Thursday, June 25, 2009
06.12.09. Cells are often thought of as machines, in the sense that they take an input and produce an output in order to perform a biological function. Upon measuring the input and output experimentally, one might hope to describe the biological function of the cell quantitatively with a mathematical function. Mathematical functions, and the notion that an equation can take an input and produce an output, had recently been introduced to the students. In this exercise, students were presented with several types of "virtual cells" on the computer, each of which took in numbers of the students' choosing and spit out corresponding numbers according to a (hidden) mathematical expression (source code in MATLAB is available by request to ajm2121(at)columbia.edu). As a class, students tried many inputs until they noticed a pattern, plotted their input/output pairs as a graph, and figured out each hidden algebraic function.
05.05.09. When only tens or hundreds of copies of a particular protein are produced in a cell, there can be large variation in the exact number per cell in a population of cells. Such variability can cause phenotypic differences among cells, even when they are genetically identical (see, e.g., Elowitz et al, Science, 2002). A histogram provides a nice way to visualize variability, as well as such staples of seventh grade statistics as mean, median, mode, and range. Students learned about histograms in the context of cell populations, then produced their own histograms from height measurements of all students in the class.
04.22.09. Bacteria swim to find nutrients and escape other organisms. Evolutionary pressures have optimized bacteria's size and structure for, among other survival requirements, the ability to swim fast. An undersized bacterium may lack flagellar strength, while an oversized bacterium may incur too much frictional drag. We quantified this intuition by devising a mathematical model from the following (somewhat contrived) considerations: (1) a bacterium's flagella are 4 micrometers longer than its radius, (2) drag is proportional to cross-sectional area, and (3) flagellar length is 6 times more important for survival than overcoming drag. With guidance, students turned these criteria into an equation for a bacterium's velocity v as a function of its radius r, v = 6*(r+4) - 3*r^2, and simplified the equation to a standard form. Students were asked "What value of r gives the largest v?", which led naturally into generating test data and plotting the equation. As shown by the plot, swimming velocity is maximal with a radius of 1 micrometer under this model, which is a biologically realistic value.
Wednesday, June 24, 2009
04.03.09. Nerves were first studied in squids, because squids have very large nerve cells (a squid's giant axon--a single nerve cell--can be millimeters thick and centimeters long). Students watched this excerpt from the film "The Squid and its Giant Nerve Fiber" (also posted here as part of a neurophysiology course at Smith College) showing the removal and activation of a squid's giant axon. As seen at minute 2:24, the axon experiences a sharp rise in voltage, then a fall, and finally a return to the original level. This voltage pattern is transmitted down the length of the nerve each time it "fires." Students were presented with a candidate mathematical model for voltage V (in millivolts) as a function of time t (in seconds), which, upon combining like terms, they simplified to V = t^3 - 15*t^2 + 50*t. Students then checked the legitimacy of the model by generating their own data points from the equation and plotting the curve. Upon comparing with the experiment in the video, it is evident that the model breaks down at large negative and positive values of t.