# Model-Based Process Optimization for the Production of Macrolactin D by Paenibacillus polymyxa

^{*}

## Abstract

**:**

_{2}concentration, base consumption, and near-infrared spectroscopy (NIR) were used for model improvement. After model extension using expert knowledge, a single superior model could be identified. Model-based state estimation with a sigma-point Kalman filter (SPKF) was based on online measurement data, and this improved model enabled nonlinear real-time product maximization. The optimization increased the macrolactin D production even further by 28% compared with the initial robust multi-model offline optimization.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Setup

_{2}. The pH value was measured using a glass electrode (405-DPAS-SC-K8S pH probe, Mettler-Toledo International Inc., Columbus, OH, USA) and controlled using sodium hydroxide and sulfuric acid to maintain a constant pH value of 7. The level of dissolved oxygen (InPro 6800, Mettler-Toledo International Inc., Columbus, OH, USA) was held at a minimum level of 50% by a PI controller that uses a stirrer speed with a minimum of 200 rpm at a sparging rate of 10 slm. The temperature was held constant at 28 °C. In the last four experiments, NIR spectra were recorded and processed online using a PGS 1.7 tc 512 (tec5 AG, Oberursel, Germany) equipped with the transmission probe Excalibur 12 (Hellma GmbH, Mullheim, Germany) with 1 mm gap width. The integration time of the sensors in the NIR spectrometer was 36 ms. Every 3 s, a spectrum was recorded and then pretreated and subsampled to a sampling frequency of $\frac{1}{6}{\mathrm{min}}^{-1}$. The same sampling frequency was applied for the off-gas carbon dioxide concentration and base consumed for pH control.

_{3}PO

_{4}to 100% acetonitril and after 2 min again in a linear gradient over 4 min back to 100% of an aqueous solution of 0.1%H

_{3}PO

_{4}. With the precolumn LiChrospher 100 RP-19 (5 μm) endcapped and the main column LiChrospher 100 RP-19 (5 μm), both made by Merck KGaA (Darmstadt, Germany), the retention time of macrolactin D was ca. 7.4 min at a detection wavelength of 230 nm (Detector PDA-100, Dionex Corporation, Sunnyvale, CA, USA).

#### 2.2. Mathematical Fermentation Modeling

#### 2.2.1. Measurement Uncertainty

_{2}) fraction in the off-gas, and the volume of added base.

#### 2.2.2. Parameter Uncertainty

**F**, which is a lower bound to ${\mathbf{C}}_{\underline{\theta}\underline{\theta}}$ [43]

**S**in parallel to the integration of the system differential equation [46]. The output sensitivities needed in Equation (4) are then readily derived.

#### 2.2.3. Model Uncertainty

_{c}) can be used for discrimination between such models. The model with the smallest AIC

_{c}shows the best fit. Based on the distance to it, all m models can be evaluated, and model “probabilities” ${w}_{\mathrm{k}}$ can be deducted.

#### 2.2.4. Parameter Subset Selection

**F**was still possible but with great uncertainties. A parameter-centered approach considers the root of the diagonal elements of ${\mathbf{C}}_{\underline{\theta}\underline{\theta}}$, i.e., the standard deviation ${\underline{\sigma}}_{\underline{\theta}}$. By excluding extremely safe parameters and extremely unsafe parameters through deleting the corresponding rows and columns from

**F**, the condition number of

**F**was improved. The bounds were set to 5000% of the parameter value as the upper bound and 0.01% of the parameter value as the lower bound. The reduced parameter covariance matrix ${\mathbf{C}}_{\underline{\theta}\underline{\theta}}$ was then used for state estimation, as described below.

#### 2.3. Multivariate Analysis of NIR Spectroscopy

#### 2.4. Process Optimization

#### 2.4.1. Robust Planning

#### 2.4.2. Online Optimization

#### 2.5. Nonlinear State Estimation

#### 2.5.1. Sigma-Point Kalman Filter

**R**, the process noise spectral density matrix

**Q**, the covariance matrix of the parameters ${\mathbf{C}}_{\underline{\theta}\underline{\theta}}$, and the initial covariance matrix of the state ${\mathbf{P}}_{xx,0}$. Heuristic design approaches [67] based on the extended Kalman filter can be used for the SPKF as well. The scaling factor for the weighing of the different sigma points h was set to $\sqrt{3}$, the optimal value for Gaussian priors [66]. Process noise was used to describe the effect of uncertain parameters. Because the covariance matrix ${\mathbf{C}}_{\underline{\theta}\underline{\theta}}$ is available via the Fisher analysis, the approach described in [67] is used here as well. To improve the numerical performance, the number of relevant parameters was reduced via subset selection, as explained above.

