Advanced Kinetic Modeling of Bio-co-polymer Poly(3-hydroxybutyrate-co-3-hydroxyvalerate) Production using Fructose and Propionate as Carbon Sources

Biopolymers are a promising alternative to petroleum-based plastic raw material. They are bio-based, non-toxic and degradable under environmental conditions. In addition to the homopolymer poly-3-hydroxybutyrate (PHB), there are a number of co-polymers that have a broad range of applications and are easier to process in comparison to PHB. The most prominent representative from this group of bio-copolymers is poly(3-hydroxybutyrate-co-3-hydroxyvalerate) (PHBV). In this article, we show a new kinetic model that describes the PHBV production from fructose and propionic acid in Cupriavidus necator ( C. necator ). The developed model is used to analyze eﬀects of process parameter variations such as the CO 2 amount in the exhaust gas and the feed rate. The presented model is a valuable tool to improve the microbial PHBV production process. Due to the coupling of CO 2 online measurements in the exhaust gas to the biomass production, the model has the potential for predicting the composition and the current yield of PHBV in the ongoing process.


Introduction
Because of their broad range of possible applications, plastics have changed our lives in the last decades. Not only the transport and shelf life of food can be guaranteed through the use of plastic packaging materials, plastics have also become indispensable in other manufacturing industries (e.g. the automotive industry) [30]. In 2019, 369 million tons of different plastics were produced worldwide, 16 % of them in Europe [38]. On the one hand, the longevity of plastic is an advantage considering a production of durable material. However, this property makes it very difficult to find suitable, save and environmental friendly disposal strategies and thus, a major problem for our environment has emerged. In the EU, only 9.4 million tons of plastic were recycled in 2018, that is approximatley 7 % of total production in Europe [38]. In the best end-of-life scenario, waste plastics are used for the production of energy. The rest must be stored in landfills for a long time. Finally, man-made plastic waste is found in different habitats (e.g. oceans, forests). In addition to efforts for avoiding plastic waste [1] and the development of improved recycling strategies at various levels [5,30,2], a large scientific community has been focusing on the improvement of production processes for alternative plastic raw material [44,3]. On the one hand, purely biomass-based alternatives have been developed, e.g. disposable cutlery and crockery made from leaves, coconut shells and fibers [20]. On the other hand, there is a wide range of research on the production of plastic alternatives using renewable feedstocks as educts or substrates of a production process. One very prominent biodegradable and bio-based plastic alternative represents the group of polyhydroxyalkanoates (PHAs) [48,26]. These polyesters stand out because of its favorable processing properties, e.g. melting behaviour or blending options [7]. Further, it is produced microbially by many bacteria and some archaea using a wide variety of non-fossil carbon sources [48]. The repertoire of possible and cheap substrates are inexpensive sugars in waste streams from manufacturing industry (juice production, sugar cane processing), volatile fatty acids (VFAs) from biogas plants and waste water in sewage treatment plants and even CO 2 [33,27,40,6,19,22,4,41]. Beside the diverse possibilities of producing PHAs microbially and bio-based, this plastic raw material has another important property: PHAs are degradable under environmental conditions [29]. All described properties make PHAs an ecologically friendly alternative in comparison to conventional petroleum-based plastic raw materials. However, from an economic point of view, the industrial production of PHAs is about five times more expensive than the production of petroleum-based polymers [21]. In addition to an improved extraction and processing of the polymers, a large part of the costs can be saved through optimized bioprocesses with increased PHA yield. This can be achieved by incorporation of sophisticated experimental investigation of different process modes or the optimization of substrates and feeding strategies with mathematical modeling [3,26,37]. Further, model approaches represent the basic component in the development of advanced process control and intensification strategies [8,32]. In the research area of PHA production a large number of models can be found that differ greatly in terms of modelling approach and complexity. Due to the complexity and variability of the bioprocess, there is no universal tool for predicting product yields regardless of the producing organism, bioreactor or process conditions such as temperature or pH [34]. Some approaches appear promising due to their simplicity and are able to reproduce the concentration curves in a qualitative manner, while they contain only little metabolic information [45,10,9]. Other approaches take the metabolism into account but, due to their complexity, they can only be used to a limited extent for model-based process control intensification and are difficult to transfer to other PHA producers [13,17,24,14,16,12]. Many of the modeling approaches focus on the microbial production of the best-known representative from the group of PHAs: PHB. From an industrial point of view, however, the experimentally well-investigated bio-co-polymer PHBV is more interesting, because of its lower melting temperature, higher elongation-to-break values and higher biocompatibility in comparison to PHB [47]. Yet, only a few model approaches already exist to investigate microbial PHBV production [9,23]. PHBV is also the target product in the present work. In order to develop a universal simulation tool, the mathematical model must contain a balanced amount of metabolic information. Such an modeling approach is rarely found in literature: In [9] the description of the metabolism was reduced to central points w.r.t. PHA production (e.g. acetyl-CoA production), which can be found in many organisms in mixed cultures. In the mathematical model presented here, we apply a time-dependent, kinetic parameter for the formation of residual biomass from fructose and propionic acid in C. necator, in order to map the dynamics of the present metabolic activity without detailed metabolic information. As the researchers at the university of Antioquia (Colombia) already have shown [18,35], the online measurement for the CO 2 content in the exhaust gas serves as excellent measure of the dynamic growth rate. Due to the coupling to the online data for CO 2 in the exhaust gas, the model is also a suitable candidate for the online estimation of the PHBV product yields and polymer composition. The present model was adapted to two datasets with aerobic PHA production in C. necator, one with only fructose as carbon source and the other with fructose and propionic acid as carbon sources. Further, the model is used in a simulation study to investigate the influence of the feed rate for the propionic acid and of constant CO 2 in the exhaust gas on the bio-co-polymer yield and the hydroxyvalerate (HV) concentration in the polymer. An application of closed-loop control approaches in a PHBV production with molasses and vinasses as substrates using a simple unstructured model in [18] and [35] shows the potential of such models as soft sensors to increase productivity. The model approach used in [35] can be used for the simulation of the dynamic evolution of the concentrations of the total bio-co-polymer, but cannot differentiate between HV and HB in the polymer. In the model presented in our manuscript, the dynamic behavior of HB and HV concentrations can be simulated and thus, it provides an important part for the soft sensor development predicting the bio-co-polymer content and HB/HV ratio in the polymer. By coupling it to polymerisation kinetics [11], it will be possible to additionally estimate the chain length distribution during the process.

