Transcript SSAC2005.QR75.LS1.1-stdnt

SSAC2005.QR75.LS1.1
Bacteria in a Flask – Spreadsheeting
Population Density vs. Time
How does the size of a bacterial
population change over time within a
closed environment?
How does the reproductive rate of
the bacteria change over time, as a
bacterial culture ages?
Photo obtained from Commonwealth of Massachusetts
Department of Public Health at:
mass.gov/dph/cdc/antibiotic/antibiotic_home.htm
Core Quantitative Concept
Rate of change
Supporting Quantitative Concepts
Unit conversions
Logarithms, orders of magnitude
Exponential growth
Graphs, XY-scatterplot
Logarithmic scale
Prepared for SSAC by
Loretta Sharma – Delta College
© The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. 2006
(Changes made on title slide by LV, 7/6/07)
1
Overview
Bacteria grow and reproduce when the nutrients they require are in plentiful supply
and their environment is hospitable. If a small number of bacteria are inoculated into
a vial containing fresh nutrient broth, the bacteria will begin to reproduce using
binary fission. At first, their rate of reproduction will be slow, as they adjust to their
new environment. This phase of slow increase in the bacterial population is known as
the lag phase. Within a short period of time, the bacteria begin to exhibit a rapid,
sustained increase in numbers; this rapid growth is known as the exponential growth
phase. As the number of bacteria increase, they begin to use up much of their food
supply. Gases necessary for their metabolic processes are also depleted if the
chamber is sealed. Metabolic wastes and toxins accumulate, along with the remains
of dead bacterial cells. These processes act together to slow the expansion of the
bacterial population, and it enters a state of equilibrium known as the stationary
phase or the plateau phase. Finally, as nutrients are depleted and waste products
accumulate, the population drops rapidly, with more bacteria dying than reproducing.
This last phase is known as the death phase.
Slide 5 describes the growth of bacteria.
Slide 6 introduces the problem, and Slides 7-12 explain
how to tackle it.
Slides 13-15 examine ways of interpreting the results.
Slides 16-17 contain end of module assignments.
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Growth of Bacteria
Within a closed environment, how does the size
of a bacterial population change over time?
What information would you need to answer this question?
What variables could affect the number (N) of bacteria over time?
In order to determine the change in the size of a population over
a time interval, Δt, you need to know N at the beginning of Δt, and
you need to know N at the end of Δt. Subtracting the first from the
second produces the change in the population over the time
interval:
Nfinal – Ninitial = ΔN.
The Greek letter Δ
(Delta) is used to
indicate “change.”
Dividing the change in population by the length of time over which
it occurred gets you the rate of change:
ΔN / Δt = Rate of Change in Population.
In the main part of this module, we will consider the changes in
population density, with units of cells/mL (number per milliliter).
We will refer to this variable as N, number of cells, because we will
always be talking about a 1-mL volume of broth.
In one of the end-ofmodule questions,
you will be asked to
calculate N*, the
number of bacteria in
the vial. What else
will you need to
know, in addition to
the population?
3
Problem
An inoculum containing a known concentration (population density)
of bacteria was placed into a flask containing sterile nutrient broth
causing the broth to have an initial concentration of 10 cells/mL.
Over the next two days, the broth was periodically sampled. The
concentration of viable bacteria was estimated at various elapsed
times through the use of serial dilution and enumeration of colonyforming units on plates. Here are the results:
5 hours – 25 cells/mL
8 hours – 90 cells/mL
12 hours – 10,000 cells/mL
19 hours – 10,000,000 cells/mL
24 hours – 600,000,000 cells/mL
30 hours – 500,000,000 cells/mL
33 hours – 1,000,000 cells/mL
36 hours – 10 cells/mL
40 hours – 1 cell/mL
PROBLEM: How did the population density change over time? During
which time interval did the maximum rate of population increase occur?
How would you organize the data to answer these questions?
4
Strategy
Strategy for determining the maximum rate of population growth, and the
time period in which it occurred:
1.
Tabulate the information concerning the bacterial concentration of each
sample and the time at which the sample was collected.
2.
Graph the tabulated information as an XY scatter plot, using Excel.
3.
4.
*
*
Find the maximum slope on the curve that best fits the data points. This is
the maximum rate of growth.
Determine the time boundaries for the period of maximum growth.
However: you have to be careful. You
need to use arithmetic, as opposed to
logarithmic, scales for your axes.
Much of this module concerns these
different scales.
By the way, arithmetic in this usage is an
adjective. To pronounce the word, accent
the third syllable. To use the same word as
a noun (for a branch of mathematics), accent
the second syllable.
