EL TRABAJO DE ALAN TURING EN MORFOGÉNESIS

8/13/2019 EL TRABAJO DE ALAN TURING EN MORFOGNESIS

1/8

ARBOR Ciencia, Pensamiento CulturaVol. 189-764, noviembre-diciembre 2013, a081 | ISSN-L: 0210-1963

doi: h p://dx.doi.org/10.3989/arbor.2013.764n6004

ALAN TURINGS WORK ONMORPHOGENESIS

EL TRABAJO DE ALAN TURINGEN MORFOGNESIS

Miguel A. HerreroDepartamento de Matem ca Aplicada, Facultad de Matem cas,

Universidad Complutense de [email protected]

Cop right: 2013 CSIC. This is an open-access ar cle distributedunder the terms of the Crea ve Commons A ribu on-NonCommercial (by-nc) Spain 3.0 License.

Cita on/Cmo citar este ar culo: Herrero, M. A. (2013). AlanTurings work on morphogenesis. Arbor , 189 (764): a081. doi: h p://dx.doi.org/10.3989/arbor.2013.764n6004

Received: 10 July 2013. Accepted: 15 September 2013.

RESUMEN: En este ar culo deba remos la contribucin hechapor Alan Turing (1912-1954) a la fundamentacin matem cade la Biologa del Desarrollo. Para ello, repasaremos brevementesu punto de vista en el nico trabajo que public sobre estetema, y describiremos su impacto tanto en las Matem cascomo en la Biologa.

Palabras clave : Formacin de patrones; Biologa delDesarrollo; Sistemas de ac vacin e inhibicin.

ABSTRACT: In this work we will discuss on the contribu onmade by Alan Turing (1912-1954) towards a mathema calfounda on of Developmental Biology. To do so, we will brie yreview the approach he laid out in his only published workon the subject, and then describe the impact of his work onMathema cs on one hand, and on Biology on the other.

KEy WORdS: Pa ern Forma on; Developmental Biology;Ac vator-Inhibitor systems.

EL LEGADO DE ALAN TURING /THE LEGACY OF ALAN TURING
http://dx.doi.org/10.3989/arbor.2013.764n6004mailto:[email protected]://dx.doi.org/10.3989/arbor.2013.764n6004http://dx.doi.org/10.3989/arbor.2013.764n6004http://dx.doi.org/10.3989/arbor.2013.764n6004http://dx.doi.org/10.3989/arbor.2013.764n6004mailto:[email protected]://dx.doi.org/10.3989/arbor.2013.764n6004


2/8

ARBORVol. 189-764, noviembre-diciembre 2013, a081. ISSN-L: 0210-1963 doi: h p://dx.doi.org/10.3989/arbor.2013.764n 6004

A l a n T u r i n g s w o r k o n m o r p h o g e n e s i s

2

a081

1. TURINGS PROPOSAL: A MATHEMATICAL THEORyOF dEVELOPMENT

Alan Turing published a single paper on what we

now call Mathema cal Biology (Turing, 1952). Hehad however a keen interest in biological problems,and le un nished a number of manuscripts on is-sues such as phyllotaxis, the way in which leavesare spa ally distributed in plants (see h p://www.turingarchive.org/ ). Interes ngly enough, the wordMathema cs does not appears in the tle of thatwork (The chemical basis of morphogenesis, see Ref.Turing, 1952). However, the approach in that ar cle isdeeply mathema cal, and some of the results derivedin it have had a considerable in uence in Physics andMathema cs, where they fueled a whole new areaknown as Pa ern Forma on. This happened, though,only a er his ideas had remained largely forgo en formany years. The impact of Turings approach in Biolo-gy is arguably far more modest, and his vision remainsout of the main stream even today.

In his work (Turing, 1952), Turing was concernedwith morphogenesis, a key issue in what it is todaycalled Developmental Biology. Roughly speaking, theterm morphogenesis is used to describe the whole setof processes by which a fully grown living being un-folds from a fer lized embryo. How such a complexobject as an adult person could develop from a quitesmall, almost homogeneous ini al stage, poses a for-midable scien c problem. Indeed, many of its crucialaspects remain largely unknown, in spite of the out-standing amount of informa on gathered on that areaduring the last century.