Mircoorganism and cultivation conditions
C. necator (H16, DSM 428) obtained from DSMZ GmbH Braunschweig was used for the fermentations. Bacteria were precultured in shake flask with 10 vol% LB medium (Carl Roth, Karlsruhe, Germnay) at 30 • C and 150 rpm. After reaching an optical density of 4 at 600 nm, the bacteria were transferred to an shake flask filled with 10 vol% of M81 medium 1 supplemented with 20 g/L fructose and 1.5 g/L ammonium chloride. The M81 preculture was grown until an OD of 4.8 and used as inoculum for the bioreactors. The fermentation were performed in a DASGIP parallel bioreactor system (Eppendorf AG, Juelich, Germany) with an inoculation OD of 0.4. During the experiments the pH was kept at 6.8 and the dissolved oxygen (DO) was 70%. In case of fructose as single carbon source, the pH-control was performed with 2 M H 2 SO 4 . During the reactor experiment with fructose and propionic acid the pH was stabilized with 20 g/L solution propionic acid as shown in [25]. The initial conditions for the reactor experiments are shown in Table 1. All bioreactor experiments were performed with M81 media at 30 • C.

Determination of total biomass
For the determination of total biomass (TBM), 1 ml culture broth was centrifuged for 10 minutes at 4000 rpm and 4 • C (VWR MicroStar 17R, Pennsylvania, USA). In a second step, the cell pellet was dried over night at 80 • C and weighted.

Enzyme Assay
By using enzymatic test kits (Kit No. 5390 and No. 10139106035, R-Biopharm AG, Darmstadt, Germany) and following the manufactures instructions ammonium and fructose concentrations were determined from supernatant of the sample.

