# Quantitative measures of disorder in biological oscillations and their implications for bioreactor operation.

код для вставкиСкачатьASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING Asia-Pac. J. Chem. Eng. 2007; 2: 640?649 Published online 13 September 2007 in Wiley InterScience (www.interscience.wiley.com) DOI:10.1002/apj.091 Research Article Quantitative measures of disorder in biological oscillations and their implications for bioreactor operation Pratap R. Patnaik* Institute of Microbial Technology, Sector 39-A, Chandigarh-160036, India Received 12 March 2007; Revised 20 July 2007; Accepted 1 August 2007 ABSTRACT: Under realistic conditions, biological oscillations show fluctuations around time-varying deterministic values. These fluctuations are often created by noise from the environment, and they vary randomly with time. With continuous flow microbial cultures, the feed stream is a major source of noise. While the detrimental effects of noise inflow are known, quantitative measures of their effects are not established. Such indexes are important because, apart from distortions, noise may displace a fermentation into an undesirable state or into chaos. Using Saccharomyces cerevisiae as an example, four measures of the effect of noise are proposed and their physical implications for the operation of continuous cultures are discussed. It is shown that, like monotonic cultures studied earlier, noiseaffected oscillating cultures too can be analyzed usefully through their fractal dimensions. ? 2007 Curtin University of Technology and John Wiley & Sons, Ltd. KEYWORDS: oscillating cultures; Saccharomyces cerevisiae; external noise; quantitative measures; fractal dimension INTRODUCTION Many biological processes exhibit oscillatory behavior under either natural or controlled conditions. Predator?prey systems (Durrett, 1993), calcium waves (Bootman et al ., 2001) and neuromotor signals observed through electroencephalograms (Baker et al ., 1999) are well-known examples of naturally occurring oscillations. Among oscillations that occur under controlled conditions, the most widely studied are those of continuous cultures of Zymomonas mobilis (Bruce et al ., 1991) and Saccharomyces cerevisiae (Satroutdinov et al ., 1992), both of which synthesize ethanol. Between them, S. cerevisiae is the more widely used and extensively studied. It is important as a model system for its rich repertoire of oscillating processes, industrially for its product, the absence of endotoxins, the ease of its downstream processing, and the availability of noninvasive methods of measurement in continuous cultures. A paradoxical aspect of both Z. mobilis and S. cerevisiae cultivations is that while their oscillations have engaged the attention of research workers, industrial fermentations try to avoid oscillations. Therefore, the study of oscillatory behavior is important from both perspectives. *Correspondence to: Pratap R. Patnaik, Institute of Microbial Technology, Sector 39-A, Chandigarh-160036, India. E-mail: pratap@imtech.res.in ? 2007 Curtin University of Technology and John Wiley & Sons, Ltd. While there are many studies of the mechanisms and models of oscillating continuous yeast cultures (see Patnaik, 2003a for a review), they have all focused on small laboratory-scale bioreactors, where spatial gradients and fluctuations in the feed stream can usually be ignored. However, production-scale operations are often ?corrupted? by such fluctuations or noise, usually present in an inflow stream. Uncontrolled noise can seriously harm a cultivation process, creating problems such as cell lysis, metabolic shifts, changes in the fermentation pattern, and run-away operation. Some of these possibilities have indeed also been observed in a laboratory-scale bioreactor subjected to deliberate perturbations (Zamamiri et al ., 2001). Therefore, it is important to be able to analyze and quantify the effect of noise on oscillating microbial cultures in order to model, optimize, and control industrial-scale bioreactors (Rohner and Meyer, 1995). Quantitative measures of the effect of noise are useful in determining optimum or safe operating conditions, and in designing noise filters and automatic control systems. In view of its importance for both research and industry, the behavior of a continuous oscillating culture of S. cerevisiae under the influence of noise in the feed stream has been analyzed here. To characterize the effect of noise, three quantitative indices of performance are proposed. The indices are: (1) degree of distortion (by noise), (2) degree of recovery (by using a noise filter), and (3) fractal dimension. They are defined and their physical significance explained in later sections. Asia-Pacific Journal of Chemical Engineering DISORDER IN