Alan Turing s chemical theory of phyllotaxis

Transcription

1 EDUCATION REVISTA MEXICANA DE FÍSICA JANUARY JUNE 04 Alan Tuing chemical theoy of phyllotaxi M.D. Rueda-Contea and J.L. Aagón Depatamento de Nanotecnología Cento de Fíica Aplicada y Tecnología Avanzada Univeidad Nacional Autónoma de México Apatado Potal Queétao México. Received 0 Novembe 0; accepted 4 Januay 04 Alan Tuing eminal 95 wo on mophogenei ] i widely nown and ecognied in the field of mathematical biology. Le nown i hi wo on the poblem of phyllotaxi which wa neve publihed at hi time but i included in Tuing collected wo ]. It conit on thee pat: the fit i a detailed mathematical deciption of the aangement of leave on the tem of plant; the econd i an application of the eaction-diffuion equation to the poblem and the thid pat i a olution of thee equation fo the cae of pheical ymmety. It i the pupoe of thi wo to peent Tuing eult contained in the econd pat in a compehenive and detailed way. Thi i motivated by the fact that thee eeache have emained obcue and ill-undetood. In paticula we focu on the mophogen equation fo an aembly of cell ince thi dicete cae may be ueful in many cicumtance whee the continuum limit i not adequate o applicable. Keywod: Phyllotaxi; Mophogenei; Tuing chemical theoy. PACS: j; e.. Intoduction The aim of Tuing The Chemical Bai of Mophogenei ] heeafte efeed to a CBM i to how that by a combination of diffuion and chemical eaction patten can aie in an oiginally homogeneou tiue. Thi highly cited wo became a mate piece of the mathematical modeling in biology and the bet nown model to explain biological patten fomation ]. Alan Tuing wa vividly inteeted in the phenomenon of Fibonacci phyllotaxi and hi wo on thi poblem i le nown ince it emained unpublihed fo many yea. Afte publihing the CBM between 95 and 954 Tuing wote unpublihed manucipt and note of hi late eeach in hi chemical theoy of mophogenei. Thee wee tudied by Tuing colleague N.E. Hoin and B. Richad 4] pieced togethe and publihed unde the title Mophogen Theoy of Phyllotaxi heeafte efeed to a MTP in 99 Tuing collected wo boo ]. Befoe decibing MTP we hould add ome wod concening phyllotaxi. The aangement of plant ogan alo called phyllotaxi ha facinated cientit and natualit fo centuie. The tudy of phyllotaxi can be taced bac to the 4th centuy B.C. in the ancient Geece o it can be conideed a the oldet banch of mathematical biology. Ancient natualit a Theophatu B.C. and Pliny - 79 A.D. ecognized and epoted ditinct patten of leaf aangement and popoed them a a tool fo plant claification. It wa by the time of Leonado Fibonacci of Pia that the elation between phyllotactic patten the Fibonacci equence and the golden mean wa ealized. A Fibonacci equence i obtained by the ule F n+ = F n + F n whee F 0 = F = and the golden mean i defined by the limit F n+ τ = lim = + 5 /. n F n The golden angle ϕ i defined a ϕ = πτ. Thee ae many diffeent type of plat ogan aangement among which the pial o helical patten a in unflowe i the mot widepead and complex 5]. In thi cae familie of pial called paatichie ae obeved and counting pial in each family eult in numbe that follow a Fibonacci equence fo a didactic deciption ee Ref.. Evenmoe the divegence angle between two conecutive plant ogan in a pial i alway appoximately ϕ. The fit fomal tudy of leaf aangement wa made by Chale Bonnet who wa able to ditinguih fou diffeent phyllotactic patten and decibe the o called genetic pial. Phyllotaxi began to be tudied in a cientific way in the 80 by a combination of obevation expeiment and theoetical hypothee. Schimpe 80 7] wa the fit one to decibe the phyllotactic pial and it elation with the Fibonacci equence. In 87 Loui and Augute Bavai 8] epeented the phyllotactic patten a point-lattice on a cylinde; thi idea wa etaen by Tuing in MTP and wa ueful to tate ome mathematical featue of phyllotactic pial. Mechanical and phyiological explanation of thi phenomenon began until the 880 9]. Late in 98 the hypothei of efficient pacing in phyllotaxi wa developed by Ridley and Aiy obtaining the Fibonacci phyllotaxi 0]. Afte the wo of Aiy ] and Hofmeite ] who etablihed the famou hypothei of inhibition which tate that the younget incipient leaf pimodium fom in the laget available pace left by the peviou pimodia the tudy of phyllotaxi tuned it attention to the hoot apical meitem whee thee pimodia emege intead of the ogan aangement on the matue tem. Chemical theoie about phyllotaxi appeaed at the ame time and Tuing contibution on the bai of hi ection-

