Dynamic genome-scale metabolic modeling of the yeast Pichia pastoris
- Francisco Saitua^{1},
- Paulina Torres^{1},
- José Ricardo Pérez-Correa^{1} and
- Eduardo Agosin^{1}Email author
https://doi.org/10.1186/s12918-017-0408-2
© The Author(s). 2017
Received: 23 August 2016
Accepted: 9 February 2017
Published: 21 February 2017
Abstract
Background
Pichia pastoris shows physiological advantages in producing recombinant proteins, compared to other commonly used cell factories. This yeast is mostly grown in dynamic cultivation systems, where the cell’s environment is continuously changing and many variables influence process productivity. In this context, a model capable of explaining and predicting cell behavior for the rational design of bioprocesses is highly desirable. Currently, there are five genome-scale metabolic reconstructions of P. pastoris which have been used to predict extracellular cell behavior in stationary conditions.
Results
In this work, we assembled a dynamic genome-scale metabolic model for glucose-limited, aerobic cultivations of Pichia pastoris. Starting from an initial model structure for batch and fed-batch cultures, we performed pre/post regression diagnostics to ensure that model parameters were identifiable, significant and sensitive. Once identified, the non-relevant ones were iteratively fixed until a priori robust modeling structures were found for each type of cultivation. Next, the robustness of these reduced structures was confirmed by calibrating the model with new datasets, where no sensitivity, identifiability or significance problems appeared in their parameters. Afterwards, the model was validated for the prediction of batch and fed-batch dynamics in the studied conditions.
Lastly, the model was employed as a case study to analyze the metabolic flux distribution of a fed-batch culture and to unravel genetic and process engineering strategies to improve the production of recombinant Human Serum Albumin (HSA). Simulation of single knock-outs indicated that deviation of carbon towards cysteine and tryptophan formation improves HSA production. The deletion of methylene tetrahydrofolate dehydrogenase could increase the HSA volumetric productivity by 630%. Moreover, given specific bioprocess limitations and strain characteristics, the model suggests that implementation of a decreasing specific growth rate during the feed phase of a fed-batch culture results in a 25% increase of the volumetric productivity of the protein.
Conclusion
In this work, we formulated a dynamic genome scale metabolic model of Pichia pastoris that yields realistic metabolic flux distributions throughout dynamic cultivations. The model can be calibrated with experimental data to rationally propose genetic and process engineering strategies to improve the performance of a P. pastoris strain of interest.
Keywords
Background
Recombinant protein production is a multibillion-dollar business, mainly comprised by therapeutic agents (i.e. recombinant biologic drugs) and industrial enzymes [1–3]. These compounds are commonly synthesized in Escherichia coli, Saccharomyces cerevisiae and Chinese Hamster Ovary cells (CHO) [1, 4–6]; however, there is strong pressure to find cost-effective alternatives to overcome technical and economic disadvantages of the aforementioned cell factories, especially in downstream processing [7].
Among the unconventional cell factories used for recombinant protein production, the methylotrophic yeast Pichia pastoris (syn. Komagataella phaffii) has received special attention thanks to its convenient physiology and easy handling [8]. There are strong promoters for this cell factory which are commercially available and that allow for the controlled expression of heterologous proteins [8]. Unlike E. coli, P. pastoris naturally performs post-translational modifications [6, 9], which are essential for most eukaryotic protein functionality [7, 10, 11]. In contrast to S. cerevisiae, P. pastoris exhibits a Crabtree-negative phenotype, showing a reduced synthesis of undesirable products, like ethanol, in glucose-limited conditions [12, 13]. It also shows a lower basal secretion of proteins when compared to other yeasts, which makes downstream processing easier [13, 14]. Finally, P. pastoris can be efficiently cultivated up to high cell densities using fed-batch technology [8], achieving high titers and productivities. For these desirable features, P. pastoris has been widely used for the expression of recombinant proteins, reaching grams per liter concentrations in several cases [9, 15–18]. Most remarkably, and as proof of its technical feasibility and adequacy, two recombinant proteins produced in this cell factory have already been approved by the FDA for medical purposes [10, 19].
Despite its growing acceptance and actual successful applications, recombinant protein production in P. pastoris can be undermined by several cellular processes, where protein folding and secretion are the most recurrent bottlenecks [14, 20, 21]. In addition, limitations may also be caused by the codon usage of the recombinant protein [22], promoter selection [23], carbon and oxygen availability in the culture [24, 25] and fed-batch operational parameters [26], seriously hampering protein yield, productivity and the economic feasibility of the process.
Industrially, P. pastoris is commonly grown in fed-batch cultures in order to maximize the titer and volumetric productivity of a desired compound, often a recombinant protein [27, 28]. This is achieved by adding a culture medium in such a way that the microorganism grows at a desired specific growth rate, which is chosen to maximize the synthesis of the target product and to limit the formation of inhibitory compounds [29]. During this and other cultivation systems, the cells adapt constantly to the changing extracellular environment and to the limited mass transfer conditions observed at high densities [30, 31]. Therefore, it is critical to understand how the cell metabolism interacts with the nutritional and environmental stresses exerted by process conditions to improve bioreactor performance [32]. This is a complex task, however, since the strain’s characteristics and process variables often require significant amounts of time and money for characterization and fine-tuning [12]. Therefore, it is desirable to have a platform to integrate different levels of information from dynamic cultivations of P. pastoris that can be used to elaborate rational hypotheses to increase process productivity.
