A biomechanical model for the study of plant morphogenesis: Coleocheate orbicularis, a 2D study species.

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1 5 th Plant Biomechanics Confeence Stockholm, August 28 Septembe A biomechanical model fo the study of plant mophogenesis: Coleocheate obiculais, a 2D study species. Lionel Dupuy1, Jonathan P. Mackenzie1 and Jim P. Haseloff 1 1 Plant Sciences depatment, Univesity of Cambidge, Downing Steet, Cambidge, UK Abstact Patten fomation in the mophogenesis in plants is a phenomenon stongly detemined by the genetic pogams of paticula species. Howeve, because ogans ae goups compised of specialized cells that ae adheent and not capable of migation, physical pocesses (cell wall expansion, diffusion) ae though to be majo aspect influencing the plant development. In the pesent study, Coleocheate obiculais, a pobable ancesto of land plant species has been used as a model species to study the mechanics of cell expansion in multicellula colonies. Image analysis methods have been developed in ode to analyze the kinematics of cell expansion and division, and a simple analytical model wee used to estimate both viscosity and cell wall division popeties. Numeical computation using the finite element method has been used to simulate the gowth of entie colonies using paametes deived fom live imaging. Compaison with live expeiments showed stong similaities between measued and simulated colonies. Also, paamete analysis showed that not only wall mechanical popeties, but also division ate and oientation ae factos stongly influencing patten fomation. Intoduction In gowing plant meistems, new ogans emege at pecise location, time and though well defined mechanisms. The detemination of models fo such behavious is, howeve, paticulaly complex due to the lage numbe of factos, eithe physical, chemical o biological, taking pat in the egulation of cell expansion though space and time. Recent advances in molecula biology have geatly contibuted to the undestanding of the egulation of gene expession in vaious plant ogans, fo example floal, apical and oot meistems, and the constuction of ealistic pedicting models of genetic egulatoy netwoks is now possible [1]. Howeve, the emegence of most complex shapes cannot be fully undestood without consideing: 1) the spatial stuctue of the poblem which detemines the natue of cell to cell signalling netwok and 2) the physical diving foces that impose expansion and defomations to that stuctue with time. In this context, simulation tools and methods ae essential to exploe and impove ou undestanding of the mechanisms of development in plants. We have been developing dynamic computational methods of plant mophogenesis at cellula level. The segmentation of live micoscopy data was used to estimate paametes of a simple viscous model. Mophogenetic, tanspot and biomechanical model of cell wall expansion have been developed in ode to analyses coupled mechanisms in mophogenesis. Most impotantly, a softwae platfom was developed to bing a flexible envionment fo the development and the

2 5 th Plant Biomechanics Confeence Stockholm, August 28 Septembe coupling of models (genetic egulation, spatio-mechanical factos, and signal tansduction) at diffeent levels of inteests of the plant stuctue (e.g Plant - Cell - wall). Live imaging and segmentation of gowing Coleocheate Coleochaete: model system Coleocheate is a small goup of fesh wate micoscopic algae of about ten species. The development of Coleocheate species shows inteesting mophological divesity, fom the geneation of iegula banched segmented filaments in C. divegens o C. nitellaum to symmetic dichotomous banching thalli in C. soluta o almost pefect symmetic centifugal expansion in C. obiculais and C. scutata. (Fig. 1). Among all Coleocheate species, C. obiculais appeaed to be the most simple of all, as its expansion emains cicula and is well adapted to a fist geneation of models. a) b) Fig. 1: a) the development of Coleocheate obiculais, when unconstained geneate 2D cicula tissues. Othe mophologies may appea when contact between seveal colonies occus o in the case of injuies b) (e.g. cell ablation). Image segmentation A cellula achitectue can be descibed as a stongly hieachical gaph object constituted of diffeent types of sub units e.g. a cell wall is constituted of two vetices (point at the cossing of thee o moe cells in a 2D system), a cell is constituted of walls, a tissue is constituted of cells etc Each of this sub-units bea specific local infomation e.g. vetices ae associated with spatial coodinates, walls ae associated with mechanical popeties and cell objects ae associated with genetic infomation (Fig. 2). Image segmentation is used to extact infomation about this cellula achitectue of live plant tissues as well as thei kinematics and can be used as template fo numeical simulations. The segmentation of images is based on a maked wateshed algoithm [2]. In maked wateshed computation, the gey-scale image is seen as a topogaphic suface. The algoithm is initiated by a finite set of points (called seeds) on the images. This topogaphy is then flooded fom below by letting each seed be a souce of wate, which ceates pogessively lage basins of pixels having identical pixel label. A pixel in contact with a paticula basin with intensity lowes than the basin is then associated to it. When the simulation eaches the maximum pixel intensity, each pixel of the image is associated to at least one initial seed label. A futhe pocessing is needed to econstitute the whole achitectue of cells. The complete achitectue of the cellula stuctue (Fig. 2) is then econstucted based only on pixels in contact with at least thee basins, in a scale inceasing ode: a wall is defined by two

