SME
Microbiology
Microbial growth
Use this for “A culture grows from 1 × 10³ to 1.024 × 10⁶ cells per mL in 5 h. How many generations, and what is the generation time?”
Chart
N(n) = 1,000·2ⁿ
A log y-axis: an exponential draws as a straight line.
This is an example. Type your own values to start a new problem.
N (cells/mL)
=
N₀ (cells/mL)
×
2
t
Time
example
g
Generation time
calculated
Formulas
N
=
N₀
×
2
n
1.024 ×
10
6
=
1,000
×
2
10
g
=
t
n
30
=
300
10
Formulas are worked in min. The step-by-step shows the conversions.
Assumptions
•
The culture is in log phase: every cell divides in two each generation, so N = N₀ × 2ⁿ.
•
The generation time g is the doubling time, the same for every generation: g = t ÷ n.
•
log₁₀(2) = 0.30103, so n = 3.3 × log₁₀(N ÷ N₀) gives the generations quickly.
Step-by-step
WE KNOW
N₀ = 1,000 cells/mL, N = 1.024 × 10⁶ cells/mL, t = 5 h
FIND
generations (n), generation time (g)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to min).
t = 5 h = 300 min (1 h = 60 min)
Step 2 · Find generations (n)
Use N = N₀ × 2ⁿ
The
number
of
doublings
from
N₀
to
N
:
the
log
of
the
fold
change
over
the
log
of
2.
n
=
log₁₀(
N
÷
N₀
)
log₁₀(2)
n
=
log₁₀(1.024 ×
10
6
÷ 1,000)
log₁₀(2)
n
=
log₁₀(1,024)
log₁₀(2)
n
=
3.01
0.301
n
=
10
Step 3 · Find generation time (g)
Use g = t ÷ n
Share the time equally among the generations.
g
=
300
10
g
=
30
min
Check
Put the numbers back in. Both sides should match.
1.024 ×
10
6
=
1,000
×
2
10
✓
30
=
300
10
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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