Systems biology offers a quantitative and comprehensive approach to address this task [33]. In particular, Genome-Scale dynamic Flux Balance Analysis (GS-dFBA) [34–36] is a modeling framework that allows the simulation of metabolism during non-stationary (batch or fed-batch) cultures. GS-dFBA models couple the dynamic mass balances of the extracellular environment of the bioreactor with comprehensive mathematical representations of cellular metabolism called Genome Scale Metabolic Models (GSMs). These structures represent the cell’s entire metabolism as a set of underdetermined constrained mass-balances [30, 37, 38]. GSMs have been employed to understand cellular behavior under different environmental conditions, to map over omics data, and to define a metabolic engineering targets [39, 40]. There are currently five published GSMs of P. pastoris [41–45] which have been developed to help the strain optimization process with a special emphasis on recombinant protein production. Moreover, one of these models has been successfully employed to improve recombinant protein production in P. pastoris [46], validating these frameworks as strain engineering tools for this particular yeast.
GS-dFBA models usually contain several parameters, whose values can be obtained by regression of experimental data. These parameters are used as inputs to obtain flux distributions throughout cultivations, so their values need to be reliable. To ensure this, pre- and post-regression diagnostics have been employed to determine if a certain parameter is supported by the observed data or not [47, 48]. These analyses consist in verifying the model’s capacity to explain the behavior of a system (goodness-of-fit) and the presence of the following parametric limitations: (i) low or no impact on the state variables (sensitivity), (ii) strong correlations with other parameters of the model (identifiability) and (iii) lack of statistical significance (significance). A model is considered robust if it has the capacity to explain different conditions, while containing only sensitive, identifiable and significant parameters.
Here, we present a robust dynamic genome-scale metabolic model of P. pastoris in glucose-limited, aerobic batch and fed-batch cultivations. To assemble the dynamic modeling framework, we started by selecting one of the available genome-scale metabolic models [43] and manually curated it to yield realistic flux distributions. Then, we included it in a set of mass balances representing the main compounds present in culture supernatant. Once assembled, the model was calibrated using experimental data from eight batch and three fed-batch cultivations. Next, we employed pre/post regression diagnostics to determine sensitivity, significance and identifiability problems in the model. In order to avoid the aforementioned statistical limitations, problematic parameters were fixed (i.e. removed from the adjustable parameter set) based on the pre/post regression diagnostics, yielding reduced and potentially robust model structures. Potentially robust model structures consisted in the original model formulation with less adjustable parameters. After evaluating these reduced models for each type of cultivation, we chose the one that presented fewer parametric limitations after being re-calibrated with the available data. These reduced models yielded no (or just a few) significance, sensitivity or identifiability problems when calibrating new data and they could predict bioreactor dynamics in conditions like the ones used for their determination. Finally, we carried out simulations to assess the potential of the model to study P. pastoris metabolism under industrially relevant conditions, and to select molecular and process engineering strategies to improve recombinant protein production.
Methods
Model construction
Kinetic block
Here, G is the glucose concentration in the medium [g/L], \( {v}_{G, Max} \) is the maximum glucose uptake rate [mmol/g_{DCW} · h] and \( {K}_G \) is the uptake half activity constant of this substrate [g/L]. Once determined, \( -{v}_G \) [mmol/g_{DCW} · h] is included as the lower bound of the corresponding exchange reaction in the model since substrate consumption is represented with a negative flux through this reaction.
These parameters are redefined during the fed-batch phase; therefore, they have two values during this type of cultivation.
Finally, the kinetic block fixes the non-growth associated maintenance ATP (m_{ATP}, a flux through the cytosolic ATP hydrolysis reaction in the model), which accounts for the energy drain caused by cellular processes not related with the generation of new cell material, such as osmoregulation, shifts in metabolic pathways, cell motility, etc. [51, 52].
Metabolic block
Where \( v \) is a vector of metabolic fluxes in [mmol/g_{DCW} · h], and \( lb \) and \( u b \) are the lower and upper bounds for each component of the flux vector.
In this formulation, α, the suboptimal growth coefficient, is an adjustable parameter from the model used to modulate the importance of the two – biologically relevant – competing objectives [48, 52, 54]. In our analysis, “optimal growth” occurs when the objective function of the cell is biomass maximization (α = 0). However, when α > 0, the calculated growth rate is lower than the theoretical maximum derived from biomass maximization, at the same glucose uptake rate. In this sense α is considered as a “suboptimal growth coefficient”; it is worthy to note that we do not refer to the optimality of the flux distribution vector, which is actually optimal, given the convexity of the problem in the metabolic block (Equation 4 - See Additional file 2 for details).
The minimization of total fluxes adds a quadratic term to the objective function, which has the practical benefit of eliminating Type III pathways [55] from the flux distribution, which arise from the multiplicity of solutions of a LP problem. These pathways appear as high fluxes (often taking the value of the upper bound of a particular flux) through closed cycles of reactions. This misleads pathway analysis because despite the mass balance around each participating metabolite is satisfied, the fluxes are thermodynamically infeasible [55]. The use of Quadratic Programing makes pathway analysis easier since these large cycling fluxes undermine the minimization of the total fluxes term in the objective function (Equation 4), so they will be forced to a minimum by the optimization software and the flux distribution will be “cleaned” from these unrealistic fluxes. This is especially significant in large networks because these cycles are more recurrent.
