Question

Plot the response of a culture to diauxic growth on glucose and lactose based on the following: \(\mu_{\mathrm{g} \text { lucose }}=1.0 \mathrm{~h}^{-1} ; \mu_{\text {lactose }}=0.6 \mathrm{~h}^{-1} ; Y_{\text {glucose }}=Y_{\text {lactose }}=0.5 ;\) enzyme induction requires 30 min to complete. Plot cell mass, glucose, and lactose concentrations, assuming initial values of \(2 \mathrm{~g} / \mathrm{l}\) glucose, \(3 \mathrm{~g} / \mathrm{l}\) lactose, and \(0.10 \mathrm{~g} / \mathrm{l}\) cells.

Short answer

Rapid growth on glucose, 30-min induction period, then slower growth on lactose.

Step-by-step solution

Understand Key Parameters

Identify the given parameters: growth rates \(\mu_{\text{glucose}} = 1.0 \, \mathrm{h}^{-1}\) and \(\mu_{\text{lactose}} = 0.6 \, \mathrm{h}^{-1}\), yield coefficients \(Y_{\text{glucose}} = Y_{\text{lactose}} = 0.5\), and the induction time for enzyme production (30 minutes or 0.5 hours). Initial concentrations are 2 g/L glucose, 3 g/L lactose, and 0.1 g/L cells.

Plot Glucose Consumption Phase

Consider that growth starts on glucose since it has a higher growth rate. The cell mass and glucose concentration change according to the differential equations: \[\frac{dX}{dt} = \mu_{\text{glucose}} \, X \text{ and } \frac{dS_{\text{glucose}}}{dt} = -\frac{\mu_{\text{glucose}}}{Y_{\text{glucose}}} \, X.\] Solve these equations over time.

Determine Time to Deplete Glucose

Using the integration \(X(t) = X_0 \, e^{\mu_{\text{glucose}} \, t}\) and the relationship that \(S(t) = S_0 - \frac{X(t) - X_0}{Y_{\text{glucose}}}\), find the time when glucose is depleted. Set \(S_{\text{glucose}} = 0\) and solve for time \[t_{\text{glucose}} = \frac{\ln((S_{\text{glucose},0} \, Y_{\text{glucose}}/X_0) + 1)}{\mu_{\text{glucose}}}.\]

Transition Period

Plot the 30-minutes enzyme induction period (0.5 hours) after glucose depletion, during which the cell growth is stalled.

Lactose Consumption Phase

Once the enzyme is induced, cells start utilizing lactose. Plot cell mass and lactose concentration using similar differential equations: \[\frac{dX}{dt} = \mu_{\text{lactose}} \, X \text{ and } \frac{dS_{\text{lactose}}}{dt} = -\frac{\mu_{\text{lactose}}}{Y_{\text{lactose}}} \, X.\] Solve these equations.

Combine Plots

Combine the plots for cell mass, glucose, and lactose concentration to show the entire process: initial rapid growth on glucose, a stationary phase during enzyme induction, and subsequent growth on lactose.

Key concepts

Microbial Growth Kinetics

Microbial growth kinetics is a study of how microorganisms grow and divide under different conditions. This field analyzes how factors like nutrient availability, temperature, and pH impact the rate at which these organisms proliferate.
The growth rate, often denoted as \(\bmu\), defines how quickly the population increases over time. It can be described mathematically using differential equations. For instance, in continuous culture systems, the growth rate can be constant, but in batch culture systems, it varies based on nutrient consumption and waste accumulation.
These kinetic principles are crucial when designing bioreactors or studying microbial behavior for industrial applications. They help predict how microorganisms will respond to changes in their environment and can optimize processes such as fermentation.

• Growth rate equations: \(\frac{dX}{dt} = \mu \cdot X\).
• Yield coefficient (\(Y\)): Reflects the efficiency of converting substrates into biomass.

Understanding these concepts helps us know why microorganisms switch from one substrate to another, like glucose to lactose, during diauxic growth.

Batch Culture

Batch culture is a method of growing microorganisms or cells. It involves placing them in a closed system where they are exposed to initial conditions only, and there is no addition or removal of substances during the process.

• Stages of batch growth include lag, exponential (log), stationary, and death phases.
• Nutrient and substrate concentrations are critical for determining growth kinetics in batch culture.

In the example provided in the exercise, the microorganisms first consume glucose because it allows for a higher growth rate. Once glucose is depleted, there is a lag during which the microorganisms induce the production of enzymes required to metabolize lactose.
Batch culture systems are widely used in research and industry due to their simplicity and the valuable kinetic data they provide. However, they also come with limitations such as the inability to maintain optimal growth conditions indefinitely.

Enzyme Induction

Enzyme induction is a biological process where the presence of a particular substance (an inducer) increases the synthesis of an enzyme required to break down that substance.

• This process is critical for microbial adaptation to different substrates.
• Induction time varies depending on the microbial species and the complexity of the required enzyme.

In the context of diauxic growth, enzyme induction explains the lag phase observed once a preferred substrate (e.g., glucose) is depleted. Microorganisms need time to synthesize enzymes for the metabolism of the next available substrate (e.g., lactose). In our example, this induction period is 30 minutes.
During this period, there is no net growth because the microorganisms are reallocating their resources to produce the necessary enzymes. Once enzyme induction is complete, growth resumes using the new substrate.
Understanding enzyme induction is crucial when analyzing microbial growth patterns and for optimizing industrial processes where multiple substrates might be present.