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.
$$N_{normal} + S_{almonella} \overset{Aw}{\rightarrow} N_{w} + S_{al\_normal}$$
$$N_{tumor} + S_{almonella} \overset{As}{\rightarrow} N_{s} + S_{al\_tumor}$$
$$S_{almonella} = S_{almonella}(t0) - N_{normal\_cell} - N_{tumor}$$
$$\dfrac {dN_{w}} {d_{t}} = A_{w} S_{almonella} N_{w}$$
$$\dfrac {dN_{s}} {d_{t}} = A_{s} S_{almonella} N_{s}$$
$$\dfrac {dS_{almonella}} {d_{t}} = - \dfrac {dN_{w}} {d_{t}} - \dfrac {dN_{s}} {d_{t}}$$
\(N_{normal\_cell}\): 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.
\(A_s\): The affinity between Salmonella and tumor cells.
\(S_{al\_normal}\): The density of infected normal cells.
\(S_{al\_tumor}\): The density of infected tumor cells.
Salmonella begins to replicate two hours after infection1 .
$$\dfrac {dN_{sal}} {d_{t}}(t) =$$
$$\dfrac {dS_{al}} {d_{t}}(t) + S_{al}(t-2) 2^{\dfrac {t-2} {T}} \ln{2}\dfrac {1} {T} $$
\(T\): Cell cycle.
According to our experimental results, we noticed that Salmonella might follow Poisson distribution in cells, so we use Matlab 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 are 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. a. Based on statistics of experimental results, we proved that the Salmonella follows Poisson distribution in normal cells. b. 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. c. Cells which are 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 cells carrying the GSDMD gene are induced by ATc, and we hope that the observed result is that the proportion of ATc-induced cell death is more than that of 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.
Figure 2. Results caused by efficiency differences of infection.
If the proportions of infection are different, the experimental results may not be able to prove that pyroptosis is induced by atc promoter. Figure 2 showed that the ATc promoter is induced and 90% cells death are caused and 70% cells die because of the promoter disclosure, but the difference of the proportions of infected cells is so big that the experimental results cannot reflect the real situation. We can reduce the difference by letting the infection proportions of the two kinds of cells both close to 100%.
We solved 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 obtained from measurement experiments can be input to predict the optimal infection time. Therefore, this App can provide guidance to the design of our experiments.
Figure 3. Salmonella infection prediction tool (for tumor cell).
A1: The predicted Salmonella numbers in tumor cell in the single cell infection experiment.A2: The predicted proportions of infected tumor cells.
A3: The concentration of added Salmonella.
A4: The density of tumor cells;
A5: Rate constant of Salmonella infecting tumor cells.
A6: The proportion of tumor cells expected to be infected.
A7: The time to reach the wanted proportion of infected cells.
Figure 4. Salmonella infection prediction tool (for mixed culture of tumor cell and normal cell).
B1: The predicted numbers of Salmonella in tumor cell (red) and normal cell (blue) in a single cell infection experiments.B2: The predicted proportions of infected cells (red for tumor cell and blue for normal cell).
B3: The concentration of added Salmonella.
B4: The density of tumor cells.
B5: Rate constant of Salmonella infecting tumor cells.
B6: The density of normal cells.
B7: Rate constant of Salmonella infecting normal cells.
B8: The predicted optimal infection time.
The parameters Nsal, Tumor and As are measured from our experiments.
Figure 5 (a part of Figure 3). Guidance for tumor cells infection experiments.
According to our experiment protocol, the MOI (multiplicity of infection) is 100, corresponding to the concentration of cells. If we want 98% of the tumor cells to be infected, the prediction result show that the infection time should be at least 2 hours to eliminate unnecessary variables.
Figure 6 (a part of Figure 4). The predicted proportions of infected cells (red for tumor
cell and blue for normal cell).
In order to reflect the affinity of Salmonella to tumor cells and to normal cells, we
hope that the difference between experimental results of tumor cells and normal cells is obvious.