2 M.D. RUEDA-CONTRERAS AND J.L. ARAGÓN diffuion theoy of mophogenei wa a pioneeing wo. Meinha and Richad 4] alo adopted the chemical appoach obtaining phyllotactic-lie patten. On the othe hand ome expeiment wee pefomed on the bai of phyical hypothee which could explain the emegence of phyllotactic aangement. A vey illutative example i found in the expeiment pefomed by Douady and Coude 5] who obtained phyllotactic pial by adding doplet of a feomagnetic mateial in a magnetic field at egula time inteval. The dicovey of the plant homone auxin and it influence on phyllotaxi 6] opened an intediciplinay way to appoach phyllotaxi by modelling an active and pola auxin tanpot in a gowing meitem 7 8] obtaining phyllotactic-lie patten on the bai of a moe ealitic et of hypothei that incopoate the main biological fact involved in plant mophogenei. The manucipt MTP i divided into thee pat. The fit one deal with geometical and deciptive phyllotaxi; the econd pat peent a chemical theoy of mophogenei and the thid pat give a olution of the mophogenetical equation fo ytem with pheical ymmety. Hee we will be concened with the econd pat and in paticula with the dicete appoach that i the fomulation fo an aembly of cell. It i woth to mention that in the CBM in Sec. 6 Tuing tudie the dicete cae of a ing of cell but the appoach in the MTP i even moe geneal. In the fit cae the dicetene i intoduced via a dicetization of the econd deivative ee fo intance Ref. but in the MTP Tuing conide the geneal poblem of diffuion of eaction of mophogen in an aembly of N cell; the tate of the oganim at any time t may be decibed by M N numbe Γ mn m =... M; n =... N whee Γ mn i the concentation of the mth mophogen in the nth cell ]. Albeit Tuing eventually conideed the limiting cae of a continuou tiue the dicete cae i inteeting by itelf and may be ueful unde many cicumtance. Fo intance contay to the dicete cae in the CTM beide the diffuion contant of each mophogen the ate of flow fom one cell to anothe i taen into account. Thi ate of flow not only depend on diffeence of concentation of the mophogen implicated but it i thought to be dependent on the geomety of the cell wall epaating the cell. Albeit baic linea algeba i involved vey oon Tuing calculation become had to follow. Thi i in pat due to the notation ued and alo becaue hi appoach i baed on hi geat intuition of the poblem. It i then the pupoe of thi wo to decibe in detail the dicete cae following a didactic point of view. Thi wo i oganized a follow: In Sec. the equation that decibe mophogen eaction and diffuion in a dicete et of cell ae etablihed and implified o that olution can be found by elementay linea algeba. Thi i done by fit conideing only diffuion of the mophogen and etting a linea ytem of equation in tem of a diffuion matix which contain all the infomation about the cell aembly it hape and geomety. Kinetic i then added to the ytem following a linea appoximation. Thi pocedue allow to expe the ytem a thee independent et of two linea equation and a olution of each et i found by claical ODE method. If the aembly conit of n cell then n diffeent et of two equation ae obtained by thi appoach. In Sec... the olution found in Sec.. ae efined by dicading the eigenvalue which do not lead to intability. That i only tem which gow fate ae ept in the olution found. Finally in Sec..4. Tuing nonlinea appoach quadatic i peented.. Mophogen equation fo an aembly of thee cell In Pat II of MTP page 88 entitled Chemical Theoy of Mophogenei Tuing tate the mophogen equation fo an aembly of cell; fit fo linea eaction ate function in Sec.. and then going beyond the linea cae in Sec... Even moe Tuing appoach upae the etiction to a ing of cell woed out in CBM by conideing abitay aangement of cell. A aleady mentioned Tuing fomulation of