C. E. Torres-Cerna et al., Batch Cultivation Model for Biopolymer Production, Chem. Biochem. Eng. Q., 31 (1) 89–99 (2017) 89 Batch Cultivation Model for Biopolymer Production This work is licensed under a a,◊ a,◊,* b Creative Commons Attribution 4.0 C. E. Torres-Cerna, A. Y. Alanis, I. Poblete-Castro, International License and E. A. Hernandez-Vargasc,d,* aCUCEI, Universidad de Guadalajara, Blvd. Marcelino Garcia Barragan 1421, Col. Olimpica, Guadalajara, Jalisco, México, C. P. 44430 bUniversidad Andrés Bello, Faculty of Biological Sciences, Center for Bioinformatics and Integrative Biology, Biosystems Engineering Laboratory, 8340176 Santiago, Chile cSystems Medicine of Infectious Diseases, Helmholtz Centre doi: 10.15255/CABEQ.2016.952 for Infection Research, Inhoffenstr. 7, 38124 Braunschweig, Germany Original scientific paper dFrankfurt Institute for Advanced Studies (FIAS), Received: July 19, 2016 Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany Accepted: March 7, 2017

This paper presents a mathematical model to evaluate the kinetics of two different Pseudomonas putida strains, wild and mutant-type for the microbial production of poly- hydroxyalkanoates (PHAs). Model parameters were estimated to represent adequately experimental data from the batch reactor using the differential algorithm. Based on the mathematical model with the best-fit parameter values, simulations suggested that the high production of PHA by the mutant strain can be attributed not only to the higher production of PHA but also to a reduction in the consumption rate of the substrates of approximately 66 %. Remarkably, the cell growth rate value is higher for the wild type than the mutant type, suggesting that the PHA increase is not only to an increase in the production rate but also to the metabolism of the cells. This mathematical model advanc- es comprehension of the PHA production capacity by P. putida paving the road towards environmentally friendly plastics. Key words: mathematical modeling, batch cultivation kinetics, biopolymers, Pseudomonas putida,

Introduction called materials. The replacement of a fraction of synthetic plastics with biodegradable Plastics are one of the most used materials produced from renewable resources offers worldwide and their production rate is increasing to solve important problems such as the overall over the years. In 2013, the worldwide plastic mate- consumption of fossil fuels, environmental pollu- rial production was approximately 299 million tons, tion, and solid waste management3. can and these numbers continue to increase, causing easily be degraded (Figure 1), and in this way, the large amounts of waste which highly contributes to plastics deposited in nature could be largely de- 1 the pollution of the environment . Plastic materials creased, serving as a carbon substrate for different have been for a long time produced through petro- . chemicals routes. Theses plastics have a high dura- Bioplastics are available in specific areas and bility and are resistant to degradation. Although applications, for example composting bags and such properties are considered important for the in- sacks, fast food tableware (cups, plastic bottles, cut- dustrial production of plastics, these are a signifi- lery, etc.), agriculture and hygiene4. Furthermore, cant source of environmental and waste manage- ment problems2. biopolymers are also used in most industries for technical applications, including medicine and phar- New avenues of research have been developed macy. Among the candidates for biodegradable to produce plastics with biodegradability properties, plastics, polyhydroxyalkanoates (PHAs) have been drawing much attention because of their similar ma-

*Corresponding authors: E-mail: [email protected] (EHV) terial properties to conventional plastics and com- [email protected] (AYA) plete biodegradability. PHAs are insoluble in water ◊These authors contributed equally to this work. and exhibit a rather high degree of polymerization

C. E. Torres-Cerna et al., Batch Cultivation Model for Biopolymer Production 89–99 90 C. E. Torres-Cerna et al., Batch Cultivation Model for Biopolymer Production, Chem. Biochem. Eng. Q., 31 (1) 89–99 (2017) 13

