Batch Culture Kinetics — Estimating μm: In a batch experiment, the maximum specific growth rate μm is equal to the slope of which plot, and over what data range?

Introduction:
The maximum specific growth rate μm is a core kinetic parameter for microbial cultures. In batch culture, μm is best estimated from the exponential phase where growth approximates first-order in biomass with respect to time at non-limiting substrate levels. The correct plotting method isolates this regime.

Given Data / Assumptions:

• Batch culture with clearly identifiable exponential phase.
• Biomass measured by optical density, cell counts, or dry mass.
• Substrate non-limiting during the selected data window.

Concept / Approach:

During exponential growth, biomass X follows X = X0 * exp(μ * t). Taking natural logarithms gives ln X = ln X0 + μ * t. Therefore, a plot of ln X versus time yields a straight line with slope μ. The maximum slope observed across all windows gives μm.

Step-by-Step Solution:

Verification / Alternative check:

Residual analysis of the fitted line should show minimal curvature in the chosen window. Parameter estimates can be cross-checked with nonlinear regression of X(t) = X0 * exp(μ * t) restricted to the exponential region.

Why Other Options Are Wrong:

B averages across non-exponential phases, biasing μ downward. C and D use biomass directly, which does not linearize exponential growth for slope extraction. E uses the wrong transformation and phase.

Common Pitfalls:

Using high OD values outside the linear range or including depletion effects will underestimate μm.

Final Answer:

Slope of ln(biomass) versus time using exponential phase data only