High pressure liquid chromatography
Concentrations of hydroxybutyrate (HB) and hydroxyvalerate (HV) were determined applying the procedure published in [15] using an Agilent 1100 high performance liquid chromatography (HPLC). For this, 1 mL of the culture broth was alkaline digested as reported in [42]. The samples were filtered through a 0.25 µm nylon membrane and 10 µL were loaded on the reverse phase column (Inertsil 100A ODS-3, 5 µm poresize, 250x4.6mm, MZ-Analysentechnik GmbH, Mainz, Germany) and and eluted isocratically with 1 mL·min −1 at 60 • C with 92% low concentrated H 2 SO 4 (0.025% solution, Carl Roth, Karlsruhe) and 8% acetonitrile (Carl Roth, Karlsruhe). The HB and HV concentrations in the polymer chains of the samples were determined by using crotonic (Carl Roth, Karlsruhe, Germany) and 2-pentenoic acid standard samples (Sigma Aldrich, St. Louis, USA), respectively. In parallel, a PHBV sample (12% HV, Sigma-Aldrich /Merck, Darmstadt, Germany) with known concentration must be measured to calculate the conversion yields Y HB and Y HB [15]: Here, the dilution ratio (D) is 2, c HB is the known HB and c HV the known HV concentration of the PHBV test sample. Due to the standard measurement of crotonic acid c CA and 2-pentenoic acid c P A , the conversion yields Y HB and Y HV can be determined, respectively. Detection takes place with a photodiode-array detector (G7115A, Agilent,Waldbronn, Germany) at 210 nm.

Kinetic modeling approach
The description of the formation and degradation in microbial PHA production is an important building block for the complete production process. There are already a number of model candidates for the formation of PHB [28,10,34,17] that can accurately reflect the development of the homopolymer concentration over time. Compared to the homopolymer PHB, the copolymer PHBV has significantly improved processing properties. However, so far only simple kinetic approaches for the formation of PHBV have been developed [9]. Further, the model fromŠpoljarić and colleagues was developed for the conversion of fatty acid methyl esters (FAMES) from biofuel to PHBV using lumped metabolic pathways [45]. The model presented here, describes the formation and degradation of HB and HV in the polymer chains using fructose and propionate, two carbon sources that frequently occur in inexpensive residues or can be produced form them, e.g. by using waste streams from juice, cheese and paper production. In our model approach, detailed metabolic reaction pathways were not taken into account to keep the model structure as simple as possible.
The following assumptions were made for the model: • A simple mass-action kinetic is assumed for the dynamics of the substrates.
• Propionate has an inhibiting effect and decelerates the growth of bacteria [25].
• The conversion of PHA into enzymatically active biomass (residual) is not affected by external propionic acid concentrations, as this is an internal process.
• C. necator begins to produce PHA already before nitrogen is depleted [34]. This behavior is considered in the model via an inhibitory term with nitrogen by a Michaelis-Menten kinetics approach (see Equation 3, term inh 2 ).
• Steric effects in the granules prevent the further production of PHA after reaching a total amount of 89 % of the total biomass (TBM, P t,max ) [28,31].
In the following, a set of ordinary differential equations for the dynamics of the system with fructose and propionic acid as substrates and residual biomass, HB and HV in the polymer chains as products is described. The dynamic state equation for the fructose concentration is given as Fructose can be metabolized for biomass production with rate parameter k 1 , the accumulation of HB in the polymer with k 4 or the conversion to CO 2 with k 7 . The growth of biomass through fructose is controlled by the activity coefficient b CO 2 (t) based on the CO 2 ratio in the exhaust gas and inhibited by the concentration of propionate with the term inh 1 . At a concentration of 1.5 g/L propionic acid (c p,inh ) the substrate uptake for biomass is completely inhibited [25]. Since CO 2 in the exhaust gas is often defined as proportion of the gas composition, we have chosen the relative CO 2 proportion to describe the metabolic activity b CO 2 (t) as follows: The metabolic activity is described by the quotient of CO 2,out in the exhaust gas and CO 2,in in the fresh inlet air. Since C. necator is a PHA producer of group 2 according to Novak and Koller [34], the build-up of HB from fructose