1. Consider the product of a constitutively expressed gene as modelled in equation 7.2 in the notes: P (t) = co-op(t) (1) For comparison, consider an autoinhibitory gone whose expression can be modelled as in equation 7.8 in the notes: d p(t ) = 1 + P( 1 ) / K - op(t). (2) a) [1] Takes = 1 (time -!) in models (1) and (2) and leth = 1 (concentration) for the autoin- hibited gene. Verify that both genes generate the same steady- state protein concentration when a = a (a + 1). (Hint: substitute ps = 4, into the autoinhibited model.) b) [1] Simulate the two models with 4, = 5 and a = 30 (concentration . time -1). Take the initial concentrations to be zero. Verify that, as a result of having a higher maximal expres- sion rate, the autoinhibited gene reaches steady state more quickly than the unregulated gone. c) (2) How would you expect the response time to be fected by cooperative binding of multiple copies of the repressor? Verify your conjecture by comparing your results from part (b) with the model d 1 -P() = "2 14 (P( ( ) / K ) 3 - op(t). Take a, = 130 (concentration . time - 1 ). 2. [3] Consider the simple model for the Collins toggle switch du du - 1. 1 + ur Recall that the analysis in the paper addressed the case when the cooperativety is identical for both repressors (i.c/ = v). This may be an unrealistic restriction, and so it is useful to understand how robust the system behaviour is to differences in these binding mechanisms. Explore this question of robustness as follows: Take nominal parameter values of, = a, = 5, 8 = 7 = 3