2Overview of ModuleSerial 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 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 show how to calculate the number of resistant bacteria and the percent resistance in a population.Slide 14 contains your homework assignment.
3Staphylococcus and methicillin resistance BackgroundStaphylococcus and methicillin resistanceStaphylococcus 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:From
4ProblemMethicillin-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.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.Source:BathroomEmergency roomMaternity ward# colonies of S. aureusLawnHow 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?
5Thinking 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.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?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:Obtaining a representative sample is one of the thorniest problems in designing an experiment.Click here to explore this issue in more detail.# male students in sample# students in sample# male students on campus# students on campus=
6Note the difference between a ratio and a fraction in this context! Thinking About the Problem: Ratios, Fractions, and PercentagesA 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.Note the difference between a ratio and a fraction in this context!Ratio = # girls : # boysFraction =# girls# girls + # boysA 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).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%?A percentage is the number of parts per It can be calculated from a fraction by dividing the numerator by the denominator, then multiplying by For example, 1/3 = = 33.33%.Hint: the answer is NOT 100 mL!
7Setting 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?If you want to review scientific notation, click here and then scroll to the top of the webpage that comes up.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.= cell with a number in it= cell with a formula in itIf 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.Theoretically, we could avoid getting a lawn by using a smaller sample volume (e.g., 1 mL = mL). But in practice it’s very hard to measure such small volumes accurately. What other approaches could we try?
8Why does Tube A have the same number of bacteria as the pipettor? Setting up a Spreadsheet: Inoculating a New TubeTake 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).Inoculum# bacteria: 5.6×109Volume: mLOriginal population# bacteria: 5.6×1011Volume: 30 mLTube A# bacteria: 5.6×109Volume: 10 mLWhy 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?
9Setting 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?
10Reversing the ProblemSo 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”).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).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.
11Reversing 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.
12Doubling 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 methicillinAgar plate WITH methicillinIn 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
13Doubling the Problem: Inferring Percent Resistance Columns B-P are part of the underlying Excel sheet; however, PowerPoint refuses to display columns Q-T of that sheet. An image of those columns has been pasted onto this slide to show the sheet’s true appearance.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?
14End of Module Assignments 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?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?Using Excel, tabulate the data you obtained from your laboratory exercise on bacterial populations. Calculate the percent resistance in each of the three populations.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?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?
15Appendix 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?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.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
16Write a formula that executes the following: Appendix 2: Logic Functions in ExcelInstead 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”.Make sure to match each “(“ with a “)”. Excel can detect parenthesis errors but doesn’t always fix the problem correctly.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”.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”.
17Appendix 2: Logic Functions in Excel 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).Continue with main presentation
18Pre-TestThis 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?