begins when there is still small amount of ammonium in the medium. This effect is modeled by the term inh 2 . As described in [28], steric effects at high polymer concentrations inhibit the conversion of substrates to PHA (term inh 3 ). According to literature values [48], the maximum achievable amount P t,max is 0.89 (89 % of the total biomass). The inhibitory steric effect is given as the ratio between overall HA concentration and total biomass concentration Finally, the dilution factor in the fed-batch process is the ratio of the feed flow rate F in and reactor volume V . For the computational study a volume balance is necessary: As for fructose, a state equation can be set up for propionate dynamics It describes the consumption of propionate for biomass with rate coefficient k 2 , CO 2 with k 8 and HB production with k 5 . In addition to the generation of HB, propionate can also be converted to HV (k 8 ). In fed-batch mode a propionate solution according to Table 1 is feed to the system with the feed flow rate F in . For the growth the organisms need ammonium. The state equation for the ammonium dynamics is In addition to the ammonium uptake for biomass growth by consuming external carbon sources (first term in Equation 9), ammonium is needed to convert the biopolymer to catalytically active biomass with the degradation rate parameter k 3 . The dynamical behavior of residual (non-PHA, catalytically active) biomass is described as follows Residual biomass is produced through the consumption of external carbon sources such as fructose and propionate and the conversion of HB and HV from the polymer chains in presence of ammonium. The following ordinary differential equations (ODEs) account for the dynamics of the monomers HB and HV in the polymer chains For the accumulation and breakdown of biopolymer (HB and HV), CO 2 formation is negligible, since the metabolic reaction pathways produce only little CO 2 compared to the breakdown of sugars and organic acids into catalytically active components of the total biomass. For the simulation of the model, the temporal evolution of the CO 2 amount in the exhaust gas from reactor experiments is required. Since the online measurement often fluctuates and does not provide a smooth curve, the data were interpolated to integrate them into the ODE model. For this, a smoothing spline interpolation was carried out with the MATLAB command csaps. For both experiments different smoothing factors were evaluated. As smoothing factor 0.2 was selected for the experiment with fructose as the only carbon source (data set 1), while the data from the reactor experiment with fructose and propionic acid as carbon sources (data set 2) achieved a smooth and well fitted curve with a smoothing factor of 0.02. For the interpolation, the splines are evaluated at the sampling points. The evaluation was carried out with the MATLAB command ppval. The curves and online data are shown in the Figure 1 for both data sets.
The feeding of the odd carbon source propionic acid was achieved by pH-control. If the pH increases, a certain amount of propionic acid with 20 g/L in the feed was added to the bioreactor to stabilize the pH at 6.8. The pre-implemented PI controller of the DASGIP parallel bioreactor system (Eppendorf AG, Jülich, Germany) was used for this purpose. As for the activity factor, frequent fluctuations are observed because of the special pHdependent feeding strategy and thus the feed rate for propionic acid was interpolated and evaluated in the same way as the CO 2 amount in the exhaust gas ( Figure 2). Here,  a factor of 0.02 delivered a smooth curve. Further, an interpolation of the volume was necessary for the fed-batch experiment with fructose and propionic acid as carbon sources (data set 2). For this purpose, a polynomial of order 10 was determined with the MATLAB command polyfit and evaluated with polyval at the sampling times. As seen in Figure 3, volume reduction by sampling was also taken into account. Hence, a decrease in volume is recorded despite an average feed rate of approx. 20 ml/h between 15 and 25 h (see Figure 2). In the experiment with fructose as single carbon source, the reactor was operated in batch mode. Since it is assumed that the system is ideally mixed, the changes in concentration are only caused by internal sinks and sources (no substrate is pumped in), the volume reduction due to sampling can be neglected in data set 1. All simulations, interpolations and evaluations were carried out with MATLAB 2019b.

