1
ESM 219
Lecture 4:
Growth and Kinetics
Concepts
•
Concerned with populations of cells and
population numbers and characteristics
•
Various ways to report population sizes
•
Unrestricted cellular division is well-
understood
•
Mass transfer-limited growth models,
including on solid media, involve more
terms
Unrestricted Cell division =
population growth
X = 2
n
X = number of cells
n = number of division events (doublings)
At the beginning, X = 2
0
= 1
2
1
= 2
2
2
= 4, etc…….
Population growth, X = 2
n
(cont.)
Let
n = (# doublings / time) * t
= growth rate constant * t
=
μ
*t
where
μ
= “specific growth rate”
Notice that equation is exponential,
so can express as X = e
μ
t
Note that starting concentration is not 1,
So X = X
o
e
μ
t
where X
o
is the initial concentration.
Unrestricted cell division is a “first
order” rate process
•
Unrestricted cell
division is a first order
rate process.
(
)
)
e
X
(
dt
d
X
dt
d
t
0
μ
=
X
dt
dX
μ
=
So, …..
X
dt
dX
μ
=
∫
∫
=
=
=
=
=
=
t
t
t
t
t
t
dt
X
dX
dt
X
dX
0
0
μ
μ
e
X
X
X
X
t
t
t
t
μ
μ
0
0
ln
=
=
Now let’s go the other direction…..
Separate variables.
Integrate over the interval 0 to t.
The result is a straight line equation….
t
X
X
μ
+
=
0
ln
ln
And if we raised e to both sides, we’d be
back where we started…

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During
exponential
phase growth, a
log-linear plot
produces a
straight line,
because
t
X
X
μ
+
=
)
ln(
)
ln(
0
Microbial Laboratory Culture
Well-mixed liquid
(flask or culture tube
or chemostat)
Solid media culture
(agar or sand culture)
In all cases
•
Prepare media with nutrients, and establish
conditions required for growth
• Inoculate
–
With a small amount of known cells for pure culture
–
With an environmental sample for enrichment culture
•
Monitor over time
–
Quantify population size at various time points
–
Or measure / observe beginning and end.
How to measure cell concentration
1.
Direct counts with a microscope
How to measure cell concentration
2.
Culturable counts
With or without a dilution series.