the poblem vey oon become cumbeome and had to follow o it i the pupoe of thi pat to tate the equation in a didactic eay to follow way. Fo thi only two mophogen u and w and thee cell with volume v v v ae conideed. Once the pocedue i claified and mathematical detail unveiled one can eaily go bac and epoduce calculation fo any numbe of cell in any configuation. Let u n and w n n = be the concentation of the mophogen in the nth cell. That i u i the concentation of mophogen u in the cell numbe one and o on. Then the flow of the mophogen u fom cell to cell i popotional to g u u whee g depend on the geomety of the cell wall epaation between thee cell. Fo example it i well nown that plant tiue ae aniotopic by vitue of the tuctue of thei cell wall 9]. The aniotopic natue of the cell wall i in tun elated to the diffeential flux of plant mophogen lie auxin 0] o the tem g might be ued to model thi effect o ome othe geometical featue that modify the diffuion of ubtance uch a the uface cuvatue ] o the pemeability of the complex plamamembane-cell-wall 7 ]. Let µ u and µ w be the diffuion coefficient of the mophogen u and w epectively and etablih the ame elation fo mophogen w that i the flow of w between cell and i popotional to g w w. With thee aumption and elation in mind we can go to the next ection and etablih the equation fo mophogen diffuion... Mophogen diffuion If we conide diffuion only the equation that decibe the concentation of the mophogen in each cell ae given by Rev. Mex. Fi

3 v du v dw = µ u g u u ALAN TURING S CHEMICAL THEORY OF PHYLLOTAXIS = µ w g w w whee v i the volume of the th cell. Notice a typo on the ight ide of Eq. II.. in MTP; Γ hould be Γ m. Since = what we actually have ae ix equation of the fom v du = µ u g u u = µ u g u u + g u u and imila expeion fo u u w w and w. The ight hand ide of can be ewitten a µ u g + g u + g u + g u o in geneal we define g = g and wite the equation fo u and w a: v du = µ u g u v dw = µ w Now let u intoduce the new vaiable g w. u = v u and w = v w. 4 By eplacing them into we obtain du dw o in matix notation u w u w u w = = µ u = µ w g v v u g v v w 5 g v vv g vv g vv g g v vv g vv g vv g g v u w u w u w µu 0. 0 µ w o that G = β β β β β β β β β β β β β β β β β β. α α α By pefoming the matix poduct the entie of the eulting matix can be witten a g = α β β 6 v v and by uing Eintein ummation convention i we have g v v = α β β. In ode to expe the ytem 5 in tem of a diagonal matix we define the new vaiable u and w a follow u w u w that i u j u w u w = = = T β β β β β β β β β u w u w u β j and w j = u w u w u w w β j. 7 Notice that the old vaiable u and w can be ecoveed by uing T t the tanpoe of T : u = j u j β j w = j w j β j. 8 Uing the ummation convention all thee bac and foth tanfomation can be witten a u = u β w = w β 9 u = u j β j w = w j β j. 0 Since β j i an othonomal et the inne poduct atifie β β = β β = β β = δ The aay G = g / ] v v i called the diffuion matix and it contain all the patial infomation of the cell aembly. Notice that G i alway ymmetic independently of the way the cell ae aanged o it i poible to diagonalize it by mean of a change of coodinate T GT whee T conit of the eigenvecto of G. By Gam-Schmi poce we can find a et of othonomal eigenvecto o that T = T t. Let α α α be the eal eigenvalue of G and β j = β j β j β j it aociated othonomal eigenvecto whee δ i the Konece delta. By uing thee popetie we can wite the ytem 5 in tem of u j and w j. Fit du We alo now that = du du g β = µ u u β. v v vv g = α β β o we wite = µ u α β β u β. Rev. Mex. Fi