BIOLOGICAL OSCILLATIONS AND THEIR IMPLICATIONS Each index was calculated both for a noise-affected culture without any intervention and for a culture for which the noise was ?filtered? as explained below. Both cultures have been studied with (1) glucose and (2) ethanol as the carbon substrates. To apply optimization and control algorithms, the distortions caused by noise have to be minimized so that acceptably noise-free data, revealing the key features of a system, are recovered from noise-affected values. This is done by employing filters, which are software devices that act on noisy data to generate data with larger signal/noise ratios. Different kinds of noise filters are available, each with its strengths and weaknesses (Nelles, 2000). For bioreactors, the extended Kalman filter (EKF) has been effective in many applications (Zorzetto and Wilson, 1996; Patnaik, 2005a), even though there are persuasive arguments in support of neural filters (Patnaik, 2003b). Unlike a neural filter, an EKF is intrinsically linked to the biological model and thus acquires greater physiological fidelity. Therefore, an EKF was applied to the noise-affected cultures studied here. FERMENTATION DESCRIPTION AND DATA GENERATION In certain ranges of operating conditions, continuous cultures of S. cerevisiae exhibit sustained oscillations with time in the concentrations of biomass, substrate (glucose or ethanol), product (ethanol), dissolved oxygen and storage carbohydrates, and in the pH, the oxygen uptake rate (OUR) and carbon dioxide evolution rate (CER). Experimental observations (Satroutdinov et al ., 1992; Keulers et al ., 1996; Zamamiri et al ., 2001) indicate that the oscillations may be controlled by regulating the dilution rate (in effect, the feed rate of the main carbon source) and the dissolved oxygen concentration. Between the two, the dilution rate is preferred as the manipulated variable for bioreactor control because it is easier to monitor and change and it leads to superior regulatory control (Dochain and Perrier, 1997). The complexity and variations of the observed oscillations have made it difficult to formulate mathematical models that are sufficiently accurate and flexible without being too complicated. Modeling efforts have approached the issue from either of two perspectives. One approach has focused on the reactions inside the cells and formulated models for their kinetics. Such models offer insight into the metabolic aspects and enable suitable genetic manipulations for strain improvement. But they do not consider external influences through transport across the cell walls, flow patterns in a reactor, and the influx of disturbances. Models that include these features as well as detailed cellular kinetics are often quite complex. Therefore, a second approach combines transport equations with judiciously lumped kinetics. Both kinds of models have ? 2007 Curtin University of Technology and John Wiley & Sons, Ltd. been reviewed recently (Patnaik, 2003a), where it has been pointed out that the second approach is more relevant for large nonideal bioreactors. Therefore, a model of the latter type was used in this study to generate data mimicking a noise-affected fermentation. Commercial and practical considerations often restrict the availability and disclosure of real industrial data. As an alternative, many investigators (Thompson and Kramer, 1994; Simutis and Lubbert, 1997; Tian et al ., 2002; Desai et al ., 2006) have relied on simulated data generated by solving a phenomenologically plausible model that incorporates experimentally validated kinetics. That approach has been followed here. The cybernetic model of Jones and Kompala (1999) was used to generate data of oscillating concentration portraits. They preferred a cybernetic approach so as to overcome some of the weaknesses of mechanistic models for the time-dependent behavior of many microbial cultures. The cybernetic approach (Dhurjati et al ., 1985) attributes the ability to learn from experience and to reach ?intelligent? decisions to living cells. These decisions are usually designed to maximize the probability of survival of the cells under varying conditions. Therefore, by manipulating the environmental conditions the cells may be induced to function in a manner that meets our requirement through their survival. The preferential synthesis of a desired product may, for instance, meet both objectives. In their cybernetic model, Jones and Kompala (1999) identified three metabolic pathways by which S. cerevisiae may utilize the available carbon sources: glucose fermentation, ethanol oxidation, and glucose oxidation. Depending on its past history and the current conditions, the culture may follow either one pathway or two or more pathways to different extents. They showed that