#### 2.5.2. Observability

## 3. Results and Discussion

#### 3.1. Automatically Generated Model (A)

_{c}and the derived model probability ${w}_{k}$. These are given in Table 2. The models are numbered by ascending AIC

_{c}, meaning that No. 1 is the most probable.

_{c}values also support this impression because no single model has a particularly high model probability. In other identification experiments (which are not shown here) the macrolactin simulation performed better. Model 9 was chosen as the best representative of its model structure, containing only a phosphate storage instead of ammonium and glucose storage. It is included in further process planning by manually assignind a model probability of 5%. It has to be noted that the decision to include model 9 was made before the extended model family B was developed. It turned out later that the phosphate storage approach is superior as will be discussed in the next subsection.

#### 3.2. Extended Model (B)

_{2}mass flow entering the gas phase of

_{3}, so the H

^{+}of the dissolved NH${}_{3}^{+}$ must be compensated by the corresponding amount of base added to keep the pH constant [71]. Further modeling approaches such as the production of organic acids in the context of an overflow metabolism are not considered. The differential equation for the mass of correction fluid base ${m}_{\mathrm{B}}$ results in the following:

_{2}, biomass, and glucose concentration. This could be because the additional measurement information is now available, while the number of parameters has increased only insignificantly. The AIC

_{c}values and the corresponding probabilities are newly calculated. It is debatable whether the measurement data of CO

_{2}and base are to be considered in the calculation of the AIC

_{c}. Both measured values have no influence on substrate consumption, biomass growth, and macrolactin production, and, therefore, are completely irrelevant for the process optimization. However, the correct description of the online measurements is relevant for the state estimation and should also be taken into account. Consequently, both approaches were pursued and are illustrated in Figure 3 in the form of AIC

_{c}values. If the model probabilities are calculated without continuous measurement data, there is an Akaike probability of almost 100% for model 11. If online measurements are considered, model 11 also shows good performance because it scores second best with only a narrow distance to the best model but a large distance to the next-best model. Hence, model 11 is used for the online optimization and state estimation.

#### NIR Spectroscopy

_{2}measurements that are more precise in the first part of the exponential growth phase at medium biomass concentrations between 0 and 2 g/L but that lack a good description of the limiting conditions.

#### 3.3. Early-Stage Robust Optimization

#### 3.3.1. Reference Nominal Process Optimization

_{c}criterion and using it for offline optimization. To avoid a limitation by the limited oxygen input, a maximum biomass concentration of 20 g/L and a maximum biomass growth of 17 g/h was determined. To prevent the depletion of glucose, even in case of uncertainties, a lower limit of 10 g/L was set for the optimization.

_{c}, models 1, 2, and 3 were almost equally likely based on the set of the four first identification experiments. Therefore, it is evident that the parametric uncertainty of model 1 does not cover the uncertainty of the model choice.

#### 3.3.2. Robust Planning and Experimental Validation

_{c}criterion was used in robust planning (multi-model trajectory planning: MMTP) with Equation (13). For model 9, a probability of 5% was defined, and the other weights were based on their Akaike probability and adjusted accordingly. The simulated courses of substrate, biomass, and macrolactin concentrations are shown together with the experimental results in Figure 9.

#### 3.4. Online Optimization

_{2}, consumed base, and NIR spectroscopy. From the extended and modified model family B, model 11 is found to be satisfactorily predictive on its own. However, besides the model uncertainty, there are other sources of deviations; for example, the initial biomass in the inoculum. Whereas offline robust optimization can only account for general deviations and leads to a very conservative design, online optimization can deal with the measured or estimated disturbances in the process hence leading to better results.

#### 3.4.1. State Estimation

_{2}concentration and the base consumption at low values, i.e., in the exponential growth phase, are a very good indicator of growth, while the model uncertainty is much more pronounced at higher concentrations. The absolute measurement uncertainty of the CO

_{2}concentration was reduced to 0.1 ${\%}_{{\mathrm{CO}}_{2}}$ to take better advantage of this effect. For the NIR spectroscopy, the opposite is the case. High uncertainty was assumed at low biomass levels, while the relative error is smaller compared with the ${\%}_{{\mathrm{CO}}_{2}}$ and base.