4 4 M.D. RUEDA-CONTRERAS AND J.L. ARAGÓN Fom 9 we have that β u = u o du = µ u α β u β = µ u α β β u. Finally fom β β = thu du = µ u α u. By applying the ame pocedue to w the ytem 5 i expeed in tem of u j and w j : du = µ u α u and dw = µ w α w which coepond to Equation II..7 in MTP. Thi change of vaiable will allow u to find a olution fo the full ytem of ix equation eaily a we will ee in the next Section. In matix fom the peviou equation ead: u u u w w w u w u w u w = α α α µu 0. 0 µ w Since G i ymmetic α R. Moeove thee eigenvalue ae poitive ii ince a negative value would mean that difeence in mophogen concentation inceae in time which ha no phyical meaning. We will ee that the olution of the mophogen equation depend entiely on the eigenvalue α and thee eigenvalue impoe ome condition on the mophogen diffuibility. In the next Section not only diffuion of the two mophogen between cell will be conideed but alo the chemical eaction between them. That i the full eactiondiffuion ytem fo plant mophogenei will be tudied... Mophogen inetic. The linea cae The following dicuion concen the detail that lead to olve II..8 in the cae of a linea chemical inetic f m. The olution fo thi cae i given by Eq. II..6 in MTP. Since only two mophogen ae conideed hee let fu w and gu w be the ate of change of u and w epectively. Then we wite the full eaction-diffuion RD ytem a follow v du v dw = µ u g u u + v f u w = µ w g w w + v g u w. 4 Uing the change of vaiable 4 and 7 the RD ytem can be witten a du dw = µ u α u + fu w and = µ w α w + gu w. In what follow we cay out a linea local analyi of aound an equilibium point u w that i f u w = 0 and g u w = 0 by following a tandad appoach ee fo intance ]. In abence of patial vaiation the ytem i Define u = du u u = fu w dw = gu w. 5 and w = w w 6 whee u and w ae mall. Thu nea the equilibium point the ytem become du whee = a u u + a w w a u = f u u w and dw a w = f w u w = b u u + b w w b u = g u u w and b w = g w u w. By incopoating the diffuion tem into the peviou equation we have the lineaized RD ytem : du dw = µ u α u + a u u + a w w 7 = µ w α w + b u u + b w w. 8 Thi ytem can be epaated into thee independent et of two coupled linea equation which can be olved by tandad method. Hee we have thee thee et: and du dw du dw du dw = α µ u + a u u + a w w = α µ w + b w w + b u u = α µ u + a u u + a w w = α µ w + b w w + b u u = α µ u + a u u + a w w = α µ w + b w w + b u u. Rev. Mex. Fi

5 ALAN TURING S CHEMICAL THEORY OF PHYLLOTAXIS 5 Fo an abitay numbe of cell n thee will be n et of uch equation. Thee et can be expeed in matix fom a du dw α µ = u + a u a w b u α µ w + b w u w =... n. 9 O even in a moe compact notation U = B U. 0 Note that ince the eigenvalue α α and α may be all ditinct it i neceay to ditinguih each matix B which define the ytem of equation fo u and w. A in the peviou Section it i poible to find a olution of 0 by mean of a change of coodinate R that diagonalize the matix B. Let p α and p α be the eigenvalue of B ; fom 9 we have that p and p atify p + α µ u a u p + α µ w b w = a w b u. The olution of 0 can be expeed in tem of p and p and the coeponding eigenvecto of B. If thee eigenvecto ae witten a S = S S and T = S S then the coodinate tanfomation R i R = which ha the invee S S S S R = S S q S S whee q = det R. Now we ae eady to find the olution of 0 which tun out to be: U t = c c R e p t 0 0 e p t R whee c = u 0 and c = w 0 ae initial condition. By intoducing the component of U in the lat equality and pefoming the matix poduct we have the olution u t = c S c S e pt S q + c S c S e p t S w t = c S c S e pt S q + c S c S e p t S. Thee olution coepond to Eq. II..0a II..0b and II.. in MPT page 9. whee The oot p and p ae explicitly p p = a u + b w α µ u + µ w ± α µ u + µ w a u + b w ] 4h α 4 h α = α µ u µ w α µ u b w + µ w a u + a u b w a w b u. It i at thi point that one of the main Tuing obevation aoe: he noticed that the tem of majo impotance in and ae thoe fo which Re p i geatet becaue they ae the one which gow fate thi i often called exponential dift; the oot p can be eithe eal o complex and thee ae many diffeent poibilitie fo the olution and but the only cae of inteet i when p i eal and maximum and α 0 i finite. Thi i decibed in Tuing CTM a the cae of tationay wave ]. A the oganim ae finite in numbe of cell and/o volume thee can only be a finite numbe of chaacteitic value α fo which Re p ha it geatet value See Appendix. In the next ubection we efine the olution and accoding to the oot p that ae of main inteet. We alo expe thee olution in tem of the oiginal vaiable u and w which ae the mophogen concentation in each cell... Tuing intability fo mophogen equation of phyllotaxi Concening phyllotaxi in page 9 of MTP Tuing tate it main