Experimental and theoretical analysis of poly (-hydroxybutyrate ...
Biochemical Engineering Journal 55 (2011) 49–58
Contents lists available at ScienceDirect
Biochemical Engineering Journal journal homepage: www.elsevier.com/locate/bej
Experimental and theoretical analysis of poly(-hydroxybutyrate) formation and consumption in Ralstonia eutropha André Franz a,d,∗ , Hyun-Seob Song b , Doraiswami Ramkrishna b , Achim Kienle a,c,d a
Max Planck Institute for Dynamic of Complex Technical Systems, 39106 Magdeburg, Germany School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, USA Otto-von-Guericke University, Chair of Automation/Modeling, 39106 Magdeburg, Germany d Magdeburg Centre for Systems Biology (MaCs), Magdeburg, Germany b c
a r t i c l e
i n f o
Article history: Received 1 July 2010 Received in revised form 28 February 2011 Accepted 18 March 2011 Available online 29 March 2011 Keywords: Ralstonia eutropha Cybernetic modeling Bifurcation analysis Biopolymers Metabolic regulation Elementary mode
a b s t r a c t In this paper a mathematical model is presented to describe poly(-hydroxybutyrate) (PHB) formation and consumption in Ralstonia eutropha. The model is based on the hybrid cybernetic modeling approach, which was introduced by Kim et al. [1] and which allows a systematic derivation of the model equations from elementary mode analysis. An extension of this approach is presented to allow for non quasistationary metabolites, i.e. PHB. The model is shown to be in good agreement with experimental data for PHB formation and consumption. The model is used afterwards to discuss the occurrence of multiple steady states in a continuous bio reactor. It is shown that the multiplicity region predicted by the model is rather small and it is argued that multiple steady states are therefore unlikely to occur in practice for this specific system. Due to various desirable features such as accounting for cellular regulation at network level and dynamics of intracellular metabolites with a moderate complexity, it is believed that the constructed model is most suitable for control, optimization and monitoring of industrial PHB production processes. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Poly(-hydroxybutyrate) (PHB) is an organic polymer, which can be synthesized by many microorganisms and which serves as internal energy and carbon reserve material. Ralstonia eutropha is a well known bacterium for producing PHB [2]. It can accumulate PHB to more than 80% of its cell dry weight [3]. R. eutropha has been re-classified several times in the past. The history of classification includes the genera Hydrogenomonas, Alcaligenes, Ralstonia, Wautersia and recently Cupriavidus necator. To avoid confusion, we will use the name Ralstonia throughout, since it is still commonly used in the literature, including very recent articles. Production of PHB is favored under limitation of key nutrients such as nitrogen, phosphate or oxygen. PHB belongs to the group of polyhydroxyalkanoates (PHA) and provides an attractive source of bioplastics that are biodegradable, biocompatible and do not depend on fossil resources. PHB production in R. eutropha has previously been modeled by a few researchers. These models can be divided into two classes:
∗ Corresponding author. E-mail address: [email protected] (A. Franz). 1369-703X/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.bej.2011.03.006
(a) models which do not consider internal regulation, e.g. [4,5] and (b) models which do consider cell internal regulation. Internal regulation can be included into modeling by the cybernetic framework, which was introduced by Ramkrishna and co-workers [6–8]. The cybernetic approach for modeling PHB production in R. eutropha was used by Yoo and Kim [9]. They have used a very simple unstructured model and compared their results with unstructured non-cybernetic models of Mulchandani et al. [5] and Asenjo and Suk [4]. Their model could successfully predict PHB production, but did not include the underlying metabolic processes. Gadkar et al. [10] addressed this discrepancy and used a structured cybernetic model to develop a model predictive control for continuous PHB production. Although they considered the metabolic pathways by which the carbon and nitrogen sources are utilized these pathways are still lumped. The cybernetic model by Pinto and Immanuel [11] is also based on a very simplified metabolic network with less complexity than the model of Gadkar et al. [10] and was used for bifurcation analysis. It was shown, that the model structure admits multiple steady states in a continuous bio reactor, depending on the parameter values. All these models are based on a more or less simplified metabolic network and either neglect PHB consumption or have fitted parameter to experimental data which do not contain any significant PHB consumption. But since PHB is an internal storage material, it is
50
A. Franz et al. / Biochemical Engineering Journal 55 (2011) 49–58
appropriate to consider its metabolism as a cycle of synthesis and consumption. In the current work the PHB consumption is included into a cybernetic model and experiments were performed which show significant consumption to gain more reliable parameters. Additionally an expanded metabolic network is considered to include more internal metabolic information. The model is based on the state of the art hybrid cybernetic approach (HCM) by Kim et al. [1]. The HCM approach allows a systematic derivation of the model equations from elementary mode analysis [12]. It is based on quasistationarity of internal metabolites, which are eliminated from the model equations. However, this approach is not suitable for PHB production, since PHB is an internal metabolite, which has to be included explicitly into the model equations. In contrast to this, the more detailed approach by Young et al. [13] takes the dynamics of internal metabolites into account and could also handle synthesis and consumption of PHB. But this approach is computationally more expensive and requires detailed information about internal kinetics, which might be hard to obtain in particular for larger networks. To overcome these limitations this article presents a compromise between the standard HCM and the cybernetic modeling approach by Young et al. [13], by taking the dynamics of a few internal metabolites explicitly into account, while for most of the internal metabolites the quasi-steady state approximation is still applied. It is shown that the model is in good agreement with experimental observations, where PHB formation and consumption are stimulated separately. In industrial production usually fed batch or continuous processes are used. While fed batch processes allow for higher biomass and product yield, a continuous process can also be advantageous due to longer periods of operating time, which can reduce production costs. However, in continuous operating mode metabolic reactions will occur simultaneously, which can lead to nonlinear phenomenas, e.g. oscillations [14,15], multiple steady states [16–18], etc. The occurrence of multiple steady states in a PHB production process with R. eutropha was discussed by Pinto and Immanuel [11] based on a simplified model. Following their idea, the model is therefore afterwards used to investigate the possibility of multiple steady states in a continuous bio reactor. It is shown that multiple steady states are unlikely to occur in practice for this specific system. Furthermore, the influence of PHB consumption is analyzed, which is usually of minor importance in a batch process, since there is usually sufficient carbon source available and PHB will not be consumed. But in continuous processes PHB consumption can have crucial influence, as shown in this study. Even in a fed batch process PHB consumption can be of great importance if other growth essential nutrients than carbon are fed to the fermenter and PHB consumption will be stimulated. It is therefore necessary to include PHB consumption into modeling, if a fed batch or continuous process is used.
2. Material and methods 2.1. Experimental section 2.1.1. Microorganisms and medium The organism used throughout this study, R. eutropha (DSM 428, ATCC 17699, NCIB 10442) was obtained from DSMZ GmbH Braunschweig, Germany, as vacuum dried culture. The strain was cultivated with the medium given in Table 1. All chemicals were from Carl Roth GmbH (Karlsruhe, Germany).
Table 1 Used medium for cultivation. Ingredient
Concentration
Fructose NH4 Cl KH2 PO4 Na2 HPO4 · 2 H2 O MgSO4 · 7 H2 O CaCl2 · 2 H2 O Fe(NH4 ) citrate Trace element solution a
20.0 g/L 1.50 g/L 2.30 g/L 2.90 g/L 0.50 g/L 0.01 g/L 0.05 g/L 5.00 ml/L
a Trace element solution (g/L): ZnSO4 · 7 H2 O 0.10, MnCl2 · 4 H2 O 0.03, H3 BO3 0.30, CoCl · 6 H2 O 0.20, CuCl2 · 2 H2 O 0.01, NiCl2 · 6 H2 O 0.02, NaMoO4 · 2 H2 O 0.03.
2.1.2. Cultivation conditions R. eutropha was grown heterotrophically in a 7 L fermenter (Biostat C, Sartorius, BBI Systems, Melsungen, Germany) with a 5 L working volume. The temperature was kept constant at 30 ◦ C and the pH was automatically maintained at pH 6.8 by adding 2 M NaOH as corrective agent. The culture broth was agitated at 400 rpm and dissolved oxygen was maintained above 50 % air saturation by changing the oxygen and nitrogen flow rate mixed with an air stream. Total flow rate was 1.5 L/min. 2.1.3. Analytical procedures Cell growth was monitored by measuring the optical density at 600 nm using a Ultrospec 500 spectrophotometer (GE Healthcare, Buckinghamshire, UK). For dilution, NaCl (0.98 % (w/v)) was used when necessary. For the determination of the cell dry weight, 3 × 10 ml of the culture broth were centrifuged in pre-weighted glass tubes for 15 min at 3000 × g. The pellets were washed in 0.98 % NaCl and subsequently dried in a freeze-dryer (Christ, Osterode am Harz, Germany). PHB content was measured as crotonic acid, formed by acid depolymerization of PHB according to Law and Slepecky [19]. Cell pellets, harvested by centrifugation, were dissolved in methylene chloride by rapid mixing and afterwards boiled for 10 min. After the samples were cooled down, they were centrifuged at 3000 × g for 15 min and the supernatants were carefully removed and collected in glass tubes. This procedure was repeated three times. The supernatants were then evaporated and the remaining PHB-containing samples were digested in 2 ml H2 SO4 (96 %) at 100 ◦ C for 30 min and subsequently diluted with concentrated H2 SO4 . UV absorbance spectra were measured with an UV–vis spectrophotometer V-560 (Jasco, Gross-Umstadt, Germany). The concentration of crotonic acid was calculated from a set of reference standards. Ammonium chloride concentration in culture supernatants broth was measured by determining NH3 with a VITROS DT60 II Chemistry System and VITROS NH3 MicroSlide from Ortho-Clinical Diagnostics (Neckargemünd, Germany) using the manufacturers instructions. Concentrations of fructose in supernatants were determined with a D-Glucose/D-Fructose test kit from R-Biopharm (Darmstadt, Germany) using the manufacturers recommended procedure. 2.2. Hybrid cybernetic model 2.2.1. Basic formulation The state y of a metabolic system can be described by the vector of external biochemical species concentrations x, the vector of specific internal biochemical species concentrations m and the biomass concentration c
y=
x m c
(1)
A. Franz et al. / Biochemical Engineering Journal 55 (2011) 49–58
where x can be divided into the vector xs of ns substrates and the vector xp of np products. The model equations for a batch reactor are then defined as 1 dxs = Ss r c dt
(2)
1 dxp = Sp r c dt
(3)
dm = Sm r − m dt
(4)
1 dc = c dt
(5)
where r is the vector of nr regulated fluxes, is the growth rate, Ss , Sp and Sm represent the (ns × nr ), (np × nr ) and (nm × nr ) stoichiometric matrices. In the hybrid cybernetic modeling approach, which was first suggested by Kim et al. [1] and more elaborated by Song and Ramkrishna [20] and Song et al. [21] it is assumed that internal metabolites are in quasi-steady state, e.g. dm = 0. dt
(6)
If in addition m is negligibly small Sm r = 0,
r≥0
(7)
is obtained from Eq. (4). A set of elementary modes (EMs) is then uniquely defined such that non negative combinations of these EMs are spanning the solution space satisfying Eq. (7), i.e. r = ZrM
(8)
where Z is the (nr × nz ) matrix in which the i th column represents the i th elementary mode and rM is the vector of fluxes through these modes. Eqs. (2)–(4) are replaced by 1 dxs = Ss Z rM c dt
(9)
1 dxp = Sp Z rM . c dt
(10)
The matrix Z can be calculated by using standard software for metabolic flux analysis of cellular networks, e.g. Metatool [22] or CellNetAnalyzer [23]. 2.2.2. Incorporation of intracellular dynamics into HCM In the original formulation of HCM it is assumed that all internal metabolites are in quasi-steady state, since usually intracellular fluxes are characterized by relatively fast dynamics in comparison to exchange fluxes. However, for some internal metabolites this assumption does not hold, e.g. internal storage materials like PHB, membranes or other cell compartments. Therefore the vector of internal metabolite concentrations m has to be divided into the vector of internal metabolite concentrations with fast dynamics mf and the vector of internal metabolite concentrations with slow dynamics ms . Similarly, the matrix Sm is divided into Sm,f and Sm,s and Eq. (4) can be rewritten to d dt
ms mf
=
Sm,s Sm,f
r−
ms mf
(11)
Only for metabolites with fast dynamics the quasi-steady state assumption holds, e.g. dmf =0 dt
(12)
and therefore Sm,f r = 0,
r ≥ 0.
