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.