aumption that can be ummaied a follow: a Thee i a homogeneou equilibium in the eaction ytem in abence of diffuion and mall deviation fom thi equilibium atify the condition fo tationay wave ]. b Deviation fom equilibium ae mall o that the influence of quadatic tem can be conideed a petubation. Nevethele thee deviation ae ufficiently lage fo the linea appoach to be inapplicable iii. c The ignificant wavelength α ae thoe fo which the eal pat of the oot p p i geatet. In the Appendix we obtain exactly the ignificant value α which ae the optimum wavelength. Accoding to thee aumption we now loo fo uitable olution fo the mophogen equation. Fit we hould etablih an algebaic elationhip between p p and α a tated in Eq. II.. and II.. in MPT page 9 and 95. Rev. Mex. Fi

6 6 M.D. RUEDA-CONTRERAS AND J.L. ARAGÓN An eigenvecto S = S S of B mut atify the equality p I B S = 0 0 and the ame fo p o we have p + α µ u a u S = a w S and p + α µ w b w S = b u S. If we pefom the ame calculation fo the econd eigenvecto T = S S all the elationhip between p p and α can be nown which tun out to be p + α µ u S = a u S + a w S p + α µ u S = a u S + a w S 5 p + α µ w S = b u S + b w S p + α µ w S = b u S + b w S. 6 In matix notation thee ae p + α µ u p S + α µ 0 u 0 S = S a u a S w. S S Define S ij ] = W α o the peviou expeion become p + α µ u p S + α µ 0 u 0 S and imilaly = a u a w W α 7 p + α µ w p S + α µ 0 w 0 S = b u b w W α. 8 The matix W α i non-ingula povided that p p o we wite the olution of 0 a u t w t whee = q X t X t W α 9 X t = c S c S e pt X t = c S c S e p t. Notice that at thi point we have olved 0 thu olution ae expeed in tem of the vaiable u and w. Since the poblem involve the vaiable u and w we hould e-wite the olution in tem of the oiginal vaiable. We defined u and w to be u and w except that they efe to diffeence fom the equilibium u and w See Eq. 6. Alo u and u wee obtained by mean of the othonomal et of eigenvecto β j of the diffuion matix G See Eq. 7. Theefoe u and u hould be obtained by mean of the tanpoe of the matix of eigenvecto a follow: u w u w u w = β β β β β β β β β u w u w u w + u u w w u w. Now Eq. 4 tate u = v u and w = v w o: u w v 0 0 u w = 0 v 0 u w 0 0 v + β β β β β β β β β u w u w. u w u w u w u w Then the elation between vaiable u and w and vaiable u and w can be witten a follow: u u = u β v w w = w β. 0 v Fom 9 we obtain the olution fo u and w : u t w t = X t X t W α. q Replacing in 0: u u = X t W α v q + X t W α β w w = X t W α v q + X t W α β which can be ewitten a u u = X l t W l α β v q l w w = X l t W l α β. v q l Rev. Mex. Fi

7 ALAN TURING S CHEMICAL THEORY OF PHYLLOTAXIS 7 Thi i then the olution of the RD ytem unde the linea appoximation and coepond to Equation II..4 in MTP. Note that thee olution depend on the function X l t which ae exponential and the numbe W ml that come fom the eigenvecto of the diffuion matix. Now fom aumption c above the only tem in and that hould be conideed ae thoe which aie fom the laget eal pat of p and p o the one containing thoe eigenvalue α that ae nea to zeo. Thoe α that yield the geatet Re p and Re p ae obtained in the Appendix by mean of a dipeion elation ]. Some ema about p and p hould be made. If p and p ae complex then we have the ocillatoy cae ] which i not of inteet fo the mophogenei phenomenon. Thu we aume that p and p ae eal an poitive. In ode to find olution 9 it i neceay that p p o fom 4: α µ u + µ w a u + b w ] 4hα > 0. A the quae oot of a poitive numbe i alway poitive we ee that p i alway lage than p ; thu we can dop all the tem that include X t and wite and imply a: u u = X t W α β v q w w = X t W α β v q whee we only chooe the tem fo which α i nea to zeo o the optimum See Appendix. We call thee tem X t 0 W 0 and X t W epectively and wite u u = v + v w w = v + v X 0 q X q X 0 q X q t W 0 α β t W α β t W 0 α β t W α β. Finally we can aume that in the two main ange of value of α nea zeo and the optimum the function W 0 and W ae contant o we wite u u = W 0 + W X 0 q t β v q v X t β = W 0 V + W U 4 w w = W 0 X 0 q t β v + W q v X t β 5 = W 0 V + W U. 6 Thu the olution of unde the linea appoximation depend entiely on the