It has been long suspected before Turing that chemicalsubstances should be instrumental in morphogenesis.Such assump on is however made precise in the veryrst lines of (Turing, 1952) with unmatched precision:

It is suggested that a system of chemical substanc-es, called morphogens, reac ng together and di us-ing through a ssue, is adequate to account for themain phenomena of morphogenesis. Such a system,although it may originally be quite homogeneous,may later develop a pa ern or structure due to aninstability of the homogeneous equilibrium, which istriggered o by random disturbances. Such reac on-di usion systems are considered in some detail in thecase of an isolated ring of cells, a mathema cally con-venient, though biologically unusual system...

And now a key technical point is stated:

The inves ga on is chie y concerned with the onsetof instability

In Turings approach simplicity is crucial. A few lines

below, our author makes clear that he only intends tomake use of a few basic physical principles to addresshis ambi ous goal:

The purpose of this paper is to discuss a possiblemechanism by which the genes of a zygote may deter-mine the anatomical structure of the resul ng organ-ism. The theory does not make any new hypotheses;it merely suggests that certain well-known physicallaws are su cient to account for many of the facts

Simple his proposal may be, but mastering it re-quires some background, which is precisely described:

...The full understanding of the paper requires a goodknowledge of mathema cs, some biology, and someelementary chemistry

These few remarkable lines encode the gist ofTurings ideas on morphogenesis. Chemical species,which mediate the shaping of unfolding living pat-terns (such as organs or limbs), are postulated to op-erate according to a number of physical mechanisms,which are reduced to a bare minimum. In fact, onlytwo such principles are discussed in detail. The rst ofthem is di usion, which accounts for random, unbi-ased molecular mo on at the microscopic scale. Thesecond is represented by chemical reac ons, wherebynew molecular substances are generated from theinterac ons between exis ng ones. It was shown in(Turing, 1952) that, from the point of view of math-ema cal modelling, reac on and di usion su ce toexplain the appearance of nontrivial pa erns whichwere ini ally absent in a featureless ini al medium.Mechanical aspects, however important they may bein prac ce, are not dealt with in the paper (Turing,1952). That makes life simpler for the modeller, buthas been a source of cri cism ever since.

As recalled before, biological pa erns are proposedto be the consequence of instabili es arising from ahomogeneous state, which was ini ally stable fromthe point of view of reac on kine cs (described math-ema cally in terms of ordinary di eren al equa ons,ODEs). Such instabili es are a result of (arbitrarilysmall) random disturbances which introduce hetero-geneity in such medium. This in turn induces di usion(represented in mathema cal terms by par al dif-feren al equa ons, PDEs). Di usion is a mass trans-port mechanism, which in an a empt to suppress
http://dx.doi.org/10.3989/arbor.2013.763n5011http://www.turingarchive.org/http://www.turingarchive.org/http://www.turingarchive.org/http://www.turingarchive.org/http://dx.doi.org/10.3989/arbor.2013.763n5011


3/8


M i g u e l A . H e r r e r o

3

a087

heterogeneity, moves chemicals from regions wheretheir concentra ons are higher (with respect to theglobal spa al average) to others where they are lower.Oddly enough, and this was one of Turings signi cantmathema cal contribu ons (Turing, 1952), this at-tempt to return to homogeneity results in the onsetof unbounded disturbance growth when at least tworeac ng chemical substances are considered; see theAppendix at the end of this paper for further details.

Turing was well aware that his sugges on that sym-metry breaking could be induced by arbitrarily smallrandom disturbances might (and actually did) looksurprising at rst glance. In (Turing, 1952), page 41,he wrote:

There appears super cially to be a di culty con- fron ng this theory of morphogenesis, or, indeed, al-most any other theory of it. An embryo in its sphericalblastula stage has spherical symmetry, or if there areany devia ons from perfect symmetry, they cannot beregarded as of any par cular importance, for the de-via ons vary greatly from embryo to embryo within aspecies, though the organisms developed from themare barely dis nguishable. One may take it thereforethat there is perfect spherical symmetry. But a systemwhich has spherical symmetry, and whose state ischanging because of chemical reac ons and di usion,will remain spherically symmetrical forever. (The same

would hold true if the state were changing according tothe laws of electricity and magne sm, or of quantummechanics.) It certainly cannot result in an organismsuch as a horse, which is not spherically symmetrical

