Overview

An alternative formulation of the population balance distribution in exponential growth was originally derived for the length distribution, f( l) [22, 37]. It is usually called the Collins-Richmond equation,

where r( l) is the rate of change of cell length l, ?( l) the probability that a newborn cell has a length between l and l + dl, and ?( l) the probability that a dividing cell has a length between l and l + dl. In this form, the equation assumes no cell death, but it can be extended to do so [100]. This equation appears to be substantially different from our previous formulation of a PBE, but can in fact be derived from this [75]. The two formulations use different probability functions, and one must derive equations between these different functions in order to transform one formulation to the other. Doing so is more than an idle exercise in variable transformation and is important because it allows one to find equations between probability functions that can be measured experimentally (at least in theory) and nonmeasurable functions that appear in a given PBM formulation. We will therefore briefly look at the derivation of a few equations between various probability functions. Experimentally obtained distributions, together with a discussion of some of the problems and issues concerning these measurements, are given elsewhere [70, 71].

It is important to understand clearly what the function ?(