Transcript SSAC2006.QR67.AEW1.1-stdnt

SSAC2006.QR67.AEW1.1
Serial Dilution
Tracking Bacterial
Population Size and
Antibiotic Resistance
From http://textbookofbacteriology.net/staph.html
Core Quantitative concept and skill
Number sense: Ratios, fractions, percentages
How can we determine the number of
bacteria in a sample? What percent
of those are antibiotic-resistant?
Supporting Quantitative concepts and skills
Number sense: Scientific notation
Modeling: Forward modeling; inverse problem
Sampling: Representative sample
Logic functions (optional)
Prepared for SSAC by
Anton E. Weisstein, Truman State University, Kirksville, MO 63501
© The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. 2006
1
Overview of Module
Serial dilution allows estimation of a range of bacterial
population sizes. Percent resistance can be calculated
by culturing bacteria on both antibiotic-free and
antibiotic-containing media and comparing the results.
Slides 3 introduces the bacterium S. aureus and discusses antibiotic resistance.
Slide 4 introduces a specific scenario of determining population size and
measuring antibiotic resistance.
Slides 5-6 introduce the key concepts of proportionality and ratios.
Slides 7-9 take you through the process of building a spreadsheet to calculate the
total number of bacteria on a plate based on a serial dilution from an initial
population of known size (forward modeling).
Slides 10-11 take you through the inverse problem: calculating the total number of
bacteria in a population based on the number of colonies on a plate.
Slides 12-13 show how to calculate the number of resistant bacteria and the
percent resistance in a population.
Slide 14 contains your homework assignment.
2
Background
Staphylococcus and methicillin resistance
Staphylococcus aureus is a gram-positive bacterium commonly found on human skin and
in the nasal cavity. While it is usually harmless, S. aureus can cause boils and lesions
(by invading the skin) or toxic shock syndrome (by entering the bloodstream directly).
Before physicians adopted sterile techniques, many patients (including women in
childbirth) died of infection caused by S. aureus carried on surgical instruments and
doctors’ hands.
The 1928 discovery and subsequent widespread use of penicillin allowed doctors to
control staph infections and saved many lives. However, by 1946, several strains of S.
aureus had evolved enzymes that neutralized penicillin. Researchers responded by
developing additional antibiotics such as methicillin. Methicillin-resistant S. aureus
(MRSA) strains were observed almost immediately and have recently become
increasingly common.
Further Reading:
http://www.wellcome.ac.uk/node5052.html
http://www.medscape.com/viewarticle/484351
From
http://images.stltoday.com/stltoday/
multimedia/staphb1031.gif
3
Problem
Methicillin-resistant S. aureus (MRSA) has recently been
detected in the hospitals of neighboring cities. Your city’s
public health department has asked you to assess the risk of
a similar outbreak in your own community. You have
collected S. aureus cultures from different areas of the local
hospital: three of these cultures are shown below.
Source:
# colonies of
S. aureus
Bathroom
Emergency
room
Maternity
ward
Lawn
0
Lawn
The term “lawn” means
that so many colonies
have grown on a plate
that we can’t tell where
one ends and another
begins. Therefore, we
can’t count the number of
colonies.
• How could you determine how many S. aureus
are present in each of these samples?
• How could you determine how many of those
bacteria are resistant to methicillin?
4
Thinking About the Problem: Proportionality
Proportionality is the property that one quantity is a constant multiple of
another property. For example, if you drive at a constant speed of 40 miles per
hour, then the distance traveled (in miles) is always exactly 40 times the time
elapsed (in hours). We therefore say that mileage is proportional to time.
Proportionality is often a crucial assumption in
biological experiments when you want to generalize
from a specific sample to a larger population. For
example, if you wanted to determine the percent of
male students on your campus, it may be difficult to
ensure you include every student in your census. A
simpler approach is to count the number of men in a
smaller sample (for example, of 100 students). If you
have a representative sample, the number of men in
the sample will be proportional to the number in the
whole student population:
# male students in sample # male students on campus
=
# students in sample
# students on campus
If you find 39 men in a
sample of 100 students, and
there are 4700 students on
your campus, how many
male students do you think
are on your campus?
