Multiplicity and stability of steady states in continuous bioreactors: dissection of cybernetic models

Microbes in industrial bioreactors generally encounter nutrient media that contain a spectrum of components satisfying various nutritional requirements with individual substrates playing roles that may be viewed as substitutable, complementary or combinations of the two to varying extents. Consequently, the pattern of uptake of di1erent nutrients (Pavlou and Fredrickson, Biotechnol. Bioengng 34(7) (1989) 971) by the organism is linked through metabolic regulation of its biochemical pathway to its environment as determined by the composition of the surrounding nutrient medium. Such a situation is indicative of an abundance of steady state multiplicity in continuous bioreactors where di1erent steady states may feature cells with notably di1erent physiological states and consequently vastly di1erent metabolic activities. A striking example of such steady state multiplicity is that observed by Hu et al. (J. Microbiol. Biotechnol. 8(1) 8), and Follstad et al. (Biotechnol. Bioengng 63(6) (1999) 675) with hybridoma cell cultures for the production of antibodies. A mathematical investigation of the foregoing steady state multiplicity must obviously involve models that incorporate metabolic regulation. In this regard, the cybernetic models developed by Ramkrishna and coworkers (Kompala et al., Biotechnol. Bioengng 28 (1986) 1044; Baloo and Ramkrishna, Biotechnol. Bioengng 38(11) (1991a,b) 1337, 1353; Straight and Ramkrishna, Biotechnol. Prog. 10(6) (1994) 574; Ramakrishna et al., Biotechnol. Bioengng 52(1) (1996) 141) represent an ideal framework of which to initiate such an investigation. These models account for regulation on suitably simpli=ed pathways by viewing it as the competition for cellular resources between di1erent enzyme systems. Such competition may produce outright victors denoted by zero enzyme levels of those vanquished or simultaneous expression and activation of all the competing enzyme systems. The cybernetic models feature the foregoing competition with cybernetic variables ui for the regulation of synthesis of the ith enzyme, and vi for its activation. The latter cybernetic variables are not di1erentiable at points where reversal of competition occurs so that the application of nonlinear bifurcation analysis based on calculation of the Jacobian must take due account of this technical di?culty. The strategy proposed here for the bifurcation analysis of cybernetic models is based on a combinatoric approach enumerating all possible consequences of competition between di1erent enzyme systems. Since the system for each combinatoric is di1erentiable, a bifurcation analysis is therefore enabled. Furthermore, a notable feature is that the procedure analytically reduces the system to one of a single dimension by purely analytical means. The bifurcation analysis is performed on the model of Kompala et al. (Kompala et al., 1986), as well as that of Ramakrishna et al. (Ramakrishna et al., 1996). The latter is a re=nement of the former, which describes sequential as well as simultaneous utilization of multiple substrates. The results of our analysis on both models demonstrate the existence of multiple, stable, steady states for certain ranges of our bifurcation parameters, D the dilution rate and the fraction of the more preferred substrate in the feed. These multiple steady states refer to di1erent levels of utilization of the more preferred substrate. This multiplicity occurs for the model of Kompala et al. (Kompala et al., 1986) at D near the maximum dilution rate and suitably small. However, multiplicity exists for the model of Ramakrishna et al. (Ramakrishna et al., 1996) even when the preferred substrate is in excess. The reasons are discussed. The methodology has potential for considerably more detailed cybernetic models and hence should be of practical signi=cance to biotechnology. ? 2001 Elsevier Science Ltd. All rights reserved.