# Lagrangian Trajectories to Predict the Formation of Population Heterogeneity in Large-Scale Bioreactors

^{*}

## Abstract

**:**

^{−1}performing multifork replication. The population showed very strong heterogeneity, as indicated by the observation that 52.9% showed higher than average adenosine triphosphate (ATP) maintenance demands (12.2%, up to 1.5 fold). These results underline the potential of CFD linked to structured cell cycle models for predicting large-scale heterogeneity in silico and ab initio.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Cell Cycle Model

^{−1}to 0.6 h

^{−1}for P. putida KT2440 were obtained by Lieder et al. [6] and processed as shown in Figure 1. The data were channeled and displayed as the frequency distribution of DNA content. The durations of cell cycle phases C (DNA replication) and D (period between replication and completed cell division) were determined iteratively by minimizing the deviation between experimental and theoretical DNA histograms. The theoretical DNA content of an asynchronous, ideal culture in which all cells have equal growth parameters was derived from the age distribution according to Cooper and Helmstetter [7]. Using this probability density function for cells of a specific cell age, Cooper and Helmstetter further calculated the theoretical chromosome content per cell at a specific cell age. This model was extended by Skarstad et al. [12] to calculate the frequency of the occurrence of a specific DNA content in an interval of ongoing DNA synthesis. The durations of phases C and D are decisive for the distribution of DNA content. Different values for C were obtained to fit the experimental histograms for various growth rates. Based on the work of Lieder [22], a function for C-phase duration, dependent on the growth rate of P. putida KT2440, was derived. A correlation for C proposed by Keasling et al. [23] was used.

_{min}is the minimal length of the C period, µ represents the growth rate and a and b are parameters that fit the experimental data. Based on the experimental data of Lieder et al. [6], the parameter estimation resulted in C

_{min}= 0.77 h, a = 1.83, and b = 4.88.

#### 2.2. Numerical Simulation

#### 2.2.1. Geometry and Reactor Setup

^{−1}was reached. The impeller Reynolds number was 1.8 × 10

^{6}, the power number 13.15, and the needed power was 226 kW.

_{glc}·${\mathrm{kg}}_{\mathrm{CDW}}^{-1}$·h

^{−1}. Aeration, gas transfer, and oxygen uptake were neglected in this study. Therefore, no gassing system was installed. A cell concentration of 10 kg

_{CDW}·m

^{−3}was assumed, and a simple Monod-like kinetic was used to simulate the substrate uptake ${\mathrm{q}}_{\mathrm{s}}$:

^{−1}. The maximum uptake rate was calculated with the biomass substrate yield Y

_{XS}= 0.40 g

_{s}·${\mathrm{g}}_{\mathrm{CDW}}^{-1}$ and the maximum growth rate μ = 0.59 h

^{−1}[22,24].

#### 2.2.2. Simulation Setup

#### 2.3. Statistical Evaluation

^{−1}, the transition area T (0.3 < μ < 0.4 h

^{−1}), and multifork replication M for μ ≥ 0.4 h

^{−1}derived by the cell cycle model (see Section 2.1.). By evaluating the cell history, further classifications were made. Six regime transitions follow when two transitions and one retention time were considered:

**STM**: transition from standard forked to multiforked with a retention time in the transition area.**STS**: standard forked, retention in the transition area, and back to standard forked**TST**: starting from the transition area with retention in a single forked area and back to transition**MTS**: multiforked replication regime to single forked replication with a retention time in the transition area**MTM**: beginning in the multifork regime with retention in the transition area and back to the multifork regime**TMT**: circulation from transition back to transition area with retention time in the multifork replication regime

## 3. Results and Discussion

^{−3}was assumed, which remained constant within the time observed. For higher biomass concentrations, stronger gradients can be expected.

#### 3.1. Gradient and Flow Field

^{−1}. The theoretical growth rate for every numerical cell was computed (Eulerian approach), resulting in an average growth rate of μ = 0.294 h

^{−1}. Ideal mixing was assured by comparing the average growth rate in the reactor (Eulerian approach) and the expected growth rate for the set feed rate μ = 0.295 h

^{−1}. In the fed batch fermentation, the feeding rate amounted to half the maximum uptake rate of P. putida. The objective of the simulation was to generate a realistic glucose gradient with concentrations for which theoretical growth rates ranging from 0.0 h

^{−1}to 0.59 h

^{−1}could be approximated. Moreover, the distribution of bacteria that were introduced from different vertical positions in the reactor at the start of the simulation is displayed.

