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Journal of Theoretical Biology 253 (2008) 363– 374

Contents lists available at ScienceDirect

Journal of Theoretical Biology

0022-51

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/yjtbi

Model for calcium dependent oscillatory growth in pollen tubes

Jens H. Kroeger a,�, Anja Geitmann b, Martin Grant a

a Ernest Rutherford Physics Building, McGill University, 3600 rue University, Montreal, Quebec, Canada H3A 2T8b Institut de Recherche en Biologie Vegetale, Departement de sciences biologiques, Universite de Montreal, Montreal, Quebec, Canada H1X 2B2

a r t i c l e i n f o

Article history:

Received 12 December 2007

Received in revised form

25 February 2008

Accepted 27 February 2008Available online 18 March 2008

Keywords:

Elasticity

Viscous flow

Tip growth

Biorhythms

Stretch-activated channels

Reaction-diffusion

93/$ - see front matter & 2008 Elsevier Ltd. A

016/j.jtbi.2008.02.042

esponding author. Tel.: +1514 398 7025; fax:

ail address: kroegerj@physics.mcgill.ca (J.H. K

a b s t r a c t

Experiments have shown that pollen tubes grow in an oscillatory mode, the mechanism of which is

poorly understood. We propose a theoretical growth model of pollen tubes exhibiting such oscillatory

behaviour. The pollen tube and the surrounding medium are represented by two immiscible fluids

separated 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 cell

wall. 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 calcium

channels which transport calcium ions into the cytoplasm and an exocytosis rate dependent on the

cytosolic calcium concentration in the apex of the cell. We find that a calcium dependent vesicle

recycling mechanism is necessary to obtain an oscillating growth rate in our model. We study the

variation in the frequency of the growth rate by changing the extracellular calcium concentration

and the density of ion channels in the membrane. We compare the predictions of our model with

experimental data on the frequency of oscillation versus growth speed, calcium concentration and

density of calcium channels.

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

1.1. Biology of the pollen tube

Pollen 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 with

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+1524 398 8434.

roeger).

a steady amplitude and frequency (Pierson et al., 1994; Weisen-seel 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

Since the first description of oscillatory growth in pollen tubesvarious models have been proposed to explain the phenomenon.

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Fig. 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 on

growth rate oscillation is not considered. The plasma membrane is located between the cytoplasm and the cell wall. Objects are not to scale.

J.H. Kroeger et al. / Journal of Theoretical Biology 253 (2008) 363–374364

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 proper-ties (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 pharmacolo-gical 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 (Holdaway-Clarke 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 apical

cytoplasm which might indicate massive exocytosis during timesof rapid growth.

1.3. Previous theoretical models for tip growth

Tip 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 theSpitzenkorper 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 Spitzenkorper 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 (Dobereiner 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

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in 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).

1.4. Contribution of this work

The goal of this work is to suggest a model describing theoscillations in pollen tube growth and cytosolic calcium concen-tration 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. Intra-cellular 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 Malho, 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 agree-ment 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.

2. General theory

Possible mechanisms have been proposed in qualitative terms(Bartnicki-Garcia et al., 2000; Dumais et al., 2006; Feijo 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 influx

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.

2.1. Fluid dynamics and viscoplastic model

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 2meþ lr � sI� apI and a continuityequation for pressure r � k � rp ¼ bqtpþ aqtr � s. Here s is thestress tensor, s the displacement field, e the linear strain, m and lare the Lame 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 (Pelce, 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 Polysticus

versicolor 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 u

of a viscous fluid driven by a pressure p

u ¼K

mrp. (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

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Fig. 2. Schematic of the Hele–Shaw experiment testing the Taylor–Saffman

relations. A fluid of viscosity m is injected with velocity Vi and pressure p into a

medium of permeability K. The Hele–Shaw cell has width Lx, length Ly and

depth Lz .

J.H. Kroeger et al. / Journal of Theoretical Biology 253 (2008) 363–374366

pressure, depending on the surface tension g of the fluid interfaceand on the interface curvature k, given by

p ¼ �gk. (2)

The assumption that the fluids are incompressible, i.e. r � u ¼ 0leads to the Laplace equation r2p ¼ 0 for the pressure. The scalingresult 2l� 1�ðg=mViÞ 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 Vi 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 Vi=l (Shraiman,1986). The two previous expressions allow to relate, usingperturbation theory, the velocity of the tip of the instability tothe elastic constant BðtÞ by

uðtÞ ¼u0

1þ ðBðtÞ=mu0Þ2=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 0:120:4mm=s forlily pollen tubes growing in vitro (1).