In the present case, PHB is considered as internal metabolite with slow dynamics. EMs are then calculated from Eq. (13) by setting metabolites with slow dynamics as external in Metatool or unbalanced in CellNetAnalyzer. One of the main advantages of HCM in comparison to the cybernetic modeling approach of Young et al. [24] is, that the use of local cybernetic variables is circumvented, since the quasi-steady state approximation for intracellular components allows the calculation of internal fluxes relative to uptake rates through exploitation of the stoichiometric coupling. Especially the application to large networks is much less computationally demanding by using HCM instead of the cybernetic model formulation by Young et al. [24]. However, this advantage has to paid for by the quasi-steady state assumption for all internal metabolites. In contrast, the approach of this article allows to include internal metabolites for which quasisteady state assumption does not hold, but still gets along with only global control. It is obvious that this approach lies between the formulation of Young et al. [24] and original HCM, but combines the advantages from both formulations. 2.2.3. Rate equations and cybernetic variables The vector rM of fluxes through EMs is controlled by the vector of cybernetic variables v rM = diag(v) diag(erel ) rkin M
(14)
and catalyzed by a vector of key enzymes e de = ˛ + rEM − diag(ˇ)e − e dt
(15)
where eirel =
ei eimax
˛i + ke,i
eimax =
with
ˇi + max i
.
(16)
˛ is the vector of constitutive enzyme synthesis rates [25], rEM is the vector of inducible enzyme synthesis rates, the term diag(ˇ)e represents enzyme degradation, the term e dilution by growth and max is the maximal growth rate of the i th EM. i The vector of inducible enzyme synthesis rates rEM is controlled by the vector of cybernetic variables u rEM = diag(u) rkin EM
(17)
and rkin and rkin are the vectors of unregulated rates which are M EM represented by kinetic equations of Monod-type kin max = kr,i rM,i
j ∈ L(i) kin rEM,i = ke,i b
xs,j
(18)
Ki,j + xs,j
j ∈ L(i)
xs,j Ki,j + xs,j
(19)
where L(i) is the set of component indices associated to the i th EM, max is the rate constant and k is the enzyme synthesis rate conkr,i e,i stant of the i th mode, Ki,j is the saturation constant of the i th mode for the j th educt and b represents the fraction of biomass ascribed to the enzyme synthesis machinery. Usually, it is assumed that the fraction of catalytically active biomass is constant, and hence, the factor b is just included into ke [21,13]. However, in the approach introduced in this study b is not a constant, since PHB can form more than 80% of the dry weight. The factor b can then be defined as b=1−
(13)
51
i
(ms,i )
(20)
52
A. Franz et al. / Biochemical Engineering Journal 55 (2011) 49–58
Fig. 1. Used metabolic network for Ralstonia eutropha. Growth precursors are underlined and external metabolites are boxed.
where i (ms,i ) is the biomass fraction, which includes all internal metabolites with slow dynamics and are assumed not to contain any proteins and nucleid acids. Note, that this is only true, if constitutive enzyme synthesis rate is much slower than inducible synthesis rate. Otherwise, a high basal enzyme level will catalyze fluxes through EMs although fraction of catalytically active biomass (b) is very low. A general form of the cybernetic control laws [24]u and v is given by u=
p , p1
v=
p p∞
(21)
where p is the return on investment (ROI) which can be calculated from a metabolic objective function. Often used metabolic objective functions are maximizing growth or maximizing substrate uptake. In this study it is assumed that the organism maximizes carbon source uptake and p can therefore be defined as [20] p = diag(fc ) diag(erel ) rkin M
(22)
is the vector where fc is the vector of uptaken carbon units and rkin M of pure kinetic rate expressions. Note, that in this case the stoichiometry matrices have to be normalized w.r.t. to the preferred carbon source and all rates are expressed as carbon uptake rates. 2.3. HCM for R. eutropha 2.3.1. Metabolic network Based on the metabolic network used by Katoh et al. [26] a network for R. eutropha was developed which includes a fructose uptake pathway, a more detailed penthose-phosphate-pathway and a PHB consumption pathway, see Fig. 1. ATP, ADP, CO2 and O2 are not shown in the figure, but are included in the network. All reaction equations can be found in A. Note, that biomass is divided into two compartments, namely PHB and non-PHB biomass. The latter one is represented by the abbreviation BIO. 2.3.2. Metabolic yield analysis From the given metabolic network 122 elementary modes have been calculated by using Metatool. However, it is not necessary to use all elementary modes, but a smaller subset which is essential for describing and predicting observed metabolic behavior. For
reducing all EMs to a smaller subset metabolic yield analysis (MYA) developed by Song and Ramkrishna [20] was used. Since internal metabolites with slow dynamics act as substrates and products one has to ensure that the yield space is convex. This was done by dividing all EMs into subgroups, where one group contains the EMs where the internal metabolite do not act as product (M−) and another group where the internal metabolite do not act as substrate (M+). These subgroups are convex in yield space and both groups include the EMs which do not produce or consume the internal metabolite. In this study PHB is the internal metabolite and although there are modes which also use PHB as substrate it is assumed that fructose is the preferred substrate and all EMs are normalized w.r.t. fructose. Hence, the yield space becomes three dimensional (YPHB/FRU , YBIO/FRU , YAMC/FRU ). However, due to the stoichiometry of the network the concentration of the non-PHB biomass compartment BIO is linearly correlated with the uptaken nitrogen source AMC. Therefore the yield space reduces to only two dimensions (YPHB/FRU , YBIO/FRU ). For each convex group the MYA was performed separately. At first so called generating modes (GM) were chosen from both groups. These are the vertices of the convex hull of the yield space. Group (M+) has five GMs and group (M−) has eight GMs. Since experimental data are available the number of GMs was further reduced to so called active modes (AM). This was done by minimizing the sum of squared weights 1 min h22 h 2
(23)
such that Zy h − ym = 0,
h ≥ 0,
h = 1
(24)
for the experimental data, which are inside the convex hull, where ym is the vector of measured yield data and Zy contains normalized GMs. For the experimental data, which are outside the convex hull, the AMs were calculated by solving the following least squares problem 1 min Zy h − ym 22 h 2
(25)
such that h ≥ 0,
h = 1
(26)
A. Franz et al. / Biochemical Engineering Journal 55 (2011) 49–58
53
Fig. 2. Yield spaces of set M− and M+ with experimental data (, , ) and chosen modes (•) for hybrid cybernetic model.
Fig. 2 shows the two dimensional yield space, the convex hulls of both groups (M+) and (M−), and the computed AMs. The AMs are then used for the HCM presented in this study. The experimental yield data YI , YII and YIII represent the yield of the three experimental stages as shown in Fig. 3. Maintenance (stage IV) is not considered in this study. The stoichiometry of all generating modes and active modes are given in Table 2 and Table 4 lists the corresponding reactions to the chosen AMs and the synthesized/consumed biomass compartments. The system equations for the hybrid cybernetic model in R. eutropha are given as follows d dt
xFRU xAMC
= Ss Z rM c
(27)
Fig. 3. Experimental data (Ammonium chloride: , Fructose: , PHB: , total biomass: ♦) and simulation results (–). Four stages can be observed in the experiment: (I) growth, (II) PHB synthesis, (III) PHB degradation, (IV) maintenance. Initial conditions: FRU0 =20.0 g/L, AMC0 =1.5 g/L, TBM0 =0.125 g/L, (PHB/TBM)0 =0.18 g/g. The experimental data in this experiment were used for parameter identification.
d mPHB = Sm,s Z rM − mPHB dt
(28)
de = ˛ + rEM b − diag(ˇ) e − e dt
(29)
where b = 1 − mPHB and the rates rM and rEM are modeled with Monod-type kinetics
dc = c dt
(30)
max rel e1 rM,1 = v1 kr,1
Table 2 Stoichiometry of generating modes (GMs) and computed subgroup of active modes (AMs), which are necessary to reproduce experimental data and hence are used for the hybrid cybernetic model. GM
1 2 3 4 5 6 7 8 9 10 11
Stoichiometry of GMs
Group
YAMC
YBIO
YFRU
YPHB
0.00 0.00 −0.25 −2.62 −0.14 −0.30 −0.38 −0.42 −0.72 −0.29 −0.11
0.00 0.00 0.54 5.55 0.30 0.64 0.80 0.89 1.53 0.62 0.24
−1.00 −1.00 −1.00 −1.00 −1.00 −1.00 −1.00 −1.00 −1.00 −1.00 −1.00
0.24 0.44 0.06 −8.91 −0.41 −0.01 −0.19 −0.30 −1.08 0.00 0.00
max rel rMi = vi kr,i ei
xFRU KFRU + xFRU
(31)
xFRU xAMC , KFRU + xFRU KAMC + xAMC
max rel rM5 = v5 kr,5 e5
i = 2, 3, 4
(32)
xFRU xAMC mPHB KFRU + xFRU KAMC + xAMC KPHB + xPHB
(33)
AM
and M+ M+ M+ M− M− M− M− M− M− M+, M− M+, M−
– 1 2 – – – – – 5 4 3
rEM,1 = u1 ke,1 rEM,i = ui ke,i
xFRU KFRU + xFRU
xFRU xAMC , KFRU + xFRU KAMC + xAMC
rEM,5 = u5 ke,5
(34) i = 2, 3, 4
(35)
xFRU xAMC mPHB KFRU + xFRU KAMC + xAMC KPHB + xPHB
(36)
Since the chosen metabolic objective function is maximizing carbon uptake the stoichiometric matrices are normalized w.r.t. to the
54
A. Franz et al. / Biochemical Engineering Journal 55 (2011) 49–58
Table 3 Chosen active modes with their corresponding metabolic reactions and the synthesized/consumed biomass compartments. AM
Corresponding reactions
BIO
PHB
1 2 3 4 5
1–3, 5–15, 17, 24, 29, 30, 34 1–14, 16, 18, 19, 24, 27–30, 34, 36 1–4, 6–15, 17–21, 24, 27–28, 33, 34, 36 1–15, 17–19, 24, 27–28, 34, 36 1, 2, 4–13, 15, 18-29, 31–32, 34–36
– synthesized synthesized synthesized synthesized
synthesized synthesized – – consumed
preferred carbon source fructose
Ss Z =
−1.00 0.00
Sm,s Z = 0.44
−1.00 −1.00 −1.00 −1.00 −0.25 −0.11 −0.29 −0.72 0.06
0.00
0.00
−1.08
0.60
0.24
0.62
0.45
rM .
