Q.83. A bacterial culture has a generation time of 34 minutes. The time taken (in minutes, rounded
off to two decimal places) for the 𝐎𝐃_𝟓𝟓𝟎 of this exponentially growing culture to increase from 0.25 to
0.85 is ____ .
Assume that 𝐎𝐃_𝟓𝟓𝟎 has a linear relationship with the cell density.

Bacterial cultures grow exponentially, and OD550 (optical density at 550 nm)
measures turbidity that is linearly proportional to cell density during the exponential phase.
For a generation time of 34 minutes, the time required for OD550 to rise from
0.25 to 0.85 is calculated using exponential growth principles.

Core Calculation

The number of generations (n) required is given by:

n = log₂(final OD / initial OD)

n = log₂(0.85 / 0.25) = log₂(3.4)

This evaluates to approximately:

n ≈ 1.766 generations

The total time (t) is then:

t = n × generation time = 1.766 × 34 ≈ 60.04 minutes

Step-by-Step Derivation

Exponential bacterial growth follows the equation:

N_t = N₀ × 2^t/g

where:

• N₀ = initial cell density
• N_t = cell density at time t
• g = generation time = 34 minutes

Since OD550 ∝ cell density during exponential growth:

OD_t / OD₀ = 2^t/g

Taking natural logarithms:

t = g × ln(OD_t / OD₀) / ln(2)

Substituting values:

t = 34 × ln(3.4) / 0.693147 ≈ 60.04 minutes

This calculation assumes ideal exponential growth with no lag or stationary phase effects.

Common Mistakes in Answer Choices

• Direct ratio (≈85 minutes): Incorrect — assumes linear growth, ignoring exponential doubling.
• One generation (34 minutes): Incorrect — OD increases more than threefold, requiring ~1.77 doublings.
• Intermediate values (e.g., 51 minutes): Incorrect — uses wrong fold-change such as OD 0.5 instead of 0.85.
• Ignoring OD linearity: OD550 directly reflects cell number only in exponential phase below saturation.

Applications in Microbiology

This type of calculation is routinely used in laboratory experiments to analyze
bacterial growth curves using spectrophotometry.
Plate readers measure OD550 in real time to track population dynamics in
biotechnology, fermentation, and antibiotic testing.

Accurate growth-time estimation helps optimize culture conditions and experimental timing.