2.2. Cell wall rheology

Whereas turgor is believed to force the expansion of the cellwall, there is no theoretical or experimental basis to supportthe hypothesis that it has a morphogenetic role given the factthat pressure is non-directional. Based on the proposition that themechanical cell wall properties at the growing tip must bedifferent from those in the shank (Green, 1969), it was suggestedthat an anisotropy in the cell wall elasticity is required to accountfor the transition between spherical and tubular shape in the tipof the cell (Dumais et al., 2004). Using a microindenter to measurethe rigidity of the tip of the pollen cell (Geitmann and Parre,2004), the rigidity of the cell was found to be an increasingfunction of the distance from the apex and then reaches a plateaubeyond a certain distance from the apex. Further experiments(Geitmann et al., 1996; Li et al., 1994; Parre and Geitmann, 2005)have established a gradient in the methyl-esterification of pectinalong the cell wall, especially in the region of the apex. Theseresults show how physical properties of the cell wall, which are

variable in space, account for the shape of the tip of the cell.Because of its dimensionality, the present model does not addressthe spatial variation of the cell wall elasticity but rather its’variation in time. We assume that the variation of the rheologicalcell wall properties can be described, as a first approximation, byone parameter that varies in time. The idea that the spatio-temporal variation can be model as a rescaling of the extensibilityprofile, given by one value that changes over time, was putforward recently (Dumais et al., 2006). We assume that thegelation of the cell wall is variable over time and depends onthe cell wall calcium concentration in a manner detailed below.The elastic constant of the cell wall BðtÞ is defined as the thicknessof the cell wall times the Young’s modulus of the cell wallcomponents (Smith et al., 2001). Another line of argumentationin favour of the elastic constant of the cell wall scaling as thethickness of the cell wall comes from the Spitzenkorper/VSCmodel (Bartnicki-Garcia et al., 2000). In this model the elongationof the fungal cell wall is exclusively parallel to the plane of the cellwall. If we model such a cell wall as parallel springs with identicalmicroscopic elasticity constants, the force necessary to elongatethese springs by a certain distance is proportional to the thicknesstðtÞ of the cell wall. The effective elastic constant BðtÞ of the cellwall is thus given by

BðtÞ ¼ tðtÞEðtÞ. (4)

Here EðtÞ is the Young’s modulus of the cell wall, related to themicroscopic spring constant of each bond. The dependence of theYoung’s modulus EðtÞ on the calcium concentration was measured(Zsivanovits et al., 2004) and an increase in Young’s modulus ofthe pectin matrix was observed upon the addition of calciumor other ions. It is important to note that BðtÞ has units of Newtonsper meter while the Young’s modulus EðtÞ has units of pressure,or Newtons per meter square. The tensile modulus per unit ofcalcium concentration was measured and found to follow thephenomenological law

EðtÞ ¼ 10:3 MPaþ 0:45MPa

mM� CwðtÞ. (5)

These measurements were performed on a pectin film of athickness of 50mm. The resulting expression for the elasticconstant BðtÞ taking into account wall thickness tðtÞ given in unitsof mm and calcium concentration Cw given in mM is

BðtÞ ¼ tðtÞ 0:2 MPaþ 0:009MPa

mM� CwðtÞ

� �. (6)

It is important to distinguish the calcium concentration in the cellwall from the calcium concentration in the cytoplasm. These twodomains are separated by the phospholipid bilayer plasmamembrane which maintains the difference in calcium concentra-tion. While the cytoplasmic calcium is involved in the control ofexo- and endocytosis (and other cellular functions), the calciumpresent in the cell wall cross-links the pectin polysaccharides torigidify the tube structure. Our formalism predicts that thegrowth rate is inversely proportional to the thickness of the cellwall to the power 2

3. A simple analysis of the viscoplastic model(Dumais et al., 2006) suggests a similar inverse proportionalitybetween growth rate and cell wall thickness. The analysis of theviscoplastic model (Dumais et al., 2006) yields a cell elongationrate _�i ¼ Fðse � syÞð1=jHjÞqH=qsi where F is the extensibility, H ¼

s2e=2 the stored elastic energy and se and sy are the effective and

constant yield stresses. The subscript i ¼ y,s,n denotes the curvi-linear coordinates. For the axisymmetric geometry, the twoprincipal curvatures are equal at the tip. Consequently, thestresses and the strain rates along these principal curvatures areequal at the tip and the effective stress at the tip is ss ¼ sy ¼ se ¼

p=2tk where k is the mean cell wall curvature, t the thickness, and

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J.H. Kroeger et al. / Journal of Theoretical Biology 253 (2008) 363–374 367

p the turgor pressure. The growth rate, or normal velocity, at thetip is un ¼ _�y=k� ut=rk where ut is the transverse velocity and r

the distance to the axis of symmetry. Replacing the effective stressand the strain rate into the previous expression yields the normalvelocity at the tip un ¼ F=kððp=2tkÞ � syÞð1=jHjÞqH=qsy � ut=rk. Inthe case of steady orthogonal growth (Bartnicki-Garcia et al.,2000), ut ¼ 0 and ð1=jHjÞqH=qsy is constant such that the normalvelocity at the tip is inversely proportional to the cell wallthickness, un / F=kðp=2tk� syÞ. This result, obtained using axial-symmetry, cannot easily be compared to the analysis of theTaylor–Saffman relations describing the flow of a finger sym-metric about two planes of symmetry rather than a central axis. Ashell theory analysis of the cell wall rheology is an important steptowards a better model of oscillatory pollen tube growth. Weargue, however, that our analysis is a very intuitive first steptowards the phenomenology of oscillatory tube growth.

