Goal

In this part we would like to answer 3 questions thanks to the stochastic modeling.

How much time do we need to wait to get a response ?

Is the sensitivity of the amplification loop given by stochastic modeling the same as in ODE modeling ?

What is the part of false positives ?
Thanks to those 3 questions we will be able to establish if our device is still performing when we take into account random variations.

Time

We would like to evaluate the time needed to get an answer. Once a bacterium has detected a Golden staph, the quorum sensing will allow the propagation of the information. Thus we will be able to get the answer really quickly and visible.
We need to establish a time not too small to avoid false negatives and not too long to avoid false positives and to be efficient. Indeed, if we take a short time we could miss some visible output signals which are longer to appear. Then we will conclude that nothing has been detected whereas there was something to detect. If we wait too long, some visible output signals can appear whereas there was nothing to detect. This phenomenon is due to the randomness inside the bacterium and espacially because of basal values. Indeed, because of the leaks at the promoter level, the production of proteins could be nonzero even when the gene is not up-regulated. We call this "basal production".

In this first part we try to determine the ideal duration of observations and in the third part we will analyse if it would still be a good scale of time with respect to false positives.

Thanks to the deterministic modeling we can have an estimation of the time needed to get an answer in different proportions. In the graph below you can observe the evolution of the output signal through time for an initial concentration of cAMP of 10^-3 mol.L-1.

After 200 minutes we get half of the maximum output signal. If we wait 200 minutes more we reach 80% of the output signal. Therefore we can assess the time needed to get a correct answer as 400 minutes for the time being. We then have to check if that time is still appropriate when we take into accout the randomness.

We would like to observe what happens after 400 minutes of waiting for several initial concentrations of cAMP. To perform that study, we use a Gillespie Algorithm to add randomness in our system. Moreover, we simulate the algorithm 100 times for each initial concentration of cAMP which interest us.

We can notice that when we wait 400 minutes we get a visible output signal as soon as we have an initial concentration of cAMP of 5,25.10^-5 mol.L-1 (10^4,5 molecules of cAMP_i). We will see in the next part that this concentration corresponds to the sensitivity of our system. Thus, our stochastic model doesn't contradict our deterministic model and we can make our further studies at a time of 400 minutes.

You can find the scripts of our simulations here.

Sensitivity

The ODE modeling gave 10^-6 mol.L-1 of CAMP_i as the sensitivity of the amplification loop. This means, if we have 10^-6 mol/L of cAMP at the initial point, the system will turn on. But this result is given by a deterministic analysis. What happens if we take into account the random phenomena of the bacterium ? Is the sensitivity still so good ?

We want to obtain the evolution of the output signal (Ca or GFP) depending on the concentration of the input signal (cAMPi) after 6h40 (400 min). To get that graph we simulate the algorithm hundreed times for each concentration of CAMPi (10, 10^1,5, 10², …, 10⁶ molecules).

We can compare the graph obtained with the one from the ODE modeling.

We can notice that, when we add stochastic variations to the model, the sensitivity of the amplification loop decreases from 10^-5,5 mol/L to 3,3.10^-5 mol.L-1. In other words, we loose a sensitivity of 2,98.10^-5 mol/L, that is 18 000 molecules of cAMP_i. However this loss of sensitivity is low and doesn't penalize our device at all.

To observe those results you can use our scripts here.

False positives

We know the time we need to wait to get a response and the sensitivity of the amplification loop. But we don't know yet if that device is reliable.
To answer this question we generate 10 000 simulations of the Gillespie Algorithm when there is no cAMP at the initial point. After 400 minutes we evaluate the number of output signals to establish the percentage of false positives. We consider as "output signal" the production of 35 molecules of Ca which represents the lower bound of molecules of Ca produced for an initial concentration of cAMP equal to the sensitivity of the amplification loop after 400 minutes according to our last modeling (part 2).

We get the results below.

The percentage of false positives is low. That means, if we wait 400 minutes after the beginning of our test, we will obtain 43 false answers out of 10 000. 43 times out of 10 000 we can see an output signal whereas there was nothing to detect.
This result is interesting regarding our device. Indeed we would like to use a 96-well plate. Thus we will be able to make 96 tests in the same time. According to our study, if no Golden staph has to be detected, no more than one sample on 96 will get a visible output signal.
However, in order to be more rigorous, we need to validate these promising results by experiments.

Finally we can study how many output signals we have when the initial concentration of cAMP is lower than the sensitivity of the amplification loop.

We can notice that if zero, one or two samples become green, we can't be sure if there was something to detect or not. But as soon as we have an initial concentration of CAMP of the order of 10³ molecules (1,66.10^-6 mol.L-1), approximately 6 samples become green. Thus, even if the initial concentration of cAMP is not higher than 10^-6 mol.L-1, we should be able to detect the Golden staph.
After the amount of 10^3,5 molecules of cAMP (5,25.10^-6 mol.L-1) at the initial pont, we are pretty sure to detect the Golden staph since almost the whole plate should become green.

You can find the scripts of our simulations here.

Conclusion

With that stochastic modeling we have confirmed and slightly modified the results from the deterministic modeling. Indeed, the 400 minutes of waiting to get a response is also true in stochastic than in deterministic modeling. The sensitivity is slightly higher than in the ODE modeling but still be correct for our device. What's more, the proportion of false positives is expected to be low. Thus, a 96-well plate will be efficient to detect the Golden staph for little concentrations and will be reliable.

References

Stochastic Simulation of Chemical Kinetics from Daniel T. Gillespie

Team:Grenoble/Modeling/Amplification/Stochastic/results

From 2012.igem.org