Model For Calcium Dependent Oscillatory Growth In Pollen

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ARTICLE IN PRESSJournal of Theoretical Biology 253 (2008) 363– 374Contents lists available at ScienceDirectJournal of Theoretical Biologyjournal homepage: www.elsevier.com/locate/yjtbiModel for calcium dependent oscillatory growth in pollen tubesJens H. Kroeger a, , Anja Geitmann b, Martin Grant aabErnest Rutherford Physics Building, McGill University, 3600 rue University, Montréal, Québec, Canada H3A 2T8Institut de Recherche en Biologie Végétale, Département de sciences biologiques, Université de Montréal, Montréal, Québec, Canada H1X 2B2a r t i c l e in foabstractArticle history:Received 12 December 2007Received in revised form25 February 2008Accepted 27 February 2008Available online 18 March 2008Experiments have shown that pollen tubes grow in an oscillatory mode, the mechanism of which ispoorly understood. We propose a theoretical growth model of pollen tubes exhibiting such oscillatorybehaviour. The pollen tube and the surrounding medium are represented by two immiscible fluidsseparated by an interface. The physical variables are pressure, surface tension, density and viscosity,which depend on relevant biological quantities, namely calcium concentration and thickness of the cellwall. The essential features generally believed to control oscillating growth are included in the model,namely a turgor pressure, a viscous cell wall which yields under pressure, stretch-activated calciumchannels which transport calcium ions into the cytoplasm and an exocytosis rate dependent on thecytosolic calcium concentration in the apex of the cell. We find that a calcium dependent vesiclerecycling mechanism is necessary to obtain an oscillating growth rate in our model. We study thevariation in the frequency of the growth rate by changing the extracellular calcium concentrationand the density of ion channels in the membrane. We compare the predictions of our model withexperimental data on the frequency of oscillation versus growth speed, calcium concentration anddensity of calcium channels.& 2008 Elsevier Ltd. All rights reserved.Keywords:ElasticityViscous flowTip growthBiorhythmsStretch-activated channelsReaction-diffusion1. Introduction1.1. Biology of the pollen tubePollen transport the male gametes from the anther of thedonor flower to the female gametes located in the ovule of thereceptor flower. For this purpose, pollen grains grow tubularprotrusions through the stigma to the ovule. These tubes growexclusively at the tip, as do fungal hyphae, root hairs and neuronalgrowth cones. The expansion of the cell wall is thought to bedriven by a hydrostatic pressure, the turgor. The building materialnecessary for elongation is provided by secretory vesicles, asshown in Fig. 1. The fusion of these vesicles to the apical plasmamembrane is assumed to be triggered by calcium ions. It has beenproposed that stretch-activated calcium channels are located inthe plasma membrane. Calcium influx through these channelsresults in a high concentration of intra-cellular calcium at the tipof the pollen tube (Pierson et al., 1994). The causal relationshipbetween the calcium influx, exocytosis, cell wall expansion andgrowth is poorly understood. However, the cooperative behaviourof these components produces a clear signal: both the calciumconcentration at the tip and the growth rate oscillate in time witha steady amplitude and frequency (Pierson et al., 1994; Weisenseel et al., 1975; Geitmann et al., 1996; Messerli and Robinson,2003; Holdaway-Clarke and Hepler, 2003). Pharmacologicalinhibition of calcium channels can induce a change in growthfrequency (Geitmann et al., 1996). The pollen tube is an idealsystem for the study of plant cell growth and morphogenesis.The understanding of the feedback mechanism governing thepulsed growth of pollen tubes might yield insight into thefluctuating growth rate of other cellular system. Althoughcontroversial due to the possible artifacts associated with themethod of data collection, it was claimed that pulsed growth wasobserved in fungal hyphae (Lopez-Franco et al., 1994). Manyfeatures of this system are shared with evolutionary very distantcells such as neurons. While the latter do not possess a cell wall,they also exhibit highly localised exo- and endocytosis events thecontrol of which is poorly understood. In pollen tubes exocytosistranslates into cell growth which in turn can be measured veryaccurately. The quantification of the pollen tube growth ratethus indirectly allows to compare and test models of exocytosisand endocytosis as well as their dependence on calciumconcentration.1.2. Review of experimental data Corresponding author. Tel.: 1 514 398 7025; fax: 1 524 398 8434.E-mail address: kroegerj@physics.mcgill.ca (J.H. Kroeger).0022-5193/ - see front matter & 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.jtbi.2008.02.042Since the first description of oscillatory growth in pollen tubesvarious models have been proposed to explain the phenomenon.