2.3. Dynamics of calcium ions

Using the mobility of calcium inside the cell wall and acrossthe plasma membrane we can formulate the equation for thedynamics of the calcium ions. Calcium, being a very small ioncompared to the other charged molecules involved, diffuses quiterapidly, although not as fast as the protons Hþ. The calciumdynamics are essentially of diffusive nature. Even though amembrane calcium pump has been identified in pollen tubes(Schiott et al., 2004), there is a need for an intra-cellular calciumsink. Indeed, given that there is positive influx of calcium into thecell and given the lowest value of the calcium concentration insidethe cell being fairly constant, there must be binding sites for in-flowing calcium. We assume that the main calcium sinks are theendoplasmic reticulum, mitochondria, and vacuole. Thus we use adiffusion–reaction equation for the dynamics of the calciumconcentration Cðx; tÞ

qCðx; tÞ

qt¼ r � Dðx; tÞrCðx; tÞ � a1Cðx; tÞ. (7)

Since the dynamics of elastic constant and material deliverydepend on the calcium concentration at specific locations and inorder to simplify notation, we specify that the cytoplasmic freecalcium is denoted CiðtÞ, the cytoplasmic calcium concentration is

Table 1Explanation, value and source of different variables and parameters used in the pollen

Parameter Symbol

Cytoplasmic diffusion constant Di ¼ 20mm2=

Wall diffusion constant Dw ¼ 0:03mm

Extracellular calcium concentration CoðtÞ ¼ 0:1 m

Intra-cellular calcium concentration CiðtÞ ¼ 0:121

Pollen tube growth rate uðtÞ ¼ 0:1� 0

Elastic constant of cell wall BðtÞ ¼ 0:04 M

Cell wall thickness tðtÞ ¼ 0:2mm

Turgor pressure p ¼ 0:120:4 M

Permeability K ¼ 10212 m

Intrinsic viscosity of pectin ½m� ¼ 532 ml g

Young’s modulus of cell wall EðtÞ ¼ 10:3 M

Calcium flux through cell wall J ¼ 30 pmol c

Rate constant for calcium binding a1 ¼ �50 s�1

Rate constant for exocytosis a2 ¼ 8:73mm

Rate constant for endocytosis a3 ¼ 1000mm

Conductance of calcium channel ~rc ¼ 15 pS

Total volume of cell wall material in apex VðtÞ ¼ 5:0mm

Density of calcium channels at apex nc ¼ 0:1m�2

Electric potential across membrane U ¼ 902130

Period of oscillation P ¼ 302100 s

All quantities whose source is denoted * were calculated in the present manuscript.

denoted CoðtÞ and the calcium concentration in the cell wall isgiven by CwðtÞ. Likewise, the diffusion constant through thecellular envelope formed by the cell wall and the plasmamembrane will be noted DwðtÞ. DwðtÞ is approximately determinedfrom the calcium flux through the cell wall (Feijo, 1999). Theymeasured a maximum flux of J ¼ 30 pmol=s cm2 across the cellwall in lily pollen tubes. The diffusion constant is given by the fluxdivided by the calcium concentration gradient across the wall, i.e.

DwðtÞ ¼J

rCðx; tÞ. (8)

The gradient of calcium across the cellular envelope rC can beapproximated as the calcium concentration difference DC ¼

1 mM ¼ 10�3 mol=L across the cellular envelope divided by thethickness of the wall t ¼ 100 nm. The diffusion constant is foundto be Dw ¼ J=rC�Jt=DC ¼ 3:0� 10�10 cm2=s or Dw ¼ 0:03mm2=s.In comparison, the diffusion constant in the cytoplasm is assumedto be Dc ¼ 20mm2=s (Falcke, 2004). Due to the presence of calciumsinks in the cytosol, calcium cannot diffuse freely but is heavilybuffered. The calcium dynamics given by Eq. (7) ignores theputative absorption of calcium ions by the cell wall. Finally, asignificant portion of the calcium flux might be absorbed by thecell wall (Holdaway-Clarke and Hepler, 2003). This effect shouldbe considered in a more careful analysis once more experimentaldata is available. The last term in Eq. (7) describes the absorptionof calcium by the sites on the endoplasmic reticulum, mitochon-dria, and vacuole. Assuming that the number of sites is constant intime, the absorption will be proportional to the concentration ofcalcium ions. Such a minimal model is necessary until a betterknowledge of the calcium sink is available. Since the first term inEq. (7) conserves the calcium concentration, the absorption mustbe equal to the calcium flux through the cell wall. The reactionrate constant a1 is given by J=Ciro where ro ¼ 6mm is the outertube radius. Using the values given in Table 1, we obtaina1 ¼ 50 s�1.