Production

P. putida ( ) Cycleofbioplastics in nature

Plasticproduct

Compost(as food formicroorganisms) Wastefor disposal

Fig. 1 – Bioplastics cycle. Bioplastics are produced by microorganisms, which are collected after being Fig. 1 Bioplastics cycle. Bioplastics usedare pr asodu solidced by wastes. microorganisms These are, wh degradedich are colle andcted used after asbein carbong used assources solid wa forstes the. Th microorganisms.ese are degraded and used as carbon sources for the microorganisms.

ranging from 105 to 107 Dalton (Da). Their non-tox- tween petroleum and bacterial-based production of icity and biodegradability properties are essential polymers. For example, the in silico driven meta- 3 Newfor theiravenues potential of research use . have been developed to producebolic plasticsengineering with approachbiodegradability developed by Po- PHAs are naturally produced as granules in the blete-Castro and coworkers, re-designed P. putida properties,cytoplasm called of cells biop bylastic numerous materials. The under replacement lim- to of enhance a fraction the productionof synthetic of mcl-PHAsplastics using glu- iting nutrients and excess carbon source3,5. Pseudo- cose as the carbon source. The metabolically engi- neered strains that were performed predicted only with monasbiode gradableputida strains polymers are the produced most used from bacteria renewable for resources offers to solve important the synthesis of medium-chain-length PHAs (mcl- one key mutation, revealed a 100 % increase in the PHAs) in the presence of different carbon sources, final PHA titer, a 50 % increase in the cellular PHA problemssuch assuch fatty as acids, the overall glucose, consumption and glycerol. ofParticular fossil fuels,content, environmental and 80 % pollution in the PHA, and yield, solid relative to its parent strain. properties of mcl-PHAs3 give them a broader appli- wastecation management range over. short-chain-lengthBioplastics can PHAseasily6 . beUsual degraded- (FAdditionallyigure 1), and to thein thisimprovements way, the through ge- ly, the production process is composed of two netic modifications, mathematical models have plasticsphases. deposited In the firstin natu phase,re couldcells arebe challengedlargely decreased, by proved serving to beas a arelevant carbon tool substrate to enhance for biopolymers growth in a minimal medium with an excess of car- production8. On the one hand, metabolic approaches differentbon microorganisms.source and a limited amount of ammonium as are focused on metabolic pathways for PHA pro- the nitrogen source. The cells rapidly consume the duction, these can reflect the intracellular dynamics. Bioplasticscarbon substrate are avai andla exhibitble in specificexponential areas growth and applications,as There isfor a particularexample interestcomposting for this bags level of model- long as ammonium is present in the medium3,6. The ling due to the high detail that can describe8. Never- second phase is characterized by the transition from theless, this detail level promotes highly4 complex and sacks,a feast fastto a food famine tableware period, which(cups, allows plastic the bottles, high cutlery,models. etc.), On agriculture the other hand, and hygienemodelling. approaches synthesis of mcl-PHA in batch reactors3,6,7. based on the population growth can help dissect the Nevertheless, the amount of bacterial produc- dynamics of different strains8. This approach could tion of mcl-PHA is insufficient compared to the de- be useful in developing different cultivation strate- mand and the reduced quantity of production, influ- gies in order to enhance the PHAs production with- encing the price of bioplastics in the market. out increasing the model complexity9,10. Systems metabolic engineering approaches of Pseu- Recently, several mathematical models, based domonas putida (P. putida) have enhanced the pro- on population growth, have been proposed138 to facil- duction of PHA, thus enabling development of a itate the dynamics of biopolymer production9–14. more efficient strain, which closes the cost gap be- Most of these works contain many parameters esti- C. E. Torres-Cerna et al., Batch Cultivation Model for Biopolymer Production, Chem. Biochem. Eng. Q., 31 (1) 89–99 (2017) 91 mated based on local optimization algorithms. Note where S is the glucose (carbon source), N is the NH4 that parameter estimation is a complex problem in- concentration (nitrogen source), X is the active bio- volving highly non-linear equations and several mass, and P is the biopolymer PHA. NH4 is as- variables. Therefore, global optimization algorithms sumed as the limiting nutrient13 and the total are needed to avoid incorrect fits and consequently (Cell Dry Mass) is composed by the catalytically incorrect interpretations. active biomass and the PHA, thus CDW = X + P. The mechanisms of self-adaption, self-organiz- Consumption rates of substrates by active bio- ing and self-learning in stochastic optimization ap- mass are presented in equations (5) and (6). m and proaches allow addressing challenging problems S m can be interpreted as the specific consumption that cannot be solved by traditional methods. In this N rates of glucose and NH by the active biomass, re- paper, a variant of the Differential Evolution (DE) 4 spectively. These are experimentally proportional to algorithm15 is considered for parameter estimation. This optimization algorithm is used in many engi- biomass concentration, and reach their maximum neering applications due to its simplicity and when the concentration of substrates is high. straightforward implementation. The substrate S is consumed by the active bio- mass at a rate (k1), which is modeled by equation (5) using the following Monod function: Materials and methods Xt( ) mS (t) = St( ) (5) Strains, medium and batch fermentations Xt( ) + K S

