# Team:UCSF/Modeling Results

### From 2012.igem.org

Though we don’t know the exact parameters in the experiments, and sometimes even cannot tune every parameter in reality, the more robust a dynamic system is, the easier the experiments could succeed.

k1 and k2 are more sensitive when they are enlarged than they are reduced.

The result of sensitivity analysis indicates that k1 or k2 should be tuned if we fail to get a steady population ratio of the two strains. But this result leaves us a dilemma in tuning the parameters in reality in experiments, because k_1 and k_2, the maximum growth rates of Escherichia coli (gDM/(Ls)), are the natural characteristics of the cells, which makes this situation intractable. We could tune α and β to get the final steady population ratio we want, but if the system couldn’t even go to a steady ratio, that doesn’t make sense in reality. That’s why a new approach should be proposed.

The same as the former computational analysis, we did Latin-cubic sampling to seek the groups of parameters which make the population ratio come to a steady state after a time period, and which enable the steady ratio to be neither too large nor too small (0.1

In Latin-cubic sampling, 1268 out of 5000 groups of parameters are selected, which indicates that this system is more robust than the original auxotroph system.

The following sensitivity analysis shows that:
u_1,u_2,z_1,z_2,D_1,D_2,K_1,K_2 are relatively sensitive, while a_1,a_2,B_1,B_2 are not sensitive. What’s more, the initial values of variables also influence the steady ratio of populations. Among all the initial values, n_1,n_2,tox_1,tox_2 are relatively sensitive. In addition, initial values of n_1 and n_2 are perhaps especially sensitive, because they can change the steady ratio of population by over ten folds, which is tested in the toxin/antitoxin simulation below.

To test our analytical result and the robustness of this system, we simulated the system in Matlab with different groups of parameters:

Setting the initial values to [0.1, 0.2, 0, 0], [0.1, 0.5, 0, 0], [0.1, 1.0, 0, 0], [0.3, 0.2, 0, 0], [0.3, 0.5, 0, 0], [0.3, 1.0, 0, 0], [0.7, 0.2, 0, 0], [0.7, 0.5, 0, 0], [0.7, 1.0, 0, 0] respectively, we follow the values of x of each line as time goes on, and a phase plane is shown below

However, when the population ratio of the two strains gets to a steady state, y (c_1/c_2 ) doesn’t necessarily be constant. As an example, with another group of parameters