getting to egg

When I was staying with friends in Boston in the early 80s, I felt the urge to make a layer cake. It must have been a birthday. A 3-decker chocolate fudge cake with orange butter icing featured in the available cook book and I duly plugged through the recipe. It was a towering success! and I added Joyce's Fudge

Cake to my repertoire. There came a time when I was fired up to make this party piece but could only locate 2 cake tins. Cranking down the ingredients to 2/3 of the original recipe should work, except for the question what is 2/3 of two eggs?  My notes indicate that I found two small eggs and used less butter; keeping the measurable quantities (flour, sugar, soured milk, cocoa) in 2/3 proportion. It was fine. But a fudge cake is supremely forgiving; it's meant to be a bit gooey and it's impossible to overcook. Overcooked fudge cake is called cookies. Heck; how big is "an egg" anyway? How long is a piece of string?

My pals at metafilter recently asked about a similar problem. The recipe requiring 3 large eggs but Cookie having only 2 large eggs and 4 medium eggs. My nerdnik answer was 

• Weigh the two L eggs you have: call that E2
• You goal is E2 * 1.5
• Weigh the 4 M eggs: call that M4
• IF |M4 - E2*1.5| > |(E2 + M) - E2*1.5| THEN DO E2+M ELSE DO M4

The shoulder bars on |M4 - E2*1.5| indicate the difference between M4 and E2*1.5 regardless of which is bigger.In words, whichever of LLM or MMMM is closer to LLL is your best guess. Others noted that this was not the obsessive hill to die on. Carrot cake will be fine so long as there is some egg in there. But I like the idea of compensating a deficiency / abundance in one ingredient by tweaking another. In the ingredient space, margarine has more in common with eggs and milk than any of them do with flour.

My current book, recommended by Tim Harford channelled by Dau.II is Math Without Numbers by Milo Beckman. Beckman's boast is that the only numerals in this math book are the page numbers. His early chapters are about LWH dimensions and beyond that to n-dimensional hyperspace of which I acquired some intuitive understanding in my synonymous codon usage days. You can lose your readers quickly in such abstractions, so Beckman talks about the dimensions of bakery. 

It cooking time and temperature are independent and quasi-complementary units of measurement. Meringues can either be made quickly at a high temperature or sloowwly in a "cool" oven. You don't get the same result but both varieties are supremmmnmmely edible. But there are the other dimensions alluded to above: flour, cocoa, eggs, butter, cream, milk, sprinkles. In this multi-dimensional bakerverse there is a region called cookies and others called cake and bread. Throw too many sultanas in the bread and you're stepping into cake territory and may need an egg for credibility.

Similar proteins

Proteins are made of amino acids stuck together in long chains which come curling out of ribosomes in an order dictated ultimately by the DNA in the genes on the chromosomes in the nucleus of every cell which makes new proteins -- which is all cells that
are still alive. I've spent the last 30 years trying to make sense of those sequences: usually comparing them by alignment. Here's part of the sequence of insulin for pigs and humans:
SP|P01315|INS_PIG   PQAGAVELGGGL--GGLQALALEGPPQKRGIVEQCCTSICSLYQLENYCN 108 SP|P01308|INS_HUMAN LQVGQVELGGGPGAGSLQPLALEGSLQKRGIVEQCCTSICSLYQLENYCN 110
                     *.* ******   *.** *****  ************************
Notes:
[Each of the 20 amino acids found in proteins has a unique 1-letter code. And a more readable 3-letter code too]
{For the first half of the 20thC, diabetics were injected with insulin extracted from thousands of pig pancreases and it worked just fine.}

You can see that the sequences, and therefore by inference, their 3-D structures, are very similar but with several significant differences: the human version has two extra amino acids GA for starters.  If the final functioning protein is in 3 dimensions, then the linear sequence, as it appears from the ribosomes, or as represented on paper and on screen as above, can be thought of as a 1-D object. In between, biochemists talk about the secondary structure, which is comprised of the intermediate building blocks or structural elements: α-helices; β-sheets; turns; ω-loops.

How similar do protein sequences have to be for us to believe that a) they have a common ancestor and b) they therefore have a similar structure and function? If you line up any pair of sequences, some of the letters will match because there are only 20 amino acids to play with. Like when I align the opening words of the first two paragraphs above:
Proteins are made of aminoacids
You can see that the sequences
3/30 = 10% of the letters appear in the same position in these random unrelated sequences. Molecular evolutionists talk about the twilight zone when ~15-20% of the amino acids are identical between two sequences. More than 20% identity and you can be reasonably confident is assigning similar function and inferring the same 3-dimensional structure. In the twilight zone you really need some independent information to make that call. You might naively think that the random probability is 5% = 1/20 because there are 20 amino acids to play with. But because you're allowed to insert gaps in one sequence [to accommodate the extra GA in the insulin above, for example] and because some amino acids eg LAG and more common than others eg HWC:

the biologically meaningful / statistically significant cut-off (and the Twilight Zone) are pitched higher than 5% at just under 20%. To be continued . . . as gene location becomes the added value independent information to match barely detectable similarities between genes in two different species.

Years ago, in the late 90s, I wrote some code to deconstruct protein sequences as if they had been hydrolysed by 1M NaOH [that's caustic soda to cooks] into a soup of their component amino acids. This might be though of as a protein's 0-dimensional structure. My program also tallied up and counted the frequency of each of the 20 amino acids. Tricking about with a test dataset, I noticed that this string of 20 numbers had quite high predictive value: the AA frequencies of Bacillus subtilis recA could predictably fish out the recA in Escherichia coli etc. I got sufficiently excited about this that I presented it as my contribution to the weekly internal lunchtime research seminar. Ken Wolfe, not yet my boss, but usually the smartest chap in the room, said he'd noticed this several years earlier but hadn't though it worth publishing. My abiding problem as a scientist is that I lack the stamina to get results, even interesting results, down on paper and through the publication process. I am thus glad to be able to publish this executive summary in The Journal of Blob Studies - they will take anything.
  Part II.