The number of uptaken carbon units fc is calculated from the molar masses of FRU and PHB and their stoichiometry factors to
fc = 1.0
1.0
1.0
1.0
Parameter
Value
max kr
[0.21 0.82 0.84 0.73 0.52]T [1/h] [0.10 0.10 0.10 0.10 0.10]T [1/h] 0.01ke [1/h] [5.0 5.0 5.0 5.0 5.0]T [1/h] 0.01 [g/L] 0.06 [g/L] [31] 0.05 [gPHB /gDW ]
ke ˛ ˇ KAMC KFRU KPHB
and growth rate of total biomass (BIO+PHB) is therefore defined as = 0.44
Table 4 Model parameters used for HCM of R. eutropha.
2.51 .
3. Results and discussion 3.1. PHB synthesis and consumption Most models in literature have neglected PHB consumption since they are usually interested only in the synthesis of PHB. However, consumption might be crucial, especially in continuous processes, since it may lower the yield of PHB and degradable PHB can be viewed as an additional substrate. Even in fed batch processes PHB consumption has to be considered if nutrients are fed to the fermenter which will stimulate growth. To identify model parameters experiments were done which include both, the synthesis and consumption of PHB, see Fig. 3. The experimental process can be divided into four stages. In the first stage both complementary substrates fructose (FRU) and ammonium chloride (AMC) are available in the medium and biomass is increasing. This ammonia rich condition is covered by the chosen active modes AM 3 and AM 4, see Table 3. Since there is sufficient nitrogen available in the medium, the NADPH synthesized in reactions (5) and (19) is completely used for synthesis of non-PHB biomass (reactions (27) and (36)). At the end of stage I AMC becomes impoverished. Under this condition the synthesized amount of NADPH cannot be used completely for synthesis of nonPHB biomass. Remaining NADPH is then used for PHB synthesis (reaction 30). This is represented by the active mode AM 2, which synthesizes both, PHB and non-PHB biomass. In stage II AMC is completely exhausted and fructose is now used to synthesize PHB as an internal storage material. This is represented by active mode AM 1. Under this ammonia limited condition NADPH, which is produced in reaction 5 cannot be invested into growth of non-PHB biomass, therefore NADPH and FRU are exclusively used for PHB synthesis. After the cells are significantly filled with PHB (approx. 45%) new AMC is added to the process (stage III) to allow investment of NADPH, internal PHB and remaining fructose into utilization of non-PHB biomass. In stage IV fructose is completely exhausted and growth has stopped. Experiments with glucose as single carbon source, which were performed additionally, show the same behavior. Since the glucose uptake in wild-type R. eutropha is still not fully understood these data are not included in this study, but will published elsewhere.
3.1.1. Model identification and validation Model parameters kr were estimated by fitting the model to the experimental data in Fig. 3. All other parameter values were fixed, since the kr values appeared to be the most sensitive parameters. As mentioned before, the constitutive enzyme synthesis rate has to be much smaller than inducible enzyme rate and ˛ was therefore chosen to be one percent of ke . The model is able to reproduce the data very well, as shown in Fig. 3. The parameter values are listed in Table 4. Although the parameters were fitted to batch data, the prediction of the washout point for a continuous system is very accurate (see next section). For model validation two independent experiments with different initial conditions were performed, which are shown in Fig. 4 left and right column. In the experiment in the left column in Fig. 4 additional ammonium chloride was added to the fermenter similar to the experiment in Fig. 3 to validate the PHB consumption. But in contrast to Fig. 3 PHB content is increasing again after 40 h, since there is still carbon source available in the medium. In contrast to the experiments in Fig. 3 and Fig. 4 left column in the experiment
Fig. 4. Experimental data for model verification (Ammonium chloride: , Fructose: , PHB: , total biomass: ♦) and simulation results (–). Left column: FRU0 =21.0 g/L, AMC0 =0.6 g/L, TBM0 =0.1 g/L, (PHB/TBM)0 =0.0 g/g. Right column: FRU0 =5.05 g/L, AMC0 =1.0 g/L, TBM0 =0.02 g/L, (PHB/TBM)0 =0.0 g/g.
A. Franz et al. / Biochemical Engineering Journal 55 (2011) 49–58
55
shown in Fig. 4 right column additional fructose was added to the fermenter instead of ammonium chloride. Both experiments show very good agreement with the hybrid cybernetic model (solid line) presented in this study. The HCM approach with the extension to a few internal metabolites which is presented in this study is very suitable to describe metabolic behavior in response to changing environmental conditions. 3.2. Nonlinear analysis of continuous cultures Continuous processes have the advantage of longer periods of operating time, which can reduce production costs. In particular the production of PHB is still much more expensive then of conventional plastics. It is therefore not surprising that there are already experimental studies available in literature which deal with continuous PHB production [27–29]. However, in continuous cultures nonlinear phenomenas such as oscillation [14,15] and multiple steady states [16–18] can occur. Therefore, following the idea of Pinto and Immanuel [11] the model is used to investigate the possibility of multiple steady states in a continuous bio reactor. The modeling equations for a continuous process are given by d dt
xFRU xAMC
=D
in xFRU in xAMC
−
xFRU xAMC
+ Ss Z rM c
(37)
d mPHB = Sm,s Z rM − mPHB dt
(38)
de = ˛ + rEM b − diag(ˇ) e − e dt
(39)
dc = ( − D) c dt
(40)
in where D is the dilution rate and xFRU
in and xAMC
are the inlet concentrations of fructose and ammonium chloride. In the remainder the parameter dependent steady state behavior is studied by numerical continuation methods provided in Diva [30]. Principal bifurcation parameters are the dilution rate D and the feed ratio =
xin
AMC
xin
FRU
in was fixed . The fructose feed concentration xFRU
at 20g/L like in the batch experiments. It is worth noting, that the cybernetic model is not differentiable at points where the return on investment of the different modes in Eq. (22) coincide due to the non differentiability of the control variables v according to Eq. (21). Such points are called catch up points and will show up as sharp corners in the solution diagram. For a proper resolution of these corners the step size of the continuation has to be adjusted accordingly. Two bifurcation points (A) and (B) and one catch up point (C) were detected, see Fig. 5. At washout point (A) the dilution rate equals the maximal growth rate (D = max ). The predicted washout point is in very good agreement with published data for the maximal growth rate of Ralstonia eutropha on fructose [31]. Although batch data were used for parameter identification the prediction of the washout point in continuous cultures is very accurate. A small hysteresis is detected over 0.060 ≤ D ≤ 0.063. The hysteresis is bordered by the points (B) and (C). A two parameter continuation of these points with respect to the substrate composition at the inlet of the reactor =
xin
AMC
xin
Fig. 5. Bifurcation diagram of total biomass (TBM) and PHB concentration w.r.t. in in = 20.0 g/L, xAMC = 1.5 g/L. Two bifurcation points were detected: dilution rate. xFRU washout point (A) and turning point (B) and a catch up point (C).
can be viewed as an additional carbon source besides fructose. But still fructose and PHB are not substitutable in the strict sense, since PHB is formed from fructose. However, once PHB is synthesized the organism can choose between fructose or PHB as growth substrate. 3.3. Influence of PHB consumption Fig. 7 illustrates the effect of PHB consumption on biomass in a continuous bio reactor. If PHB consumption is neglected, then the concentrations of total biomass and PHB (not shown) are significantly higher at low dilution rates (D ≤ 0.1 h−1 ) and the hysteresis disappears. As mentioned before, this kind of hysteresis is observed if there are two or more substitutable substrate. If PHB consumption is neglected, then there is only one carbon substrate, namely
and dilution rate D shows
FRU
that the hysteresis unfolds from = 0 and vanishes for > 0.17, see Fig. 6. Such kind of hysteresis is usually observed if there are two or more substitutable substrates available as shown by Namjoshi and Ramkrishna [32] and by Kumar et al. [17]. However in this study there are only two complementary substrates (FRU and AMC), but no substitutable. But as mentioned before, due to the introduction of internal metabolites and their dynamics into the model, new products and substrates are added. The internal metabolite PHB
Fig. 6. Two parameter continuation with respect to dilution rate and .
56
A. Franz et al. / Biochemical Engineering Journal 55 (2011) 49–58
model is most suitable for control, optimization and monitoring of industrial PHB production processes. Nonlinear analysis of a continuous reactor reveals that PHB consumption can reduce the PHB yield significantly at low dilution rates. Depending on the feeding strategy in fed batch processes, PHB consumption can also be stimulated. Therefore PHB consumption has to be included into modeling of continuous and fed batch production processes, which was previously neglected in available literature. Further, the occurrence of multiple steady states was predicted. However, the multiplicity region predicted by the model is rather small and it is argued that multiple steady states are therefore unlikely to occur in practice for this specific system. Acknowledgments A. Franz and A. Kienle appreciate the financial support of the German Federal Ministry for Education and Research (BMBF) under the FORSYS program. The technical support of Ruxandra Rehner during the experiments is greatly acknowledged. Appendix A. Metabolic reactions
Fig. 7. Comparison of biomass steady states with and without PHB consumption.
fructose, and therefore the hysteresis disappears. Furthermore, at low dilution rates and < 0.17 AMC is growth limiting and therefore there is a surplus of fructose. Due to the chosen metabolic objective function of maximizing substrate uptake, PHB concentration and therefore concentration of total biomass are higher if PHB consumption is not considered in the model. At high dilution rates (D > 0.1 h−1 ) the consumption of PHB has no significant effect. Besides continuous processes, the production of PHB in fed batch fermentation was also studied by many researchers [33,29,34]. They have shown that optimal feeding strategies can lead to higher PHB yield and productivity. Lopez et al. [34] used dynamic optimization techniques to obtain optimal feeding profiles for carbon and nitrogen sources. They found, that there are two different phases: biomass growth followed by PHB production. In the first phase carbon and nitrogen are fed into the fermenter to maximize biomass. In the second phase only carbon is fed into the fermenter to maximize PHB. In this scenario PHB consumption is not stimulated. However, if nitrogen source is added to the process, PHB consumption will be stimulated and has therefore to be included into the model. In an industrial process, where a process controller compensates disturbances, this is very likely to occur, if for instance biomass is lower than a certain set point. 3.4. Conclusion In this study, a hybrid cybernetic model based on a detailed metabolic network was developed to describe PHB synthesis as well as PHB consumption. The latter one is usually neglected in available literature. The hybrid cybernetic model approach was extended to include internal metabolites with slow dynamics, since PHB, which is an internal metabolite, cannot be characterized by fast dynamics. To identify reasonable parameters experiments were performed which include both: synthesis and consumption of PHB. The developed model shows very good agreement with experimental data sets. It was validated with independent experiments. Due to various desirable features such as accounting for cellular regulation at network level and dynamics of intracellular metabolites with a moderate complexity, it is believed that the constructed
No.