dynamic competition among the pathways was the driving force for oscillations. By contrast, some mechanistic models identify this (incorrectly) as a consequence of oscillatory metabolism. Both types of models, however, suggest manipulating the dilution rate to control the nature of the oscillations. The equations of the model are presented in the Appendix. These equations were fitted by Jones and Kompala to data from a number of sources (von Meyenburg, 1973; Satroutdinov et al ., 1992; Keulers et al ., 1996) and the values they obtained for the parameters are listed in Table 1. Both Satroutdinov and coworkers (1992) and Keulers et al . (1996) used the same strain of S. cerevisiae (a wild type diploid strain IFO 0233) but different carbon substrates. Whereas the former authors fed glucose, Keulers et al . supplied ethanol. Since ethanol is a product of glucose metabolism, this is a crucial difference, whose effect on the impact of noise has been evaluated here. Once the data have been generated, they are considered to represent a real bioreactor and the model then becomes redundant. After the parameters were determined from the noise-free studies, the model Asia-Pac. J. Chem. Eng. 2007; 2: 640?649 DOI: 10.1002/apj 641 642 P. R. PATNAIK Asia-Pacific Journal of Chemical Engineering Table 1. Values of the parameters (Jones and Kompala, 1999). Parameter ? ?? ? ?1 ?2 ?3 �max �max �max ?1 ?2 ?3 ?4 D G0 kL a K1 K2 K3 KO2 KO3 O? Y1 Y2 Y3 Units Value h?1 g h?1 h?1 g g?1 g g?1 g g?1 h?1 h?1 h?1 g g?1 g g?1 g g?1 g g?1 h?1 g l?1 h?1 g l?1 g l?1 g l?1 mg l?1 mg l?1 mg l?1 g g?1 g g?1 g g?1 1.0 0.1 0.2 6.0 6.0 0.3 0.44 0.32 0.31 0.27 1.067 2.087 0.95 0.16 28.0 1200.0 0.1 0.02 0.001 0.0001 0.0001 7.5 0.16 0.74 0.50 was solved again with 1/f noise added to the flow rate of the carbon substrate because this kind of noise is the most prevalent in biological processes (Yates, 1992). Details of the model are omitted here to conserve space. It is important to point out that addition of noise to the inflow stream before the model is solved embeds the noise in the process. This is more realistic than adding noise, as done by Desai et al . (2006), to the concentrations obtained from a solution of the deterministic model. Points on the solution curves for different concentrations denoted data mimicking a noise-affected bioreactor. The choice of the sampling interval is an important aspect of data selection. Although common and convenient, a constant interval may not be a good choice because a small interval generates an unnecessarily large number of data points from shallow regions of a concentration profile whereas too widely spaced data may omit important variations from steeply changing regions. These considerations have been discussed by Simutis et al . (1997) and by Chen and Rollins (2000), who have referred to two kinds of nonconstant sampling periods. One is random sampling, where the time interval between any two points varies randomly. While practical convenience may favor this method in some plant operations, it is not always an optimum choice ? 2007 Curtin University of Technology and John Wiley & Sons, Ltd. and may have some of the weaknesses of a constant sampling period. A more meaningful choice would require the sampling period to vary according to the requirements of the process. This results in the second kind of sampling variation, in which the interval varies inversely as the local gradient of the concentration profile. Thereby, closely spaced data are sampled from sharply changing regions and widely spaced values from mildly changing regions. This kind of sampling preserves the relevant features without generating superfluous data; its usefulness has been demonstrated in previous applications (Patnaik, 2001, 2003b). THEORY AND APPLICATION OF THE KALMAN FILTER The EKF has been shown to be effective in attenuating the flow of noise into S. cerevisiae fermentations under both monotonic and oscillating conditions (Zorzetto and Wilson, 1996; Patnaik, 2005a). A detailed account of its theory is available in Grewal and Andrews (1993). Briefly, the basic Kalman filter is a set of mathematical equations that provides an efficient recursive solution of the least-squares type. The filter can provide estimations of past, present, and future states of a system even when a precise model is not known. This feature is useful for microbial processes under nonideal (realistic) conditions because models developed on laboratory data may become inapplicable or imprecise under the influence of disturbances and spatial gradients (Gillard and Tragardh, 1999; Liden, 2002). The basic Kalman filter addresses the problem of trying to estimate the state x of a discrete-time controlled process that is governed by the linear difference equation: xk = Axk ?1 + Buk + wk ?1 (1) with a measurement vector that follows: zk = Hxk + vk (2) In these and later equations, lower case letters with overbars denote vectors while similar capital letters denote matrices. Scalars do not have overbars. (k ? 