S-Shaped Growth Curves In Fermentations And Golden Ratio

A model of the growth curve of microorganisms was proposed, which reveals a relationship with the number of a 'golden section', 1.618..., for main parameters of the growth curves. The treatment mainly concerns the ratio of the maximum asymptotic value of biomass in the phase of slow growth to the real value of biomass accumulation at the end of exponential growth, which is equal to the square of the 'golden section', i.e., 2.618. There are a few relevant theorems to explain these facts. New, yet simpler, methods were considered for determining the model parameters based on hyperbolic functions. A comparison was made with one of the alternative models to demonstrate the advantage of the proposed model. The proposed model should be useful to apply at various stages of fermentation in scientific and industrial units. Further, the model could give a new impetus to the development of new mathematical knowledge regarding the algebra of the 'golden section' as a whole, as well as in connection with the introduction of a new equation at decomposing of any roots with any degrees for differences between constants and/or variables.