For this work, considered was the parent strain where KS is the saturation or affinity constant of the P. putida KT2440 (DSM 6125), which was obtained substrate S. from the German Collection of Microorganisms and NH4 dynamics are represented by equation (6). Cell Cultures (DSMZ, Braunschweig, Germany). Experimental data6 indicate that ammonium is com- To generate single deletion mutant of P. putida pletely consumed in the growth phase. Different KT2440, genome editing was applied16. P. putida functions were considered for the NH4 consump- strains were grown in a defined mineral medium. tion, however, the best estimations were achieved The experiments were performed in a 2 L top-bench by the following rate equation: BIOSTAT bioreactor with working volume of 1.5 L at 30 °C and 180 rpm. The dissolved oxygen was Xt( ) mN (t) = Nt( ) (6) kept at 20 % of air saturation by control and the Xt( ) + K pH-value was maintained at 7. N The basic structure of PHA was confirmed by where KN is the saturation or affinity constant of the NMR analysis. Thirty mL of culture was harvested substrate N. and the pellet was washed with deionized water, Active biomass dynamics of strains are pre- frozen at –20 °C and lyophilized. The lyophilizate sented in equation (7). The specific growth rate of was then suspended in 10 mL chloroform and kept biomass with glucose is defined according to the for 3 h at 80 °C to extract the PHA. The PHA com- “two substrates” kinetic equation (known as “dou- position, as well as the cellular PHA concentration, ble Monod” relation). The limiting substrate ammo- were determined by gas chromatography (GC) and nium (N) is essential to produce the active biomass mass spectrometry (MS). A more detailed descrip- X and limits its synthesis at low concentrations. The 6 tion of the experiments can be found in . active biomass growth m writes as follows:

