Wednesday, February 11, 2009
02.11.09. Bacteria take in nutrient molecules through chemoreceptor patches on their cell walls (image from Berg and Purcell, Biophys J, 1977). The efficiency of this process depends in part on the fraction of the bacterium's surface that is covered by receptors. Using toy models of bacteria with receptors represented by different shapes, students used their own measurements and area calculations to answer the question: if a nutrient molecule hits the surface of the bacterium, what is the probability that it gets absorbed?
02.04.09. If a disease is rare, even a very accurate test will produce results in which a significant fraction of the positives are false positives--cases where the test reports that a patient has the disease when he or she actually does not. For example, if 1% of a population has a disease, half of the positive results from a test that is 99% accurate will be false positives! We simulated this phenomenon with a game in which students had a 1/10 chance of having a disease (one of ten tiles was drawn from a bag), they were subjected to a test that was accurate 5/6 of the time (a die was rolled), and we computed the fraction of the positive test results that was due to false positives. We then went through a calculation of the theoretical fraction using the formula p = x(1-d)/[x(1-d)+d(1-x)], where p is the false positive fraction, d is the disease prevalence (1/10 in the game), and x is the test error rate (1/6 in the game). This led to an interesting discussion of the difference between experimental and theoretical probabilities.
01.28.09. When lambda phage (a virus) infects a host cell, it either undergoes lysis, in which it exploits cell resources to make copies of itself and kills the cell, or lysogenesis, in which it incorporates its DNA into the cell's and awaits replication (see, for example, this animation from McGraw-Hill). This decision is a probabilistic one, like flipping a weighted coin. Students made probability trees detailing the possible outcomes and their probabilities after several such decisions for a virus and a host cell.
12.17.08. JFK airport in Queens is located right next to Jamaica Bay Wildlife Refuge, and "bird strikes"--when birds collide with airplanes--are a common problem. In the early 1990s attempts were made to reduce the number of bird strikes by shooting birds; then in 1996 falcons were introduced in an attempt to scare birds away (see, for example, a 1997 article in the New York Times). Both tactics resulted in a noticeable decrease in the number of bird strikes, as seen in data on the number of bird strikes per year at JFK (the graph shown here is from Garber, SD. “Effectiveness of falconry in reducing risk of bird strikes under study at JFK International.” International Civil Aviation Organization Journal. 51(7):5-7, 1996). Students from Ms. Utton's Independent Projects Week (IPW) team and from Mr. Seymour's math classes were given the data and asked to make a plot like the one shown here. In addition to providing practice with making graphs, the plot helped illustrate the story behind the data and helped connect the data with events at JFK. Incidently, shortly after this lesson was taught a plane from Laguardia was forced to make an emergency landing in the Hudson River due to birds getting sucked into the engine. Bird strikes happen!
12.10.08. The ability to make quick estimates of quantities is a powerful skill to have, in science and in everyday life. Students used their knowledge of ratios to perform long unit conversions, obtaining estimates of quantities such as the total number of students at MS 88, and the population of Brooklyn. Starting from considerations like "there are about 4 people per household," "about 30 households per building," "about 10 buildings per block," etc., students estimated the population of Brooklyn to roughly within a factor of 2!
11.26.08. Sometimes the best way to get a sense of the size of things in this universe is to take it one order of magnitude at a time. Students watched FSU's "Secret Worlds: The Universe Within" java tutorial to get a sense of what objects exist at what lengthscale (see also the American Museum of Natural History's "Scales of the Universe" exhibit). Students then got practice in converting cumbersome decimals to scientific notation, and writing tiny or huge length measurements in more appropriate units such as nanometers or lightyears.
11.19.08. Competition between different strains of bacteria can lead to spacial segregation at the population level (see, for example, Hallatschek et al, PNAS, 2007). Students simulated the competition between a "good" and "bad" strain of bacterium using a number game. Each student started with a certain number of good and bad bacteria; good bacteria were positive and bad bacteria were negative, yielding a total "colony score" that indicated the harmfulness of the population. Colony scores changed by the addition of nutrients (scores doubled) or antibiotics (scores were halved if negative) and by "competition" rounds, in which students combined their scores with neighors' scores. After several rounds, groups of adjacent students tended to converge upon the same score, with some groups finding positive scores and some finding negative scores, illustrating the segration effect.
11.12.08. Although often in the news in the context of infection, the staphylococcus bacterium lives in roughly 20 percent of human beings in a harmless form, most prevalently in the mucus. Ten students in each class swabbed the insides of their noses to see if they were part of the lucky 20 percent. Cassie Fairchild at the Columbia Medical Center analyzed the samples by adding them to a particular salt that, when fermented by staph, turns from red to yellow. Results were delivered to students in the form of a sum of positive and negtive integers--if the sum was negative, the student was negative for staph, and vice versa!
11.05.08. E. coli are rod-shaped bacteria. Using a scale model, students calculated the volume, in cubic micrometers, occupied by a single E. coli bacterium. The less-than-seamless construction of the model gave a hint that the volume is that of a cylinder plus two halves of a sphere.
10.21.08. It's always good to measure something in more than one way. A second way to estimate the number of bacteria in a colony is to use areas. We brought in petri dishes on which bacteria had grown for 8 hours (courtesy of Laura Wingler at Columbia), forming circular colonies. By measuring the area of a bacterial colony, and dividing by the area of a single bacterium, students obtained a second estimate of the population size of an 8-hour-old colony of bacteria.
10.17.08. The focus of this collaboration is to use math to explore microbiology, and bacteria have proven a rich model organism. We started by showing students a video of E. coli bacteria undergoing exponential growth (the video was taken by Mr. Mugler at Caltech's Bootcamp). Students used exponents figure out, if one bacterium divides every 20 minutes, how many bacteria there will be after 8 hours.