Consideration of Death rate

We can gain additional insight into inhibition from a common model for microbial growth:

where x = organism concentration
µ = specific growth rate coefficient
Kd = specific death rate coefficient
When x is set to unity, an apparent specific growth rate = µ - Kd. The death term is effectively incorporated into µ when data for mass or numbers of cells are used to calculate growth rates because there is no distinguishing of live cells from dead. With a death term added, the inhibition equation becomes (Bungay, 1992):

The logic behind this is that inhibitory substrates must affect death rates to some extent. Unfortunately, this equation allows µ to assume minus values at high values of S. This is prevented in a computer program by simply adding an IF statement to restore µ to zero if it goes negative. When Ki is zero, the denominator of the first term becomes that of the Monod equation. There is a profound effect of Ks in all cases. The equation can be modified to emphasize death by making the second term Kd S2.

Graph Comparing Andrews Equation with Equation Based on Death

There are some actual data that fit the Andrews equation well, but there are other cases where growth stops completely at high concentrations of an inhibitory substance. The equation based on death can fit more data sets but at the expense of adding another coefficient.

References

Bungay, H.R. (1992) BASIC Environmental Engineering, BiLine Associates, Troy, NY
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