A combined metabolic/polymerization kinetic model on the microbial production of poly(3-hydroxybutyrate)

Transcription

1 RESEARCH PAPER New Biotechnology Volume 27, Number 4 September 2010 A combined metabolic/polymerization kinetic model on the microbial production of poly(3-hydroxybutyrate) Giannis Penloglou 1,2, Avraam Roussos 2, Christos Chatzidoukas 2 and Costas Kiparissides 1,2 1 Department of Chemical Engineering, Aristotle University of Thessaloniki, P.O. Box 472, Thessaloniki, Greece 2 Chemical Process Engineering Research Institute, P.O. Box 60361, Thermi, Thessaloniki, Greece Abstract In the present work, an integrated dynamic metabolic/polymerization kinetic model is developed for the prediction of the intracellular accumulation profile and the molecular weight distribution of poly(3- hydroxybutyrate) (P(3HB) or PHB) produced in microbial cultures. The model integrates two different length/time scales by combining a polymerization kinetic model with a metabolic one. The bridging point between the two models is the concentration of the monomer unit (i.e. 3-hydroxybutyryl-CoA) produced during the central aerobic carbon metabolism. The predictive capabilities of the proposed model are assessed by the comparison of the calculated biopolymer concentration and number average molecular weight with available experimental data obtained from batch and fed-batch cultures of Alcaligenes eutrophus and Alcaligenes latus. The accuracy of the proposed model was found to be satisfactory, setting this model a valuable tool for the design of the process operating profile for the production of different polymer grades with desired molecular properties. Introduction Polyhydroxyalkanoates (PHAs) are microbial polyesters produced in a variety of microorganisms, under nutrient limiting conditions, as intracellular carbon and energy storage compounds [1]. PHAs exhibit significant advantages compared to conventional polymeric materials as they are produced from renewable sources, they are non-toxic and 100% biodegradable [2]. Poly(3-hydroxybutyrate) (PHB) represents the most important member of the PHAs, since it was the first PHA discovered and is still the most studied one. It is a biopolymer with an extensive range of applications since its mechanical properties are similar to conventional commercial polymers, such as polypropylene [3]. In spite of the potential of the PHAs, their introduction to the world-wide market is currently limited due to a series of economic and engineering considerations [4]. Presently, commercially available biopolymers are significantly more expensive than their synthetic alternatives Corresponding author: Kiparissides, C. (cypress@eng.auth.gr), (cypress@certh.gr) [5] and, therefore, represent only a small portion of the total polymer volume capacity. The increased production cost of PHAs as well as their efficient separation and subsequent downstream processing are some of the problems that hinder their applicability. Therefore, there is a growing need for the development of novel microbial processes in order to maximize the overall process efficiency and reduce the total production cost [6]. To this end, advanced mathematical models can provide the means first to understand and then to control the underlying biochemical phenomena, leading to the production of biopolymers with desirable molecular and end-use properties, in a competitive way. The traditional approach for modeling a PHA-producing microbial process is based on the assumption that an unstructured, nonsegregated biophase is growing in a spatially homogeneous environment [7 9]. The specific biomass growth rate is commonly described by an empirical equation and, consequently, these so-called macroscopic models cannot predict the process behavior under a wide range of fermentation conditions. Despite the /$ - see front matter ß 2010 Elsevier B.V. All rights reserved. doi: /j.nbt