Reaction
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
FRU + PEP + ATP → F16P + PYR + ADP F16P → F6P F16P → 2G3P AMC → NH3 G6P + 2NADP → Rl5P + CO2 + 2NADPH Rl5P ↔ R5P Rl5P ↔ X5P X5P + R5P ↔ S7P + G3P S7P + G3P ↔ E4P + F6P X5P + E4P ↔ G3P + F6P F6P → G6P G3P + NAD + ADP ↔ 3PG + NADH + ATP 3PG ↔ PEP PEP + ADP → PYR + ATP OXA + ATP → PEP + ADP + CO2 PYR → AcCoA + Form PYR + NAD → AcCoA + NADH + CO2 AcCoA + OXA → ISC ISC + NADP → ˛KG + NADPH + CO2 ˛KG + NAD → SucCoA + NADH + CO2 SucCoA + ADP ↔ SUC + ATP SUC + FAD ↔ MAL + FADH MAL + NAD → OXA + NADH PYR + ATP → OXA + ADP ISC → SUC + GOX AcCoA + GOX → MAL NH3 + ˛KG + NADPH → GLUT + NADP GLUT + NH3 + ATP → GLUM + ADP 2AcCoA ↔ AcAcCoA AcAcCoA + NADPH → PHB + NADP PHB + NAD → ACE + NADH ACE + SucCoA → AcAcCoA + SUC SUC → SUCx 2NADH + O2 + 4ADP → 2NAD + 4ATP 2FADH + O2 + 2ADP → 2FAD + 2ATP 0.21G6P + 0.07F6P + 0.9R5P + 0.36E4P + 0.13G3P + 1.53PG + 0.52PEP + 2.83PYR + 3.74AcCoA + 1.79OXA + 8.32GLUT + 0.25GLUM + 41.1ATP + 8.26NADPH + 3.12NAD → BIO + 7.51 ˛KG + 2.61CO2 + 41.1ADP + 8.26NADP + 3.12NADH
All stoichiometry coefficients are given in mmol, except BIO is given in g. Appendix B. Nomenclature
b c
fraction of catalytically active parts in biomass biomass concentration
A. Franz et al. / Biochemical Engineering Journal 55 (2011) 49–58
D e emax erel fc h ke kr KAMC KFRU KPHB m mf ms mPHB p r rM rkin M rEM rkin EM Sm Sm , f Sm , s Sp Ss u v x xs xp xAMC xFRU in xAMC in xFRU y ym Z Zy M− M+ ˛ ˇ
dilution rate vector of enzyme levels vector of maximum enzyme levels vector of relative enzyme levels vector of uptaken carbon units weight vector for the modes of Zy vector of enzyme synthesis rate constants vector of rate constants saturation constant for ammonium chloride saturation constant for fructose saturation constant for PHB vector of specific internal concentrations vector of specific internal concentrations with fast dynamics vector of specific internal concentrations with slow dynamics specific concentration of PHB vector of ROI vector of regulated fluxes vector of regulated fluxes through EMs purely kinetic part of rM vector of regulated enzyme synthesis rates of EMs purely kinetic part of rEM stoichiometry matrix of internal metabolites stoichiometry matrix of internal metabolites with fast dynamics stoichiometry matrix of internal metabolites with slow dynamics stoichiometry matrix of products stoichiometry matrix of substrates vector of cybernetic variables which control enzyme synthesis vector of cybernetic variables which control enzyme activity vector of external concentrations vector of substrate concentrations vector of product concentrations concentration of ammonium chloride concentration of fructose concentration of ammonium chloride at inlet concentration of fructose at inlet state vector of a metabolic system vector of measured yield data EM matrix normalized EM matrix set of internal metabolite consuming EMs set of internal metabolite producing EMs vector of constitutive enzyme synthesis rates vector of enzyme consumption constants feed composition growth rate
B.1. Abbreviations
AM CM EM HCM ROI PHA TBM MYA YS
Active mode Cybernetic model Elementary mode Hybrid cybernetic model Return on investment Polyhydroxyalkanoates Total biomass (dry weight) Metabolic yield analysis Yield space
57
B.2. Metabolites
3PG 3-phosphoglycerate ˛KG Alpha-ketoglutarate AcAcCoA Acetoacetyl CoA AcCoA Acetyl CoA ACE Acetoacetate ADP Adenosine diphosphate ATP Adenosine triphosphate AMC Ammonium chloride BIO Non-PHB biomass Carbon dioxide CO2 E4P Erythrose-4-phosphate F16P Fructose-1,6-Bisphosphate F6P Fructose-6-Phosphate FAD Flavin adenin dinucleotide, oxidized FADH Flavin adenin dinucleotide, reduced Form Formiate FRU Fructose G3P Glyceraldehyde-3-phosphate G6P Glucose-6-phosphate GLUM Glutamine GLUT Glutamate GOX Glyoxylate ISC Isocitrate MAL Malate NAD Nicotinamide adenine dinucleotide, oxidized NADH Nicotinamide adenine dinucleotide, reduced NADP Nicotinamide adenine dinucleotide phosphate, oxidized NADPH Nicotinamide adenine dinucleotide phosphate, reduced NH3 Ammonia O2 Oxygen OXA Oxaloacetate PEP Phosphoenol pyruvate PHB Poly(-hydroxybutyrate) PYR Pyruvate R5P Ribose-5-phosphate Rl5P Ribulose-5-phosphate S7P Sedoheptulose-7-phosphate SUC Succinate SucCoA Succinate CoA SUCx Succinate, external X5P Xylulose-5-phosphate References [1] J.I. Kim, J.D. Varner, D. Ramkrishna, A hybrid model of anaerobic E. coli GJT001: combination of elementary flux modes and cybernetic variables, Biotechnol. Prog. 24 (2008) 993–1006. [2] F. Reinecke, A. Steinbüchel, Ralstonia eutropha strain H16 as model organism for PHA metabolism and for biotechnological production of technically interesting biopolymers, J. Mol. Microbiol. Biotechnol. 16 (2009) 91–108. [3] A.J. Anderson, E.A. Dawes, Occurrence, metabolism, metabolic role, and industrial uses of bacterial polyhydroxyalkanoates, Microbiol. Rev. 54 (1990) 450–472. [4] J.A. Asenjo, J.S. Suk, Kinetics and models for the bioconversion of methane into an intracellular polymer poly-beta-hydroxybutyrate, Biotechnol. Bioeng. Symp. 15 (1986) 225–234. [5] A. Mulchandani, J.H.T. Luong, C. Groom, Substrate-inhibition kinetics for microbial-growth and synthesis of poly-beta-hydroxybutyric acid by Alcaligenes eutrophus ATCC-17697, Appl. Microbiol. Biotechnol. 30 (1989) 11–17. [6] D.S. Kompala, D. Ramkrishna, G.T. Tsao, Cybernetic modeling of microbial growth on multiple substrates, Biotechnol. Bioeng. 26 (1984) 1272–1281. [7] D. Ramkrishna, A cybernetic perspective of microbial growth, ACS Symp. Ser. 207 (1983) 161–178. [8] D. Ramkrishna, D.S. Kompala, G.T. Tsao, Are microbes optimal strategists, Biotechnol. Prog. 3 (1987) 121–126. [9] S. Yoo, W.S. Kim, Cybernetic model for synthesis of poly-beta-hydroxybutyric acid in Alcaligenes eutrophus, Biotechnol. Bioeng. 43 (1994) 1043–1051.
58
A. Franz et al. / Biochemical Engineering Journal 55 (2011) 49–58
[10] K.G. Gadkar, F.J. Doyle, T.J. Crowley, J.D. Varner, Cybernetic model predictive control of a continuous bioreactor with cell recycle, Biotechnol. Prog. 19 (2003) 1487–1497. [11] M.A. Pinto, C.D. Immanuel, Sensitivity of bifurcation traits to model parameters in poly-beta-hydroxy butyrate production, in: ADCHEM. [12] J. Stelling, S. Klamt, K. Bettenbrock, S. Schuster, E.D. Gilles, Metabolic network structure determines key aspects of functionality and regulation, Nature 420 (2002) 190–193. [13] J.D. Young, K.L. Henne, J.A. Morgan, A.E. Konopka, D. Ramkrishna, Integrating cybernetic modeling with pathway analysis provides a dynamic, systems-level description of metabolic control, Biotechnol. Bioeng. 100 (2008) 542–559. [14] J. Weber, A. Kayser, U. Rinas, Metabolic flux analysis of Escherichia coli in glucose-limited continuous culture. ii. Dynamic response to famine and feast, activation of the methylglyoxal pathway and oscillatory behaviour, Microbiology-SGM 151 (2005) 707–716. [15] K.D. Jones, D.S. Kompala, Cybernetic model of the growth dynamics of Saccharomyces cerevisiae in batch and continuous cultures, J. Biotechnol. 71 (1999) 105–131. [16] I. Disli, A. Kremling, A. Kienle, A model based analysis of multiple steady states in continuous cell cultures, in: Proceedings MATHMOD 09 Vienna - Full Papers CD Volume. [17] M.V. Kumar, K.P. Zeyer, A. Kienle, S. Pushpavanam, Conceptual analysis of the effect of kinetics on the stability and multiplicity of a coupled bioreactorseparator system using a cybernetic modeling approach, Ind. Eng. Chem. Res. 48 (2009) 10962–10975. [18] A.A. Namjoshi, W.S. Hu, D. Ramkrishna, Unveiling steady-state multiplicity in hybridoma cultures: The cybernetic approach, Biotechnol. Bioeng. 81 (2003) 80–91. [19] J.H. Law, R.A. Slepecky, Assay of poly-beta-hydroxybutyric acid, J. Bacteriol. 82 (1961) 33. [20] H.S. Song, D. Ramkrishna, Reduction of a set of elementary modes using yield analysis, Biotechnol. Bioeng. 102 (2009) 554–568. [21] H.S. Song, J.A. Morgan, D. Ramkrishna, Systematic development of hybrid cybernetic models: application to recombinant yeast co-consuming glucose and xylose, Biotechnol. Bioeng. 103 (2009) 984–1002.
[22] A. von Kamp, S. Schuster, Metatool 5.0: fast and flexible elementary modes analysis, Bioinformatics 22 (2006) 1930–1931. [23] S. Klamt, J. Saez-Rodriguez, E.D. Gilles, Structural and functional analysis of cellular networks with CellNetAnalyzer, BMC Syst. Biol. 1 (2007). [24] J.D. Young, D. Ramkrishna, On the matching and proportional laws of cybernetic models, Biotechnol. Prog. 23 (2007) 83–99. [25] B.G. Turner, D. Ramkrishna, Revised enzyme-synthesis rate expression in cybernetic models of bacterial-growth, Biotechnol. Bioeng. 31 (1988) 41–43. [26] T. Katoh, D. Yuguchi, H. Yoshii, H.D. Shi, K. Shimizu, Dynamics and modeling on fermentative production of poly (beta-hydroxybutyric acid) from sugars via lactate by a mixed culture of Lactobacillus delbrueckii and Alcaligenes eutrophus, J. Biotechnol. 67 (1999) 113–134. [27] S.T. Yu, C.C. Lin, J.R. Too, Phbv production by Ralstonia eutropha in a continuous stirred tank reactor, Process. Biochem. 40 (2005) 2729–2734. [28] S.T. Wu, Y.C. Lin, J.R. Too, Continuous production of poly(3-hydroxybutyrateco-3-hydroxyvalerate): Effects of c/n ratio and dilution rate on hb/hv ratio, Korean J. Chem. Eng. 26 (2009) 411–416. [29] S. Khanna, A.K. Srivastava, Optimization of nutrient feed concentration and addition time for production of poly(beta-hydroxybutyrate), Enzyme Microb. Technol. 39 (2006) 1145–1151. [30] M. Mangold, A. Kienle, E.D. Gilles, K.D. Mohl, Nonlinear computation in DIVA–methods and applications, Chem. Eng. Sci. 55 (2000) 441–454. [31] C. König, H.G. Schlegel, Prolonged organotrophic growth of hydrogenomonas H16 in chemostat, Arch. Mikrobiol. 62 (1968) 41–55. [32] A.A. Namjoshi, D. Ramkrishna, Multiplicity and stability of steady states in continuous bioreactors: dissection of cybernetic models, Chem. Eng. Sci. 56 (2001) 5593–5607. [33] S. Shahhosseini, Simulation and optimisation of PHB production in fed-batch culture of Ralstonia eutropha, Process. Biochem. 39 (2004) 963–969. [34] J.A. Lopez, V. Bucala, M.A. Villar, Application of dynamic optimization techniques for poly(beta-hydroxybutyrate) production in a fed-batch bioreactor, Ind. Eng. Chem. Res. 49 (2010) 1762–1769.