Turings answer to his own carefully worded state-ment goes as follows (Turing, 1952, p. 42):

There is a fallacy in this argument. It was assumedthat the devia ons from spherical symmetry in theblastula could be ignored because it makes no par-cular di erence what form of asymmetry there is.It is, however, important that there are some devia-

ons, for the system may reach a state of instabilityin which these irregulari es, or certain components ofthem, tend to grow. If this happens a new and stableequilibrium is usually reached, with the symmetry en-rely gone

We have already no ced that the seemingly coun-terintui ve fact that symmetry breaking could be in-duced by any small, random perturba on is supportedin (Turing, 1952) by means of a simple mathema calmodel (cf. Appendix). However, the nal statement inthe previous excerpt, namely the fact that eventu-

ally a new and stable equilibrium is usually reached,with the symmetry en rely gone would require forits jus ca on of addi onal, and highly sophis cated,mathema cal tools of a nonlinear nature. We shallpresently return to this point in Sec on 2 below.

Turings descrip on of the ini al stages of Pa ernForma on in an embryo as a di usion-induced destabi-liza on of an ini ally stable, homogeneous steady stateinstantly appeals to a mathema cally-minded readerfor various reasons. It is elegant, simple, looks quitegeneral and can be illustrated by means of elementaryarguments. However, it just looks too good to be true,and the limita ons of his approach were apparent toTuring himself. As a ma er of fact, in page 37 in (Turing,1952), a few lines a er his already quoted excerpt atthe beginning of his note, he plainly declared:

In this Sec on a mathema cal model of the growingembryo will be described. This model will be a simpli-

ca on and an idealiza on, and consequently a fal-si ca on. It is to be hoped that the features retained

for discussion are those of greatest importance in the present state of knowledge

Indeed, the Mathema cs behind Turings argu-ments were known to their author to be insu cientto fully achieve his purpose. A key technical limita-on in his work is that signi cant results could only

be obtained under the assump on of linearity in theequa ons involved. This is plainly declared in a nalSec on 13 of (Turing, 1952, cf. p. 71 there), where afew conclusions are gathered:

The wave theory which has been developed heredepends essen ally on the assump on that the reac-on rates are linear func ons of the concentra ons,an assump on which is jus able in the case of asystem just beginning to leave a homogeneous condi-on. Such systems certainly have a special interest asgiving the rst appearance of a pa ern, but they arethe excep on rather than the rule. Most of an organ-

ism, most of the me, is developing from one pa erninto another, rather than from homogeneity into a

pa ern. One would like to be able to follow this moregeneral process mathema cally also

The last goal is acknowledged to be out of reach bythe author at that me (1952).

The di cul es are, however, such that one cannothope to have any very embracing theory of such proc-esses, beyond the statement of the equa ons
http://dx.doi.org/10.3989/arbor.2013.764n6004http://dx.doi.org/10.3989/arbor.2013.764n6004


4/8



4

a081

(Turing, 1952, loc. cit.). Nevertheless, Turing con-ceived a possible way out of this stalemate by embrac-ing what we would today call Computa onal Biology:

It might be possible, however, to treat a few par cu-lar cases in detail with the aid of a digital computer.This method has the advantage that it is not so nec-essary to make simplifying assump ons as it is whendoing a more theore cal type of analysis. It mighteven be possible to take the mechanical aspects of the

problem into account as well as the chemical, whenapplying this type of method. The essen al disadvan-tage of the method is that one only gets results for

par cular cases

It may well be guessed from the previous sentencesthat Turing clearly foresaw the impending power ofcomputers, a technology that he himself contributedto develop, as a tool to address biological problems,developmental and otherwise. However, state-of-the-art computers in 1952 where unable to yield any sci-en c breakthrough in a mathema cally-based theo-ry of Pa ern Forma on.

2. THE IMPACT OF TURINGS IdEAS IN MATHEMATICSANd BIOLOGy

How was Turings work received at his me? Itwould be fair to say that his proposals remained large-

ly forgo en during the ies and the six es of pre-vious century. Several facts conspired to that e ect.To begin with, his un mely death in 1954, only twoyears a er (Turing, 1952) was published, preventedhim from further developing his ideas and methods.Moreover, some momentous events happened at thatme which contributed to diver ng biologists a en-on away from Turings work.

