THE USE OF MODELS IN FERMENTATION CONTROL

THE USE OF MODELS IN FERMENTATION CONTROL T. Yoshida and H. Taguchi Department of Fermentation Technology Faculty of Engineering, Osaka University Yamada-kami, Suita-shi, Osaka 565, Japan Abstract Recently there has been a strong interest in the direct digital control of fermentation processes. More effort on identification or modeling of the processes is indispensable to accomplish this highly sophisticated control. This paper was written to present a fundamental view on model construction, process identification, parameter estimation, analysis of model and examples of uses of model. Models in the papers surveyed in the last five years are presented classifying them into three categories: subculture, subcellular and submolecular models. Assessment of model by means of sensitivity analysis and several approaches to modify models for process control are discussed. 94 T. Yoshida, H. Taguchi 1. Introduction Continuous developments in fermentation technology supported by tremendous advances in molecular biology and other related fields have resulted in more precise understanding of microbial systems. More quantitative expressions of fermentation processes have accumulated as a result of progress in studies of fermentation kinetics by various kinds of approaches, some of which have been developed in related fields, mainly chemical engineering. The scale of fermentation production is growing and more sophisticated technology of control engineering might be applied to our field in consequence. A computer aided control system is the most craved aid for fermentation systems, because the fermentation system has a multiplicity of variables. Process control means not only the practice of an optimal policy but also solving the optimum problems. The identification of the model is essential for process control. Takamatsu [1] reported the results of an inquisition about the way to raise the efficiency of a computer coupled system. 27 of 83 answers from the chemical engineers working, for practical use of a computer, requested improvement or development of the mathematical model, and another 17 wanted a multiple variable control system. The investigation done with chemical engineers may not agree with one with biochemical engineers. The result is, however, very interesting and the authors expected a stronger request for modeling of fermentation processes. Up to now many models have been presented, analysed and applied. First, a general concept of modeling for fermentation processes, is described and examples of models are shown according to a classification into three categories. Some technical methods for process identification and parameter estimation are listed. After applications of sensitivity analysis is introduced, modification of simple models and refinement of complex models are discussed. Finally, a couple of examples of the uses of models in actual fermentation control and in developing control algorithms are shown. 2. A General Concept of Fermentation Models The performance of fermentation processes is based on the biological activity, which is an exhibition of interactive enzyme action. Main organisms employed for industrial purposes are microorganisms such as bacteria, fungi and streptomyces. Many techniques for modeling and simulation developed in chemical engineering can be applied. However more efforts are still necessary for the quantitative understanding of the response of microorganisms to environmental factors. Recently there has been a tremendous development of the instrumentation to detect and control environmental factors and this advance has allowed more development of modeling. A great variety of models have been developed, the famous Monod expression [2] for T. Yoshida, H. Taguchi 95 microbial cell growth is the first example. According to the literatures (e.g. Tsuchiya [3]) models were classified into two types, namely unstructured models in which a uniformly distributed cell mass is considered along with the relationship between the biomass and production and structured models in which the cell structure is considered, the total nitrogen content is differentiated (DNA, RNA, protein), and furthermore enzyme activities in metabolic pathways are possibly taken into account. Fermentation systems show complex nonlinear characteristics caused by several hundred interactive enzymatic reactions. It might be impossible to describe the system involving all enzyme reactions. Therefore one needs to extract a master rule from the many observations on the responses of the system to environmental factors or to pick up a rate-limiting one out of the many reactions in the system. With this background in mind we like to divide the models of fermentation processes into three categories in the same way as Koga et al. [4] suggested, namely: subculture, subcellular and submolecular models. | (1) Subculture model In subculture models the microbiological system is dealt with at an averaged macroscopic level and many of them are accepted as the process model for practical use in design and control of fermentation processes. It is assumed