Obtaining a representative
sample is one of the
thorniest problems in
designing an experiment.
Click here to explore this
issue in more detail.
5
Thinking About the Problem: Ratios, Fractions, and Percentages
A ratio is a comparison of the relative
size of two or more numbers. For
example, in a family with 2 daughters
and 4 sons, the ratio of girls to boys is
2:4 (or equivalently 1:2). A family with 5
daughters and 10 sons would also have
a 1:2 ratio.
A fraction is the portion of a group with
a specific property. For example, the
fraction of girls in the above family is
2/6 (or equivalently 1/3).
A percentage is the number of parts per
100. It can be calculated from a fraction
by dividing the numerator by the
denominator, then multiplying by 100.
For example, 1/3 = 0.3333 = 33.33%.
Note the difference between
a ratio and a fraction in this
context!
Ratio = # girls : # boys
Fraction =
# girls
# girls + # boys
Problems:
• If we expect a 9:7 ratio of red:
white flowers from a particular
cross, what fraction of the
flowers will be red?
• How much water should you
add to 10 mL of 50% sucrose
solution to dilute it to 5%?
Hint: the answer is
NOT 100 mL!
6
Setting up a Spreadsheet: Sampling from a Population
Imagine that 30 mL of waste water from the hospital
bathroom contains 5.6×1011 S. aureus. If we pipette
out 0.1 mL of this water, how many bacteria would
you expect to find in that sample?
2
3
B
Sample 1:
C
Bathroom
If you want to review scientific
notation, click here and then
scroll to the top of the webpage
that comes up.
D
Original
Sample for
4
population
serial dilution
5 # Bacteria:
5.60E+11
1.87E+09
6 Volume (mL):
30
0.1
= cell with a number in it
7
Sample for
8
Tube A
serial dilution
= cell with a formula
in it
9 # Bacteria:
1.87E+09
1.87E+07
10 Volume (mL):
10
0.1
11
If you were to plate this sample
on a for
Sample
12
Tube B
serial
Petri dish, each bacterium
in
thatdilution
13 # Bacteria:
1.87E+07
sample
would reproduce
to form 1.87E+05
a
14 Volume (mL):
10
single colony. Because there are so 0.1
15
many colonies, they would quickly
Sample for
start
16 to overlap and form
Tube Ca continuous
serial dilution
17 # Bacteria:
lawn.
So we need to1.87E+05
pipette out 1.87E+03
a
18 smaller
Volume (mL):
10
0.1
number of bacteria.
19
Recreate this portion of the
spreadsheet. In Cell D5, enter a
formula that will calculate the number
of bacteria in the 0.1 mL sample.
Theoretically, we could avoid getting a
lawn by using a smaller sample
volume (e.g., 1 mL = 0.001 mL). But in
practice it’s very hard to measure such
small volumes accurately. What other
approaches could we try?
7
Setting up a Spreadsheet: Inoculating a New Tube
Take the 0.1 mL pipette sample
(“inoculum”) from the previous slide.
What would happen if we added it to a
test tube containing 9.9 mL of water
instead of adding it to a Petri dish?
Enter Rows 8 through 10 into
your spreadsheet. In Cell C9,
enter a formula that calculates
the number of bacteria in tube A
(the test tube described above).
4
5
6
7
# Bacteria:
Volume (mL):
8
9 # Bacteria:
10 Volume (mL):
11
C
Bathroom
D
Original
Sample for
population
serial dilution
5.60E+11
1.87E+09
30
0.1
Sample for
Tube A
serial dilution
1.87E+09
1.87E+07
10
0.1
Sample for
12
Tube B
serial dilution
Why does Tube A
13 # Bacteria:
1.87E+05
have the same 1.87E+07
14 Volume
10
0.1
number(mL):
of bacteria
as the pipettor?