Contents lists available at ScienceDirect
Biochemical Engineering Journal journal homepage: www.elsevier.com/locate/bej
Experimental and theoretical analysis of poly(-hydroxybutyrate) formation and consumption in Ralstonia eutropha André Franz a,d,∗ , Hyun-Seob Song b , Doraiswami Ramkrishna b , Achim Kienle a,c,d a
Max Planck Institute for Dynamic of Complex Technical Systems, 39106 Magdeburg, Germany School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, USA Otto-von-Guericke University, Chair of Automation/Modeling, 39106 Magdeburg, Germany d Magdeburg Centre for Systems Biology (MaCs), Magdeburg, Germany b c
a r t i c l e
i n f o
Article history: Received 1 July 2010 Received in revised form 28 February 2011 Accepted 18 March 2011 Available online 29 March 2011 Keywords: Ralstonia eutropha Cybernetic modeling Bifurcation analysis Biopolymers Metabolic regulation Elementary mode
a b s t r a c t In this paper a mathematical model is presented to describe poly(-hydroxybutyrate) (PHB) formation and consumption in Ralstonia eutropha. The model is based on the hybrid cybernetic modeling approach, which was introduced by Kim et al. [1] and which allows a systematic derivation of the model equations from elementary mode analysis. An extension of this approach is presented to allow for non quasistationary metabolites, i.e. PHB. The model is shown to be in good agreement with experimental data for PHB formation and consumption. The model is used afterwards to discuss the occurrence of multiple steady states in a continuous bio reactor. It is shown that the multiplicity region predicted by the model is rather small and it is argued that multiple steady states are therefore unlikely to occur in practice for this specific system. Due to various desirable features such as accounting for cellular regulation at network level and dynamics of intracellular metabolites with a moderate complexity, it is believed that the constructed model is most suitable for control, optimization and monitoring of industrial PHB production processes. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Poly(-hydroxybutyrate) (PHB) is an organic polymer, which can be synthesized by many microorganisms and which serves as internal energy and carbon reserve material. Ralstonia eutropha is a well known bacterium for producing PHB [2]. It can accumulate PHB to more than 80% of its cell dry weight [3]. R. eutropha has been re-classified several times in the past. The history of classification includes the genera Hydrogenomonas, Alcaligenes, Ralstonia, Wautersia and recently Cupriavidus necator. To avoid confusion, we will use the name Ralstonia throughout, since it is still commonly used in the literature, including very recent articles. Production of PHB is favored under limitation of key nutrients such as nitrogen, phosphate or oxygen. PHB belongs to the group of polyhydroxyalkanoates (PHA) and provides an attractive source of bioplastics that are biodegradable, biocompatible and do not depend on fossil resources. PHB production in R. eutropha has previously been modeled by a few researchers. These models can be divided into two classes:
∗ Corresponding author. E-mail address: [email protected] (A. Franz). 1369-703X/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.bej.2011.03.006
(a) models which do not consider internal regulation, e.g. [4,5] and (b) models which do consider cell internal regulation. Internal regulation can be included into modeling by the cybernetic framework, which was introduced by Ramkrishna and co-workers [6–8]. The cybernetic approach for modeling PHB production in R. eutropha was used by Yoo and Kim [9]. They have used a very simple unstructured model and compared their results with unstructured non-cybernetic models of Mulchandani et al. [5] and Asenjo and Suk [4]. Their model could successfully predict PHB production, but did not include the underlying metabolic processes. Gadkar et al. [10] addressed this discrepancy and used a structured cybernetic model to develop a model predictive control for continuous PHB production. Although they considered the metabolic pathways by which the carbon and nitrogen sources are utilized these pathways are still lumped. The cybernetic model by Pinto and Immanuel [11] is also based on a very simplified metabolic network with less complexity than the model of Gadkar et al. [10] and was used for bifurcation analysis. It was shown, that the model structure admits multiple steady states in a continuous bio reactor, depending on the parameter values. All these models are based on a more or less simplified metabolic network and either neglect PHB consumption or have fitted parameter to experimental data which do not contain any significant PHB consumption. But since PHB is an internal storage material, it is
50
A. Franz et al. / Biochemical Engineering Journal 55 (2011) 49–58
appropriate to consider its metabolism as a cycle of synthesis and consumption. In the current work the PHB consumption is included into a cybernetic model and experiments were performed which show significant consumption to gain more reliable parameters. Additionally an expanded metabolic network is considered to include more internal metabolic information. The model is based on the state of the art hybrid cybernetic approach (HCM) by Kim et al. [1]. The HCM approach allows a systematic derivation of the model equations from elementary mode analysis [12]. It is based on quasistationarity of internal metabolites, which are eliminated from the model equations. However, this approach is not suitable for PHB production, since PHB is an internal metabolite, which has to be included explicitly into the model equations. In contrast to this, the more detailed approach by Young et al. [13] takes the dynamics of internal metabolites into account and could also handle synthesis and consumption of PHB. But this approach is computationally more expensive and requires detailed information about internal kinetics, which might be hard to obtain in particular for larger networks. To overcome these limitations this article presents a compromise between the standard HCM and the cybernetic modeling approach by Young et al. [13], by taking the dynamics of a few internal metabolites explicitly into account, while for most of the internal metabolites the quasi-steady state approximation is still applied. It is shown that the model is in good agreement with experimental observations, where PHB formation and consumption are stimulated separately. In industrial production usually fed batch or continuous processes are used. While fed batch processes allow for higher biomass and product yield, a continuous process can also be advantageous due to longer periods of operating time, which can reduce production costs. However, in continuous operating mode metabolic reactions will occur simultaneously, which can lead to nonlinear phenomenas, e.g. oscillations [14,15], multiple steady states [16–18], etc. The occurrence of multiple steady states in a PHB production process with R. eutropha was discussed by Pinto and Immanuel [11] based on a simplified model. Following their idea, the model is therefore afterwards used to investigate the possibility of multiple steady states in a continuous bio reactor. It is shown that multiple steady states are unlikely to occur in practice for this specific system. Furthermore, the influence of PHB consumption is analyzed, which is usually of minor importance in a batch process, since there is usually sufficient carbon source available and PHB will not be consumed. But in continuous processes PHB consumption can have crucial influence, as shown in this study. Even in a fed batch process PHB consumption can be of great importance if other growth essential nutrients than carbon are fed to the fermenter and PHB consumption will be stimulated. It is therefore necessary to include PHB consumption into modeling, if a fed batch or continuous process is used.
2. Material and methods 2.1. Experimental section 2.1.1. Microorganisms and medium The organism used throughout this study, R. eutropha (DSM 428, ATCC 17699, NCIB 10442) was obtained from DSMZ GmbH Braunschweig, Germany, as vacuum dried culture. The strain was cultivated with the medium given in Table 1. All chemicals were from Carl Roth GmbH (Karlsruhe, Germany).
Table 1 Used medium for cultivation. Ingredient
Concentration
Fructose NH4 Cl KH2 PO4 Na2 HPO4 · 2 H2 O MgSO4 · 7 H2 O CaCl2 · 2 H2 O Fe(NH4 ) citrate Trace element solution a
20.0 g/L 1.50 g/L 2.30 g/L 2.90 g/L 0.50 g/L 0.01 g/L 0.05 g/L 5.00 ml/L
a Trace element solution (g/L): ZnSO4 · 7 H2 O 0.10, MnCl2 · 4 H2 O 0.03, H3 BO3 0.30, CoCl · 6 H2 O 0.20, CuCl2 · 2 H2 O 0.01, NiCl2 · 6 H2 O 0.02, NaMoO4 · 2 H2 O 0.03.
2.1.2. Cultivation conditions R. eutropha was grown heterotrophically in a 7 L fermenter (Biostat C, Sartorius, BBI Systems, Melsungen, Germany) with a 5 L working volume. The temperature was kept constant at 30 ◦ C and the pH was automatically maintained at pH 6.8 by adding 2 M NaOH as corrective agent. The culture broth was agitated at 400 rpm and dissolved oxygen was maintained above 50 % air saturation by changing the oxygen and nitrogen flow rate mixed with an air stream. Total flow rate was 1.5 L/min. 2.1.3. Analytical procedures Cell growth was monitored by measuring the optical density at 600 nm using a Ultrospec 500 spectrophotometer (GE Healthcare, Buckinghamshire, UK). For dilution, NaCl (0.98 % (w/v)) was used when necessary. For the determination of the cell dry weight, 3 × 10 ml of the culture broth were centrifuged in pre-weighted glass tubes for 15 min at 3000 × g. The pellets were washed in 0.98 % NaCl and subsequently dried in a freeze-dryer (Christ, Osterode am Harz, Germany). PHB content was measured as crotonic acid, formed by acid depolymerization of PHB according to Law and Slepecky [19]. Cell pellets, harvested by centrifugation, were dissolved in methylene chloride by rapid mixing and afterwards boiled for 10 min. After the samples were cooled down, they were centrifuged at 3000 × g for 15 min and the supernatants were carefully removed and collected in glass tubes. This procedure was repeated three times. The supernatants were then evaporated and the remaining PHB-containing samples were digested in 2 ml H2 SO4 (96 %) at 100 ◦ C for 30 min and subsequently diluted with concentrated H2 SO4 . UV absorbance spectra were measured with an UV–vis spectrophotometer V-560 (Jasco, Gross-Umstadt, Germany). The concentration of crotonic acid was calculated from a set of reference standards. Ammonium chloride concentration in culture supernatants broth was measured by determining NH3 with a VITROS DT60 II Chemistry System and VITROS NH3 MicroSlide from Ortho-Clinical Diagnostics (Neckargemünd, Germany) using the manufacturers instructions. Concentrations of fructose in supernatants were determined with a D-Glucose/D-Fructose test kit from R-Biopharm (Darmstadt, Germany) using the manufacturers recommended procedure. 2.2. Hybrid cybernetic model 2.2.1. Basic formulation The state y of a metabolic system can be described by the vector of external biochemical species concentrations x, the vector of specific internal biochemical species concentrations m and the biomass concentration c
y=
x m c
(1)
A. Franz et al. / Biochemical Engineering Journal 55 (2011) 49–58
where x can be divided into the vector xs of ns substrates and the vector xp of np products. The model equations for a batch reactor are then defined as 1 dxs = Ss r c dt
(2)
1 dxp = Sp r c dt
(3)
dm = Sm r − m dt
(4)
1 dc = c dt
(5)
where r is the vector of nr regulated fluxes, is the growth rate, Ss , Sp and Sm represent the (ns × nr ), (np × nr ) and (nm × nr ) stoichiometric matrices. In the hybrid cybernetic modeling approach, which was first suggested by Kim et al. [1] and more elaborated by Song and Ramkrishna [20] and Song et al. [21] it is assumed that internal metabolites are in quasi-steady state, e.g. dm = 0. dt
(6)
If in addition m is negligibly small Sm r = 0,
r≥0
(7)
is obtained from Eq. (4). A set of elementary modes (EMs) is then uniquely defined such that non negative combinations of these EMs are spanning the solution space satisfying Eq. (7), i.e. r = ZrM
(8)
where Z is the (nr × nz ) matrix in which the i th column represents the i th elementary mode and rM is the vector of fluxes through these modes. Eqs. (2)–(4) are replaced by 1 dxs = Ss Z rM c dt
(9)
1 dxp = Sp Z rM . c dt
(10)
The matrix Z can be calculated by using standard software for metabolic flux analysis of cellular networks, e.g. Metatool [22] or CellNetAnalyzer [23]. 2.2.2. Incorporation of intracellular dynamics into HCM In the original formulation of HCM it is assumed that all internal metabolites are in quasi-steady state, since usually intracellular fluxes are characterized by relatively fast dynamics in comparison to exchange fluxes. However, for some internal metabolites this assumption does not hold, e.g. internal storage materials like PHB, membranes or other cell compartments. Therefore the vector of internal metabolite concentrations m has to be divided into the vector of internal metabolite concentrations with fast dynamics mf and the vector of internal metabolite concentrations with slow dynamics ms . Similarly, the matrix Sm is divided into Sm,f and Sm,s and Eq. (4) can be rewritten to d dt
ms mf
=
Sm,s Sm,f
r−
ms mf
(11)
Only for metabolites with fast dynamics the quasi-steady state assumption holds, e.g. dmf =0 dt
(12)
and therefore Sm,f r = 0,
r ≥ 0.