Indeed, only one year a er (Turing, 1952) appeared,Watson and Crick published their seminal work on thestructure of DNA (Watson and Crick, 1953) that ledgenera ons of scien sts to dig deeper and deeper intogene cs. On the other hand, in the same year 1952A. Hodgkin and A. Huxley published the rst of a se-ries of ar cles (Hodgkin and Huxley, 1952) concernedwith a key issue in neuroscience, that of modellingthe propaga on of signals in nerves. Oddly enough,Mathema cs was paramount to the techniques usedin Watson and Crick (1953) (crystallography) and inHodgkin and Huxley (1952) (numerical simula on ofODEs) systems. Why it was that mathema cal model-ling failed to appeal to developmental biologists?

It could well be that Crystallography had beengrounded on Mathema cs and Physics since its very

beginning, and that Neurosciences were ripe enoughto bene t from mathema cal tools while Develop-mental Biology was not. In a way, the ques on was re-lated to that of selec ng a problem which was at thesame me important and within reach of already ex-is ng mathema cal techniques. It is interes ng in thisrespect to read what Francis Crick, a physicist turnedbiologist, revealed to Lewis Wolpert, a dis nguisheddevelopmental biologist, during a scien c interview.Their dialogue went as follows:

Lewis Wolpert : How do you choose your theore cal work?

Francis Crick : Well, what I do is, not choose a problem,but to choose a subject, and then try to move aroundin this subject un l I nd an idea that yields: some-

thing that clicks together.

I dont say that I am going to try and solve such andsuch problem, because it may turn out, especially inbiology, that its insoluble. The problem of how pro-teins fold up, for example. We showed great discrimi-na on in not choosing that problem. It has not beensolved to this day.

The way I work is to take a given area and to look at problems which look as if they might be tractable, aswe thought the gene c code would be.

(see Wolpert and Richards, 1988 for the whole in-terview). One may wonder if a mathema cal theoryof developmental was tractable at all in the absenceof su cient computer power. This came soon a er-wards, though, and its staggering increase since thelate six es has de nitely shi ed the balance in mod-ellers minds from theory to large-scale simula ons, atrend that is clearly dominant nowadays. As a ma erof fact, even the compara vely low compu ng level inthe early seven es was enough to take a step furtherform (Turing, 1952), as we will see next.

2.1. Be on linearit . Ac vator-inhibitor s stems

In the context of morphogenesis, one had to waitfor twenty years un l a further signi cant step withrespect to (Turing, 1952) was provided. This wasachieved by the nonlinear ac vator-inhibitor theorydue A. Gierer and H. Meinhardt (1972). The authorssummarized their approach in that work as follows:

The hypothesis is being stated that the elementary process in pa ern forma on may be the forma on ofa primary pa ern of two morphogens, one ac ng as


5/8



5

a087

ac vator, and one with inhibitory e ect, the inhibi onbeing derived from, and extending into a wider area.

Ac vator and inhibitor react auto -and cross cataly -cally on their sources

A key point is now immediately made:

Since linear rela ons will not su ce, non-linearequa ons have to be postulated

As a striking di erence with the situa on in 1952,the basic nonlinear ac vator-inhibitor proposed inGierer and Meinhardt (1972) could be easily simu-lated in 1972, and the results of such simula ons re-vealed the emergence of pa erns that are renderedstable by the e ect of the nonlinearity retained in themodel. Actually, since the publica on of Gierer andMeinhardt (1972), the study of systems of this type(see for instance Koch and Meinhardt, 1994) has beenmostly done by means of numerical simula ons. Fullyrigorous analysis (proving in par cular existence anduniqueness of solu ons) has indeed become a rarityin this context. See however (Rothe, 1984), wherethe systems proposed in Gierer and Meinhardt (1972)were rigorously studied.

