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

Abstract

Successful scale-up of bioprocesses requires that laboratory-scale performance is equally achieved during large-scale production to meet economic constraints. In industry, heuristic approaches are often applied, making use of physical scale-up criteria that do not consider cellular needs or properties. As a consequence, large-scale productivities, conversion yields, or product purities are often deteriorated, which may prevent economic success. The occurrence of population heterogeneity in large-scale production may be the reason for underperformance. In this study, an in silico method to predict the formation of population heterogeneity by combining computational fluid dynamics (CFD) with a cell cycle model of Pseudomonas putida KT2440 was developed. The glucose gradient and flow field of a 54,000 L stirred tank reactor were generated with the Euler approach, and bacterial movement was simulated as Lagrange particles. The latter were statistically evaluated using a cell cycle model. Accordingly, 72% of all cells were found to switch between standard and multifork replication, and 10% were likely to undergo massive, transcriptional adaptations to respond to extracellular starving conditions. At the same time, 56% of all cells replicated very fast, with µ ≥ 0.3 h^-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.

Keywords: Lagrange trajectory; cell cycle model; computational fluid dynamics; energy level; population dynamics; scale-up; stirred tank reactor.

Figure A1

Schematic diagram of reactor geometry derived from Haringa et al. [21]. The stirred tank reactor contains four baffles and two Rushton turbines with eight blades (bottom) and six blades (top). Dimensions indicated by capital letters are explained in Table A1.

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$·{\mathrm{g}}_{\mathrm{CDW}}^{-1}·$h⁻¹. Range of the x-axis from q_ATP,min = 5.57 mmol_ATP·${\mathrm{g}}_{\mathrm{CDW}}^{-1}·$h⁻¹ to q_ATP,max = 52.98 mmol_ATP$·{\mathrm{g}}_{\mathrm{CDW}}^{-1}·$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.