In this exercise, treat the components as simple (reactive) chemical species and perform the appropriate (dynamic) material balance. Assume that a messenger RNA mRNA is produced by a constant (basal) expression rate from a particular gene. In addition, assume that the mRNA degrades according to a frst-order decay rate. 

(a) Write the equation for the dynamics of the mRNA concentration as a function of the expression rate G₀ and the decay rate constant k_d ^mRNA.

(b) Assume that each mRNA molecule is translated to form p copies of a protein product, P. Furthermore, the protein is subject to first-order degradation, with a decay rate constant k_P ^mRNA. Write the equation for the dynamics of the protein concentration.

(c) Assume that the system has been operating for some time at a constant gene expression rate G₀, and then the expression rate changes instantaneously to a value G₁. Derive an analytical expression for the transient responses for mRNAand P.