Sum of squares of residuals = 0.096
Fig. 14.6. How regression analysis is used to determine the most appropriate kinetic equation and the values of the parameter of this equation that give the best fit to the experimental data. In this case the logistic equation would be selected since it gives a better fit, as indicated by the smaller sum of squares of residuals. The residuals are the vertical lines that represent, for a particular time, the difference between the experimental value and the value predicted by the equation
Take care to select the appropriate interval for the regression analysis. The kinetic equations in Table 14.1 may apply to only part of the overall kinetic profile. There might be lag and stationary phases not described by these equations, in which case it is necessary to select carefully the region of the growth profile to which the kinetic equation will be fitted. For example:
• None of these equations explicitly describe a lag phase. However, the exponential and logistic equations may give apparent lag phases on a linear-linear plot if the initial biomass concentration is very low.
• The exponential and linear equations do not describe any limitation on growth. Of course if the growth curve is followed for long enough, the biomass profile must eventually show a maximum concentration (Cm). For these equations it may be appropriate to define a separate stationary phase. The logistic and deceleration equations can describe a stationary phase, which occurs at Cm for the logistic equation and at Co.eA for the deceleration equation. These equations make no assumptions about the mechanism of limitation. In different systems limitations on the maximum amount of growth might be related to the exhaustion of essential nutrients, to the accumulation of inhibitory end products of metabolism, or to steric considerations (i.e., through the biomass "filling" the physical space available, noting that, even at their maximum packing density, fungal hyphae occupy only about 34% of the available volume (Auria et al. 1995)). Therefore the significance of Cm may vary from system to system. Typically it will be treated as a simple empirical parameter.
• There may even be a decline or death phase, which is not described by any of these equations. The modeling of death kinetics is discussed in Chap. 16.4.
Keep the environmental conditions constant. The parameters of the equation will change for cultures grown in different conditions, for example, at different temperatures, on different substrates, or with different O2 concentrations in the gas phase. Therefore, during the fermentation the conditions should be held as constant as possible. This may not be simple, even at small scale. Difficulties in maintaining constant conditions and experimental strategies to minimize deviations are discussed in Chap. 15. Note that in more sophisticated studies, in which the effects of varying conditions on growth are investigated, it may actually be desirable to vary the conditions in a deliberate manner during the fermentation.
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