In this study, we employed a curated version of the iPP668 model developed by Chung and collaborators [43], called iFS670 (Additional files 3 and 4). In this updated version, we incorporated the arabitol biosynthesis pathway and the stoichiometric reactions for the production of three recombinant proteins (FAB fragment, Human Serum Albumin and Thaumatin). The arabitol synthesis pathway was included because it was a major compound in the culture supernatant of our experiments. Moreover, the reversibility of cytosolic reactions involving redox cofactors and mitochondrial symporters was checked according to Pereira et al. [56] in order to obtain a more realistic flux distribution through the central metabolism. This was done because the initial flux distributions obtained with the un-modified iFS670 model presented the exact same problems as the iMM904 model of Saccharomyces cerevisiae on Pereira’s work, suggesting that the central metabolism structure of the iPP668 model was based upon the aforementioned S. cerevisiae model. These problems were caused by: (i) the lack of a flux through the oxidative branch of the Pentose Phosphate Pathway; (ii) the presence of a flux of a cytosolic NAPDH dependent isocitrate dehydrogenase (which was the responsible of producing cytosolic NADPH); (iii) an unrealistic flux through mitochondrial symporters; and (iv) almost no mitochondrial formation of α-ketoglutarate. These model limitations are inconsistent with previous P. pastoris fluxomic studies in glucose-limited aerobic conditions [24, 57, 58].
FBA problems were solved using the Constraint-Based Reconstruction and Analysis (COBRA) toolbox [59, 60], which employs the programming library libSBML [61] and the SBML toolbox [62]. Finally, we used Gurobi 6.0.2 as an optimization solver.
Dynamic block
Where \( V \) is volume [L], \( t \) is time [h], \( F(t) \) is the feed function for the fed-batch phase in [L/h]. \( S R \) is a constant sampling rate [L/h] determined from each cultivation to emulate the remaining volume of the culture considering sampling, since this value is used for the calculation of the feeding profile during the feed phase. During the batch phase of the fed-batch cultures, we collected between 15 and 20% of the reactor volume in samples. For batch cultivations, \( F(t) \) was eliminated from the mass balances. \( X \) is the biomass concentration [g/L], μ is the specific growth rate [h^{−1}] (obtained from equation 4), G is the extracellular concentration glucose [g/L], G_{F} is the feed’s glucose concentration [g/L], P_{K} is the k-th extracellular product concentration in [g/L], \( {v}_{P_k} \) is the corresponding production rate [mmol/g_{DCW} · h] and MW accounts for the corresponding molecular weight [g/mmol].
The set of equations was solved in Matlab 2013a (Mathworks, USA) using the solvers ode113 and ode15s for batch and fed-batch cultures respectively.
Model parameters
Parameters of the model
Symbol | Name | Units | LB | Initial value | UB |
---|---|---|---|---|---|
v _{ G,max } | Maximum glucose uptake rate | mmol/g _{ DCW } ·h | 0 | 2.5 | 10 |
K _{ G } | Half saturation constant for glucose uptake | g/L | 0 | 10^{−4} | 10^{−3} |
v _{ EtOH,B } | Ethanol minimum secretion rate (batch) | mmol/g _{ DCW } ·h | 0 | 0.5 | 3 |
v _{ Pyr,B } | Pyruvate minimum secretion rate (batch) | mmol/g _{ DCW } ·h | 0 | 0.1 | 2 |
v _{ Arab,B } | Arabitol minimum secretion rate (batch) | mmol/g _{ DCW } ·h | 0 | 0.2 | 2 |
v _{ Cit,B } | Citrate minimum consumption rate (batch) | mmol/g _{ DCW } ·h | 0 | 0 | 2 |
v _{ EtOH,FB } | Ethanol minimum consumption rate (fed-batch) | mmol/g _{ DCW } ·h | 0 | 0 | 2 |
v _{ Pyr,FB } | Pyruvate minimum consumption rate (fed-batch) | mmol/g _{ DCW } ·h | 0 | 0 | 2 |
v _{ Arab,FB } | Arabitol minimum consumption rate (fed-batch) | mmol/g _{ DCW } ·h | 0 | 0 | 2 |
v _{ Cit,FB } | Citrate minimum consumption rate (fed-batch) | mmol/g _{ DCW } ·h | 0 | 0 | 2 |
α _{ B } | Sub-optimal growth coefficient (batch) | [−] | 0 | 0 | 10^{−3} |
α _{ FB } | Sub-optimal growth coefficient (fed-batch) | [−] | 0 | 0 | 10^{−3} |
m _{ ATP } | Non-growth associated ATP | mmol/g _{ DCW } ·h | 0 | 2 | 10 |
T _{ Fed } | Time when secondary metabolite consumption starts in fed-batch cultures | h | 20 | 25 | 32 |
Model calibration with experimental data
Strains
Four P. pastoris strains were employed in this study: a parental GS115 strain (Invitrogen) and three recombinant strains constructed according to the instructions of the manufacturers harboring respectively one, five and eight copies of the gene encoding for the sweet protein thaumatin. Even though the strains were transformed, thaumatin was not detected at concentrations higher than 100 μg/L in the cultivations. Therefore, due to its small contribution to the overall mass balance, thaumatin production was left out of the analysis and none of the parameters of the model were associated with it. Nevertheless, a mass balance for a recombinant protein can be easily added to the framework.