poible value of p which in tun depend on the value of α µ u and µ w and the value of the tability matix au a w. b u b w Some etiction fo the tability matix and the diffuion coefficient µ u µ w ae alo etablihed in the Appendix though the analyi of the condition fo Tuing intability..4. Mophogen inetic. The quadatic cae We now conide the cae when the eaction ate ae quadatic function of the mophogen concentation. The quadatic appoach i neceay fo two eaon. Fit the linea analyi i not ufficient fo patten fomation becaue it only detemine a table tate of the ytem. Second if ome eigenvalue α i zeo the linea appoximation i not applicable. Thu it become neceay to analye the quadatic cae. Recall that the full eaction-diffuion ytem i du dw = µ u v = µ w v g u + f u w g w + g u w. Auming that f and g ae quadatic the ytem can then be witten a du dw = µ u v g u + a u u u + a w w w + K u u + K u u w w + K u u = µ w v g w + b u u u + b w w w + L u u + L u u w w + L u u whee K ij L ij R. Thee coepond to Eq. II..7 in MPT page 96. We aim to expe thee equation in tem Rev. Mex. Fi

8 8 M.D. RUEDA-CONTRERAS AND J.L. ARAGÓN of the vaiable X l t. By uing 9 we can obtain thee vaiable in tem of the invee of W α a X t X t = q u t w t W α. Then X l can be witten in tem of W ml α a follow X t = q u W + w W 7 X t = q u W + w W. 8 Notice that the lat equalitie ae expeed in tem of the vaiable u w o it i neceay to obtain thee in tem of the oiginal vaiable u w. Since u w u w u w = v v we can ewite 7 a X t = q + X t = q + v β β β β β β β β β u u u u w w w w u u w w v u u β W α v w w β W α v u u β W α v w w β W α o uing the ummation convention we wite fo each l and each X l = q v β W l α u u + q v β W l α w w. 9 Now by calculating the time deivative of the lat equality we obtain Equation II..8 in MPT page 96: dx l = q v β W l α du + W l α dw ]. 40 Thi lat equation i witten fo each o thee i a double ummation ove and ove l. We now ubtitute du / and dw / in 40 in tem of the quadatic appoach intoduced above and uing we get u u = W 0 V + W U w w = W 0 V + W U. Thu du / dw / can be witten a du dw = µ u v g u + a u u u + a w w w + K W 0 V + W U + K W 0 V + W U + K W 0 V + W U W 0 V + W U = µ w v g w + b u u u + b w w w + L W 0 V + W U + L W 0 V + W U + L W 0 V + W U. W 0 V + W U We now will wite the lat equation in a moe manageable way by fit witing down the diffuion and linea pat in tem of X l and W ml. The diffuion tem fo du / ae µu v g u which can be witten uing the ummation convention a µu v g u fo each. Now fom 0 we ee that fo each u = v q X l W l α β + u w = v q X l W α β + w. 4 Alo fom 6 we have g = v v α β β. Thu the diffuion pat fo du / i µ u g u = µ u v v α β β v v ] X l W l α β + u v q = µ u α β β β X l W l α + ] v β u v q = µ u v α β q X l W l α + v β β u ]. Rev. Mex. Fi

9 ALAN TURING S CHEMICAL THEORY OF PHYLLOTAXIS 9 By evaluating dw / and noticing that β β = 0 we get µ u g u = µ ] u α β X l W l α 4 v v q µ w v g w = µ w v α β q X l W l α ]. 4 Analogouly by vitue of 4 we wite the linea tem fo du / and dw / a follow a u u u + a w w w = v q X l β a u W l α + a w W l α ] 44 b u u u + b w w w = v q X l β b u W l α + b w W l α ]. 45 Then the ubtitution of the diffuion and linea tem in 40 yield dx l =q v β µ ] u W l α α β X l W l α v q + q v β + a w W l α W l α X l β a u W l α v q ] + q v β µ w v W l α ] α β X l W l α + q v β W l α q ] X l β b u W l α + b w W l α. v q By implification and eaangement of tem we have dx l = X l µu α + a u W l α W l α + a w W l α W l α + X l µw α + b u W l α W l α + b w W l α W l α. 46 Fom 5 we have and fom 6 p = µ u α + a u W α W α + a w W α W l α p = µ w α + b u W α W α + b w W α W α. Then by ubtituting thee expeion in 46 and including the quadatic tem one get dx l = p X l + p X l + q v β W l α K W 0 V + W U + W l α K W 0 V + W W 0 V + W U U + W l α K W 0 V + W U + W l α L W 0 V + W U + W l α L W 0 V + W U W 0 V + W U ] +W l α L W 0 V + W U. By witing down thi lat equation in tem of X 0 l and X l one get Eq. II..9 in MPT page 96. Finally by expanding the quadatic expeion and gouping imila tem we can wite thi equation in a moe abbeviated way: dx l = p X l + p X l + q v β ] W l α F V + F V U + F U + q v β W l α ] F V + F V U + F U which coepond to II..0 in MPT page 96. Hee the value fo F j ae F = K W 0 F = K W 0 W + K + K W 0 W F = K W + K W 0 W 0 + K W 0 W 0 W + W W 0 + K W W + K W F j can be imilaly defined by eplacing K ml by L ml.. Rev. Mex. Fi

