Wednesday, December 16, 2009
12.07.09. A variety of systems can be understood fruitfully and quantitatively through the language of energy conservation, including a living cell, a falling book, or, as with Mr. Seymour and Mr. Wasylyk's Integrated Project Design (IPD) this week, a model car. In teams students constructed cars out of paint stirrers (chassis), CDs (wheels), and mousetraps (the power source), focusing on design concerns such as weight minimization, the tradeoff between distance and speed, and the dual role of friction as both helpful (wheel grip) and harmful (axle rubbing). Energy conservation proved a useful lens as we discovered with students how the initial potential energy stored in the mousetrap spring is converted into the translational kinetic energy of the car, the rotational kinetic energy of the wheels, and the eventually dominant heat energy lost due to axle friction. A visual demonstration of this energy transfer was presented and is included here as a video.
11.24.09. The development of a living embryo, i.e. the positioning and differentiation of thousands of cells, relies on information encoded in the numbers of proteins within these cells. These numbers are small--typically tens or hundreds--and since the proteins themselves are products of probabilistic reactions, these numbers fluctuate. Fluctuations have the most severe effect when the averages they fluctuate around are small, a point well illustrated by bags of M&Ms. On average, one sixth of the M&Ms in a any bag are red, meaning a large bag containing, say, 600 M&Ms, has roughly 100 red ones. Even 10% fluctuations in this number are hardly noticeable (a pile of 110 red M&Ms is not easily distinguished from a pile of 100). A fun-sized bag, on the other hand, contains only 15 or so M&Ms, meaning that 2.5 of them (on average) are red. Fluctuations of only a few M&Ms can mean the difference between five and none, a dramatically noticeable effect, especially if reds are one's favorite (read: are favored by environmental pressures). Cells are fun-sized: their proteins are few in number, and fluctuations in these numbers can produce markedly different phenotypes (the green spots in the picture here, from Golding et al, Cell, 2005, are individual mRNAs--molecules that produce proteins--present in only several copies per bacterial cell). These fluctuations place a physical limit on, e.g., the precision with which a collection of embryonic stem cells can differentiate into specific cell types (see, e.g., Tkacik et al, PNAS, 2008). Using fun-sized M&M packages as models for cells, students explored the concepts of (1) variation, measuring and plotting as histograms the numbers across packages of total and specifically colored M&Ms, and (2) number sense, interpreting data on the numbers of M&Ms per package (nonnegative integers or "whole numbers"), package weights (real numbers), and M&M circumferences (irrational numbers if using pi).
11.16.09. Microorganisms come in many shapes, from spherical and rod-shaped bacteria to egg-shaped budding yeast to mutably-shaped amoeba. Shape can be an important factor in an organism's survival; for example, it is often advantageous (on an evolutionary scale) for a microorganism to maximize its surface area (in order to increase the amount of nutrients taken in) and minimize its volume (in order to reduce the amount of nutrients needed). We studied the analog of this problem in two dimensions, measuring for various 2-D cutouts the ratio of perimeter to length (length, a proxy for area or volume, was defined as the longest straight line spanning the shape). Ratios for amoebic cross-sections were by far the largest and most variable across the population, while ratios for circles (representing cross-sections of, e.g., spherical Staphylococcus aureus bacteria) all neared the famous, irrational pi.