(675c) Parameter Identification for Cybernetic Models of Bioprocesses

Within the chemical industry, there is a growing trend towards the use of biochemical routes for the production of chemical products. Consequently, there has been a tremendous increase in interest among chemical engineers in the development of detailed mathematical models of biological processes as they can aid in improving process design, operation and control [1]. Developments in the biological sciences are continually increasing our knowledge of the underlying biology, thus enabling the formulation of detailed mechanistic models of these processes, often containing a large number of parameters. The accurate estimation of these parameters poses a challenging task as (a) measurements of key intracellular species may be unavailable and (b) exhaustive experimentation may be expensive. In this paper, systems engineering tools are used to demonstrate how these challenges can be overcome in the parameter estimation of a cybernetic model of the manufacture of poly-â-hydroxybutyrate (PHB) , a biopolymer.

Biopolymers are gaining importance due to the potential for the production of biodegradable products through the use of renewable raw materials. Poly-â-hydroxybutyrate (PHB) is one such biopolymer whose production has been studied in detail (e.g., [2-4]). PHB belongs to the class of bacterial polyesters collectively called polyhydroxyalkanoates (PHAs). PHAs have properties similar to polypropylene and are important due to their complete biodegradability, with recognised potential applications in reducing disposable waste problems and in certain medical applications [3]. As mentioned above, the metabolic processes underlying the production of PHB are now reasonably well understood [2-4]. The biological function of PHAs in bacteria is similar to that of glycogen in mammals and starch in plants [2]. When subject to a large excess of carbon source (glucose) in relation to a second source such as nitrogen or phosphorous, most bacteria channel the excess carbon source to accumulate PHAs as a carbon and energy storage material [4]. When the carbon energy sources are exhausted, the accumulated PHAs will then be degraded to sustain cell growth. This phenomenon is exploited as a means of industrial production of PHAs, by subjecting microbial cells to growth under a large carbon source and a limiting secondary source. In practice, though the limiting nutrient could also be phosphorous, sulphur or oxygen, PHB production is normally induced with nitrogen as the second substrate, and by limiting its supply in comparison to that of glucose [3]. The bacterium Alcaligenes eutrophus is the most widely used organism for the production of PHAs as it is easy to grow, its physiology leading to PHA synthesis is well understood, and it accumulates large amounts of PHB (up to 80% of cell dry weight) in a simple medium under nitrogen-limited conditions [2]. The detailed information on the metabolic mechanisms leading to PHB synthesis in bacteria has led to the development of detailed models that consider the intracellular aspects of this process [5,6,7] . These have included mainly batch and fed-batch operating conditions [2,5,8,9,10], but also continuous conditions [3,6].

A simple ten-state model of PHB production was formulated based on the cybernetic modelling framework, a powerful methodology for describing the complex phenomena observed in biological cells. This technique hypothesises that cells have evolved optimal goal-oriented strategies as a result of evolutionary pressures [11]. Thus, unlike modelling based purely on kinetic considerations, cybernetic models consider biological cells to be optimal control systems that seek to maximise a specific performance index or goal [12]. The optimality hypothesis implies that cells direct the synthesis and activity of enzymes such that a nutritional objective (the goal) is achieved in an optimal manner [11]. This optimal resource allocation is attained by the introduction of the so-called cybernetic variables that modify the rates of enzyme production and of enzyme activation to tailor the metabolic reaction rates. Optimality is achieved by defining the cybernetic variables in accordance with the law of diminishing returns which states that, given a number of resources to be allocated among a certain number of alternatives, the amount of a resource allocated to a particular alternative is proportional to the ratio of the yield from that alternative to the resources allocated to it [13]. Depending on the nature of the metabolic pathway being examined, the definition of the cybernetic variables varies and this has been addressed in detail [11]. The model formulated here considers the metabolic pathways - the glycolytic pathway and the TCA cycle - by which glucose and ammonium sulphate are assimilated by the cell and channelled into the production of precursors required for cell growth. It also accounts for the storage and degradation of excess carbon source in the form of PHB through the PHB synthesis and degradation pathways respectively.

The eleven parameters in this model were estimated using experimental data published in the literature [14]. This data pertained to measurements of glucose, ammonium sulphate, biomass and PHB in a batch process favouring the production of PHB. Three sets of parameter values were obtained, all fitting the data extremely well. In order to ascertain the reason for this multiplicity in the parameters, two studies were undertaken. First, simulation-based sensitivity studies were undertaken to determine the influence of individual parameters upon the measured species. It was observed that a majority of the eleven parameters showed strong sensitivity. Only the parameters pertaining to PHB degradation were insensitive to the experiment, an observation attributable to the fact that the experiment favoured PHB accumulation. The problem of parameter identifiability [15] was then solved. The solution to the parameter identifiability gives the largest difference between each parameter in two potential parameter sets that give essentially the same output prediction. If this difference is large, then it is unlikely that unique parameter values can be obtained. The solution to this identifiability problem for the present model indicates that the batch experiment utilised for parameter estimation was not amenable to identification of unique values of all parameter values.

Although the three identified parameter sets give close prediction of the experimental data, they predict considerably different dynamic and steady-state behaviour under other operating conditions. This observation can be used to distinguish and discriminate through carefully designed experiments. This is done by solving the parameter distinguishability problem [15], which involves the determination of an experiment that maximises the difference between the predicted outputs of two given sets of parameters. The solution to this problem points to fed-batch experiments that would give significantly different outputs for the three parameter sets. Further, bifurcation analyses of the three parameter sets showed that the asymptotic predictions of the three parameter sets were very different indicating that carefully designed continuous experiments could also be used to uniquely identify all the parameters.

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