Team:UCSF/Modeling Results

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.

To test the robustness of this dynamic system, we sampled the parameters using Latin-cubic sampling, and searched the suitable groups of parameters which could make the population ratio of the two strains (x=n1/n2) get to a constant number, and at the same time could also enable the final steady ratio to be neither too large nor too small, say, not larger than 10, and not smaller than 0.1.

Finally we found 261 out of 10000 groups of parameters that meet our requirement. This number gives an evaluation of the robustness.

What’s more, to give guidance on how to tune parameters to make experiments work out, we should find out which parameter is sensitive, say, which one to tune and how to tune it to make an experiment successful.

Due to the reasons above, we did sensitivity analysis using C++, the results are as following:

k1 and k2 are more sensitive than other parameters;
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.

Result of Computational Analysis: Toxin/Antitoxin System

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 (x between 0.1 and 10)

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: u1, u2, z1, z2, D1, D2, K1, K2 are relatively sensitive, while a1, a2, B1, B2 are not sensitive. What’s more, the initial values of variables also influence the steady ratio of populations. Among all the initial values, n1, n2, tox1, tox2 are relatively sensitive. In addition, initial values of n1 and n2 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.

Result of Simulation

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

k1=4, k2=3.9, K1=1, K2=1, α1=1, α2=4, β1=1, β2=1
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

k1=101,k2=100,K1=1e^(-5),K2=1e^(-5),α1=1e^(-4),α2=1e^(-4),β1=1e^(-6),β2=1e^(-6),
the behavior of this dynamic system changes. Though the ratio of the two strains and c_1 come to a steady state finally, c_2 increases continuously as time goes on. 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 ( The initial values are the same as the former situation.), we follow the values of x of each line as time goes on, and a phase plane is shown below.

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 <regulartext><br>
 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.
+u1, u2, z1, z2, D1, D2, K1, K2  are relatively sensitive, while  a1, a2, B1, B2 are not sensitive. What’s more, the initial values of variables also influence the steady ratio of populations. Among all the initial values, n1, n2, tox1, tox2 are relatively sensitive. In addition, initial values of n1 and n2 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.