2.4. Stretch-activated calcium channels

The stretch-activated channels affect the dynamics of thecalcium concentration in our model by changing the diffusion ofthe ions through the membrane. The relation that allows to relate

growth model

Source

s Falcke (2004)2=s Feijo (1999)

M Messerli and Robinson (2003)

:1mM Holdaway-Clarke and Hepler (2003)

:4mm=s Pierson et al. (1994)

Pamm *

Holdaway-Clarke and Hepler (2003)

Pa Benkert et al. (1997)2=s Pa Skotheim and Mahadevan (2004)

Zsivanovits et al. (2004)

Pa Zsivanovits et al. (2004)

m2 Feijo (1999)

Feijo (1999)3=mM s�1 *2=s *

Dutta and Robinson (2004)3 Holdaway-Clarke and Hepler (2003)

Feijo (1999), Dutta and Robinson (2004)

mV Holdaway-Clarke and Hepler (2003)

Pierson et al. (1994)

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J.H. Kroeger et al. / Journal of Theoretical Biology 253 (2008) 363–374368

the diffusion constant and the properties of the stretch-activatedchannels is the Einstein relation DwðtÞ ¼ kBTmðtÞ=q. The mobilitymðtÞ of the calcium ions is related to the conductivity rðtÞ of amembrane containing gated channels by mðtÞ ¼ rðtÞ=qCwðtÞ. HereqðtÞ is the charge of the calcium ions and CwðtÞ is the ionicconcentration in the cell wall. The conductivity of membranes hasbeen studied extensively since Hudgkin and Huxley performedvoltage–current measurements on squid giant axons. Rewritingthe Einstein relation in terms of the surface density of channels inthe membrane nc , the surface area of the membrane S, theconductivity of a single open channel rc and the probability thatthe channels are open Po yields

DwðtÞ ¼kBTncSrcPo

q2CwðtÞ. (9)

For one open channel, this relation reduces to the expressionfound in the context of electroporation (Barnett, 1990). Theopen probability Po is usually described by a sigmoid functionof voltage for voltage-gated ion channels (Noble, 1960, 1962;Shibasaki, 1987) and a sigmoid function of pressure for stretch-activated ion channels (Alexandre and Lassalles, 1991). Using theGibbs–Thompson relation (Sagui and Grant, 1999), we approx-imate the density of channels as nc ¼ N � k=l. Here N is the numberof channels in the apex, l is a characteristic tube length (Skotheimand Mahadevan, 2005) and k is the curvature of the fluid interfacewhich obeys Eq. (2). Combining Eqs. (1), (2), (9) and using asigmoid function of pressure Po ¼ 1=ð1þ ea5ð�pþp0ÞÞ for the openprobability, we obtain

DwðtÞ ¼kBTNpSrc

lgq2CwðtÞð1þ ea5ð�pþp0ÞÞ. (10)

The value of the constant a5 ¼ 0:49=kN1 m�2 (Alexandre andLassalles, 1991) determines the steepness of the sigmoid function.For the purpose of our feedback mechanism, we write the openprobability in terms of the growth rate uðtÞ, a dynamic variableof our model. From Eq. (1), we know that the pressure p exertedon the cell wall and thus on the calcium channels by the turgoris proportional to the elongation rate of the protruding tube p ¼

mluðtÞ=K (Skotheim and Mahadevan, 2005). Our final expressionfor the diffusion constant through the membrane is given by

DwðtÞ ¼a4uðtÞ

CwðtÞð1þ ea6ð�uðtÞþu0ÞÞ. (11)

Here a4 ¼ kBTNmSrc=Kgq2 and a6 ¼ a5lm=K:nc can be approximatedby multiplying the flux through the cell wall J ¼ 30 pmol=cm2 s(Feijo, 1999) by the valence of the calcium z ¼ 2 and Faraday’sconstant F ¼ 96 485 A s=mol to yield the current density acrossthe membrane I ¼ zFJ ¼ 5:8mA=cm2. The conductance ~r of themembrane, different from the conductivity r, is obtained bydividing this current density by the potential difference U andmultiplying by the surface area S of the cell apex, to give ~r ¼ IS=U.The ratio between the concentration of calcium inside and outsidethe membrane is in the order of 100 (Feijo, 1999). Thus we obtaina membrane potential of

U ¼RT

zFlnðCo=CiÞ ¼ 59:5 mV. (12)

By approximating the apex by a hemisphere of radius r0 ¼ 3:5mmand surface area S ¼ 154mm2, we obtain a cell membraneconductance of ~r ¼ 150 pS. This number can be compared to theconductance of the stretch-activated calcium channel measuredto be ~rc ¼ 15 pS (Dutta and Robinson, 2004) and yields a value of10 for the ratio of the conductance of the tip of the pollen tube andthe conductance of one calcium channel. From this comparisonwe estimate that the number of active calcium channels in the tip

area of the pollen cell at any given time is approximately 10 andnc ’ 0:1mm�2.

2.5. Temporal change in cell wall thickness

Another role of calcium in the pollen tube is to triggersecretory vesicle fusion with the plasma membrane. The mainfunction of this phenomenon is to bring new pectin, enzymes andother components to the cell wall. Exocytosis is concentrated atthe apex where elongation occurs. Much research has beendevoted to the relation between cytosolic or intra-cellular calciumconcentration and exocytosis since this phenomenon is ubiqui-tous in eucaryotic cells. One finding is the vesicle fusion rateincrease with intra-cellular calcium concentration (Blank et al.,2001) and the spatial correlation between high calcium concen-tration and vesicle fusion (Roy et al., 1999). A second key result isthe observation of a time delay between the arrival of the calciumsignal and the peak of the vesicle fusion rate. The delay can beexplained by the fact that a certain threshold amount of calciumhas to be absorbed by the secretory vesicles before they can fuse(Matese and McClay, 1998). The amount of vesicle fusion, or rateat which exocytosis occurs, should thus be proportional to thesum of the available calcium divided by the number of ionsnecessary to reach the threshold. The chemical reaction of thecalcium binding protein X with four calcium ions to produceactivated calcium binding proteins X� (Destexhe et al., 1994;McCormick and Huguenard, 1992) is described by