that the rate of growth and metabolic activity are explicit functions only of the state of the system, and not of time as follows, r= rCC) (1) where € is a variable vector. When the state in the model refers to population density, the growth rate is expressed as a function of population density: us r(x) (2) where vr, is the growth rate, namely the rate of production of cell mass per unit volume of culture and X is the population density, or the cell mass per unit volume of culture. Nyiri [5] has sorted out many unstructured models for the growth rate and the production rate. The first type of the growth model shows exponential growth, r= ux (3) x | where LU is a kinetic coefficient termed the specific growth rate, which is a function : of environmental factors as discussed later. Khan and Ghose [6] presented the same | kind of model that Kono and Asai [7, 8] developed. Their model is based on the fact that the value of coefficient U changes according to the phase in a batch culture, namely, induction, transient, exponential growth and declining growth. The next is the logistic law model: 96 T. Yoshida, H. Taguchi Bam HX(1 - BX) (4) This simple model seems to be useful for the simulation of a batch culture at constant environmental conditions. Constantinides tried to predict the optimal temperature profiles for batch fermentation of penicillin using the logistic model. Monod assumed the following stoichiometric relation between growth and consumption of the substrate which limits growth: E --+Y (5) where rs is the consumption rate of the limiting substrate and Y is termed the yield constant. Next Monod recognized that the growth rate could have a maximum value, and he postulated that the substrate dependence of growth rate followed the MichaelisMenten form; sure 2 (6) where S is the concentration of the limiting substrate and un is the maximum specific growth rate obtained when S is much greater than the constant Ky: which is termed the half-velocity constant. Eqs. (5) and (6) together are a complete statement of Monod's model, and they may be applied to various cases, for optimization, process analysis and process control. In the continuous fermentor, u_Ss dx m qe ht gaa ev” s us as _ Lam (8) de an StR _ where Sn is the concentration of limiting substrate in the feed. Usability of Monod's model for fermentation processes has been investigated by many authors. They recognize the acceptability of the model for the analysis of steady state, though with some modifications, e.g. consideration of endogenous term in equations. Extensive application of the model for the prediction of transient behaviour has been tried. Many authors [9-12] claimed, as described in a later item, that modifications should be made to the model for analysis of transient state. Recently, multi-substrate limiting systems were subjected to study. Sinclair and Ryder [13] developed models for continuous culture under both carbon and oxygen limiting conditions. It was found that Monod kinetics were applicable to the growth rate dependence on oxygen concentration but Contoi's kinetics were superior for the corresponding dependence on carbon substrate concentration. Fujio et al. [14] also presented a model for fed-batch culture under carbon and oxygen limiting conditions. Spitzer [15] analysed the system under control of substrate and pH by means of the model which includes the equation of change in concentration of hydrogen ions. T. Yoshida, H. Taguchi 97 Some products cause a reduction in growth rate. Several relationships have been developed to characterize this reduction, one is an exponential reduction in growth rate [16]: = UL exp (-kP,) (9) Eganberdiev et al. [17] consider that a rectanglar hyperbola fits the data more correctly bed me (10) Hinshelwood [18] has shown linear inhibition of the form I w= u, G.0-4 P,) (11) Zines [19] investigated the steady state and the dynamic properties of the fermentative bacterium, K. aerogenes, using the Hinshelwood model. Frequency response analysis, using sinusoidal variations in the dilution rate showed that ethanol increased the time constant of the metabolic parameter. (2) Subcellular model Microbial products have been broadly classified into two groups, "growth associated" and "nongrowth associated". This classification was made according to whether the product is formed while the organism is in the exponential or in the stationary phase. Antibiotics (e.g., penicillin, novobiocin, streptomycin), toxin, and antigens, which are secondary metabolites, have been included in the group of nongrowth associated products. Models for secondary metabolite production must describe the variations of properties, in other words, chemical composition and metabolic activities of individual cells. Sawada and Kojima [20] utilized the structured model developed by Tsuchiya et al. [3] for the investigation of the effect of micromixing on a continuous operation. In the model, cells are divided into two kinds, viable cells and inactive cells. The latter were viable cells before the production of an inhibitory substance by the cells. This model can predict the growth curve including