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A Kinetic Model for Phototropic Responses of Oat Coleoptiles

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A Kinetic Model for Phototropic Responses of Oat Coleoptiles
  ZIMMERMAN   BRIGGS-PHOTOTROPICDOSAGE-RESPONSE CURVES Literature Cited 1. ASOMANING, E. J. A.   A. W. GALSTON. 1961. Comparative study of phototropic response   pig- ment content in oat   barley seedlings. Plant Physiol. 36: 453-464. 2. BLAAUW, A. H. 1909. Die Perzeption des Lichtes. Rec. Trav. Botan. Neerl. 5: 209-372. 3. BLAAUW-JANSEN, G. 1959. The influenceof red   far red light on growth   phototropism ofthe Avena seedling. Acta Botan. Neerl. 8: 1-39. 4. BRIGGS, W. R. 1960. Lightdosage   the photo- tropic responses of corn   oat coleoptiles. Plant Physiol. 35: 951-962. 5. BRIGGS, WV. R. 1963. Red light, auxin relation- ships,   the phototropic responses of corn   oat coleoptiles. Am. J. Botany (in press). 6. BUNSEN, R.   H. RoscoE. 1862. Photochemische Untersuchungen. Ann. Phys. Chem. 117: 529-562. 7. CURRY, G. M. 1957.Studies on the spectralsen-sitivityof phototropism. Ph.D. Dissertation. Harvard University, Cambridge, Mass. 8. CURRY, G. M.,K. V. THIMANN,   P. M. RAY. 1956. The basecurvatureresponse of Avena seedlings to the ultraviolet. Physiol. Plantarum 9: 429-440. 9. Du Buy,H. G.   E. NUERNBERGK. 1934. Photo-tropismus und Wachstum derPflanzen. II. Ergeb. Biol. 10: 207-322. 10. FR6SCHEL, P.1908. Untersuchungen iiberdie heliotropische Prasentationszeit. I. Sitzber, Math- naturwiss. Kl. Kais. Akad. Wiss. 107: 235-256. 11. THIMANN, K.V.   G. M. CURRY. 1960. Photo- tropism   phototaxis. In: Comparative Bio- chemistry I: 243-306. Academic Press, N.Y. 12. THIMANN, K.V.   G. M. CURRY. 1961. Photo- tropism. In: Lightand Life,WV. D. MIcElroy and B. Glass, eds. pp 646-670. Johns Hopkins Press, Md. 13. ZIMMERMAN, B. K.   W. R.BRIGGS. 1963. A kinetic model forthe phototropic responses of oat coleoptiles. Plant Physiol. 38:253-261. A Kinetic Model for Phototropic Responses of Oat Coleoptiles1 2 Burke K. Zimmerman3   Winslow R. Briggs4 BiophysicsLaboratory   Department of BiologicalSciences, Stanford University, Stanford, California The present paper is an attempt to account ac- curately for phototropic tip curvature in the Avena coleoptile as a function of the intensityofunilateral illumination and duration of exposure to monochro- matic light (4358 A). Detailed dosage-response curves on which this analysis is based are presented in the preceding paper (22). It haslongbeen as- sumed that the direction andmagnitude of the light gradient across the coleoptile tip determine respec- tively thedirection and magnitude ofthe phototropic response (14,16). Experimentsby Buder (9) con- firm that thedirection of the gradient does indeed deterimine thedirection of the phototropic response. 1 Received July 6, 1962. 2 This work was supported by a National Institutes of Health Predoctoral Fellowship to the senior author and by grants G-8688andG-21530 from the National Science Foundation and a grant from ResearchCorpora- tion to the junior author. 3 Biophysics Laboratory.Present address: Commit- tee on Biophysics, University of Chicago,5640 Ellis Avenue,Chicago 37, Illinois. 4Department of Biological Sciences. By using narrow light pipes and illuminating the interior of the tip as well as illuminating small por- tions of the outside of the tip with narrow beams of light, Buder found that thedirection of curvature was always in thedirection of the maximum light gradient, not in the direction of the incident radia- tion. Hisexperiments also produced gradients many times larger than normally occur when the entire tip is placed in a beam of light, but approximately the same amount of curvature was always observed. The role of the light gradient in determining the magnitude of the phototropic response has been questioned by Curry (12). He pointed out that the light gradient across the tip shouldbe a function of the