#### 3.2. Lagrangian Trajectory

#### 3.3. Statistical Evaluation

#### 3.3.1. Regime Transition Frequency

^{−1}and μ > 0.4 h

^{−1}) were located in the upper half of the reactor. Rushton turbines usually cause flow patterns moving away from the blades to the wall, where they circulate up or down, thereby forming large eddies for each stirrer set (Figure 2B). Consequently, cells will often circulate in this segment and do not pass other areas of the reactor. The lower part of the reactor, which does not provoke a regime transition and, therefore, badly supplies the organisms with substrate, consisted of three segments. As a result, the average retention time in the TST transition was the longest (${\overline{\tau}}_{\mathrm{TST}}$ = 8.54 s). All other average and maximum retention times are listed in Table 1. The shapes of the distributions follow a Poisson distribution. The maximal retention time was defined as the limit, within which 99% of the values were located.

^{−1}before they adapt to growth rates of less than 0.3 h

^{−1}. During the observation window of 260 s, 72.6% of all cells were expected to carry out this move at least once and to linger more than 30 s in regime S. About 14.7% of all cells were expected to stay more than 70 s in regime S after experiencing higher glucose concentrations in regime T. Furthermore, if a regime transition from maximal to moderate growth conditions (MTS) with the retention time in regime T and S is assumed, 55.5% of all cells performed this move for more than 30 s. A retention time of 70 s was calculated for 10.4% of all cells. The time scales of 30 s and 70 s were shown to significantly influence the transcriptional response of E. coli [26], leading to the assumption that changes in adenosine triphosphate (ATP) and guanosine triphosphate (GTP) levels of P. putida KT2440 could also be expected.

#### 3.3.2. Energy and C-Phase Duration Distribution

^{−1}was expected. Using the Lagrangian approach, an average growth rate of µ = 0.269 h

^{−1}was computed, indicating an adequate deviation of 8.5% compared to the Eulerian approach with µ = 0.294 h

^{−1}(see Section 3.1).

_{ATP}is presented in Figure 5A. The growth rate µ and q

_{ATP}were not evenly distributed compared to the mean value, but exhibited individual distributions according to the gradient. The ATP consumption rate was calculated applying Pirt’s law (see Equation (3)). While only 6.3% of all cells had a mean ATP consumption rate of q

_{ATP,mean}= 29.31 ± 2 mmol

_{ATP}·${\mathrm{g}}_{\mathrm{CDW}}^{-1}$·h

^{−1}, 40.8% showed a reduced consumption rate of less than 27.31 mmol

_{ATP}·${\mathrm{g}}_{\mathrm{CDW}}^{-1}$·h

^{−1}, and 52.9% showed an increased energy demand of 31.31 mmol

_{ATP}·${\mathrm{g}}_{\mathrm{CDW}}^{-1}\xb7$h

^{−1}in comparison to the average consumption rate. Moreover, 12.2% show an energy demand that was more than 1.5 times that of the mean value in the reactor.

_{min}= 0.86 h to C

_{max}= 2.05 h. Clearly, the bacteria were not evenly distributed according to the mean value, and there was a large heterogeneity in the reactor. Although only 22.3% of all cells had a replication phase of 1.21 ± 0.2 h, about 30% possessed a C-period of more than 1.41 h. In contrast, 47.7% displayed a shorter replication phase than the average time for replication (less than 1.01 h). Moreover, approximately 56.1% of the cells were rapidly replicating cells with a growth rate higher than µ = 0.3 h

^{−1}. For these cells, it can be assumed that they already started to completely adjust their metabolism to achieve multifork replication. As shown in Figure 5B, the bioreactor population was strongly heterogeneous, characterized by a nonequal distribution of bacteria in different cell cycle states. Three different growth phenotypes are shown: C-phase durations of (i) 0.94 ± 0.08 h, (ii) 1.68 ± 0.1 h, and (iii) a transition state of C-phases ranging from 1.1 to 1.5 h. Previously, subpopulations resulting from chemostat experiments have been categorized in populations containing one, two, or more than two chromosomes [27]. With this simulation setup, a model-based superposition of subpopulations containing different growth rates to mimic the scenario in a (fed)batch fermentation was shown. For the underlying gradient, new categories of subpopulations according to the C-phase durations mentioned above can be formulated.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Table A1.**Dimensions of the reactor setup pictured in Figure A1.

Description | Symbol | Relation |
---|---|---|

Reactor diameter | D_{R} | 3.00 m |

Impeller diameter | D_{I} | 0.43 D_{R} |

Impeller height | H_{I} | 0.21 D_{I} |

Bottom clearance | C_{1} | 0.30 D_{R} |

Impeller spacing | ΔC | 1.00 D_{R} |

Upper clearance | C_{2} | 1.27 D_{R} |

Baffle width | B | 0.10 D_{R} |

Liquid height | H_{L} | C_{1} + ΔC + C_{2} |

## Appendix B

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**Figure 1.**Approach for the cell cycle dynamics model. (

**A**) Representative flow cytometry scatter plot for deoxyribonucleic acid (DNA) content of the growth rate µ = 0.3 h

^{−1}. (

**B**) DNA content over counts for growth rates ranging from µ = 0.1 h

^{−1}up to µ = 0.6 h

^{−1}. A single genome is indicated by 1, and double chromosomes by 2. Black lines present experimental data, and blue dashed lines present the calculation of the cell cycle model. (

**C**) Approximated C-phase duration over growth rate estimated by the cell cycle model (1% parameter covariance). Black dashed lines indicate the transition regime from single-forked to multiforked replication. Flow cytometry data obtained by Lieder et al. [6].