3 5 th Plant Biomechanics Confeence Stockholm, August 28 Septembe vetices shaing the same two basins. A cell is constituted of all the walls having the same basin in common. a) b) Fig. 2: a) Plant cellula achitectue has a stongly hieachical stuctue which entities ae defined at specific spatial scales (vetex-wall-cell-tissue-ogan ). b) The cellula stuctue of Coleocheate using wateshed segmentation and econstucted with the algoithms we have developed. Mechanics of cell wall expansion Fom a mechanical point of view, the cell can be seen as a closed thin walled stuctue, maintained in tension by tugo pessue (Fig. 3). The mechanism of cell expansion is then geneally believed to esult fom a diving foce, tugo, acting on a yielding cell wall mateial, whee both of these factos ae ultimately contolled by the plant metabolic activity [3]. Moe ecent papes have poposed vaious othe yielding behaviou ([4], [5], [6]), but in the absence of stong expeimental evidence, we chose to assume the simple viscous hypothesis to deive an analytical fomulation of cell expansion (Fig. 3). Kinematics: as it is obseved expeimentally that adial expansion and division concen exclusively the fist laye of cells at the peiphey of the colony [7]. Cell size (adial l and tangential l t ) in this laye is assumed emain constant. Finally, cell expansion in assumed to be linea in the laye of cell at the edge, and null fo inteio cell. Theefoe, velocity field can easily be expessed as a function of the ate of adial expansion R/dt and the adial position R + l v = e ; [ R l, R] l (1) v = 0; < R l The stain ates on [ R l, R] in the cylindical coodinate ae ε = R / R and ε t = R / l. Tensile foces in cell walls: Stain in cell walls in theefoe constant and depend only on oientation. Theefoe, assuming a pefect viscous mateial popety, it is possible to elate expansion ate of the colony with tensile foces in cell walls: t = µ. St. R / R tt = µ. S. R / l (2) whee t t is the tensile foce in wall, is the viscosity coefficient (Pa.s) of the wall, S t and S ae coss sections of cell wall in tangential and adial diection. Wall coss sections wee

4 5 th Plant Biomechanics Confeence Stockholm, August 28 Septembe assumed to emain constant thoughout defomation i.e. no explicit cell wall synthesis is consideed in the model. Expansion of the colony: Fom the equilibium of foces acting on cell wall, it is possible to deive an expession of adial gowth of the system: ( P P ) hrl l R 0 t = (3) ( St. llt / R + S R)µ When the adius becomes lage elative to the dimensions of the cell, the expession of educes to ( P P0 ) h. llt /( S R ) and can easily be used fo the detemination of the viscosity coefficient fom expeimental data. a) b) Fig. 3: a) Biomechanical model fo cell expansion in mophogenesis: Cell wall esponse to tugo pessue though a viscous yielding of cell wall. b) the analytical model consist of a pefectly cicula colony of adius R, with a constant adial l and tangential l t cell length. The velocity field v is centifuge and linea fo bounday cells. Foces in the stuctue consists of pessue inside (P) and outside (P 0 ) the cell, tensile foces in walls in the adial (t ), and tangential diection (t t ). Multicellula computations Softwae fo the simulation of plant cell polifeation has been developed on the basis of pevious wok. [8]. Cells consist of individual automatons, chaacteized by a genetic activation state, whose behaviou is detemined as a function of the signals it eceives fom its neighbouhood. Computation of cellula wall expansion was pefomed using the finite element method. Because the stuctues being analysed ae two dimensional, Eule Benoulli beam hypothesis has been adopted. Time integation was caied accoding to a fowad-eule finite diffeence scheme and the scaled gadient conjugate method is used fo solving the system of linea equations. The finite element mesh used fo each wall consists of the two vetices defining its bounday and a mid point between them. Inteio cells wee consideed to be igid. The extenal foce at the oigin of expansion is tugo pessue, applied at each node of the finite element mesh. The division of a cell occus along one of two pincipal diections, paallel and pependicula to the edge of the colony. A cell divides when its volume eaches a theshold, in the plane of minimal distance between existing walls. Theshold values fo cell division whee based on fixed cell size dimensions. Duing the simulation, the volume thesholds ae dawn andomly using a nomal distibution detemined fom expeimental cell size distibution.