wavelength of theincident light. Thus if the magnitude of curvature depends upon the magni- tude of the light gradient, it should also be a func- tion of wavelength, something not observed (17). His attempt to account for the shape of the dosage- response curves on thebasis of the light gradient was also unsuccessful. Detailed studies of tip curvaturebased on the assumption that both the direction and magnitude of the gradient are necessary for determining thephoto- 253  PLA-\N-T PHYSIOLOGY tropic response (21) did not successfully account for the (losage-response curves previously described (21, 22 ). The assumption that the magnitude of the gra(lient (leterlniines the magnitude of the phototropic response wvas thits rejected for the following reasons. The measured gradienit across the ti) was too small to account for the width of the log dosage-response curves. EIven assuming arbitrary gradients much lar-ger or imtuclh smiialler than that observed, the dosage-resp)olnse curves beyond peak first positive curvature could not beaccounted for. Twto extremes were considered, a uniforml (listribution of pigment over tlle surface of the coleoptile til. and a homogge- neous (listributioni of pigmiient thlouglhouit the volume of the tip, xxitlh theassumptiointhatthe response is proportional to the dlifference between the numiiber of pigmiient mlolecules excite(d on lighted and(I shaded sides. Several kinetic mechanismiis for pigment ac- tivation xvere assumedI with negative results in all cases. Thu.. it will be assumlled1 inthis paper that xvhile the gradient (letermiineis the (lirection of the response. it has no bearing- on tlle mnagnituide. The approach used in this paper is to (letermine x-hether- the (losage-responise curves previ-utsly de- scribedl fol- oatts   22) can be dlescrile(l 1v an algebraic sum of simiplemiiathematical fuiictions. The problenm is thein to findl a kinetic mlo(lel x-viich yields futictionsof the forlml obtained. AnIy finite, continuouts, single-valuied funiction canI be dlescribedl by a Fourier- series (a suIml of trigonometric functions) or 1y a poxver series (a suI of algdebraic functions) if one takes a sufficient numlhbel- of term-ls and a(djusts tlle constants appropri- atel]v. Ther-efore, any combilnation of funlctions N-hlich cani be fitted to the dosage-resp)oise curxves nmust be kept sim)ple. i.e., xvith as fexv termls and eimipirical conistants as possible. If suichl curve-fitting is successful. one imiust then obtain this comnbination of simple fulnctiolns as the soluitions of (lifferential equations arising from thekinetic mo(lel. There is goo(d evidence that tip curvatuire may be the result ot three independent or at least distinct mechanism.s, corresponding roughly to first positive, first negative, andl second positive curvature. Each of these medliates the lateral translocation of auxin (1. 7, 19, 20). The portion of the dosage-response curve involv ing first positive and( first negativecurvature )beys the BuInsen1-Roscoe recil)rocitv law of photoclhemiiical equivaleince (10), i.e., that it is a functioniof the total amount of radiation absorbed alone. The..e points are consideredl in cletail else- where  22 . Exposure of the plants to red light de- creases the sensitivity of first positive and first nega- tive curvature to blue light, therebyproducing a shift of the enitire (losage-response curves to higher dosages (log einsteiins cm 2 at 4358 A). Secon(d positive curvature, oni theother hand, seems to be a function only of the length of exposuire. Thus, at low in-tensities the curvature that one observes in certain regions is the sum of the dosage-respoinse curve for first positive and first negative curvature plus that for secoind positive curvature. Red light imakes thislattersysteimi more sensitive to phototropicinduc- tion, shifting its dosage-response curve toward lower dosages (22). Fromii the above, it is clearthat at least secoindl positive curxature is quite distinct