Parameter Space Analysis

Recall that the equations of our system are as follows (see Equations page for parameter definitions):

We wrote a program which takes the equations for our system, calculates the nullclines and records whether or not bistability is achieved. Whether or not bistability is achieved depends strongly on the parameters λ_i, K_i, μ_i and n_i.

The values for the parameters λ_i, K_i and μ_i were taken from literature^[1]-[13] but have a large uncertainty attached to them. To take this uncertainty into account we selected each parameter value randomly from a range of two orders of magnitude around the literature value for each parameter. We knew that n₁ was between 1 and 3 and n₂=1. This is due to the number of stem loops on the two DNA strands (see Fig 1 in Equations). The strand associated with GFP has two stem loops. A protein can bind to one stem loop and this can then hinder the binding of a protein to the second stem loop (thus making n₁ less than 2 but more than 1) or encourage the binding of a protein to the second stem loop (thus making n₁ more than 2 but less than 3). The strand associated with the CFP has only one stem loop and thus n₂=1. Please see Curve Fitting for an explanation on how we narrowed down the value for n₁.

We ran the program for various combinations of Hill coefficients (n_i) between 1 and 5. Each time the program was run, the Hill coefficients remained constant but the other parameters varied.

Choosing the parameters in this way meant that each time the program was run we would sometimes achieve a combination of parameters that allowed bistability, and sometimes not.

The program was run 100 times and the number of times bistability was achieved was output to the screen as a percentage.

We then decided to run the program for various different parameter ranges. We cannot change individual parameters specifically, but in the lab it is possible to vary the order of magnitude of a parameter. We wanted to know, if we varied our parameters and Hill coefficients, what is the best percentage we can possibly get? These results are shown in scenarios 2-4.

Scenario 1 – unmodified parameters

These parameters were taken from, or derived from, the information contained within various sources of literature^[1]-[13]. The parameters were converted from concentrations to numbers of molecules.

We did not know what concentration of galactose and methionine was optimal, so we ran various tests and found the value for which the percentages were maximised. We found that the optimal percentages occurred if we have 100000 times more galactose than methionine. See table 1.

Table 1. The percentage of times our system will work for various combinations of Hill coefficients for the original parameters. n1 is the Hill coefficient for the GFP rate-reaction equation and n2 is the Hill coefficient for the CFP rate-reaction equation.

Here we can see that if n₂=1, then the best scenario we can hope for is that n₁=5. In this situation, 3.29% of the parameter combinations tested will give bistability and the possibility of switching. However, with n₁ estimated to be between 1 and 3, we can say that at most 2.03% of the parameter combinations tested gave bistability.

Scenario 2 – modified parameters 1

Percentages are maximised if we have 100000 times more galactose than methionine. See table 2.

Table 2. The percentage of times our system will work for various combinations of Hill coefficients for the modified parameters.

The best scenario is if we have n₁=5 and n₂=5.

Scenario 3 – modified parameters 2

Percentages are maximised if there is 100000 times more galactose than methionine. See table 3.

Table 3. The percentage of times our system will work for various combinations of Hill coefficients for the modified parameters.

Here we find that the optimal scenario is n₁=3 and n₂=5. The trend which has developed in this table will be discussed at the end of scenario 4.

Scenario 4 – modified parameters 3

Percentages are maximised if we have 10000000 times more galactose than methionine. See table 4.

Table 4. The percentage of times our system will work for various combinations of Hill coefficients for the modified parameters.

An interesting trend is seen in scenarios 3 and 4. The percentages increase to a maximum at n₁=2 and n₂=5 (n₁=3 and n₂=5 in scenario 3) and then decrease, quite rapidly, towards Hill coefficients of n₁=5 and n₂=5. In the parameter space, there is a region where bistability is possible. For the smaller Hill coefficients, the randomly chosen parameters (as described above) would allow the nullclines to have a shape such that they would cross three or more times and bistability is achieved. The higher Hill coefficients, however, for the same range of parameters, would have nullclines which were shaped so that they did not cross three or more times – in parameter space, they fall outside the region of bistability.

Conclusions

The ideal scenario is to change the parameters to that in scenario 4 and have Hill coefficients of n₁=2 and n₂=5. Here 98.17% of the parameter combinations tested gave bistability. Contrary to our initial assumptions, depending on the range of parameters, higher Hill coefficients are not always better. The parameters will dictate the shape and position of the nullclines and whether or not bistability is possible.

It is possible to determine the optimal values of the parameters for an ideal working of the switch. However, to change the parameters in the experimental construct so much would require much more time than we had available.

Return to Stability

Continue to Determination of the the Hill Coefficient n1

References

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