4Ca2þþ X2X�. (13)

The fusion of the vesicles proceeds by reaction of vesicles Ve

with the activated binding protein X� according to the dynamics

X� þ Ve2V�e . (14)

We will assume that the concentration of vesicles ½Ve� is largecompared to the cytosolic calcium concentration CiðtÞ. In otherwords, the calcium concentration, rather than the amount ofsecretory vesicles, is the limiting factor in the reaction. Thisassumption is supported by the measurements of Parton et al.(2001) who observed that the number of secretory vesicles in thevicinity ð123mmÞ of the apex is constant in time even though thefusion rate changes (Holdaway-Clarke et al., 2003). By making thisassumption, we render the fusion rate independent of the numberof vesicles and set it equal to one quarter of the sum of calciumions absorbed by the vesicles. By denoting the intra-cellularcalcium concentration by CiðtÞ and the exocytosis rate (volume ofcell wall material added by the secretory vesicles fusion) by qV=qt,we have

qVðtÞ

qt¼ a2CiðtÞ. (15)

Here a2 is the equilibrium constant of the fusion reaction givenabove. Its value can be estimated by using an experimental valuefor the change in cell wall volume qV=qt (Steer and Steer, 1989)and an average value of 1mM for the calcium concentration at theapex:

a2 ¼qVðtÞ=qt

CiðtÞ¼ 0:524

mm3

mM �min. (16)

We can derive the time dependent thickness of the apex, denotedby tðtÞ, as follows. We approximate the volume of pectin in thecylindrical part of the cell by Vc ¼ pðr2

o � r2i ÞL where L is the length

of the tube. Likewise, we approximate the volume of pectin in theapex by Va ¼ ð4p=6Þr2

otðtÞ. Here r0 and ri are the outer and innerradii, respectively, of the pollen tube which typically have valuesof approximately 6 and 5:9mm. Assuming that there are nochanges in volume per mass during the insertion of cell wall

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J.H. Kroeger et al. / Journal of Theoretical Biology 253 (2008) 363–374 369

material into the existing cell wall, the rate at which the sum ofthese two volume changes equals the rate at which vesicles fuseto the wall

qVðtÞ

qt¼

qqt

pðr2o � r2

i ÞLðtÞ þ4p6

r2otðtÞ

� �

¼ pðr2o � r2

i ÞuðtÞ þ4p6

r2o

qtðtÞqt

, (17)

where uðtÞ ¼ qL=qt is the growth rate of the pollen tube and qt=qt

is the rate of change of the thickness of the apex. Solving Eq. (17)for qt=qt yields

qtðtÞqt¼ �

3

2r2o

ðr2o � r2

i ÞuðtÞ þ3

2pr2o

qVðtÞ

qt

¼ �3

2r2o

ðr2o � r2

i ÞuðtÞ þ3a2

2pr2o

CiðtÞ, (18)

where we used Eq. (15).We now introduce into our model a mechanism for vesicle

recycling, namely endocytosis. Endocytosis is the budding andrelease of clathrin-coated or non-clathrin coated endocytoticvesicles from the plasma membrane (Derksen et al., 1995; Partonet al., 2001; Battey et al., 1999). The process of exocytosis andendocytosis in plant cells has been reviewed by Battey et al. Theendocytotic internalisation of dye, observed directly in pollentubes (Camacho and Malho, 2003), was argued to be due to a fastcalcium dependent mechanism. Although the endocytotic inter-nalisation of pectin material could not be established by thisstudy, it was observed in other plant cells (Volkmann and Baluska,2006). The calcium dependence for our endocytosis term is basedon the observation (Antov et al., 2004, 2005; Rosemberg andKorenstein, 1997) that endocytosis is directly proportional to theelectric field jEnj normal to the plasma membrane. The trans-membrane potentials encountered in pollen tubes are of the sameamplitude ð�50 mVÞ as those required to enhance endocytosis inthe animal cells (Antov et al., 2004, 2005; Rosemberg andKorenstein, 1997). The universal mechanism invoked to explainthis effect is the electric field dependence of membrane elasticproperties, namely spontaneous curvature and Gaussian modulus.These parameters affect the ability of the membrane to bend andbud an endocytotic vesicle (Siegel and Kozlov, 2004). Whereasmany other factors such as dynamin (Cousin and Robinson, 2000)or GTPase are known to affect endocytotic activity, increasing theosmolarity DC proved to increase endocytosis in hippocampalsynapses (Stevens and Williams, 2000). Increasing intra-cellularcalcium was found to inhibit endocytosis in the active zone ofnerve terminals (Cousin and Robinson, 2000) or in synapticterminals of retinal bipolar neurons (Gersdorff and Matthews,1994). All these observations are consistent with the mechanismdetailed by Antov et al.