fromii the othermechanismi,s. In furtlher support ofthis contentioni. auxin trainslocationl experinments (7, r,. Gillespie. K. V. ThliiaiI I)e rsoaIlal comimiiluinicationI) have shoxvn that the capacity for lateral transport of atuxini ex- tends at least3 limi basally fronm the tip for liglht dosages eliciting second positixve curvature. It oc- curs only in the top fe i ntildre(d miicrons for first positive curvature. Stuclies oIn the course of developmnent of the vari- Ouis tv pes of curvature and theirfinal appearaince also suggest the presenice of tlhree separate mlechani- isImls. This exidence is presentedl and (liscussed else- xvhere ( 22 ). Developmiieint of a Kiinetic AMolel for Phototr-op- ismii. On the basis of the foregoing exvidence, it shall be asstlume(d that three (listiict systemis contribute to the observed (losage-response curves, anid the nota tion, systemiis I. II, ailcl III, slhall replace the sonme- tilmies confusinlg terminology of first positive, first negative, anid second positive curv atuire. The ascendiing portion of the (losage-respoinse curxes obtai ned xvith and( ithoutred liglht treatmiient, resemiiblesthe curve one would expect if pigmllentactivation obeys first order kinetics, i.e., represenlts the simiiple conxversion of one molecular species to anotlher ith tlle rate proportional to the conicetrti-a- tion of the first. Suclh a curve is expresse(d b y aIn equation of the formii R K(1 - e-kIt) xvhere R is phototropic r espolnse (or concentr-ation of active species), K aid k empirical constants, I the intensity of blue radiation,  Iand t the (luraItionl of the exposure. This is the nmost likely process as it is tlle simplest mechainismi by xvhich light can convert an inactix e molecular species to an active forml. It wvould tlhein appear reasonable to assumtiie that the entil-e dosage-resp)onse curve for systenms I and II could be (lescribedl by a combiinatioin of similar functionis correspoin(liing to the fornmation of active formsand to the subsequent inactivation of these activespecies. Such a sequence of events hlas been propose(l earlier oIn strictly qualitative groutn(ls (6). Fiunctioins of tli s forml \vere successfullxy fitted to the existing dosage-iresponse curves (froml 2 2 for svsteuis I and (fig 1, 2, lowermllost curves) with thie folloxillg result: R K, { ( 1 _e -klIt)  1 e-kIt)] RK K ](1 kIt (1 ek t ] } I This equation is entirely eiiipirical, yet corresponds exactly to the forml one would expect for a positive curvature mleclhanismii consisting of fornmation of an active species folloxved by inactivatioin at higher doses, anid( a separate but similar negative curvature svsteni.  I lTer-e are only four enmpirical conistaIlts, sinlceK1 iS   normallization constanlt an(l of the 254  ZIMMERMAN   BRIGGS-KINETIC MODEL FOR PHOTOTROPIC RESPONSES LLJ Un z 0 0- Uf) Lii   0r LI 0 0 0 I a- LOG (Ixt),EINSTEINS CM-2 at 4358 A 1.6 Lu;_ I.TA xIu I=1. +XIU- U) zO  2I 880 ,, 0 U) a_ l 0 4 r 0 a-.8 ~ LO  It,INTIS M2at45 olnt gie ordlgtteatet xprmna H H   I 1o.4 x 10 00 <~ 0 ------------------------------- Plus Red Light -.4 -13 -12 -ii -10 -9 -8 LOG (Ixt),EINSTEINS CM2 at 4358 IG. 1 (top). Theoretical phototropic dosage-response curves at three intensities of monochromatic blue light. Plantts given nso red light treatment. Experimental pointsforthese intensities (22) shown for comparison. intensities (I) in einsteins CMn 2 seca 1 at 4358A. FIG. 2 (bottohe). Theoretical phototropic dosage- respoisecurves at three intensities of monochromatic blue light. Plants given 2 hr red light. Experimental points (22) shown for comparison. Lowermost curve, systems I and II alone. Intensities (I) in einsteins CM2 sec1 at 4358 A. snall k s is arbitrary. The reason onek is arbitrary is that-the absolute position of the dosage-response curve depends on the k s which are in part determined by theeffectiveness of the light interacting with the pigment. While this effectiveness might be of con- siderable