Gene regulation A gene produces protein by first creating an intermediate molecule called mRNA. High gene expression leads to a lot of mRNA molecules which, in turn, leads to a lot of protein. High levels of protein can suppress the production of mRNA, however, thereby regulating gene expression. Suppose m(t) and p(t) are the amounts of mRNA and protein in a cell (in hundreds of copies) at time t (in hours). A simple model of gene regulation is: dm/dt = 1 - p - m dp/dt = m - p Find the steady state of this system of differential equations; Use a change of variable to reduce this system to a homogeneous system of linear differential equations (Similar to Problem #1, use the steady state); Classify the steady state; Suppose that initially there is no mRNA or protein: m(0) = p(0) = 0. Solve the initial-value problem from part (b) (You should get a stable spiral for your fixed point. So you will have complex eigenvalues, eigenvectors, and coefficients. Once you have found the coefficients for the initial value problem, use Euler's Formula (e^it = cos t + i sin t) to simplify your solution); In the mp-plane, plot solution from part (d); Sketch the plots of m(t) and p(t) from part (d) as functions of time. What does this plot tell you and does it agree with the results from parts (c), (d), and the plot in part (e).