(245c) Convenient Prediction of Steady State Multiplicity Patterns - Multiple Autocatalytic Reactions

Chemical and biological reactions with nonlinear kinetics when carried out in well-mixed reactors give rise to the possibility of reactor operation at multiple operating points at steady state. When multiple steady states are physically realizable, identification of various multiplicity regions in the space of reactor operating conditions is important for design, optimization and controlled operation of chemical and biological reactions. Prediction of steady state multiplicity patterns in terms of profiles of concentrations of species influencing reaction kinetics is important for better understanding of the reaction behavior and design, optimization and control of reactors used to conduct these reactions. These patterns are simpler with respect to certain operating variables, such as the feed concentration of a reactant, and complex with respect to other operating variables, such as the reactor space time. With reactor space time as the variable parameter, exotic patterns, such as isolas, mushrooms, and multi-stability, have been reported for isothermal and non-isothermal reactor operation. In prior studies, these have been obtained via extensive iterations involving sweeps through parameter spaces. In the present work, a convenient procedure for analyzing and predicting steady state multiplicity patterns is developed and illustrated considering one or more autocatalytic reactions as specific examples. Autocatalytic reactions and processes are commonly encountered in growth of all living cells, processes involving free radicals, polymerization processes, many inorganic and organic reactions, and crystallization processes. An analysis of steady state multiplicity of two (parallel, series and mutating) autocatalytic reactions occurring in a well-mixed reactor is presented. The generation of an autocatalyst (single for parallel/series autocatalysis and two for mutating autocatalysis) from competing or consecutive resources by cubic autocatalysis is followed by its decay. Parallel and series autocatalysis is observed in growth of living cells on multiple substitutable resources (nutrients) and co-metabolism of primary and secondary nutrients (simultaneous utilization of multiple nutrients via independent metabolic pathways inside living cells). A secondary autocatalyst may be generated from a primary autocatalyst by mutation, with both autocatalysts competing for a common resource. Examples of primary autocatalyst are (i) plasmid-bearing (recombinant) cells subject to instability and (ii) healthy cells in the case of variety of cancers. A single well-mixed reactor may operate at up to five steady states for two parallel/series autocatalytic reactions and seven steady states for mutating autocatalysis. For each example, steady state multiplicity patterns are predicted in a non-iterative fashion by a judicious choice of parameter combinations. The space of ratios of the kinetic parameters for the autocatalytic reactions and ratios of supply of resources is divided into different regions depending on the maximum number of steady states admissible. The space of the remaining kinetic and operating parameters is divided into multiple regions based on the number and identity of physically realizable steady states. This division allows exact determination of appearance and disappearance of particular steady states. Continuation curves for limit point (LP) bifurcation are identified in the multi-dimensional kinetic and operating parameter space. There is a wide variety of dispositions of these for each example of multiple autocatalytic reactions. Concentration profiles of species participating in the reactions are predicted conveniently using the LP continuation diagrams. These are simple when feed composition is varied and the same is the case when reactor space time is varied in the absence of autocatalyst decay. With autocatalyst subject to decay, exotic patterns such as single and multiple isolas and mushrooms of different varieties, are possible with reactor space time as the variable parameter. The LP continuation diagrams enable convenient and precise prediction of emergence and extinction of isolas and mushrooms. A reversible sweep in a parameter combination as reactor space time is varied leads to these exotic patterns. Two or more consecutive reactions are therefore necessary for admission of isolas and mushrooms. The reaction systems exhibit rich varieties of steady state patterns, specific illustrations being provided for some of these. The present approach enables a much easier identification of the multitude of rich steady state multiplicity patterns.