interest, it has no bearing -on the kinetics. Hence, one of the k s may be adjusted until the positions of the experimental and theoretical dosage- response curves along the abcissa coincide, theothersthen being determiined in relation to it. One may now writereasonable kinetic schemes for systems I and II. System I: hv lhv x y ->z, k1I kI hvhv System II: x > y z, k3I k41 where y is a phototropically active pigmentform for lateral transport of auxin away from thelighted side and   is an active form for translocation in the op- positedirection. Inactive forms of y and y are denoted by z and z . The unexcited pigments are x and x . Thus the total response for systems I and II would be R = y - y . Reverse reactions, recovery following inactiva- tion, and (legradation and synthesis of new pigment molecules have been assumed to be unimportant over the period of the experimentand thus neglected, al- though all miglht w-ell occur. Briggs (6) exposed corn coleoptiles to a high dosage of unilaterallight which producedno curvature (presumably because the pigments for systems I and II had beenconverted entirely toz and z respectively, and exposure times were too short for induction of significant second positive curvature). He followed this exposure at various time intervals with a second exposurewhich,by itself, normally resulted in maximum first positive curvature. If the second exposure followed the first immediately, no curvature resulted. But, as the de- lay between exposures was increased, more and more curvature occurred, with 20minutes required to obtain maximum first positive curvature. Curry (12)observed the same phenomenon for oatcoleop- tiles. These observations stronglysuggest thepres- ence of either a recovery mechanism or de novo synthesis of pigment. In either case, the overall rate constant is small enough to be neglectedwithout in- troducing significant errors. Hence, systems I and II canbothbe treated as closedorconservativesystems. i.e., x + y + z = constant. Forsystem I, one obtains the following set of simultanieous differential equations: dx --=- klIx, (it = k1Ix - k2Iy, and dIt dz   = k..Iy. dt II III IV Solutions are readily obtained by standard procedures (17). * (t) = x(0) [k1/(k,-k,)] (e-k2It - e-k,It) = x(O) [kl/(kl-k2)] [(1 - e-klIt) - (1 - e-k,It)] V where y(t) is the concentration of y at time t after the light is turned on, and x(0) is theconcentration of x at t = 0. Since the phototropic response is as- sumed to beproportional to the concentration of y, the aboveequation is precisely the form dlesired from 2 55  PLANT PHYSIOLOGY theempirical curve (I). A similar set of e(quations is obtained for v (t). Vith a curvatureof 300 nornlalizedl to 1.0, miieasuredin einsteins cm-2 sec- , an(l t in seconds,the following values of the constants providle the best fit of y an(d   to the experimental losage-response curves. No re(d light treatmiient: x(0) = 0.92, k, 4.47 X 1011, k., 2.40 X 109, x'(0) = 0.82, k,= 3.73 x 109, k4 1.18 X 10 . Two hours redl light precediilg blue exposure: x(0) = 0.92, k, 1.12 X 1011, k., 6.0 x 108. x'(0) 0.82, k3 = 9.33 x 108, k4 2.95 x 108. The third phototropic system will inow be con- si(leredl. If one subtractsthe theoretical curves for svstemis I and( II alone (i.e., lowermost curve, fig 2) fromii an experimlental curve in wlhiclh system III oc- cuirs, onie obtains for system III siplyI a linear func- tion of timle, i.e., R Kt. This dlependence becomes app)arent when the subtracte(d curves for several in- tenisities are plotte(d asa funiction of t (fig 3). The value of K may vary slightly with intensity. but the dlata are n,Ot sufficiently goo(d to w arrant this con- eluision.If the plants are given re(d light before ex- posture to blue, K is approximately (louble(l. It is implortant to note that since tlis system can result in very large curvatures to thepointwhere factors other than l)hototropic in(luctioni of tlle translocation syvstem are limiting, the formii Kt