As a ma er of fact, there are understandable reasonsfor this shi from theory to simula ons. It is beyondques on that biological processes are exceedinglycomplex, and therefore most models, if manageable,have to keep to a few assump ons and disregard therest, including obviously those that remain unknown atthe me. To compare various modelling alterna ves,there seems to be no be er way than to check whattheir predic ons are, and computa onal methods helpin this respect more than any other mathema cal tech-nique. In par cular rigorous analysis, with its charac-teris cally long me scale to achieve results, and thedi cul es inherent to its own method, is no match foreducated computer guesses when it comes to explor-ing a largely unknown eld. This has contributed towidening the gap between theory and simula ons that

shows no signs as yet of being lled.In the years since Gierer and Meinhardt (1972) was

published, a large number of ar cles using reac on-di usion equa ons to address biological (and notmerely developmental) problems has been produced;see for instance Murray (2003) and its references. Inthis way, the subject of Mathema cal Biology has es-tablished itself as a branch of contemporary Mathe-ma cs that a racts a large following. Its status amongbiologists is however quite a di erent ma er, as weshall discuss below.

2.2. The Impact of Turings work in Biolog

While the interest in using mathema cal modelsin Biology has been steadily increasing among math-

ema cians and physicists ever since (and arguably be-fore) the publica on of (Turing, 1952), the situa onis quite di erent as far as biologists are concerned.In fact, Turings approach, and mathema cal model-ling in general, has yet to be accepted in most Biologyquarters, with the single possible excep on of Ecologyand related elds. It is beyond the scope of this ar cleto provide a full account, not to say a reasonable ex-plana on, of the reasons behind biologists misgivingswith respect to Mathema cs. Instead, we shall con-tent ourselves with sketching a few points which willhopefully shed some light onto that situa on.

To begin with, many biologists consider living beingsto be simply too complex to be amenable to anythingclose to mathema cal modelling, whose realm shouldbe accordingly con ned to that of inorganic ma er.This belief runs deep both in science and philosophysince the very origins of scien c thought. No less amaster than Aristotle (ca 384 BC - 322 BC) pronouncedMathema cs valid only to deal with immaterial thingswith these terse words:

The minute accuracy of mathema cs is not to bedemanded in all cases, but only in the case of thingswhich have no ma er. Hence (mathema cal) meth-od is not that of natural science; for presumably thewhole of nature has ma er

(see Aristotle, 1998; see also the web link: h p://clas-sics.mit.edu/Aristotle/metaphysics.html ). It has takenMathema cs almost two thousand years to gain a sol-id grasp as a key method in Physics, and only a er suchminds as Galileo and Newton have made their markthere. In our days no one would seriously challengethe principle that Mathema cs is the tool to under-stand ma er, from atoms to stars. When it comes toliving beings, however, the situa on is very di erent.

An illumina ng view on the stance taken by manybiologists with respect to Mathema cs is given byEvelyn Fox Keller in (Keller, 2002). She provides therea penetra ng account of what she refers to as un-mely births of Mathema cal Biology. An example ofthat situa on is provided by the career of Nicholas Ra-shevsky (1899-1972), founder of a journal that even-tually became the Bulle n of Mathema cal Biology,currently one of the leading publica ons in that eld.In 1934 Rashevsky, presented at Cold Spring Harbor(CSH), then at the ini al stages of its current role as a
http://dx.doi.org/10.3989/arbor.2013.764n6004http://classics.mit.edu/Aristotle/metaphysics.htmlhttp://classics.mit.edu/Aristotle/metaphysics.htmlhttp://classics.mit.edu/Aristotle/metaphysics.htmlhttp://classics.mit.edu/Aristotle/metaphysics.htmlhttp://dx.doi.org/10.3989/arbor.2013.764n6004


6/8



6

a081

beacon for cu ng-edge Biology worldwide, his math-ema cal model of cell division. Rashevskys approachwas a classical one in physics, where simpli ed situa-ons are considered rst (for instance, cells may as-sumed spherical to start with) to be later generalizedonce a few relevant conclusions have been derivedfrom such simpli ed cases. The answer found in CSHto his approach was disappoin ng. In par cular, as re-called in (Keller, 2002), E. B. Wilson, a leading biologistof his genera on, prepared a short paper discussingRashevskys proposals, which contained statementsas the following:

Science need not be mathema cal,Simply because a subject is mathema cal it need nottherefore be scien c