15
Sample for
Tube
C A serial dilution
Tube
Inoculum 16
17 9# Bacteria:
1.87E+05
# bacteria:
5.6×1091.87E+03
# bacteria: 5.6×10
8 0.1
18 Volume (mL):
Volume: 10
10 mL
Volume: 0.1 mL
19
Why does the
pipettor have the
same concentration
of bacteria as the
original population?
Original population
# bacteria: 5.6×1011
Volume: 30 mL
2
3
B
Sample 1:
Setting up a Spreadsheet: Full Experimental Design
A full dilution series may include many steps
in which we pipette a small amount (0.1 mL in
this example) from one tube into a tube of
distilled water and mix thoroughly. We will
model a series with four dilution steps.
The last step in serial dilution is to pipette a
small amount (again, 0.1 mL in this example)
from the final tube onto an agar plate. Over
several days, each bacterium pipetted onto the
agar will then grow into a distinct colony. If
you have fewer than five or more than 500
colonies on your plate, you might want to
change the number of dilution steps.
2
3
4
5
6
7
B
Sample 1:
# Bacteria:
Volume (mL):
8
9 # Bacteria:
10 Volume (mL):
11
12
13 # Bacteria:
14 Volume (mL):
15
16
17 # Bacteria:
18 Volume (mL):
19
C
Bathroom
D
Original
Sample for
population
serial dilution
5.60E+11
1.87E+09
30
0.1
Sample for
Tube A
serial dilution
1.87E+09
1.87E+07
10
0.1
Sample for
Tube B
serial dilution
1.87E+07
1.87E+05
10
0.1
Sample for
Tube C
serial dilution
1.87E+05
1.87E+03
10
0.1
Inoculum for
Enter the remaining
rows into your
20
Tube D
agar plate
spreadsheet. Based
on
the
initial
population
21 # Bacteria:
1.87E+03
1.87E+01
11
22 Volume
10
0.1
of 5.6×10 bacteria,
how(mL):
many colonies
9
would you expect on the agar plate?
Reversing the Problem
So far, we have worked with an example in which we know the number of bacteria in the
original population and want to predict how many colonies we will see after serial dilution
(“forward modeling”). In practice, this is usually reversed: we know the number of
colonies and want to determine the original population size (the “inverse problem”).
Re-create
this
worksheet.
Use
Use
this
this
version
version
if you
if you
plated
plated
from
from
Tube
Tube
DB
(four
(two
dilution
dilution
steps).
steps).
Use this version if you
plated from Tube C
(three dilution steps).
Note that we
are solving
three slightly
different
versions of
the same
problem.
Click here to
learn a
cleaner way
to set up this
sheet.
10
Reversing the Problem: Calculating Population Size
2
3
4
5
6
7
8
9
10
B
C
Agar plate WITHOUT streptomycin
Plate inoculated
from:
# colonies
on plate:
# Bacteria:
Volume (mL):
11
12 # Bacteria:
13 Volume (mL):
14
15
16
17
18
19
20
21
22
23
24
25
D
35
Inoculum from
tube C into
tube D
3500
0.1
# Bacteria:
Volume (mL):
Inoculum from
tube B into
tube C
350000
0.1
# Bacteria:
Volume (mL):
Inoculum from
tube A into
tube B
35000000
0.1
Inoculum from
vial into tube A
3500000000
0.1
G
H
I
J
Plate inoculated
from:
# colonies
on plate:
Tube C
35
K
L
Tube B
35
Tube D
3500
10
Tube C
350000
10
Tube B
35000000
10
Tube A
3500000000
10
# Bacteria:
Volume (mL):
Inoculum
from tube C
onto plate
35
0.1
# Bacteria:
Volume (mL):
Inoculum
from tube B
into tube C
3500
0.1
# Bacteria:
Volume (mL):
Inoculum
from tube A
into tube B
350000
0.1
# Bacteria:
Volume (mL):
Inoculum
from vial into
tube A
35000000
0.1
Vial
3.5E+11
Using what you learned
10
working through Slides 8-10,
enter formulas into the orange
cells that will calculate the
number of bacteria in each
inoculum and each tube.