may., in fact. be the 1.6 z 0 a- PLUS RED LIGHT 1.2 Kz8.27xI04 SEC |   .8 0~   0~~~~~ 0 MINUS RED LIGHT z 0 K-2.95xlO 4 SEC- z 500 1000 1500 EXPOSURE TIME, SECONDS FIG. 3. Net phototropic response obtained wx-hen curvature predicted for systems I and II is subtracted from total experimental curvature (from 22) and plotted as a function of exposure time.Liniear dependence of system III on length of exposure is now apparent. In- tensities (I) in einisteins cm -2 sec- 1 at 4358A. Open circles: I = 1.4 X 10-12,2hrred light. Solid circles: I = 1.4 X 10- 1, 2 hrred light. Open squares: I = 1.4 X 10- 12, Ino red light. Solid squares: I = 1.4 X 10-11, no red light. K = u(0) Laa,a/(a, + a-1)] liniting case of A(1 - e- at), vlicll for small (at becomes Aat as readilydetermiinied by a j)over serlies expansion. It could also be an apl)roximation of a variety of other functions. Howvever, tlle exl)olen- tial formii above can be (lerive(l fromii a kineticallv reasonable milodel wvhich cani accounit for thle red liglht- in(luce(l sshifts of all three systems. CoInsider now the followinIg kinetic miiodlel: a1i a., u t---  v -- NV, a-1I where wv is a phototropically active componenit aln(d u and v areprecursors. Again, recovery, synthesis, an(l (legradation are neglectedl. The light-induce(d conversion of u to v is considered strongly reversible, whereas wN, is formiied byv means of a (lark reaction (le- pending upon the concentration of v. This miiodel leadls (lirectly tothe follow-ilng set of (lifferential eqtua- tiolls: (III = a-,Iv - a1Ju, (It = allu - (a-T1 + av. (It  1 (It a ., . VI VII VTIII Solutions are again readily obtainedl by stan(lar(l pro- ce(lures. Since both v and wN are of interes-t. they are g,iven below v(t) =u 0 l   )  et tut I iX antIl w-( t) = u   0)t   ) A ,I, a n I N t u I I l e A- t.   l ,\t w-here A = - 32 (a,I + a-,I + a., 4-  12 r  I + a-1I + a, )2 - 4a,a.1J] ad( A. -  2 (a, I + a.-,I + .- ) - Y2 [(alI + a-1I + a. )2 _ 4a,ai1] Nowv let us assumle that a..<<aI. a-,I for all experimentallymeaningful values of I. Qualitative- ly it is now clear that an equilibriumi will be estab- lished betwreen u(t) and v(t), the ratioof u to v remlainiing relatively constant while v is converted exponentially to w. This becomes apparent in the followinIg approximate solutionl for v for small t, imiiposing the above conditioni. 256  ZIMMERMAN   BRIGGS-KINETIC MODEL FOR PHOTOTROPIC RESPONSES v(t) = u(O)[al/(al + a-,)] [1 - e- (a, + a- )It] xi Thus when t becomes sufficiently large, v(t) ap- proaches aconstant which is then slowly diminished according to the size of the a s, but governed by a.,. A further assumption must now be made and that is that the equilibrium concentration between u and v is far in the direction of tu. Ifnot, even a relatively short flashof light (but long enough to establish equilibrium) would convert a substantial part of u to v which would thenbeconverted spon- taneously to w even after thelight is turned off. Thus, relatively large curvatures would be obtained over a variety of light conditions and the empirical linear time dependence of system III curvature would not be observed. This assumption is also necessary to account for the flashing light experiments to be disctussed later. Thus. it is specified that the amount of v remaining immediately after the light is turned off ( w hich is subsequently converted to v) is sufficiently small to contribute little to phototropic curvature. Under these conditions, the amount of w formedduring an experiment can be represented by u (t) where t is the duration of time the light is left on. The approximate solution of w(t) after equilibrium between uand v has been established [t > 1 l/(al + a_1)I] is as follows: w(t) = u(O) [1 - e -a1a.t/(aj + a1) u [a,aC,/(al + a-,)]t XII Thus w(t), the quantity observed as phototropic response, has the behavior of a singleexponential term and is entirely independent of intensity, as re- quired by the empirical curves. If u(O) is