Moreover, to summarize the exchanges betweenRashevsky and Wilson, E. Ponder, director of CSH atthat me wrote:

One point upon which there seems to be pre y gen-eral agreement is that there is li le rela on betweenthe amount of which has been done on the mathe-ma cs of (biological) growth and the clari ca on ofthe subject which has resulted

It is fu le to conjure up in the imagina on a system of dif- ferenal equa ons for the purpose of accoun ng for facts

which are not only very complex, but largely unknown

What we require at the present me is more meas-urement and less theory

These words, wri en less than twenty years before the publica on of (Turing, 1952) represent an an ci-pated and complete rebu al of Turings approach.There is an ominous ring in a sentence declaringthat what we need is more measurement and lesstheory. Indeed, theories, so appreciated in Physics,are looked down upon by many dis nguished devel-opmental biologists, to keep to the eld that Turinghimself selected to study. This statement is plainly ex-plained by Hans Meinhardt in (Gordon and Beloussov,2006). There he says:

Why do models (in Biology) have only a limited rep-uta on? My own experience is that experimentalistsare not very enthusias c if it turns out that a processwas correctly predicted. They worked hard to nd thebasic principles by themselves. Frequently the predic-on is then handled more as a specula on, if not com-

pletely ignored.

This is very di erent to the habit in physics where anexperimental observa on would be in no way dimin-ished if it is preceded by a theore cal predic on, onthe contrary

A bit later in Gordon and Beloussov (2006), Mein-hardt provides a pungent comment that could be ex-tended to the di cul es that new ideas usually ndwhen they are formulated:

The recep on of some of my models had a strangehistory. First they were regarded as unrealis c ormisleading: cannot be. More or less abruptly thischanged later into: that is trivial, how else should itbe? This switch had di erent me constants in di er-ent communi es. Both a tudes provide the freedomto ignore the theore cal work

Despite a general unfriendly welcome, the use ofReac on-Di usion models to address biological prob-lems has slowly gained ground as me has elapsed.Even hardliners nowadays consider it appropriate torefer to it, although not without due reserva ons.For instance, in the book Principles of Development (Wolpert et al ., 1999), whose principal author is LewisWolpert (already men oned here), Reac on-Di usionmechanisms are dealt with in page 317 in Chapter 10(devoted to organogenesis) in the 1999 edi on. Aninteres ng descrip on of stripe forma on in shes isprovided as a possible example of applica on of thesemathema cal methods. The text referred concludes,though, with the following cau onary statement:

Nevertheless, there is as yet no direct evidence fora reac on-di usion system pa erning any developingorganism (chapter 10, p. 317).

However, a number of molecular agents have beeniden ed as candidates to mediate such processes, andtheir number con nues to increase (Meinhardt, 2008).

3. CONCLUdING REMARKS

Despite the obstacles found, some of which havebeen brie y addressed here, there is currently a largerinterac on between biomedical scien sts, mathema-cians and physicists that has ever been in the past.It is beyond the scope of this note to report on someof the exci ng scien c fron ers where this collabo-ra on is already bearing fruit. Of course, the sheercomplexity of living beings represents a formidablebarrier that has to be overcome case by case for thiscollabora on to succeed. That was certainly foreseenin the concluding sentence in (Turing, 1952) which ac-curately summarized what was achieved there:


7/8



7

a087

It must be admi ed that the biological exampleswhich it has been possible to give in the present paperare very limited. This can be ascribed quite simply tothe fact that biological phenomena are usually very

complicated. Taking this in combina on with the rela-vely elementary mathema cs used in this paper onecould hardly expect to nd that many observed bio-logical phenomena would be covered. It is thought,however, that the imaginary biological systems whichhave been treated, and the principles which havebeen discussed, should be of some help in interpre ngreal biological forms

There is a long way to go following the path laid outin Turings celebrated paper. This seems to be a taskfor many genera ons of scien sts, and it appears tobe currently well on its way.

ACKNOWLEdGEMENTS

This work originated from a lecture given at thesymposium The Alan Turing legacy, held at Fun-dacin Ramn Areces in Madrid, in October 2012. Theauthor is grateful to the organizers for their invita onto par cipate in that event. Par al support from MI-NECO Grant MTM2011-22656 is also acknowledged.