Hidden in plain sight

There are some narratives in science which everyone knows to be true. That Rosalind Franklin was robbed by Wilkins, Crick and Watson, not only of credit for discovering the structure of DNA, but also of a share in the subsequent 1962 Nobel Prize. That the supervisors of Jocelyn Bell Burnell elbowed her aside to get their No-Bell Prize in 1974. Neither of these stories really stand up to scrutiny. JBB  never let her No-Bell rankle (her exclusion was a consequence of the unconsidered norms of the time) or hold her back from carving her own career in science. Her contributions to the existence of Open University have arguably had more impact than her discovery of pulsars. Franklin was a really good pair of hands, completely focused on the task of making and photographing perfect microscopic crystals. She wasn't allowed enough time to put her findings into context, partly because her standards were too high and she was still perfecting the technique. But partly because the vaunting ambition and striding egos of Crick and Watson steam-rollered right over her, snagging a good-enough picture of the DNA structure with them.

Not all missing the Nobel Prize boat stories are as well known. There's a war-story  of early molecular biology How Hard It Is Seeing What Is in Front of Your Eyes in the 28 June 2018 volume of Cell. It concerns Lisa Steiner [R 50 years later] a student in Yale Medical School in the 1950s who discovered that the enzyme she was working on, RNase-A, could be denatured by an artificially hostile chemical environment and would spontaneously refold itself into a working enzyme if the conditions were re-normalised. That was interesting but just another experimental detail on the way towards her MD thesis. Her boss didn't think it was worth writing up as a separate paper; but in those days there was less pressure to churn out the LPUs [least publishable unit]. A couple of years later in 1961 Christian Anfinsen carried out a similar experiment on the same enzyme and decided that the spontaneous folding of protein into its native, functional state must be the norm for all proteins. If that was true, then it explained how any 1-dimensional sequence of amino acids, translated from a specific 1-D sequence of nucleic acids [DNA via RNA], became a 3-dimensional, living, working, protein. The structure and function were an intrinsic property of the sequence itself and didn't need any external shaping. There is/was, I find this week in Wikipedia, a concept called Anfinsen's Dogma, although it was never referred to as such when I was an undergraduate in Genetics in the 1970s - its synonym the thermodynamic hypothesis being preferred in the Old World. It takes a certain chutzpah confidence to allow your name to be attached to your scientific finding and launch it as a universal religious truth. And present it with such assurance (and with sufficient existential context) that people believe you have revealed the holy grail. The Nobel Prize [1972] falls into your lap in due course.

When the author of the Cell essay about cloaks of invisibility finally gets to talk to Steiner about her missed opportunity, she remembers being delighted that her findings had been replicated. As for how she felt about being scooped, her response is typically sanguine: ‘‘That was only right, David. He understood the significance. We did not.’’  Part of the reason she and her supervisor didn't understand the significance is because A Lot happened in the intervening couple of years. The field of molecular biology was on fire at the time. Francis "Modesty" Crick had floated his own Central Dogma [prev] in 1958. As well as Anfinsen's RNase paper, 1961 was also the year that Nirenberg and Matthaei made the first assignment UUU = Phe for the universal genetic code AND Jacob and Monod's paper on the Lac Operon was published. All these new data started to form a wall against which you could bounce new ideas, new ways of seeing [Berger]. 40 years later, by contrast, the human genome didn't seem to change much the same-old same-old certainties and existing hierarchies.

But it was all good: like Jocelyn Bell Burnell going on to have a life after her major early discovery, Lisa Steiner moved sideways: making significant contributions to our understanding of the evolution of the immune system in vertebrates. The adaptive immune system is all about cells: B-cells, and T-cells or -lymphocytes, a class of white blood cells: where they are produced and what they do. B cells were were first identified in the Bursa of chickens. Mammals don't have a bursa, but the equivalent cells develop in the bone-marrow and then migrate to the spleen for further development. Danilova & Steiner (2002) showed that this development occurs in the pancreas of zebrafish Danio rerio. That opened our minds about the multiplicity of functions that are carried out by tissues: it's usually more complicated that you read in textbooks.

Another 2007 retrospective event about Steiner's 40 years in science was captured on film. Going back over her life, it is clear that she could have been a contender in a number of the fields through which she passed: a math professor at Wellesley was one possibility. Her message to her young pre-med advisees in MIT is to paraphrase Thomas Edison "to have a good career, try a lot of careers". What a loss it would have been to immunology, if Steiner had gotten stuck folding and unfolding proteins like a molecular laundress in Yale. I like that recognition that, rather than two roads diverging, there are hundreds of paths weaving back and forth in the yellow wood of life, dozens of which will be fulfilling and lead to success (however that ephemeral quality might be defined).  The film is interesting: you can skip the first 8 minutes which is a rambling technical introduction by Herman Eisen her boss from the 1960s. Almost all the questions come from old men until, 36 minutes in, Steiner responds to a young woman who asks if she has ever experienced discrimination. Her answer is statistical: about how few were the women in each cohort of her early career. I wonder if like for Jocelyn Bell Burnell "At this time it was traditional for male students to cat-call, heckle and stamp their feet when a female student entered a lecture theatre." But it seems that Steiner treated that sort of nonsense as the inevitable invisible background and got on with her work. The opportunity to explore this issue is lost because the next question from one of the old men is about the old amino acid analyzer and the memories are back on track about science rather than bias. Thirty years after she was hired, in the late 1990s, Steiner was approached by two other female MIT faculty, Nancy Hopkins and  Mary-Lou Pardue, about gender bias in their place of work. A little research revealed that there were only 17 female Faculty with tenure compared to 197 tenured men. Until they talked, this manifest asymmetry had been part of the invisible normal background. Their report was leaked to the Boston Globe and the New York Times and attracted a chorus of me too from female academics across the country.

In 2008, her colleagues invited Steiner in for lunch at MIT, to celebrate her 40 years of contributing to their Institute. The report of the event noted that, from being the second female faculty member when she started, now 28% of the teaching staff were women; as were 50% graduate students and 65% of undergraduates. "The department has come a long way since she was first hired" claimed the author. Sure, mate, but that age pyramid says that you've a long way to go. And who was asked for a comment on Steiner's career? Chris, Uttam, Jack, Graham, Boris, Robert: all blokes! Where were the [28% after all] women who, because of Steiner's trail-blazing example and active advocacy and presence as a role-model, were now members of the Faculty Club and might have liked to add their 2c-worth of thanks?

