1 Serial Dilution Tracking Bacterial Population Size and Antibiotic Resistance How can we determine...

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Serial Dilution

Tracking Bacterial Population Size and

Antibiotic Resistance

How can we determine the number of bacteria in a sample? What percent

of those are antibiotic-resistant?

Prepared for SSAC byAnton E. Weisstein, Truman State University, Kirksville, MO 63501

© The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. 2006

Core Quantitative concept and skillNumber sense: Ratios, fractions, percentages

From http://textbookofbacteriology.net/staph.html

Supporting Quantitative concepts and skillsNumber sense: Scientific notationModeling: Forward modeling; inverse problemSampling: Representative sampleLogic functions (optional)

SSAC2006.QR67.AEW1.1

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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.

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.

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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.htmlhttp://www.medscape.com/viewarticle/484351

Background

From http://images.stltoday.com/stltoday/multimedia/staphb1031.gif

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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.

Problem

• 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?

Source: BathroomEmergency

roomMaternity

ward

# colonies of S. aureus

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.

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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.

Thinking About the Problem: Proportionality

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

# students in sample

# male students on campus

# 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.

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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%.

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!

Note the difference between a ratio and a fraction in this

context!

Ratio = # girls : # boys

Fraction = # girls

# girls + # boys

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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?

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 L = 0.001 mL). But in practice it’s very hard to measure such small volumes accurately. What other approaches could we try?

= cell with a number in it

= cell with a formula in it

If you want to review scientific notation, click here and then

scroll to the top of the webpage that comes up.

If you were to plate this sample on a Petri dish, each bacterium in that

sample would reproduce to form a single colony. Because there are so many colonies, they would quickly

start to overlap and form a continuous lawn. So we need to pipette out a

smaller number of bacteria.

B C D2 Sample 1: Bathroom3

4Original

populationSample for

serial dilution5 # Bacteria: 5.60E+11 1.87E+096 Volume (mL): 30 0.17

8 Tube ASample for


12 Tube BSample for


16 Tube CSample for


20 Tube DInoculum for agar plate

21 # Bacteria: 1.87E+03 1.87E+0122 Volume (mL): 10 0.1

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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).

Original population# bacteria: 5.6×1011

Volume: 30 mL

Inoculum# bacteria: 5.6×109

Volume: 0.1 mL

Tube A# bacteria: 5.6×109

Volume: 10 mL

Why does the pipettor have the

same concentration of bacteria as the

original population?

Why does Tube A have the same

number of bacteria as the pipettor?


4Original



8 Tube ASample for


12 Tube BSample for


16 Tube CSample for




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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.

Enter the remaining rows into your spreadsheet. Based on the initial population of 5.6×1011 bacteria, how many colonies would you expect on the agar plate?


4Original



8 Tube ASample for


12 Tube BSample for


16 Tube CSample for




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Re-create this worksheet.

Re-create this worksheet.

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”).

Note that we are solving

three slightly different

versions of the same problem.

Use this version if you plated from Tube D (four dilution steps).

Use this version if you plated from Tube C

(three dilution steps).

Use this version if you plated from Tube B (two dilution steps).

Click here to learn a

cleaner way to set up this

sheet.

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Reversing the Problem: Calculating Population Size

Using what you learned working through Slides 8-10,

enter formulas into the orange cells that will calculate the number of bacteria in each inoculum and each tube.

Now use this spreadsheet to calculate the original population size if you had 164 colonies on a plate inoculated from Tube C.

B C D E F G H I J K L2 Agar plate WITHOUT streptomycin3

4Plate inoculated from: Tube D

Plate inoculated from: Tube C

Plate inoculated from: Tube B

5# colonieson plate: 35

# colonieson plate: 35


6

7

Inoculum from tube D onto

plate Tube D8 # Bacteria: 35 35009 Volume (mL): 0.1 10

10

11

Inoculum from tube C into

tube D Tube C

Inoculum from tube C onto plate Tube C

12 # Bacteria: 3500 350000 # Bacteria: 35 350013 Volume (mL): 0.1 10 Volume (mL): 0.1 1014

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Inoculum from tube B into

tube C Tube B

Inoculum from tube B into tube C Tube B

Inoculum from tube B onto plate Tube B

16 # Bacteria: 350000 35000000 # Bacteria: 3500 350000 # Bacteria: 35 350017 Volume (mL): 0.1 10 Volume (mL): 0.1 10 Volume (mL): 0.1 1018

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Inoculum from tube A into

tube B Tube A

Inoculum from tube A into tube B Tube A

Inoculum from tube A into tube B Tube A

20 # Bacteria: 35000000 3500000000 # Bacteria: 350000 35000000 # Bacteria: 3500 35000021 Volume (mL): 0.1 10 Volume (mL): 0.1 10 Volume (mL): 0.1 1022

23Inoculum from vial into tube A Vial

Inoculum from vial into

tube A Vial

Inoculum from vial into

tube A Vial24 # Bacteria: 3500000000 3.5E+11 # Bacteria: 35000000 3500000000 # Bacteria: 350000 3500000025 Volume (mL): 0.1 10 Volume (mL): 0.1 10 Volume (mL): 0.1 10

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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.

Tube containing mixture of

susceptible & resistant bacteria

Agar plate WITHOUT methicillin

Agar plate WITH 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.