2 New Biotechnology Volume 27, Number 4 September 2010 RESEARCH PAPER importance of the central carbon metabolism, responsible for the accumulation of biopolymers in bacterial cells, only a limited number of publications have dealt with the detailed (i.e. structured) mathematical modeling of metabolic pathways in cells. The main reasons are the lack of information regarding the reaction kinetics, the metabolic regulation as well as the variations introduced by changes in the physiological conditions of the culture. Leaf and Srienc [10], presented a metabolic model for the description of the intracellular PHB synthesis in Alcaligenes eutrophus that accounted for the regulatory behavior of the cells. The authors showed that complex, mechanistic reaction rate expressions (i.e. derived from a ping-pong Bi-Bi reaction mechanism) resulted in significantly more accurate results than the simplified, irreversible Michaelis Menten kinetic model. A similar kinetic model was employed by van Wegen et al. [11] for the investigation of PHB accumulation in Escherichia coli. The authors concluded that the PHB accumulation rate was highly sensitive to parameters such as the culture s ph and the intracellular concentrations of the acetyl-coenzyme A and coenzyme A. Dias et al. [12] developed a mathematical model for the prediction of PHB and residual biomass concentrations in mixed microbial cultures based on a simplified metabolic network comprising seven reactions. In consistency with experimental observations, the model predicted that the specific PHB productivity rate increased under nitrogen-limited conditions as well as that the PHB accumulation exhibited a self-inhibiting behavior. In a follow-up work, the authors presented a segregated model, accounting for a number of cell populations characterized by different kinetic growth rates [13]. Despite the extensive studies on the metabolism of biopolymerproducing bacteria [14 16], the detailed mechanisms of polymer accumulation and the associated molecular weight distribution (MWD) are not sufficiently understood [17]. It should be noted that the MWD is, to a large extent, responsible for the end-use properties of biopolymers, including their physical, chemical, mechanical and rheological characteristics. For example, the mechanical properties of biopolymers considerably deteriorate when the weight average molecular weight (M w ) is lower than Da [18]. Moreover, for thermoplastic applications the value of M w should be higher than Da [19]. The molecular weight distribution of biopolymers is affected by a variety of variables, including the host microorganism, the substrate type and concentration, the nutritional (e.g. nitrogen source concentration) and operating conditions (e.g. ph and temperature), as well as the downstream polymer separation and processing. Typical values of the number average molecular weight (M n ) of PHB, range from to Da [3,6,20] although extreme values as high as have also been reported in mutant strains [21,22]. Due to a number of unknown factors (e.g. the lack of information on key aspects of the intracellular polymerization-degradation mechanisms) and limited number of accurate experimental measurements on MWD, only few studies have dealt with the prediction of the molecular weight of biopolymers in bacterial cultures. Bradel and Reichert [23] developed a mathematical model for the prediction of the molecular weight distribution of PHB produced in flask cultures of A. eutrophus under different ph values. The applicability of the proposed model was limited to the special case where the fructose (as carbon source) concentration remained constant, assuming in addition a constant monomer concentration. Moreover, chain transfer and polymer degradation reactions were assumed to be negligible, an assumption that does not hold true in wild-type bacteria. Mantzaris and coworkers [24,25] presented a rigorous population balance model in continuous form based on steady-state experimental data that accounted for either a constant or a chain length dependent propagation rate. However, their calculations were based on a subjective and arbitrary assumption for the average propagation rate of PHB chains (i.e. 2 monomer units/chain/s). Moreover, the kinetic model did not include a biopolymer degradation mechanism. The present paper deals with the development of an integrated metabolic/polymerization kinetic model for the prediction of the concentration and molecular weight distribution of PHB in bacterial cultures. The proposed approach combines a realistic description of cell metabolism (i.e. monomer concentration is not assumed to be constant) with a polymerization kinetic model comprising initiation, propagation, chain transfer and degradation reactions. The proposed modeling framework is developed and tested in a system of A. eutrophus utilizing fructose as carbon source; however, it can be adjusted accordingly to describe the behavior of different microbial systems. This means that the proposed approach is generic enough to be applied to any PHA production system, regardless of the choice of bacterial strain or operating conditions. Demonstrating further the