Mathematical model St( )  Nt( )  mm= (t) max    Xt( ) (7) The mathematical model proposed here is StK( ) ++ SN11  NtK( )  based on the following differential equations: where m is the maximum growth rate, K and K dSt( ) max S1 N1 =− ktm ( ) (1) represents the affinity constants of X. dt 1 S The biopolymer PHA (P) is produced at a rate dNt( ) k by active biomass dynamics of P. putida (X), =− ktm ( ) (2) 4 dt 2 N which is triggered by a limited amount of ammoni- um provided in the growth medium, this can be ex- dXt( ) = m (t) (3) pressed by the following equation: dt K N dPt( ) mP (t) = Xt( ) (8) = kmm( tk) + ( t) (4) Nt( ) + K dt 34p N 92 C. E. Torres-Cerna et al., Batch Cultivation Model for Biopolymer Production, Chem. Biochem. Eng. Q., 31 (1) 89–99 (2017) m Note that P(t) term will increase when the val- ulation, the difference is scaled and added to a third ∉ ue of ammonium decreases. A growth term based individual r3 {r1, r2} chosen randomly to create a m on the biomass growth k3 (t) was introduced to new mutant vector vi. The individuals are defined as 12 p provide better fitting for the PHA. This signifies Xig,= [ xx ig ,,, ig , …, x ig ,], where i = 1, 2, …, NP, that small amounts of PHA are produced during g = 1, 2, …, G. The variable G represents the maxi- biomass growth. mum number of generations, and NP the population number. Parameter estimation The first step is the initialization of the Parameter estimation for mathematical models population within a search space for each element can be understood as the search for a set of values of the individuals, defined by model restrictions = … for the parameters θ that minimize the difference as Xmin[ xx1, min , 2, min, , x p, min ] , and Xmax = []xx1,max, 2, max,…, x p, max . between the outcome of model yi and experimental data yi as close to zero as possible. This search is The second step is the mutation, for which dif- restricted to system dynamics, algebraic restrictions, ferent strategies exist. In this work, the following and system constraints. In this paper, the Sum of the 17 mutation strategy is considered: Square of Weighed Residues (SSWR) function is used for parameter estimation, and is defined as fol- v=+− x Fx ( x ) i, g best, g r1, g r 2, g (11) lows: where x is the individual with the best fit in the 2 best,g mnjj current generation. x and x are different vectors yy− r1,g r2,g SSWR (θ ) = ii (9) chosen randomly among the population, and F ∈ ∑∑j ji=11 = max ( y ) [0,1] is the scaling factor. where j and i represent the number of variables and The third step is the crossover, a new trial vec- tor is generated recombining the mutant v and experimental data points, respectively, y is the set of i,g 6 target x vectors. For this paper, the exponential experimental data points , and y is the outcome of i,g the mathematical model. approach (12) was considered. In this approach, s ∈ [1,p] and L ∈ [1,p] are random integer numbers. To know the behavior of each parameter within s represents the starting element in target vector x an interval of given values, in each ODE the likeli- i,g to create a new trial vector u recombining the L hood profile for all parameters is estimated. Using i,g components donated by the mutant vector v . equation (10), each parameter was fixed while other i,g parameters in the equation were estimated. Since vj if j∈∪ A B = ig, the ODEs integration routine requires dense data uig,  j (12)  xig, otherwise sets at different times depending on the adaptive  step size, inputs in each estimation are approximat- ed by piecewise linear interpolation. For these esti- where A:= {s, …, min(p,s + L – 1)}, and B : = {1, …, mations, the Residual Sum of Squares (RSS) is used s + L – p – 1)} as minimization function. The last step is the selection, a cost function is n 2 used to determine which vector survives for the RSS (θ ) =∑( yii − y ) (10) new generation (g + 1), the mutant ui,g and target xi,g i=1 vectors are evaluated to find which has the best The differential equations are solved using yield. The basic idea is shown as follows: solver ode45 from MATLAB to obtain the model outcome. u if f u ≤ f x  ig,( ig ,,) ( ig) x + =  ig,1 (13) Differential evolution algorithm xig, if f( xig ,,) < f( u ig)  The minimization of equations (9) and (10) im- where f (⋅) is the evaluation of the objective func- plies a non-linear optimization problem with sever- tion. al variables, which can be solved using a global optimization algorithm. Differential evolution (DE) algorithm15 is among the most popular global opti- Numerical results mization algorithms. DE is designed for optimizing p-dimensional continuous functions. Each individu- The experimental work assessed the growth al in the population is a p-dimensional vector that and PHA production performance by the wild and represents a candidate solution to the problem. mutant-type P. putida strains in minimal medium In DE algorithm, two individual mutually ex- using glucose as a carbon source (18.5 g L−1) and a ≠ cluding r1 r2 are picked randomly among the pop- limiting amount of ammonium as nitrogen source 14 develop a model as simple as possible without compromising accuracy, in the proposed

model, the formation and consumption of gluconate were not considered.

