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.

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.

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.

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.

energy conservation in mousetrap cars

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.

cells are fun-sized

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).

shape and survival

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.

droplets for laser-driven fusion

11.09.09. Nuclear fusion is promising as a potential source of sustainable energy. When two deuterium nuclei (a deuterium nucleus consists of a proton bound to a neutron) fuse, for example, they can produce a Helium isotope (two protons bound to one neutron) and a bare high-energy neutron. The energy of these bare neutrons can be converted to consumable energy (for example by using the neutrons to heat water, produce steam, and drive turbines). Deuterium exists naturally, albeit rarely, in the form of deuterized water in the oceans, so the problem consists of extracting the deuterized water from the natural water and imbuing the deuterium nuclei with enough initial energy to fuse. A promising way of achieving this initial energy is by irradiating small droplets of the deuterized water with a powerful, fast-pulsed laser; in this case the diameter of droplet that maximizes fusion yield is in the 0.01 to 1 micron range. As an undergraduate I (Mr. Mugler) worked with Professor Tom Donnelly and other students at Harvey Mudd College to develop a controllable source of droplets of this size, an experience which featured prominently in my "path" (see previous post) from middle school to the present day. We used a piezoelectric oscillator to vibrate (at Megahertz frequencies) a column of fluid at its base, which produced from its surface an aerosol of micron-scale droplets whose diameter we characterized by measuring the scattering pattern of a laser shone through the aerosol (the diagram shown is from Donnelly et al, Phys Fluids, 2004). More recently, current Harvey Mudd undergraduates have brought the droplet source to the University of Texas at Austin, which houses one of the most powerful lasers in the world, and have successfully achieved laser-driven fusion with deuterized water droplets. For more information see a recent article in the Harvey Mudd College Bulletin.

path to college and beyond

11.09.09. Great scientific discoveries are often made rather incidentally by people blindly following their most passionate interests. The same can be true for great careers, scientific and otherwise. We took some time to encourage the students to think critically and specifically about their own interests, and how these interests might influence their choice of high school (many New York City public high schools are highly specialized and students must apply), college, and beyond. Students made flow charts showing their own projected paths through middle school, high school, college, and (potentially) grad school. Mr. Mugler and Mr. Seymour then shared their own charts detailing their windy paths. This activity generated many interesting questions from students, on topics ranging from green energy to laser-driven fusion to graduate study in dance.

laser tweezers

11.04.09. An optical trap (also called "laser tweezers") is an experimental setup in which the motion of a small (on the order of a micron in diameter) plastic bead is controlled by a laser beam. The refraction of light through the bead exerts an attractive force (on the order of several piconewtons) on the bead directed toward the center of the beam. Optical traps are useful for a variety of biophysical applications in which one wishes to move or manipulate a single molecule. For example, when the bead is affixed to the cargo end of a motor protein, the laser can be used to control the force against which the protein pulls, which can affect the protein's step frequency or step size (see previous posts). As a visual aid to the lessons below, students were presented with this video, taken by Columbia physics majors Dan Amrhein and Alex Kaz, which shows a fixed laser beam (lower-left corner, directed into the screen) alternately "trapping" and losing a 2-micron bead as the microscope stage is moved from side to side.

motor protein syncrony and LCM

11.04.09. The step length of a motor protein (see previous posts) is affected by external factors both deterministic, such as the force exerted by its cargo, and random, such as collisions with other particles in the cell (see, e.g., Clemen et al, Biophys J, 2005). We imagined two motor proteins with different step lengths starting at the same point on a filament, and we asked at what distance would they coincide once more? Viewing each protein as stepping along a number line, it is clear that the answer is the least common multiple (LCM) of their step sizes. Students practiced finding LCMs in the context of stepping proteins, and this led to a comparison of two very different techniques for finding the LCM: (1) listing out the multiples of each number explicitly and (2) finding the prime factorization of each number and multiplying the greatest common factor (GCF) by the unshared factors.