**Figure 2.**Simulation of gradients and bacterial lifelines. (

**A**) Averaged substrate gradient calculated for 150 s, colored by regime classification: standard replication S (μ < 0.3) in light gray, transition regime T (0.3 ≤ μ ≤ 0.4) in gray, and multifork replication M (μ > 0.4) in dark gray. (

**B**) Average flow field estimated for 150 s. (

**C**) Representative magnified bacteria particles (around 2000) at a certain time step (colored by particle ID; low numbers in dark gray represent a starting point close to the reactor bottom, high numbers in light gray represent a starting point close to the reactor top). Horizontal section planes are indicated by dashed red lines; otherwise, the top view is shown.

**Figure 3.**Bacterial lifeline and regime transition classification. (

**A**) Two-dimensional (2D) bacterial lifeline for different growth rates μ over time. The black line represents raw data, and the red line represents filtered data (moving average filter to correct discrete random walk (DRW) fluctuations). Black dashed lines indicate the transition regime from single-forked to multiforked replication. (

**B**) Translation of filtered (one-dimensional (1D) filter) growth rate curves for the three regimes: multifork replication regime M, transition between standard forked and multiforked T, and standard replication S. Examples for two bacterial lifelines L1 and L2 are depicted. For L1, five regime transitions (STS, TST, STM, TMT, and MTS; see Section 2.3) were analyzed. (

**C**) Bacterial movement patterns for two bacterial lifelines (L1 in gray and L2 in black). Starting positions are indicated by black circles.

**Figure 4.**Regime transition frequency as a function of the retention time $\tau $. Regime transition classifications are indicated in the left corner of each panel. The second capital letter always indicates the area, in which the retention time $\tau $ was measured. The regime transition count for each retention time was scaled logarithmically.

**Figure 5.**Distribution of C-phase duration and energy level. (

**A**) Frequencies of cells with a specific adenosine triphosphate (ATP) consumption rate (q

_{ATP}) tracked for 20 s. Average value of q

_{ATP,mean}= 29.31 mmol

_{ATP}$\xb7{\mathrm{g}}_{\mathrm{CDW}}^{-1}\xb7$h

^{−1}. Range of the x-axis from q

_{ATP,min}= 5.57 mmol

_{ATP}·${\mathrm{g}}_{\mathrm{CDW}}^{-1}\xb7$h

^{−1}to q

_{ATP,max}= 52.98 mmol

_{ATP}$\xb7{\mathrm{g}}_{\mathrm{CDW}}^{-1}\xb7$h

^{−1}. (

**B**) Frequency of cells having a specific duration of replication (C-phase). Average C-phase duration of C

_{mean}= 1.21 h. Range of the x-axis from C

_{min}= 0.86 h to C

_{max}= 2.05 h. Counts were divided into 300 bins.

**Table 1.**Average and maximal retention time in a specific regime. For the six regimes (STS, TST, TMT, MTM, STM, and MTS), the average ($\overline{\tau}$) and maximal retention times (${\tau}_{\mathrm{max}}$) are displayed in seconds. The maximum $\tau $ was defined as the limit, within which 99% of the values were located.

Regime Transition | $\overline{\mathit{\tau}}$ [s] | ${\mathit{\tau}}_{max}$ [s] |
---|---|---|

STS | 0.99 | 3.7 |

TST | 8.54 | 73.5 |

TMT | 3.53 | 16.25 |

MTM | 2.45 | 13 |

STM | 0.95 | 6.6 |

MTS | 0.88 | 5.5 |

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**MDPI and ACS Style**

Kuschel, M.; Siebler, F.; Takors, R.
Lagrangian Trajectories to Predict the Formation of Population Heterogeneity in Large-Scale Bioreactors. *Bioengineering* **2017**, *4*, 27.
https://doi.org/10.3390/bioengineering4020027

**AMA Style**

Kuschel M, Siebler F, Takors R.
Lagrangian Trajectories to Predict the Formation of Population Heterogeneity in Large-Scale Bioreactors. *Bioengineering*. 2017; 4(2):27.
https://doi.org/10.3390/bioengineering4020027

**Chicago/Turabian Style**

Kuschel, Maike, Flora Siebler, and Ralf Takors.
2017. "Lagrangian Trajectories to Predict the Formation of Population Heterogeneity in Large-Scale Bioreactors" *Bioengineering* 4, no. 2: 27.
https://doi.org/10.3390/bioengineering4020027