Fun With Shapes and Numbers

In the diagram below, in between two circles, each with a radius of 1 meter, three smaller circles are squeezed in, centred at C1 , C2 and C3. If you imagine it was possible to continue drawing and squeezing in more circles until you had a billion of them, how much space would be left between the top of the smallest circle and point A? And what ubiquitous thing’s diameter is very close to that distance?

What could fit into the tiny space between point A and the smallest of a billion circles squeezed into the space between the two large circles, one of which is centered at B? Diagram by the author.

This is a variation of an old problem, and although the situation is often used as an example of a non-geometric series in calculus books, it could also be solved with some basic algebra and induction. And for that reason, versions of the puzzle have appeared in math competitions for bright kids who have yet to venture into calculus.

Let’s begin by solving for the radius of the first circle (the red one centered at C1). If we come with expressions or numbers for all three sides,  we could apply the Pythagorean theorem to triangle ABC1.

Notice that AC1 ‘s length can simply be obtained by subtracting the radius of the red circle from the 1-meter length of the square’s side, which we obtain from the radius of the large, identical circles.

AC1 = 1 – r1.

Since AB² + AC1 ² = BC1 ²,  where the hypotenuse, BC1 , is simply the sum of the radii, we obtain:

1² + (1 – r1) ² = (1 + r1) ²

Solving for r1 yields  r1 = 1/4 so the red circle’s diameter = d1  = 2 * 1/4 =  1/2 .

Not to bore you, we will walk through the steps only one more time to obtain the diameter, d2, of the blue circle centered at C2.

We add the red circle’s obtained diameter of  1/2 to the radius of  the blue circle, r2 , and then subtract the sum from 1 meter to obtain AC2.

AC2 = 1 – (1/2 + r2 )=   1/2 – r2

Then we apply the Pythagorean theorem to the second triangle, ABC2,

AB² + AC2 ² = BC2 ²,  or:

1² + ( 1/2 – r2) ² = (1 + r2) ²

1 + 1/4 – r2 + r2 ² = 1 + 2r2 + r2 ²

1/4 = 3r2

r2 = 1/12

so the blue circle’s diameter = 2 * 1/12 = d2 = 1/6.

As promised, without going through the similar details, d3 = 1/12 and for a fourth circle that we squeeze in, d4 will be 1/20.

Now a pattern becomes apparent. For the diameter of the nth circle, dn,  dn = 1/n(n+1). But if you did not realize that, you could still solve the problem. After all, we’re interested in the leftover distance between the top of the billionth circle and point A.

After we squeezed in one circle , we had a diameter of  1/2 . After squeezing in a pair, the sum of the two circles’ diameters 1/21/6 = 2/3. Squeeze in three circles and the sum is  1/21/6 + 1/12 = 3/4 .  A fourth insertion yields 1/21/6 + 1/12 + 1/20 = 48/60 = 4/5.

So for squeezing in an n-number of  circles, the sum of the diameters is n/n+1 and since the side of the square is 1 meter in length, the remaining distance will be 1 – n/n+1. Using a common denominator, that expression is equal to n+1 -n/n+1 = 1/n+1.  For a billion circles then, the leftover distance would be

1/1 000 000 000 +1 = 1/1 000 000 001 meters,

which is extremely close to being to a nanometer.

α-D-Glucopyranose, the hexagonal form of glucose. Image from Wikimedia commons.

1/1 000 000 000 meters or 1 nanometer, is a little more than three times the size of a single water molecule, and just slightly wider than the ring of one glucose molecule, the fuel for all brains, including those that devised and solved this puzzle. And use up just an extra iota of  glucose to realize that the leftover space would be long enough but not wide enough to accommodate the entire molecule. Finally, imagine a circle for every person alive on Earth in 2050, about 9.4 billion, and the distance remaining between the smallest circle and point A will only as big as the diameter of the smallest and most common atom in the universe.

Perfectly Natural: Gardeners’ Love of Chemistry and Chemists’ Love of Plants

When citrus trees are grown from seed, they revert to a wild state and they can take decades to flower. About 15 years after germination I’m still waiting to see grapefruit blossoms in our kitchen, and of course the short days of long winters at high latitudes don’t help the matter. But a few years ago I was lucky enough to visit an organic citrus farm in the Orlando, Florida region, which turned out to be a far more magical treat than any ride at nearby DisneyWorld.

On a single citrus tree we not only see fruits at various stages of ripening but also new flowers. Picture by the author.

Unlike most plants, those of the genus Citrus can simultaneously bloom while fruit is ripening elsewhere on the same tree. And the blossoms include some of the most delightful fragrances one will ever experience. And understanding how some of the compounds are synthesised by tangerines, lemons, grapefruits and plants in general makes one appreciate them even more.

