Question

Consider an industrial-scale batch fermentation. A 10,0001 fermenter with \(5 \times 10^{10}\) cells \(/ \mathrm{ml}\) is the desired scale-up operation. Inoculum for the large tank is brought through a series of seed tanks and flasks, beginning with a single pure colony growing on an agar slant. Assume that a colony \(\left(10^{6}\right.\) plasmid-containing cells) is picked and placed in a test tube with \(1 \mathrm{ml}\) of medium. Calculate how many generations will be required to achieve the required cell density in the \(10,000 \mathrm{l}\) fermenter. What fraction of the total population will be plasmid-free cells if \(\mu_{+}=1.0 \mathrm{~h}^{-1}, \mu_{-}=1.2 \mathrm{~h}^{-1}\), and \(P=0.0005\) ?

Short answer

Approximately 39 generations are required. The fraction of plasmid-free cells will be 0.05%.

Step-by-step solution

- Determine initial and final cell densities

Initial cell density starts with a colony containing \(10^6\) plasmid-containing cells in \(1 \, \text{ml}\) of medium. Final cell density is required as \(5 \times 10^{10} \, \text{cells/mL}\) in a total tank volume of \(10,000 \, \text{L} = 10,000,000 \, \text{mL}\).

- Calculate total cell density required

Calculate the total cell density required:\[ \text{Total cells required} = 5 \times 10^{10} \times 10,000,000 = 5 \times 10^{17} \text{cells} \]

- Calculate generations required

The number of generations (n) required can be determined using the formula\[ N = N_0 \times 2^n \]Rearrange to solve for n:\[ n = \frac{\log(N/N_0)}{\log(2)} \]From above:\[ N_0 = 10^6 \text{ cells} \]\[ N = 5 \times 10^{17} \text{cells} \]\[ n = \frac{\log(5 \times 10^{17} / 10^6)}{\log(2)} = \frac{\log(5 \times 10^{11})}{\log(2)} \]\[ n \approx \frac{11.69897}{0.3010} \ \approx 38.86 \]

- Calculate the fraction of plasmid-free cells

Given growth rates \( \mu_+ = 1.0 \text{h}^{-1} \) and \( \mu_- = 1.2 \text{h}^{-1} \), and plasmid loss rate \( P = 0.0005 \). The equilibrium fraction of plasmid-containing cells \( \theta \) is given by:\[ \theta = \frac{\mu_+}{\mu_+ + P} = \frac{1.0}{1.0 + 0.0005} \approx 0.9995 \]Then, fraction of plasmid-free cells \(1 - \theta \) = \[ 1 - 0.9995 = 0.0005 \] or 0.05%.

Key concepts

Cell Density Calculation

In industrial-scale batch fermentation, calculating cell density is crucial to ensure the desired number of cells are present in the fermenter. Imagine starting with a single pure colony of cells, which includes around 1 million plasmid-containing cells (\(10^6\text{ cells}\)) in 1 milliliter of medium. Now, consider that the end goal is to achieve a density of \(5 \times 10^{10} \text{ cells/mL}\) in a massive fermenter that holds 10,000 liters of medium.
To break it down:

• The initial cell density is determined. We start from 1 million cells per milliliter.
• The final cell density desired is \(5 \times 10^{10} \text{ cells/mL}\) across a volume of \(10,000 \text{ liters} = 10,000,000 \text{ mL}\).

Total cells required calculates thus:
\[ \text{Total cells required} = 5 \times 10^{10} \text{ cells/mL} \times 10,000,000 \text{ mL} = 5 \times 10^{17} \text{ cells} \] This calculation ensures that the desired number of cells will be available to carry out the fermentation process efficiently.

Generations Required for Scale-Up

The next step in the process is to determine how many generations of cell division are required to achieve the desired cell density.
We use the formula:
\ N = N_0 \times 2^n \
Where:

• \(N\) is the final number of cells
• \(N_0\) is the initial number of cells
• \(n\) is the number of generations

Rearranging the formula to solve for \(n\), we get:
\[ n = \frac{\text{log}(N / N_0)}{\text{log}(2)} \] Plugging in our values:

• \(N_0 = 10^6 \text{ cells} \)
• \(N = 5 \times 10^{17} \text{ cells} \)

\[ n = \frac{\text{log}(5 \times 10^{17} / 10^6)}{\text{log}(2)} = \frac{\text{log}(5 \times 10^{11})}{\text{log}(2)} \] \[ n \text{ approximately } = \frac{11.69897}{0.3010} \text{ which is about } 38.86 \] This means approximately 39 generations are required to scale up from the initial inoculum to the final desired cell density.

Plasmid Loss Rate

Maintaining the plasmid within the bacterial cells is a constant challenge in fermentation processes. Plasmid loss happens because not all plasmid-containing cells retain their plasmids after each cell division.
When considering the rates

• \( \mu_+ = 1.0 \text{h}^{-1} \)
• \( \mu_- = 1.2 \text{h}^{-1} \)
• Plasmid loss rate: \( P = 0.0005 \)

Our aim is to calculate the fraction of plasmid-free cells.
The equilibrium fraction of plasmid-containing cells \( \theta \) is given by:
\[ \theta = \frac{\mu_+}{\mu_+ + P} = \frac{1}{1 + 0.0005} \text{ which is about } 0.9995 \] So, the fraction of plasmid-free cells is:
\[ 1 - \theta \text{ which equals } 1 - 0.9995 = 0.0005 \text{ or } 0.05\text{%} \] This small fraction indicates that only 0.05\text{%} of the cells will be plasmid-free, a vital consideration in maintaining the efficacy of the fermentation process.

Fermentation Scale-Up Process

The scaling-up process in fermentation is critical to maximize productivity and ensure consistency in industrial applications. Scaling up from laboratory to industrial scale involves several steps to transition from a small-scale culture to a full-scale batch fermentation.
Key points in the process include:

• Seed Tanks and Flasks: Begin by growing the initial cell colony in a small flask or test tube, typically with 1 ml of medium.
• Incremental Steps: Gradually increase the culture volume through several seed tanks of increasing size.
• Monitoring: Carefully monitor cell growth and conditions (temperature, pH, nutrient availability).
• Consistency: Ensure that conditions and growth rates remain consistent throughout the scaling-up process.

Fermentation scale-up requires proficient planning and precise monitoring at each step.
It involves not only growing a larger volume but also maintaining the quality and characteristics of the original cell culture.
Proper scale-up ensures that the desired cell density and productivity are achieved in the final large-scale fermentation.