Parameter identification
For parameter identification the following objective function was minimized: Here, the error between simulated x sim and experimental data x exp at time point t i is determined and weighted with the maximum value in the experimental data set. The kinetic parameters were determined using the algorithm fmincon in MATLAB 2019b. The ODEs were numerically solved with the algorithm ode15s with a relative tolerance of 10 −9 . To avoid a sub-optimal initial parameter set a multi-start approach with N = 10000 was applied. To further validate the resulting parameter set obtained by the local optimization strategy, the global optimization algorithm Differential Evolution (DE) was selected [46]. The resulting set of parameters can be found in Table 1.
To proof parameter identifiability, profile likelihoods are determined for the parameter set [39]. Supplement B shows all profile likelihoods and the corresponding likelihood-based confidence intervals. All profile likelihoods show a distinct minimum at the estimated parameter set and thus all parameters and the model itself are (locally) identifiable. The confidence intervals are obtained by calculating the value of a χ 2 -distribution with a confidence level α=0.95 and one degree of freedom as proposed in [39].

Identification using different data sets
The kinetic model was adapted to two data sets from bioreactor experiments with a working volume of 1.2 L. In the first data set, fructose was the only carbon source that was metabolized under aerobic conditions. The second data set was also obtained under aerobic conditions with fructose and propionic acid as carbon sources. Here, propionic acid was added via a pH-regulated feed as proposed by Kim and coworkers [25] aiming on a constant pH value of 6.8. From the available data of these sets, the online data for the CO 2 content in the inflow and in the exhaust gas, the feed rate for the propionic acid and the exact volume considering the sampling volume were used in case of data set 2. In case of data set 1 online data for the CO 2 content in the inflow and in the exhaust gas were used too, but a constant volume and a batch mode (F in =0) were considered. The CO 2 content in the exhaust gas, the feed stream for the propionic acid and the reactor volume were approximated as described in section 3.1.1. The smoothed measurement data were used for the model simulation. Furthermore, the concentrations for total biomass, biopolymer, fructose and propionic acid were determined offline in both data sets (see section 2). The dynamic behavior for the conversion of the substrates from data set 1 (only fructose) can be reproduced well with the present model (solid lines, Figure 4(a)). The model shows larger deviations for the substrates from data set 2 (fructose and propionic acid), especially in the last time segment from 25 h (dashed lines, Figure 4(b)). On the  one hand, this is due to the approximation of the inflow rate for propionic acid (see Figure  2), and on the other hand, measurement of the propionic acid in the medium becomes more difficult. It seems to be, that there are more and more apoptosis fragments in the culture supernatant that disrupt the signal obtained by HPLC (own experimental findings).
The model for the case of fructose as a single substrate can reproduce the production and depletion of total biomass and HB with sufficient accuracy (Figure 5(a)). The same applies to the case with fructose and propionic acid as carbon sources ( Figure 5(b)). In particular, the degradation of HB and HV in the polymer chains after an NH 4 Cl shot at 24 h can be mapped very well by the model. This property is important when working with waste streams which also contain nitrogen sources and which are added during the ongoing process in order to keep the carbon in excess. Since the model can reproduce the product concentrations very well, it can be further used fora simulation study.

Computational study
In the following, the identified model was used to investigate the maximum product concentrations applying different constant CO 2 and propionic acid feed profiles. A constant feed rate can be easily implemented on every bioreactor system with continuous pumps. The constant CO 2 in the exhaust gas represents a complex control task, as CO 2 is produced by the bacteria themselves (autogenous CO 2 ). However, the research work in [43] shows the feasibility of this control task which can also be used in future validation experiments.
In the simulation study ( Figure 6), a constant CO 2 proportion in the exhaust gas and a constant feed rate for propionate were assumed for the process time. The propionic acid concentration in the feed was set to 20 g/L as in data set 2. Figure 6(a) shows the maximum total HB and HV in the polymer chains P t * as a function of CO 2 in the exhaust gas and the feed rate for propionic acid. The inhibitory area is clearly visible on the left side. In this range, the propionic acid concentration in the medium becomes too high so that growth is inhibited. In case of increased CO 2 values in the exhaust gas, the feed rate can be increased too without triggering growth inhibition. This behavior can be justified as follows: A higher exhaust gas value for CO 2 produced by the microorganisms (autogenous CO 2 ) is triggered by an increased uptake of substrates from the medium. In C. necator the degradation of pentoses and hexoses takes place via the Entner-Doudoroff (ED) pathway. Furthermore, the tricarboxylic acid cycle (TCA) is mainly responsible for the generation of energy and precursor molecules for biomass synthesis from precusors of metabolic sugar degradation and organic acids. Both, the ED pathway and the TCA, generate CO 2 as a by-product, which can be found in the exhaust gas values of the simulation study. With other words, an increased autogenous CO 2 in the exhaust gas can be translated to stronger residual biomass growth with increased substrate uptake. As a result, the growth inhibition with increased CO 2 in the exhaust gas only occurs at higher feeding rates for propionic acid. Beside this beneficial effect of higher exhaust CO 2 , the total biopolymer concentration is decreasing in the non-inhibitory area. This effect occurs, because the CO 2 output, is related to the higher residual biomass growth and hence, the substrates were less translated into biopolymers. The highest total biopolymer concentration of 12.5 g/L within 60 h simulation time can be achieved without feeding propionic acid to the system and with a very low CO 2 content in the exhaust gas. But without an active feed rate no HV will be produced and the HV content is decisive for improved processing of copolymers compared to their homopolymers. The maximum HV concentration in the polymer is shown in Figure 6(b) for different constant autogenous CO 2 and propionic acid feeding rates. Here, the inhibiting region can be seen again due to an high propionate concentration in the medium. For a high HV concentration our model predicts feed rates between 12 mL/h and 40 mL/h depending on the CO 2 in the exhaust. In general, less than 2.5 % CO 2 in the exhaust gas, leads to higher HV concentrations. As our simulation results show, the feed rate for propionic acid must be chosen very carefully because of it's strong correlation to the CO 2 value: Too high feed rates at higher CO 2 in the exhaust gas lead to a less high residual biomass and further a decreased HV concentration (see 8). Further characteristic values of the simulation study are shown in Figure 8 (e.g. residual biomass at maximum total HB and HV concentration, total monomer/total biomass ratio). Three exemplary time courses for different production goals were illustrated in Figure 7. In case A the dynamic behaviour to achieve a high total polymer concentration (HB+HV) with the given initial conditions is shown. For this purpose the feed rate was set to 0 mL/h and the CO 2 amount to 1 %. For case B the same CO 2 value as in case A was applied together with a feed rate of 25 mL/h to show an example time course for an high HV concentration. In addition to the two preferred fermentation results (case A and B, Figure 7), the inhibitory case is also shown (case C, , Figure 7) by increasing the feed rate to 105 mL/h.