more than three or fourtimes x(0), then only small values of [caia./(al + a-1)]t can be important, and thus a linear time-dependence will be observed. The constant u(0) (ala2)/(a1l + aCY) may be readily foundfrom the experimental curves for sys- tem III (fig 3). The portion of the dosage-response curves resulting from system III may thus be pre- dicted at any intensity. As a further experimental test of the model pro- posed for system III, it is a direct consequence of equations IX through XII that flashing light would enhance the response for a given dose and intensity over the responseobtained from continuous exposure. Since dw/dt is equal to a9v(t). then w(t) is equal to the a.) f0tv(t)dt. It is clear that if the light is left on long enough for v to reach equilibrium con- centration and thenturned off. v will notrevert to u butcan only disappear by spontaneous conversion to w via the dark reaction governed by a.,. While this also occurs with thelight on, it is not necessary to leavethelight on for the concentration ofv (t) to remain close to equilibrium during the course of an experiment. Hence, the integral of v, and thus v, vouldbe much larger for agiven dosage than if the light were applied continuously. The condition im- posed above, that equilibrium betweenuandv be far in the direction of u must hold in order for the enhancement to be substantial. Otherwise one flash wouldl be sufficientto convert most of u to v, and essentially the same response would be obtained over a variety of conditions. Graphically the situation is illustrated in figure 4. The total area under thecurve is proportional to the phototropic response. In principle, one shouldbe able to evaluatethe a  s from flashing light experiments based on equa- tions IX andl X. In practice, many difficulties areencountered. It may be possible, however, to deter- mine the constant a, without elaborate experiments. Suppose thatthe length and number of light flashes are fixed,that the length ofa single light flash is sufficiently long for equilibrium to be reached, and that total illumination is high enough for systems I and II to be inactivated. If the length of the dark period is nov varied, one would expect enhance- ment of curvature which eventually levels off as the clark period is lengthened. The response one would expect wvhile the light is on vill benearly equal whether or not the light is flashed provided that the length of the experiment is sufficiently short so that the equilibriumconcentration of v will not be seri- ously altered. Howvever, it must be long enough for an accurately measurable amount of curvature to be obtained, say 5°. The magnitude of enhance- ment is now given as follows: E = u(0)n   a _) f T e a2,t a, + a- , u(0)n  a a,l -f-a-1 ) (1 - e -a,) (It = XIIIwhere E is the magnitude of enhancement, u (0) the initial concentration of u (t), n the number of flashes, and T the duration of the dark period. While this is only an approximation, it shouldgive an enhancement curve similar to that in figure 5. Let Emax be the maximum possible amount of en- hancement obtainable with a particular number and length of flashes. Thus. a., and u(O) a1 (a, + a-1) may be determined both from Emax and the time constant by xvhiclh the curve rises.It is clear that a plotof log(E - Emax) versus t will give a., as the slope of a straight line. It shouldbe borne in mind that n, the number offlashes, must be kept small. If n is too large. one may still obtain a curve similar to that in figure 5, but looking more like a straight line thanan ex- ponential. This is because so much curvature is ob- tained that factors other than the concentration of w become limiting long before the dark period may be made long enough for the concentration of v to ap- proach zero after each flash. Thus, in order tore- duce the number of flashes and still remain beyond the range of systems I and II, one may increase 257
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