Experiments
The batch model was calibrated with aerobic glucose limited cultivations of the four strains available; each cultivation was performed twice. On the other hand, the fed-batch model was calibrated with data from three cultures of the strain with one copy the recombinant gene, under the same environmental conditions of the batch cultivations.
Cultivation conditions
Where μ_{MAX} = 0.1 [1/h], μ_{MIN} = 0.07 [1/h] and C = 0.07 [1/h]. Therefore, μ_{SET}(t) decays exponentially from 0.1 to 0.07 [1/h], which has been found to increase (in contrast to constant growth rates in the feed phase) the final biomass concentration in fed-batch cultivations of E. coli and S. cerevisiae performed in our laboratory [66].
Culture media
The culture media employed in these studies were based on Tolner et al. [67]. Inoculum: Glucose 10 [g/L], (NH_{4})_{2}SO_{4} 1.8 [g/L], MgSO_{4} · 7H2O 2.3 [g/L], K_{2}SO_{4} 2.9 [g/L], trace elements solution 0.8 [ml/L], histidine 0.08 [g/L], sodium hexametaphosphate 5 [g/L] and biotin 0.32 [mg/L]. Batch cultures: Glucose 50 [g/L], (NH_{4})_{2}SO_{4} 9 [g/L], MgSO_{4} · 7H_{2}O 11.7 [g/L], K_{2}SO_{4} 14.7 [g/L], trace elements solution 4 [ml/L], histidine 0.4 [g/L], sodium hexametaphosphate 25.1 [g/L] and biotin 1.6 [mg/L] and sodium hydroxide NaOH 1 [g/L]. Feeding medium: Glucose 500 [g/L], MgSO_{4} · 7H2O 9 [g/L], trace solution 12.5 [g/L], histidine 4 [g/L] and biotin 0.1 [g/L]. Sodium hydroxide was added to all the media until a pH of 6 was reached.
Analytical procedures
Sampling and biomass determination
Samples of ~6 mL were periodically collected (every 2–3 h) from all fermentations. Biomass was measured by optical density (OD) at 600 nm using an UV-160 UV-visible spectrophotometer (Shimadzu, Japan). Biomass concentration was determined using the linear relationship: 1 OD_{600} = 0.72 [g/L] using the methodology from [68]. Then, samples were centrifuged at 10.000 rpm for 3 min and the supernatant stored at −80 °C for further analysis.
Extracellular metabolite concentration analyses
Glucose, ethanol, arabitol, citrate and pyruvate extracellular concentrations were quantified in duplicate by High-Performance Liquid Chromatography (HPLC), as detailed in Sánchez et al. [48], with the exception of the working temperature of the Anion-Exchange Column (Bio-Rad, USA), which was lowered from 55 °C to 35 °C for better resolution.
Objective Function
With θ representing the parameter space, m the number of measured variables, n the number of measurements per variable, X_{ij} ^{mod} the dFBA output of variable i and measurement j, X_{ij} ^{exp} the corresponding experimental value and \( \underset{j}{ \max}\left({X}_{ij}^{exp\;}\right) \) the maximum value measured for variable i.
Pre/Post regression analysis
Once the initial calibration of the model was completed, statistical tests were performed in order to determine if the initial model formulation had sensitivity, identifiability or significance problems [47].
Where X_{i}(t) is the ith state variable in time t and θ_{k} is the kth parameter. With all g_{ik} values, we formed a sensitivity matrix g(t) for each experimental time, in which the kth column denotes the sensitivity of the kth parameter on the state variables. These matrices were averaged to obtain a single normalized score of the sensitivity of parameter k on the state variable i during the cultivation. Furthermore, if the score of each variable was under 0.01 for a given parameter, this parameter was considered insensitive and a candidate to be fixed (or left out of the adjustable parameter set) in the reparametrization stage.
Identifiability refers to the possibility of unambiguously determining the parameter values by fitting a model to experimental data. If parameter identifiability is not properly assessed, misleading parameter values can be obtained after model calibration. To calculate identifiability, we determined the correlation between the columns of the sensitivity matrix using the corrcoef function from Matlab, which yielded a correlation coefficient matrix (C). A pair of parameters j and k was considered to be correlated (therefore not-identifiable) if the absolute value of the number at the (j, k) position in the correlation coefficient matrix was higher than 0.95 (\( \left(\left|{C}_{jk}\right|\ge 0.95\right) \)).
Δ(CI_{k}) is the CI’s length. A parameter was not significant if the confidence interval contained zero, i. e. if the absolute value of the CC was equal or larger than 2.
Reparametrization
A reparametrization procedure called HIPPO [75] (Heuristic Iterative Procedure for Parameter Optimization, http://www.systemsbiology.cl/tools/) was applied to overcome parametric statistical limitations in the model.
First, HIPPO performed sensitivity and identifiability tests on the initial calibration results for each dataset. Then, model parameters were fixed one by one until the non-fixed subset presented none of the statistical limitations. Finally, significance was determined for the remaining parameter set, also called the reduced model structure. If all the remaining parameters were significantly different from zero, the resulting structure is considered to be an a priori robust candidate for cross calibration with the available data.