# Bacteria:
Volume (mL):
F
Plate inoculated
from:
# colonies
on plate:
Tube D
Inoculum from
tube D onto
plate
35
0.1
E
Tube C
3500
10
Tube B
350000
10
Tube A
35000000
10
Vial
3500000000
10
# Bacteria:
Volume (mL):
Inoculum
from tube B
onto plate
35
0.1
Tube B
3500
10
# Bacteria:
Volume (mL):
Inoculum
from tube A
into tube B
3500
0.1
Tube A
350000
10
# Bacteria:
Volume (mL):
Inoculum
from vial into
tube A
350000
0.1
Vial
35000000
10
Now use this spreadsheet to
calculate the original population
size if you had 164 colonies on a
plate inoculated from Tube C.
11
Doubling the Problem: Inferring Percent Resistance
So far, we have been plating our bacteria onto agar that contains nutrients but no
antibiotic. To find how many antibiotic-resistant bacteria are present, we must also
plate onto agar that contains the antibiotic (in this example, methicillin).
For simplicity, let’s assume that the antibiotic kills all and only the susceptible
bacteria. Each colony on a methicillin plate therefore represents a single resistant
bacterium in the inoculum. We can then calculate the number of resistant bacteria in
the original population the same way we calculated the total number of bacteria.
Agar plate WITHOUT methicillin
In reality, you can’t tell the difference
between susceptible and resistant
colonies just by looking at them. All
we can do is count the number of
colonies on each plate.
Tube containing
mixture of
susceptible &
resistant bacteria
12
Agar plate WITH methicillin
Doubling the Problem: Inferring Percent Resistance
B
C
2 Agar plate WITH streptomycin
3
Plate inoculated
4 from:
# colonies
5 on plate:
6
7
8 # Bacteria:
9 Volume (mL):
10
11
12 # Bacteria:
13 Volume (mL):
14
15
16 # Bacteria:
17 Volume (mL):
18
19
20 # Bacteria:
21 Volume (mL):
22
23
24 # Bacteria:
25 Volume (mL):
D
E
F
Plate inoculated
from:
# colonies
on plate:
Tube D
35
Inoculum from
tube D onto plate
35
0.1
Tube D
3500
10
Inoculum from
tube C into tube D
3500
0.1
Tube C
350000
10
Inoculum from
tube B into tube C Tube B
350000 35000000
0.1
10
Inoculum from
tube A into tube B
35000000
0.1
Tube A
3.5E+09
10
Inoculum from
vial into tube A
3500000000
0.1
Vial
3.5E+11
10
G
H
I
J
Plate inoculated
from:
# colonies
on plate:
Tube C
35
K
L
Tube B
35
M
N
Plate inoculated
from:
# colonies
on plate:
O
P
Q
R
Plate
inoculated
from:
# colonies
on plate:
Tube A
35
S
T
U
V
2 Agar plate
3
Plate inocu
4 from:
# colonies
5 on plate:
6
Original vial
35
7
8 # Bacteria:
9 Volume (m
10
# Bacteria:
Volume (mL):
Inoculum from
tube C onto plate Tube C
35
3500
0.1
10
# Bacteria:
Volume (mL):
Inoculum from
tube B into tube
C
Tube B
3500
350000
0.1
10
# Bacteria:
Volume (mL):
Inoculum from
tube B onto plate Tube B
35
3500
0.1
10
# Bacteria:
Volume (mL):
Inoculum from
tube A into tube
B
Tube A
350000 35000000
0.1
10
# Bacteria:
Volume (mL):
Inoculum from
tube A into tube B Tube A
3500 350000
0.1
10
# Bacteria:
Volume (mL):
Inoculum from
vial into tube A
35000000
0.1
# Bacteria:
Volume (mL):
Inoculum from vial
into tube A
Vial
350000 35000000
0.1
10
Vial
3.5E+09
10
11
12 # Bacteria:
13 Volume (m
14
Under Excel’s Edit menu, choose “Move/Copy
Sheet” to make a copy of your previous
worksheet. Then add Columns N through T to
model the cases of plating from Tube A or directly
from the vial with the original population.