1) is the current instant of time and k is the point one time-step ahead. wk and vk represent the process noise and measurement noise respectively. Previous studies (Montague and Morris, 1994; Rohner and Meyer, 1995; Gillard and Tragardh, 1999) show that wk and vk may be represented as white noise with normal probability distributions: p(w ) ? N (0, Q) (3) p(v ) ? N (0, R) (4) where Q and R are the respective covariance matrices. Asia-Pac. J. Chem. Eng. 2007; 2: 640?649 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering DISORDER IN BIOLOGICAL OSCILLATIONS AND THEIR IMPLICATIONS Since Eqn. (1) applies to linear equations whereas many fermentation (and other biological) processes follow nonlinear models, the EKF was developed. It applies to any nonlinear difference equation of the form: xk = f(xk ?1 , uk , wk ?1 ) (5) zk = h(xk , vk ) (6) In principle, the EKF solves the problem of determining the current estimates of a set of variables by expressing them as linear functions centered around the partial derivatives of the process and measurement functions evaluated at the (known) previous instant of time. Note that both Eqns (1) and (2) and Eqns (5) and (6), in pairs, are in discrete form whereas most biological processes are described by continuous models. This is not an impediment because, in practice, data are sampled at discrete points in time. Since the EKF allows any arbitrary variation in the sampling interval, this may be varied according to the nature of the process. For instance, the interval may be made inversely proportional to the current concentration gradient, thus generating closely spaced data when the variations are steep and more widely separated points during mild variations. As explained before, data simulating a noise-influenced oscillating culture were generated by solving the Jones and Kompala (1999) model, first without noise and then with 1/f noise in the feed stream. In an experimental application, the measurement covariance R is usually measured prior to the operation of the filter since it relates to the filter and not the process. The process noise covariance Q is more difficult to determine since typically we do not have the ability to observe the process we are estimating. So, based on previous studies (Zorzetto and Wilson, 1996; Soroush, 1998),Q was set initially to Qd = ([0.0001 . . . . . . ..0.0001]T ) and R to 0.003 I, where I is the identity matrix and Qd is a diagonal matrix. Since the glucose and oxygen feed streams are the only inflows to the bioreactor, environmental noise was considered to be present in these two flow rates, thus making Q a (2*2) matrix. Both Q and R get updated recursively as shown in Fig. 1. Apart from its applicability to nonlinear processes, an important distinction between the EKF and the basic discrete Kalman filter is that in the former case the Jacobian Hk in the equation for the Kalman gain Kk also gets updated with each iteration, thereby speeding up convergence and improving the accuracy of estimations. As explained before, data from the concentration profiles were sampled at intervals of time inversely proportional to the local gradients and the tuning of the EKF was updated progressively over successive intervals. ? 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Figure 1. Computation procedure of the EKF. In the left box the ?corrected? state variables x?k?1 at each sampling instant t = k ? 1 are used to estimate their values at the next point t = k, i.e. x?. These values are again corrected in the right box and the results fed back. The process begins with known initial values, provided from outside as shown. CALCULATION OF THE FRACTAL DIMENSIONS Under realistic conditions, many biological processes have a degree of self-similar randomness that may be characterized by fractal geometry. In some cases the randomness may be inherent, as in the surface structure of proteins (Lewis and Rees, 1985), the morphology of mycelial flocs (Logan and Wilkinson, 1991) and antibody?antigen interactions (Sadana and Beelaram, 1994). Randomness may also be induced by external influences, mainly noise or disturbances. Recent work with Escherichia coli (Patnaik, 1999, 2000) has shown that the randomness introduced by noise entering through a feed stream can be quantitatively expressed and interpreted through fractal dimensions calculated from the concentration-time profiles. Measurements of microbial fermentations usually generate data of concentrations or activities as functions of time. Dubuc et al . (1989) compared a number of methods to compute fractal dimensions from such timeseries data, and recommended the variation method. Their work may be consulted for the theory; only the computer implementation is briefly described here. Suppose a one-dimensional profile (such as concentration vs time) is discretized into a set of (N + 1) points ?