Concluding Remarks
In this manuscript a model approach is presented, which enables integration of online data for the estimation of the yield and the composition of the copolymer PHBV in C. necator. Compared to other approaches, no genome-scale metabolic networks or reduced variants are necessary [36,17], since it is a pure kinetic approach that describes changes in the metabolism due to a CO 2 -dependent biomass production rate that changes over time. Despite the lack of detailed metabolic information, our kinetic model can display the data sets with fructose as single substrate and fructose and propionate as substrate with high accuracy. The model approach has a similar complexity as the models presented in Koller [28] and Dias [9], whereby the metabolic activity that controls the biomass growth can be described by the CO 2 profile in the exhaust gas. Another unstructured model approach successfully applied as soft sensor for the prediction of biomass, substrates and total biopolymer in a vinasse-molasse PHA process needs some extension to give information about the HV and HB ratios in the polymer during the process [18,35]. Hence, in comparison to the model approach used in [18,35] our more complex model represents an excellent basis to predict HV and HB concentrations of the  system from available online measurements. Further the model approach can be used for the design of observers and state estimators for the reconstruction of non-measurable states. A first example in this direction was presented recently by Carius and coworkers [8]. Here, a Unscented Kalman Filter and a Moving Horizon Estimation based on a hybrid cybernetic PHB model [17] was designed and evaluated. Future work should focus on the design of model-based soft sensor approaches, since it will enable reliable online estimation of the PHBV content using measurements of the exhaust gas online without the need of additional expensive hardware sensors. Furthermore, the transferability of the kinetic model to other PHA producers must be researched. Here, the focus should be on PHA producers, which are already producing PHA under growth conditions, e.g. C. necator DSM 515, as the model was designed for this group of bacteria. Finally, a control concept should be developed which is able to keep CO 2 in the exhaust gas on a desired level over a longer period of time. The work of Shang and colleagues [43] shows that it is in principle possible to control a process  parameter that is strongly influenced or caused by the bacteria. The CO 2 amount in the exhaust gas was adjusted in the work of Shang and coworkers in order to investigate the inhibitory effects of CO 2 . Such an effect has not yet been taken into account in the model presented in this manuscript, but should be considered in future model extensions. Finally, the model was used to investigate the influence on the polymer yield and the HV proportion in a simulation study with different CO 2 fractions in the exhaust gas and constant feed rates for propionic acid. The lower the feed rate was set and the less CO 2 was in the exhaust gas, the more PHA was produced but with less HV content in the polymer. A suitable feed rate for propionic acid input was predicted to be between 12 mL/h and 40 mL/h in order to achieve high HV concentration in the final co-biopolymer. Future work should focus on the prediction of optimal CO 2 profiles to increase both biomass growth and PHA accumulation. All in all, our model approach provides the basis for a broad range of possible future applications and will help to make the production process of bio-polymers more reliable and less expensive.

Author contributions
SD and RD conceived and designed research. SD and JB conducted experiments. SD analyzed and visualized the data and wrote the original draft of the manuscript. RD and AK supervised the work and edited the original draft. All authors have read and agreed to the published version of the manuscript.
(a) HV/total biopolymer ratio at maximum total biopolymer concentration after 60 h.
(b) Residual biomass concentration at highest total biopolymer concentration after 60 h.
(c) Biopolymer concentration/total biomass ratio at maximum total biopolymer concentration after 60 h.