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Doubling the Problem: Inferring Percent Resistance

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.

Why might you want to use fewer dilution steps in plating onto agar with methicillin than you used in plating onto agar

without methicillin?

B C D E F G H I J K L M N O P2 Agar plate WITH streptomycin3

4Plate inoculated from: Tube D



Plate inoculated from: Tube A





6

7Inoculum from

tube D onto plate Tube D8 # Bacteria: 35 35009 Volume (mL): 0.1 10

10

11Inoculum from

tube C into tube D Tube CInoculum from

tube C onto plate Tube C12 # Bacteria: 3500 350000 # Bacteria: 35 350013 Volume (mL): 0.1 10 Volume (mL): 0.1 1014

15Inoculum from

tube B into tube C Tube B

Inoculum from tube B into tube

C Tube BInoculum from

tube B onto plate Tube B16 # Bacteria: 350000 35000000 # Bacteria: 3500 350000 # Bacteria: 35 350017 Volume (mL): 0.1 10 Volume (mL): 0.1 10 Volume (mL): 0.1 1018

19Inoculum from

tube A into tube B Tube A

Inoculum from tube A into tube

B Tube AInoculum from

tube A into tube B Tube AInoculum from

tube A onto plate Tube A20 # Bacteria: 35000000 3.5E+09 # Bacteria: 350000 35000000 # Bacteria: 3500 350000 # Bacteria: 35 350021 Volume (mL): 0.1 10 Volume (mL): 0.1 10 Volume (mL): 0.1 10 Volume (mL): 0.1 1022

23Inoculum from vial into tube A Vial

Inoculum from vial into tube A Vial



24 # Bacteria: 3500000000 3.5E+11 # Bacteria: 35000000 3.5E+09 # Bacteria: 350000 35000000 # Bacteria: 3500 35000025 Volume (mL): 0.1 10 Volume (mL): 0.1 10 Volume (mL): 0.1 10 Volume (mL): 0.1 10

Q R S T U V W X Y Z [ \ ] ^ _ `2 Agar plate WITH streptomycin3

Plate inoculated from: Original vial 4

Plate inoculated from: Tube D



# colonieson plate: 35 5




6

7Inoculum from

tube D onto plateTube D8 # Bacteria: 35 35009 Volume (mL): 0.1 1010

11Inoculum from

tube C into tube DTube CInoculum from

tube C onto plateTube C12 # Bacteria: 3500 350000 # Bacteria: 35 350013 Volume (mL): 0.1 10 Volume (mL): 0.1 1014

15Inoculum from

tube B into tube CTube B

Inoculum from tube B into tube

C Tube BInoculum from

tube B onto plateTube B16 # Bacteria: 350000 35000000 # Bacteria: 3500 350000 # Bacteria: 35 350017 Volume (mL): 0.1 10 Volume (mL): 0.1 10 Volume (mL): 0.1 1018

19Inoculum from

tube A into tube BTube A

Inoculum from tube A into tube

B Tube AInoculum from

tube A into tube BTube A20 # Bacteria: 35000000 3.5E+09 # Bacteria: 350000 35000000 # Bacteria: 3500 35000021 Volume (mL): 0.1 10 Volume (mL): 0.1 10 Volume (mL): 0.1 1022

Inoculum from vial onto plate Vial 23




# Bacteria: 35 3500 24 # Bacteria: 3500000000 3.5E+11 # Bacteria: 35000000 3.5E+09 # Bacteria: 35000035000000Volume (mL): 0.1 10 25 Volume (mL): 0.1 10 Volume (mL): 0.1 10 Volume (mL): 0.1 10

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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?

End of Module Assignments

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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%.

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?

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.

This discussion is abridged from the Fallacy Files

website (click here for link).

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.Back

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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”.

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”.

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.

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Appendix 2: Logic Functions in Excel

Continue with main presentation

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 (in the Format menu, choose Cells,

then choose the Numbers tab).

B C D E F G H2 Agar plate WITHOUT streptomycin Agar plate WITH streptomycin3

4


Plate inoculated from: Vial



6

7Inoculum from

tube D Tube DInoculum

from tube D Tube D8 # Bacteria: n/a n/a # Bacteria: n/a n/a9 Volume (mL): 0.1 10 Volume (mL): 0.1 10

10

11Inoculum from

tube C Tube CInoculum

from tube C Tube C12 # Bacteria: 35 3500 # Bacteria: n/a n/a13 Volume (mL): 0.1 10 Volume (mL): 0.1 1014

15Inoculum from

tube B Tube BInoculum

from tube B Tube B16 # Bacteria: 3500 350000 # Bacteria: n/a n/a17 Volume (mL): 0.1 10 Volume (mL): 0.1 1018

19Inoculum from

tube A Tube AInoculum

from tube A Tube A20 # Bacteria: 350000 35000000 # Bacteria: n/a n/a21 Volume (mL): 0.1 10 Volume (mL): 0.1 1022

23Inoculum from

vial VialInoculum from vial Vial

24 # Bacteria: 35000000 3.5E+09 # Bacteria: 72 720025 Volume (mL): 0.1 10 Volume (mL): 0.1 102627 Proportion of resistant bacteria in original population: 2.06E-0628 Percentage of resistant bacteria in original population: 0.000206%

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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?

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.

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