The electric field can be calculated using the Nernst–Planckequation (Feijo, 1999) where z ¼ 2 is the valence of calcium ions, F

is Faraday’s constant and R is the universal gas constant

jEnjðtÞ ¼RTrCðx; tÞ

zFCiðtÞ. (19)

The rate of change of thickness, Eq. (18), becomes

qtðtÞqt¼ �

3

2r2o

ðr2o � r2

i ÞuðtÞ þ3a2

2pr2o

CiðtÞ � a3rCðx; tÞ

CiðtÞ. (20)

Since Antov et al. measured a dye uptake relative to a control, itis impossible to extract the value of a3 from their observation andwe choose to fit a3 such that the oscillation period obtained isclose to experimental values. The third term is necessary tostabilise the oscillating growth in our model. Endocytosis withsuch a calcium dependence prevents the cell wall from becomingtoo rigid. From Eq. (18) we know that the rate of change of the

thickness depends on the growth rate. As long as the growth rateremains below ð1=ðr2

o � r2i ÞncÞCiðtÞ, then qt=qt is positive and the

thickness increases. This situation can lead to a complete halt inthe elongation of the cell and disrupt the oscillation mechanism.To maintain oscillating growth rate in our model, we need a termthat softens the cell wall when the growth rate is very small. Wehypothesise that one role of endocytosis in apical growth is toremove calcium from the tip-located cell wall. The reduction ofcalcium prevents cell wall pectin to gelate, thus softening the cellwall and maintaining the cell wall anisotropy necessary for apicalgrowth (Dumais et al., 2004).

3. Methods

(i)

We modelled the dynamics of the pollen tube in a mediumas two liquids separated by an interface as seen in Fig. 2. Thecell wall and the plasma membrane represent the interfacebetween the cytoplasm (inside liquid phase) and the growthmedium (outside liquid phase). A hydrodynamic equationrelates the pressure inside the pollen tube, the elasticproperties of the cell wall and the growth rate. We solvethe dynamics in 1-D thus taking advantage of the unidirec-tional growth of this cylindrical cell. Albeit a simplification,we believe that this description retains the essentialbehaviour of the pollen tube.

(ii)

We take into account the dynamics of the cell wall at the tipby considering the physical properties of the wall buildingmaterial to be in balance with those of the cell wallincluding the tube shank and those of the material suppliedby fusion of vesicles at the apex.

(iii)

We couple the elasticity of the cell wall to the calcium flow.The dynamics of the calcium ions moving through the ionchannels is given by a diffusion equation. We model thecalcium flow through ion channels in the membrane by aneffective diffusion constant that varies in time Dðx; tÞ. Thediffusion constant in the cell wall x 2 ½325� will be denotedDð3; tÞ ¼ Dð4; tÞ ¼ Dð5; tÞ ¼ DwðtÞ.

(iv)

We describe the dynamics of the ion channels using thegenerally accepted idea that some types of ion channelsare stretch-activated gates. We model the opening andclosing of these channels in terms of effective diffusion. Theconformation of the channels depends on the tensile forceapplied on the membrane and yields a conductivity whichchanges in time.

(v)

From the mathematical point of view, all differentialequations are solved by the Euler finite-difference method.Since the diameter of a typical pollen tube is of the order of10mm, and our mesh consists of 100 grid points, we set themesh size to Dx ¼ 10mm=100 ¼ 0:1mm. The calcium con-centration is fixed at both ends of the integration domain. Theintra-cellular distal end of the domain is maintained at acalcium concentration of 0:01mM while the extracellular endof the domain is maintained at a fixed value in the range of0:121:1mM depending on the simulation run. These valuesfor the cytosolic calcium concentration are drawn experiment(Holdaway-Clarke and Hepler, 2003). The free calciumconcentration is denoted by Cðx; tÞ and shown in Fig. 3. Wespecify that the cytoplasmic free calcium is denoted CiðtÞ

and the cytoplasmic calcium concentration is denoted CoðtÞ.We obtain the cell wall calcium concentration from thedynamics of the model. We let the calcium diffuse accordingto Eqs. (7) and (11). At each time step, we evaluatethe calcium concentrations at their respective positions(i.e. CwðtÞ ¼ Cð4; tÞ, CiðtÞ ¼ Cð8; tÞ and CoðtÞ ¼ Cð1; tÞ) and put

ARTICLE IN PRESS

0

0.5

1

0 50 100

C [μ

M]

Distance [μm]

Fig. 3. Cytoplasmic calcium concentration along the growth axis of the pollen

tube.

0.095

0.096

0.097

0.098

0.099

0.1

u [μ

m/s

]

0.098

0.099

0.1

s]

0 2000 4000t [s]

J.H. Kroeger et al. / Journal of Theoretical Biology 253 (2008) 363–374370

these values into Eq. (6) of the cell wall elasticity. Thenumerical value for the diffusion constant of calcium is set toequal 1.0 everywhere except in the cell wall. Its value in thecell wall was discussed in the section on calcium channels.We simultaneously solve the coupled differential Eqs. (3), (6),(7), (11) and (20). Numerical values for time step, spatialmesh size and numerical parameters are ðDx;Dt;u0; a1;

a2; a3:a4Þ ¼ ð0:7;0:0001;2:0E� 6;0:1;42:0;20:0;1:5E7Þ.