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Creating Your Spreadsheet
Recreate this
spreadsheet
2
3
4
5
6
7
8
9
10
11
12
B
Elapsed
Time
(hours)
0
5
8
12
19
24
30
33
36
40
C
Population
Density, N
(cells/mL)
10
25
100
10,000
10,000,000
600,000,000
500,000,000
1,000,000
10
1
D
Phase
Be certain to include
units in your labels at
the top of each
column!
For each of these green cells,
indicate the phase that
corresponds to the time interval
between the preceding row and
the row of the cell. For example,
Cell D4 refers to the interval of
time between Row 3 and Row 4.
A = Lag phase
B = Exponential-growth phase
C = Stationary phase
D = Death phase
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Creating Your Graph
Copy this much of your
spreadsheet , and create a
scatter plot of Population
Density vs. Time
2
3
4
5
6
7
8
9
10
11
12
B
Elapsed
Time
(hours)
0
5
8
12
19
24
30
33
36
40
C
Population
Density, N
(cells/mL)
10
25
100
10,000
10,000,000
600,000,000
500,000,000
1,000,000
10
1
Steps for making a scatter plot:
1. Highlight both columns, including the
labels.
2. Go to ChartWizard.
3. Under “Standard Types,” select “XY
(scatter),” then select the chart subtype
with points connected by lines (NOT
smoothed lines).
4. Hit “Next.”
5. Hit “Next,” again. Enter a chart title and
titles for your x and y axes. Don’t forget
to indicate units!
6. Select “Gridlines,” and unselect the box
for “Major Gridlines.”
7. Select “Finish” and your graph will
appear.
8. Right-click on the legend box that
identifies the line connecting the points
on the graph, select clear, and it will
disappear.
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Here is the graph 
Creating Your Graph and Not Liking It
Improving Your Graph
Population Density (cells/mL)
Population Density vs. Time
2
3
4
5
6
7
8
9
10
11
12
700,000,000
600,000,000
500,000,000
400,000,000
300,000,000
200,000,000
B
Elapsed
Time
(hours)
0
5
8
12
19
24
30
33
36
40
C
Population
Density, N
(cells/mL)
10
25
100
10,000
10,000,000
600,000,000
500,000,000
1,000,000
10
1
100,000,000
0
0
10
20
30
40
50
Time (hours)
Is it easy to read the information collected at the
early and late stages?
How can you rescale the y-axis to make the
information more understandable?
How many orders
of magnitude is
this range?
When your data span
orders of magnitude, it is
often useful to plot the
data on a logarithmic
scale.
Add Column D to your
spreadsheet and
calculate the log of the
population density. We
will graph that instead of
the population density
itself.
Click here if you would
like help creating
Column D.
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Expanding Your Spreadsheet
Here is the result of taking the
logarithm of the population
density in cells/mL.
2
3
4
5
6
7
8
9
10
11
12
B
C
Elapsed
Time
(hours)
Population
Density, N
(cells/mL)
0
5
8
12
19
24
30
33
36
40
10
25
100
10,000
10,000,000
600,000,000
500,000,000
1,000,000
10
1
D
Log
Population
Density in
cells/mL
Log N cells/mL
1.0
1.4
2.0
4.0
7.0
8.8
8.7
6.0
1.0
0.0
= value is given.
= calculate; use cell equation.
Beware: The log of N would be
something different if N were
expressed in units of cells per
cubic inch, for example. That
is why the label says Log of
the population density when
the population density is
expressed in cells/mL One
must be careful and not forget
what units are being used.
Now plot the Log of
the Population
Density vs. Elapsed
Time   
9
Changing your Graph
Log of Population
Density in cells/mL
New Graph
Compare how the
information is
represented on
the two different
types of graphs.
LOG of Population Density vs. Time
10
8
6
4
2
0
0
5
10
15
20
25
30
35
40
45
Time (hours)
Another way to convert
the y-axis to a log scale
is to right-click on it and
then select “Format
Axis.” Select “Scale,”
then check the box for
“Logarithmic Scale,”
and then hit “OK.”
Population Density (cells/mL)
Old Graph
Population Density vs. Time
700,000,000
600,000,000
500,000,000
400,000,000
300,000,000
200,000,000
100,000,000
0
0
10
20
30
Time (hours)
40
50
10
Interpreting Your Results
Log of Population
Density in cells/mL
LOG of Population Density vs. Time
Which segment
of the graph
represents the
time interval with
the maximum
increase in
population?
10
B3
8
B2
6
B1
4
2
0
0
5
10
15
20
25
30
35
40
45
Time (hours)
Many people would say B1, because it
has the steepest slope. They would be
wrong. Why?
The slope in this graph shows the rate of
change of the logarithm of population
density. The question asks for ΔN/Δt, not
ΔLog N/Δt. To see ΔN/Δt, the rate of
change of the population, you have to
look at the graph of N vs. t.
With that in mind, which
segment does have the
greatest rate of population
growth?