Cross calibration of robust structure candidates derived from the reparametrization stage using the available datasets
Values at which problematic parameters were fixed in the cross-calibration stage
Parameter | Fixation value | Units | Reference |
---|---|---|---|
v _{ G,max } | 6 | mmol/g _{ DCW } ·h | [63] |
K _{ G } | 0.0027 | g/L | [63] |
v _{ EtOH,B } | 0 | mmol/g _{ DCW } ·h | - |
v _{ Pyr,B } | 0 | mmol/g _{ DCW } ·h | - |
v _{ Arab,B } | 0 | mmol/g _{ DCW } ·h | - |
v _{ Cit,B } | 0 | mmol/g _{ DCW } ·h | - |
v _{ EtOH,FB } | 1.21 | mmol/g _{ DCW } ·h | * |
v _{ Pyr,FB } | 0.14 | mmol/g _{ DCW } ·h | * |
v _{ Arab,FB } | 0.15 | mmol/g _{ DCW } ·h | * |
v _{ Cit,FB } | 0.008 | mmol/g _{ DCW } ·h | * |
α _{ B } | 0 | [−] | [85] |
α _{ FB } | 0 | [−] | [85] |
m _{ ATP } | 2.18 | mmol/g _{ DCW } ·h | [43] |
T _{ Fed } | 22 | h | * |
- I.Relative difference between calibration objective functions (J _{ DIFF } ):$$ {J}_{DIFF}=\frac{1}{n}\cdot {\displaystyle \sum_{i=1}^n}\frac{J_{i, Reduced}-{J}_{i, Original}}{J_{i, Original}} $$(17)
Where n corresponds to the number of cultures of each type, \( {J}_{i, Original} \) is the calibration objective function (Equation 11) achieved for dataset i using the original model structure and \( {J}_{i, Reduced} \) is the calibration objective function achieved in dataset i using a reduced, a priori robust, modeling structure.
- II.
Percentage of Significance issues; refers to the number of times a parameter is found to be non-significant out of the total of significance determinations performed for a structure. For instance, if a model structure had 6 parameters and 8 datasets were used to calibrate it, a total of 48 significance determinations were performed for that particular model.
- III.
Percentage of Sensitivity issues; refers to the number of times one of the estimated parameters shows low or no impact over state variables (average relative sensitivity ≤ 0.01) out of the total sensitivity determinations performed.
- IV.Percentage of Identifiability issues; corresponds to the number of times a pair of parameters presents a strong correlation (≥0.95), out of the total parameter pairs of a modeling structure. If p is the number of parameters of the model and n is the number of datasets used for its calibration, the total of parameter pairs for which identifiability was determined is:$$ Total\ pairs=\frac{p\cdot \left( p-1\right)}{2}\cdot n $$(18)
Finally, the modeling structure that presented the lowest J_{DIFF} and fewest statistical limitations was used as a robust structure candidate for the corresponding type of culture.
Robustness check of the chosen modeling structure
Once a candidate for a robust structure was determined for the batch and fed-batch configurations, we tested its robustness (absence of parametric problems) by calibrating it with new experimental data. For the batch model, we employed fermentation data from P. pastoris GS115 strain grown with 40 [g/L] of glucose as carbon source at T° = 25 °C and pH = 6. The robustness of the fed-batch model was assessed with a glucose-limited cultivation consisting of a 60 [g/L] glucose batch phase and an exponential feed using 500 [g/L] of glucose. The medium was added in the feeding phase in order to achieve an exponentially decreasing growth rate from 0.1 to 0.07 [1/h].
Model validation
Finally, the predicting capability of the model was evaluated for conditions similar to the ones used in the initial calibrations (training set).
The robust batch model was first calibrated with the two cultivations of the strain harboring one copy of the thaumatin gene, obtaining a characteristic parameter set for that strain. Then, these parameters were used to predict the course of a different batch cultivation performed in the same conditions (30 °C and pH 6).
This procedure was also applied for the fed-batch model. Here, the bioreactor dynamics was simulated using the parameters obtained in the best calibration within the training dataset (the one in which the calibration objective function was minimal compared to the rest of the calibrations) using the robust modeling structure obtained previously. This prediction was compared with experimental data of a different fed-batch cultivation.
Goodness of fit
The Anderson-Darling test was used to verify if the residuals between simulations and experimental data \( \left({X}_{ij}^{mod}-{X}_{ij}^{exp}\right) \) were normally distributed. If they were, the differences between them can be attributed to measurement noise and not to model inadequacy. The failure of this test by one of the model’s state variables (p-value < 0.05) indicates that a different mathematical relation than the one used in the model may underlie its dynamics. Therefore, the results of this test may be used to confirm or update the kinetic expressions associated with the consumption and production of compounds.
Simulation
Analysis of the metabolic flux distribution during key stages of a dynamic cultivation
After the calibration of the fed-batch model with the dataset used for checking its robustness, we evaluated the central metabolic flux distributions at three different stages of the cultivation: exponential growth during the batch phase (~20 h), ethanol and arabitol consumption during glucose starvation phase (~27.5 h) and controlled growth during the feeding phase (~45 h).
Discovery of beneficial knock-out targets for the overproduction of recombinant Human Serum Albumin (HSA)
We simulated one batch cultivation for each gene in the model and compared their final protein and biomass concentrations with those of the parental strain. The candidates that reached a higher HSA concentration than the parental strain were manually analyzed and some of them were proposed as candidates to improve HSA production. It is important to mention that we used a set of parameters derived in this study to characterize the growth kinetics of the HSA producing strain used in the simulations. Therefore, the predictions derived from this work should be assessed carefully and considered only as an example of the applicability of our modeling framework.