In the present case, PHB is considered as internal metabolite with slow dynamics. EMs are then calculated from Eq. (13) by setting metabolites with slow dynamics as external in Metatool or unbalanced in CellNetAnalyzer. One of the main advantages of HCM in comparison to the cybernetic modeling approach of Young et al. [24] is, that the use of local cybernetic variables is circumvented, since the quasi-steady state approximation for intracellular components allows the calculation of internal fluxes relative to uptake rates through exploitation of the stoichiometric coupling. Especially the application to large networks is much less computationally demanding by using HCM instead of the cybernetic model formulation by Young et al. [24]. However, this advantage has to paid for by the quasi-steady state assumption for all internal metabolites. In contrast, the approach of this article allows to include internal metabolites for which quasisteady state assumption does not hold, but still gets along with only global control. It is obvious that this approach lies between the formulation of Young et al. [24] and original HCM, but combines the advantages from both formulations. 2.2.3. Rate equations and cybernetic variables The vector rM of fluxes through EMs is controlled by the vector of cybernetic variables v rM = diag(v) diag(erel ) rkin M
(14)
and catalyzed by a vector of key enzymes e de = ˛ + rEM − diag(ˇ)e − e dt
(15)
where eirel =
ei eimax
˛i + ke,i
eimax =
with
ˇi + max i
.
(16)
˛ is the vector of constitutive enzyme synthesis rates [25], rEM is the vector of inducible enzyme synthesis rates, the term diag(ˇ)e represents enzyme degradation, the term e dilution by growth and max is the maximal growth rate of the i th EM. i The vector of inducible enzyme synthesis rates rEM is controlled by the vector of cybernetic variables u rEM = diag(u) rkin EM
(17)
and rkin and rkin are the vectors of unregulated rates which are M EM represented by kinetic equations of Monod-type kin max = kr,i rM,i
j ∈ L(i) kin rEM,i = ke,i b
xs,j
(18)
Ki,j + xs,j
j ∈ L(i)
xs,j Ki,j + xs,j
(19)
where L(i) is the set of component indices associated to the i th EM, max is the rate constant and k is the enzyme synthesis rate conkr,i e,i stant of the i th mode, Ki,j is the saturation constant of the i th mode for the j th educt and b represents the fraction of biomass ascribed to the enzyme synthesis machinery. Usually, it is assumed that the fraction of catalytically active biomass is constant, and hence, the factor b is just included into ke [21,13]. However, in the approach introduced in this study b is not a constant, since PHB can form more than 80% of the dry weight. The factor b can then be defined as b=1−
(13)
51
i
(ms,i )
(20)
52
A. Franz et al. / Biochemical Engineering Journal 55 (2011) 49–58
Fig. 1. Used metabolic network for Ralstonia eutropha. Growth precursors are underlined and external metabolites are boxed.
where i (ms,i ) is the biomass fraction, which includes all internal metabolites with slow dynamics and are assumed not to contain any proteins and nucleid acids. Note, that this is only true, if constitutive enzyme synthesis rate is much slower than inducible synthesis rate. Otherwise, a high basal enzyme level will catalyze fluxes through EMs although fraction of catalytically active biomass (b) is very low. A general form of the cybernetic control laws [24]u and v is given by u=
p , p1
v=
p p∞
(21)
where p is the return on investment (ROI) which can be calculated from a metabolic objective function. Often used metabolic objective functions are maximizing growth or maximizing substrate uptake. In this study it is assumed that the organism maximizes carbon source uptake and p can therefore be defined as [20] p = diag(fc ) diag(erel ) rkin M
(22)
is the vector where fc is the vector of uptaken carbon units and rkin M of pure kinetic rate expressions. Note, that in this case the stoichiometry matrices have to be normalized w.r.t. to the preferred carbon source and all rates are expressed as carbon uptake rates. 2.3. HCM for R. eutropha 2.3.1. Metabolic network Based on the metabolic network used by Katoh et al. [26] a network for R. eutropha was developed which includes a fructose uptake pathway, a more detailed penthose-phosphate-pathway and a PHB consumption pathway, see Fig. 1. ATP, ADP, CO2 and O2 are not shown in the figure, but are included in the network. All reaction equations can be found in A. Note, that biomass is divided into two compartments, namely PHB and non-PHB biomass. The latter one is represented by the abbreviation BIO. 2.3.2. Metabolic yield analysis From the given metabolic network 122 elementary modes have been calculated by using Metatool. However, it is not necessary to use all elementary modes, but a smaller subset which is essential for describing and predicting observed metabolic behavior. For
reducing all EMs to a smaller subset metabolic yield analysis (MYA) developed by Song and Ramkrishna [20] was used. Since internal metabolites with slow dynamics act as substrates and products one has to ensure that the yield space is convex. This was done by dividing all EMs into subgroups, where one group contains the EMs where the internal metabolite do not act as product (M−) and another group where the internal metabolite do not act as substrate (M+). These subgroups are convex in yield space and both groups include the EMs which do not produce or consume the internal metabolite. In this study PHB is the internal metabolite and although there are modes which also use PHB as substrate it is assumed that fructose is the preferred substrate and all EMs are normalized w.r.t. fructose. Hence, the yield space becomes three dimensional (YPHB/FRU , YBIO/FRU , YAMC/FRU ). However, due to the stoichiometry of the network the concentration of the non-PHB biomass compartment BIO is linearly correlated with the uptaken nitrogen source AMC. Therefore the yield space reduces to only two dimensions (YPHB/FRU , YBIO/FRU ). For each convex group the MYA was performed separately. At first so called generating modes (GM) were chosen from both groups. These are the vertices of the convex hull of the yield space. Group (M+) has five GMs and group (M−) has eight GMs. Since experimental data are available the number of GMs was further reduced to so called active modes (AM). This was done by minimizing the sum of squared weights 1 min h22 h 2
(23)
such that Zy h − ym = 0,
h ≥ 0,
h = 1
(24)
for the experimental data, which are inside the convex hull, where ym is the vector of measured yield data and Zy contains normalized GMs. For the experimental data, which are outside the convex hull, the AMs were calculated by solving the following least squares problem 1 min Zy h − ym 22 h 2
(25)
such that h ≥ 0,
h = 1
(26)
A. Franz et al. / Biochemical Engineering Journal 55 (2011) 49–58
53
Fig. 2. Yield spaces of set M− and M+ with experimental data (, , ) and chosen modes (•) for hybrid cybernetic model.
Fig. 2 shows the two dimensional yield space, the convex hulls of both groups (M+) and (M−), and the computed AMs. The AMs are then used for the HCM presented in this study. The experimental yield data YI , YII and YIII represent the yield of the three experimental stages as shown in Fig. 3. Maintenance (stage IV) is not considered in this study. The stoichiometry of all generating modes and active modes are given in Table 2 and Table 4 lists the corresponding reactions to the chosen AMs and the synthesized/consumed biomass compartments. The system equations for the hybrid cybernetic model in R. eutropha are given as follows d dt
xFRU xAMC
= Ss Z rM c
(27)
Fig. 3. Experimental data (Ammonium chloride: , Fructose: , PHB: , total biomass: ♦) and simulation results (–). Four stages can be observed in the experiment: (I) growth, (II) PHB synthesis, (III) PHB degradation, (IV) maintenance. Initial conditions: FRU0 =20.0 g/L, AMC0 =1.5 g/L, TBM0 =0.125 g/L, (PHB/TBM)0 =0.18 g/g. The experimental data in this experiment were used for parameter identification.
d mPHB = Sm,s Z rM − mPHB dt
(28)
de = ˛ + rEM b − diag(ˇ) e − e dt
(29)
where b = 1 − mPHB and the rates rM and rEM are modeled with Monod-type kinetics
dc = c dt
(30)
max rel e1 rM,1 = v1 kr,1
Table 2 Stoichiometry of generating modes (GMs) and computed subgroup of active modes (AMs), which are necessary to reproduce experimental data and hence are used for the hybrid cybernetic model. GM
1 2 3 4 5 6 7 8 9 10 11
Stoichiometry of GMs
Group
YAMC
YBIO
YFRU
YPHB
0.00 0.00 −0.25 −2.62 −0.14 −0.30 −0.38 −0.42 −0.72 −0.29 −0.11
0.00 0.00 0.54 5.55 0.30 0.64 0.80 0.89 1.53 0.62 0.24
−1.00 −1.00 −1.00 −1.00 −1.00 −1.00 −1.00 −1.00 −1.00 −1.00 −1.00
0.24 0.44 0.06 −8.91 −0.41 −0.01 −0.19 −0.30 −1.08 0.00 0.00
max rel rMi = vi kr,i ei
xFRU KFRU + xFRU
(31)
xFRU xAMC , KFRU + xFRU KAMC + xAMC
max rel rM5 = v5 kr,5 e5
i = 2, 3, 4
(32)
xFRU xAMC mPHB KFRU + xFRU KAMC + xAMC KPHB + xPHB
(33)
AM
and M+ M+ M+ M− M− M− M− M− M− M+, M− M+, M−
– 1 2 – – – – – 5 4 3
rEM,1 = u1 ke,1 rEM,i = ui ke,i
xFRU KFRU + xFRU
xFRU xAMC , KFRU + xFRU KAMC + xAMC
rEM,5 = u5 ke,5
(34) i = 2, 3, 4
(35)
xFRU xAMC mPHB KFRU + xFRU KAMC + xAMC KPHB + xPHB
(36)
Since the chosen metabolic objective function is maximizing carbon uptake the stoichiometric matrices are normalized w.r.t. to the
54
A. Franz et al. / Biochemical Engineering Journal 55 (2011) 49–58
Table 3 Chosen active modes with their corresponding metabolic reactions and the synthesized/consumed biomass compartments. AM
Corresponding reactions
BIO
PHB
1 2 3 4 5
1–3, 5–15, 17, 24, 29, 30, 34 1–14, 16, 18, 19, 24, 27–30, 34, 36 1–4, 6–15, 17–21, 24, 27–28, 33, 34, 36 1–15, 17–19, 24, 27–28, 34, 36 1, 2, 4–13, 15, 18-29, 31–32, 34–36
– synthesized synthesized synthesized synthesized
synthesized synthesized – – consumed
preferred carbon source fructose
Ss Z =
−1.00 0.00
Sm,s Z = 0.44
−1.00 −1.00 −1.00 −1.00 −0.25 −0.11 −0.29 −0.72 0.06
0.00
0.00
−1.08
0.60
0.24
0.62
0.45
rM .