The exponential growth of PHA started at 10 hours, this could be attributed to the

shortage of nitrogen source in the medium. The PHA content exhibits two phases, the first is

proportional to cell growth, when PHA production is slow. The second phase is the

exponential growth of PHA, which is triggered when the nitrogen source present in the

− medium is less than 0.1 g L 1.

The development of new avenues of thinking to decrease bioplastics costs are urgently

needed. To this end, mathematical models can help interpret experimental results on

quantitative grounds. Here, the reactor dynamics6 are represented with the system (1)-(4). C. E. Torres-Cerna et al., Batch Cultivation Model for Biopolymer Production, Chem. Biochem. Eng. Q., 31 (1) 89–99 (2017) 93

Fig. 2 – Experimental data for the wild (black) and mutant-type (gray) strains of P. putida. S is the Fig. 2 Experimental data for the wild (black) and mutant-type (gray) strains of P. putida. S is the glucose concentration, N is the NH4 concentration, glucose concentration, N is the NH concentration, CDW is the total biomass. CDM is the total biomass. 4

(350 mg L−1). The experimental data6 are shown in e.g., least value, steepest descendent, inner and out- Figure 2. The exponential growth of CDW initiates er approximation, etc. Innovative optimization ap- at 5Before hours and mathematical continues while models glucose can in the be medi considered- proaches for reliable help tackle predict challengingions, thproblemse unknow thatn can - um exists. The stationary phase is reached when the not be solved by traditional methods. Here, the DE glucose concentration is less than 0.1 g L–1. In Po- algorithm15 is considered and the fine tuning of the blete-Castro work6, the authors showed gluconate algorithm is based on18–20. Furthermore, the novel production by the wild type strain, and at the same approach proposed by Raue et al.21 exploits the time possible consumption of the gluconate when likelihood profile to determine both structurally and the glucose is depleted in the medium. However, to practically non-identifiability. In addition, the itera- 14722 develop a model as simple as possible without com- tive fitting routine proposed by Binder et al. is promising accuracy, in the proposed model, the for- used to decompose the problem of fitting a system mation and consumption of gluconate were not con- of coupled ordinary differential equations (1)–(4) sidered. into smaller sub-problems. Using equation (10) the likelihood profile for all parameters are estimated to The exponential growth of PHA started at 10 check the identifiability. hours, this could be attributed to the shortage of ni- Results of likelihood profiles are displayed in trogen source in the medium. The PHA content ex- Figure 3. Likelihood profile graphics suggests KP is hibits two phases, the first is proportional to cell structurally non-identifiable. In addition, parame- growth, when PHA production is slow. The second ters KS1, k4, and k3 are practically non-identifiable phase is the exponential growth of PHA, which is since their confidence regions do not converge to- triggered when the nitrogen source present in the wards a specific value21. Parameters k , k and K −1 3 4 S1 medium is less than 0.1 g L . could take any value close to zero and do not en- The development of new avenues of thinking hance the model fit significantly. In the same way,

to decrease bioplastics costs are urgently needed. To KP could take any value inside the interval without this end, mathematical models can help interpret enhancing the model fit. On the other hand, likeli-

experimental results on quantitative grounds. Here, hood profile graphics suggest that parameters k1, k2, 6 the reactor dynamics are represented with the sys- KS and KN1 are identifiable. Even though a threshold tem (1)–(4). cannot be determined, the graphics offer an over- Before mathematical models can be considered view of which parameters can be identifiable or not. for reliable predictions, the unknown parameters To simplify the parameters estimations based should be estimated by comparing model output to on likelihood profile, graphics parameters detected experimental data (maximum likelihood principle). as non-identifiable were fixed. Values for fixed pa- However, the mathematical model equations (1)–(4) rameters were chosen to minimize the SSWR func- possess parameters that could be difficult to esti- tion, while the remaining parameters were simulta- m mate using common model optimization strategies, neously estimated. Furthermore, values for max 94 C. E. Torres-Cerna et al., Batch Cultivation Model for Biopolymer Production, Chem. Biochem. Eng. Q., 31 (1) 89–99 (2017)

