An Application Of Cybernetic Principles To The Modeling And Optimization Of Bioreactors
Abstract
The word cybernetics has its roots in the Greek word \kybernetes" or \steersman" and was coined by Norbert Wiener in 1948 to describe \the science of control and communication, in the animal and the machine". The discipline focuses on the way various complex systems (animals/machines) steer towards/maintain their goals utilizing information, models and control actions in the face of various disturbances. For a given animal/machine, cybernetics considers all the possible behaviors that the animal/machine can exhibit and then enquires about the constraints that result in a particular behavior. The thesis focuses on the application of principles of cybernetics to the modeling and optimization of bioreactors and lies at the interface of systems engineering and biology. Specifically, it lies at the interface of control theory and the growth behavior exhibited by microorganisms. The hypothesis of the present work is that the principles and tools of control theory can give novel insights into the growth behavior of microorganisms and that the growth behavior exhibited by microorganisms can in turn provide insights for the development of principles and tools of control theory.
Mathematical models for the growth of microorganisms such as stoichiometric, optimal and cybernetic assume that microorganisms have evolved to become optimal with respect to certain cellular goals or objectives. Typical cellular goals used in the literature are the maximization of instantaneous/short term objectives such biomass yield, instantaneous growth rate, instantaneous ATP production rate etc. Since microorganisms live in a dynamic world, it is expected that the microorganisms have evolved towards maximizing long term goals. In the literature, it is often assumed that the maximization of a short term cellular goal results in the maximization of the long term cellular goal. However, in the systems engineering literature, it has long been recognized that the maximization of a short term goal does not necessarily result in the maximization of the long term goal. For example, maximization of product production in a fedbatch bioreactor involves two separate phases: a first phase in which the growth of microorganisms is maximized and a second phase in which the production of product is maximized. An analogous situation arises when the bacterium E. coli passes through the digestive tract of mammals wherein it first encounters the sugar lactose in the proximal portions and the sugar maltose in the distal portions. Mitchell et al. (2009) have experimentally shown that when E. coli encounters the sugar lactose, it expresses the genes of maltose operons anticipatorily which reduces its growth rate on lactose. This regulatory strategy of E. coli has been termed asymmetric anticipatory regulation (AAR) and is shown to be beneficial for long term cellular fitness by Mitchell et al. (2009). The cybernetic modeling framework for the growth of microorganisms, developed by Ramakrishna and coworkers, is extended in the present thesis for modeling the AAR strategy of E. coli. The developed model accurately captures the experimental observations of the AAR phenomenon, reveals the inherent advantages of the cybernetic modeling framework over other frameworks in explaining the AAR phenomenon, while at the same time suggesting a scope for the generalization of the cybernetic framework.
As cybernetics is interested in all the possible behaviors that a machine (which is, in the present case, microorganism) can exhibit, a rigorous analysis of the optimal dynamic growth behavior of microorganisms under various constraints is carried out next using the methods of optimal control theory. An optimal control problem is formulated using a generalized version of the unstructured Monod model with the objective of maximization of cellular concentration at a fixed final time. Optimal control analysis of the above problem reveals that the long term objective of maximization of cellular concentration at a final time is equivalent to maximization of instantaneous growth rate for the growth of microorganisms under various constraints in a two substrate batch environment. In addition, reformulation of the above optimal control problem together with its necessary conditions of optimality reveals the existence of generalized governing dynamic equations of the structured cybernetic modeling framework.
The dynamic behavior of the generalized equations of the cybernetic modeling framework is analyzed further to gain insights into the growth of microorganisms. For growth of microorganisms on a single growth limiting carbon substrate, the analysis reveals that the cybernetic model exhibits linear growth behavior, similar to that of the unstructured Contois model at high cellular concentrations, under appropriate constraints. During the growth of microorganisms on multiple substitutable substrates, the analysis reveals the existence of simple correlations that quantitatively predict the mixed substrate maximum specific growth rate from single substrate maximum specific growth rates during simultaneous consumption of the substrates in several cases. Further analysis of the cybernetic model of the growth of S. cerevisiae on the mixture of glucose and galactose reveals that S. cerevisiae exhibits suboptimal dynamic growth with a long diauxic lag phase and suggests the possibility for S. cerevisiae to grow optimally with a significantly reduced diauxic lag period.
Since cybernetics is interested in understanding the constraints under which a particular machine (microorganism) exhibits a particular behavior, a methodology is then developed for inferring the internal constraints experienced by the microorganisms from experimental data. The methodology is used for inferring the internal constraints experienced by E. coli during its growth on the mixture of glycerol and lactose.
An interesting question in the study of the growth behavior of microorganisms concerns the objective that the microorganisms optimize. Several studies aim to determine these cellular objectives experimentally. A similar question that is relevant to the optimization of fedbatch bioreactors is \what are the objectives that are to be optimized by the feed flow rate in various time intervals for the optimization of a final objective?" It was mentioned previously that the maximization of product production in a fedbatch bioreactor involves maximization of growth of microorganisms first and the maximization of product production later. However, such guidelines can only be stated for relatively simple bioreactor optimization problems and no such guidelines exist for sufficiently complex problems. For complex problems, the answer to the above question requires the formulation and solution of a genetic programming problem which can be quite challenging. An alternative numerical solution methodology is developed in the present thesis to address the above question. The solution methodology involves the specification of bioreactor objectives in terms of the bioreactor trajectory in the state space of substrate concentrationvolume. The equivalent control law of the sliding mode control technique is used for finding the inlet feed ow rate that tracks the bioreactor trajectory accurately. The search for the best bioreactor trajectory is carried out using the stochastic search technique genetic algorithm. The effectiveness of the developed solution methodology in determining the optimal bioreactor trajectory is demonstrated using three challenging bioreactor optimization problems.
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