Existence of a scaling relation in continuous cultures of : the steady states are completely determined by the ratio of carbon and oxygen uptake rates.

Background: Recently, we showed that steady-state continuous cultures of   follow the principles of growth on mixture of two complementary substrates. More precisely, when such cultures are fed with progressively higher concentrations of glucose at fixed dilution rate  = 0.1 h, oxygen mass-transfer coefficient  = 50 h, and oxygen solubility , they transition from glucose- to oxygen-limited growth through an intermediate dual-limited regime in which both glucose and oxygen are limiting, and ethanol is produced without loss of glucose. It is, therefore, of considerable interest to characterize the dual-limited regime. We found that the dual-limited regime occurs precisely when the operating parameters , , and satisfy the relation , where and denote g of glucose consumed per g of oxygen consumed in the carbon- and oxygen-limited regimes. In this work, our goal was to determine if the above characterization of the dual-limited regime holds over a wider range of , , and to understand why the dual-limited regime is determined by the dimensionless ratio .

Results: To this end, we performed the foregoing experiments at three additional dilution rates ( = 0.07, 0.15, and 0.20 h) and one additional mass-transfer coefficient (  = 100 h). We find that the above characterization of the dual-limited regime is valid for these conditions as well. Furthermore, the boundaries of the dual-limited regime are determined by the dimensionless ratio , because the steady-state concentrations are completely determined by this ratio. More precisely, if the steady-state concentrations of biomass, glucose, oxygen, and ethanol are suitably scaled, they collapse into a single curve with as the independent variable.

Conclusion: The dual-limited regime is characterized by the relation over the entire range of operating condition 0.07 h ≤  ≤ 0.20 h and . Since the effect of all operating parameters is embedded in the single parameter , the dimensionless plot provides a powerful tool to compare, with only a handful of data, various ethanol-producing strains over a wide range of operating conditions.