However, the numbers of bacteria in different host cells are difficult to count, and we can only count the
number of infected cells and calculate their the proportions. Therefore, we need to
predict the time when the difference is most obvious. Our App just can do this for us.
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 similar molecules like ATc (anhydrotetracycline) diffuse into bacteria, they will bind to tetR protein and unleash the tetR protein from DNA, and thus relieve the inhibition 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: first, how fast does the circuit react to ATc; second, how many 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 between the promoter inhibition and the tetR protein expression.
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 facts, we give the following hypotheses:
1. Regard two tetO operons as one equivalently.
2. Ignore the diffusion of ATc.
3. The reaction time between ATc and tetR, tetR and DNA is much shorter than transcription and
translation.
$$tetR + [tetR - ATc_2] = tetR_{total}$$
$$tetR + 2 \times ATc = [tetR - ATc_2]$$
$$P_{tet} + [tetR_2 - P_{tet}] = [P_{tet}]_{total}$$
$$2 \times tetR + P_{tet} = [tetR_2 - P_{tet}]$$
$$$$
Based on Hill function, we can determine the amount of activated tetR, tetRact:
$$ tetR_{act} + n \times S_x(t) \rightarrow [tetR - S_x(t)_n] $$
$$ K_X = \dfrac {tetR_{act} \times S_x^n (t)} {[tetR - S_x(t)_n]} $$
$$ tetR = tetR_{act} + [tetR - S_x(t)_n] $$
$$ tetR_{act} = \dfrac {tetR} {1 + \dfrac {S_x^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} - (tetR_{act})_n] $$
$$ P_{tet\_copy} = P_{tet\_act} + [P_{tet} - (tetR_{act})_n] $$
$$ K_d = \dfrac {P_{tet\_copy} \times tetR^n_{act} } {[P_{tet} - (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}}} $$
Kinetic equations of transcription and translation:
$$ \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 $$
Growth curve of bacteria based on logistics model from P. F. Verhulst:
$$ N(t) = \dfrac {K_{max}} {1 + C \cdot e^{-rt}} $$
Total GSDMD expressed in bacteria community:
$$ GSDMD_{total-amount} = N(t) \cdot GSDMD\cdot{CFU}\cdot{diluted-ratio}\cdot{V_{Bacteria volume}}$$
\(S_x(t)\): 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 (unrepressed) 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 units 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 a useful instruction to wet lab.
To gain the parameters in bacterial growth curve, we carried out an experiment to measure the growth of Salmonella. Then we fitted the obtained data into a logistics model. By doing these we figured out that \(r = 60min^{-1} \), \(K_{max} = 0.9997 \) and \(C = 7.2319 \). Results and diagram are shown below (Figure 7):
Figure 7. Bacterial growth curve.
After complete the work above, we used MATLABTM to solve the equations above and acquired a series of diagrams which visually demonstrated the relationships and helped the wet lab group get an overall view of how ATc influences 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). In this action we didn’t take the growth of bacteria into account. From the diagrams, we can figure out that as ATc added increase, the concentration of GSDMD will also rise. Results are shown below (Figures 8, 9, 10.):
Figure 8. Concentration of tetR (nM) - time (s).
Figure 9. Concentration of GSDMD (nM) - time (s).
Figure 10. Max concentration of GSDMD (nM) - ATc Concentration (nM).
With this software, one can adjust all the parameters needed in the equations above and attain the diagrams which indicates the relations between concentration of GSDMD and time, concentration tetR and time and the maximum value of GSDMD and the initial concentration of ATc. The program will also generate a function describing the relationship between the maximum concentration of GSDMD and the concentration of ATc. With the help of this program, members in wet lab group can conveniently decide how much ATc should be added into cultivation environment according to their requirements (Figures 11, 12). Be advised that users must multiply the CFU number, bacteria cell volume and diluted ratio to the data obtained from this app to gain a final result.
Figure 11. Software parameters.
Figure 12. Software 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 relationship between concentration of target gene and concentration of inducer added, which can instruct the researchers modulate 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 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 our 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 the features, we think that the cytokine storm caused by pyroptosis is controllable.
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