15
16 # Bacteria:
17 Volume (m
18
# Bacteria:
Volume (mL):
# Bacteria:
Volume (mL):
Inoculum from
tube A onto plate Tube A
35
3500
0.1
10
Inoculum from vial
into tube A
3500
0.1
Vial
350000
10
19
20 # Bacteria:
21 Volume (m
22
Inoculum from
vial onto plate
# Bacteria:
35
Volume (mL):
0.1
Why might you want to use
fewer dilution steps in plating
onto agar with methicillin than
you used in plating onto agar
without methicillin?
Vial
23
3500 24 # Bacteria:
10 25 Volume (m
13
End of Module Assignments
1.
Proportionality. Thousand Hills State Park covers 3215 acres. A thorough survey of
one representative acre yields a count of 83 shagbark hickory trees. About how many
shagbark hickories would you expect to find in the entire park?
2.
Ratios & fractions. A careless professor accidentally adds 500 mL of alcohol to a
fishbowl containing 4 L of water and one unlucky goldfish. What is the ratio of alcohol
to water in the fishbowl? What is the proportion of alcohol in the fishbowl?
3.
Using Excel, tabulate the data you obtained from your laboratory exercise on bacterial
populations. Calculate the percent resistance in each of the three populations.
4.
Why do you think the lab guide asks you to use only plates that have fewer than 300
colonies? Why do you think it asks you to use only plates that have more than 5
colonies?
5.
The dilution factor is the ratio between the final volume in a dilution step and the
volume added from the previous step in the dilution series. For example, this
experiment used a dilution factor of 100 (= 10 mL / 0.1 mL). With this dilution factor,
how many dilution steps did it take to reduce the concentration by a factor of 106? If
each tube had contained only 0.3 mL of distilled water instead of 9.9 mL, how many
dilution steps would it have taken to reduce the concentration by 106?
14
Appendix 1: Representative and Unrepresentative Samples
In 1936, the Literary Digest magazine held a straw poll for the upcoming
Presidential election. The magazine mailed out millions of mock ballots; of the
two million returned, Republican Alf Landon received about 60%. The
incumbent, Democrat Franklin D. Roosevelt, received only about 40%.
This discussion
is abridged from
the Fallacy Files
website (click
here for link).
Based on these results, the Digest concluded that Landon would win. However,
Roosevelt won the actual election with 61% of the popular vote (the 2nd-largest
margin in U.S. history). Why was the Digest’s prediction so far off?
The Digest had compiled their mailing list from
directories of car owners and phone subscribers as
well as their own readership. In 1936, the country
was in the midst of the Great Depression: many
Americans could not afford cars or magazine
subscriptions. The Digest’s poll was therefore
biased toward prosperous voters, who historically
were more likely to vote Republican.
That same year, another pollster named George
Gallup predicted a Roosevelt victory. Gallup’s
prediction was based on a much smaller sample
(only 50,000 people), but his sample was more
representative. His later reputation for accurate
polling was built on this correct prediction.
Imagine that you wanted to take a
political poll of students on your
campus. Could you simply poll
the students in your biology
class, or those on your dorm
floor?
Identify at least three specific
steps you could take to ensure
that your sample was as
representative as possible.
Back
15
Appendix 2: Logic Functions in Excel
Instead of setting up a separate column for plating from each tube, we
can use Excel’s built-in logic functions. This takes longer to set up
but is more intuitive and easier to interpret.
The first logic function we need is the IF
function. This function checks whether a
particular statement is true. It then gives
one output if the statement is true and a
different output if the statement is false.
For example, the formula
=IF($B$3>0, “Big”, “Small”)
yields “Big” if the value in Cell B3 is
positive and “Small” otherwise.
We will also need the OR function. This
function checks two or more statements to
see if they are true. It then gives the result
“TRUE” if any of the statements are true
and the result “FALSE” if all the statements
are false. For example, the formula
=OR(2+1=4, 3*7=21)
yields “TRUE”.