0.095

0.096

0.097

u [μ

m/

0.095

0.096

0.097

0.098

0.099

0.1

u [μ

m/s

]

0 2000 4000t [s]

0 2000 4000t [s]

Fig. 4. Oscillation of tip growth rate u as function of time for different numbers of

active calcium channels. Top: nc ¼ 1� 10�5 mm�2, middle: nc ¼ 3� 10�5 mm�2,

bottom: nc ¼ 4:5� 10�5 mm�2.

4. Results

Our model presents a mechanism for oscillatory pollen tubegrowth and the maintenance of a steep calcium concentrationgradient at the tube apex. In fact, the oscillation in pollen tubegrowth rate and membrane conductivity is responsible for anincreasing gradient in the calcium concentration profile as oneapproaches the tube apex. The profile of the cytosolic calciumconcentration along the growth axis of the pollen tube is shownin Fig. 3. We find oscillatory behaviour for growth-rate, wallthickness and intra-cellular calcium concentration. The growthrate at the tip uðtÞ is shown as function of time in Fig. 4, fordifferent calcium channel densities. In Fig. 5 we show thecytosolic calcium concentration at the apex CiðtÞ as a function oftime. The evolution in time of the wall thickness tðtÞ (Fig. 6) isshown to display oscillatory behaviour. One observes that wallthickness and growth rate are phase delayed. They have the samefrequency, but they are out-of-phase by 203 � (Fig. 7). The resultthat wall thickness and growth rate is phase delayed is a directconsequence of our model, expressed in Eq. (3), stating that thegrowth rate uðtÞ is maximal when the elastic constant BðtÞ isminimal. The phase-delay is consistent with the findings ofHoldaway-Clarke and Hepler (2003) who report that the maximaof wall thickness and growth rate are phase-delayed by2252240 �. The cytosolic calcium concentration shows oscillatorybehaviour and is found to lag by 150 � behind the growth rate inour model (Fig. 7). Holdaway-Clarke and Hepler found a delay of30245 � between the maximum growth rate and the maximumcalcium concentration. While there is a quantitative discrepancybetween the theoretical and experimental data, it is important tonote that both agree upon a time lag between cytosolic calciumpeaks and the growth rate. This is remarkable, as the discoveryof this lag from experimental data was very puzzling initially asa causal relationship between calcium concentration and growth

ARTICLE IN PRESS

0.57

0.572

0.574

0.576

0.578

0.58

0.582

0.584

0 2000 4000

Ci [

μM]

0.57

0.572

0.574

0.576

0.578

0.58

0.582

0.584

Ci [

μM]

0.57

0.572

0.574

0.576

0.578

0.58

0.582

0.584

Ci [

μM]

t [s]

0 2000 4000t [s]

0 2000 4000t [s]

Fig. 5. Oscillation of cytosolic calcium concentration C as function of time for

different numbers of active calcium channels. Top: nc ¼ 1� 10�5 mm�2, middle:

nc ¼ 3� 10�5 mm�2, bottom: nc ¼ 4:5� 10�5 mm�2.

0.375

0.38

0.385

0.39

0.395

0 2000 4000

τ [μ

m]

0.375

0.38

0.385

0.39

0.395

τ [μ

m]

0.375

0.38

0.385

0.39

0.395

τ [μ

m]

t [s]

0 2000 4000t [s]

0 2000 4000t [s]

Fig. 6. Oscillation of cell wall thickness T as function of time for different numbers

of active calcium channels. Top: nc ¼ 1� 10�5 mm�2, middle: nc ¼ 3� 10�5 mm�2,

bottom: nc ¼ 4:5� 10�5 mm�2.

J.H. Kroeger et al. / Journal of Theoretical Biology 253 (2008) 363–374 371

ARTICLE IN PRESS

-1.5

-1

-0.5

0

0.5

1

1.5

0 50 100 150 200t [s]

u(t)τ (t)C(t)

Fig. 7. Correlation between growth rate, cytosolic calcium concentration and

apical cell wall thickness. All curves have been normalised to have zero mean and a

maximum of one.

110

115

120

125

130

135

140

145

150

155

160

0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1

Per

iod

[s]

Co [μM]

Fig. 8. Dependence of oscillation period on external calcium concentration.

160

162

164

166

168

170

172

174

1e-05 2e-05 3e-05 4e-05 5e-05 6e-05

Per

iod

[s]

nc [μm-2]

Fig. 9. Dependence of oscillation period on the density of calcium channels in the

apical plasma membrane.

160

162

164

166

168

170

172

174

0.097 0.0975 0.098 0.0985 0.099

Per

iod

[s]

<u> [μm/s]

Fig. 10. Period of oscillation in growth rate at the tip versus average growth rate at

the tip. We varied the density of calcium channels from 1:026:0� 10�5 mm�2.

110

115

120

125

130

135

140

145

150

155

160

0.0992 0.0993 0.0994 0.0995 0.0996 0.0997 0.0998

Per

iod

[s]

<u> [μm/s]

Fig. 11. Period of oscillation in growth rate at the tip versus average growth rate at

the tip. We varied the extracellular calcium concentration from 0.96 to 1:0mM.

J.H. Kroeger et al. / Journal of Theoretical Biology 253 (2008) 363–374372

rate was thought to require calcium peaks to precede peaks in thegrowth rate. Our model shows that this does not necessarily needto be the case.

