Thursday, March 25, 2010

bacterial fluorescence vs time

03.24.10. A major contributor to the explosion of quantitative data in microbiology in the past 10-20 years has been the advent of the fluorescent protein. Originally extracted from jellyfish, fluorescent proteins can be designed to be coupled to (either produced along with or literally bound to) other intracellular molecules of interest. Under a microscope the brightness then provides a proxy for molecular concentration. This fluorescent "tagging" procedure has elucidated scads of time-, space-, and concentration-dependent molecular processes inside individual living cells. As a class we observed two movies of bacterial cells whose fluorescence intensity changed as a function of time: (1) a movie taken by Mr. Mugler at Caltech's Bootcamp, in which bacteria were photobleached, causing the fluorescence to decrease exponentially with time, and (2) the movie shown here, from Stricker et al, Nature, 2008, in which bacteria were synthetically engineered to oscillate in fluorescence over time. Students were then presented with two sets of fluorescence vs time data and asked which data set corresponded to which movie, a question that they answered definitively by constructing a double-line graph. This exercise reinforced the preceding class discussion that line graphs are appropriate for visualizing and comparing changes over time.

Wednesday, February 24, 2010

exponential vs additive growth

02.24.10. Given ample nutrients, warmth, and space, bacteria divide regularly with division times on the order of an hour, and are thus a canonical model for exponential growth (see 10.17.08 post). The proliferation of viruses, on the other hand, is host-dependent, and therefore an additive model might be more appropriate. The rhinovirus, for example, a causative agent of the common cold, can be present in as many as 1 million copies per milliliter of nasal mucus (Web MD). Since we each produce about 1 liter of mucus per day (Dr. Oz), this amounts to approximately 1 billion viruses per cold victim. Assuming that an average of one person per day in a local community contracts a cold, the number of viruses grows additively: 1 billion, 2 billion, 3 billion, etc. Students were given this information and posed the question "Will there be more bacteria or viruses, and when?". Answering the question led students to recall (1) the conversion between hours and days, (2) the representation of repeated multiplication as exponentiation, and (3) the use of scientific notation and exponential notation, both on paper and in the calculator. As demonstrated by the plot above, exponential growth is much more rapid than linear growth, and under these models the bacterial population handily overtakes the viral population in about 30 hours.

Wednesday, February 10, 2010

outrunning diffusion

01.25.10. Bacteria swim toward higher nutrient concentrations using a "run and tumble" strategy (see 03.20.09 post and this animation from ClearScience, from which a screen shot is shown here). It is important to recognize, however, that regions of high nutrient concentration are constantly changing, since nutrient particles are diffusing throughout the medium (as Kool-Aid mix diffuses throughout a glass of water). A bacterium must reach the nutrients before they diffuse away. As a class, we realized that this criterion can be made quantitative using an inequality: the bacterium's swimming properties (which students quantified as its velocity v and its run length L) must be greater than the nutrient's diffusive properties (which students quantified as the recently discussed diffusion constant D). Examining the units (or 'dimensions') of the three quantities as a class, we realized multiplication of v and L yields the only dimensionally consistent result, giving vL > D. Students isolated L, then substituted D = 1,000 um^2/s and v = 30 um/s (bacteria can swim about 30 times their body length in 1 second!) to compute the minimum run length necessary to "outrun" diffusion: L > D/v ~ 33 um (micrometers), a biologically realistic value.

Thursday, January 21, 2010

a limit on the size of a bacterium

01.21.10. The way a bacterium breathes limits how large it can grow. Unlike a human, who actively takes in oxygen and expels carbon dioxide, a bacterium passively takes in oxygen and other nutrients, absorbing through its membrane only those molecules that strike its surface while diffusing throughout the surrounding medium. The maximum metabolic intake M (molecules per time per mass) of such a passively "breathing" organism is readily calculated (see, e.g., Philip Nelson's Biological Physics, Sec. 4.6.2, p. 138) as M = 3Dc/(nR^2), where c and D are the concentration and diffusion constant, respectively, of the external molecule (e.g. oxygen), and n and R are the density and radius, respectively, of the bacterium. Solving this equation for R yields an upper bound for the size of the bacterium, i.e. R < sqrt[3Dc/(nM)]. Students, armed with values of D, c, n, and M, solved for and graphed the result R < 5 microns (which is actually a quite reasonable limit for bacterial size). As a "challenge" question, students were also asked to figure out how algebraically to get from the M equation to the R equation by understanding the steps in an analogous problem.