Aristotle (1998). Metaphysics . Englishtransla on (1998), Penguin.

Gierer, A. and Meinhardt, H. (1972). Atheory of biological pa ern forma on.Kyberne k, 12, pp. 30-39.

Gordon, R. and Beloussov, L. (2006). Fromobserva ons to paradigms: the impor-tance of theories and models. An inter-view with Hans Meinhardt. Int. J. Dev.Biol., 50, pp. 103-11.

Hodgkin, A. L. and Huxley, A. F. (1952). Aquan ta ve descrip on of membranecurrent and its applica on to conduc-on and excita on in nerve. Journal ofPhysiology 117 (4), pp. 500-544.

Keller, E. F. (2002). Making sense of life .Harvard University Press.

Koch, A. J. and Meinhardt, H. (1994). Bio-logical pa ern forma on: from basicmechanisms to complex structures.Rev. Mod. Phys. , 66 (4), pp. 1481-1507.

Meinhardt, H. (2008). Models of biologi-

cal pa ern forma on: from elementarysteps to the organiza on of embryonicaxes. Curr. Top. Dev. Biol., 81, pp. 1-63.

Murray, J. D. (2003). Mathema cal BiologyII: spa al models and biomedical appli-ca ons . Springer Series on Interdiscipli-nary Applied Mathema cs.

Rothe, F. (1984). Global solu ons of reac-on-di usion systems . Lecture Notes inMathema cs 1072, Springer.

Turing, A. (1952). The chemical basis ofmorphogenesis. Phil. Trans. Royal Soc.London , 237 (641), pp. 37-72.

Watson, J. D. and Crick, F. H. C. (1953).Molecular structure of nucleic acids: a

structure of deoxyribose nucleic acid.Nature 171, pp. 737-738.

Wolpert, L. and Richards, A. (1988). JustGossiping: An interview with FrancisCrick. A Passion for Science , Oxford Uni-versity Press.

Wolpert, L. et al. (1999 ). Principles of deve-lopment . Oxford University Press.

REFERENCES


8/8



8

a081

APPENdIX: TURINGS INSTABILITy IN A NUTSHELL

The following argument, taken from pages 42 and

43 in Turing (1952), illustrates the manner in whichbreakdown of symmetry or homogeneity sets inthe case of a pair of con guous cells, having ini allynearly the same contents.

Consider two cells I and II, and two substances (mor-phogens) X, Y present on them (c.f. Figure A below).

Chemical reac ons will be assumed among X and Yof a linear nature and given by

(A1)

(A2)

Note that if both morphogens have concentra onsX = Y = 1 in both cells, there is equilibrium of a stablenature, as can be seen from the study of the eigenva-lues associated to (A1), (A2). Suppose now that, due

Figure A: Two similar cells (I and II) where two morpho-gens X,Y are present. The respec ve concentra ons inI and II are denoted by means of indexes I and II.

perhaps to some uctua on, the ini al values in I andII turn out to be:

(A3)

According to (A1), (A2), X and Y are produced bychemical ac on at rates 0.18 and 0.22 in the rst cell,and destroyed at the same rate in the second. As aconsequence of the heterogeneity introduced by (A3),a di usion mechanism sets in. Suppose that ow dueto di usion from the rst cell to the second occurs ata rate 0.5 for the rst morphogen , and 4.5 for thesecond (Note that the ra o of the second di usivityto the rst one is 9, about one order of magnitude).The combined reac on-di usion process may now berepresented by the following equa ons:

(A4)

(A5)

Let ( t ) be the perturba on induced in each mor-phogen star ng from an ini al value (0)=0.02 in (A3).It then follows from (A4), (A5) that for any t >0,

(A6)

so that an exponen al dri away from equilibriumis induced by the ini al perturba on (A3). Note thatnega ve values for morphogen concentra ons shouldbe discarded. Actually, when applica on of the pre-vious formulae result in the concentra on of a mor-phogen in a cell becoming nega ve, it should be un-derstood that it is instead removed only at the rate atwhich it is reaching that cell by di usion.

EL TRABAJO DE ALAN TURING EN MORFOGÉNESIS

Documents

Transcript of EL TRABAJO DE ALAN TURING EN MORFOGÉNESIS