Fig. 3 – Likelihood profiles for parameters of both strains, wild type – black line, and mutant type – gray line were taken from the work of Poblete-Castro6. Table 1 – Model parameters. These are the best fitting result using the iterative fitting routine22. In Figure 3, parameters k3 and k4 are non-identifi- able. Thus, the parameter k4 was fixed, while the Parameter Wild Type Mutant type parameter k was estimated. The best-fit parameter 3 k (h–1) 0.210 0.071 values can be found in Table 1. 1 k (h–1) 0.712 5.719 Model simulation results in Figure 4 reveal 2 k (h–1) 0.051 0.146 good fitting to the experimental data depicted by 3 * –1 empty circles. The model presents a better fit for k4 (h ) 0.007 0.0105 –1 wild type strain than mutant type strain. This differ- KS (g L ) 0.434 0.206 ence is clearer on PHA graphics, where there is a * –1 KN (g L ) 0.739 14.789 logistic growth in the first 10 hours, after that the * –1 KS1 (g L ) 0.001 0.001 growth is linear. This behavior is also presented in –1 KN1 (g L ) 0.047 0.028 CDW graphics and is consistent with NH4 con- m 1 –1 0.53 0.450 sumption. max (h ) * –1 As discussed previously, the production and KP (g L ) 3 0.216 consumption of gluconate by the wild type strain (*) parameters were fixed. described in6 were not considered in the presented (1) parameter was taken from experimental data6 15

15 C. E. Torres-Cerna et al., Batch Cultivation Model for Biopolymer Production, Chem. Biochem. Eng. Q., 31 (1) 89–99 (2017) 95 a)

b)

Fig. 4Fi –g. Simulation4 Simulation results.results. Th Thee resu resultslts are diaresp ladisplayed,yed, a) for wia)ld for stra wildin, an dstrain, b) for m andutant b) st raforin . mutantThe simula strain.tions weThere desimulationsveloped usin gwere parameter developed valuusinges from parameter Table 1 values from Table 1. Fig. 4 Si mulation results. The results are displayed, a) for wild strain, and b) for mutant strain. The simulations were developed using parameter values from Table 1 model. However, the gluconate production and con- fected by the high variability of experimental mea- 25 sumption could Confidence explain intervthe growthals we ofre activecalculated bio- usingsurements the Weight or byed theBootstrap erroneous method assumption. Results in the mass and PHA after the glucose is consumed. statistical distributions. Confidence intervals were calculated using theThe Weight Weighteded Bootstrap method method is25 an. Resultseffective Confidence interval tool for semiparametric estimation23, which allows the study of asymptotic properties of the proposed Due to the highly variable data in biological solution when there are measurement and numerical systems, further validation of estimated parameters errors24. The Weighted Bootstrap method can be is necessary. Even though the likelihood profile of- written as follows: 152 fers a visual insight for parameters validation, due 2 mn jj to a lack of experimental data it cannot be deter- yyii− θˆ = argmin w (14)152 mined clearly whether a parameter is identifiable or Bi∑∑ j ji=11 = max ( y ) not. Confidence intervals help determine the degree of validity for each parameter estimated. Bootstrap where w is a vector with independent, identically methods are used to provide statistical inferences of distributed positive random weights with mean a set of parameters. However, these results are af- [E (w) = 1] and variance [Var (w) = 1]. Statistical dis- 96 C. E. Torres-Cerna et al., Batch Cultivation Model for Biopolymer Production, Chem. Biochem. Eng. Q., 31 (1) 89–99 (2017) Table 2 – The confidence interval for each parameter. After 25 θˆ tribution of w is independent of y and y . B is the 1000 weighted bootstrap, confidence interval with vector of estimated parameters after each bootstrap. 95 % of confidence for both strains is shown. Confidence intervals were calculated using the Confidence interval Weighted Bootstrap method25. Results after 1000 Parameter Wild type Mutant type bootstrap trials are displayed in Table 2, for each constant parameter, the 2.5 % and 97.5 % quantiles k (h–1) 0.1790–0.3405 0.0600–0.0910 1 of the estimates were selected to compute the 95 % K (g L–1) 0.3070–0.8900 0.0995–0.4575 S confidence interval. –1 k2 (h ) 0.6235–0.8300 4.7625–6.0970 The model was refitted in each repetition to ob- –1 tain the corresponding parameter distribution. Pa- KN1 (g L ) 0.0320–0.0620 0.0220–0.0450 –1 rameter distribution is displayed in Figure 5. Pa- k3 (h ) 0.0381–0.0685 0.1020–0.1930 rameters KS for the wild strain and k2 for the mutant

