Watching the daisies grow: Turing and Biology

There’s a sketch drawn by Turing’s mother while he was still a child, showing a hockey game at school. The boys in the background are playing hockey, and Turing in the foreground is not playing, instead he’s leaning over to inspect a daisy growing in the field. The title of the sketch is “Hockey or watching the daisies grow”.

Turing continued this interest in biology into his adult life. In 1952 Turing wrote what became his most cited paper “The Chemical Basis of Morphogenesis”. This work looked at the question of how structure in nature comes about. How do we start from single cells and end up with complex patterns and shapes? For example, the black and white patterning on a cow, the patterns on a sea shell, the dappling on a fish.

He came up with the idea of having two interacting chemicals, which he called “morphogens”. These would diffuse through a space, and inhabit or promote each other as they met. He modelled this system with two equations, showing how the amount of chemicals would vary over time across the space. He demonstrated that his model could provide convincingly life-like patterns, and suggested that this might be how nature does it. Not only that but he programmed his computer to help him to calculate the results of the equations for certain cases.

His work was not particularly well understood at first. Then in more recent times, biologists did experiments on the fruit fly Drosophilia and showed that although the fly had these patterns, they were definitely not because of two morphogens. So Turing’s ideas fell out of fashion. However, recently scientists have taken up the ideas again and shown that there are many species in nature that work in exactly this way.

Back to that daisy in the hockey field – it had been noticed by other scientists that Fibonacci numbers were often found in nature’s spirals. The number of spirals found in a daisy head, or a sunflower head, or at the base of a pine cone, or in cauliflowers, rose petals, leaves on stalks, all were Fibonacci numbers.

Fibonacci numbers come from a series where each number is produced by adding the previous two numbers. So we have:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,…

Turing wondered why this was, and how it was that as  sunflower head grew larger, the number of spirals would get larger, but would jump to the next Fibonacci number each time. And would his models that investigated the formation of structure in Nature be of help in interpreting this problem? This is the work that he was investigating just before he died. In Turing’s centenary year Prof Jonathan Swinton in Manchester University thought it would be a fitting tribute to Turing to encourage as many people as possible to grow sunflowers, collect and count the spirals in the heads, and use this data to finish Turing’s work.

Aberystwyth University Computer Science department grew sunflowers, here are the spiral counts: 55 in the clockwise direction and 89 in the anticlockwise direction. So Turing was right.

Turing was interested in such a wide range of topics. Over his lifetime he looked at what it meant to compute something, what is a computer, how to break encryption, what intelligence and artificial intelligence might mean, and the potential for computers to change the way we think and work. Near the end of his life he also tackled biological problems, set computers to work to investigate biological problems, and if only he had lived longer (to understand the structure of DNA, for example), we can only guess at the fascinating things we would have discovered.

By Amanda Clare, Lecturer in Computer Science at Aberystwyth University

Gwylio Llygaid y Dydd: Turing a Bioleg

Lluniwyd sgets gan fam Alan Turing tra’r oedd dal yn blentyn. Dengys y sgets gem hoci yn yr ysgol. Tra bo’r bechgyn yn y cefndir yn chwarae hoci, mae Alan yn plygu drosodd i edrych ar lygedyn y dydd yn y cae. Teitl y sgets yw “Hoci neu wylio llygedyn y dydd yn tyfu.”

Parhaodd diddordeb Turing mewn bioleg wedi iddo dyfu’n oedolyn. Yn 1952 ysgrifennodd erthygl, o’r  enw“The Chemical Basis of Morphogenesis”. Dyma’r darn o waith gan Turing sydd wedi ei ddyfynnu amlaf gan academyddion. Yn yr erthygl hon, mae’n cwestiynu sut daw strwythur mewn natur i fodolaeth. Sut ydym yn cychwyn gydag un gell, ac o hynny yn creu patrymau a siapiau cywrain megis y patrymau du a gwyn ar groen gwartheg, y patrymau ar gregyn mor neu’r patrymau ar gen pysgod?