Evaluation of different feeding policies in silico to improve recombinant protein production considering specific information about the strain and process setup
Simulations were run using the parameters obtained in the calibration used for intracellular flux analysis and adding the q_{P} vs μ relation for HSA biosynthesis in the mass balances. The process limitations (based on our setup) were a maximum reactor volume of 1 L, and a maximum oxygen transfer rate of 10.9 [g/L · h]. If any of these limits were violated by either the feeding rate of medium or the oxygen uptake rate (extracted from the model), the integration stopped.
We assessed 13 exponential feeding policies. Five of them maintained a constant growth rate during the feeding phase and the rest considered a decreasing growth rate throughout the culture (Additional file 7). After the simulation, we ranked the strategies according to the volumetric productivity of recombinant HSA and chose the best one as a cultivation strategy that could potentially improve bioreactor performance.
Results and discussion
The batch and fed-batch models were developed in four steps: (i) determination of initial parametric problems, (ii) reparametrization and cross calibration, (iii) robustness evaluation and (iv) validation of predictive potential under the studied conditions.
Once the models were developed, three applications were proposed to improve recombinant |protein production using Human Serum Albumin as a case study.
Initial parametric problems
Batch model
Potential Robust Structures Tested in the Cross-Calibration Stage for the batch model
Structure | Parameters included |
---|---|
Original | v _{ G, Max }, K _{ G } v _{ EtOH,B }, v _{ Pyr,B }, v _{ Arab,B }, v _{ Cit,B }, m _{ ATP } and α _{ B } |
1 | v _{ G,Max }, v _{ EtOH,B }, v _{ Pyr,B }, v _{ Arab,B }, v _{ Cit,B } and α _{ B } |
2 | v _{ G, Max }, v _{ Cit,B } and α _{ B } |
3 | K _{ G }, v _{ EtOH,B }, v _{ Pyr,B }, v _{ Arab,B } and v _{ Cit,B } |
4 | v _{ EtOH,B } and v _{ Cit,B } |
5 | v _{ G,Max }, v _{ Pyr,B }, v _{ Arab,B } |
6 | v _{ G,Max }, v _{ EtOH,B }, v _{ Pyr,B }, v _{ Arab,B }, v _{ Cit,B } |
7 | v _{ G,Max }, K _{ G }, v _{ EtOH,B }, v _{ Pyr,B }, v _{ Cit,B } |
8 | K _{ G }, v _{ Pyr,B }, v _{ Arab,B }, α _{ B } and m _{ ATP } |
Fed batch model
Data from three aerobic, glucose-limited fed-batch cultivations was successfully calibrated with the initial model of fourteen parameters. As in the batch model, several statistical parametric limitations arose (Additional file 8). The most frequent correlation (in two out of the three calibrations) was between \( {v}_{G, Max} \) and the \( {v}_{EtOH, B} \) during the batch phase. Also, \( {v}_{EtOH, B} \) and \( {v}_{Arab, FB} \) showed 5 and 6 strong correlations with other parameters of the model, respectively.
Finally, the citrate minimum secretion rate during the fed-batch phase and the suboptimal growth during the feeding phase (\( {\alpha}_{FB} \)) were the parameters that presented more sensitivity and significance limitations.
Reparametrization and cross calibration
After model calibration and the subsequent determination of the parametric problems for each dataset, the non-relevant parameters were fixed (left out of the adjustable set) using HIPPO [75] to achieve robust modeling structures.
Batch model
The reduced batch models derived from the initial calibrations (Table 3) were recalibrated with the available data (eight batch cultivations) to determine if they could reproduce P. pastoris behavior appropriately. The persistence of parametric problems in the reduced models was compared to the original model.
Structure 6 lacks the sub-optimal growth parameter \( {\alpha}_B \), which forces the solution of a linear programming (LP) problem of specific growth rate maximization in the metabolic block. This is because this parameter was assumed to be zero if it was left out of the adjustable parameter set (Table 2), which eliminates the total flux minimization term from the objective function. This structure showed a significant increase in significance and sensitivity compared to the original model; however, identifiability was a major problem (Table 4). Probably, the multiple solutions associated with an underdetermined LP problem may hamper the possibility to unambiguously infer parameter values from the data.
Therefore, due to the recurrent identifiability issues found in Structure 6, it was preferable to apply Structure 1 to fit a different dataset to check its robustness in aerobic, glucose-limited batch cultures of P. pastoris.