10 0 M.D. RUEDA-CONTRERAS AND J.L. ARAGÓN. Concluion In thi wo we peented Alan Tuing mathematical theoy fo plant patten fomation in a detailed and didactic way. The impotance of thi analyi i evident given the fundamental ole that Tuing publihed wo fo animal mophogenei ha played fo undetanding a numbe of mophogenetic phenomena ] ince it contitute a imple mechanim that can lead to patten fomation in living oganim 4]. The eult detailed hee unpublihed in Tuing time contitute a dicete fomulation fo plant mophogenei and can a well give inight and a deepe undetanding of biological phenomena in which the cell-cell inteaction ae of main impotance. We believe that the dicete theoy ha not deeved enough attention and it i the pupoe of thi pape to alleviate thi ituation. Thu we woed out Tuing theoy of phyllotaxi fo a imple cae in which thee ae only thee cell and two mophogen u and w. Thi pocedue allowed to claify and follow the whole calculation and mathematical manipulation needed to etablih the RD ytem in a olvable way and thu find it olution. Thi imple appoach i not howeve limiting becaue the analyi can be eaily extended fo an abitay numbe of cell and mophogen. Thi i clea fom equation 0 whoe deivation how how to epaate the whole n m ytem of equation into n et of m linea equation fo the cae of n cell and m ditinct mophogen. Tuing model fo phyllotaxi alo allow to etablih the equation fo any geometical configuation of cell by mean of the diffuion matix which tun out to be a vey imple and ingeniou idea. The diffuion matix mae up a vey ueful tool to exploe how the olution ae affected by the geomety of the domain. A ummay of the pocedue needed to apply the dicete model popoed by Tuing i a follow. Afte lineaiing and intoducing the appopiate vaiable the olution fo the mophogen equation ae given in Sec... Eq.. The behaviou of thee olution depend on the oot p which ae the eigenvalue of the full lineaied RD ytem 0. Thee eigenvalue depend on the value fo the wavelength α. By mean of the dipeion elation Eq. 5 in the Appendix one can chooe only thoe wavelength that will dive the olution to the fatet exponential gowth. Deivation of thi optimum wavelength give a numbe of etiction on the diffuion coefficient µ u and µ w Eq. 54 and 58 and the component of the tability matix a u b u a w b w 50. Thu the analyi we peent in the Appendix give all the condition fo Tuing intability fo a dicete ytem that wa not woed out by Tuing in MTP no a fa a we now by anyone ele. On the bai of the eult obtained by the linea appoach we finally give the mophogen equation fo the cae in which the eaction ate ae quadatic function of u and w Sec..4.. Though the linea cae it wa poible to et the quadatic ytem in an eay to olve way which depend entiely on the oot p. The Tuing eult peented hee might be vey ueful fo thoe inteeted in modelling patten fomation phenomena fom a dicete point of view. Acnowledgement We would lie to than Fautino Sánchez Gaduño and Octavio Miamonte both fom the National Autonomou Univeity of Mexico UNAM fo citical eading of the manucipt. Computational uppot fom Beatiz Millán and Alejando Gómez i alo gatefully acnowledged. Thi wo wa financially uppoted by CONACYT though gant 6744 and Appendix In thi ection we deive the neceay and ufficient condition fo Tuing intability of the RD ytem. That i we obtain condition fo the ytem to be table in abence of patial petubation but untable when diffuion i peent. The optimum wavelength α fo exponential dift will be detemined. In abence of patial vaiation the ytem i du = fu w dw = gu w. 47 By lineaiation about the equilibium u w 47 become du dw whee = au a w b u b w u w = A u w u = u u w = w w and A i the tability matix. The equilibium point u = w = 0 hould be table o the olution λ of deta λi = 0 hould have negative eal pat. By computing the deteminant we have o deta λi = λ λt A + det A 48 λ = t A ± t A 4 det A. 49 Thu Re λ < 0 equie du dw t A = a u + b w < 0 and det A = a u b w a w b u > Now conideing the full lineaied RD ytem α µ = u + a u a w b u α µ w + b w which can be witten a u w U = B U. 5 Rev. Mex. Fi