At the level of citrus tissues, we see among different species variations upon a theme, the theme being what all plants of the genus have in common: bee-pollinated, bisexual flowers with usually 5 petals and 5 sepals and superior ovaries which become the fruit’s rind and flesh. Their leaves are evergreen and being part of subtropical trees, even the hardiest among them, the Meyer’s lemon, cannot survive temperatures below -5 °C.  At the biochemical level, in order to produce their special fragrance, there are variations upon themes as well.  Like many other plants they use a common precursor, the  molecule isopentenyl pyrophosphate, and with subtle changes they produce their beautiful characteristic bouquets which diffuse into our nasal receptors.

Using a process relying on thiamine(vitamin B1), among others, plants make isopentenyl pyrophosphate from acetyl coenzyme A, a metabolic byproduct of glucose known as pyruvate.   Isopentenyl pyrophosphate is a useful and ubiquitous molecule containing a 5-carbon atom-building block known as an isoprene unit. Some isopentenyl pyrophosphate molecules isomerize, meaning they rearrange with the same atoms, essentially shifting the double bond from the tail end over to the next pair of carbon atoms. The stable pyrophosphate group then leaves the molecule, leaving behind a reactive positive charge on a carbon atom at the head of the isopentenyl molecule. This is incidentally one of the many reasons plants need to absorb phosphorus from their environment.

Simplified reaction mechanism(enzyme is not shown): electron flow reveals how isopentenyl pyrophosphate bonds to a charged molecule derived from isopentenyl pyrophosphate itself. The product, geranyl pyrophosphate is then used by lemons to make geraniol. Many other fragrant molecules are also derived from geranyl pyrophosphate. See diagram below. Diagram by the author.
The backbone of geranyl pyrophosphate is highlighted in red and is found in three citrus scents.

Notice that I emphasized that only some isopentenyl pyrophosphate isomerize and become ionized. The rest then attack and stick to the positively charged molecules, creating  geranyl pyrophosphate (C10H20O7P2). This molecule can eventually be used to make an important component of cell membranes known as cholesterol, but lemons and some non-citrus plants also use the right enzyme to react it with water to produce geraniol (C10H18O). Although roses produce far more of the scent, it is also a minor component of citrus peels. In addition, lemon plants can oxidize the alcohol group of geraniol to an aldehyde to produce two isomers of citral, A and B. The former has a strong lemon scent and the latter, which has the same formula but with an aldehyde group on the other side of the carbon-double-bond, has a less intense but sweeter odor.

One 2009 study by Citrus Research and Education Center in Florida analyzed the flower scents of 15 species of citrus plants including lemons, limes, sweet oranges and mandarins. Using a relatively new solvent-less technique known as solid phase microextraction (SPME) along with mass spec-gas chromatagrophy, they identified 70 compounds, of which 29 were identified for the first time. The compounds belonged to four different groups of terpenes, compounds that are all derived from previously mentioned isoprene units. One of those oxygenated terpenes, linalool, is an alcohol derivative of geranyl pyrophosphate. Of the two possible isomers of the compound, oranges produce the R-version of linalool, which smells like lavender blended with citrus. It attracts bees and me.

A bee in a lemon flower. Linalool is one of the compounds that attracts the pollinator. Picture by the author.
geranyl pHosp deriv
More variation upon a theme at the molecular level: geranyl pyrophosphate, which is itself built up from two isoprene units, goes on to be the precursor of four other key compounds found in the floral essence of several citrus plants. Structures by the author.

Twenty four and forty five percent of  the blossom-volatiles of sweet oranges and mandarins, respectively, consist of  ß -myrcene. This compound is yet another variation upon the theme of geranyl pyrophosphate. Instead of having an allylic alcoholic head, it has a pair of conjugated double-bonded carbons and a pleasant fragrance. The same fruit blossoms and those of certain limes and lemons also produce of yet another geranyl pyrophosphate-derivative known as E-ocimene. Its aroma has been described as woody, green and tropical, an indication of how difficult it is for humans to describe smells. The difficulty becomes more pronounced when the isolated compounds of the labs force us to perceive “solo performances” while the reality of nature’s citrus blossoms present us with a symphony.


The Botanical Garden. Ryx and Phillips. Firefly.

A comparison of citrus blossom volatiles. Phytochemistry 70 (2009) 1428–1434

Principles of Biochemistry. Lehninger.