Fed-batch model
Potential robust structures for a fed-batch model
Structure | Parameters included |
---|---|
Original | v _{ G,Max }, K _{ G }, v _{ EtOH,B }, v _{ Pyr,B }, v _{ Arab,B }, v _{ Cit,B }, v _{ EtOH,FB }, v _{ Pyr,FB }, v _{ Arab,FB }, v _{ Cit,FB }, α _{ B }, α _{ FB }, m _{ ATP }, T _{ Cons } |
1 | v _{ G,Max }, K _{ G }, v _{ Pyr,B }, v _{ Cit,B }, v _{ EtOH,FB }, v _{ Pyr,FB }, v _{ Arab,FB }, v _{ Cit,FB }, α _{ B }, m _{ ATP }, T _{ Cons } |
2 | K _{ G }, v _{ EtOH,B }, v _{ Pyr,B }, v _{ Arab,B }, v _{ Cit, B }, v _{ EtOH,FB }, v _{ Pyr,FB }, α _{ B }, m _{ ATP } |
3 | v _{ G,Max }, K _{ G }, v _{ Pyr,B }, v _{ Arab,B }, v _{ Cit,B }, v _{ Pyr,FB } α _{ B }, α _{ FB }, m _{ ATP }, T _{ Cons } |
Robustness check
Batch model
Parameter values achieved in the validation of the batch model structure
Parameter | Value | Units |
---|---|---|
v _{ G,Max } | 6 | mmol/g_{DCW} · h |
v _{ EtOH,B } | 1.47 ± 0.07 | mmol/g_{DCW} · h |
v _{ Pyr,B } | 0.13 ± 0.05 | mmol/g_{DCW} · h |
v _{ Arab,B } | 0.14 ± 0.06 | mmol/g_{DCW} · h |
v _{ Cit,B } | 0.09 ± 0.04 | mmol/g_{DCW} · h |
α _{ B } | 4.1 ± 0.9 · 10^{−4} | [−] |
Despite the sensitivity problem associated with \( {v}_{G, Max} \) for this particular dataset, we included this parameter in the proposed robust modeling structure. This is because for some calibrations, e.g. the batch cultivations of strains harboring 8 copies of the thaumatin gene, the state variables were very sensitive to this parameter (average sensitivity > 0.7, recall that the sensitivity threshold is 0.01); hence, it should be included to achieve a close fit to the data. Therefore, if this parameter is found insensitive in future calibrations, it could be easily fixed at reported values.
We achieved a robust modeling structure for glucose-limited, aerobic batch cultivations of Pichia pastoris, composed of six parameters that estimate specific consumption and production rates of all the species involved in the mass balances. The modeling structure also allows us to determine the specific growth rate by solving a bi-objective optimization problem, which reduces the identifiability issues arising between parameters (comparison between candidate batch model robust structures 1 and 6).
Fed-batch model
Parameter values achieved in the calibration to check the robustness of the fed-batch model. The confidence interval on the time where the consumption of secondary metabolites started T_{CONS}, could not be determined due to the stiffness of the solution caused by a sudden consumption of arabitol and ethanol
Parameter | Value | Units |
---|---|---|
v _{ MAX } | 2.09 ± 0.46 | mmol/g_{DCW} · h |
K _{ S } | 5.55 · 10^{−2} ± 0.0000004 · 10^{−2} | g/L |
v _{ Pyr,B } | 0 | mmol/g_{DCW} · h |
v _{ Arab,B } | 0.42 ± 0.17 | mmol/g_{DCW} · h |
v _{ Cit,B } | 0.04 ± 0.00 | mmol/g_{DCW} · h |
v _{ Pyr,FB } | 0 | mmol/g_{DCW} · h |
α _{ B } | 2.6 · 10^{−4} ± 0.4 · 10^{−4} | [−] |
α _{ FB } | 2.455 · 10^{−5} ± 0.003 · 10^{−5} | [−] |
m _{ ATP } | 7.0 ± 1.4 | mmol/g_{DCW} · h |
T _{ Cons } | 25.73 | H |
The chosen model structure showed a strong fitting capacity and a limited occurrence of parametric identifiability, sensitivity and significance problems. Therefore, we selected it as the most robust model structure for fed-batch cultivations of P. pastoris.
Model validation
Batch model
Fed-batch model
Potential applications of the model
Analysis of the metabolic flux distribution at different stages of a dynamic cultivation
During exponential growth in the batch phase, the carbon reaching the glucose-6-phosphate node is split between carbohydrate production (11%), glycolysis (63%) and the oxidative branch of the PPP (24%). Furthermore, the latter is the main source of cytosolic NADPH. Cytosolic ATP is formed by the activity of the ATP synthase and substrate-level phosphorylation (glycolysis and synthesis of arabitol and ethanol) (data not shown). In the iPP618 model, which is the basis of the iFS670, cytosolic NADPH was produced by a NADP dependent isocitrate dehydrogenase, and no flux appeared through the oxidative branch of the PPP. Using the proposals from Pereira et al. [56], the flux through this pathway was restored and overall agreement in directionality to fluxomic studies performed in similar conditions was achieved (Additional file 3).
During the starvation phase, ethanol and arabitol are co-consumed with limited formation of biomass (μ = 0.02 h^{−1}). As indicated by the negative fluxes, both compounds are directed towards the TCA cycle in order to synthesize the necessary reducing equivalents to fuel oxidative phosphorylation. The ATP formed in this pathway - ~ 7 mmol/g_{DCW}·h -, is mostly employed for maintenance. Even though this m_{ATP} is high compared to other reported values for P. pastoris (2.2 – 5 mmol/g_{DCW}·h) [43], it is required to account for the fast consumption of both secondary metabolites under limited cellular growth. The use of a recombinant strain for model calibration, which might have higher maintenance requirements, could further explain this result.
Finally, during controlled growth at the feed phase, neither ethanol nor arabitol are produced. All the carbon is directed towards biomass formation and the energy necessary for its synthesis and maintenance. This result agrees with previous fluxomic studies carried out in aerobic, glucose-limited chemostats [57, 58], where significant carbon fluxes through the oxidative and non-oxidative branches of the PPP were found, without arabitol formation. Furthermore, the model shows significant oxaloacetate transport from the cytosol to the mitochondria, which was also observed in the cited studies. The most distinguishable feature of this phase is the high activity of the TCA cycle, which almost doubles the flux through this pathway reported under glucose limited conditions in chemostats ([24, 57, 58]). This higher activity in the TCA is probably associated with the need to cope with maintenance and growth-associated energy requirements under stressful conditions, such as high cell density, especially when no significant substrate level phosphorylation besides glycolysis occurs.
