Parameter Space Analysis

What is the Likelihood of our Switch Working?

Recall that the equations of our system are as follows:

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=1 and estimated n₂ to be between 1 and 5. Therefore, we ran the program for various combinations of Hill coefficients 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. This percentage indicated how probable it was that our system would perform under those conditions.

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 n₁=1, which is suggested in the literature, then the best scenario we can hope for is that n₂=5. In this situation, 3.29% of the parameter ranges tested will give bistability and the possibility of switching.