg\_ATc} = 0.0007 \), (All unites are combined of nM and s) Results are shown below (Figure 6,7,8):
Figure 6. Concentration of tetR(nM) - time(s)
Figure 7. Concentration of GSDMD (nM) - time(s)
Figure 8. Max concentration of GSDMD (nM) - time(s)
MATLABTM. With this app in hand, you can adjust all the parameters needed in the equations above and attain the diagrams which indicates the relation between concentration of GSDMD and time, concentration tetR and time and the summit value of GSDMD and the initial concentration of ATc. The app will also generate a function describing the relationship between the max concentration of GSDMD and the concentration of ATc. With the help of this app, members in wet lab group can conveniently decide how much ATc should be added into cultivation environment according to their requirements. (Figure 9,10)
Figure 9. APP Parameters
Figure 10. APP Diagrams
The model of ATc induced circuit is very common and well-known to biology researchers. The common-known significance to this model is that it can demonstrate the relation between concentration of target gene and concentration of inducer added, which can instruct the researchers regulate their circuit precisely. In our project, this model will tell the members in wet lab group that how much GSDMD will be expressed under a certain concentration of ATc in the Salmonella community formed in the tumor cell. Another significance for this model is that, the response time of is very short and the response speed is extremely fast. We anticipate that just minutes are needed to induce the fluorescence. This phenomenon is also verified in the experiment. In less than 10 minutes, fluorescence can be detected under fluorescence microscope. Especially, a remarkable significance to our project is that it’s a self-destructive system, which means, without any further operation, the process of induction can be self-terminated. As ATc degrades, the expression of GSDMD will significantly decreases, thus the process of pyroptosis can be inhibited. Based on these considerations, we think that the cytokine storm caused by pyroptosis is controllable.
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5. Alon, U. An Introduction to Systems Biology: Design Principles of Biological Circuits (Chapman & Hall/CRC Mathematical and Computational Biology).Pdf.
6. William and Mary iGEM 2016. A Kinetic Model of Molecular Titration. 1–11 (2016).