(ti ), i = 0, 1, 2, . . . . . . , N . Choose a set of integers ki , i = 1, 2, . . . . . . , imax with imax < N . The values of the ki are to be so chosen that ?i = ki ? ki ?1 ? ki ?1 and ?i = ki /N 1. Then Dubuc et al . (1989) have shown that two curves, called the upper envelope u(t) and the lower envelope b(t), can be defined on either side of the original fractal curve. For convenience in programming, we identify the discrete times ti , i = 0, 1, 2, . . . . . . , N simply by the indices 0, 1, 2, . . . . . . , N . Then the values of u(t) and b(t) are calculated as follows. u1 (j ) = max[?(j ? 1), ?(j ), ?(j + 1)] (7) b1 (j ) = min[?(j ? 1), ?(j ), ?(j + 1)] (8) Asia-Pac. J. Chem. Eng. 2007; 2: 640?649 DOI: 10.1002/apj 643 Asia-Pacific Journal of Chemical Engineering ui (j ) = max[ui ?1 (j ? ?i ), ui ?1 (j + ?i )] (9) bi (j ) = min[bi ?1 (j ? ?i ), bi ?1 (j + ?i )] (10) where i = 2, 3, . . . . . . , imax and j = 1, 2, . . . . . . , N . If we define wi = ?1 P. R. PATNAIK N [ui (j) ? bi (j )], j =1 i = 1, 2, . . . . . . , imax then it has been shown that: ln(wi /ki 2 ) = d ln(1/ki ) + p (11) where d and p are constants. This equation yields a straight line whose slope d is the fractal dimension of the curve. APPLICATION AND DISCUSSION Satroutdinov et al . (1992) and Keulers et al . (1996) studied the oscillatory behavior of a continuous fermentation with wild-type diploid strain IFO 0233 of S. cerevisiae in a Bioflo fermenter (New Brunswick, NJ) with a working volume of 1350 ml. Satroutdinov et al . (1992) used glucose feed and observed oscillations in the concentrations of glucose, ethanol (the product), the OUR, the CER, and glycogen (the principal storage carbohydrate). They suggested that oscillations were regulated by changes in the rate of glycolysis inside the cells. To test this hypothesis, Keulers et al . (1996) eliminated glucose by employing ethanol as the carbon source. Ethanol concentration, the OUR, and the CER oscillated but in a manner different from those with glucose feed. In the absence of glucose, glycogen was not detected but another carbohydrate, trehalose, was present in small concentrations in both glucose- and ethanol-fed cultures. To evaluate the efficiency of the EKF, the cybernetic model of Jones and Kompala (1999) was solved, as explained earlier, first without any noise and then with 1/f noise in the feed stream. To obtain data representing different situations, the model containing noise was solved separately without and with the filter, for both glucose and ethanol as the carbon substrate in either case. The dilution rate was maintained at 0.16 h?1 in all cases. This value was chosen for consistency with the deterministic results of Jones and Kompala (1999), who obtained sustained oscillations at this dilution rate. The effect of a change in dilution rate is discussed later as part of the present analysis. Figures 2?7 show the deterministic profiles, the unfiltered noise-affected profiles, and the filtered profiles for three key variables, each with either glucose or ethanol as the feed ? 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Figure 2. Oscillations in the concentration of ethanol with glucose feed at a dilution rate of 0.16 h?1 . The plot without noise is from Jones and Kompala (1999) while the two noise-affected plots were obtained in this study. ?1 644 Figure 3. Oscillations in the concentration of ethanol with ethanol feed at a dilution rate of 0.16 h?1 . The plots have the same connotations as in Fig. 2. Note the contrast between the nature of the plots in Figs 2 and 3, especially the distortions created by noise in Fig. 3. stream and at the dilution rates used by the investigators (Satroutdinov et al ., 1992; Keulers et al ., 1996). Ethanol concentration is an important variable because it is both the principal product and is utilized by the cells when there is insufficient glucose, as at low dilution rates (Satroutdinov et al ., 1992; Keulers et al ., 1996). Thus, ethanol provides a critical physiological feedback Asia-Pac. J. Chem. Eng. 2007; 2: 640?649 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering DISORDER IN BIOLOGICAL OSCILLATIONS AND THEIR IMPLICATIONS Figure 4. Oscillations in the oxygen uptake rate with glucose feed at a dilution rate of 0.16 h?1 . The plots have the same connotations as in Fig. 2. Gas-to-liquid mass transfer resistance and the low concentrations of dissolved oxygen contribute to the variations in this figure being sharper than in Fig. 2. Figure 5. Oscillations in the oxygen uptake rate with ethanol feed at a dilution rate of 0.16 h?1 . The plots have the same connotations as in Fig. 2. As for ethanol, noise causes large distortions in the smooth oscillations but an EKF has been able to restore nearly noise-free behavior. control. In fact, Keulers et al . (1996) performed experiments with ethanol as the sole carbon source to resolve the uncertainty in Satroutdinov et al .?s (1992) suggestion that either changes in glycolysis rate caused by ? 