It doesn't look like discrimination has handicapped Lisa Steiner in her life in science, but I think it's because she would not let it grind her down. It is important to stress that her contributions to science and society have been as a person of integrity, decency and modesty - a mensch.

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Eurogene - the map

I told y'all that you should go to Dublin on Darwinday to hear Dan Bradley talk about the Genetic Origins of the Irish. But I know that some things you can't delegate: you just have to do them yourself. Accordingly, I leapt into the Little Red Yaris at 1705hrs and drove to Dublin to hear the news from the frontiers of biogeography. But the news is always based on the olds and the most beautiful and informative picture of my 2018 [at top: far better copy] was published in 2008! I may well have been entranced by that map when it came out ten years ago, but I've since forgotten all about it. Heck, I've forgotten my car-keys and where I left my glasses as well.

That map is Fig 1 in a paper in Nature: Genes mirror geography within Europe which sampled the sequenced genomes of 3000 Europeans (and four Turks) and tallied up each person's state at 500,000 different variable sites in their DNA sequence. That's a shed-load of data and you can't make much headway by ticking off (3000 x 3000)/2 x 500,000 cases of Sean is different from Jean here but the same there, while Giovanni is different again. Well actually you can, and that's what John Novembre et al. did in 2008. They put the whole dataset into a hopper called Principle Components Analysis and gave it all a good shake and a jiggle. PCA reconciles all the internal inconsistencies, and calculates the position of each person in N-dimensional hyperspace. No, I too only have a hazy notion of what that really means but in practice it calculates how near or far each person is from each other person in the dataset. It will come as no surprise when it turned out that the quartet of Turks looked really similar to each other genetically and rather different from the Europeans . . . and the Irish too: like each other, quite similar to Brits and Scots and less like Poles and Greeks. A lot of that difference will smooth itself out over the next 100 years as our 200,000 resident Poles make babies with their Irish neighbours.

You can do these sort of studies because the cost of generating the primary data has collapsed over the last 30 years. The first ever chunk of genomic DNA, yeast Saccharomyces cerevisiae chromosome III, was contracted by the EEC (=EU) 30 years ago at 320,000 ecus (=€) or €1 /base. We carried out the first non-trivial added-value analysis of that data - one of my three big ideas in science. With that stepping stone achieved, planners looked to sequencing The Human Genome: it cost €300,000,000 (10c per base) and took ten years. Now you can sequence A human genome for €1,000; it will take a day; and there is enough server power to do many genomes in parallel. So 3,000 genomes is quite affordable in a big science sort of way.

What is most striking about the distribution of genomes across the most explanatory axes of the PCA landscape is how closely it maps onto the geography of Europe.  The pale blue of Greece and the Balkans is nearest to Turkey over on the right; the grey Italian peninsula runs parallel and a little more distant; and further away again is a purple peninsula of Iberian genomes. At the opposite end of the continent, the Irish intercalate with the Brits; the Scandinavians have both shared and separate identity etc. etc. If you look closely, you can see Paddy-No-Pals off on his own in the sea like as sort of Uber-Irish outlier. Maybe she is not Paddy at all but Caitlín Ní Uallacháin. Also note the five rogue ITs in the sea at bottom left of the diagram; they do indeed have Italian passports but they are actually Sardinians. There is no evidence here that the compatriots of Szilárd, Wigner, von Neumann and Teller come from Mars.

I was doing a similar analysis waaaay back in 1980s. I took me 2 years of tramping the streets of towns and cites in New England and the Canadian Maritimes, scoring genetic variation in domestic cats Felis catus to gather a sample of 10,000 cats in 35 different populations diagnosed for 7 genetic variants. (35 x 35)/2 x 7 is quite a bit smaller than (3000 x 3000)/2 x 500,000 !! But it was all my own work. One finding was that genetic distance was correlated (highly significant statistically) with geographic distance but that relationship only explained 16% of the variability in the sample. 84% of the variation was noise - some of which could be accounted for by the history of the patterns of French, English and Dutch colonisation in the 1600s. That was what my PhD thesis concluded aNNyway.

When you cough up your $100 to get your DNA sequenced, 23andMe will compare your DNA to a database like this one and place your genome on the map. Unless you are truly and incestuously descended from the Pharaohs, your genome will be a mess of fragments from the miscegenation of your ancestors. 23andMe will give you a summary sound-byte like "50% Irish; 25% English; 20% French; a toe from the Maghreb and a Neanderthal fingernail". You may take that assessment with a huge pinch of salt because the data will be inherently noisy.

Prochlorococcus

Q. What's the most abundant living thing on this our blue planet?

I recently read, through tears of frustration, Atlantic by Simon Winchester.  I found it worthy, erudite and informative . . . but scrappy and ultimately dissatisfying. But 430/490ths of the way through the book, he devotes a couple of pages to the - to me -wholly unexpected researches and calculations of Dr Penny Chisholm [R at the time when she made her remarkable discoveries] which sent me off down the rabbit-hole to fill in the details. I didn't go further than last year at the University of Illinois at Urbana-Champaign for a symposium Looking in the Right Direction: Carl Woese and the New Biology tribbing [bloboprev] UIUC's most famous scientific faculty member. Dr Chisholm led off the event with a stunning talk about her life on the ocean wave. If you can spell evolution, this is definitely worth the hour of your time (and stick around for the post-talk questions).

I spent 12 years of my life discovering genes in an immunology lab and being bemused by the instruments, experiments and mind-set of my immunology colleagues. Eventually I got to understand what they meant when they talked about CD4⁺CD8⁺ T-cells, CD4dims, side-scatter and gating. This is the language flow-cytometry, an instrument that immunologists use to characterise individual cells and count them under different experimental conditions. In 1985, Chisholm and her collaborator Rob Olson took one of these delicate monsters out into the wide Sargasso Sea and ran samples of salt water through it. They were looking for a blue-green alga BGA aka cyanobacteria called Synechococcus which had recently been discovered in marine samples. What they found as well was a smaller [0.6μm] amazingly abundant [10⁵/ml] photosynthetic bacterium that eventual got the name Prochlorococcus marinus. Even if you only count the upper 200m of the oceans where sunlight penetrates there are a helluva a lot of mls of sea-water and the best estimate is that there are a Carl "Billions" Sagan number of Prochlorococcus - 3 billion billion billion = 10²⁷ - out there; making them probably the most abundant species of which we know.