11 ALAN TURING S CHEMICAL THEORY OF PHYLLOTAXIS By taing into account the patial tem we now loo fo the condition necceay to dive the ytem to intability. Fom the deteminant det pi B = 0 we obtain the eigenvalue p α = p a function of the wavelength α a the oot of p p α µ u + µ w a u + b w ] + h α = 0 5 whee h α = α µ u µ w α µ u b w + µ w a u + det A. The olution p of 5 mut atify Re p α > 0 fo ome α 0. Thee ae p = α µ u + µ w a u + b w ] ± α µ u + µ w a u + b w ] 4h α. 5 Since ta < 0 we have fom 5 that Re p α > 0 can be achieved only if h α < 0 fo ome α 0. Since det A hould be poitive the only poibility fo h α to be negative i that µ u b w + µ w a u > Taing into account 50 we conclude that µ u µ w. Thee ae the neceay condition fo intability but they ae not ufficient; fo h α to be negative it minimum mut be negative too o by diffeentiation of h with epect to α we have h α = α µ u µ w µ u b w + µ w a u o the minimum of h i attained at α m = µ ub w + µ w a u µ u µ w 55 and it i equal to i h min = µ ub w + µ w a u 4µ u µ w + det A. 56 Since h min mut be negative the condition fo hα < 0 deta < µ ub w + µ w a u 4µ u µ w. 57 At the onet of intability bifucation h min = 0 o det A = µ u b w + µ w a u /4µ u µ w. Thu by defining µ = µ w /µ u we can obtain the citical diffuion coefficient µ uc µ wc and µ c = µ wc /µ uc a the appopiate oot of µa u + b w 4µ deta = µ a u + µ a u b w deta + b w. 58 The citical wavelength α c i then α c = µ u c b w + µ wc a u µ uc µ wc. 59 Then fo µ > µ c thee exit a ange α < α c < α fo which h α < 0. Hee α and α ae the two diffeent oot of hα povided that µ > µ c : α = µ u b w +µ w a u ± µ u b w +µ w a u 4µ u µ w deta. 60 µ u µ w Thu Re p α > 0 fo all α α α and thee exit α 0 in thi ame ange fo which the polynomial p p α µ u + µ w a u + b w ]+h α ha a maximum that i Re p α 0 i maximum. We call α 0 to the optimum wavelength. Solution ae then expeed only in tem of the optimum wavelength and the wavelength nea zeo. i. The ummation convention tate that the epetition of an index in a tem denote ummation with epect to that index ove it ange. Fo example the expeion a ix i = p mean a x + a x a nx n = p. ii. It could be that one of thee eigenvalue i zeo. If thi i the cae it i howeve poible to follow a quadatic appoach which i peented in Sec..4.. iii. Fo the patten to aie the ytem cannot tay in a tate of equilibium but it athe need to come upon Tuing intability.. A.M. Tuing Phil. Tan. R. Soc. London B A.M. Tuing in Collected Wo of A.M. Tuing: Mophogenei edited by P.T. Saunde Noth Holland Amtedam 99.. S. Kondo and T. Miua Science J. Swinton in Alan Tuing: Life and Legacy of a Geat Thine edited by C. Teuche Spinge N.Y. 005 pp I. Adle D. Baabe and R.V. Jean Ann. Bot F. Sánchez-Gaduño Micelánea Matemática C.F. Schimpe Magazin fü Phamacie L. Bavai and A. Bavai Annale de Science Natuelle Botanique F. Delpino Alti della R. Univeita di Genova J.N. Ridley Math. Bioci H. Aiy Poc. Roy. Soc. London W. Hofmeite in Handbuch de Phyiologichen Botani Rev. Mex. Fi

12 M.D. RUEDA-CONTRERAS AND J.L. ARAGÓN edited by W. Hofmeite A. de Bay Th. Imich and J. Sach Velag Von Wilhem Engelmann Lepzig 868 pp H. Meinha in Poitional contol in plant development edited by P.W. Balow and D.J. Ca Cambidge Univeity Pe Cambidge 984 pp F.J. Richad Phil. Tan. R. Soc. London B S. Douady and Y. Coude J. Theo. Biol D. Reinha et al. Natue H. Jönon M. Heile B. Shapio E. Meyeowitz and E. Mjolne P. Natl. Acad. Sci. USA R. Smith S. Guyomac h T. Mandel D. Reinha C. Kuhlemeie and P. Puiniewicz P. Natl. Acad. Sci. USA C. Lloyd and J. Chan Nat. Rev. Mol. Cell. Bio E. Feau M. Feau J. Kleine-Vehn A. Matiniee G. Mouille S. Venhette S. Vannete J. Runion and J. Fiml Cu. Biol R. Plaza F. Sánchez-Gaduño P. Padilla R. Baio and P.K. Maini J. Dyn. Diffe. Equ S. Wyatt and N. Capita Tend Cell. Biol J.D. Muay Mathematical Biology. II: Spatial Model and Biomedical Application thid edition Spinge New Yo A. Giee and H. Meinha Kybeneti Rev. Mex. Fi