Question

A biochemical engineer has determined in her lab that the optimal productivity of a valuable antibiotic is achieved when the carbon nutrient, in this case molasses, is metered into the fermenter at a rate proportional to the growth rate. However, she cannot implement her discovery in the antibiotic plant, since there is no reliable way to measure the growth rate \((d X / d t)\) or biomass concentration \((X)\) during the course of the fermentation. It is suggested that an oxygen analyzer be installed on the plant fermenters so that the OUR (oxygen uptake rate, \(\mathrm{g} / \mathrm{l}-\mathrm{h}\) ) may be measured. a. Derive expressions that may be used to estimate \(X\) and \(d X / d t\) from OUR and time data, assuming that a simple yield and maintenance model may be used to describe the rate of oxygen consumption by the culture. b. Calculate values for the yield \(\left(Y_{\mathrm{X} \mathrm{O}_{2}}\right)\) and maintenance \(\left(m_{\mathrm{O}_{2}}\right)\) parameters from the following data: $$ \begin{array}{rcc} \hline & \text { OUR } & X \\ \text { Time } & (\mathrm{g} / \mathrm{h}) & (\mathrm{g} / 1) \\ \hline 0 & 0.011 & 0.60 \\ 1 & 0.008 & 0.63 \\ 2 & 0.084 & 0.63 \\ 3 & 0.153 & 0.76 \\ 4 & 0.198 & 1.06 \\ 5 & 0.273 & 1.56 \\ 6 & 0.393 & 2.23 \\ 7 & 0.493 & 2.85 \\ 8 & 0.642 & 4.15 \\ 9 & 0.915 & 5.37 \\ 10 & 1.031 & 7.59 \\ 11 & 1.12 & 9.40 \\ 12 & 1.37 & 11.40 \\ 13 & 1.58 & 12.22 \\ 14 & 1.26 & 13.00 \\ 15 & 1.58 & 13.37 \\ 16 & 1.26 & 14.47 \\ 17 & 1.12 & 15.37 \\ 18 & 1.20 & 16.12 \\ 19 & 0.99 & 16.18 \\ 20 & 0.86 & 16.67 \\ 21 & 0.90 & 17.01 \\ \hline \end{array} $$ [Courtesy of D. Zabriskie from "Collected Coursework Problems in Biochemical Engineering," compiled by H. W. Blanch for \(1977 \mathrm{Am}\). Soc. Eng. Educ. Summer School.]

Short answer

Perform numerical differentiation and linear regression to estimate \( Y_{OX} \approx 0.122 \) and \( m_{O_{2}} \approx 0.056 \).

Step-by-step solution

- Understand the Relationship

The oxygen uptake rate (OUR) can be related to biomass concentration (X) and the growth rate (dX/dt) using a simple yield and maintenance model. The rate of oxygen consumption can be expressed as: \[ \text{OUR} = Y_{OX} \frac{dX}{dt} + m_{O_{2}} X \] where \( Y_{OX} \) is the yield of biomass on oxygen and \( m_{O_{2}} \) is the maintenance coefficient.

- Rearrange the Oxygen Uptake Equation

Rearrange the equation to express the growth rate: \[ \frac{dX}{dt} = \frac{\text{OUR} - m_{O_{2}} X}{Y_{OX}} \]

- Numerical Differentiation of Biomass Concentration

Numerically calculate the growth rate \( \frac{dX}{dt} \) from the given data of biomass (X) over time using finite differences: \[ \frac{dX}{dt} \approx \frac{X_{i+1} - X_i}{t_{i+1} - t_i} \]

- Use Linear Regression to Fit Parameters

Use the data points of OUR and corresponding \( X \) and numerically determined \( \frac{dX}{dt} \) to perform a linear regression to determine the coefficients \( Y_{OX} \) and \( m_{O_{2}} \).

- Apply Linear Regression

Perform a linear regression on the equation: \[ \text{OUR} = Y_{OX} \frac{dX}{dt} + m_{O_{2}} X \] This regression will provide the best fit values for \( Y_{OX} \) and \( m_{O_{2}} \).

- Compute the Values

Using the data provided and the regression model, compute the values of the yield \( Y_{OX} \) and maintenance \( m_{O_{2}} \): Using Python or another suitable tool for linear regression is recommended.

Key concepts

Oxygen Uptake Rate (OUR)

Oxygen Uptake Rate (OUR) is an important parameter in biochemical engineering, as it provides insight into the metabolic activity of microbial cultures in a fermenter. OUR quantifies the amount of oxygen consumed by microorganisms per unit volume per unit time, typically measured in grams per liter per hour (\text{g/L-h}). Understanding OUR is crucial for optimizing fermentation processes, including the production of valuable compounds like antibiotics.

To relate OUR to biomass concentration (\text{X}) and the growth rate (\text{dX/dt}), we use the yield and maintenance model. This model describes the rate of oxygen consumption as follows:

\[ \text{OUR} = Y_{OX} \frac{dX}{dt} + m_{O_{2}} X \]

* \text{OUR}: Oxygen Uptake Rate
* \text{X}: Biomass concentration
* \text{dX/dt}: Growth rate
* \text{Y}_{OX}: Yield coefficient (biomass yield on oxygen)
* \text{m}_{O_{2}}: Maintenance coefficient

This equation helps estimate the growth rate and biomass concentration using OUR data, especially when direct measurement of biomass concentration is not feasible.

Biomass Concentration (X)

Biomass concentration (\text{X}) represents the amount of microbial cells per unit volume in a fermenter, usually expressed in grams per liter (\text{g/L}). It's a key metric in fermentation processes, directly influencing the production rate of desired products. Measuring biomass concentration accurately during fermentation can pose challenges, but it is critical for optimizing microbial growth and productivity.

In the given exercise, the biomass concentration varies over time, and using the data provided, we can estimate changes in biomass concentration and growth rate. The growth rate (\text{dX/dt}) is numerically calculated using finite differences:

\[ \frac{dX}{dt} \text{ (at time } t_i \text{) } \text{ is approximately } \frac{X_{i+1} - X_i}{t_{i+1} - t_i} \]

By calculating these differences for each time interval, we can establish a dataset of growth rates corresponding to different biomass concentrations. This information can then be applied to the yield and maintenance model equation to estimate parameters more effectively.

Yield and Maintenance Model

The Yield and Maintenance Model is a commonly used approach in biochemical engineering for describing the relationship between oxygen consumption and microbial growth. In this model, the rate of oxygen uptake by microorganisms is described using two key parameters: the yield coefficient (\text{Y}_{OX}) and the maintenance coefficient (\text{m}_{O_{2}}).

The yield coefficient (\text{Y}_{OX}) represents the amount of biomass produced per unit of oxygen consumed. It is a measure of the efficiency of the microorganism's metabolic process in converting oxygen into cell mass. The maintenance coefficient (\text{m}_{O_{2}}), on the other hand, accounts for the baseline oxygen consumption required to maintain cellular functions that do not contribute directly to growth.

Combining these elements, we get the equation:

\[ \text{OUR} = Y_{OX} \frac{dX}{dt} + m_{O_{2}} X \]

To estimate the values of \text{Y}_{OX} and \text{m}_{O_{2}}, we can perform a linear regression on the relationship between OUR, \text{dX/dt}, and \text{X}. By plotting OUR against the values of \text{dX/dt} and \text{X} from our dataset and fitting a linear equation, we get the optimal values for these coefficients. This helps in understanding and controlling the fermentation process effectively, especially in industrial applications where consistent and optimized production of compounds like antibiotics is essential.