Math aside - there are lots of Prochlorococcus but they are very small and I think Homo sapiens has about the same biomass and rats Rattus norvegicus probably out-weigh us both.  Calculating the mass of all humanity is an exercise I do with my 1st Year Quant Methods. If we take the average weight (the clinically obese balanced by the starving infants) as 50kg and 7.4 billion of us in August 2016 [more today!] then there are 0.370 billion tonnes = 4 x 10⁸ tonnes available to carnivorous aliens from the Planet Zorg.  otoh, each Pcc weighs 2 x 10^-19 tonnes and there are 3 x 10²⁷ of them. So they tip the scales at 6 x 10⁸ tonnes. Not so tasty as humans and much harder to catch: got a sieve, Zorgistas?

Finding so many so tiny photosynthetic organisms in the ocean made us radically re-think our view of the carbon cycle, which up until then had been focused, with a sense of awe, on tropical rain-forest which includes some of the largest living things on the planet. The annual amount 5 Gtonnes of carbon fixed by Prochlorococcus is in the same ball-park as the excess 6 Gtonnes we are adding to the atmosphere by burning fossils fuels. Our carbon footprint is still increasing but the density of Prochlorococcus is remarkably stable year after year, despite doubling about once a day, so we must conclude that half of each day's reproductive bounty is promptly scarfed up by something else for dinner. As food, its carbon-based structures - protein, glucose and lipid - are quite quickly metabolised back into CO₂ and so are unlikely to be a major player in locking up the carbon which we are hosing into the atmosphere as we drive to work and heat homes in ludicrously unsuitable living spaces like New England, the whole of Canada, Scandinavia, and much of Russia.

One of the things that makes Prochlorococcus suitable for a lifetime's scientific research is that some strains can be cultivated in the lab while they can also be studied in their natural environment. The first such tame strain was SS120 [Sargasso Sea 120m down]. By flipping focus between the data generated by these different approaches, Chisholm has built up a startling picture of this tiny organism's place in the wide wet world. Small as it is, they have discovered it shipping fluid-filled vesicles out from the surface membrane for purposes intriguing but not yet fully understood. These vesicles contain DNA, RNA and proteins and could be regarded as mobile workshops spreading out from the minuscule mother-ship. It seems that Prochlorococcus lives in a complex commensal relationship with other bacteria that have only a tenuous physical connexion - it's really dilute in the sea. These other bacteria are able to scavenge the limiting elements, mainly nitrogen, phosphorus and iron, in a marine environment. It's almost as if Prochlorococcus is paying its taxes in fixed carbon and receiving the key N, P, Fe building blocks for making more DNA and protein as it doubles its numbers every day.

And please, please be extremely skeptical of people, even 'scientists' who propose that, by seeding the ocean with iron or one of the other limiting elements we can boost the carbon-holding capacity of oceanic life. That reductionist single-factor analysis is likely mad bad and dangerous to implement because it will boost some unknown and unculturable microbes at the expense of others and upset the equilibrium of the n-dimensional [Fe, P, N, pH, salinity, conductivity, day-length, light intensity, depth, unknown-but-important-1, unmeasurable-but-important-2, intangible-but-vital-3 etc.] hyperspace that is an ecological niche. That niche is just one species characterised for what we can measure. But an ecosystem is an interlocked set of niches involving thousands of interacting and inter-dependent species. To the nearest whole number, the best estimate of how much we know about how an ecosystem works is . . . nothing.

A final word on genetic diversity. What they call Prochlorococcus marinus is a sort of universe in itself. Small cell has a small genome - 2,000 genes = about half that of E.coli - but the genetic complement = biochemical capability is wildly different between any tow randomly selected individuals from the "same species". When the US Dept Energy were planning to sequence the genome of Prochlorococcus marinus Chisholm insisted that they sequence two different genomes, to get a handle on 'allowable variability' in the species. Now there are 40 ecotypes fully sequenced. They all have a core genome of about 1,300 genes which make the proteins to do the 'housekeeping' tasks all living cells do every day. The rest - the flexi-genome - do different things, presumably related to the peculiar niche in which they serve/thrive. 200 m down is really different from 2 m down and the concentration of phosphorus is 10x different between the Atlantic and the Pacific.  If you sequence a genome and find nitrate reductase, then you know that bug will do well near an estuary where nitrate is flooding off the agricultural land upstream. With Prochlorococcus every time they sequence another isolate they find dozens or hundreds of new genes with novel functions. Collectively they have a "pan-genome" of 80,000 genes making them 4x more complex than, say, humans.

But then, my mentor Lynn Margulis [prev] often said that, given the astonishing diversity of life on Earth,  all mammals are 'essentially the same', each of these 5,000 furry lactating viviparous species are trifling variants on the same fundamental system. Molecular genetic comparisons have shown that there are several extra species of giraffe, rhinoceros and elephant and we now lose sleep over the likelihood that we cannot save them all from poaching and habitat destruction. On the Margulis 'essentially the same' argument we could lose all members of all three mammals and it will have less impact [biochemical, ecological, evolutionary] than the loss of a small fraction of the variants that are currently binned into species Prochlorococcus marinus. Bonnets off to Dr Chisholm for comtinuing to bowl this hoop along the road!

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Interplanetary comms

I was on about the science-bard Christian Bök and his cunning plan to send his poetry out for appreciation by the plain people of a planet orbiting Alpha Centauri. One of the points in that essay is that we're quite possibly 'at nothing' to have been a) sending out radio-traffic or b) listening to incoming radio traffic at SETI the Search for ExtraTerrestial Intelligence. That made me think about how we can possibly expect ET and his pals to make sense of a string of binary digits. Carl Sagan and Frank Drake did it all with graphics in the famous NASA gold-plate that was attached to the outside of the Pioneer spacecraft:

 

It makes you wonder what they were thinking to imagine that this was an efficient way to reach other forms of intelligent life.  Frank Drake is after all the originator of the Drake Equation:

which allows us to guesstimate how many intelligent communicative civilisations there are in our Galaxy. The answer is not very many anywhere near here. Sure if 'someone' happens upon a small object lost in interstellar space, the plaque might be a reasonable way of telling where/who we are.  The star-burst is centred on our Sun and indicates the distance and direction of the nearest stars and it's important to communicate that people don't wear clothes and that as a species there is a relatively mild sexual dimorphism. A radio broadcast is more efficient because it goes out in all directions and travels a lot faster/further than a steampunk space-ship. 