5 5 th Plant Biomechanics Confeence Stockholm, August 28 Septembe Results and discussion The evolution of the size of the colonies, as measued fom live imaging expeiments, showed a linea incease of the adius with time (e.g numbe of cell has a quadatic gowth). As expansion takes place at the peiphey of the colony, the incease in adius of the colony is detemined by the individual cells in the adial diection. The pedicted evolution of adius with time fom the analytical models and the finite elements simulations was compaed to expeimental data (Fig. 4) and epoduced simila linea adial expansion ate. Investigations into the key mechanical behaviou occuing in mophogenesis have been caied out by simulating some stongly contasting scenaios (Fig. 4), emphasizing potential mechanisms tiggeed by genetic activation of cells. Bending defomations wee shown to be majo mechanism influencing the way stuctues ae established, as it has been emphasized by [9]. When moment of Inetia is deceased, the symmety of expansion disappeas because foces in adial walls tended to be homogenized and thei expansion to vay with cell size. Fo the lowest values, howeve, bending of outside walls became the main facto diving the establishment of the stuctue. Each outside wall gives bith to an outgowth, stating a new diection of expansion. In the simple analytical model deived in the pevious section, tugo pessue, cell size and viscosity coefficient wee found to influence expansion ate identically. Howeve, when such paametes ae modified locally as it is found in eal oganisms, diffeent types of mechanisms could be obseved: locally inceased tugo pessue was most efficient to geneate outgowth and initiated a second colony. The eason is that only an incease in tugo pessue can geneate lateal foces at the ealy stage of the simulation. Lateal foces induce highe anticlinal division ate. An inceased numbe of cells had highe tugo pessue which then acceleated the fomation of the outgowth. In all othe cases, vetical foces on both sides of the wall ae in equilibium at the beginning of the simulation, until the outside walls ae defomed enough to geneate lateal foces. Fo the same eason, cell ablation geneates seconday cicula outgouth as it is found on live injued tissues: The ablation of cell induced a decease of tugo pessue in the neighbouhood of cells being isolated. It esulted in local isotopic expansion as opposed to dominant adial expansion in matue Coleocheate. I) II) Fig. 4: I) Gowth cuve of the adius of the colonies as obseved expeimentally (+), as pedicted by the analytical model (--) and simulated ( ) and the 95% confidence inteval in shaded gey. II) Cell polifeation pattens in vaious conditions: a-c) cicula gowth with vaious wall bending popeties ( moment of inetia of m m m 4 ). d-f) outgowth esulting fom local vaiation in tugo pessue (d), viscosity (e) and cell size (f). g-h) outgowths esulting fom the foces eleased fom ablated cells. i) simulations of colonies getting in contact

6 5 th Plant Biomechanics Confeence Stockholm, August 28 Septembe Conclusion In this study, we have been developing a mechanical computational famewok fo the simulation of mophogenesis in plant at the cellula level. A fist set of simulations showed a ange of possible foces and coesponding mechanisms can dive a biological system to establish its stuctue. Using models and compute simulations, it will be possible in the futue to complement expeimental investigations by isolating cetain factos o visualising hypotheses to exclude incompatible biological assumptions. Integating self-oganised models fo the egulation cell polifeation is now essential to go futhe in the undestanding of the mechanisms, and paticulaly how the genetic pogams inteact with the key physical vaiables esponsible fo emegence of geometical pattens. Coleocheate obiculais was used in this study as a fist expeimental system. The colonies adhee to a plastic o a glass substate and gow as a cell monolaye. This simplistic stuctue and kinematics of gowth makes C. obiculais a unique and compelling new system fo the study of the biomechanics of mophogenesis: High thoughput time lapse micoscopy can be implemented using compute contolled micoscope with a obotic stage, automatic segmentation techniques can be used to extact and analyse kinematics of gowth, simple analytical models can be developed and used to compaed with numeical simulations. Acknowledgements This eseach was suppoted by the Biotechnology and Biological Sciences Reseach Council (BBSRC) eseach gant (BBS/B/16720). Refeences 1. Espinosa-Soto, C., P. Padilla-Longoia, and E. Alvaez-Buylla, R., A gene Regulatoy Netwok Model fo Cell-Fate Detemination duing Aabidopsis Flowe Development That is Robust and Recoves Expeimental Gene expession Pofiles. The Plant Cell, (p Meye, F. and S. Beuche, Mophological Segmentation. Jounal of Visual Communication and Image Pesentation, (1): p Geen, P., B., Gowth Physics in Nitella: a Method fo Continuous in Vivo Analysis of Extensibility Based on a Mico-manomete Technique fo Tugo Pessue. Plant Physiology, (8): p Geen, P., B., R.O. Eickson, and J. Buggy, Metabolic and Physical Contol of Cell Elongation Rate. Plant Physiology, (p Doington, K., L. The theoy of viscoelasticity in biomateials. in The Mechanical popeties of Biological Mateials. 1980: The Symposia of the Society fo Expeimental Biology. 6. Wei, C. and P. Lintilhac, M., Loss of stability - a new model fo stess elaxation in plant cell walls. Jounal of Theoetical Biology, (p Machant, H., J. and J. Pickett-Heaps, D., Mitosis and Cytokinesis in Coleocheate Scutata. Jounal of Physiology, Rudge, T. and J. Haseloff. A computational model of cellula mophogenesis in plants. in Lectue Notes in Compute Science: Advances in Atificial Life Coen, E., A.-G. Rolland-Lagan, M. Matthews, A. Bangham, J., and P. Pusinkiewicz, The genetics of geomety. Poceedings of the National Academy of Sciences of the United States of Ameica, (14): p