The number of uptaken carbon units fc is calculated from the molar masses of FRU and PHB and their stoichiometry factors to
fc = 1.0
1.0
1.0
1.0
Parameter
Value
max kr
[0.21 0.82 0.84 0.73 0.52]T [1/h] [0.10 0.10 0.10 0.10 0.10]T [1/h] 0.01ke [1/h] [5.0 5.0 5.0 5.0 5.0]T [1/h] 0.01 [g/L] 0.06 [g/L] [31] 0.05 [gPHB /gDW ]
ke ˛ ˇ KAMC KFRU KPHB
and growth rate of total biomass (BIO+PHB) is therefore defined as = 0.44
Table 4 Model parameters used for HCM of R. eutropha.
2.51 .
3. Results and discussion 3.1. PHB synthesis and consumption Most models in literature have neglected PHB consumption since they are usually interested only in the synthesis of PHB. However, consumption might be crucial, especially in continuous processes, since it may lower the yield of PHB and degradable PHB can be viewed as an additional substrate. Even in fed batch processes PHB consumption has to be considered if nutrients are fed to the fermenter which will stimulate growth. To identify model parameters experiments were done which include both, the synthesis and consumption of PHB, see Fig. 3. The experimental process can be divided into four stages. In the first stage both complementary substrates fructose (FRU) and ammonium chloride (AMC) are available in the medium and biomass is increasing. This ammonia rich condition is covered by the chosen active modes AM 3 and AM 4, see Table 3. Since there is sufficient nitrogen available in the medium, the NADPH synthesized in reactions (5) and (19) is completely used for synthesis of non-PHB biomass (reactions (27) and (36)). At the end of stage I AMC becomes impoverished. Under this condition the synthesized amount of NADPH cannot be used completely for synthesis of nonPHB biomass. Remaining NADPH is then used for PHB synthesis (reaction 30). This is represented by the active mode AM 2, which synthesizes both, PHB and non-PHB biomass. In stage II AMC is completely exhausted and fructose is now used to synthesize PHB as an internal storage material. This is represented by active mode AM 1. Under this ammonia limited condition NADPH, which is produced in reaction 5 cannot be invested into growth of non-PHB biomass, therefore NADPH and FRU are exclusively used for PHB synthesis. After the cells are significantly filled with PHB (approx. 45%) new AMC is added to the process (stage III) to allow investment of NADPH, internal PHB and remaining fructose into utilization of non-PHB biomass. In stage IV fructose is completely exhausted and growth has stopped. Experiments with glucose as single carbon source, which were performed additionally, show the same behavior. Since the glucose uptake in wild-type R. eutropha is still not fully understood these data are not included in this study, but will published elsewhere.
3.1.1. Model identification and validation Model parameters kr were estimated by fitting the model to the experimental data in Fig. 3. All other parameter values were fixed, since the kr values appeared to be the most sensitive parameters. As mentioned before, the constitutive enzyme synthesis rate has to be much smaller than inducible enzyme rate and ˛ was therefore chosen to be one percent of ke . The model is able to reproduce the data very well, as shown in Fig. 3. The parameter values are listed in Table 4. Although the parameters were fitted to batch data, the prediction of the washout point for a continuous system is very accurate (see next section). For model validation two independent experiments with different initial conditions were performed, which are shown in Fig. 4 left and right column. In the experiment in the left column in Fig. 4 additional ammonium chloride was added to the fermenter similar to the experiment in Fig. 3 to validate the PHB consumption. But in contrast to Fig. 3 PHB content is increasing again after 40 h, since there is still carbon source available in the medium. In contrast to the experiments in Fig. 3 and Fig. 4 left column in the experiment
Fig. 4. Experimental data for model verification (Ammonium chloride: , Fructose: , PHB: , total biomass: ♦) and simulation results (–). Left column: FRU0 =21.0 g/L, AMC0 =0.6 g/L, TBM0 =0.1 g/L, (PHB/TBM)0 =0.0 g/g. Right column: FRU0 =5.05 g/L, AMC0 =1.0 g/L, TBM0 =0.02 g/L, (PHB/TBM)0 =0.0 g/g.
A. Franz et al. / Biochemical Engineering Journal 55 (2011) 49–58
55
shown in Fig. 4 right column additional fructose was added to the fermenter instead of ammonium chloride. Both experiments show very good agreement with the hybrid cybernetic model (solid line) presented in this study. The HCM approach with the extension to a few internal metabolites which is presented in this study is very suitable to describe metabolic behavior in response to changing environmental conditions. 3.2. Nonlinear analysis of continuous cultures Continuous processes have the advantage of longer periods of operating time, which can reduce production costs. In particular the production of PHB is still much more expensive then of conventional plastics. It is therefore not surprising that there are already experimental studies available in literature which deal with continuous PHB production [27–29]. However, in continuous cultures nonlinear phenomenas such as oscillation [14,15] and multiple steady states [16–18] can occur. Therefore, following the idea of Pinto and Immanuel [11] the model is used to investigate the possibility of multiple steady states in a continuous bio reactor. The modeling equations for a continuous process are given by d dt
xFRU xAMC
=D
in xFRU in xAMC
−
xFRU xAMC
+ Ss Z rM c
(37)
d mPHB = Sm,s Z rM − mPHB dt
(38)
de = ˛ + rEM b − diag(ˇ) e − e dt
(39)
dc = ( − D) c dt
(40)
in where D is the dilution rate and xFRU
in and xAMC
are the inlet concentrations of fructose and ammonium chloride. In the remainder the parameter dependent steady state behavior is studied by numerical continuation methods provided in Diva [30]. Principal bifurcation parameters are the dilution rate D and the feed ratio =
xin
AMC
xin
FRU
in was fixed . The fructose feed concentration xFRU
at 20g/L like in the batch experiments. It is worth noting, that the cybernetic model is not differentiable at points where the return on investment of the different modes in Eq. (22) coincide due to the non differentiability of the control variables v according to Eq. (21). Such points are called catch up points and will show up as sharp corners in the solution diagram. For a proper resolution of these corners the step size of the continuation has to be adjusted accordingly. Two bifurcation points (A) and (B) and one catch up point (C) were detected, see Fig. 5. At washout point (A) the dilution rate equals the maximal growth rate (D = max ). The predicted washout point is in very good agreement with published data for the maximal growth rate of Ralstonia eutropha on fructose [31]. Although batch data were used for parameter identification the prediction of the washout point in continuous cultures is very accurate. A small hysteresis is detected over 0.060 ≤ D ≤ 0.063. The hysteresis is bordered by the points (B) and (C). A two parameter continuation of these points with respect to the substrate composition at the inlet of the reactor =
xin
AMC
xin
Fig. 5. Bifurcation diagram of total biomass (TBM) and PHB concentration w.r.t. in in = 20.0 g/L, xAMC = 1.5 g/L. Two bifurcation points were detected: dilution rate. xFRU washout point (A) and turning point (B) and a catch up point (C).
can be viewed as an additional carbon source besides fructose. But still fructose and PHB are not substitutable in the strict sense, since PHB is formed from fructose. However, once PHB is synthesized the organism can choose between fructose or PHB as growth substrate. 3.3. Influence of PHB consumption Fig. 7 illustrates the effect of PHB consumption on biomass in a continuous bio reactor. If PHB consumption is neglected, then the concentrations of total biomass and PHB (not shown) are significantly higher at low dilution rates (D ≤ 0.1 h−1 ) and the hysteresis disappears. As mentioned before, this kind of hysteresis is observed if there are two or more substitutable substrate. If PHB consumption is neglected, then there is only one carbon substrate, namely
and dilution rate D shows
FRU
that the hysteresis unfolds from = 0 and vanishes for > 0.17, see Fig. 6. Such kind of hysteresis is usually observed if there are two or more substitutable substrates available as shown by Namjoshi and Ramkrishna [32] and by Kumar et al. [17]. However in this study there are only two complementary substrates (FRU and AMC), but no substitutable. But as mentioned before, due to the introduction of internal metabolites and their dynamics into the model, new products and substrates are added. The internal metabolite PHB
Fig. 6. Two parameter continuation with respect to dilution rate and .
56
A. Franz et al. / Biochemical Engineering Journal 55 (2011) 49–58
model is most suitable for control, optimization and monitoring of industrial PHB production processes. Nonlinear analysis of a continuous reactor reveals that PHB consumption can reduce the PHB yield significantly at low dilution rates. Depending on the feeding strategy in fed batch processes, PHB consumption can also be stimulated. Therefore PHB consumption has to be included into modeling of continuous and fed batch production processes, which was previously neglected in available literature. Further, the occurrence of multiple steady states was predicted. However, the multiplicity region predicted by the model is rather small and it is argued that multiple steady states are therefore unlikely to occur in practice for this specific system. Acknowledgments A. Franz and A. Kienle appreciate the financial support of the German Federal Ministry for Education and Research (BMBF) under the FORSYS program. The technical support of Ruxandra Rehner during the experiments is greatly acknowledged. Appendix A. Metabolic reactions
Fig. 7. Comparison of biomass steady states with and without PHB consumption.
fructose, and therefore the hysteresis disappears. Furthermore, at low dilution rates and < 0.17 AMC is growth limiting and therefore there is a surplus of fructose. Due to the chosen metabolic objective function of maximizing substrate uptake, PHB concentration and therefore concentration of total biomass are higher if PHB consumption is not considered in the model. At high dilution rates (D > 0.1 h−1 ) the consumption of PHB has no significant effect. Besides continuous processes, the production of PHB in fed batch fermentation was also studied by many researchers [33,29,34]. They have shown that optimal feeding strategies can lead to higher PHB yield and productivity. Lopez et al. [34] used dynamic optimization techniques to obtain optimal feeding profiles for carbon and nitrogen sources. They found, that there are two different phases: biomass growth followed by PHB production. In the first phase carbon and nitrogen are fed into the fermenter to maximize biomass. In the second phase only carbon is fed into the fermenter to maximize PHB. In this scenario PHB consumption is not stimulated. However, if nitrogen source is added to the process, PHB consumption will be stimulated and has therefore to be included into the model. In an industrial process, where a process controller compensates disturbances, this is very likely to occur, if for instance biomass is lower than a certain set point. 3.4. Conclusion In this study, a hybrid cybernetic model based on a detailed metabolic network was developed to describe PHB synthesis as well as PHB consumption. The latter one is usually neglected in available literature. The hybrid cybernetic model approach was extended to include internal metabolites with slow dynamics, since PHB, which is an internal metabolite, cannot be characterized by fast dynamics. To identify reasonable parameters experiments were performed which include both: synthesis and consumption of PHB. The developed model shows very good agreement with experimental data sets. It was validated with independent experiments. Due to various desirable features such as accounting for cellular regulation at network level and dynamics of intracellular metabolites with a moderate complexity, it is believed that the constructed
No.