**R**was hence a diagonal matrix with the corresponding variances. For the uncertainties of the initial values, a heuristic approach [67] yielded a relative deviation of ${\sigma}_{{\underline{\widehat{x}}}_{0}}=40\%$ for the biomass components and a value of ${\sigma}_{{\underline{\widehat{x}}}_{0}}=15\%$ for the remaining states. For the creation of sigma points, the parameter covariance matrix ${\mathbf{C}}_{\underline{\theta}\underline{\theta}}$ was used after the subset selection described above instead of

**Q**. The corresponding standard deviations of the parameters are given in Table A2.

#### Observability

#### 3.4.2. Experimental Application of Online Optimization

## 4. Conclusions

_{2}concentration in the off-gas. The use of process NIR spectroscopy with PLS regression was incorporated in the SPKF and led to an even more robust estimation because of its complementary measurement information. The use of real-time product maximization was successfully tested for the production of macrolactin D by Paenibacillus polymyxa. Besides reducing the amount of glucose feed by 28% compared with the offline planning, the optimization led to the largest mass of macrolactin compared with all other performed cultivations in the same setting.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

MMTP | multi-model trajectory planning (robust) |

TP | Trajectory planning |

RTO | Real-time optimization |

PLS | Partial least squares |

NIR | Near-infrared (spectroscopy) |

P.p. | Paenibacillus polymyxa |

Am | Ammonium |

Ph | Phosphate |

C | Glucose |

Xa | Active biomass |

X | Biomass |

V | Volume |

ML | macrolactin D |

St | Storage component |

## Appendix A. Model Equations and Parameters for the Automatically Generated Model

Parameter | Value Mod. 1 | Value Mod. 2 | Value Mod. 3 | Value Mod. 9 | Unit | |
---|---|---|---|---|---|---|

${r}_{\mathrm{Xa}}$ | ${\mu}_{\mathrm{Xam}}$ | 0.0130 | 0.0406 | 0.0472 | 1.3274 | 1/h |

${K}_{\mathrm{XaAm}}$ | 0.0542 | 0.0359 | 0.0475 | g/L | ||

${K}_{\mathrm{XaPh}}$ | 0.0099 | 0.0139 | 0.0158 | g/L | ||

${K}_{\mathrm{XaC}}$ | 0.00029 | 0.00023 | 0.00021 | 0.0001 | ||

${K}_{\mathrm{XaPhSt}}$ | 0.0002 | g/L | ||||

${r}_{\mathrm{zXa}}$ | ${\mu}_{\mathrm{dXam}}$ | 0.0561 | 0.0485 | 0.0497 | 0.0013 | 1/h |

${r}_{\mathrm{AmSt}}$ | ${\mu}_{\mathrm{AmStm}}$ | 0.0554 | 0.0342 | 0.0370 | 1/h | |

${K}_{\mathrm{AmStAm}}$ | 0.3815 | 0.2844 | 0.3614 | g/L | ||

${K}_{\mathrm{AmStPh}}$ | 3.2159 | 0.6581 | 0.0300 | g/L | ||

${K}_{\mathrm{AmStC}}$ | 0.0001 | g/L | ||||

${r}_{\mathrm{zAmSt}}$ | ${\mu}_{\mathrm{zAmStm}}$ | 0.2489 | 0.1467 | 2.2940 | 1/h | |

${K}_{\mathrm{zAmStAm}}$ | 0.0060 | g/L | ||||

${K}_{\mathrm{zAmStPh}}$ | 16.6722 | 0.7386 | 31.0420 | g/L | ||

${K}_{\mathrm{zAmStC}}$ | 0.1443 | g/L | ||||

${r}_{\mathrm{CSt}}$ | ${\mu}_{\mathrm{CStm}}$ | 0.1608 | 0.1352 | 0.1185 | 0.1608 | 1/h |

