Derivation of single-cell mass distribution functions. Consider anorganism for which division occurs at the cell mass M, birth at the cell mass M/2,and the single-cell mass growth rate follows first-order kinetics, i.e.,rðmÞ = km.
(a) Find the normalized, steady-state cell mass distribution for this organismin a chemostat with dilution rate D.
(b) Find the relation between D and k, and use it to eliminate from the expressionforfðmÞ.
(c) Find the distribution function for zero th-order kinetics (i.e.,rðmÞ = k), and compare the distribution functions for, respectively, zero and first-order kinetics.
(d) Assuming a constant yield Y sx of mass per individual cell from the substrate, write a steady-state substrate balance and simplify this to obtain an equation among D,k, and n(the total cell number).
(e) Assume that the substrate dependence of k follows a Monod-type expression. Find the substrate and cell number concentrations as functions of the dilution rate.