Fig. 5 – Parameter distributions. The histograms show the distribution for parameters of both wild (black) and mutant (gray) strains. Data taken from bootstrap results. C. E. Torres-Cerna et al., Batch Cultivation Model for Biopolymer Production, Chem. Biochem. Eng. Q., 31 (1) 89–99 (2017) 97

Fig. 6 – Parameter dependency. The scatter plots show the parametric dependence between parameters for both wild (black) and mutant (gray) strains. Data taken from bootstrap results.

strain have the larger confidence intervals, respec- ters (Figure 6). It is the case of parameters k1 – KS tively, this allows the model to be more flexible to which are linearly dependent, if k1 increases, KS in- changes of these parameters. creases too; this behavior was expected because The parametric dependence was obtained using both parameters belong to the same equation. The the results from Weighted Bootstrap method, each same behavior is presented in parameters k2 – KN parameter was plotted against another in scatter (Figure 7). Parameter KN was fixed due to always plots, which show the correlation between parame- reaching the upper limits, this behavior is consistent 15 98 C. E. Torres-Cerna et al., Batch Cultivation Model for Biopolymer Production, Chem. Biochem. Eng. Q., 31 (1) 89–99 (2017) some models have described fed-batch and batch cultures9,10,13, and these approaches use the fed- batch cultures model to increase the PHA yield. The model presented in this work is simpler, which helps to obtain a better parameter estimation. Dif- ferent terms were considered for substrates equa-

tions (1)–(2), e.g. bilinear terms (k1SX) or more complex like Hill functions (Xn/(Xn + KS)). Never- theless, a better estimation was reached using the equations (1)–(2). As future work, to enhance the accuracy of the model, the dynamics of gluconate presented by Poblete-Castro et al. work6 will be in- cluded.

Fig. 7 – Linear dependence between parameters k2 – KN. A Fig. 7 Linear dependence between padirectrame terelationrs k2 – KexistsN. A betweendirect relatio parametersn exists bek2 twandee nK paN. rameters k2 and KN. Acknowledgments This work was supported by CONACyT Mexico with their likelihood profile (Figure 3) where a and DAAD Germany through the program PROAL- threshold cannot be determined. MEX funding the project “OPTREAT”. Alma Y. Al- For the glucose equation (1), theFor theconsumption glucose equation rate (1), ( the) consumptionand the affinity anis constant acknowledges ( )CONACyT of for their financial rate (k1) and the affinity constant (KS) of mutant support provided by the project CB-256769, and type strain are smaller than those of wild strain. Ad- Ignacio Poblete-Castro acknowledges the financial mutant type strain are smallerditionally, than those the proportion of wild between strain. bothAdditionally, parameters the supportproportion by the between program Convenio de Desempeño