${K}_{\mathrm{CStC}}$ | 0.0001 | 0.0001 | 0.0001 | 0.0001 | g/L | |

${K}_{\mathrm{CStAm}}$ | 0.0386 | 0.0257 | 0.0309 | 0.0386 | g/L | |

${K}_{\mathrm{CStPh}}$ | 0.2572 | 0.3142 | 0.2797 | 0.2572 | g/L | |

${r}_{\mathrm{zCSt}}$ | ${\mu}_{\mathrm{zCStm}}$ | 0 | 0.0022 | 0 | 0 | 1/h |

${K}_{\mathrm{zCStAm}}$ | 0.1003 | 0.1003 | 0.1003 | g/L | ||

${K}_{\mathrm{zCStPh}}$ | 0.1001 | g/L | ||||

${K}_{\mathrm{zCStC}}$ | 0.0095 | 0.0097 | 0.0095 | g/L | ||

${r}_{\mathrm{PhSt}}$ | ${\mu}_{\mathrm{PhStm}}$ | 0.2880 | ||||

${K}_{\mathrm{PhStPh}}$ | 16.098 | |||||

${r}_{\mathrm{zPhSt}}$ | ${\mu}_{\mathrm{zCStm}}$ | 0.0862 | ||||

${r}_{\mathrm{Ml}}$ | ${\mu}_{\mathrm{Mlm}}$ | 0.0141 | 0.0173 | 0.0079 | 0.0068 | 1/h |

${K}_{\mathrm{MlAm}}$ | 9.999 | 100 | 33.825 | 2.607 | g/L | |

${K}_{\mathrm{MlPh}}$ | 0.0244 | 0.0152 | 0.0439 | 0.3427 | g/L | |

${K}_{\mathrm{MlC}}$ | 20.902 | 8.919 | 39.521 | 100.0 | g/L | |

${r}_{\mathrm{zMl}}$ | ${\mu}_{\mathrm{zMlm}}$ | 0.0087 | 0.0116 | 0.0100 | 0.1668 | 1/h |

${r}_{\mathrm{M}}$ | ${Y}_{\mathrm{M}}$ | 0.0488 | 0.0436 | 0.0423 | 0.0810 | 1/h |

${K}_{\mathrm{M}}$ | 0.01 | 0.01 | 0.01 | 0.01 | g/L | |

${Y}_{\mathrm{Xa}}$ | 31.402 | 9.133 | 8.233 | 5.578 | g/g | |

${Y}_{\mathrm{PhXa}}$ | 3.6319 | 0.9241 | 0.7598 | 0.2646 | g/g | |

${Y}_{\mathrm{CXa}}$ | 0.3083 | 3.1094 | 0.4393 | 13.7042 | g/g | |

${Y}_{\mathrm{CCSt}}$ | 4.831 | 4.515 | 5.015 | g/g |

## Appendix B. Modified Model for Continuous Measurement Data

#### Appendix B.1. Model B 11

Name | $\mathit{\theta}$ | $\mathit{\sigma}$ | $\mathit{\sigma}/\mathit{\theta}$ | Name | $\mathit{\theta}$ | $\mathit{\sigma}$ | $\mathit{\sigma}/\mathit{\theta}$ |
---|---|---|---|---|---|---|---|

${\mu}_{\mathrm{dXam}}$ | 0.14 | 0.0015 | 1.1% | ${K}_{\mathrm{XaAm}}$ | 0.0001 | 0.0002 | 184.1% |

${\mu}_{\mathrm{MIm}}$ | 0.010 | 0.00030 | 3.1% | ${K}_{\mathrm{XaC}}$ | 0.0003 | 0.0001 | 33.6% |

${\mu}_{\mathrm{PhStm}}$ | 0.043 | 0.00039 | 0.9% | ${K}_{\mathrm{XaPh}}$ | 0.0343 | 0.0005 | 1.6% |

${\mu}_{\mathrm{Xam}}$ | 0.079 | 0.0004 | 0.5% | ${K}_{\mathrm{zPhStC}}$ | 0.0001 | 0.00017 | 166.0% |

${\mu}_{\mathrm{zMlm}}$ | 0 | - | - | ${K}_{\mathrm{M}}$ | 0.01 | - | - |

${\mu}_{\mathrm{zPhStm}}$ | 0.0116 | 0.00017 | 1.5% | ${Y}_{\mathrm{BAm}}$ | 0.024 | 0.00001 | 0.1% |

${g}_{0,\mathrm{PhSt}}$ | 0.75 | 0.007 | 0.9% | ${Y}_{\mathrm{M}}$ | 0.09 | 0.00054 | 0.6% |

${K}_{\mathrm{MIC}}$ | 100 | 9.77 | 9.8% | ${Y}_{{\mathrm{CCO}}_{2}}$ | 0.65 | 0.003 | 0.5% |

${K}_{\mathrm{MIPh}}$ | 0.084 | 0.007 | 8.1% | ${Y}_{\mathrm{PhXa}}$ | 0 | - | - |

${K}_{\mathrm{MlAm}}$ | 5.66 | 0.36 | 6.4% | ${Y}_{\mathrm{Xa}}$ | 5.53 | 0.019 | 0.3% |

${K}_{\mathrm{PhSt}}$ | 0.077 | 0.0014 | 1.8% | ${Y}_{\mathrm{CXa}}$ | 14.58 | 0.044 | 0.3% |

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**Figure 1.**Identification experiment for model family A; the models used for further planning are highlighted in color. Concentrations of a substance i are given as ${c}_{i}$, and ${u}_{j}$ denotes a feed stream.

