Steady-state multiplicity in bioreactors: bifurcation analysis of cybernetic models

Biological systems have an additional level of complexity compared to other chemical systems because of the effects of metabolic regulation, a defining feature of biosystems. Metabolic regulation in the form of control of enzyme synthesis and activity leads to non-linear behavior in bioreactors. Mathematical models that take into account these control mechanisms can be very successful in capturing the peculiarities of bioreactors such as multiple steady states and periodic phenomena. Cybernetic models model the expression and activation of enzymes by the use of cybernetic control variables and have been used to explain multiplicities in hybridoma reactors. In particular Namjoshi et al. (Biotechnol. BioEng. (2002), accepted) have been able to predict the transition from batch and fed batch to continuous culture in hybridoma experiments (Biotechnol. BioEng. 67(1) (2000) 25). The resulting multiple steady states vary widely in cell mass and waste metabolites. The model captures this multiplicity and its bifurcation analysis has revealed additional steady-state branches, three unstable and one stable. The stability of the additional steady state (steady state 4) was confirmed by dynamic simulations using fed-batch strategy prior to initiation of continuous operation. Steady state 4, in view of its close proximity to steady state 3, appears to have little practical significance. The likelihood of additional steady states that may be significantly different can, however not be ignored. Thus, it seems possible to envisage other states of metabolic activity, displaying alternative flux distributions that could lead to steady states notably different from those already determined. Calculation of the most singular point of the system herein is rendered difficult by both its size and possession of non-differentiable variables. The bifurcation analysis reveals the steady-state behavior under a range of operating conditions and can be used to plan optimum bioreactor operation.