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Figure 6. Oscillations in the carbon dioxide evolution rate with glucose feed at a dilution rate of 0.16 h?1 . Since carbon dioxide evolution is a respiratory process whereas ethanol formation is fermentative, the carbon dioxide concentrations are much smaller and the differences between ethanol and glucose substrates are diametrically opposite between Figs 2 and 3 in one pair and Figs 6 and 7 in the other. Figure 7. Oscillations in the carbon dioxide evolution rate with ethanol feed at a dilution rate of 0.16 h?1 . The plots here and in Fig. 6 may be contrasted with those of Figs 2 and 3 as explained for Fig. 6. switching of glycogen metabolism or feedback regulation of the tricarboxylic acid (TCA) cycle regulated the oscillations. Since glycolysis is negligible with ethanol Asia-Pac. J. Chem. Eng. 2007; 2: 640?649 DOI: 10.1002/apj 645 646 P. R. PATNAIK medium, the observed oscillations supported TCA regulation as the controlling factor. These inferences about oscillations on glucose and ethanol encapsulate the more complex connections within the metabolic framework. Such a summary is intentional but useful. It is intentional because this study addresses the macroscopic manifestations of the effects of disturbances on oscillatory behavior vis-a?-vis bioreactor operation, and it is not an investigation into the mechanisms per se. Nevertheless, a brief account of the mechanistic basis of oscillatory performance is useful because the occurrence and the type of oscillations depend on the operating conditions (Satroutdinov et al ., 1992; Jones and Kompala, 1999; Patnaik, 2003a). While the metabolic ramifications of sustained oscillations, even under noise-free conditions, are not yet fully understood (Murray et al ., 2001; Zamamiri et al ., 2001), they do reveal that internal feedback controls have a major role in the regulatory processes (Satroutdinov et al ., 1992; Duboc et al ., 1996; Lloyd, 1998). These may be monitored and controlled through two measurable variables, the OUR and the CER, whose variations are portrayed in Figs 4?7, whereas those of ethanol are shown in Figs 2 and 3. It may be clarified here that the time spans of Figs 2, 4, and 6 differ from those of Figs 3, 5, and 7 because the former set were generated from the deterministic profiles of Satroutdinov et al . (1992) and the latter from those of Keulers et al . (1996). This difference does not matter for studies of the effects of noise since any sufficiently long slice of time from a persistent oscillating profile provides an adequate starting basis for such an analysis. Although the profiles for ethanol, OUR, and CER are visually similar, there are quantitative differences. Without noise, Satroutdinov et al .?s (1992) and Keulers et al .?s (1996) data showed that a change in the carbon source affected the amplitudes of the oscillations but not the frequencies. To evaluate the effect of noise on these variables, three indices may be defined. One is the span of the amplitude: a = 100 (maximum amplitude) ? (minimum amplitude) � (minimum amplitude) (12) The second is the degree of distortion caused by noise; this may be expressed as a sum of squares of normalized deviations: 100 d= N N (noise-affected value) - (noise-free value) 2 � (noise-free value) (13) ? 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Asia-Pacific Journal of Chemical Engineering where N is the number of data sampled. The third index is the degree of recovery of noise-free oscillations from noise-affected oscillations by using an EKF: ? (degree of distortion without an EKF) ? ?(degree of distortion with an EKF) ? r = 100 ? ? (degree of distortion without an EKF) ? (14) The values of these indices are presented in Table 2, and they show interesting trends with respect to the span of the amplitude. As the span decreases, implying weaker oscillations, the degree of distortion, both without and with an EKF, increases and the degree of recovery decreases. These results pertain to a dilution rate of 0.08? 