The benefit in growth rate is proportional to the rate at which LacZ ( ) breaks down its substrate, lactose, which is approximately proportional to the number of copies of the protein

where denotes a constant factor. Note that the expression of LacZ ( ) is only beneficial in the absence of glucose, i.e. when cAMP ( ) is produced. On the other hand the expression of LacZ consumes energy and reduces the growth rate approximately by a constant factor

Therefore only for longer intervals of expression enough proteins accumulate to compensate for the cost of protein expression. The fitness function to be maximized is given by

The dynamics of the proteins and is modeled by

with constant maximum production rate and degradation rate . This results in a maximum expression rate and decay time constant of . The production rates are given by

(1) | |||

(2) |

where denotes the Heaviside step function and matrix denotes activation coefficients. is in this example is activated instantaneously by input , i.e. . This model is already implemented in MATLAB and only the activation coefficients have to be modified for the following analysis.

For this task the input has the shape of a pulse with amplitude 1 and a duration of either 20 ms or 200 ms occurring with probability and ( ), respectively. The activation coefficients should be optimized with genetic algorithms to maximize the growth rate for different settings of and .

- a)
- Download the example code and the Genetic algorithm toolbox.
^{1} - b)
- Complete the code for the evaluation of the fitness value in the file
`calc_fitness.m`. - c)
- Optimize
consisting of only positive elements (
) for
,
,
and
using a population size of 5000 and 500 generations. Describe and explain the mechanisms of the solution. Average results over several runs of the genetic algorithm to average out outliers. Hand in a plot illustrating the dynamic of the gene regulatory network (GRN).
- d)
- Repeat the optimization with parameters
,
. Describe different solutions (mechanisms of the GRN) found by the genetic algorithm. Hand in a plot of the gene regulatory network dynamics for each solution.
- e)
- Hand in a two dimensional plot with axes
and
which shows for the results obtain in d) which solution is selected in different parameter ranges.
- f)
- [2* P] Repeat point d) with
consisting of positive and negative elements (
). Describe one of the new solutions that didn't emerge for
for each of the parameter ranges found in e).