Reaction
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
FRU + PEP + ATP → F16P + PYR + ADP F16P → F6P F16P → 2G3P AMC → NH3 G6P + 2NADP → Rl5P + CO2 + 2NADPH Rl5P ↔ R5P Rl5P ↔ X5P X5P + R5P ↔ S7P + G3P S7P + G3P ↔ E4P + F6P X5P + E4P ↔ G3P + F6P F6P → G6P G3P + NAD + ADP ↔ 3PG + NADH + ATP 3PG ↔ PEP PEP + ADP → PYR + ATP OXA + ATP → PEP + ADP + CO2 PYR → AcCoA + Form PYR + NAD → AcCoA + NADH + CO2 AcCoA + OXA → ISC ISC + NADP → ˛KG + NADPH + CO2 ˛KG + NAD → SucCoA + NADH + CO2 SucCoA + ADP ↔ SUC + ATP SUC + FAD ↔ MAL + FADH MAL + NAD → OXA + NADH PYR + ATP → OXA + ADP ISC → SUC + GOX AcCoA + GOX → MAL NH3 + ˛KG + NADPH → GLUT + NADP GLUT + NH3 + ATP → GLUM + ADP 2AcCoA ↔ AcAcCoA AcAcCoA + NADPH → PHB + NADP PHB + NAD → ACE + NADH ACE + SucCoA → AcAcCoA + SUC SUC → SUCx 2NADH + O2 + 4ADP → 2NAD + 4ATP 2FADH + O2 + 2ADP → 2FAD + 2ATP 0.21G6P + 0.07F6P + 0.9R5P + 0.36E4P + 0.13G3P + 1.53PG + 0.52PEP + 2.83PYR + 3.74AcCoA + 1.79OXA + 8.32GLUT + 0.25GLUM + 41.1ATP + 8.26NADPH + 3.12NAD → BIO + 7.51 ˛KG + 2.61CO2 + 41.1ADP + 8.26NADP + 3.12NADH
All stoichiometry coefficients are given in mmol, except BIO is given in g. Appendix B. Nomenclature
b c
fraction of catalytically active parts in biomass biomass concentration
A. Franz et al. / Biochemical Engineering Journal 55 (2011) 49–58
D e emax erel fc h ke kr KAMC KFRU KPHB m mf ms mPHB p r rM rkin M rEM rkin EM Sm Sm , f Sm , s Sp Ss u v x xs xp xAMC xFRU in xAMC in xFRU y ym Z Zy M− M+ ˛ ˇ
dilution rate vector of enzyme levels vector of maximum enzyme levels vector of relative enzyme levels vector of uptaken carbon units weight vector for the modes of Zy vector of enzyme synthesis rate constants vector of rate constants saturation constant for ammonium chloride saturation constant for fructose saturation constant for PHB vector of specific internal concentrations vector of specific internal concentrations with fast dynamics vector of specific internal concentrations with slow dynamics specific concentration of PHB vector of ROI vector of regulated fluxes vector of regulated fluxes through EMs purely kinetic part of rM vector of regulated enzyme synthesis rates of EMs purely kinetic part of rEM stoichiometry matrix of internal metabolites stoichiometry matrix of internal metabolites with fast dynamics stoichiometry matrix of internal metabolites with slow dynamics stoichiometry matrix of products stoichiometry matrix of substrates vector of cybernetic variables which control enzyme synthesis vector of cybernetic variables which control enzyme activity vector of external concentrations vector of substrate concentrations vector of product concentrations concentration of ammonium chloride concentration of fructose concentration of ammonium chloride at inlet concentration of fructose at inlet state vector of a metabolic system vector of measured yield data EM matrix normalized EM matrix set of internal metabolite consuming EMs set of internal metabolite producing EMs vector of constitutive enzyme synthesis rates vector of enzyme consumption constants feed composition growth rate
B.1. Abbreviations
AM CM EM HCM ROI PHA TBM MYA YS
Active mode Cybernetic model Elementary mode Hybrid cybernetic model Return on investment Polyhydroxyalkanoates Total biomass (dry weight) Metabolic yield analysis Yield space
57
B.2. Metabolites
3PG 3-phosphoglycerate ˛KG Alpha-ketoglutarate AcAcCoA Acetoacetyl CoA AcCoA Acetyl CoA ACE Acetoacetate ADP Adenosine diphosphate ATP Adenosine triphosphate AMC Ammonium chloride BIO Non-PHB biomass Carbon dioxide CO2 E4P Erythrose-4-phosphate F16P Fructose-1,6-Bisphosphate F6P Fructose-6-Phosphate FAD Flavin adenin dinucleotide, oxidized FADH Flavin adenin dinucleotide, reduced Form Formiate FRU Fructose G3P Glyceraldehyde-3-phosphate G6P Glucose-6-phosphate GLUM Glutamine GLUT Glutamate GOX Glyoxylate ISC Isocitrate MAL Malate NAD Nicotinamide adenine dinucleotide, oxidized NADH Nicotinamide adenine dinucleotide, reduced NADP Nicotinamide adenine dinucleotide phosphate, oxidized NADPH Nicotinamide adenine dinucleotide phosphate, reduced NH3 Ammonia O2 Oxygen OXA Oxaloacetate PEP Phosphoenol pyruvate PHB Poly(-hydroxybutyrate) PYR Pyruvate R5P Ribose-5-phosphate Rl5P Ribulose-5-phosphate S7P Sedoheptulose-7-phosphate SUC Succinate SucCoA Succinate CoA SUCx Succinate, external X5P Xylulose-5-phosphate References [1] J.I. Kim, J.D. Varner, D. Ramkrishna, A hybrid model of anaerobic E. coli GJT001: combination of elementary flux modes and cybernetic variables, Biotechnol. Prog. 24 (2008) 993–1006. [2] F. Reinecke, A. Steinbüchel, Ralstonia eutropha strain H16 as model organism for PHA metabolism and for biotechnological production of technically interesting biopolymers, J. Mol. Microbiol. Biotechnol. 16 (2009) 91–108. [3] A.J. Anderson, E.A. Dawes, Occurrence, metabolism, metabolic role, and industrial uses of bacterial polyhydroxyalkanoates, Microbiol. Rev. 54 (1990) 450–472. [4] J.A. Asenjo, J.S. Suk, Kinetics and models for the bioconversion of methane into an intracellular polymer poly-beta-hydroxybutyrate, Biotechnol. Bioeng. Symp. 15 (1986) 225–234. [5] A. Mulchandani, J.H.T. Luong, C. Groom, Substrate-inhibition kinetics for microbial-growth and synthesis of poly-beta-hydroxybutyric acid by Alcaligenes eutrophus ATCC-17697, Appl. Microbiol. Biotechnol. 30 (1989) 11–17. [6] D.S. Kompala, D. Ramkrishna, G.T. Tsao, Cybernetic modeling of microbial growth on multiple substrates, Biotechnol. Bioeng. 26 (1984) 1272–1281. [7] D. Ramkrishna, A cybernetic perspective of microbial growth, ACS Symp. Ser. 207 (1983) 161–178. [8] D. Ramkrishna, D.S. Kompala, G.T. Tsao, Are microbes optimal strategists, Biotechnol. Prog. 3 (1987) 121–126. [9] S. Yoo, W.S. Kim, Cybernetic model for synthesis of poly-beta-hydroxybutyric acid in Alcaligenes eutrophus, Biotechnol. Bioeng. 43 (1994) 1043–1051.
58
A. Franz et al. / Biochemical Engineering Journal 55 (2011) 49–58
[10] K.G. Gadkar, F.J. Doyle, T.J. Crowley, J.D. Varner, Cybernetic model predictive control of a continuous bioreactor with cell recycle, Biotechnol. Prog. 19 (2003) 1487–1497. [11] M.A. Pinto, C.D. Immanuel, Sensitivity of bifurcation traits to model parameters in poly-beta-hydroxy butyrate production, in: ADCHEM. [12] J. Stelling, S. Klamt, K. Bettenbrock, S. Schuster, E.D. Gilles, Metabolic network structure determines key aspects of functionality and regulation, Nature 420 (2002) 190–193. [13] J.D. Young, K.L. Henne, J.A. Morgan, A.E. Konopka, D. Ramkrishna, Integrating cybernetic modeling with pathway analysis provides a dynamic, systems-level description of metabolic control, Biotechnol. Bioeng. 100 (2008) 542–559. [14] J. Weber, A. Kayser, U. Rinas, Metabolic flux analysis of Escherichia coli in glucose-limited continuous culture. ii. Dynamic response to famine and feast, activation of the methylglyoxal pathway and oscillatory behaviour, Microbiology-SGM 151 (2005) 707–716. [15] K.D. Jones, D.S. Kompala, Cybernetic model of the growth dynamics of Saccharomyces cerevisiae in batch and continuous cultures, J. Biotechnol. 71 (1999) 105–131. [16] I. Disli, A. Kremling, A. Kienle, A model based analysis of multiple steady states in continuous cell cultures, in: Proceedings MATHMOD 09 Vienna - Full Papers CD Volume. [17] M.V. Kumar, K.P. Zeyer, A. Kienle, S. Pushpavanam, Conceptual analysis of the effect of kinetics on the stability and multiplicity of a coupled bioreactorseparator system using a cybernetic modeling approach, Ind. Eng. Chem. Res. 48 (2009) 10962–10975. [18] A.A. Namjoshi, W.S. Hu, D. Ramkrishna, Unveiling steady-state multiplicity in hybridoma cultures: The cybernetic approach, Biotechnol. Bioeng. 81 (2003) 80–91. [19] J.H. Law, R.A. Slepecky, Assay of poly-beta-hydroxybutyric acid, J. Bacteriol. 82 (1961) 33. [20] H.S. Song, D. Ramkrishna, Reduction of a set of elementary modes using yield analysis, Biotechnol. Bioeng. 102 (2009) 554–568. [21] H.S. Song, J.A. Morgan, D. Ramkrishna, Systematic development of hybrid cybernetic models: application to recombinant yeast co-consuming glucose and xylose, Biotechnol. Bioeng. 103 (2009) 984–1002.
[22] A. von Kamp, S. Schuster, Metatool 5.0: fast and flexible elementary modes analysis, Bioinformatics 22 (2006) 1930–1931. [23] S. Klamt, J. Saez-Rodriguez, E.D. Gilles, Structural and functional analysis of cellular networks with CellNetAnalyzer, BMC Syst. Biol. 1 (2007). [24] J.D. Young, D. Ramkrishna, On the matching and proportional laws of cybernetic models, Biotechnol. Prog. 23 (2007) 83–99. [25] B.G. Turner, D. Ramkrishna, Revised enzyme-synthesis rate expression in cybernetic models of bacterial-growth, Biotechnol. Bioeng. 31 (1988) 41–43. [26] T. Katoh, D. Yuguchi, H. Yoshii, H.D. Shi, K. Shimizu, Dynamics and modeling on fermentative production of poly (beta-hydroxybutyric acid) from sugars via lactate by a mixed culture of Lactobacillus delbrueckii and Alcaligenes eutrophus, J. Biotechnol. 67 (1999) 113–134. [27] S.T. Yu, C.C. Lin, J.R. Too, Phbv production by Ralstonia eutropha in a continuous stirred tank reactor, Process. Biochem. 40 (2005) 2729–2734. [28] S.T. Wu, Y.C. Lin, J.R. Too, Continuous production of poly(3-hydroxybutyrateco-3-hydroxyvalerate): Effects of c/n ratio and dilution rate on hb/hv ratio, Korean J. Chem. Eng. 26 (2009) 411–416. [29] S. Khanna, A.K. Srivastava, Optimization of nutrient feed concentration and addition time for production of poly(beta-hydroxybutyrate), Enzyme Microb. Technol. 39 (2006) 1145–1151. [30] M. Mangold, A. Kienle, E.D. Gilles, K.D. Mohl, Nonlinear computation in DIVA–methods and applications, Chem. Eng. Sci. 55 (2000) 441–454. [31] C. König, H.G. Schlegel, Prolonged organotrophic growth of hydrogenomonas H16 in chemostat, Arch. Mikrobiol. 62 (1968) 41–55. [32] A.A. Namjoshi, D. Ramkrishna, Multiplicity and stability of steady states in continuous bioreactors: dissection of cybernetic models, Chem. Eng. Sci. 56 (2001) 5593–5607. [33] S. Shahhosseini, Simulation and optimisation of PHB production in fed-batch culture of Ralstonia eutropha, Process. Biochem. 39 (2004) 963–969. [34] J.A. Lopez, V. Bucala, M.A. Villar, Application of dynamic optimization techniques for poly(beta-hydroxybutyrate) production in a fed-batch bioreactor, Ind. Eng. Chem. Res. 49 (2010) 1762–1769.