ARTICLE IN PRESS364J.H. Kroeger et al. / Journal of Theoretical Biology 253 (2008) 363–374Fig. 1. Schematic profile of a pollen tube. Intensity of gray indicates the concentration of calcium ions. The cytoskeleton is not included in the schematics since its impact ongrowth rate oscillation is not considered. The plasma membrane is located between the cytoplasm and the cell wall. Objects are not to scale.It has been speculated that variations in turgor pressure may bethe cause for these oscillations (Messerli and Robinson, 2003;Harold, 2005); reviewed by Chebli and Geitmann (2007).Measurements of turgor have shown, however, that overall theturgor does not change in a measurable manner in time (Benkertet al., 1997). Although recent results from micro-indentation(Geitmann et al., unpublished) support the absence of significantturgor oscillations, the analysis of Benkert et al. does not excludepressure oscillations with amplitude or frequency below theresolution of the measuring technique. One of us has proposedearlier that such oscillatory growth phenomenon is an effect ofthe changes in the mechanical properties of the cell wall and, inparticular, a result of alterations in its elastic/rheological properties (Geitmann, 1999). Furthermore, the thickness of the apical cellwall was found to change over time (Holdaway-Clarke and Hepler,2003). Other physical properties of the cell wall might alsoundergo fluctuations (Bosch and Hepler, 2005).The concentration in the cytosol and the fluxes across themembrane of various ions change over time and the temporalrelationship between these events and the changes in the growthrate is considered to provide a key for understanding theoscillation mechanism (Holdaway-Clarke and Hepler, 2003).Experiments have shown that the oscillation frequency of thecytosolic calcium concentration at the pollen tube tip is identicalwith that of the growth rate (Pierson et al., 1994). However, thetwo signals are phase delayed with calcium lagging behind by10240 (Messerli et al., 2000; Holdaway-Clarke and Hepler,2003). Also, oscillation in the flux of Hþ ions and in pH-value ofthe cytosol have been observed to occur in the tip of the apex(Holdaway-Clarke and Hepler, 2003). These observations supportthe idea that the oscillation mode plays a fundamental role in thegrowth of pollen tubes. To study the cause-effect relationshipbetween ion concentration and growth rate, various pharmacological approaches were carried out. It was found that inhibitingcalcium channels leads to a decrease in oscillation frequency(Geitmann and Cresti, 1998). The effect of changes in theextracellular calcium concentration was investigated and it wasobserved that they affect the frequency of oscillations (HoldawayClarke et al., 2003). One also saw that an increase in overall tubegrowth corresponds to a shorter period between growth ratepeaks. Interesting in this context is also an estimate of the densityof vesicles present at the apex (Parton et al., 2001). The number ofvesicles was found to oscillate in time with the same frequency asthe growth rate. The phase was shifted such that growth peakscoincide with a decline in the numbers of vesicles in the apicalcytoplasm which might indicate massive exocytosis during timesof rapid growth.1.3. Previous theoretical models for tip growthTip growth is defined as the process that tube-shaped cellsemploy to elongate by depositing wall material exclusively at thetip, rather than by adding material over the entire length of thetube (as is the case in the more common diffuse growth). Tipgrowth is therefore characterised by the fact that only a highlyconfined area of cellular surface expands. This type of growth hasbeen the subject of many theoretical studies (Reinhardt, 1892;Ricci and Kendrick, 1972). The most recent models include theSpitzenkörper model or vesicle supply centre (VSC) model forfungal hyphae (Bartnicki-Garcia et al., 2000), which predictsthe distribution of vesicles inside the fungal tip region, and theviscoplastic model for root hair growth (Dumais et al., 2006).These models accurately predict the shape of tip growing cells.However, they are unable to predict the oscillation in calciumconcentration or the change in mean growth rate under externalinfluences like calcium channel inhibitors or changes in theexternal calcium concentration. A model of tip growth based onthe Henschel-fine equation (Denet, 1996). This model is able topredict the time evolution of cell shape upon interaction with amorphogen. One recent advance is a 2-D model describingcontinuous growth involving a form factor called extensibility,which is a function in space describing the ratio of strain rate andstress (Dumais et al., 2006; Bernal et al., 2007). Their model issimilar to an elastic model (Goriely and Tabor, 2003) whichpredicts shape formation under a given stress. While the Dumaismodel relates strain rate and stress which makes it a fluiditymodel (hence viscoplastic), Goriely and Tabor’s model is anelasticity model and thus considers the cell wall to be a solid.Finally, the polarity of the hyphae was related to experimentalobservations of the cytoskeleton and the Spitzenkörper vesiclecluster at the apex of the cell (Harris, 2006). However, none of theabove models account for calcium dependent oscillatory growth.Growth of cells and mobility of membranes has beeninvestigated also for other types of cells such as mammaliancells. The spreading behaviour of mouse embryonic fibroblasthas been measured (Döbereiner et al., 2004) which led toclassification into ‘‘phases’’ of (a) fast continuous spreading and(b) periodic membrane retraction. The dynamics of the neuronalgrowth cone was measured (Betz et al., 2006), and an oscillation

