Final answer:
To construct the differential equation model, we write rate equations for each reaction using the law of mass action and the concentrations of the reactants. Setting the rates of change of concentration to zero allows us to determine the steady-state concentrations as functions of the mass-action constants.
Step-by-step explanation:
To construct a differential equation model of the given reaction network, we need to write the rate equations for each reaction. Using the law of mass action, we can express the rate of change of concentration for each species in terms of the rate constants and the concentrations of the reacting species.
For the given network:
d[A]/dt = -k1[A]
d[B]/dt = k1[A] - k2[B] - k3[B] + k4[C]
d[C]/dt = k2[B] + k3[B] - k4[C] - 2k5[C] + k6[D]
d[D]/dt = k4[C] - k6[D]
To determine the steady-state concentrations of all species as functions of the mass-action constants, we need to set the rates of change of concentration for each species to zero. Solving the resulting system of equations will give us the steady-state concentrations.
Let's assume the steady-state concentrations of A, B, C, and D are denoted as [A]s, [B]s, [C]s, and [D]s respectively:
k1[A]s = 0
k1[A]s - k2[B]s - k3[B]s + k4[C]s = 0
k2[B]s + k3[B]s - k4[C]s - 2k5[C]s + k6[D]s = 0
k4[C]s - k6[D]s = 0
Solving this system of equations will give us the steady-state concentrations as functions of the mass-action constants.