lag, exponential and declining growth and stationary phase and the effect of age of innoculum on the growth of culture. Another example of a subcellular model is based on the age approach associating the product biosynthesis rate with the age of the culture. Shu [21] expressed the rate of production as a function of the age of the cell (6). eg z A, exp (-k, 9) (12) Blanch and Rogers [22] developed a model for gramicidin S according to the concept that each cell in a culture follows a predetermined genetic order in carrying out its 98 T. Yoshida, H. Taguchi metabolic functions. The formation of products by an organism can therefore be considered as a function of cell age. The cell age is arbitrarily divided into two groups, viz., "immature" and "mature", where mature cells form the product. Brown and Vass [23] also proposed a method to determine the maturing time and applied it to several antibiotic productions. Matsumura et al. [24] analysed cell behaviour in the transient state of a continuous culture with a sophisticated "apparent age" model considering the residence time distribution in a fermenter. (3) Submolecular model Progress in studies on cell physiology and metabolism allows the construction of more precise models which describe changes of chemicals in cells. Peringer et al. [26] presented a mathematical model using the concept of an equilibrium between the ATP requirements and the ATP produced by catabolic processes. An analytical expression for the time course of growth could be derived which provides a quantitative expression of the Pasteur effect, predicts the value of Y instantaneous values of P/O ratios, ATP? the quantity of substrate and oxygen consumed and of biomass produced. Kremer [27] derived a kinetic model exhibited by acetyl Co-A metabolism and ATP regeneration to investigate possible control mechanisms in tetracycline production. Monod [25] has developed the most favored hypothesis on induction and repression of enzyme production in cells. His operon hypothesis is a splendid example of recent advances in molecular biology and very helpful to investigate microbial behaviour even from a macroscopic point of view. The hypothesis is constructed with the theories of genetical regulation of enzyme production with chemical effectors, namely, repressor and inducer. Models based on the operon hypothesis have been developed for quantitative expression of industrial fermentation processes. Preferential synthesis of a enzyme, a-galactosidase, was described in submolecular model by Imanaka et al. [28]. Enzyme production rate is proportional to the concentration of mRNA in cells (M): dE eae keM - LE (13) where E denotes enzyme content in cells and the second term on the right hand side of the equation represents dilution by growth. m-RNA is synthesised with negative correlation to the content of repressor (R) and is subject to first order decomposition: dM der ke (R, - R) - k7M - uM (14) R. is the critical concentration of repressor to stop the synthesis of m-RNA. The change of repressor content, R, and a complex of repressor and inducer, Rön; > were expressed as follows: dR _ —— ae denen ue eas kuRS,, + ksRS, - uR (15) T. Yoshida, H. Taguchi 99 BL a er Or Da HE = kuRS, KsBS,. URS, (16) where Sai is the intracellular concentration of inducer, galactose. This model gave a good prediction of the batch and continuous culture of a-galactosidase production [29] and was utilized for the optimization of the enzyme production process [30]. Van Dedem [31] proposed the same kind of model describing the degree of repression or induction of extracellular enzyme synthesis as a function of nutrient concentration. 3. Process Identification and Parameter Estimation Identification of dominant biological or chemical rate processes is essential for the optimal operation and the best performances of the fermentation process, although information about mass and heat transfer and mixing characteristics is sometimes necessary. The responses of the processes to the change of circumstances could be investigated in two ways: experiments with the actual processes or simulations of the processes by means of models. (1) Simulation of fermentation processes Simulation is defined in the present article as the description of real systems of fermentation processes by use of mathematical equations; computer methods are generally required for their solution. Blanch and Dunn [32] presented a full commentary on modeling and simulation in biochemical engineering. Several important steps can be recognized in any simulation development as shown inthe figure. First, the purpose of Theoretical Background ee eee — | Chemical Kinetics N | Enzymology | Mathematical] „| Parameter Cell Physiology Model Estimation Population Dynamics Transport Phenomena A Determination of Experimental Conditions Steps for simulation of fermentation processes simulation must be clearly defined and a good insight into real fermentation phenomena is quite helpful for modeling. The second step is to write the equations. There are two starting points for this step: one is theoretical analysis and the other is analysis of