0.085 h?1 , employed by Satroutdinov et al . (1992) and Keulers et al . (1996). Since the dilution rate strongly influences the nature of oscillations (Beuse et al ., 1998; Jones and Kompala, 1999; Patnaik, 2003a), the effect of inflow noise at different dilution rates was analyzed. Since a number of variables have to be compared across a range of dilution rates, it is cumbersome to use the indices a, d, and r defined above. Fractal indices offer a more convenient way to characterize disorder in biological oscillations (Pincus, 1991; Patnaik, 1999, 2000). For each dilution rate, the concentration plot of each variable yields one fractal dimension, computed according to the variation method described above (Dubuc et al ., 1989). These dimensions have been plotted in Fig. 8. At small dilution rates, the fractal dimensions are close to the unit value of a smooth curve. This is expected since there is little influx of noise. As the dilution rate increases, so does the influx of noise; consequently, the fractal dimensions also increase and approach the limiting value of 1.75, which is characteristic of 1/f noise. Consistent with the inferences from the indices a, d, and r above, strong oscillations are less perturbed by noise than are weak oscillations, resulting in smaller fractal dimensions and a slower approach to the limiting value. Table 2. Span of the amplitude, degree of noise-induced disorder, and degree of recovery by EKF for different variables. Degree of distortion Variable OUR (G)a Ethanol (G) OUR (E) CER (E) CER (G) Ethanol (E) a Span of amplitude Without EKF With EKF Degree of recovery 234.0 72.5 59.0 47.4 7.2 1.5 0.0392 0.540 0.671 0.715 1.193 1.582 0.0066 0.159 0.241 0.301 0.699 1.049 83.2 70.6 64.1 57.9 41.4 33.7 G = glucose feed; E = ethanol feed. Asia-Pac. J. Chem. Eng. 2007; 2: 640?649 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering DISORDER IN BIOLOGICAL OSCILLATIONS AND THEIR IMPLICATIONS the inflow of noise may also drive a smoothly oscillating process to chaotic behavior (Patnaik, 2005b). Such changes obviously have serious implications for cultivation processes, thus emphasizing the importance of quantitative measures of the effects of noise on microbial oscillations. CONCLUSIONS Figure 8. Variations in the fractal dimensions for different variables over a range of dilution rates and with glucose and ethanol, separately, as the substrate. 1?Ethanol (Ethanol); 2?CER (Glucose); 3?CER (Ethanol); 4?OUR (Ethanol); 5?Ethanol (Glucose); 6?OUR (Glucose). The names in brackets denote the feed streams. These trends pose two interesting practical questions for large-scale cultivations. First, does noise affect the metabolic basis of deterministic oscillations? Secondly, how best can the smaller sensitivities to disturbances and the better filtering efficiencies for large oscillations be balanced against the greater complexity of their control policies? Although measurements of intracellular metabolites under noise-affected conditions have not yet been reported, results from experiments in the absence of noise suggest that with glucose as the medium the oscillations are caused by changes in glycolytic flux (Satroutdinov et al ., 1992), whereas with ethanol these are driven by flux changes in the TCA cycle (Keulers et al ., 1996). Within either system, the oscillations of different metabolites differ in their amplitudes, phases, and time constants (Duboc et al ., 1996; Keulers et al ., 1996; Wittmann et al ., 2005). We recall at this point that at low glucose concentrations S. cerevisiae switches to ethanol as its main carbon source, and reverts to glucose again when its level rises sufficiently (Satroutdinov et al ., 1992; Duboc et al ., 1996; Beuse et al ., 1998; Jones and Kompala, 1999). Thus, the driving force and the mechanistic basis of the observed oscillating outputs may shift over a length of time. This possibility and the fact that under a particular set of conditions both oscillating states and nonoscillating stationary states can exist (Zamamiri et al ., 2001) imply that noise can displace a culture from an oscillating state to a stationary state or the other way. Indeed, ? 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Under realistic conditions representative of large bioreactors, oscillating microbial cultures in fed-batch and continuous operations are affected by noise carried by the feed streams. Mild noise distorts the oscillations, thereby rendering it difficult to identify the true features of the process. Stronger noise may drive a fermentation into another metabolic state, indicated by changes in the oscillating characteristics or by a shift to a nonoscillating state. More seriously, noise may even cause smooth oscillations to degenerate into chaos. Therefore, for optimal operation of a cultivation process, it is important to have good estimates of the disorder induced on the oscillations by the influx of noise. In this study, the effects of noise on an oscillating continuous culture of S. cerevisiae were investigated, both