applicability of the model, the latter is employed for the dynamic simulation of the Alcaligenes latus fermentative production of PHB, using sucrose as carbon source. In what follows, the metabolic mechanism for the synthesis and degradation of PHB is described and the differential equations regarding the mass conservation of the various intracellular species are derived. Subsequently, model predictions are compared with experimental measurements on PHB concentration and number average molecular weight obtained from batch and fedbatch cultures of A. eutrophus (later known as Ralstonia eutropha, Wautersia eutropha, Cupriavidus necator, etc.). Model development Depending on the bacterium type and carbon source utilized, different metabolic pathways can be established to describe the microbial production of PHAs [2]. Carbohydrates constitute a very common carbon source for the fermentative production of PHB in Alcaligenes species bacteria. The central aerobic carbon metabolism that leads to the production of PHB in Alcaligenes species bacteria is depicted in Fig. 1 [26,27]. The carbon source (e.g. a carbohydrate, fructose in this case) is initially converted into acetyl-coenzyme A (AcCoA) through the Entner Doudoroff pathway. The AcCoA is an intermediate metabolite for the synthesis of PHB via a sequence of three enzymatic reactions. In the first reaction, two AcCoA molecules are condensed by the catalytic action of the enzyme 3- ketothiolase (phaa) to form one molecule of acetoacetyl-coenzyme A (Ac-AcCoA). Subsequently, Ac-AcCoA is reduced by acetoacetyl-coa reductase (phab) to 3-hydroxybutyryl-CoA (3- HBCoA) at the expense of NADH. In the third enzymatic reaction, the monomer unit 3-HBCoA is polymerized into PHB following a polymerization mechanism catalyzed by synthase (phac). Finally, under the action of depolymerase (phaz) the accumulated PHB is hydrolyzed into 3-hydroxybutyrate (3-HB). Subsequently, 359

3 RESEARCH PAPER New Biotechnology Volume 27, Number 4 September 2010 FIG. 1 Metabolic pathway for the intracellular synthesis and degradation of PHB in Alcaligenes species. 3-hydroxybutyrate is converted to AcCoA which is utilized as carbon and energy source under carbon starvation conditions (i.e. substrate depletion). Notice that while a simplified pathway of the central aerobic carbon metabolism of A. eutrophus is considered here, any other metabolic pathway belonging to a different strain can be employed. By the selection of the appropriate detailed metabolic model it is possible to apply the proposed mathematical framework to different microbial systems or/and different carbon sources (e.g. carbohydrates or fatty acids). The last two enzymatic steps described above, can be further analyzed into a set of comprehensive reactions catalyzed by the presence of PHA synthase and depolymerase that control the molecular formation and degradation of PHB polymer chains (see Fig. 2). The above kinetic mechanism is representative of both micellar and budding models have been identified in bacteria regarding the formation of PHB granules. In both theories, two PHA synthase molecules, via their two thiol groups (i.e. active sites), derived from two cysteine residues of the enzyme subunits, form a homodimer with two catalytic active sides (i.e. two thiol groups) [28 30]. The catalytic mechanism of the PHB synthesis is initiated by the addition of a monomer (3-HBCoA) molecule to one of the two active sites. Chain propagation occurs via the reaction of the anchored PHB chain on one thiol group of a synthase dimer with to the 3- HBCoA molecule, which is bound to the other active site of the same homodimer. Active polymer chains undergo a chain transfer reaction to an agent (X) (e.g. water in this case or an enzyme with a water molecule in its active from) resulting in the formation of an inactive (dead) polymer chain with simultaneous release of a synthase molecule. Finally, inactive polymer chains may undergo chain-end degradation catalyzed by depolymerase. The cyclic nature of the PHB metabolism (i.e. simultaneous accumulation and turnover) in bacterial cells under nitrogen limitation conditions has been demonstrated by Doi et al. [31] and Taidi et al. [32]. Notice that by appropriate modifications to the polymerization kinetic mechanism, it is possible to apply the proposed framework to systems with more than one carbon sources in order to predict the formation of copolymer chains. In the present study, the following polymerization depolymerization kinetic scheme was employed based on the work of