You can send the Golden Plate out digitally as a string of 1s and 0s repeated endlessly. If ET can read radiowaves they will have reasonably sophisticated mathematics, the reasoning goes, and so will know what is a prime number.  If the picture is constructed as N lines of M pixels each, where both N and M are primes, then there is only one way to assemble the string into a two dimensional picture of black and white pixels.  Actually there are two ways (M x N) or (N x M) but that shouldn't try the patience of the recipients. Alan Turing and his pals at Bletchley Park worked days and nights to crack the Enigma code. Actually there are other possibilities: like the lines could be written L to R or boustrophedon [from the Greek βουστροφηδόν = as the ox ploughs = lines running backwards and forwards like a knitting machine].  It turns out that prime numbers are primes regardless of how many fingers you have.  We count in 10s, computers do binary, ET would presumably count in octal because, like Mickey Mouse, he only has four digits on each hand.

For the actuality of  making 2-D pictures on a rectangular grid, check out Matt Parker's Stand-up Maths routine of programming Excel to do this.

Scovilles

Hellish hot is how The Boy and his mother like their sauce. But they are nothing compared to their [grand]mother, who sprinkles chili pepper on her cornflakes with a knife.

When it comes to measuring the whoomph of chili-peppers Capsicum spp. you could crush some up, filter it and run it through an HPLC to determine the precise quantity of capsaicin - 8-methyl-N-vanillyl-6-nonenamide- [structure above] and then say a particular chilli is this hot in moles per gram.  It's complicated by the fact that there are several derivatives of the base capsaicin called capsaicinoids which vary as to their pungency. But chili-bums [aka ring-burners?] had developed a comparative scale at least 100 years ago which is independent of HPLC technology. In 1912, a pharmacist called Wilbur Scoville extracted the peppery volatiles by steeping a precise weight of pepper in alcohol.  This essence is then serially diluted in a sugar water until half of a panel of expert tasters cannot detect it. This allows a Scoville Heat Unit scale from 0 for regular bell peppers to much more the 1,000,000 SHUs for chilis like the Carolina Reaper, Bhut Jolokia or Naga Viper. That's a measurement of similarity that is fairly well reproducible even if a dimensional analysis [SI Units?] of what you are actually measuring is not possible.

There is a lot of theatricality, not to say machismo, about the consumption of hot peppers as we you can hear with the hilarity off-stage in Good Mythical Morning, [GMM previously]. If that reminds you of Withnail and Matter, I'm with you. That's all mildly interesting . . . not really funny unless you find boxing or cock-fighting [hmmm haven't covered that yet] amusing. What's more interesting is the variability among people as to their capacity to taste/tolerate hot pepper.  We had an experiment in this line on a visit to Dublin's Taj Mahal forty years ago.

Metafilter had an interesting compendium recently about 'snacksandsuch' Nate a young feller with a shockin' nonchalance as he chews down on a Chocolate Bhut Jolokia and records his subjective impressions for science. I'm afraid it just made me think of people who wake up from a paracetamol over-dose feeling alright but already doomed to die. But his actions also have a taste of JBS Haldane ruthlessly experimenting on himself for science. If you read the comments on the metafilter post, you'll see that, for several people, chili-chewing is a young man's business. As you pass 30 your tolerance of them changes: not so much in the appetite but in the processing. There is a suggestion that the intestinal flora loses the ability to digest capsaicin as you get older - that's interesting and should be followed up.  If your intestinome loses and gains biochemical capability as you age, that may have profound effects in pharmacology. It's sad that Nate adds a note to his Chocolate Bhut Jolokia jaunt "This is my last video. This is because people keep saying my videos are fake." that's a loss for science. But then again, if science post-graduate students burst into tears and went home when a referee dissed their papers, then there would be a lot less science. Then again then again that would possibly be a good thing because 45% of scientific papers have contributed nothing to science.  Here's a suggestion that eating chilis burns fat and reduces weight in the obese, young Nate could follow that up and really help his community.

Units

One of the weirdly wonderful things about science is the naming of units.  It's a bit more limited than the naming of species or the naming of genes because there are fewer things that can be measured. I don't mean different ways of measuring the same fundamental thing: cubits, versts, ells, miles, metres, ångströms, fathoms, chains and parsecs all measure length.  It makes a difference as when two contributing engineering firms used newtons and pounds-force in the same space ship with costly results.

According the Système International d'Unités SI system there are 7 base units and you can measure everything else by combining these. Velocity is measured in m/s or ms^-1 acceleration as ms^-2 and so forth.  The seven fundamentals are illustrated in ring [R] are as follows

• time second s
• length meter m
• mass kilogram kg
• luminosity candela cd
• molecular qnt mole mol
• current ampere A
• temperature kelvin K

You measure energy in Joules for convenience but that can be reduced to kg.m²s^-2. Conversely the 'derived unit' for electrical resistance is the ohm which is kg.m²A^-2s^-3.  Conductance is the opposite/reciprocal and so is measured in mhos or siemens. The list is long with tributes to the great physicists, engineers and chemists of times past:

• volt[a] for electromotive force; farad[ay] for electrical capacitance
• henry for inductance; units kg⋅m²⋅s⁻²⋅A⁻²
• hertz for frequency
• pascal  for pressure [our weather-people give us the pressure in hectoPascals]
• sievert, bequerel and gray for various aspects of radiation we're not meant to use Curies any more because One Ci is the same as 3.7 × 10¹⁰ Bq, so Curies are redundant and sooooo yesterday.

