Back-of-the-Envelope-Calculations (Should be on the Front of Our Minds)

Science is an incredibly precise profession.  Adding two, ten, or twenty times the amount of a reagent could be plenty to ruin an experiment.  Yet despite the importance of details, calculating everything to the third decimal place can cause us to lose focus of the bigger picture – and losing perspective in science can be a very dangerous proposition.

The term “back-of-the-envelope calculation” still causes me to break out in cold sweats, a Pavlovian response leftover from studying for my qualifying exam in grad school.  It refers to a quick approximation of a more complex calculation, with the goal of determining the correct order of magnitude rather than the exact number.  Beyond science, it turns out we use back-of-the-envelope calculations all of the time in everyday life.  If a friend said, “I just bought a new Mini Cooper and I can park it anywhere – it’s only seven meters long!”  We’d do a quick back-of-the-envelope calculation and realize that they slipped up – it’s likely our friend meant to say “seven feet” – not “extended Hummer limousine.”

At the bench, building in this instinctual internal verification system is critically important.  Developing a “gut sense” can help generate reasonable hypotheses and set off alarm bells when the wheels have started to come off (“My 25kD protein is active as a tetramer, which means it is the same size as the nucleus…”).

In fact, we probably perform more back-of-the-envelope calculations in lab than we realize.  If we accidentally set the pipetman to “10” instead of “100” microliters, we’d probably notice the error when we looked at how much liquid was pulled up into the tip.  Something just wouldn’t seem right.  And that little internal warning would save us from potentially destroying an experiment.

It’s true that between Google and Wikipedia, it feels like memorizing numbers is a thing of the past.  However, there are still a few numbers that are worth committing to memory, that will make us better scientists.  They’re guidelines and approximations to help give our scientific perspective reasonable boundaries, just like the rest of our lives.  Here are a few examples to get started:

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Length of a Carbon-Carbon:

single bond: 1.54 Angstroms (Å)

double bond: 1.33 Å

triple bond: 1.21 Å

Diameter of :

mammalian cell: ~10 microns (micrometers)

bacterial cell: ~1 micron

Vaccinia/Smallpox virus: ~300 nanometers

Rhinovirus capsid: ~30 nanometers

Dissociation/Rate constants:

Very strong K_d (Biotin-Avidin) = 10^-15 M

Weak K_d = 10^-6 M

Diffusion-controlled reaction rate limit: ~10⁹ M^-1s^-1

Measurements:

1 nanometer = 10 Angstroms

For H₂O, 1 gram = 1 cm³ = 1 mL

RPM = 300 * sqr(RCF/R)    [sqr=square root]

RCF = 11 * radius(cm) * (thousands of RPM)^2

Light-related:

UV Light: <400nm

Visible Light: 400 – 700nm

Infrared light: >700nm

Speed of light: ~1 foot/nanosecond

Protein/DNA-related:

Half-life of a peptide bond in water: 7 years

Number of amino acids per turn of an α-helix: 3.3

Avg. weight of an amino acid in a protein (for quick mass calculations): ~110 daltons

Avg. size of a protein: 300 amino acids

Avg. size of ssDNA base: 330 grams/mol

Avg. size of dsDNA base pair: 660 grams/mol

Mammalian Cell-related:

Visual guide for mammalian cell concentrations when doing cell culture:

100,000 cells/ml = pretty much clear

1 million cells/ml = faintly cloudy solution

10 million cells/ml = nearly opaque

Avg. volume: 0.5 picoliters (assuming 5 micron radius and 1.0 g/mL density of cytoplasm)

Other:

pH blood plasma: 7.4

Avg. doubling time of E. coli: 20-30 minutes

Thermal voltage: ~25 millivolts @ room temperature

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Have any other numbers you find helpful?  Please add them to the comments section below and we’ll add them to the post!

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