Kawaguchi and Doi [33]: Initiation E-SH þ M #! k m1 E-SH-M #! k i Propagation P 1 -ES þ CoA-SH (1) P n -ES þ M #! k m2 P n -ES-M #! k p P nþ1 -ES þ CoA-SH (2) Chain Transfer P n -ES þ H 2 O! kt D n þ E-SH (3) Degradation D n þ E-OH! k d D n 1 þ D 1 þ E-OH (4) where E-SH, M #, CoA-SH and E-OH denote the concentrations of synthase dimer, monomer coenzyme A complex (M-SCoA), coenzyme A and depolymerase, respectively. Furthermore, P n -ES (P n ), P n -ES-M # (P n ) and D n are the corresponding concentrations of active, intermediate and inactive polymer chains with a degree of polymerization equal to n. In contrast to the original polymerization model proposed by Kawaguchi and Doi [33], initiation is assumed to occur in two steps with the formation of an intermediate synthase monomer complex (E-SH-M # ). Similarly, a two-step reaction is also considered for polymer chain propagation where an intermediate active polymer monomer complex (P n - ES-M # ) is initially formed. In the following mathematical model developments, it is assumed that the polymerase (PhaC), depolymerase (PhaZ) and chain transfer agent concentrations are constant throughout the course of polymerization [33,34]. Since the enzyme activity is known to depend on the total cell concentration [35], the above 360

4 New Biotechnology Volume 27, Number 4 September 2010 RESEARCH PAPER FIG. 2 Detailed kinetic mechanism of the intracellular polymerization-degradation of PHB in Alcaligenes species. assumption will be realistic when the concentration of residual biomass remains constant (e.g. under nitrogen limitation conditions). Moreover, it is considered that the rate-limiting step in the degradation mechanism is the binding of the inactive polymer chain to depolymerase. Finally, the effects of population heterogeneity (i.e. segregation of cells) and mass-transfer limitation phenomena are not taken into account. Based on the postulated kinetic scheme and model assumptions, the net production rates for the various intracellular molecular species will be given by the following equations: Active polymer chains of length n d½p n Š ¼ k i E-SH-M # dðn 1Þ km2 ½P n Š M # þ k p P n 1 Hðn 1Þ k t ½P n Š n ¼ 1; 2; :::; 1 (5) Intermediate polymer chains of length n d P n ¼ k m2 ½P n Š M # k p P n n ¼ 1; 2; :::; 1 (6) Inactive polymer chains of length n 361

5 RESEARCH PAPER New Biotechnology Volume 27, Number 4 September 2010 d½d n Š Monomer d M # ¼ k t ½ P n Š k d½ D nšþk d½ D nþ1š n ¼ 2; 3; :::; 1 (7) ¼ J M ðþ k t X 1 m1 M# km2 M # n¼1 ½P n Š (8) Synthase Monomer complex d E-SH-M # ¼ k m1 M# ki E-SH-M # (9) where k m1 ¼ k m1½e-shš, k d ¼ k d½e-ohš and k t ¼ k t½h 2 OŠ. The Kronecker delta function, d(x), and the Heaviside step function, HðxÞ, are defined by the following equations: 1; if x ¼ 0 dðxþ ¼ (10) 0; otherwise 1; if x > 0 HðxÞ ¼ (11) 0; if x 0 Finally, J M ðtþ denotes the monomer production rate (flux) from upstream metabolic steps. Notice that polymer chains with length n equal to one (i.e. 3-hydroxybutyrate) are assumed to be instantaneously transformed to AcCoA and, therefore, are not included in the respective population balance (i.e. Eq. (7)). The system of differential Eqs. (5) (9) can be integrated in time provided that the monomer production rate term, J M ðtþ, is known. This rate can be calculated by either a kinetic model of cell metabolism or by metabolic flux analysis. In general, J M ðtþ will depend on the metabolic pathway, the assimilated carbon source and, moreover, the vectors k, Y, C and J denoting the reaction kinetic constants, the yield coefficients and the upstream metabolite concentrations and fluxes, respectively: J M ðþ¼g t ðk; Y; C; J; tþ (12) It should be noted that the number of population balance equations for the active, intermediate and inactive polymer chains in Eqs. (5) (7) depend on the maximum degree of polymerization that typically ranges from 10 6 to Consequently, the computational effort associated with the solution of the complete set of differential equations becomes prohibitively high even for the contemporary high-end processors. To deal with this limitation and reduce the dimensionality of the problem a numerical method (i.e. the fixed pivot technique) was employed. Following the developments of Kumar and Ramkrishna [36], polymer chains can only exist at predefined discrete chain lengths x i, called pivots. If a polymer chain is formed at a length u between two pivots then it is assigned to the two adjacent pivots with appropriate fractions so that any two moments of the chain length