without any filter and with an EKF to reduce the inflow of noise. Four indexes of the effect of noise were introduced: (1) the span of the amplitude, (2) the sum of squares of normalized deviations, (3) the degree of recovery, and (4) the fractal dimension. It was observed that weak oscillations are distorted more and recover poorly upon removal (or filtering) of the noise. While offering quantitative measures of disorder introduced by noise, these inferences also suggest that the common preference for fermentations with monotonic profiles may not be good under noiseinfluenced conditions. Operational difficulties also discourage strongly oscillating cultures. Thus, the optimal choice involves a balance between the sensitivity to noise for monotonic profiles and weak oscillations, on the one hand, and the difficulties of controlling more robust fermentations with strong oscillations. NOMENCLATURE C D ei E G G0 kL a intracellular storage carbohydrate concentration (g l?1 ) dilution rate (h?1 ) key enzyme concentration for i-th pathway (g g?1 biomass) ethanol concentration in the bioreactor (g l?1 ) glucose concentration in the bioreactor (g l?1 ) glucose concentration in the feed stream (g l?1 ) oxygen mass transfer coefficient (h?1 ) Asia-Pac. J. Chem. Eng. 2007; 2: 640?649 DOI: 10.1002/apj 647 648 P. R. PATNAIK Asia-Pacific Journal of Chemical Engineering Ki Michaelis constant for i-th pathway (g l?1 )9 KO2 , KO3 oxidative pathway oxygen saturation constants (mg l?1 ) O dissolved oxygen concentration in the bioreactor (mg l?1 ) ? dissolved oxygen solubility limit (mg l?1 ) O biomass growth rate on i-th pathway (h?1 ) ri carbon substrate concentration for i-th pathSi way (g l?1 ) t elapsed time (h) cybernetic variable controlling key enzyme ui synthesis for i-th pathway (?) cybernetic variable controlling key enzyme vi activity for i-th pathway (?) X biomass concentration in the bioreactor (g l?1 ) yield coefficient for i-th pathway (g biomass Yi g?1 substrate) GREEK LETTERS ? ?? ? ?i ?i 礽 礽 ,max specific enzyme synthesis rate (h?1 ) constitutive enzyme synthesis rate (g h?1 ) specific enzyme degradation rate (h?1 ) stoichiometric coefficient for i-th carbon substrate (?) stoichiometric coefficients for storage carbohydrate synthesis and degradation (?) specific growth rate of biomass on i-th substrate (h?1 ) maximum specific growth rate on i-th substrate (h?1 ) The pathways are not mutually exclusive and, at a given instant, the organism may follow two or more pathways at different rates. Each pathway is controlled by a key enzyme ei , with synthesis rate ui and activity vi , which follow: ri ui = (A4) rj j vi = ri max rj (A5) j With Eqns (A1) ? (A5), the mass balances for a continuous flow bioreactor may be written as follows: dX = (ri vi )-D X (A6) dt i r1 v1 r3 v3 dG = (G0 ? G)D ? + X dt Y1 Y3 dC dX ? ?4 C +X (A7) dt dt dE r1 v1 r2 v2 = ?DE + ?1 ? X (A8) dt Y1 Y2 dO r2 v 2 r3 v3 ? = kL a(O ? O) ? ?2 + ?3 X (A9) dt Y2 Y3 dei Si = ?ui dt Ki + Si ? ? (A10) ? ? (rj vj ) + ? ? ei + ? ? APPENDIX j The model of Jones and Kompala (1999) Depending on the prevailing conditions, S. cerevisiae may follow any one of three metabolic pathways. The rate of growth ri along each pathway follows modified Monod kinetics, as given below. Glucose fermentation r1 = �e1 Ethanol oxidation r2 = �e2 Glucose oxidation r3 = �e3 E K2 + E G K3 + G G K1 + G O KO2 + O O KO3 + O (A1) (A2) (A3) ? 2007 Curtin University of Technology and John Wiley & Sons, Ltd. dC = ?3 r3 v3 ? (?1 r1 v1 + ?2 r2 v2 )C dt (rj vj )C ? (A11) j Inclusion of the term ? ? in the enzyme synthesis Eqn (A10) is based on study by Turner and Ramkrishna (1988), who have shown its importance in predicting the induction of enzymes that have been repressed for long durations. The specific growth rates thus also include ? ? in the model: 礽,max + ? 礽 = 礽,max (A12) ? + ?? Equation (A11) expresses the rate of change of internal storage carbohydrates that are an integral part of the metabolism (Satroutdinov et al ., 1992; Duboc et al ., 1996). Asia-Pac. J. Chem. Eng. 2007; 2: 640?649 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering DISORDER IN BIOLOGICAL OSCILLATIONS AND THEIR IMPLICATIONS The ?i are the stoichiometric coefficients for different substrates Si , and ?i are similar coefficients for carbohydrate synthesis and consumption by the cells. Jones and Kompala (1999) may be consulted for a full discussion of the model. A point not clarified there is the identification of S1 , S2 , and S3 . Reference to Eqns (A1)?(A3) shows that S1 = G, S2 = E , and S3 = G. This identification is needed to solve the model. REFERENCES Baker SN, Kilner JM, Pinches EM, Lemon RN. The role of synchrony and oscillations in the motor output. Exp. 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