ARTICLE IN PRESSJ.H. Kroeger et al. / Journal of Theoretical Biology 253 (2008) 363–374365in growth velocity between protrusion and retraction (positiveand negative velocity) was found. Motility was interpreted asa stochastic resonance phenomenon, which is a mechanism toamplify weak signals with the help of noise. The protrusion offilopodial and lamellipodial in migrating cells was modelled asa balance between the merging and the drift-diffusion of thefilopodia (Mogilner, 2006; Mogilner and Rubenstein, 2005).Finally, mechanotransduction was modelled as the effect ofchange of lateral intra-cellular space as a function of theconcentration of ligands in the extracellular space (Kojic andTschumperlin, 2006).and thus material delivery never corresponds to the amountneeded for elongation and growth. The over-compensation ofcalcium influx due to a sudden opening of the membrane gatingchannels leads to growth rate oscillations. Section 2 is divided inthe following subsections. The growth of the pollen tube as aconsequence of the elongation of the cell wall is treated in Section2.1. The elastic properties of the cell wall and its dependence oncalcium concentration are the subject of Section 2.2. The flow ofcalcium ions in the cell and across the cell wall is discussed inSections 2.3 and 2.4. Section 2.5 discusses the calcium triggereddelivery of cell wall material which changes the cell wall rheology.1.4. Contribution of this work2.1. Fluid dynamics and viscoplastic modelThe goal of this work is to suggest a model describing theoscillations in pollen tube growth and cytosolic calcium concentration and to make quantitative predictions which can be testedby experiments. In a first step towards the construction of ourmodel, we considered hydrodynamic equations for the growthand elongation of the cell wall combined with channel gatedcalcium influx and a calcium dependent mechanism for thedeposition of cell wall material. The cell elongation is modelled bya viscous, pressure driven flow, as was done to describe theextension of cell walls (Lockhart, 1965) and the continuous growthof root hairs (Dumais et al., 2006). An important component of ourmodel is the dynamics of the cell wall material deposition. Intracellular transport of relatively soft cell wall material and cellmembrane material is achieved through the action of secretoryvesicles: exocytosis. This process is regulated and triggered bycalcium ions (Camacho and Malhó, 2003) and proteins such asthe SNARE complex. In our model, the amount of calcium ionsinside the cell is increased by influx through gated channels in themembrane and diminished by absorption on available sites. Thegating of the calcium influx induces an imbalance betweenthe cell wall material secreted and the cell wall material necessaryfor elongation. With these ingredients, an unstable oscillatorygrowth was produced. On the other hand, we found that ourmodel can generate stable oscillatory growth if we invoke thehypothesis of endocytosis: a calcium dependent increase indeformability of the cell wall through reduction in net deliveryrate of cell wall material.To the best of our knowledge this is the first time that atheoretical model for oscillations in pollen tip growth includescalcium dynamics. Furthermore, we can make quantitativepredictions, e.g., for the dependency of the growth frequency onparameters like the inhibition of calcium channels and theextracellular calcium concentration. We find qualitative agreement between experimental measurements of oscillatory growthand predictions of our model. Such agreement can be viewed asindirect support for the calcium dependence of endocytosis inpollen tubes.Since the motion of plant cells is believed to be driven by theturgor pressure and cell wall material is considered to beviscoelastic, the elongation of plant cell walls has traditionallybeen modelled as a viscous, pressure driven, flow (Skotheim andMahadevan, 2004; Dumais et al., 2006). In particular, (Skotheimand Mahadevan, 2004) plant cells were modelled as poroelasticfilaments filled with a liquid and submitted to a pressure p. Thisanalysis starts from the continuity and conservation requirementsof the Navier–Stokes equations and a Hookean stress–strainrelation. Following the derivation for poroelastic media (Biot,1941; Skotheim and Mahadevan, 2004), the governing equationsare a stress–strain relation s ¼ 2me þ lr sI apI and a continuityequation for pressure r k rp ¼ bqt p þ aqt r s. Here s is thestress tensor, s the displacement field, e the linear