distribution are conserved. A more detailed description of the fixed pivot technique is presented in Appendix A and in the original work of Saliakas et al. [37]. In this work, a mixed uniform-logarithmic discretization rule was adopted so that the number of continuous-discrete differential equations per polymer chain population was reduced to Results and discussion The proposed metabolic/polymerization kinetic model of this study was validated against experimental data reported by Kawaguchi and Doi [33] that correspond to the fermentation of Alcaligenes eutrophus H16 (ATCC 17699), in non-growth conditions. According to that work, a two-stage cultivation profile was employed. Firstly, the bacterial cells were grown in a nutrient-rich medium without accumulating any amount of PHB. Subsequently, once a high-density cell population was obtained, cells were harvested and transferred into a nitrogen source free mineral medium that favored the PHB accumulation, while the residual biomass remained constant. In the case that the biomass concentration is constant, the monomer production rate, J M ðtþ, can be assumed to be proportional to the fructose consumption rate, J F ðtþ: J M ðþ¼y t M=F J F ðþ t (13) where Y M=F (mol of monomer produced/mol of fructose consumed) is a monomer to substrate yield coefficient. The latter is considered constant throughout the second stage of cultivation. The kinetic rate constants (i.e. k i, k m1, k m2, k p, k t, k d ) and the yield coefficient Y M=F appearing in the system of differential and algebraic Eqs. (5) (13) were estimated based on a set of experimental data of Kawaguchi and Doi [33], using a general non-linear parameter estimator (M. Caracotsios, Model parametric sensitivity analysis and nonlinear parameter estimation. Theory and applications, PhD thesis, University of Wisconsin, Madison, WI, 1986). The values of the estimated parameters as well as their respective 95% confidence intervals are reported in Table 1. Note that the values of these parameters are strongly affected by the metabolic pathway and the experimental conditions (i.e. the Entner Doudoroff pathway occurring under non-growth conditions due to nitrogen limitation). Subsequently, the model was employed for the calculation of the PHB concentration, MWD and M n for three different cases, specifically: (i) a batch culture with an initial fructose concentration S F (0) = 5 g/l, (ii) a batch culture with an initial fructose concentration S F (0) = 10 g/l and (iii) a fedbatch culture with initial fructose concentration S F (0) = 5 g/l and a pulse feeding of 5 g fructose per liter of culture volume at time t =24h. Batch fermentation: cases (i) and (ii) The dynamic evolution of the fructose concentration for cases (i) and (ii) is depicted in Fig. 3. As can be seen, the fructose concentration decreases linearly with time, at a rate equal to J F 0:85 g=l=h, independently of its initial concentration. As a result, the carbon source is completely depleted in 6 and 12 h (twofold time in case (ii)), respectively. In Fig. 4, the model predictions are compared with experimental measurements on PHB production. Apparently, there is a good agreement between TABLE 1 Point estimates and 95% confidence limits for the parameters of the polymerization depolymerization model. Parameter Point Estimate 95% confidence interval k i (h 1 ) k p (h 1 ) kt (h 1 ) km1 (h 1 ) k m2 (l/mol/h) kd (h 1 ) Y M=S

6 New Biotechnology Volume 27, Number 4 September 2010 RESEARCH PAPER FIG. 3 fructose concentration for cases (i) and (ii). FIG. 5 Dynamic evolution of the molecular weight distribution of PHB for case (i). model and experimental results. Notice that during the feast phase (i.e. the first 6 and 12 h of cultivation, for cases (i) and (ii), respectively), the rate of polymer accumulation is approximately independent of the initial fructose concentration. However, by the time that the fructose concentration has been depleted, the monomer production is terminated and the PHB concentration starts decreasing due to the dominant action of the PHB depolymerase. The inactive monomer (3-hydroxybutyrate), produced by the PHB degradation reaction, is rapidly transformed into AcCoA. It is pointed out that the maximum PHB concentrations, achieved in both cases at the time point where the carbon source is depleted, are 1.50 and 3.10 g/l, resulting in a polymer to fructose yield, Y P/S, equal to and g/g, respectively. The predicted molecular weight distributions of PHB for the two different initial fructose concentrations are depicted in Figs. 5 and 6. As can be seen, the MWDs evolve rapidly to a maximum peak value (i.e. at 7 h and 13 h for cases (i) and (ii), respectively) that corresponds to the respective maximum value for the number average molecular weight, M n. From this point onward, the peak value of the MWD decreases reflecting the respective decreases in the PHB concentration and M n.infig. 