You can learn a lot by
calculating both rates
of change
11
Interpreting Your Results, 2
First, add a column that calculates the rate of change of population
2
3
4
5
6
7
8
9
10
11
12
B
C
Elapsed
Time
(hours)
Population
Density, N
(cells/mL)
0
5
8
12
19
24
30
33
36
40
10
25
100
10,000
10,000,000
600,000,000
500,000,000
1,000,000
10
1
D
E
Log
Rate of
Population
Population
Density in
cells/mL Log Change ΔN /Δt,
cells /mL/hr),
N cells/mL
1.0
1.4
3
2.0
25
4.0
2,475
7.0
1,427,143
8.8
118,000,000
8.7
-16,666,667
6.0
-166,333,333
1.0
-333,330
0.0
-2
The maximum
rate of change
of the
population.
This is
Segment B3
of the graph.
12
Interpreting Your Results
Next, add a column that calculates the rate of change of the log of the population
2
3
4
5
6
7
8
9
10
11
12
B
C
Elapsed
Time
(hours)
Population
Density, N
(cells/mL)
0
5
8
12
19
24
30
33
36
40
10
25
100
10,000
10,000,000
600,000,000
500,000,000
1,000,000
10
1
For true exponential
growth, ΔLogN/Δt is a
constant. What is
going on here?
D
E
F
G
Log
Rate of
Population
Population
Rate of change
Density in
cells/mL Log Change ΔN /Δt, of Log N cells/mL Segment
cells /mL/hr), Δlog N cells/mL /Δt on Graph
N cells/mL
1.0
1.4
3
0.080
A
2.0
25
0.201
A
4.0
2,475
0.500
B1
7.0
1,427,143
0.429
B2
8.8
118,000,000
0.356
B3
8.7
-16,666,667
-0.013
C
6.0
-166,333,333
-0.900
D
1.0
-333,330
-1.667
D
0.0
-2
-0.250
D
Max
ΔLogN/Δt
A value of ΔLogN/Δt = 0.5 means that N
increases by a factor of 100.5 = 3 every minute.
In other words, the population density triples as
each minute passes. That’s the maximum rate 13
of relative change in this experiment.
End of Module Assignments
1.
Print out the two graphs and label the four stages mentioned in the
explanatory slide(s).
2.
Write a short paragraph on the advantages and disadvantages of logarithmic
as opposed to arithmetic scale for charting bacterial populations.
3.
Using Excel, tabulate and graph the data you obtained from your laboratory
exercise in bacterial populations. Be sure to identify the growth phases on
your graph. How is the graph of your lab data similar or different from the
graph you created in this module?
4.
Predict how the shape of the graph would differ, if fresh nutrient broth was
introduced into the system each day, and used broth was filtered off. (Assume
that whole bacteria, viable and nonviable, would not be removed by this
system.)
5.
The population curve model presumes a “closed system”; do you think the
human population on the Earth would follow the dynamics of this model? Why
or why not? HINT: In what ways is the Earth like a closed system, and in what
ways is it not?
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End of Module Assignments
6.
INTERNET RESEARCH: Find the website for the US Census Bureau. At that
site, there are tables listing the estimated human population of the Earth (as
determined by various sources) from pre-historic times through 1950. There
are also more accurate and precise tables listing the population from 1950 to
the present, along with projections of population growth for the future, through
2050. Use this data to create an Excel graph of the human population from
pre-history through the present. In what phase of the population growth curve
is humanity, at this time?
7.
Calculate the absolute population size of the bacteria at 24 hours, using the
data given in this Module. Assume the chamber size was 4 liters, and that the
aliquots of broth removed for population testing were so small that they did
not affect this volume appreciably. Using Excel, tabulate and graph the data
you obtained from your laboratory exercise in bacterial populations. Be sure
to identify the growth phases on your graph. How is the graph of your lab data
similar or different from the graph you created in this module?
15
Log Transformation
2
3
4
5
6
7
8
9
10
11
12
B
C
Elapsed
Time
(hours)
Population
Density, N
(cells/mL)
0
5
8
12
19
24
30
33
36
40
10
25
100
10,000
10,000,000
600,000,000
500,000,000
1,000,000
10
1
D
Log
Population
Density in
cells/mL Log
N cells/mL
1.0
1.4
2.0
4.0
7.0
8.8
8.7
6.0
1.0
0.0
To transform the data in
column “C” into log
data:
1.
In the first cell in the log
column, type:
=log(
2.
A new box appears.
Replace the contents of
the parentheses in this
box with: (C3,10)
3.
This represents the
number of the first cell to
be transformed (C3) and
the log base (10). Notice
how the formula appears
in the formula box.
4.
Hit “Enter.”
5.
Highlight the D3 cell,
and drag the lower right
corner down to create
the rest of the log table.
= value is given.
= calculate; use cell equation.
Return to Slide 8
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