experimental results. Fundamental knowledge about fermentation and related 100 T. Yoshida, H. Taguchi fields is a need for both cases. The third is the parameter estimation by experiments with the aid of mathematical or numerical techniques. The experimental conditions depend on the method of parameter estimation. Finally, the validity of the model should be justified with experiments, and reconstruction of model should be done if necessary. Chi and Howell [10] presented a typical example of rate process identification comparing two competing models. They compared a modified Powell-Ierusalenskii bottle neck model to a Monod-Haldane model for the prediction of the transient behaviour of a continuous culture. They used Pseudomonas which has a characteristic of substrate inhibition. Applying Monod-Haldane model, us le ONE an Ss 4; Following Powell [33], they introduced Q substrate which is a quantitative expression of the physiological state of the microorganisms and governs the growth as follows: lL = Q YG (S) (18) and the change of Q was expressed in the following equation: aQ _ s Y s dt Um K #8487 7K, ar CR_+5+057/R, (19) where C is the ratio of the lower limit of Q, Q, to the upper limit A The observed transient behaviour was predicted by the new model, but could not be predicted by a Monod-Haldane model. (2) Techniques for parameter estimation Parameter estimation is one of the most important problems in the identification of processes. Values of parameters must be estimated by analysis of experimental results for the most rational explanation of the behaviour of the process. Observations or Measurements obtained by experiments are rather random rather than determistic, since under actual conditions, process noise, cycling, signal noise and other phenomena interfere with all measurements. Linear least squares estimation method has been employed for parameter estimation of rate process in fermentation. Many parameter estimation methods have been developed for the nonlinear parameter system. The least squares parameter estimation does not require prior knowledge of the distribution of unobserved errors, and it yields unbiased estimates, and results in the minimum variance among all the linear unbiased estimation methods, the least squares technique is used extensively. Values of parameters are estimated to minimize n o=,2, [¥-G@lwiy-6] (20) where Y is the matrix of observation and G is the matrix of prediction. If Wis a diagonal matrix, becomes a "weighted least squares" criterion, if W is a unit matrix is the "ordinary least squares" criterion. Box and Draper's criterion [34] was applied by Johnson and Berthovex [35] to estimate two parameters in a Monod model for continuous T. Yoshida, H. Taguchi 101 culture. They showed that the confidence region for estimated parameters with both the data of cell concentration and substrate concentration is much smaller than with the data of any of the two alone. Sundstrom et al. [36] studied response of biological reactors to the sinusoidal variations of substrate concentration and employed the following criterion. Ss - oN 2 xX - X, 2 o=2 ica ee er rl, (21) e ie where subscripts p and e denote prediction and experiment, respectively. Bode diagram has been utilized to estimate parameters or analyse the response to sinusoidal variation of input signal in a fermentation system [19, 37, 38]. 4. Reassessment of Conventional Models for Process Control A large number of models have been presented as surveyed in previous items, but it is a big problem to choose which model should be employed for "process control". Criteria for choice of model may be listed as follows: 1. Object of control: what kind of control is desired; holding control or state variable constant, program control or optimal adaptive control? More sophisticated models must be prepared for more fine control. 2. Availability of equipment or facilities for control: available model may be limited with extent of instrumentation, although every effort should be made to search for a detector or controller. 3. Availability of an algorithm for control: a control algorithm is obviously essential for actual computer control, therefore a complex model can not be employed. However, it is not so difficult to develop a control algorithm if there is adequate technical talent. Any models developed previously for other purposes, (e.g., kinetic study, fundamental analysis of microbial behaviour or design of plant) might be analysed and modified with a view to application for process control. (1) Application of sensitivity analysis to assess models The technique of sensitivity analysis has been applied to study the effect of parameter perturbation on state variables and objective functions for optimal control. Yoshida and Wada [39] analysed the behaviour of a continuous culture for implementation of process control by means of a sensitivity analysis. In the continuous fermentation system where the performance equation is expressed as Eqs. (7) and (8), the following equations can be obtained for the small perturbations, x1 and xz of the two state variables X and S: dxı _ 42 = Yabxz (22) L —