Being precise in the naming of parts is important in science, so that everyone is on the same page and we can make direct comparisons of things that are distant but too massive or delicate to shift so that they stand side-by-side.   There is a branch of elementary maths called dimensional analysis so you know what units are involved.  An engineer might want to trade off the stiffness of some material [in Newtons per metre kg⋅s⁻²] against its thermal conductivity [in watts per meter Kelvin m⋅kg⋅s⁻³⋅K⁻¹] things cancel out so you're measuring in m⋅s⁻¹⋅K⁻¹. It was an inventory game that I was obsessively good at in school.  Words which in normal everyday speech are very similar in meaning, have precise definitions and differentiation in dimensional analysis:  Force, power and work for example:

• Force: measured in newtons N = kg·m·s⁻²
• Work: measured in joules J = kg·m²·s⁻²
• Power: measured in watts W = kg·m²·s⁻³

Biology is a bit noisier and so much more interesting. The woolliness of living systems [and not just sheep] makes precision of measurement more difficult and it's even hard to establish the units in which to measure things.  Francis Galton, somewhat facetiously coined a millihelen for the amount of beauty that will launch a single ship. For vitamins and vaccines we are often reduced to IUs [international units] so that two messy preparations from different sources cane be compared as to their efficacy. The effectiveness of [artificial] insulin is determined by a "mouse convulsion test" which is quite as disturbing as it sounds - especially for the mouse.

Fermat

Ritual Random Rabbit Refuge . . . Refmat? Well not quite. But today we're on Fermat and his Last Theorem. Why?  Beacuse 11th April 1953 is the birthday of my near contemporary Andrew Wiles.  Even my Grade 'D' in Pass (i.e. not Honours)  Maths students at the Institute know about Pythagoras' Theorem and the square of the hypotenuse: a² + b² = c² especially such integer solutions as 3² + 4² = 5² which was known to the Egyptians and used to square off the base of their pyramids. Pierre de Fermat (1601-1665) was one of the greatest mathematicians of his day and did solid work on probability and number theory but he is known, even widely outside the world of mathematics, for a throw away line where he broke the two dimensional bounds of Pythagoras and moved on to larger exponents. He tried to find solutions to formulae like aⁿ + bⁿ = cⁿ where the exponent n is greater than 2 and a, b and c are greater than 1 and came up with none. Then, with a rush of blood to the head, he proved to himself that there were NO solutions to those equations and wrote a note-to-self in the margin of his copy of Diophantus' Arithmetica: "Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet" Which roughly translates as "It is impossible for a cube to be the sum of two cubes, or a fourth power to be the sum of two fourth powers.  I have discovered a marvellous proof of this idea but the margin here is not big enough to contain it."  Hanc marginis exiguitas non caperet was a gauntlet thrown down to mathematicians for the next 350 years in its implication that, if he had a sheet of clean foolscap paper, Fermat would show everyone the answer to the conundrum.  You could also try to disprove Fermat's proof by finding a single counter-example and many a schoolchild doodled away on that sort of problem - to no avail. Lots of mathematicians rose to the challenge and came up with partial proofs. For example, Sophie Germain (1776-1831) proved that Fermat's Conjecture was true where the exponent was any odd prime less than 100 and others built on that for any odd prime of whatever size. Many a marvellous proofs and intriguing mathematical relationship was revealed along the way.  Many folk were convinced that a wrong-wrong-nearly-right solution declared in the 19thC was what Fermat had discovered but nobody came up with a proof than convinced the mathematical community . . .

. . . until 1993 when British mathematician Andrew Wiles [R with assertion] published his solution to the conundrum. It is too complex to describe here: Wiles' proof runs to 150 pages of closely reasoned arguments, dense with Greek letters and weird symbols but it hinges on making a connexion between problems in geometry and problems in number theory.  The way I visual this is that mathematics had built up two teetering towers of proof upon proof extending from something really solid like Euclid's statement that there is a line, infinitely thin, that connects two points . . .  Mathematicians had followed their noses to show that IF that is true THEN this is also true. They did this unconstrained by any sense of utility; it was just beautiful.  Along the way since 1637, those mathematical ideas have put a man on the moon, put Lara Croft into the pants heart of teenage boys, and developed heart pace-makers. Governments across the world try to fund science that will do good things for the economy or society.  If the pay-off is within the lifetime of that government, then so much the better.  They have no desire to fund ideas that may, in 350 years, bear fruit.  But they are wrong - as many useful ideas are generated from pure science as from applied science - nobody has any idea where the next great leap forward will develop.

What Wiles achieved was finding a one-to-one mapping that completed the circle between the tentacles of two towers of proof that had gone off from the baseline in completely different directions.  The two unstable chains of connected ideas became stronger because of this link and have doubtless served as the jumping off point for many other wonderful mathematical advances. Shortly after Wiles declared his hand, it was shown that there was a glitch in his proof, over which he tore his hair for another 450 days until he was struck by an insight "so indescribably beautiful... so simple and so elegant" that he knew he'd cracked it - and the mathematical community agreed. Another thing that is important in this story is that Wiles had his field of expertise where, after years of cogitation on a series of related ideas, he knew a lot - possibly all there was to know about about that area of mathematics.  One day, he went to a seminar presented by a visiting speaker from another area of mathematics entirely - well outside of Wiles' comfort zone.  At that lecture Wiles had the revelation that he could fulfill his boyhood ambition and solve Fermat's Last Theorem. We should all go to any and all opportunities to listen to new ideas from visiting speakers.

Spreadsheet grunts

On Saturday last I drove down to Cork to pick up Dau.II and her Christmas clobber and deliver a 20kg Belfast sink (which is another story); all of which was hard to handle by public transport.  We also gave a lift home to another Co.Carlow girl and her Christmas clobber which included a cello in a hard case. The Toyota Yaris was brim full.

Dau.II is in the process of extending her skill-set from being a plongeur, burrito-piler and health&safety liaison officer at her cafe in Cork into more of a management role.  It will require her to learn the rudiments of ExCel, the Microsoft spreadsheet programme.  This is one of the many things that is more or less beyond my competence which I am nevertheless required to teach at The Institute. I make no bones about this ignorance, and believe that having me "in charge" is probably a better learning experience for the students than having an 'expert' rambling off with the fairies into the subtleties of the software, while assuming that everyone knows the basics.  ANNyway, I am in a position to give the daughter some pointers about how to best use ExCel to produce: accounts for the accountant; product lists for the suppliers; rosters for the workers and optimistic graphs for the financial backers. As we motored out of Cork towards Waterford, I was telling my two young passengers about what ExCel can do because I couldn't open a laptop to show them.  And I realised that my experience with spreadsheets goes back a long, long way to before spreadsheet programmes existed and it was all done on large sheets of paper spread out over a desk.