Ei ddatrysiad i hyn oedd dau gemegyn a alwai yn forffogenau a fyddai’n cydgyffwrdd. Byddai’r rhain yn gwasgaru drwy’r gofod a byddant naill ai yn atal neu yn annog ei gilydd wrth gyfarfod. Modelodd y system hon gan ddefnyddio dau hafaliad, gan ddangos sut byddai lefelau y cemegion yn newid gydag amser a phellter. Dangosodd y gallai’r modwl hwn greu patrymau tebyg iawn i fyd natur, ac awgrymodd felly mai dyma sut oedd y broses yn gweithio mewn natur. Yn fwyfwy, dyfeisiodd raglen ar gyfer ei gyfrifiadur er mwyn ei gynorthwyo i ganfod canlyniadau’r hafaliadau mewn rhai sefyllfaoedd.

Ni chafodd ei waith fawr o sylw i ddechrau, gan nad oedd pobl yn ei ddeall yn iawn. Fodd bynnag yn fwy diweddar, cyflawnodd fiolegwyr arbrofion ar y pryfyn Drosophilia gan ddangos er bod gan y pryfyn y patrymau hyn, ni chrëwyd m’ohonynt gan ddau forffogen. Gan hynny, daeth syniadaeth Turing yn anffasiynol. Fodd bynnag yn ddiweddar mae gwyddonwyr wedi dechrau ail werthuso ei waith, ac maent wedi dangos fod rhai rhywogaethau yn ymddwyn fel a ddisgrifia Turing.

I ddychwelyd at y llygedyn y dydd ar y cae hoci. Roedd gwyddonwyr eraill wedi canfod fod troellianau mewn natur yn aml yn cyd-fynd a rhifau Fibonacci . Y mae’r nifer o droellianau ar lygedyn y dydd neu flodyn haul neu gôn pinwydden neu flodfresych neu betalau rhosod neu ddail ar goesyn blodyn i gyd yn rifau Fibonacci.

Daw rhifau Fibonacci o gyfres lle mae pob rhif yn gyfuniad o’r ddau rif blaenorol. Felly mae gennym :

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,…

Meddyliodd Turing paham oedd hyn yn wir, a paham mae blodyn haul fel mae’n tyfu yn datblygu mwy o droellianau, ond mae’r nifer o droellianau bob amser yn cyd-fynd a rhifau Fibonacci. A fyddai ‘r modelau ar gyfer canfod strwythurau mewn natur o gymorth wrth ddadansoddi’r broblem hon? Dyma’r gwaith a oedd yn mynd a’i fryd pan fu farw. Gan hynny, ar gyfer dathlu canmlwyddiant ei enedigaeth, ystyriodd yr Athro Jonathan Swinton ym Mhrifysgol Manceinion y byddai’n deyrnged addas i annog gymaint o bobol a phosib i dyfu blodau haul, ac yna cyfrif y troellianau yn y blodyn a defnyddio’r data hwn i gyflawni gwaith Turing.

Tyfwyd blodau haul yn Adran Cyfrifiadureg Prifysgol Aberystwyth; dyma gyfrifiadau’r troellianau : roedd 55 i gyfeiriad y cloc ac 89 mewn cyfeiriad gwrthgloc.  Felly, roedd Alan Turing yn gywir.

Roedd Turing yn ymddiddori mewn ystod eang o bynciau. Yn ystod ei fywyd edrychodd ar ystyd cyfrifo, beth yw cyfrifiadur, cêl-ddadansoddi, beth yw ystyr deallusrwydd a deallusrwydd artiffisial a gallu cyfrifiaduron i newid ein meddylfryd a’n modd o weithio. Tua diwedd ei fywyd, edrychodd hefyd ar broblemau biolegol a datblygodd gyfrifiaduron a allai ei datrys. Petai ond wedi byw yn hirach – er mwyn gallu deall strwythur DNA er enghraifft, gallwn ond dychmygu sawl peth arall y gallai fod wedi ganfod.

Gan Amanda Clare