Froth! by Mark Denny

Author:Mark Denny
Language: eng
Format: epub, mobi
Publisher: Johns Hopkins University Press
Published: 2009-04-05T16:00:00+00:00

which is the exponential behavior described in the main text. Now substitute equation (3.7) into (3.2) to obtain a differential equation for p, and integrate with the initial condition p(0) = p0 to obtain equation (3.4). Incidentally, this equation looks very much like the extreme value distribution known to statisticians.

We can fix the two independent model parameters α and β (or, equivalently, r and β) as follows. We see from (3.4) that the yeast cells initially grow exponentially in numbers: this is Malthusian growth. Observation of real brewer’s yeast under such conditions shows that they double in number every 2 hours, which means that the fecundity parameter must be given by β = 1/2ln(2), i.e., β = 0.35 hr–1. The second model parameter, r, is obtained by noting that the peak yeast cell density is observed to be about 108 mL–1. So the ratio of pitching rate to peak density is about 0.004 for the pitching rate of figure 3.4a (and is 0.1 for the pitching rate of figure 3.4b). It is easy to show from equation (3.4) that this ratio is given algebraically by er, where e=2.7182818285 is the base of natural logarithms, and so r = 0.00147 (r = 0.037). Hence the two parameters of the model are fixed.

It is worth noting here that the time at which cell population peaks can be calculated from equation (3.4). It is tpeak = ln(r)/β. For times exceeding tpeak, or perhaps a little earlier, real yeast cells switch to anaerobic respiration and so the value of β will change. A more detailed model of yeast cell population dynamics would include this effect.

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