The Merck Index. Twelfth Edition

Myrcene as a Natural Base Chemical in Sustainable Chemistry: A Critical Review

Life at the End of Quantum Tunnels

Recently a biochemistry student told me that her classmates looked like they had seen a ghost when their professor seemingly took a left turn from a lecture on cellular respiration and started to discuss quantum tunnelling. But this 90-year discovery keeps surfacing in different contexts, reminding us that without the tunnelling effect, there would be no life in the universe.

Part of the lecture focused on iron–sulfur clusters, which play a role in the oxidation-reduction reactions of mitochondrial electron transport. The clusters are part of four protein complexes that sequentially shuttle electrons. The latter are ultimately gained from the breakdown of food molecules and are destined for oxygen. In so doing, protons are consumed inside the mitochondrial membrane while others are pumped out, creating a potential difference that helps motor the synthesis of adenosine triphosphate (ATP). Then ATP goes on to facilitate a host of energy-requiring reactions that keep an organism alive.

Each green arrow represents an electron jump due to quantum tunnelling.

But each time an iron cluster transfers an electron, it does so against a potential energy barrier. How does it do it? Because of the wave-like properties of a tiny particle like the electron, when it’s up against a thin-enough barrier, such as the 2.2 to 3.0 angstrom gaps (0.22 to 0.30 nanometers) shown in the diagram, there is a small but non-zero probability that the electron will be in the gap, and more importantly, also beyond it.  The best way to convince yourself that quantum tunnelling is physically possible is to go through the math and physics, and if you’re interested, it’s found here.  The author does not show every tedious algebraic step, but if you get stuck, I will gladly help in the comments section. It’s great fun while the laundry is being done.

Life involves a struggle against entropy made possible by a continuous energy source. For the planets and presumably moons that harbour life, the most important energy source is fusion from the sun. If you are like me in that you once assumed that the prodigious gravitational force at the core of a sun could provide hydrogen atoms with sufficient energy to overcome Coulombic repulsion and bring about fusion,

Image credit: E. Siegel

then you were also incorrect. It turns out that the kinetic energies are too small by a factor of 1000. So how does fusion take place? Like electrons in iron clusters, hydrogen atoms, although more massive, are small enough, and thanks to gravity, close enough to overcome the thousandfold barrier working against them. So quantum tunnelling is ultimately working with gravity to make stars shine.

The fact that tunnelling probability decreases steeply with lower thermal velocities extends the duration of smaller stars, those weighing less than 1.5 solar masses. This is important in that it gives life enough time to evolve in solar systems with appropriate conditions. One of the prerequisites of life, we imagine, is the presence of water on the surface of a moon or planet. Whether water is out-gassed or brought in via a comet or asteroid, it has to be first synthesized in molecular clouds according to this reaction between molecular hydrogen and hydroxyl radicals:

OH + H2  →  H + H2O

The extremely cold temperatures combined with adsorption on dust particles create boundaries small enough for quantum tunnelling to allow the production of molecular hydrogen from its atomic counterparts. There is even evidence that the hydroxyl reaction itself benefits from the same phenomenon.

From deep space back to our bodies, can tunnelling cause unwelcome changes in the DNA molecule? In the double helix or “twisted ladder” of DNA, each nucleotide of one strand of the ladder is attracted to its complement on the other strand by means of a hydrogen bond. A hydrogen bond consists of a lone pair of electrons from one nucleotide attracted to the hydrogen bonded to an oxygen or nitrogen atom of the nucleotide on the other side of the strand.

from Modelling Proton Tunnelling in the Adenine–Thymine Base Pair
A. D. Godbeer , J. S. Al-Khalili * and P. D. Stevenson

But there is a small possibility that the proton (hydrogen without electrons) can overcome the potential energy barrier and end up bonded to the hydrogen-less atom on the other strand. If the effect would be common enough, it could lead to a mutation. It should be noted that this a very active area of research and these authors have concluded that, at least in the adenine-thymine base pair, tunnelling does not occur. Less controversial is the ideas that quantum tunnelling plays a key role in the repair of DNA from ultraviolet damage, specifically in the electron-transfer needed to undo the dimerization of pyrimidines.

If those shocked biochemistry students read this blog, I am not sure that it would erase the “seen-a-ghost” expression from their faces. As educators we don’t often empathize enough with their survival-mode of trying to focus on the “essentials” that will get them through a given course. Quantum tunnelling and quantum phenomena are central ideas, but grasping them rests on an above average foundation of mathematics, physics and chemistry concepts. Is it realistic to assume that most biochemistry freshmen have already acquired that? We have to be patient, fuel them with enthusiasm and make sure that we don’t muddy the waters of key concepts with too much content in our courses.

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