(k1/KS) is smaller for mutant strain than that one of Apoyo a la Inovación en Educación Superior (PMI- UAB 1301). both parameters ( / ) is smalthe wildler strain. for mutant On the other strain hand, than in the th atNH one4 equa of- the wild strain. On the tion (2), despite the higher values in parameters k2 and KN for mutant strain with respect to those of other hand, in the NH4 equationthe wild (2 strain), despi (seete Table the 1),higher their valuesproportion in ispa rameteReferencesrs and for smaller between parameters for mutant type strain 1. URL: http://www.worldwatch.org/global-plastic-produc- mutant strain with respect tothan those wild oftype the strain. wild These strain proportions (see Table suggest 1), that their proportiontion-rises--lags-0 is smaller (21.11.2016) the mutant strain consumes the substrates slowly in 2. Zhao, R., Torley, P., Halley, P. J., Emerging biodegradable comparison to the wild strain. Furthermore, the mu- materials: - and -based bio-nanocomposites, J. between parameters for mutantant tstrain type grows strain slower than than wild the wildtype strain. strain. These proportionsMater. Sci. 43 (2008)suggest 3058. doi: https://doi.org/10.1007/s10853-007-2434-8 3. Lee, S. Y., Bacterial polyhydroxyalkanoates, Biotechnol. that the mutant strain consumes the substrates slowly in comparison toBioeng. the 49wild (1996) strain. 1. Conclusions doi: https://doi.org/10.1002/(SICI)1097-0290(19960105)49: 1<1::AID-BIT1>3.3.CO; 2-1 Furthermore, the mutant strain growsThis work slower presents than a mathematicalthe wild strain. model for 4. Bastioli, C., Global status of the production of biobased biopolymer production in a batch reactor, which can packaging materials, Starch/Staerke 53 (2001) 351. assess the performance of two different P. putida doi: https://doi.org/10.1002/1521-379X(200108)53:8<351:: strains, wild and mutant type strains. The model pa- AID-STAR351>3.0.CO; 2-R 4 Conclusions rameters were evaluated using a variant of DE algo- 5. Wältermann, M., Hinz, A., Robenek, H., Troyer, D., Re- rithm to minimize the difference between experi- ichelt, R., Malkus, U., Galla, H. J., Kalscheuer, R., Stövek- en, T., Von Landenberg, P., Steinbüchel, A., Mechanism of mental data and the model output. These parameters -body formation in prokaryotes: How bacteria fatten were estimated through computing their likelihood up, Mol. Microbiol. 55 (2005) 750. This work presents a mathemprofile aticaland confidence model forintervals. biopolymer The likelihood production indoi: a https://doi.org/10.1111/j.1365-2958.2004.04441.xbatch reactor, profiles helped determine the parameter identifiabil- 6. Poblete-Castro, I., Binger, D., Rodrigues, A., Becker, J., which can assess the performanceity. This analysis of two was different complemented P. putida with the strains, confi- wildMartins and mutantDos Santos, type V. a P., Wittmann, C., In-silico-driven dence interval estimations. metabolic engineering of Pseudomonas putida for enhanced production of poly-hydroxyalkanoates, Metab. Eng. 15 strains. The model parameters Estimationswere evaluated suggested using that the a highervariant PHA of pro DE- algorit(2013)hm 113.to minimize duction by the mutant strain, in comparison with the doi: https://doi.org/10.1016/j.ymben.2012.10.004 wild strain, may be related to the adequate con- 7. Poblete-Castro, I., Escapa, I. F., Jäger, C., Puchalka, J., the difference between experimentalsumption of substrates,data and which the canmodel be clearlyoutput. ob - These Chiparame Lam, C.,ters Schomburg, were D., Prieto, M., Martins dos San- served in the glucose and nitrogen consumption tos, V. A., The metabolic response of P. putida KT2442 pro- ducing high levels of polyhydroxyalkanoate under single- graphics (Figure 4). The presented model contains and multiple-nutrient-limited growth: Highlights from a fewer parameters than previous mathematical mod- multi-level omics approach, Microb. Cell Fact. 11 (2012) 34. els for biopolymer productions9–14. Furthermore, doi: https://doi.org/10.1186/1475-2859-11-34

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