kinesin to and fro

10.19.09. With enough external force, kinesin (see previous post) can be forced to take backwards steps as well as forward steps. In fact, the fraction of forward to backwards steps can be measured as a function of the external force applied by the optical trap (the plot shown is from Shao et al, PNAS, 2006). Forward and backward motion fit nicely with students' understanding of positive and negative numbers, and we imagined kinesin stepping along a number line. Students represented individual kinesin paths both visually as trajectories along a number line and numerically with an equation showing addition of positive and negative integers. Comparing the absolute value of all forward steps to that of all backward steps provides a quick way to determine the sign of the kinesin's net displacement.

motor proteins

10.14.09. Kinesin is a well-studied example of a "motor protein," a protein that steps along microtubules to transport cargo from one side of a cell to another (see an animation of kinesin by the Vale lab at UCSF here). Recent experiments (the plot shown is from Visscher et al, Nature, 1999) have been able to track the movement of a single kinesin molecule by attaching its cargo end to a plastic bead whose displacement is controlled by a laser beam (a so-called "optical trap"). At low (or no) cargo load, the velocity of kinesin depends on two parameters: its step size and the diffusion constant. Students practiced turning statements into equations by translating sentences such as "kinesin’s velocity is the quotient of twice the diffusion constant and the step size." Students then calculated the velocity for the approximate values D = 600 nm^2/s for the diffusion constant and L = 8 nm for the step size and compared with the slope of the plot.

a new year

10.14.09. Mr. Seymour and Mr. Mugler are back for a new year with a whole new batch of students hungry to learn! This year we'll be combining the most successful lessons from last year with new lessons to create a curriculum that is more cohesive, exciting, and beneficial to the students' understanding of seventh grade math.

cells as function machines

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.

variation in a population

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.

optimized bacterial swimming

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.

squid nerve response

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.

run and tumble

03.20.09. Many bacteria forage for nutrients using a "run and tumble" strategy: they swim straight ("run") until they detect a decrease in nutrient concentration, then they spin around ("tumble") to face a new direction and try again. Students were shown a demo of a running-and-tumbling bacterium whose jagged path was labeled with length measurements, then asked, "How far would the bacterium have had to swim if it went in a straight line instead?" With a bit of guidance ("Do you see the triangles?"), students recognized this as an application of the Pythagorean theorem, and that the answer lay in summing several hypotenuse calculations.

networks and patterns

03.13.09. Cells perform their functions by using networks of interacting proteins. For example, the network shown here (from Alon et al, Nature, 1999) describes the protein interactions occuring inside a bacterium that turn the detection of a nutrient into the activation of the bacterium's flagellar motor (the "propeller" it uses to swim toward more nutrients). Mathematically, a network is just a set of dots ("nodes") connected by lines ("edges"). Students drew fully-connected networks with 3, 4, 5,... nodes and counted the number of possible edges. By noticing the pattern, students derived as a class an equation to compute the number of possible edges x from the number of nodes n: x = n(n-1)/2.

there are bugs in my dessert!

02.25.09. The Food and Drug Administration (FDA) is the governing body charged with, among other things, preventing contamination in the food that we eat. Many times, however, the amounts of contaminants that they allow in foods is surprisingly high (the often stomach-churning data are available freely on the FDA's website here). For example, the FDA allows 30 insect fragments per 100 grams of peanut butter and 75 insect fragments per 50 grams of cocoa powder before recalling these foods. Students used ratios and proportions to calculate from these data the allowed number of insect fragments in a Reese's peanut butter cup. Only after obtaining their result were they permitted to (not without trepidation) enjoy their treat. (For those interested, a peanut butter cup can contain roughly 30 insect fragments before the FDA deems it unfit to eat--not to mention a few rodent hairs...)

how do bacteria eat?

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?

false positives

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.

virus probability tree

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.

bird strikes at JFK

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!

ratios and estimation

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!

powers of ten

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.

bacteria competition

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.

swabbing for staph

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!

bacteria volume

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.

bacterial colonies

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.

bacterial growth

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.

launch

Welcome! Mr. Seymour and Mr. Mugler have created this site as a way to share our work and that of our students with other educators, administrators, students, and anyone else who's interested! We'll start by recapping with some posts on past lessons.