Back in 1980, I was making steady progress through graduate school but perennially short of cash.  My sister was working in London for Time-Life Books on their Good Cook series.  She blagged me a job in Time-Life European Head Office in Amsterdam as a Summer intern helping formulate the budget for the next financial year. And yes, in those days interns were paid, even by Megacorp. We would generate tables like this:

	Jan	Feb	Mar	. . .	Nov	Dec
Africa	$10m	$10m	$12	. . .	$12	$20
MidEast	$25m	$20m	$20m	. . .	$20	$35m
N.Amer	$100	$120m	$140m	. . .	$180	$240m
Etc.	$	$	$	. . .	$	$

. . . then make photocopies and send faxes, so that the management could have an informed discussion.  They would come back with changes: "Reduce the spend in Africa over the Summer by 10%, but keep the row and column totals the same by allocating to surplus equally between the Middle East and South America."  The geographical spreadsheet would have to be reconciled with the product spreadsheet so eventually the Vice-President of Sales in Singapore would know how much she had available to promote the Great Generals series across AsiaPac. Large sums of money would be bandied about the boardroom and four of us quants would have to recalculate everything from the broad-brush scribbles, changes and counter-changes that the big knobs were thrashing out. It was quite good fun with stressful bursts of activity interspersed with cubicle-world idleness and chatter.  Our line-manager, a very nice woman called Bonnie answered to a fearsome buzz-cut Texan who had been to Business School - you could tell because he could operate a desk-calculator with three fingers. But to do this he had to take a deep breath . . . tense his jaw . . . position his hand just so . . . and . . . key-punch! We could beat him one-fingered if there were fewer than 5 numbers to sum.

Round about the same time, Dan Bricklin was a student in Harvard Business School taking "Three-Finger-Calculator 101" and watching one of his professors generating the numbers for a table that grew to fill the blackboard. When the Prof had to make a change, it required a flush of interrelated changes spreading out across the board. Bricklin's insight was to realise that this could be done far better by the computers like the Apple II which were coming into homes and offices across the Western World in the last years of the 1970s.  He created the first brilliantly intuitive and idiot-proof spreadsheet program called Visicalc, which was launched in June 1979. The reason why ExCel columns are labelled A B C, while the rows are 1 2 3 is because Bricklin thought it would be a good idea for Visicalc. First-comer Visicalc dominated the market, seeing off several knock-offs and alternatives and over the next 4 years sold 1 million copies. Then it was abruptly knocked off its perch byLotus 1-2-3, whose developers had learned from the mistakes made by Visicalc and its competitors. When Bricklin's company was consumed by Lotus in April 1985 he got a job there as a developer. In the 1980s the market was far from saturated and there was lots of market to share, so it was a field day for competent and creative programmers.

Lotus 1-2-3 was in its turn annihilated by ExCel: developed by Microsoft  and compatible with the standards - MS-Word and Powerpoint - that we still recognise 20+ years later as being the essential tools for academia and business. I'm not stupid but I still can't reliably persuade ExCel to draw a 2-dimensional scatterplot - so I'm inclined to label that software poorly designed and counter-intuitive.  Visicalc and the spreadsheet concept could have gone any direction in 1985 and it went to hell in a bucket.  But it still beats tricking about with a pencil and a desk-calculator.

Effortless Grace

In the WEA, we hand out PhDs if you sit at the feet of your mentor for 3 years, or a little more, carry out some original research and write it up as a thesis.  When I did my PhD in the USA, we were required to attend two years of formal courses with other experts in relevant fields, then sit a Comprehensive Exam before officially starting our research work.  Of course, this is not how it happened in reality.  None of my grad school pals sat on the thumbs of their curiosity while studenting for two years and then metamorphosed into a research butterfly.  I was quite complacent when I arrived in Boston: with a marginal degree from the Best Department in the Best University in Ireland, I thought I knew it all.  Or at least knew enough.  Those two years were an eye-opener and I learned a lot about things that were just not acknowledged in the narrow and over-specialised bachelor's degree course back home.  And my Comps were Comprehensive Exams, my thesis committee of five set me a day-long examination paper on whatever they thought was important - history, ecology, genetics, evolution, mathematics, anthropology.  It was gruelling, challenging and a little inspiring.  At about tea-time, I threw down my pen, shook out the claw that my hand had become, and vowed that it was the last examination I'd ever take; and so far I've stuck to that promise.

I was thinking about this when I was reading about Grace Chisholm Young's viva after submitting her 1895 PhD thesis ("Algebraisch-gruppentheoretische Untersuchungen zur sphärischen Trigonometrie" as you ask) at Göttingen "which consisted of probing questions, in German, by several professors on conic sections, geometry, differential equations, physics, astronomy, and . . .  the area of her dissertation."  It was the first PhD thesis ever awarded to a woman in Germany.  You may ask why a woman with such a British sounding name was doing a PhD in Germany.  It was because the UK was still decades from according women in academia parity of esteem.  I was writing recently about Clara Immerwahr, who obtained her PhD a few years later in Breslau.  Immerwahr was a chemist who married another chemist and I think she became deeply unhappy in her choice: she finally shot herself out of her misery.

Contrariwise, the life of Grace Chisholm has the ring of happiness.  Her older brother was given the full monty as regards education - boarding school, then a scholarship to Christ Church Oxford (which didn't admit women until 1978!) finishing with a First in Literae Humaniores.  He went on to be editor of Encyclopædia Britannica, so he was clearly no slouch intellectually.  Grace and her sister didn't go to school but were educated at home and groomed for "good works and a good marriage".  Problem was that she was a formidable mind with a particular and outstanding grasp of mathematics.  She went to the women-only Girton College, Cambridge where she fell in with another really sharp mathematician called Isabel Maddison.  At one point, for a jape, they took the exams for Final Honours School in mathematics at the University of Oxford and out-gunned all the blokes. After Cambridge, as I say, she romped off to Göttingen for her PhD.  Shortly thereafter she returned to England to look after her aging parents and married William Henry Young who had tutored her in mathematics when she was going through Girton. They were remarkably prolific academically, knocking off more than 200 mathematical  papers and several books.  Their biographers have been exercised as to who contributed how much to this steamer-trunk full of maths and the consensus seems to be that, while she deserves some credit for several of his single author papers, the reciprocal probably isn't true.  But partitioning the credit is not really the point: they were a good team and must have struck creative sparks off each other as flint and steel. They were prolific in other ways as well, having 6 children together in short order before WWI.  Not content with excelling in mathematics, Grace also took all the courses for a medical degree and learned six languages as her family tooled round Europe in the 1920s and 30s.  She taught each of her children a musical instrument in between making formal contributions to geometry and set theory.  And she wrote a well-regarded book on sexual reproduction for children called "Bimbo and the Frogs".