This analysis could have been performed using the genome-scale model in static conditions by deriving instantaneous exchange rates from contiguous samples and determining the flux distributions by specific growth rate maximization. Nevertheless, the inspection of flux distributions after model calibration has the advantage of considering the overall behavior of the cells during the cultivations. This provides more experimental support for the determination of parameters such as \( {m}_{ATP} \), \( {K}_G \), that cannot be directly estimated but that have a strong impact on the model output.
Discovery of single knock-outs to improve recombinant Human Serum Albumin production using Minimization of Metabolic Adjustment (MOMA) as the objective function to simulate mutant behavior
We performed 670 (number of genes in the model) batch simulations of single knock-out strains to discover beneficial deletions for the production of recombinant Human Serum Albumin (HSA), a 66 kDa protein with 16 disulfide bridges, that comprises about one half of the total blood serum protein [81].
We decided to leave Cluster I out of the analysis because of the impaired growth observed in the simulations, mainly due to the deletion of reactions associated to lipid biosynthesis. However, candidates from Cluster II (32 in total) were manually analyzed to identify the cause of HSA overproduction (Additional file 11).
Bioprocess optimization for HSA overproduction
Feeding policies evaluated to improve the production of Serum Albumin in a particular bioreactor setup
Strategy | μ_{MAX} | Rate | μ_{MIN} | q_{P} [mg/g · h] | X_{FINAL} [g/L] | P_{FINAL} [mg/L] | Limitation |
---|---|---|---|---|---|---|---|
1 | 0.14 | - | - | 2.85 | 164.8 | 138 | Oxygen |
2 | 0.12 | - | - | 2.59 | 187.8 | 135 | Oxygen |
3 | 0.1 | - | - | 2.32 | 195.3 | 130 | Volume |
4 | 0.08 | - | - | 2.29 | 191.3 | 138 | Volume |
5 | 0.06 | - | - | 2.28 | 184.7 | 154 | Volume |
Best | 0.14 | 0.1 | 0.08 | 2.83 | 197.5 | 150 | Volume |
The improvement in process productivity by modifying substrate addition during the feed phase is less efficient than the one attained by genetic modifications. However, other process variables such as reactor volume and oxygen transfer may be modified to further improve HSA production.
Conclusions
Current GSMs of P. pastoris have been employed to address cellular behavior in stationary conditions. They have been successfully used for predicting production and consumption rates of different compounds and even achieving a 40% improvement of recombinant protein production by model-discovered knock-outs [42]. However, little attention has been given to the actual metabolic flux distribution that these reconstructions yield and how they evolve in a dynamic environment. Resulting flux distributions are important for two reasons: (i) they help to understand the cellular response to the different stresses to which the cell is subjected to and (ii) they can serve as input for several algorithms whose aim is to find metabolic engineering targets to improve the production of a certain compound.
In this work, we developed a robust dynamic GSM of glucose-limited aerobic cultivations of P. pastoris, linking and showing the impact that the model formulation process has over flux balance analysis. The assembled platform can fit several datasets with minimum significance, sensitivity and identifiability problems in its parameters. Moreover, if properly trained, it can be used to predict bioreactor dynamics. The model could also be employed to obtain realistic flux distributions throughout dynamic cultivations and to determine metabolic and process engineering strategies to improve the production of a target compound.
To broaden its applications to other relevant conditions for P. pastoris, the model could be calibrated with data from cultures with different carbon sources and feeding strategies, such as glycerol batch phase followed by a methanol induction phase. Also, the model could be used to study perturbations such as oxygen limitation, which is a common problem in industrial P. pastoris cultivations [84]. Moreover, it would be desirable to calibrate the model with data from a strain capable of producing high concentrations of a recombinant protein to understand and quantify the metabolic burden caused by this production.
Finally, it is expected that the incorporation of more curated metabolic reconstructions [44], gas mass balances and the knowledge derived from testing the hypotheses proposed using the model would improve its accuracy and broaden its applicability.
Declarations
Acknowledgements
We would like to acknowledge Alexandra Lobos for her support during the experiments and for facilitating the P. pastoris strains. We are also grateful to Dr. Dong-Yup Lee for facilitating the iPP668 model for COBRA, from which our version was built upon. English edition of the final manuscript by Lisa Gingles is highly appreciated.
Funding
This project was funded by CORFO (Project 11CEII-9568). F.S. was recipient of a M.Sc. scholarship from CONICYT N° 22140230 and P.T. obtained a Ph.D scholarship from the same institution, N° 21140759.
Availability of data and materials
All data generated or analyzed during this study are included in this published article (Additional files 4, 5, 6, 7, 8, 9, 10 and 11)
Authors’ contributions
FS, PT, RP and EA conceived the experiments and simulations. FS and PT performed the experiments. FS assembled the model, performed the parametric analysis and simulations. All authors read and approved the final manuscript.
Competing interest
The authors declare that they have no competing interests.
Consent for publication
Not Applicable
Ethics approval and consent to participate
Not Applicable
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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