We studied the frequency f (related to the period P byf ¼ 2p=P) of oscillation under variation of the following para-meters. We varied the extracellular calcium concentration andobserved that increasing the extracellular calcium concentrationincreases the period of growth rate oscillation (Fig. 8). Then wevaried the density of ion channels (number per unit surface area)and observed that inhibiting calcium channels, i.e. decreasing thenumber of active channels per area, increases the period of growthrate oscillation (Fig. 9). Finally, we determined the average growthrates for the simulations with variable channel density andplotted the period as a function of average growth rate (Fig. 10).We observe that the period of growth rate oscillation increaseswhen the average growth rate increases. Such a relation betweenthe average growth rate hui and the period P has never beenmeasurement experimentally for variable channel densities andthus constitutes a verifiable prediction of our model.

The period as a function of average growth rate for changingvalues of extracellular calcium concentration is plotted in Fig. 11.

We observe that the period decreases as the average growth rateincreases. This behaviour is qualitatively similar to the findings ofHoldaway-Clarke et al. (2003). The slope of the period versusgrowth rate curve from our model is significantly lower than thatof �1 measured experimentally by Holdaway-Clarke et al. We canalso compare our results to the findings of Messerli and Robinson(2003) who observed a 43% decrease in average growth rate upon

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J.H. Kroeger et al. / Journal of Theoretical Biology 253 (2008) 363–374 373

a 100-fold increase in extracellular calcium concentration. Bycombining the results of Figs. 11 and 8, we obtain a 1–2% decreasein average growth rate upon a 4% increase in extracellular calciumconcentration. Even though our curves fit the experimental dataqualitatively, there is a discrepancy in the numerical value of therelations. The period of oscillation of our model varies much morethan what was observed by Holdaway-Clarke et al. (2003) orMesserli and Robinson (2003) for a given variation in extracellularcalcium concentration.

Our model is clearly more sensitive to the effect of changingextracellular calcium concentration. This may be due to the factthat we did not take into account the absorption of calcium ionsby the cell wall in our model (Holdaway-Clarke and Hepler, 2003;Feijo, 1999). Unesterified sites on the pectin polymers in the cellwall bind calcium ions. These negatively charged sites have astrong affinity (Somers, 1973) for the free calcium ions and thisaffinity maintains a constant ratio of bound and free calcium ions.This affinity may explain the cell wall’s ability to buffer the flow ofcalcium ions and explain the lower sensitivity to changes inextracellular calcium concentration observed in experiments.

5. Discussion

This work is the first mathematical model of tip growththat predicts calcium dependent oscillating growth. It is based onEqs. (3), (6), (7), (11) and (20) describing the balance of pressureand cell wall elongation as well as the controlled flow of calciumions across the cell membrane and their reaction with secretoryvesicles. Furthermore, our model predicts a changing period whenthe extracellular calcium concentration or the number of activecalcium channels are changed. The dependence of the periodon these parameters as well as its correlation with the averagegrowth rate all agree qualitatively with experimental obser-vations (Holdaway-Clarke et al., 2003; Geitmann and Cresti,1998; Messerli and Robinson, 2003). The phase delays betweenthe maxima of intra-cellular calcium concentration, growth rateand cell wall thickness broadly fit the accepted values (Holdaway-Clarke and Hepler, 2003; Chebli and Geitmann, 2007), except forthe time lag of tip calcium influx. This delay is not explained byour model since we neglect important aspects of the calciumdynamics such as its absorption by the cell wall (Somers, 1973).Finally, we offer testable predictions of the elastic properties ofthe cell wall as a function of time.

This work offers a unified framework, derived from varioussources, that couples many different aspects of pollen tubegrowth. It offers the possibility to study the mechanism of cellularfeatures, such as the dynamics of calcium channels, which is fastand difficult to study, but has a considerable impact on the growthprocess. In our model, we have assumed that the turgor pressuresremains constant in time. The relation between turgor pressure,rheology of the cell wall and growth rate of the tip is given by theTaylor–Saffman fluid relations. The only term not having a solidbackup by experiments is the cell wall uptake due to endocytosis.Issues related to pollen tube growth that were not addressed inthis model include pH oscillations, changes in cell wall methyl-esterification and the change of apex surface area due to apexdeformation and insertion of vesicle material.

Our model offers a possible calcium concentration dependencefor vesicle recycling, i.e. endocytosis. It presents a set of coupledequations explaining how pollen tubes maintain a cytosoliccalcium gradient at the tip. It has been shown experimentallythat dissipating this gradient inhibits growth (Pierson et al., 1994).An important step to improve this model of tip growth willrequire a transformation into a 3-D model that is coupled torheological parameters of the cell wall variable in space and time.

A better knowledge of the vesicle dynamics must be achieved inorder to get a better estimate of cell wall secretion. One keyelement of such a program is a measurement and a theoreticalstudy of the rates of exocytosis and endocytosis as functions of theintra-cellular calcium concentration.

Acknowledgements

We are thankful for the generous support of this research bythe Natural Sciences and Engineering Research Council of Canadaand by Le Fonds quebecois de la recherche sur la nature et lestechnologies.

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