7, model predictions are compared with experimental measurements for M n. Apparently, for both cases there is a satisfactory agreement of model results with the corresponding experimental data. Notice that for case (ii), the final value of M n is slightly higher than the respective one for case (i) (i.e. approximately 640,000 and 850,000 g/mol, respectively). This is attributed to the early cease of the molecular development of the polymer chains in case (i), due to the faster monomer exhaustion under lower initial fructose concentration. Moreover, the predicted maximum values of M n (i.e. approximately 900,000 and 950,000 g/mol) appear in both cases when the fructose concentration is depleted. FIG. 4 PHB concentration for cases (i) and (ii). FIG. 6 Dynamic evolution of the molecular weight distribution of PHB for case (ii)

7 RESEARCH PAPER New Biotechnology Volume 27, Number 4 September 2010 FIG. 7 PHB number average molecular weight of PHB for cases (i) and (ii). FIG. 9 PHB concentration in case (iii). Fed-batch fermentation: case (iii) The time evolution of fructose concentration for case (iii) is depicted in Fig. 8. Note that the consumption rate of both the initially loaded fructose and the fructose pulse (occurred after the complete consumption of the first one) is the same in the batch case (i.e. equal to 0.85 g/l). After the second pulse, the concentration of fructose reached its initial value (5 g/l) and once again the consumption rate was identical. In Fig. 9, a good agreement between the model predictions and the experimental measurements on the PHB concentration is depicted. The maximum amount of PHB, 2.55 g/l, is attained at about 7 h after the introduction of the fructose pulse, corresponding to a polymer to fructose yield, Y P/S, equal to g/g. It should be noted that the PHB concentration in this case did not increase beyond the maximum value attained in case (ii), since the total PHB accumulation for a given amount of substrate depends on the amount of the residual biomass. Moreover, the bacterial culture remained in famine conditions for approximately 17 h, where the accumulated PHB was intracellularly consumed (around 0.3 g/l) as carbon and energy source for maintenance. The dynamic evolution of the respective molecular weight distribution is presented in Fig. 10. As can be seen, the MWD reaches an initial maximum peak value at about t = 7 h. At that time, the substrate concentration is approximately equal to zero. Subsequently, polymer degradation dominates over the development of the polymer chains. However, after the introduction of the fructose pulse feeding at t = 24 h, the polymer chains start growing again and the MWD reaches a new peak value at, approximately t = 31 h. As can be seen in Fig. 11, the maximum value for M n, 950,000 g/mol, is attained at about t = 7 h. Notice that this maximum value is not exceeded after the introduction of the second fructose pulse and is almost the same to the maximum M n value obtained under batch fermentation conditions (see case (ii)). FIG. 8 fructose concentration in case (iii). FIG. 10 Dynamic evolution of the molecular weight distribution of PHB for case (iii)

8 New Biotechnology Volume 27, Number 4 September 2010 RESEARCH PAPER the metabolic pathway toward the PHB production starts from sucrose. The mathematical model was expanded to account for the simultaneous growth of the residual biomass in parallel with the PHB accumulation and the consumption of ammonium sulfate, while the part of the model that describes the polymerization kinetics remained the same. In Fig. 12(a), the dynamic evolution of the nutrient concentrations (i.e. sucrose and ammonium sulfate) and the respective production of the residual biomass are presented. Along the time evolution of the culture, the PHB accumulation rate and the built up of the PHB are shown in Fig. 12(b). The model parameters were properly tuned to make the model consistent with the new conditions. The excellent agreement of the model predictions with the respective experimental measurements proves the general applicability of the model to various PHAs production systems. FIG. 11 PHB number average molecular weight