**Figure 2.**Identification experiment for model family B; models used for planning are highlighted in color.

**Figure 3.**AIC

_{c}values for the modified model family (B) with and without consideration of online measurement data.

**Figure 5.**Cross-validation prediction of biomass concentrations based on NIR spectra using PLS regression.

**Figure 8.**Comparison of the experiment planned with model 1 and simulation of it with models 2, 3, and 9.

**Figure 9.**Robust planning using models 1, 2, 3, and 9 with offline values of the experimental validation. Optimization constraints for the volume and glucose are depicted as gray lines.

**Figure 10.**SPKF using online data compared with the preceding planning and offline measurements for a cultivation with large deviations of the initial biomass, where TP A1, A2, A3, and A9 represent the simulated evolution with the respective model based on the offline robust planning (MMTP).

**Figure 11.**Comparison of the preceding offline planning with online optimization together with offline measurements.

**Table 1.**Coefficients to describe the measurement uncertainty according to Equation (2).

b in g/L | b in % | b in mL | |||||
---|---|---|---|---|---|---|---|

X | Am | Ph | C | Ml | CO_{2} | Base | |

a in % | 3 | 5.9 | 3.7 | 2.6 | 3.1 | 2 | 2 |

b | 0.03 | 0.004 | 0.02 | 0.202 | 0.004 | 0.3 | 3 |

Model No. | Stored Components | AIC_{c} | ${\mathit{w}}_{\mathit{k}}$ |
---|---|---|---|

1 | Am, C | −971.1 | 0.39 |

2 | Am, C | −971.0 | 0.37 |

3 | Am, C | −970.2 | 0.24 |

4 | Am, C | −941.2 | 0 |

5 | Am, C | −929.8 | 0 |

6 | Am, C | −929.6 | 0 |

7 | Am, C | −921.1 | 0 |

8 | Am, C | −809.7 | 0 |

9 | Ph | −715.8 | 0 |

10 | Ph | −712.9 | 0 |

11 | Ph | −680.7 | 0 |

**Table 3.**Simulated makrolactin quantity at the end of fermentation; comparison of nominal and robust planning.

Model No. | ||||||
---|---|---|---|---|---|---|

${\mathit{m}}_{\mathbf{ML}}({\mathit{t}}_{\mathbf{end}})$ in g | 1 | 2 | 3 | 9 | min | Weighted Mean |

Nom. TP | 15.0 | 1.1 | 2.5 | 1.8 | 1.1 | 6.6 |

MMTP | 9.8 | 8.3 | 8.7 | 4.2 | 4.2 | 8.7 |

**Table 4.**Modified coefficients of Equation (2) for use in the state estimation.

b in ${\%}_{{\mathbf{CO}}_{2}}$ | b in mL | b in g/L | |
---|---|---|---|

CO_{2} | Base | X NIR | |

a in % | 7.5 | 7.5 | 5 |

b | 0.1 | 3 | 0.5 |

Feeding Volume (L) | Offline TP | Online TP | Saving |
---|---|---|---|

Am | 1.9 | 1.4 | 23% |

Ph | 0.7 | 0.6 | 9% |

C | 3.2 | 2.3 | 28% |

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## Share and Cite

**MDPI and ACS Style**

Krämer, D.; Wilms, T.; King, R.
Model-Based Process Optimization for the Production of Macrolactin D by *Paenibacillus polymyxa*. *Processes* **2020**, *8*, 752.
https://doi.org/10.3390/pr8070752

**AMA Style**

Krämer D, Wilms T, King R.
Model-Based Process Optimization for the Production of Macrolactin D by *Paenibacillus polymyxa*. *Processes*. 2020; 8(7):752.
https://doi.org/10.3390/pr8070752

**Chicago/Turabian Style**

Krämer, Dominik, Terrance Wilms, and Rudibert King.
2020. "Model-Based Process Optimization for the Production of Macrolactin D by *Paenibacillus polymyxa*" *Processes* 8, no. 7: 752.
https://doi.org/10.3390/pr8070752