"Effortless Grace" is not quite right as a title.  She worked damned hard and put in tremendous hours with her children, her mathematics, and her many other commitments.  But I get the feeling that, all along the way she was supported and encouraged.  By her parents initially and then by such key people as W.H. Young who taught and much later married her; Arthur Cayley who sent her to Felix Klein in Göttingen. Klein who took her on and supervised her PhD thesis, and was famous for the Klein Bottle (shown right full of beer) a sort of n-dimensional Möbius strip. Even the enormous Georg Cantor had this to say of one of her books "Es ist eine Freude für mich, um zu sehen, was mit Fleiß, Geschick und Erfolg, den Sie gearbeitet haben, und ich wünsche Ihnen in Ihrer weiteren Forschungen in diesem Bereich auch die besten Ergebnisse, die mit einer solchen Tiefe und Schärfe des Geistes auf beiden Teilen Sie Ihre Sie können nicht umhin zu erreichen. (It is a pleasure for me to see with what diligence, skill and success you have worked and I wish you, in your further researches in this field as well, the finest results, which, with such depth and acuteness of mind on both your parts, you cannot fail to attain).

When the phony war of 1939-1940 hotted up with the invasion of France, Grace decided to repatriate two of her grand-children from Switzerland to England.  She was almost too late, having to bundle the two kids and herself onto the last plane from Paris to London.  She never saw her husband again, because both of them died before the war ended.  From reading some of her letters, she must have been good fun, you'd like to have had lunch with her, and it's her birthday (1868) today!

Bonnets OFF (women in science series).

Cantor counting

A month ago I had occasion to use Terence's all-embracing aphorism homo sum - humani nihil a me alienum puto, in the context of what should and should not be allowable in discourse.  Because something makes you feel uncomfortable is not sufficient reason to close up shop on its depiction or discussion.  I'd like to be able to lay claim to a narrower constituency which I won't affect to render in Latin, so Yoda-speak will have to do: "Scientist I am, beyond me nothing within its boundaries is".  Which makes it sufficiently obscure and oracular, but I've already admitted that wide chunks of physics send my brain teetering over the abyss or at least skittering down the inclined plane.

I'm teaching elementary math to our Yr1 biologists, or at least I am hosting a forum whereby the youngsters can use Khanacademy to practice their skills in geometry and algebra.  It reminds me that, when I was their age, I was quite nifty at mathematics and could knock off a bit a calculus, or trick about with set theory.  But I know that there are limits to what I can get my brain to accept.

Today is Georg Cantor's birthday. He was born on 19th February 1845 in St Petersburg Санкт-Петербург Petrograd Leningrad St Petersburg (they've changed the name a few times).  The 19th February was an example of the holdout that the Russians kept from embracing the Gregorian calendar until long after the rest of Europe.  19/02/45 was called 03/03/45 by everyone to the West of the cultural capital of Russia.  Cantor was smart and forced us to think on the nature of infinity in mind-melting ways so that since his time we recognise that there are some infinities that are bigger than others.

He showed first that the infinity of odd numbers is the same size as the infinity of all countable numbers although reason tells us that there must be twice as many of the latter.  He did this by explaining s l o w l y his idea of mapping.  He said that you could pair off the two sets of numbers two at a time
1=1 2=3 3=5 4=7 5=9 6=11 . . . ad infinitum
until they had all gone, so they must be equivalent in size.  Qualified Shazzam!?

If you can accept that you can push yourself to accept that rational numbers (fractions+integers to us) are also countable in the same way.  All you need to do is organise your data in a particular way so that you can tick the numbers off sequentially and know you're not missing any out. Cantor's insight was to set up all the fractions (which includes all the integers because 1 = 1/1; 2 = 2/1 etc.) in a grid and count them off along the blue diagonal against the counting numbers.  So far, so good?  Cantor then went on to apply similar reasoning to show that the points on a line were equivalent to the points in n-dimensional space.  This wrecked even Cantor's head and he famously wrote to his correspondent Dedekind "Je le vois, mais je ne le crois pas!" ("I see it, but I don't believe it!").

You might think that you (or at least mathematicians) can count any set of anything - but you can't.  And Cantor elegantly showed that the real numbers are not countable, that there must be some real numbers that fall between the cracks of the rationals . . . and so the infinity of reals must be bigger than the infinity of the countable.  Real numbers are the rationals (integers and fractions) PLUS the irrational numbers that cannot be represented by a fraction.  Although 22/7 is a damned good approximation for the ratio between the  circumference and diameter of a circle, any fule kno that this ratio is represented by π Pi 3.1415926...  Another beautiful real is the golden ratio ϕ Phi 1.6180339...  Last year, I laid out a bizarre connexion between the three most famous irrationals.

As with the diagonalised fractions, Cantor challenged us to give him any ("big as you like, big as you can imagine, ALL of them") set of real numbers and he would write them in a list one above the other (for convenience just dealing with the digits after the deci-point):
71828 18284 59045 23536 ..... (e)
14159 26535 89793 23846 ..... (π)
61803 39887 49894 84820 ..... (ϕ)
77777 77777 77777 77777 ..... (7/9)
.....
.....
Cantor then showed that we couldn't have included ALL the real numbers in the list because he could imagine/create/write a number that was different from the first number in its first digit AND different from the second number in its second digit AND different from the third number in its third digit etc.  That argument could be re-employed in the new N+1 list, with another number.  Sooooo, the infinity of real numbers is bigger than the infinity of rationals.

So far my head isn't piled up in a heap at the bottom of the cliffs of insanity, but beyond this much of mathematics quickly wings off towards the horizon leaving me behind in a pedestrian grounded world knowing my limits.