strain, m and lare the Lamé coefficients, a is a non-dimensional number relatedto the volume fraction and I is the identity tensor. Finally, k is thefluid permeability tensor and b is the bulk compliance of thematerial. If the solid and liquid components of the material aretreated as being incompressible, the mixture must also beincompressible, and b ¼ 0 (Skotheim and Mahadevan, 2004). Thisapproximation reduces the latter equation to Darcy’s law: Eq. (1).This analysis, and thus Darcy’s law, was used to model the timeand length scales involved in the motion of plant and fungi(Skotheim and Mahadevan, 2005). It allowed to classify theelongation rate and growth rate of 26 species of plants and fungi.In order to recover the shape of a growing pollen tube, Darcy’slaw and other fluid models need to be supplemented byconstraints or boundary conditions. The viscoplastic model(Dumais et al., 2006) assumes zero extensibility, equivalent toan infinite viscosity, in the tube shank cell wall. We assume aconstant pressure at the base of the pollen tube and, due to therigidity of the cell wall, zero flow on a surface parallel to thepollen tube which is situated at finite distance from those walls.The viscous flow obeying Eq. (1) and constrained by the conditiongiven above, expressed by Eq. (2), is known to adopt a tubularprofile called a Taylor–Saffman finger (Pelcé, 2000). Hence Eqs. (1)and (2), when taken together, are called the Taylor–Saffmanrelations. Their solution, a rhizoid of equation y ¼ cotðx dÞ, can beseen in Fig. 2. The cell walls of the filamentous fungi Polysticusversicolor and Pythium aphantdermatum modelled by the VSCmodel (Bartnicki-Garcia et al., 2000) display precisely such ahypoid curved profile. In the VSC model, d is the distance betweenthe tip of the hypoid and the VSC. The Taylor–Saffman relationswill yield a scaling relation between the growth rate u and thephysical properties of the cell wall. Darcy’s law gives the velocity uof a viscous fluid driven by a pressure p2. General theoryPossible mechanisms have been proposed in qualitative terms(Bartnicki-Garcia et al., 2000; Dumais et al., 2006; Feijó et al.,2001) to account for the oscillation of the tube growth rate. Thebasic elements that are agreed upon are a constant hydrostaticpressure, calcium triggered exocytosis and stretch-activatedcalcium channels. The dynamics that induce oscillations in ourmodel are a balance equation between pressure and growth rate,obtained from fluid dynamics, and a continuity equation for thedelivery of cell wall material. The material delivery, exocytosis andendocytosis, is triggered by in-flowing calcium ions. However, dueto the gating properties of the cell membrane, the calcium influxu¼Krp.m(1)Here K denotes the permeability of the infiltrated mediumand m stands for the viscosity of the injected viscous fluid.The Taylor–Saffman relations are completed by a constraint on

ARTICLE IN PRESS366J.H. Kroeger et al. / Journal of Theoretical Biology 253 (2008) 363–374Fig. 2. Schematic of the Hele–Shaw experiment testing the Taylor–Saffmanrelations. A fluid of viscosity m is injected with velocity V i and pressure p into amedium of permeability K. The Hele–Shaw cell has width Lx , length Ly anddepth Lz .pressure, depending on the surface tension g of the fluid interfaceand on the interface curvature k, given byp ¼ gk.(2)The assumption that the fluids are incompressible, i.e. r u ¼ 0leads to the Laplace equation r2 p ¼ 0 for the pressure. The scalingresult 2l 1 ðg mV i Þ was obtained for the relation between thewavelength l, given as a fraction of the experimental setup widthLx , and the surface tension g of the fluid (Shraiman, 1986). In thisrelation m is the fluid viscosity and V i is the injection velocity ofthe fluid given in cm/s which is different from the velocity u incm/s of the tip of the instability. u is related to the injectionvelocity and the wavelength by the relation u ¼ V i l (Shraiman,1986). The two previous expressions allow to relate, usingperturbation theory, the velocity of the tip of the instability tothe elastic constant BðtÞ byuðtÞ ¼u01 þ ðBðtÞ mu0 Þ2 3.(3)However, its rigorous derivation (Shraiman, 1986; Hong andLanger, 1986; Combescot et al., 1986) is beyond the scope of thispaper. Here we replace the surface tension g of the originalnotation of Shraiman by the elastic constant BðtÞ which can beestimated for polymer matrices such as the pollen cell wall. Fromnow on, u0 is the maximum growth rate of the pollen tube, theexperimental value of which is approximately 0:120:4 mm s forlily pollen tubes growing in vitro (1).2.2. Cell wall rheologyWhereas turgor is believed to force the expansion of the cellwall, there is no theoretical or experimental basis to supportthe hypo

Model for calcium dependent oscillatory growth in pollen tubes Jens H. Kroegera,, Anja Geitmannb, Martin Granta a Ernest Rutherford Physics Building, McGill University, 3600 rue University, Montre al, Quebec, Canada H3A 2T8 b Institut de Recherche en Biologie Ve getale, Departement de sciences biologiques, Universite de Montreal

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