Journal of Theoretical Biology 222 (2003) 485–494 A theoretical and empirical investigation of delayed growth response in the continuous culture of bacteria Sean Ellermeyer*, Jerald Hendrix, Nariman Ghoochan Department of Mathematics, Kennesaw State University, 1000 Chastain Road, Box 1204, Kennesaw, GA 30144-5591, USA Received 27 August 2002; received in revised form 2 December 2002; accepted 15 January 2003 Abstract When the growth of bacteria in a chemostat is controlled by limiting the supply of a single essential nutrient, the growth rate is affected both by the concentration of this nutrient in the culture medium and by the amount of time that it takes for the chemical and physiological processes that result in the production of new biomass. Thus, although the uptake of nutrient by cells is an essentially instantaneous process, the addition of new biomass is delayed by the amount of time that it takes to metabolize the nutrient. Mathematical models that incorporate this ‘‘delayed growth response’’ (DGR) phenomenon have been developed and analysed. However, because they are formulated in terms of parameters that are difficult to measure directly, these models are of limited value to experimentalists. In this paper, we introduce a DGR model that is formulated in terms of measurable parameters. In addition, we provide for this model a complete set of criteria for determining persistence versus extinction of the bacterial culture in the chemostat. Specifically, we show that DGR plays a role in determining persistence versus extinction only under certain ranges of chemostat operating parameters. It is also shown, however, that DGR plays a role in determining the steady-state nutrient and bacteria concentrations in all instances of persistence. The steady state and transient behavior of solutions of our model is found to be in agreement with data that we obtained in growing Escherichia coli 23716 in a chemostat with glucose as a limiting nutrient. One of the theoretical predictions of our model that does not occur in other DGR models is that under certain conditions a large delay in growth response might actually have a positive effect on the bacteria’s ability to persist. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: Chemostat; Continuous culture; Delayed growth response; Microbial growth 1. Introduction The standard mathematical models for microbial growth in batch and continuous (chemostat) culture that were developed during the 1940s and 1950s provide a framework for the study of microbial growth rate as a function of the concentration of a single nutrient (usually a carbon source) that is necessary for growth. These models were first proposed and studied by Monod (1950), Novick and Szilard (1950), Herbert et al. (1956), and others. The underlying idea is that microbial growth rate is determined by the concentration of growth– limiting nutrient in the medium and that the growth rate adjusts itself instantaneously to changes in nutrient concentration. Consequently, the standard models (system ð6; 7Þ for batch culture and system ð8; 9Þ for continuous culture) are coupled systems of two ordinary differential equations—one equation describing the change in nutrient concentration and the other equation describing the change in microorganism concentration. Because of its applicability in many areas such as, for example, wastewater treatment and the operation of industrial fermenters, the continuous culture model ð8; 9Þ has received a great deal of attention since it was first introduced and a complete mathematical theory of this model has been developed. In addition, the model has undergone numerous modifications to account for various phenomena that are relevant in specific applica- tions but that are not incorporated into the standard model. When such modifications are made, it is always a central question of interest to find criteria under which the new model predicts that the microorganism will be able to persist at a steady state in the culture vessel for an indefinitely long period of time. An accompanying *Corresponding author. Tel.: +1-770-4236129; Fax: +1-770- 4236629. E-mail address: sellerme@kennesaw.edu (S. Ellermeyer). 0022-5193/03/$-see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0022-5193(03)00063-8