Logic functions (like many others) can be nested to build
more complex formulas. For example, the formula
=IF(OR(2+1=4, 3*7=21), “Good”, “Not so good”)
yields “Good”.
Make sure to match each “(“
with a “)”. Excel can detect
parenthesis errors but
doesn’t always fix the
problem correctly.
Write a formula that executes the following:
• If Cell $C$4 contains the text “Tube D”, then return the value in Cell D8.
• If Cell $C$4 contains the text “Tube C”, then return the value in Cell $C$5.
• If Cell $C$4 contains any other value, then return the text “n/a”.
16
Appendix 2: Logic Functions in Excel
2
3
4
5
6
B
C
Agar plate WITHOUT streptomycin
Plate
inoculated
from:
# colonies
on plate:
7
8 # Bacteria:
9 Volume (mL):
10
11
12 # Bacteria:
13 Volume (mL):
14
15
16 # Bacteria:
17 Volume (mL):
18
19
20 # Bacteria:
21 Volume (mL):
22
23
24
25
26
27
28
# Bacteria:
Volume (mL):
D
E
Plate
inoculated
from:
# colonies
on plate:
Tube C
35
Inoculum from
tube D
Tube D
n/a
n/a
0.1
10
Inoculum from
tube C
35
0.1
Tube C
3500
10
Inoculum from
tube B
3500
0.1
Tube B
350000
10
Inoculum from
tube A
Tube A
350000 35000000
0.1
10
Inoculum from
vial
35000000
0.1
F
G
Agar plate WITH streptomycin
Vial
72
# Bacteria:
Volume (mL):
Inoculum
from tube D
Tube D
n/a
n/a
0.1
10
# Bacteria:
Volume (mL):
Inoculum
from tube C
Tube C
n/a
n/a
0.1
10
# Bacteria:
Volume (mL):
Inoculum
from tube B
Tube B
n/a
n/a
0.1
10
# Bacteria:
Volume (mL):
Inoculum
from tube A
Tube A
n/a
n/a
0.1
10
Inoculum
from vial
Vial
3.5E+09
10
# Bacteria:
Volume (mL):
Proportion of resistant bacteria in original population:
Percentage of resistant bacteria in original population:
H
2.06E-06
0.000206%
72
0.1
Vial
7200
10
Now build the spreadsheet at
left. You will need to use IF
and OR statements to
calculate the number of
bacteria in each inoculum.
Think carefully about the
experimental design: each
inoculum will need a different
formula!
Express the frequency of
resistant bacteria in the original
population as both a proportion
and a percentage. This is
easiest to do by assigning these
cells different number formats
Continue with main presentation
(in the Format menu, choose Cells,
then choose the Numbers tab).
17
Pre-Test
This test is to determine whether the following spreadsheet module helps you understand the
following mathematical concepts. Please answer the following questions as best you can,
working with other members of your lab group. If you can’t answer, do not become stressed;
instead, write down specific questions you would like to ask to help you answer the prompt.
Your responses will help me assess whether the module contributes to your understanding.
1. One-third of the entering freshmen at East Dakota University are biology majors.
If the freshman class is representative of all 27,000 students on campus, how
many East Dakota students are majoring in biology?
2. Hereditary esophageal dysfunction (HED) is a rare condition seen in miniature
Schnauzers. A cross between two adults with HED yielded a 13:3 ratio of healthy
to affected pups. What fraction of the pups in this cross would you expect to be
affected?
3. A 0.1 mL sample from a vial containing S. aureus bacteria is added to a test tube
containing 9.9 mL of distilled water and mixed thoroughly. A 0.1 mL sample from
this tube is then added to another tube containing 9.9 mL of distilled water and
again mixed thoroughly. Finally, a 0.1 mL sample from this second tube is plated
onto an agar plate. Over the next two days, each bacterium on the plate then
grows into a distinct colony. If 162 such colonies form on the plate, how many
bacteria were present in each 1 mL of the original vial?
18