for case (iii). Application to other PHA production systems In order to make evident that the present modeling approach can be efficiently applied to other microbial systems under different conditions, the proposed model was revised for the growth-associated PHB production by Alcaligenes latus and tested against experimental data produced in a 2-l batch culture (BioFlo 110- New Brunswick Scientific). The fermentation conditions are described in the original work of Wang and Lee [38]. In this case, Conclusions An integrated metabolic/polymerization model was developed for the prediction of the PHB concentration and molecular weight distribution in bacterial cultures. A cell metabolism model was employed to calculate the monomer unit production rate from the carbon substrate. The calculated rate was then employed as input to the polymerization kinetic model to predict the PHB concentration and MWD. The combined model was applied to the investigation of Alcaligenes species fermentation under different operating policies. It was found that the fed-batch operating policy did not result in an increase of either the total PHB concentration or the corresponding maximum value of M n, as compared to the batch operation for the same total fructose consumption. This behavior can be explained by the absence of residual biomass growth. In view of the above conclusions and the model-based interpretation of the experimental results, it is evident that the present metabolic/kinetic modeling approach can be further improved to capture additional phenomena of different length and time scales in order to successfully address the problem of the optimal operation of the microbial processes for the production of desired PHA grades. Acknowledgements This work was carried out with the financial support of EC under the IP-project titled Sustainable Microbial and Biocatalytic Production of Advanced Functional Materials (BIOPRODUCTION/NMP-2-CT ). Appendix A In the present work the fixed pivot technique (FPT) of Kumar and Ramkrishna [36] was properly adapted for solving Eqs. (5) (7). Assuming that the active, intermediate and inactive polymer chain concentrations h i remain constant in the discrete chain length domain x j 1 ; x j, one can define the following lumped active, intermediate and inactive polymer chain concentrations, P j, P j and D j, corresponding to the j element: FIG. 12 Comparison of model predictions with experimental measurements of (a) sucrose, ammonium sulfate and residual biomass concentrations and (b) PHB concentration and number average molecular weight for the cultivation of Alcaligenes latus. P j ¼ Z x j x j 1 P n ½ Šdx; P j ¼ Z x j x j 1 Z P x j n dx; D j ¼ ½ Šdx (A1) x j 1 D n Accordingly, the chain length domain is discretized into a number of nt node points (pivots) using a logarithmic discretization rule

9 RESEARCH PAPER New Biotechnology Volume 27, Number 4 September 2010 If the length of a new polymer chain, u, lies in a position between two pivots, it is assigned to the two adjacent pivots with the appropriate weights so that two moments of the chain length distribution (i.e. number and mass of polymer chains) are conserved. Notice that polymer chains formed via chain initiation and chain transfer reactions always correspond to the defined node points. From the application of the FPT to Eqs. (5) (7), the following system of continuous-discrete differential equations is obtained: Lumped molar balance equations for the active polymer chains dp j ¼ k i½e-sh-m # Šdð j 1Þ k m2 P j ½M # Šþk p Hð j 1Þ Xj j ¼ 1; 2;...; nt k¼1 P k A j;k k t P j (A2) Lumped molar balance equations for the intermediate polymer chains dp j ¼ k m2p j M # k p P j j ¼ 1; 2;...; 1 (A3) Lumped molar balance equations for the inactive polymer chains dd j ¼ k t P j k d D j þ k d X jþ1 k¼ j D k B j;k j ¼ 2; 3;...; nt (A4) The matrices A j;k and B j;k are defined as follows: 8 x jþ1 x t >< x j x t x jþ1 x A j;k ¼ jþ1 x j x t x j 1 where x t ¼ x k þ x 1 (A5) >: x j 1 x t x j x j x j 1 8 x jþ1 x t >< x B j;k ¼ jþ1 x j x t x j 1 >: x j x j 1 x j x t x jþ1 x j 1 x t x j where x t ¼ x k x 1 (A6) The total polymer chain length distribution of polymer chains is calculated from the sum of the individual active, intermediate and inactive polymer chain distributions: T j ¼ P j þ P j þ D j (A7) Finally, the number average molecular weight was calculated by the following equation: P nt j¼1 M n ¼ x! jt j P nt j¼1 T MW j where MW is the molecular weight of the repeating unit. 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